Financial modelling and business forecasting
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financialmodellingandbusinessforecasting1.zipdata.docx
(1)
Means, standard deviations and correlations
The dataset is: new09.in7
The sample is: 1973(4) - 1990(2)
Means
LBelgium LFrance LGermanyLFrance/BelgiumLFrance/GermanyLBelgium/Germany
3.8105 3.7169 3.9794 -1.9465 0.92657 2.8731
Standard deviations (using T-1)
LBelgium LFrance LGermanyLFrance/BelgiumLFrance/GermanyLBelgium/Germany
0.29047 0.42521 0.17195 0.10754 0.23542 0.13881
Correlation matrix:
LBelgium LFrance LGermanyLFrance/Belgium
LBelgium 1.0000 0.99593 0.99555 0.92029
LFrance 0.99593 1.0000 0.99782 0.91758
LGermany 0.99555 0.99782 1.0000 0.91119
LFrance/Belgium 0.92029 0.91758 0.91119 1.0000
LFrance/Germany 0.97657 0.98038 0.97257 0.94242
LBelgium/Germany 0.94326 0.95181 0.94353 0.82359
LFrance/GermanyLBelgium/Germany
LBelgium 0.97657 0.94326
LFrance 0.98038 0.95181
LGermany 0.97257 0.94353
LFrance/Belgium 0.94242 0.82359
LFrance/Germany 1.0000 0.96585
LBelgium/Germany 0.96585 1.0000
Normality tests and descriptive statistics
The dataset is: new09.in7
The sample is: 1973(5) - 1990(2)
Normality test for DLBelgium
Observations 202
Mean 0.0049006
Std.Devn. 0.0040206
Skewness 0.55869
Excess Kurtosis -0.35971
Minimum -0.0029746
Maximum 0.016097
Asymptotic test: Chi^2(2) = 11.597 [0.0030]**
Normality test: Chi^2(2) = 23.525 [0.0000]**
Normality test for DLFrance
Observations 202
Mean 0.0067183
Std.Devn. 0.0037830
Skewness 0.28292
Excess Kurtosis -0.22797
Minimum -0.0024968
Maximum 0.019186
Asymptotic test: Chi^2(2) = 3.1322 [0.2089]
Normality test: Chi^2(2) = 3.9005 [0.1422]
Normality test for DLGermany
Observations 202
Mean 0.0028933
Std.Devn. 0.0030606
Skewness 0.59041
Excess Kurtosis 0.35519
Minimum -0.0033465
Maximum 0.012716
Asymptotic test: Chi^2(2) = 12.798 [0.0017]**
Normality test: Chi^2(2) = 12.800 [0.0017]**
Normality test for DLFrance/Belgium
Observations 202
Mean 0.0017844
Std.Devn. 0.011918
Skewness 0.48711
Excess Kurtosis 4.8743
Minimum -0.055216
Maximum 0.043456
Asymptotic test: Chi^2(2) = 207.95 [0.0000]**
Normality test: Chi^2(2) = 83.018 [0.0000]**
Normality test for DLFrance/Germany
Observations 202
Mean 0.0037082
Std.Devn. 0.012603
Skewness 1.0110
Excess Kurtosis 3.1420
Minimum -0.038558
Maximum 0.050134
Asymptotic test: Chi^2(2) = 117.50 [0.0000]**
Normality test: Chi^2(2) = 28.248 [0.0000]**
Normality test for DLBelgium/Germany
Observations 202
Mean 0.0019238
Std.Devn. 0.0078273
Skewness 4.2544
Excess Kurtosis 31.142
Minimum -0.018404
Maximum 0.069908
Asymptotic test: Chi^2(2) = 8772.2 [0.0000]**
Normality test: Chi^2(2) = 415.61 [0.0000]**
(2) Unit-root test
Stationary
EQ( 1) Modelling DLFrance by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant -0.0160450 0.007638 -2.10 0.0369 0.0217
Trend -0.000107278 1.858e-005 -5.77 0.0000 0.1435
LFrance_1 0.00908918 0.002558 3.55 0.0005 0.0597
sigma 0.00279924 RSS 0.00155930985
R^2 0.460593 F(2,199) = 84.96 [0.000]**
log-likelihood 902.324 DW 1.08
no. of observations 202 no. of parameters 3
mean(DLFrance) 0.00671834 var(DLFrance) 1.43108e-005
// Batch code for EQ( 1)
module("PcGive");
package("PcGive", "Single-equation");
usedata("new09.in7");
system
{
Y = DLFrance;
Z = Constant, Trend, LFrance_1;
}
estimate("OLS", 1973, 5, 1990, 2);
EQ( 2) Modelling DLFrance by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0269034 0.001874 14.4 0.0000 0.5076
LFrance_1 -0.00543452 0.0005012 -10.8 0.0000 0.3702
sigma 0.00301714 RSS 0.00182062088
R^2 0.370198 F(1,200) = 117.6 [0.000]**
log-likelihood 886.676 DW 0.911
no. of observations 202 no. of parameters 2
mean(DLFrance) 0.00671834 var(DLFrance) 1.43108e-005
EQ( 3) Modelling DLFrance by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
LFrance_1 0.00171584 8.072e-005 21.3 0.0000 0.6921
sigma 0.00428889 RSS 0.00369731806
log-likelihood 815.124 DW 0.452
no. of observations 202 no. of parameters 1
mean(DLFrance) 0.00671834 var(DLFrance) 1.43108e-005
EQ( 4) Modelling DLBelgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0215108 0.01391 1.55 0.1235 0.0119
Trend -1.75107e-005 2.083e-005 -0.841 0.4015 0.0035
LBelgium_1 -0.00388990 0.004198 -0.927 0.3553 0.0043
sigma 0.00343212 RSS 0.00234411488
R^2 0.282144 F(2,199) = 39.11 [0.000]**
log-likelihood 861.15 DW 1.34
no. of observations 202 no. of parameters 3
mean(DLBelgium) 0.00490063 var(DLBelgium) 1.61655e-005
EQ( 5) Modelling DLBelgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0328908 0.003186 10.3 0.0000 0.3476
LBelgium_1 -0.00734906 0.0008341 -8.81 0.0000 0.2796
sigma 0.00342961 RSS 0.00235244174
R^2 0.279594 F(1,200) = 77.62 [0.000]**
log-likelihood 860.792 DW 1.33
no. of observations 202 no. of parameters 2
mean(DLBelgium) 0.00490063 var(DLBelgium) 1.61655e-005
EQ( 6) Modelling DLBelgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
LBelgium_1 0.00123717 7.802e-005 15.9 0.0000 0.5557
sigma 0.00423554 RSS 0.00360589874
log-likelihood 817.653 DW 0.874
no. of observations 202 no. of parameters 1
mean(DLBelgium) 0.00490063 var(DLBelgium) 1.61655e-005
EQ( 7) Modelling DLGermany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0218191 0.01885 1.16 0.2484 0.0067
Trend -6.70422e-006 1.501e-005 -0.447 0.6557 0.0010
LGermany_1 -0.00458449 0.005113 -0.897 0.3710 0.0040
sigma 0.00284988 RSS 0.00161623889
R^2 0.145862 F(2,199) = 16.99 [0.000]**
log-likelihood 898.702 DW 1.26
no. of observations 202 no. of parameters 3
mean(DLGermany) 0.00289333 var(DLGermany) 9.36755e-006
EQ( 8) Modelling DLGermany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0299740 0.004654 6.44 0.0000 0.1718
LGermany_1 -0.00680706 0.001169 -5.82 0.0000 0.1450
sigma 0.00284417 RSS 0.00161785847
R^2 0.145006 F(1,200) = 33.92 [0.000]**
log-likelihood 898.601 DW 1.26
no. of observations 202 no. of parameters 2
mean(DLGermany) 0.00289333 var(DLGermany) 9.36755e-006
EQ( 9) Modelling DLGermany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
LGermany_1 0.000713342 5.508e-005 13.0 0.0000 0.4549
sigma 0.00311743 RSS 0.00195338812
log-likelihood 879.567 DW 1.05
no. of observations 202 no. of parameters 1
mean(DLGermany) 0.00289333 var(DLGermany) 9.36755e-006
EQ(10) Modelling DLFrance/Belgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant -0.0820533 0.04060 -2.02 0.0446 0.0201
Trend 5.27450e-005 3.517e-005 1.50 0.1353 0.0112
LFrance/Belgium_1 -0.0402793 0.01914 -2.10 0.0366 0.0218
sigma 0.0118449 RSS 0.0279202324
R^2 0.0269562 F(2,199) = 2.756 [0.066]
log-likelihood 610.928 DW 1.23
no. of observations 202 no. of parameters 3
mean(Y) 0.00178439 var(Y) 0.000142048
EQ(11) Modelling DLFrance/Belgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant -0.0255781 0.01521 -1.68 0.0943 0.0139
LFrance/Belgium_1 -0.0140523 0.007802 -1.80 0.0732 0.0160
sigma 0.0118819 RSS 0.0282357082
R^2 0.0159616 F(1,200) = 3.244 [0.073]
log-likelihood 609.793 DW 1.25
no. of observations 202 no. of parameters 2
mean(Y) 0.00178439 var(Y) 0.000142048
EQ(12) Modelling DLFrance/Belgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
LFrance/Belgium_1 -0.000956051 0.0004306 -2.22 0.0275 0.0239
sigma 0.0119357 RSS 0.0286347099
log-likelihood 608.376 DW 1.25
no. of observations 202 no. of parameters 1
mean(Y) 0.00178439 var(Y) 0.000142048
EQ(17) Modelling DLFrance/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0192151 0.009474 2.03 0.0439 0.0203
Trend 7.54858e-005 7.164e-005 1.05 0.2933 0.0055
LFrance/Germany_1 -0.0251260 0.01782 -1.41 0.1600 0.0099
sigma 0.0125617 RSS 0.0314015384
R^2 0.0213727 F(2,199) = 2.173 [0.117]
log-likelihood 599.06 DW 1.24
no. of observations 202 no. of parameters 3
mean(Y) 0.00370822 var(Y) 0.000158848
EQ(18) Modelling DLFrance/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.00998048 0.003598 2.77 0.0061 0.0370
LFrance/Germany_1 -0.00678005 0.003770 -1.80 0.0736 0.0159
sigma 0.0125652 RSS 0.0315767197
R^2 0.0159132 F(1,200) = 3.234 [0.074]
log-likelihood 598.498 DW 1.25
no. of observations 202 no. of parameters 2
mean(Y) 0.00370822 var(Y) 0.000158848
EQ(19) Modelling DLFrance/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
LFrance/Germany_1 0.00335708 0.0009417 3.57 0.0005 0.0595
sigma 0.0127727 RSS 0.0327915104
log-likelihood 594.686 DW 1.22
no. of observations 202 no. of parameters 1
mean(Y) 0.00370822 var(Y) 0.000158848
EQ(20) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0324338 0.03354 0.967 0.3346 0.0047
Trend 1.68881e-005 3.010e-005 0.561 0.5753 0.0016
LBelgium/Germany_1 -0.0112248 0.01269 -0.885 0.3774 0.0039
sigma 0.00785526 RSS 0.0122793188
R^2 0.00780208 F(2,199) = 0.7824 [0.459]
log-likelihood 693.893 DW 1.34
no. of observations 202 no. of parameters 3
mean(Y) 0.00192383 var(Y) 6.12667e-005
EQ(21) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
Constant 0.0147556 0.01147 1.29 0.1998 0.0082
LBelgium/Germany_1 -0.00446743 0.003989 -1.12 0.2641 0.0062
sigma 0.00784179 RSS 0.0122987469
R^2 0.00623224 F(1,200) = 1.254 [0.264]
log-likelihood 693.734 DW 1.34
no. of observations 202 no. of parameters 2
mean(Y) 0.00192383 var(Y) 6.12667e-005
EQ(22) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
LBelgium/Germany_1 0.000657903 0.0001922 3.42 0.0007 0.0551
sigma 0.00785456 RSS 0.0124005016
log-likelihood 692.901 DW 1.34
no. of observations 202 no. of parameters 1
mean(Y) 0.00192383 var(Y) 6.12667e-005
H0 unit root
H1 STATIONARY
ALMOST OF THEM SUGGEST I0
ADF TEST
H0 I(1)
H1 I(2)
EQ(23) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium/Germany_1 0.336949 0.06742 5.00 0.0000 0.1125
LBelgium/Germany_1 -0.0179784 0.01208 -1.49 0.1383 0.0111
Constant 0.0494398 0.03191 1.55 0.1229 0.0120
Trend 3.40206e-005 2.873e-005 1.18 0.2377 0.0071
sigma 0.00743136 RSS 0.0108793353
R^2 0.119947 F(3,197) = 8.95 [0.000]**
log-likelihood 702.125 DW 1.82
no. of observations 201 no. of parameters 4
mean(Y) 0.00194223 var(Y) 6.15032e-005
EQ(24) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium/Germany_1 0.326982 0.06696 4.88 0.0000 0.1075
LBelgium/Germany_1 -0.00439956 0.003809 -1.16 0.2495 0.0067
Constant 0.0139460 0.01096 1.27 0.2047 0.0081
sigma 0.00743891 RSS 0.0109567886
R^2 0.113682 F(2,198) = 12.7 [0.000]**
log-likelihood 701.412 DW 1.82
no. of observations 201 no. of parameters 3
mean(Y) 0.00194223 var(Y) 6.15032e-005
EQ(25) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium/Germany_1 0.329561 0.06703 4.92 0.0000 0.1083
LBelgium/Germany_1 0.000441060 0.0001882 2.34 0.0201 0.0269
sigma 0.00745046 RSS 0.0110463712
log-likelihood 700.594 DW 1.82
no. of observations 201 no. of parameters 2
mean(Y) 0.00194223 var(Y) 6.15032e-005
EQ(26) Modelling DLFrance/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLFrance/Germany_1 0.394828 0.06590 5.99 0.0000 0.1541
Constant 0.0253629 0.008845 2.87 0.0046 0.0401
LFrance/Germany_1 -0.0411394 0.01669 -2.46 0.0146 0.0299
Trend 0.000146035 6.721e-005 2.17 0.0310 0.0234
sigma 0.011599 RSS 0.0265035905
R^2 0.17352 F(3,197) = 13.79 [0.000]**
log-likelihood 612.638 DW 1.94
no. of observations 201 no. of parameters 4
mean(Y) 0.00373002 var(Y) 0.000159542
EQ(27) Modelling DLFrance/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLFrance/Germany_1 0.370646 0.06556 5.65 0.0000 0.1390
Constant 0.00761390 0.003423 2.22 0.0273 0.0244
LFrance/Germany_1 -0.00568251 0.003554 -1.60 0.1115 0.0127
sigma 0.0117074 RSS 0.0271387011
R^2 0.153715 F(2,198) = 17.98 [0.000]**
log-likelihood 610.258 DW 1.92
no. of observations 201 no. of parameters 3
mean(Y) 0.00373002 var(Y) 0.000159542
EQ(28) Modelling DLFrance/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLFrance/Germany_1 0.390919 0.06556 5.96 0.0000 0.1516
LFrance/Germany_1 0.00196840 0.0009030 2.18 0.0304 0.0233
sigma 0.011823 RSS 0.0278166762
log-likelihood 607.779 DW 1.92
no. of observations 201 no. of parameters 2
mean(Y) 0.00373002 var(Y) 0.000159542
EQ(29) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium/Germany_1 0.336949 0.06742 5.00 0.0000 0.1125
Constant 0.0494398 0.03191 1.55 0.1229 0.0120
Trend 3.40206e-005 2.873e-005 1.18 0.2377 0.0071
LBelgium/Germany_1 -0.0179784 0.01208 -1.49 0.1383 0.0111
sigma 0.00743136 RSS 0.0108793353
R^2 0.119947 F(3,197) = 8.95 [0.000]**
log-likelihood 702.125 DW 1.82
no. of observations 201 no. of parameters 4
mean(Y) 0.00194223 var(Y) 6.15032e-005
EQ(30) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium/Germany_1 0.326982 0.06696 4.88 0.0000 0.1075
Constant 0.0139460 0.01096 1.27 0.2047 0.0081
LBelgium/Germany_1 -0.00439956 0.003809 -1.16 0.2495 0.0067
sigma 0.00743891 RSS 0.0109567886
R^2 0.113682 F(2,198) = 12.7 [0.000]**
log-likelihood 701.412 DW 1.82
no. of observations 201 no. of parameters 3
mean(Y) 0.00194223 var(Y) 6.15032e-005
EQ(31) Modelling DLBelgium/Germany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium/Germany_1 0.329561 0.06703 4.92 0.0000 0.1083
LBelgium/Germany_1 0.000441060 0.0001882 2.34 0.0201 0.0269
sigma 0.00745046 RSS 0.0110463712
log-likelihood 700.594 DW 1.82
no. of observations 201 no. of parameters 2
mean(Y) 0.00194223 var(Y) 6.15032e-005
EQ(32) Modelling DLBelgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium_1 0.326990 0.06678 4.90 0.0000 0.1085
Constant 0.0218492 0.01332 1.64 0.1024 0.0135
LBelgium_1 -0.00481865 0.004012 -1.20 0.2312 0.0073
Trend -1.78575e-006 1.997e-005 -0.0894 0.9288 0.0000
sigma 0.00323924 RSS 0.00206705122
R^2 0.366874 F(3,197) = 38.05 [0.000]**
log-likelihood 869.03 DW 2.07
no. of observations 201 no. of parameters 4
mean(DLBelgium) 0.0049045 var(DLBelgium) 1.62429e-005
EQ(33) Modelling DLBelgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium_1 0.327704 0.06613 4.96 0.0000 0.1103
Constant 0.0229921 0.003726 6.17 0.0000 0.1613
LBelgium_1 -0.00516766 0.0009288 -5.56 0.0000 0.1352
sigma 0.00323111 RSS 0.00206713512
R^2 0.366848 F(2,198) = 57.36 [0.000]**
log-likelihood 869.025 DW 2.07
no. of observations 201 no. of parameters 3
mean(DLBelgium) 0.0049045 var(DLBelgium) 1.62429e-005
EQ(34) Modelling DLBelgium by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLBelgium_1 0.563880 0.05875 9.60 0.0000 0.3165
LBelgium_1 0.000537264 9.763e-005 5.50 0.0000 0.1321
sigma 0.00351932 RSS 0.00246473478
log-likelihood 851.345 DW 2.26
no. of observations 201 no. of parameters 2
mean(DLBelgium) 0.0049045 var(DLBelgium) 1.62429e-005
EQ(35) Modelling DLFrance by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLFrance_1 0.465581 0.06391 7.28 0.0000 0.2122
Constant -0.00635720 0.006970 -0.912 0.3628 0.0042
LFrance_1 0.00411569 0.002388 1.72 0.0864 0.0148
Trend -5.22226e-005 1.823e-005 -2.86 0.0046 0.0400
sigma 0.00249668 RSS 0.00122797857
R^2 0.57419 F(3,197) = 88.55 [0.000]**
log-likelihood 921.365 DW 2.13
no. of observations 201 no. of parameters 4
mean(DLFrance) 0.00670529 var(DLFrance) 1.43476e-005
EQ(36) Modelling DLFrance by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLFrance_1 0.541023 0.05928 9.13 0.0000 0.2961
Constant 0.0125788 0.002251 5.59 0.0000 0.1363
LFrance_1 -0.00256031 0.0005327 -4.81 0.0000 0.1045
sigma 0.00254171 RSS 0.00127913695
R^2 0.556451 F(2,198) = 124.2 [0.000]**
log-likelihood 917.263 DW 2.2
no. of observations 201 no. of parameters 3
mean(DLFrance) 0.00670529 var(DLFrance) 1.43476e-005
EQ(37) Modelling DLFrance by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLFrance_1 0.774790 0.04509 17.2 0.0000 0.5974
LFrance_1 0.000377414 9.312e-005 4.05 0.0001 0.0763
sigma 0.00272799 RSS 0.00148094365
log-likelihood 902.541 DW 2.45
no. of observations 201 no. of parameters 2
mean(DLFrance) 0.00670529 var(DLFrance) 1.43476e-005
EQ(38) Modelling DLGermany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLGermany_1 0.365669 0.06631 5.51 0.0000 0.1337
Constant 0.0197762 0.01776 1.11 0.2667 0.0063
LGermany_1 -0.00453649 0.004816 -0.942 0.3474 0.0045
Trend 9.17538e-007 1.418e-005 0.0647 0.9485 0.0000
sigma 0.00266156 RSS 0.00139552576
R^2 0.254234 F(3,197) = 22.39 [0.000]**
log-likelihood 908.511 DW 2.03
no. of observations 201 no. of parameters 4
mean(DLGermany) 0.00287059 var(DLGermany) 9.30977e-006
EQ(39) Modelling DLGermany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLGermany_1 0.365186 0.06572 5.56 0.0000 0.1349
Constant 0.0186701 0.004802 3.89 0.0001 0.0709
LGermany_1 -0.00423450 0.001187 -3.57 0.0005 0.0604
sigma 0.00265486 RSS 0.00139555543
R^2 0.254218 F(2,198) = 33.75 [0.000]**
log-likelihood 908.509 DW 2.03
no. of observations 201 no. of parameters 3
mean(DLGermany) 0.00287059 var(DLGermany) 9.30977e-006
EQ(40) Modelling DLGermany by OLS
The dataset is: new09.in7
The estimation sample is: 1973(6) - 1990(2)
Coefficient Std.Error t-value t-prob Part.R^2
DLGermany_1 0.468260 0.06223 7.52 0.0000 0.2215
LGermany_1 0.000374458 6.579e-005 5.69 0.0000 0.1400
sigma 0.00274739 RSS 0.00150208012
log-likelihood 901.116 DW 2.11
no. of observations 201 no. of parameters 2
mean(DLGermany) 0.00287059 var(DLGermany) 9.30977e-006
Unit-root tests
The dataset is: new09.in7
The sample is: 1974(5) - 1990(2)
LFrance: ADF tests (T=190, Constant; 5%=-2.88 1%=-3.47)
D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob
12 -1.357 0.99925 0.002062 3.358 0.0010 -12.30
11 -1.542 0.99913 0.002121 0.04993 0.9602 -12.25 0.0010
10 -1.553 0.99912 0.002115 0.7887 0.4314 -12.26 0.0042
9 -1.613 0.99909 0.002112 1.623 0.1063 -12.26 0.0090
8 -1.762 0.99901 0.002122 0.03910 0.9689 -12.26 0.0067
7 -1.782 0.99901 0.002116 -1.004 0.3165 -12.27 0.0142
6 -1.693 0.99906 0.002116 4.666 0.0000 -12.28 0.0183
5 -2.310 0.99866 0.002233 2.328 0.0210 -12.17 0.0000
4 -2.774 0.99840 0.002260 -2.795 0.0057 -12.15 0.0000
3 -2.300 0.99867 0.002301 5.080 0.0000 -12.12 0.0000
2 -3.405* 0.99797 0.002450 2.586 0.0105 -12.00 0.0000
1 -4.317** 0.99751 0.002487 9.193 0.0000 -11.98 0.0000
0 -10.13** 0.99434 0.002988 -11.62 0.0000
LBelgium: ADF tests (T=190, Constant; 5%=-2.88 1%=-3.47)
D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob
12 -1.597 0.99792 0.003028 3.324 0.0011 -11.53
11 -1.894 0.99747 0.003112 -0.2390 0.8114 -11.48 0.0011
10 -1.884 0.99751 0.003104 -0.7801 0.4364 -11.49 0.0046
9 -1.817 0.99761 0.003101 1.347 0.1798 -11.50 0.0097
8 -2.001 0.99739 0.003108 1.148 0.2525 -11.50 0.0103
7 -2.202 0.99715 0.003111 -0.7425 0.4588 -11.50 0.0125
6 -2.115 0.99730 0.003107 1.692 0.0924 -11.51 0.0193
5 -2.457 0.99691 0.003122 2.009 0.0460 -11.50 0.0122
4 -2.943* 0.99635 0.003148 3.500 0.0006 -11.49 0.0051
3 -3.999** 0.99511 0.003242 0.9936 0.3217 -11.44 0.0001
2 -4.581** 0.99471 0.003242 0.9753 0.3307 -11.44 0.0001
1 -5.312** 0.99430 0.003242 4.105 0.0001 -11.45 0.0002
0 -8.407** 0.99199 0.003376 -11.37 0.0000
LGermany: ADF tests (T=190, Constant; 5%=-2.88 1%=-3.47)
D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob
12 -0.9617 0.99863 0.002338 3.426 0.0008 -12.05
11 -1.337 0.99805 0.002408 3.889 0.0001 -11.99 0.0008
10 -1.903 0.99715 0.002502 0.7668 0.4442 -11.92 0.0000
9 -2.039 0.99698 0.002499 1.961 0.0514 -11.93 0.0000
8 -2.395 0.99648 0.002518 1.040 0.2996 -11.92 0.0000
7 -2.650 0.99618 0.002519 1.808 0.0723 -11.92 0.0000
6 -3.180* 0.99554 0.002535 -2.974 0.0033 -11.91 0.0000
5 -2.549 0.99643 0.002588 -0.3619 0.7179 -11.88 0.0000
4 -2.535 0.99654 0.002582 -0.8986 0.3700 -11.89 0.0000
3 -2.411 0.99677 0.002581 1.527 0.1286 -11.89 0.0000
2 -2.785 0.99634 0.002590 0.9681 0.3343 -11.89 0.0000
1 -3.062* 0.99607 0.002590 5.313 0.0000 -11.90 0.0000
0 -4.730** 0.99385 0.002771 -11.77 0.0000
LFrance/Belgium: ADF tests (T=190, Constant; 5%=-2.88 1%=-3.47)
D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob
12 -1.110 0.99116 0.01047 -0.8751 0.3827 -9.048
11 -1.126 0.99105 0.01046 -1.072 0.2852 -9.054 0.3827
10 -1.142 0.99092 0.01047 -2.186 0.0301 -9.058 0.3861
9 -1.188 0.99045 0.01058 1.229 0.2208 -9.042 0.0864
8 -1.165 0.99063 0.01059 2.479 0.0141 -9.044 0.0884
7 -1.143 0.99068 0.01074 0.3712 0.7109 -9.021 0.0152
6 -1.144 0.99069 0.01072 -0.2431 0.8082 -9.031 0.0270
5 -1.149 0.99067 0.01069 -1.023 0.3075 -9.041 0.0454
4 -1.158 0.99060 0.01069 -0.8568 0.3926 -9.046 0.0517
3 -1.161 0.99058 0.01068 -2.119 0.0355 -9.052 0.0636
2 -1.226 0.98997 0.01078 -0.9129 0.3625 -9.039 0.0247
1 -1.289 0.98948 0.01078 5.761 0.0000 -9.045 0.0296
0 -1.090 0.99037 0.01166 -8.892 0.0000
LFrance/Germany: ADF tests (T=190, Constant; 5%=-2.88 1%=-3.47)
D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob
12 -0.6070 0.99789 0.01055 -2.396 0.0176 -9.033
11 -0.5990 0.99789 0.01069 -1.805 0.0728 -9.011 0.0176
10 -0.5818 0.99794 0.01076 -0.3928 0.6949 -9.003 0.0119
9 -0.5804 0.99795 0.01073 1.063 0.2891 -9.013 0.0288
8 -0.5674 0.99800 0.01073 2.019 0.0450 -9.017 0.0376
7 -0.5464 0.99805 0.01083 2.034 0.0434 -9.005 0.0145
6 -0.5394 0.99806 0.01092 0.1321 0.8950 -8.993 0.0057
5 -0.5412 0.99806 0.01089 -0.6065 0.5449 -9.004 0.0107
4 -0.5307 0.99810 0.01087 -0.7124 0.4771 -9.012 0.0165
3 -0.5201 0.99814 0.01086 -0.9540 0.3413 -9.020 0.0231
2 -0.5204 0.99814 0.01085 -0.5035 0.6152 -9.026 0.0274
1 -0.5291 0.99812 0.01083 5.395 0.0000 -9.035 0.0390
0 -0.5403 0.99794 0.01161 -8.901 0.0000
LBelgium/Germany: ADF tests (T=190, Constant; 5%=-2.88 1%=-3.47)
D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob
12 -0.6166 0.99782 0.006538 0.5495 0.5834 -9.990
11 -0.6083 0.99785 0.006525 -1.173 0.2423 -9.998 0.5834
10 -0.6252 0.99779 0.006532 -1.808 0.0722 -10.00 0.4350
9 -0.6861 0.99756 0.006573 0.9361 0.3505 -9.993 0.1806
8 -0.6261 0.99778 0.006571 0.6478 0.5180 -9.999 0.2177
7 -0.5868 0.99793 0.006560 1.005 0.3160 -10.01 0.2881
6 -0.5306 0.99813 0.006560 -0.2897 0.7724 -10.01 0.3028
5 -0.5494 0.99807 0.006544 1.481 0.1402 -10.02 0.3982
4 -0.4962 0.99825 0.006565 1.109 0.2690 -10.02 0.3054
3 -0.4448 0.99843 0.006569 1.607 0.1097 -10.02 0.2985
2 -0.3763 0.99867 0.006597 -1.197 0.2328 -10.02 0.2126
1 -0.4202 0.99852 0.006605 4.826 0.0000 -10.02 0.2005
0 -0.2665 0.99900 0.006985 -9.917 0.0003
(3) 黄色部分是重要的数据
SYS( 1) Estimating the system by OLS
The dataset is: new09.in7
The estimation sample is: 1974(4) - 1990(2)
URF equation for: LFrance/Belgium
Coefficient Std.Error t-value t-prob
LFrance/Belgium_1 1.27793 0.07985 16.0 0.0000
LFrance/Belgium_2 -0.309790 0.1287 -2.41 0.0172
LFrance/Belgium_3 -0.135067 0.1286 -1.05 0.2954
LFrance/Belgium_4 0.0602494 0.1277 0.472 0.6377
LFrance/Belgium_5 -0.0144114 0.1285 -0.112 0.9109
LFrance/Belgium_6 -0.00309561 0.1268 -0.0244 0.9806
LFrance/Belgium_7 0.0294940 0.1275 0.231 0.8174
LFrance/Belgium_8 0.162431 0.1272 1.28 0.2037
LFrance/Belgium_9 -0.0270322 0.1271 -0.213 0.8319
LFrance/Belgium_10 -0.267770 0.1229 -2.18 0.0309
LFrance/Belgium_11 0.111255 0.1227 0.906 0.3661
LFrance/Belgium_12 -0.00155779 0.07546 -0.0206 0.9836
LFrance_1 0.385940 0.4200 0.919 0.3596
LFrance_2 -1.31872 0.7486 -1.76 0.0801
LFrance_3 1.97146 0.7607 2.59 0.0105
LFrance_4 -1.04391 0.7477 -1.40 0.1647
LFrance_5 0.324587 0.7649 0.424 0.6719
LFrance_6 -0.567520 0.7572 -0.750 0.4547
LFrance_7 0.460296 0.7576 0.608 0.5444
LFrance_8 -0.100251 0.7546 -0.133 0.8945
LFrance_9 -0.223160 0.7339 -0.304 0.7615
LFrance_10 0.760773 0.7369 1.03 0.3035
LFrance_11 -1.23509 0.7159 -1.73 0.0865
LFrance_12 0.597129 0.4041 1.48 0.1416
LBelgium_1 -0.0722244 0.2914 -0.248 0.8046
LBelgium_2 0.337063 0.4337 0.777 0.4382
LBelgium_3 -0.410626 0.4445 -0.924 0.3570
LBelgium_4 -0.450874 0.4508 -1.00 0.3188
LBelgium_5 0.432919 0.4426 0.978 0.3295
LBelgium_6 -0.191279 0.4382 -0.437 0.6631
LBelgium_7 0.115131 0.4390 0.262 0.7935
LBelgium_8 -0.250199 0.4402 -0.568 0.5706
LBelgium_9 0.637659 0.4348 1.47 0.1445
LBelgium_10 -0.503691 0.4372 -1.15 0.2511
LBelgium_11 0.479278 0.4390 1.09 0.2766
LBelgium_12 -0.111923 0.2770 -0.404 0.6867
Constant -0.307948 0.1164 -2.65 0.0090
sigma = 0.0101403 RSS = 0.01583507807
URF equation for: LFrance
Coefficient Std.Error t-value t-prob
LFrance/Belgium_1 0.0104870 0.01579 0.664 0.5075
LFrance/Belgium_2 -0.0111617 0.02544 -0.439 0.6614
LFrance/Belgium_3 -0.0237979 0.02544 -0.936 0.3509
LFrance/Belgium_4 0.0589219 0.02525 2.33 0.0209
LFrance/Belgium_5 -0.00695515 0.02541 -0.274 0.7847
LFrance/Belgium_6 -0.0373424 0.02508 -1.49 0.1385
LFrance/Belgium_7 0.0230060 0.02521 0.912 0.3630
LFrance/Belgium_8 -0.0271489 0.02516 -1.08 0.2822
LFrance/Belgium_9 0.0329476 0.02514 1.31 0.1919
LFrance/Belgium_10 -0.0419706 0.02430 -1.73 0.0862
LFrance/Belgium_11 0.0288738 0.02427 1.19 0.2360
LFrance/Belgium_12 -0.00523507 0.01492 -0.351 0.7262
LFrance_1 1.47084 0.08304 17.7 0.0000
LFrance_2 -0.493216 0.1480 -3.33 0.0011
LFrance_3 0.361839 0.1504 2.41 0.0173
LFrance_4 -0.638682 0.1478 -4.32 0.0000
LFrance_5 0.316810 0.1512 2.09 0.0378
LFrance_6 0.290642 0.1497 1.94 0.0540
LFrance_7 -0.320522 0.1498 -2.14 0.0339
LFrance_8 -0.0522679 0.1492 -0.350 0.7266
LFrance_9 0.0848384 0.1451 0.585 0.5596
LFrance_10 0.150978 0.1457 1.04 0.3017
LFrance_11 -0.118942 0.1415 -0.840 0.4020
LFrance_12 -0.0425372 0.07991 -0.532 0.5953
LBelgium_1 -0.103993 0.05762 -1.80 0.0730
LBelgium_2 0.205379 0.08574 2.40 0.0178
LBelgium_3 -0.254146 0.08788 -2.89 0.0044
LBelgium_4 0.0752933 0.08913 0.845 0.3995
LBelgium_5 0.0359132 0.08750 0.410 0.6821
LBelgium_6 0.110359 0.08664 1.27 0.2047
LBelgium_7 -0.131423 0.08680 -1.51 0.1320
LBelgium_8 0.0330297 0.08703 0.380 0.7048
LBelgium_9 0.0910438 0.08597 1.06 0.2912
LBelgium_10 -0.224310 0.08645 -2.59 0.0104
LBelgium_11 0.124484 0.08679 1.43 0.1535
LBelgium_12 0.0203587 0.05477 0.372 0.7106
Constant 0.0354463 0.02301 1.54 0.1255
sigma = 0.00200488 RSS = 0.0006190098657
URF equation for: LBelgium
Coefficient Std.Error t-value t-prob
LFrance/Belgium_1 0.00357489 0.02257 0.158 0.8743
LFrance/Belgium_2 -0.0126261 0.03636 -0.347 0.7289
LFrance/Belgium_3 -0.00383292 0.03636 -0.105 0.9162
LFrance/Belgium_4 0.0264039 0.03609 0.732 0.4655
LFrance/Belgium_5 -0.0208393 0.03632 -0.574 0.5670
LFrance/Belgium_6 -0.00161438 0.03585 -0.0450 0.9641
LFrance/Belgium_7 -0.0247817 0.03604 -0.688 0.4927
LFrance/Belgium_8 0.0423463 0.03596 1.18 0.2408
LFrance/Belgium_9 -0.00238368 0.03593 -0.0663 0.9472
LFrance/Belgium_10 -0.0369844 0.03474 -1.06 0.2887
LFrance/Belgium_11 0.0293605 0.03469 0.846 0.3986
LFrance/Belgium_12 -0.0180046 0.02133 -0.844 0.3998
LFrance_1 -0.0995052 0.1187 -0.838 0.4032
LFrance_2 0.165785 0.2116 0.784 0.4345
LFrance_3 0.101434 0.2150 0.472 0.6377
LFrance_4 -0.177123 0.2113 -0.838 0.4032
LFrance_5 0.0245123 0.2162 0.113 0.9099
LFrance_6 0.340829 0.2140 1.59 0.1133
LFrance_7 -0.374297 0.2141 -1.75 0.0824
LFrance_8 -0.0700255 0.2133 -0.328 0.7431
LFrance_9 0.177214 0.2074 0.854 0.3942
LFrance_10 0.180473 0.2082 0.867 0.3875
LFrance_11 -0.394862 0.2023 -1.95 0.0528
LFrance_12 0.173936 0.1142 1.52 0.1298
LBelgium_1 1.06640 0.08235 12.9 0.0000
LBelgium_2 -0.160496 0.1226 -1.31 0.1923
LBelgium_3 -0.178947 0.1256 -1.42 0.1563
LBelgium_4 0.305609 0.1274 2.40 0.0176
LBelgium_5 -0.0198356 0.1251 -0.159 0.8742
LBelgium_6 -0.126473 0.1238 -1.02 0.3087
LBelgium_7 -0.0291057 0.1241 -0.235 0.8148
LBelgium_8 0.151122 0.1244 1.21 0.2263
LBelgium_9 -0.0326786 0.1229 -0.266 0.7906
LBelgium_10 -0.0869635 0.1236 -0.704 0.4826
LBelgium_11 0.0450370 0.1241 0.363 0.7171
LBelgium_12 -0.00605258 0.07829 -0.0773 0.9385
Constant 0.0622459 0.03289 1.89 0.0603
sigma = 0.00286568 RSS = 0.001264666523
log-likelihood 2436.99964 -T/2log|Omega| 3250.05141
|Omega| 1.6600113e-015 log|Y'Y/T| -6.84461615
R^2(LR) 1 R^2(LM) 0.990616
no. of observations 191 no. of parameters 111
F-test on regressors except unrestricted: F(111,456) = 35967.7 [0.0000] **
F-tests on retained regressors, F(3,152) =
LFrance/Belgium_1 84.5353 [0.000]**LFrance/Belgium_2 1.98731 [0.118]
LFrance/Belgium_3 0.620620 [0.603] LFrance/Belgium_4 1.84291 [0.142]
LFrance/Belgium_5 0.119660 [0.948] LFrance/Belgium_6 0.767952 [0.514]
LFrance/Belgium_7 0.572443 [0.634] LFrance/Belgium_8 1.77795 [0.154]
LFrance/Belgium_9 0.649431 [0.584] LFrance/Belgium_10 2.61722 [0.053]
LFrance/Belgium_11 0.826255 [0.481] LFrance/Belgium_12 0.242228 [0.867]
LFrance_1 112.831 [0.000]** LFrance_2 5.30298 [0.002]**
LFrance_3 3.91118 [0.010]* LFrance_4 6.60333 [0.000]**
LFrance_5 1.52990 [0.209] LFrance_6 1.87779 [0.136]
LFrance_7 2.16555 [0.094] LFrance_8 0.0662745 [0.978]
LFrance_9 0.317530 [0.813] LFrance_10 0.823492 [0.483]
LFrance_11 2.34789 [0.075] LFrance_12 1.90022 [0.132]
LBelgium_1 64.1130 [0.000]** LBelgium_2 3.25593 [0.023]*
LBelgium_3 3.14657 [0.027]* LBelgium_4 2.19467 [0.091]
LBelgium_5 0.372815 [0.773] LBelgium_6 1.27165 [0.286]
LBelgium_7 0.801519 [0.495] LBelgium_8 0.578631 [0.630]
LBelgium_9 1.09842 [0.352] LBelgium_10 2.56105 [0.057]
LBelgium_11 1.01961 [0.386] LBelgium_12 0.117583 [0.950]
Constant 3.88549 [0.010]*
correlation of URF residuals (standard deviations on diagonal)
LFrance/Belgium LFrance LBelgium
LFrance/Belgium 0.010140 0.051944 -0.040747
LFrance 0.051944 0.0020049 0.24802
LBelgium -0.040747 0.24802 0.0028657
correlation between actual and fitted
LFrance/Belgium LFrance LBelgium
0.99549 0.99999 0.99995
I(1) cointegration analysis, 1974(4) - 1990(2)
eigenvalue loglik for rank
2409.911 0
0.15264 2425.728 1
0.081940 2433.893 2
0.032007 2437.000 3
H0:rank<= Trace test [ Prob]
0 54.177 [0.000] **
1 22.542 [0.022] *
2 6.2133 [0.181]
Asymptotic p-values based on: Restricted constant
Restricted variables:
[0] = Constant
Number of lags used in the analysis: 12
beta (scaled on diagonal; cointegrating vectors in columns)
LFrance/Belgium 1.0000 0.80740 -5.6671
LFrance -1.2433 1.0000 1.7194
LBelgium 1.7382 -1.8156 1.0000
Constant -0.18262 4.8069 -22.299
alpha
LFrance/Belgium -0.064213 -0.067818 -0.00028364
LFrance -0.0069044 0.0027978 -0.00092994
LBelgium -0.028709 0.012456 0.00012874
long-run matrix, rank 3
LFrance/Belgium LFrance LBelgium Constant
LFrance/Belgium -0.11736 0.011531 0.011234 -0.30795
LFrance 0.00062463 0.0097832 -0.018011 0.035446
LBelgium -0.019382 0.048371 -0.072388 0.062246
// Batch code for SYS( 1)
module("PcGive");
package("PcGive", "Multiple-equation");
usedata("new09.in7");
system
{
Y = "LFrance/Belgium", LFrance, LBelgium;
Z = Constant, "LFrance/Belgium_1", "LFrance/Belgium_2",
"LFrance/Belgium_3", "LFrance/Belgium_4", "LFrance/Belgium_5",
"LFrance/Belgium_6", "LFrance/Belgium_7", "LFrance/Belgium_8",
"LFrance/Belgium_9", "LFrance/Belgium_10", "LFrance/Belgium_11",
"LFrance/Belgium_12", LFrance_1, LFrance_2, LFrance_3,
LFrance_4, LFrance_5, LFrance_6, LFrance_7, LFrance_8,
LFrance_9, LFrance_10, LFrance_11, LFrance_12, LBelgium_1,
LBelgium_2, LBelgium_3, LBelgium_4, LBelgium_5, LBelgium_6,
LBelgium_7, LBelgium_8, LBelgium_9, LBelgium_10, LBelgium_11,
LBelgium_12;
}
estimate("OLS", 1974, 4, 1990, 2);
dynamics();
SYS( 2) Cointegrated VAR
The dataset is: new09.in7
The estimation sample is: 1974(4) - 1990(2)
Cointegrated VAR (12) in:
[0] = LFrance/Belgium
[1] = LFrance
[2] = LBelgium
Restricted variables:
[0] = Constant
Number of lags used in the analysis: 12
beta
LFrance/Belgium 1.0000
LFrance -1.2433
LBelgium 1.7382
Constant -0.18262
alpha
LFrance/Belgium -0.064213
LFrance -0.0069044
LBelgium -0.028709
Standard errors of alpha
LFrance/Belgium 0.023864
LFrance 0.0046458
LBelgium 0.0066255
Restricted long-run matrix, rank 1
LFrance/Belgium LFrance LBelgium Constant
LFrance/Belgium -0.064213 0.079837 -0.11162 0.011726
LFrance -0.0069044 0.0085843 -0.012001 0.0012609
LBelgium -0.028709 0.035694 -0.049902 0.0052427
Standard errors of long-run matrix
LFrance/Belgium 0.023864 0.029670 0.041480 0.0043579
LFrance 0.0046458 0.0057762 0.0080753 0.00084840
LBelgium 0.0066255 0.0082376 0.011517 0.0012099
Reduced form beta
LFrance/Belgium -1.0000
LFrance 1.2433
LBelgium -1.7382
Constant 0.18262
Standard errors of reduced form beta
LFrance/Belgium 0.00000
LFrance 0.28616
LBelgium 0.45912
Constant 0.72170
Moving-average impact matrix
0.96145 4.6128 -3.2599
1.7958 37.208 -12.965
0.73138 23.960 -7.3983
log-likelihood 2425.72842 -T/2log|Omega| 3238.7802
no. of observations 191 no. of parameters 105
rank of long-run matrix 1 no. long-run restrictions 0
beta is not identified
No restrictions imposed
LFrance/Belgium: Portmanteau(12): 1.48753
LFrance : Portmanteau(12): 8.94026
LBelgium : Portmanteau(12): 8.36552
LFrance/Belgium: Normality test: Chi^2(2) = 38.144 [0.0000]**
LFrance : Normality test: Chi^2(2) = 22.000 [0.0000]**
LBelgium : Normality test: Chi^2(2) = 2.9930 [0.2239]
LFrance/Belgium: ARCH 1-7 test: F(7,142) = 3.9842 [0.0005]**
LFrance : ARCH 1-7 test: F(7,142) = 0.28833 [0.9576]
LBelgium : ARCH 1-7 test: F(7,142) = 0.62275 [0.7365]
LFrance/Belgium: Hetero test: F(72,81) = 0.91774 [0.6438]
LFrance : Hetero test: F(72,81) = 0.50705 [0.9982]
LBelgium : Hetero test: F(72,81) = 0.51161 [0.9979]
Vector Portmanteau(12): 45.4787
Vector Normality test: Chi^2(6) = 61.569 [0.0000]**
Vector Hetero test: F(432,463)= 0.81429 [0.9848]
SYS( 3) Estimating the system by OLS
The dataset is: new09.in7
The estimation sample is: 1974(4) - 1990(2)
URF equation for: LFrance/Germany
Coefficient Std.Error t-value t-prob
LFrance/Germany_1 1.18243 0.08104 14.6 0.0000
LFrance/Germany_2 -0.254381 0.1257 -2.02 0.0447
LFrance/Germany_3 -0.0111361 0.1263 -0.0882 0.9299
LFrance/Germany_4 0.0268588 0.1254 0.214 0.8307
LFrance/Germany_5 -0.0811245 0.1244 -0.652 0.5153
LFrance/Germany_6 0.0385568 0.1183 0.326 0.7450
LFrance/Germany_7 0.138214 0.1182 1.17 0.2442
LFrance/Germany_8 -0.0552427 0.1194 -0.463 0.6442
LFrance/Germany_9 0.0907171 0.1157 0.784 0.4344
LFrance/Germany_10 -0.0938167 0.1118 -0.839 0.4028
LFrance/Germany_11 -0.105044 0.1098 -0.957 0.3401
LFrance/Germany_12 0.00748530 0.07084 0.106 0.9160
LFrance_1 0.568700 0.3908 1.46 0.1476
LFrance_2 -1.66185 0.6607 -2.52 0.0129
LFrance_3 2.12237 0.6768 3.14 0.0021
LFrance_4 -1.31341 0.6667 -1.97 0.0506
LFrance_5 0.906003 0.6936 1.31 0.1935
LFrance_6 -1.40503 0.6899 -2.04 0.0434
LFrance_7 1.08005 0.6955 1.55 0.1225
LFrance_8 -0.128726 0.6988 -0.184 0.8541
LFrance_9 -0.665868 0.6794 -0.980 0.3286
LFrance_10 0.497608 0.6871 0.724 0.4701
LFrance_11 -0.330165 0.6828 -0.484 0.6294
LFrance_12 0.384502 0.3981 0.966 0.3356
LGermany_1 -0.399822 0.3462 -1.15 0.2500
LGermany_2 0.428888 0.5277 0.813 0.4176
LGermany_3 -0.113032 0.5192 -0.218 0.8279
LGermany_4 -0.306531 0.5163 -0.594 0.5536
LGermany_5 0.349434 0.5177 0.675 0.5007
LGermany_6 0.475014 0.5024 0.945 0.3459
LGermany_7 -1.39212 0.5044 -2.76 0.0065
LGermany_8 0.817985 0.5289 1.55 0.1240
LGermany_9 0.545546 0.5297 1.03 0.3046
LGermany_10 -0.227258 0.5266 -0.432 0.6666
LGermany_11 0.284420 0.5199 0.547 0.5851
LGermany_12 -0.450999 0.3254 -1.39 0.1677
Constant -0.131695 0.2478 -0.531 0.5959
sigma = 0.00994372 RSS = 0.01522714336
URF equation for: LFrance
Coefficient Std.Error t-value t-prob
LFrance/Germany_1 0.00237054 0.01727 0.137 0.8910
LFrance/Germany_2 -0.0222292 0.02678 -0.830 0.4079
LFrance/Germany_3 0.0109498 0.02691 0.407 0.6847
LFrance/Germany_4 0.0263148 0.02672 0.985 0.3263
LFrance/Germany_5 0.0127706 0.02651 0.482 0.6306
LFrance/Germany_6 -0.0559919 0.02521 -2.22 0.0278
LFrance/Germany_7 0.0377117 0.02519 1.50 0.1365
LFrance/Germany_8 -0.0132246 0.02544 -0.520 0.6039
LFrance/Germany_9 0.0120699 0.02466 0.489 0.6252
LFrance/Germany_10 -0.0146932 0.02383 -0.617 0.5384
LFrance/Germany_11 -0.00135161 0.02339 -0.0578 0.9540
LFrance/Germany_12 0.00345209 0.01509 0.229 0.8194
LFrance_1 1.39022 0.08326 16.7 0.0000
LFrance_2 -0.322240 0.1408 -2.29 0.0234
LFrance_3 0.167007 0.1442 1.16 0.2486
LFrance_4 -0.556010 0.1421 -3.91 0.0001
LFrance_5 0.372277 0.1478 2.52 0.0128
LFrance_6 0.308417 0.1470 2.10 0.0375
LFrance_7 -0.399356 0.1482 -2.69 0.0078
LFrance_8 -0.0658622 0.1489 -0.442 0.6589
LFrance_9 0.193218 0.1448 1.33 0.1840
LFrance_10 -0.0535239 0.1464 -0.366 0.7152
LFrance_11 -0.0500518 0.1455 -0.344 0.7313
LFrance_12 0.0131828 0.08482 0.155 0.8767
LGermany_1 0.0410234 0.07377 0.556 0.5790
LGermany_2 -0.00344278 0.1124 -0.0306 0.9756
LGermany_3 0.0141478 0.1106 0.128 0.8984
LGermany_4 0.0697912 0.1100 0.634 0.5268
LGermany_5 -0.137718 0.1103 -1.25 0.2138
LGermany_6 0.0525577 0.1071 0.491 0.6242
LGermany_7 -0.0388818 0.1075 -0.362 0.7180
LGermany_8 0.00803967 0.1127 0.0713 0.9432
LGermany_9 -0.0590020 0.1129 -0.523 0.6019
LGermany_10 0.0424012 0.1122 0.378 0.7060
LGermany_11 -0.0357749 0.1108 -0.323 0.7472
LGermany_12 0.0536652 0.06932 0.774 0.4401
Constant -0.0141731 0.05281 -0.268 0.7888
sigma = 0.00211877 RSS = 0.0006913324248
URF equation for: LGermany
Coefficient Std.Error t-value t-prob
LFrance/Germany_1 -0.00995883 0.01876 -0.531 0.5962
LFrance/Germany_2 0.00806314 0.02909 0.277 0.7820
LFrance/Germany_3 0.0364057 0.02923 1.25 0.2149
LFrance/Germany_4 -0.0432459 0.02903 -1.49 0.1383
LFrance/Germany_5 0.00823761 0.02879 0.286 0.7752
LFrance/Germany_6 -0.00739354 0.02738 -0.270 0.7875
LFrance/Germany_7 -0.000402614 0.02737 -0.0147 0.9883
LFrance/Germany_8 0.00640793 0.02763 0.232 0.8169
LFrance/Germany_9 -0.0113554 0.02679 -0.424 0.6722
LFrance/Germany_10 0.00921922 0.02588 0.356 0.7222
LFrance/Germany_11 0.00209335 0.02541 0.0824 0.9344
LFrance/Germany_12 -0.00116559 0.01640 -0.0711 0.9434
LFrance_1 0.116560 0.09044 1.29 0.1994
LFrance_2 -0.0817372 0.1529 -0.535 0.5937
LFrance_3 0.0121145 0.1566 0.0773 0.9385
LFrance_4 0.0130135 0.1543 0.0843 0.9329
LFrance_5 -0.117251 0.1605 -0.730 0.4663
LFrance_6 0.157006 0.1597 0.983 0.3270
LFrance_7 -0.109813 0.1610 -0.682 0.4961
LFrance_8 0.0848636 0.1617 0.525 0.6006
LFrance_9 0.267655 0.1573 1.70 0.0908
LFrance_10 -0.349672 0.1590 -2.20 0.0294
LFrance_11 0.0257251 0.1580 0.163 0.8709
LFrance_12 0.0127173 0.09213 0.138 0.8904
LGermany_1 1.16865 0.08013 14.6 0.0000
LGermany_2 -0.244820 0.1221 -2.00 0.0468
LGermany_3 0.118195 0.1202 0.984 0.3269
LGermany_4 -0.108472 0.1195 -0.908 0.3655
LGermany_5 0.0930923 0.1198 0.777 0.4384
LGermany_6 -0.404054 0.1163 -3.47 0.0007
LGermany_7 0.393746 0.1167 3.37 0.0009
LGermany_8 -0.144701 0.1224 -1.18 0.2390
LGermany_9 0.0697007 0.1226 0.569 0.5705
LGermany_10 -0.00537249 0.1219 -0.0441 0.9649
LGermany_11 0.211525 0.1203 1.76 0.0807
LGermany_12 -0.219975 0.07530 -2.92 0.0040
Constant 0.174402 0.05736 3.04 0.0028
sigma = 0.00230147 RSS = 0.0008156992738
log-likelihood 2474.65444 -T/2log|Omega| 3287.70622
|Omega| 1.11910951e-015 log|Y'Y/T| -6.29263643
R^2(LR) 1 R^2(LM) 0.986345
no. of observations 191 no. of parameters 111
F-test on regressors except unrestricted: F(111,456) = 49332.7 [0.0000] **
F-tests on retained regressors, F(3,152) =
LFrance/Germany_1 72.3906 [0.000]**LFrance/Germany_2 1.56110 [0.201]
LFrance/Germany_3 0.530140 [0.662] LFrance/Germany_4 1.39341 [0.247]
LFrance/Germany_5 0.271015 [0.846] LFrance/Germany_6 1.80232 [0.149]
LFrance/Germany_7 1.10053 [0.351] LFrance/Germany_8 0.190291 [0.903]
LFrance/Germany_9 0.369824 [0.775] LFrance/Germany_10 0.419550 [0.739]
LFrance/Germany_11 0.312637 [0.816] LFrance/Germany_12 0.0250941 [0.995]
LFrance_1 95.1567 [0.000]** LFrance_2 3.29445 [0.022]*
LFrance_3 3.43553 [0.019]* LFrance_4 6.12358 [0.001]**
LFrance_5 3.07888 [0.029]* LFrance_6 3.47117 [0.018]*
LFrance_7 3.72339 [0.013]* LFrance_8 0.215050 [0.886]
LFrance_9 1.79252 [0.151] LFrance_10 1.88604 [0.134]
LFrance_11 0.127775 [0.944] LFrance_12 0.307600 [0.820]
LGermany_1 75.0178 [0.000]** LGermany_2 1.70258 [0.169]
LGermany_3 0.353433 [0.787] LGermany_4 0.668355 [0.573]
LGermany_5 1.15449 [0.329] LGermany_6 5.03881 [0.002]**
LGermany_7 7.14852 [0.000]** LGermany_8 1.38611 [0.249]
LGermany_9 0.651169 [0.583] LGermany_10 0.133081 [0.940]
LGermany_11 1.30487 [0.275] LGermany_12 4.20461 [0.007]**
Constant 3.57993 [0.015]*
correlation of URF residuals (standard deviations on diagonal)
LFrance/Germany LFrance LGermany
LFrance/Germany 0.0099437 0.15995 0.098497
LFrance 0.15995 0.0021188 0.25420
LGermany 0.098497 0.25420 0.0023015
correlation between actual and fitted
LFrance/Germany LFrance LGermany
0.99918 0.99999 0.99991
I(1) cointegration analysis, 1974(4) - 1990(2)
eigenvalue loglik for rank
2451.615 0
0.15287 2467.458 1
0.056413 2473.003 2
0.017142 2474.654 3
H0:rank<= Trace test [ Prob]
0 46.080 [0.002] **
1 14.393 [0.269]
2 3.3025 [0.536]
Asymptotic p-values based on: Restricted constant
Restricted variables:
[0] = Constant
Number of lags used in the analysis: 12
beta (scaled on diagonal; cointegrating vectors in columns)
LFrance/Germany 1.0000 -0.33556 0.052729
LFrance 0.044039 1.0000 -0.31341
LGermany -1.3365 -2.0026 1.0000
Constant 4.1392 4.5402 -2.9665
alpha
LFrance/Germany -0.097026 0.057534 -0.0029306
LFrance -0.0030697 -0.0046797 -0.0066679
LGermany 0.0070976 0.029879 -0.0031577
long-run matrix, rank 3
LFrance/Germany LFrance LGermany Constant
LFrance/Germany -0.11649 0.054180 0.011527 -0.13170
LFrance -0.0018510 -0.0027252 0.0068069 -0.014173
LGermany -0.0030948 0.031181 -0.072480 0.17440
SYS( 4) Cointegrated VAR
The dataset is: new09.in7
The estimation sample is: 1974(4) - 1990(2)
Cointegrated VAR (12) in:
[0] = LFrance/Germany
[1] = LFrance
[2] = LGermany
Restricted variables:
[0] = Constant
Number of lags used in the analysis: 12
beta
LFrance/Germany 1.0000
LFrance 0.044039
LGermany -1.3365
Constant 4.1392
alpha
LFrance/Germany -0.097026
LFrance -0.0030697
LGermany 0.0070976
Standard errors of alpha
LFrance/Germany 0.019970
LFrance 0.0042744
LGermany 0.0047086
Restricted long-run matrix, rank 1
LFrance/Germany LFrance LGermany Constant
LFrance/Germany -0.097026 -0.0042729 0.12968 -0.40161
LFrance -0.0030697 -0.00013519 0.0041029 -0.012706
LGermany 0.0070976 0.00031257 -0.0094863 0.029379
Standard errors of long-run matrix
LFrance/Germany 0.019970 0.00087944 0.026691 0.082659
LFrance 0.0042744 0.00018824 0.0057129 0.017692
LGermany 0.0047086 0.00020736 0.0062932 0.019490
Reduced form beta
LFrance/Germany -1.0000
LFrance -0.044039
LGermany 1.3365
Constant -4.1392
Standard errors of reduced form beta
LFrance/Germany 0.00000
LFrance 0.37250
LGermany 0.90508
Constant 2.2089
Moving-average impact matrix
-0.18798 16.848 4.7169
-0.62231 34.388 6.3657
-0.16115 13.738 3.7389
log-likelihood 2467.45781 -T/2log|Omega| 3280.50959
no. of observations 191 no. of parameters 105
rank of long-run matrix 1 no. long-run restrictions 0
beta is not identified
No restrictions imposed
LFrance/Germany: Portmanteau(12): 6.85112
LFrance : Portmanteau(12): 8.11577
LGermany : Portmanteau(12): 4.3378
LFrance/Germany: Normality test: Chi^2(2) = 33.424 [0.0000]**
LFrance : Normality test: Chi^2(2) = 38.314 [0.0000]**
LGermany : Normality test: Chi^2(2) = 3.1750 [0.2044]
LFrance/Germany: ARCH 1-7 test: F(7,142) = 1.1036 [0.3641]
LFrance : ARCH 1-7 test: F(7,142) = 0.12194 [0.9967]
LGermany : ARCH 1-7 test: F(7,142) = 0.44855 [0.8698]
LFrance/Germany: Hetero test: F(72,81) = 0.72380 [0.9184]
LFrance : Hetero test: F(72,81) = 0.70373 [0.9352]
LGermany : Hetero test: F(72,81) = 0.47543 [0.9992]
Vector Portmanteau(12): 58.9484
Vector Normality test: Chi^2(6) = 74.927 [0.0000]**
Vector Hetero test: F(432,463)= 0.60106 [1.0000]
SYS( 5) Estimating the system by OLS
The dataset is: new09.in7
The estimation sample is: 1974(4) - 1990(2)
URF equation for: LBelgium/Germany
Coefficient Std.Error t-value t-prob
LBelgium/Germany_1 1.24411 0.08057 15.4 0.0000
LBelgium/Germany_2 -0.429889 0.1303 -3.30 0.0012
LBelgium/Germany_3 0.242191 0.1348 1.80 0.0744
LBelgium/Germany_4 -0.0840452 0.1334 -0.630 0.5296
LBelgium/Germany_5 0.0963068 0.1260 0.764 0.4458
LBelgium/Germany_6 -0.184540 0.1258 -1.47 0.1446
LBelgium/Germany_7 0.102924 0.1285 0.801 0.4244
LBelgium/Germany_8 -0.0151956 0.1278 -0.119 0.9055
LBelgium/Germany_9 0.0137378 0.1228 0.112 0.9111
LBelgium/Germany_10 -0.0475944 0.1206 -0.395 0.6937
LBelgium/Germany_11 -0.0848517 0.1137 -0.746 0.4567
LBelgium/Germany_12 0.0888137 0.06890 1.29 0.1993
LBelgium_1 0.175739 0.1940 0.906 0.3664
LBelgium_2 -0.200036 0.2639 -0.758 0.4497
LBelgium_3 0.125802 0.2635 0.477 0.6337
LBelgium_4 -0.0591721 0.2655 -0.223 0.8239
LBelgium_5 0.160206 0.2678 0.598 0.5506
LBelgium_6 -0.228600 0.2653 -0.862 0.3902
LBelgium_7 -0.00589070 0.2654 -0.0222 0.9823
LBelgium_8 0.613126 0.2710 2.26 0.0250
LBelgium_9 -0.814964 0.2786 -2.93 0.0040
LBelgium_10 0.0607708 0.2843 0.214 0.8310
LBelgium_11 -0.0625811 0.2797 -0.224 0.8233
LBelgium_12 0.217344 0.1686 1.29 0.1992
LGermany_1 -0.373746 0.2162 -1.73 0.0859
LGermany_2 0.267891 0.3412 0.785 0.4336
LGermany_3 -0.00824147 0.3465 -0.0238 0.9811
LGermany_4 0.0151444 0.3493 0.0434 0.9655
LGermany_5 0.450110 0.3569 1.26 0.2092
LGermany_6 -0.376543 0.3550 -1.06 0.2905
LGermany_7 -0.351132 0.3493 -1.01 0.3164
LGermany_8 0.249408 0.3406 0.732 0.4651
LGermany_9 0.134627 0.3275 0.411 0.6816
LGermany_10 0.370252 0.3264 1.13 0.2584
LGermany_11 -0.204470 0.3263 -0.627 0.5319
LGermany_12 -0.0934401 0.2143 -0.436 0.6634
Constant -0.0804209 0.1089 -0.739 0.4613
sigma = 0.00614438 RSS = 0.00581402083
URF equation for: LBelgium
Coefficient Std.Error t-value t-prob
LBelgium/Germany_1 0.0564882 0.03530 1.60 0.1116
LBelgium/Germany_2 -0.0506352 0.05710 -0.887 0.3766
LBelgium/Germany_3 0.0121875 0.05907 0.206 0.8368
LBelgium/Germany_4 0.0364642 0.05844 0.624 0.5336
LBelgium/Germany_5 -0.0796239 0.05520 -1.44 0.1512
LBelgium/Germany_6 0.101231 0.05513 1.84 0.0683
LBelgium/Germany_7 -0.0832351 0.05629 -1.48 0.1413
LBelgium/Germany_8 0.0265802 0.05599 0.475 0.6356
LBelgium/Germany_9 -0.0570059 0.05381 -1.06 0.2911
LBelgium/Germany_10 0.0869395 0.05285 1.65 0.1020
LBelgium/Germany_11 -0.0532367 0.04982 -1.07 0.2870
LBelgium/Germany_12 0.0395195 0.03019 1.31 0.1924
LBelgium_1 0.951219 0.08498 11.2 0.0000
LBelgium_2 -0.140726 0.1156 -1.22 0.2255
LBelgium_3 -0.131885 0.1154 -1.14 0.2550
LBelgium_4 0.276431 0.1163 2.38 0.0187
LBelgium_5 -0.118454 0.1173 -1.01 0.3143
LBelgium_6 0.161914 0.1162 1.39 0.1656
LBelgium_7 -0.338599 0.1163 -2.91 0.0041
LBelgium_8 0.199237 0.1187 1.68 0.0953
LBelgium_9 0.0412499 0.1220 0.338 0.7358
LBelgium_10 -0.105923 0.1245 -0.850 0.3964
LBelgium_11 0.0981605 0.1225 0.801 0.4243
LBelgium_12 -0.00651130 0.07386 -0.0882 0.9299
LGermany_1 0.222759 0.09472 2.35 0.0199
LGermany_2 -0.293356 0.1495 -1.96 0.0515
LGermany_3 0.463101 0.1518 3.05 0.0027
LGermany_4 -0.370651 0.1531 -2.42 0.0166
LGermany_5 0.0287706 0.1564 0.184 0.8543
LGermany_6 1.07279e-005 0.1555 0.00 0.9999
LGermany_7 0.271382 0.1530 1.77 0.0781
LGermany_8 -0.00624830 0.1492 -0.0419 0.9667
LGermany_9 -0.128807 0.1435 -0.898 0.3707
LGermany_10 0.0195920 0.1430 0.137 0.8912
LGermany_11 -0.151840 0.1430 -1.06 0.2899
LGermany_12 0.0880479 0.09389 0.938 0.3498
Constant -0.229299 0.04770 -4.81 0.0000
sigma = 0.00269192 RSS = 0.001115951271
URF equation for: LGermany
Coefficient Std.Error t-value t-prob
LBelgium/Germany_1 -0.0362026 0.03033 -1.19 0.2344
LBelgium/Germany_2 0.0568306 0.04906 1.16 0.2485
LBelgium/Germany_3 0.0147421 0.05075 0.290 0.7718
LBelgium/Germany_4 -0.0108972 0.05021 -0.217 0.8285
LBelgium/Germany_5 -0.116697 0.04742 -2.46 0.0150
LBelgium/Germany_6 0.120452 0.04737 2.54 0.0120
LBelgium/Germany_7 -0.0318466 0.04837 -0.658 0.5112
LBelgium/Germany_8 -0.0253422 0.04810 -0.527 0.5991
LBelgium/Germany_9 0.0294442 0.04623 0.637 0.5252
LBelgium/Germany_10 -0.00683282 0.04541 -0.150 0.8806
LBelgium/Germany_11 -0.0270554 0.04281 -0.632 0.5283
LBelgium/Germany_12 0.0284262 0.02594 1.10 0.2748
LBelgium_1 0.0404842 0.07301 0.554 0.5801
LBelgium_2 -0.00894351 0.09935 -0.0900 0.9284
LBelgium_3 -0.168750 0.09917 -1.70 0.0909
LBelgium_4 0.288801 0.09992 2.89 0.0044
LBelgium_5 -0.238411 0.1008 -2.36 0.0193
LBelgium_6 0.133391 0.09987 1.34 0.1836
LBelgium_7 -0.105906 0.09988 -1.06 0.2907
LBelgium_8 -0.0980220 0.1020 -0.961 0.3380
LBelgium_9 0.289230 0.1049 2.76 0.0065
LBelgium_10 -0.164248 0.1070 -1.53 0.1268
LBelgium_11 -0.0314415 0.1053 -0.299 0.7656
LBelgium_12 0.0501439 0.06346 0.790 0.4306
LGermany_1 1.29590 0.08138 15.9 0.0000
LGermany_2 -0.311902 0.1284 -2.43 0.0163
LGermany_3 0.225749 0.1304 1.73 0.0855
LGermany_4 -0.310123 0.1315 -2.36 0.0196
LGermany_5 0.123547 0.1344 0.920 0.3592
LGermany_6 -0.224938 0.1336 -1.68 0.0943
LGermany_7 0.295004 0.1315 2.24 0.0263
LGermany_8 -0.103835 0.1282 -0.810 0.4192
LGermany_9 0.0448156 0.1233 0.364 0.7167
LGermany_10 -0.0346024 0.1229 -0.282 0.7786
LGermany_11 0.298141 0.1228 2.43 0.0164
LGermany_12 -0.274759 0.08066 -3.41 0.0008
Constant -0.0233588 0.04099 -0.570 0.5696
sigma = 0.00231282 RSS = 0.0008237694326
log-likelihood 2525.68367 -T/2log|Omega| 3338.73545
|Omega| 6.55862423e-016 log|Y'Y/T| -8.14095441
R^2(LR) 1 R^2(LM) 0.992103
no. of observations 191 no. of parameters 111
F-test on regressors except unrestricted: F(111,456) = 31811.5 [0.0000] **
F-tests on retained regressors, F(3,152) =
LBelgium/Germany_1 84.4832 [0.000]**LBelgium/Germany_2 4.84693 [0.003]**
LBelgium/Germany_3 1.19021 [0.315] LBelgium/Germany_4 0.302757 [0.823]
LBelgium/Germany_5 2.17739 [0.093] LBelgium/Germany_6 2.83358 [0.040]*
LBelgium/Germany_7 0.829163 [0.480] LBelgium/Germany_8 0.253797 [0.859]
LBelgium/Germany_9 0.740966 [0.529] LBelgium/Germany_10 1.08987 [0.355]
LBelgium/Germany_11 0.703625 [0.551] LBelgium/Germany_12 1.53880 [0.207]
LBelgium_1 47.3689 [0.000]** LBelgium_2 0.824504 [0.482]
LBelgium_3 1.08271 [0.358] LBelgium_4 3.50152 [0.017]*
LBelgium_5 1.88295 [0.135] LBelgium_6 1.02400 [0.384]
LBelgium_7 2.88040 [0.038]* LBelgium_8 3.81197 [0.011]*
LBelgium_9 5.08111 [0.002]** LBelgium_10 0.814791 [0.488]
LBelgium_11 0.339140 [0.797] LBelgium_12 0.857423 [0.465]
LGermany_1 87.0156 [0.000]** LGermany_2 2.45571 [0.065]
LGermany_3 3.33343 [0.021]* LGermany_4 2.87626 [0.038]*
LGermany_5 0.906398 [0.440] LGermany_6 1.52991 [0.209]
LGermany_7 2.15385 [0.096] LGermany_8 0.388259 [0.762]
LGermany_9 0.465400 [0.707] LGermany_10 0.483026 [0.695]
LGermany_11 3.39496 [0.020]* LGermany_12 5.56514 [0.001]**
Constant 8.75168 [0.000]**
correlation of URF residuals (standard deviations on diagonal)
LBelgium/Germany LBelgium LGermany
LBelgium/Germany 0.0061444 -0.16532 -0.11966
LBelgium -0.16532 0.0026919 0.34182
LGermany -0.11966 0.34182 0.0023128
correlation between actual and fitted
LBelgium/Germany LBelgium LGermany
0.99917 0.99996 0.99991
I(1) cointegration analysis, 1974(4) - 1990(2)
eigenvalue loglik for rank
2492.611 0
0.21606 2515.858 1
0.081106 2523.936 2
0.018136 2525.684 3
H0:rank<= Trace test [ Prob]
0 66.145 [0.000] **
1 19.651 [0.059]
2 3.4957 [0.504]
Asymptotic p-values based on: Restricted constant
Restricted variables:
[0] = Constant
Number of lags used in the analysis: 12
beta (scaled on diagonal; cointegrating vectors in columns)
LBelgium/Germany 1.0000 0.39584 0.030822
LBelgium -2.6147 1.0000 -0.57142
LGermany 3.1370 -1.8566 1.0000
Constant -5.2325 2.3655 -1.9104
alpha
LBelgium/Germany -0.027470 -0.078750 0.019827
LBelgium 0.040111 -0.010946 -0.0033899
LGermany 0.0018460 -0.016235 -0.012932
long-run matrix, rank 3
LBelgium/Germany LBelgium LGermany Constant
LBelgium/Germany -0.058031 -0.018254 0.079858 -0.080421
LBelgium 0.035674 -0.11389 0.14276 -0.22930
LGermany -0.0049789 -0.013672 0.023001 -0.023359
SYS( 6) Cointegrated VAR
The dataset is: new09.in7
The estimation sample is: 1974(4) - 1990(2)
Cointegrated VAR (12) in:
[0] = LBelgium/Germany
[1] = LBelgium
[2] = LGermany
Restricted variables:
[0] = Constant
Number of lags used in the analysis: 12
beta
LBelgium/Germany 1.0000
LBelgium -2.6147
LGermany 3.1370
Constant -5.2325
alpha
LBelgium/Germany -0.027470
LBelgium 0.040111
LGermany 0.0018460
Standard errors of alpha
LBelgium/Germany 0.015225
LBelgium 0.0064922
LGermany 0.0056482
Restricted long-run matrix, rank 1
LBelgium/Germany LBelgium LGermany Constant
LBelgium/Germany -0.027470 0.071826 -0.086175 0.14374
LBelgium 0.040111 -0.10488 0.12583 -0.20988
LGermany 0.0018460 -0.0048267 0.0057910 -0.0096593
Standard errors of long-run matrix
LBelgium/Germany 0.015225 0.039808 0.047761 0.079665
LBelgium 0.0064922 0.016975 0.020366 0.033970
LGermany 0.0056482 0.014768 0.017719 0.029554
Reduced form beta
LBelgium/Germany -1.0000
LBelgium 2.6147
LGermany -3.1370
Constant 5.2325
Standard errors of reduced form beta
LBelgium/Germany 0.00000
LBelgium 0.47463
LGermany 0.72964
Constant 1.0775
Moving-average impact matrix
1.3452 0.54091 8.2649
0.61395 -0.35713 16.896
0.082895 -0.47009 11.448
log-likelihood 2515.85803 -T/2log|Omega| 3328.90981
no. of observations 191 no. of parameters 105
rank of long-run matrix 1 no. long-run restrictions 0
beta is not identified
No restrictions imposed
LBelgium/Germany: Portmanteau(12): 5.96898
LBelgium : Portmanteau(12): 3.39841
LGermany : Portmanteau(12): 6.49229
LBelgium/Germany: Normality test: Chi^2(2) = 121.37 [0.0000]**
LBelgium : Normality test: Chi^2(2) = 6.3458 [0.0419]*
LGermany : Normality test: Chi^2(2) = 2.1111 [0.3480]
LBelgium/Germany: ARCH 1-7 test: F(7,142) = 0.42037 [0.8884]
LBelgium : ARCH 1-7 test: F(7,142) = 0.66839 [0.6985]
LGermany : ARCH 1-7 test: F(7,142) = 0.12698 [0.9963]
LBelgium/Germany: Hetero test: F(72,81) = 0.66325 [0.9616]
LBelgium : Hetero test: F(72,81) = 0.37190 [1.0000]
LGermany : Hetero test: F(72,81) = 0.49807 [0.9986]
Vector Portmanteau(12): 52.0286
Vector Normality test: Chi^2(6) = 120.49 [0.0000]**
Vector Hetero test: F(432,463)= 0.69746 [0.9999]
（4）AR
图acf 和Pasf
---- Maximum likelihood estimation of ARFIMA(4,0,0) model ----
The estimation sample is: 1973(5) - 1990(2)
The dependent variable is: DLFrance/Belgium
The dataset is: new09.in7
Coefficient Std.Error t-value t-prob
AR-1 0.382891 0.06996 5.47 0.000
AR-2 -0.0181099 0.07466 -0.243 0.809
AR-3 -0.0856939 0.07447 -1.15 0.251
AR-4 -0.0745809 0.06953 -1.07 0.285
Constant 0.00177940 0.0009659 1.84 0.067
log-likelihood 625.927726
no. of observations 202 no. of parameters 6
AIC.T -1239.85545 AIC -6.13789827
mean(DLFrance/Belgium) 0.00178439 var(DLFrance/Belgium) 0.000142048
sigma 0.0109093 sigma^2 0.000119013
BFGS using numerical derivatives (eps1=0.0001; eps2=0.005):
Strong convergence
Used starting values:
0.38482 -0.018468 -0.086797 -0.075880 0.0017844
---- Maximum likelihood estimation of ARFIMA(3,0,0) model ----
The estimation sample is: 1973(5) - 1990(2)
The dependent variable is: DLFrance/Belgium
The dataset is: new09.in7
Coefficient Std.Error t-value t-prob
AR-1 0.391643 0.06968 5.62 0.000
AR-2 -0.0169831 0.07482 -0.227 0.821
AR-3 -0.114939 0.06940 -1.66 0.099
Constant 0.00178156 0.001040 1.71 0.088
log-likelihood 625.354221
no. of observations 202 no. of parameters 5
AIC.T -1240.70844 AIC -6.142121
mean(DLFrance/Belgium) 0.00178439 var(DLFrance/Belgium) 0.000142048
sigma 0.0109409 sigma^2 0.000119704
BFGS using numerical derivatives (eps1=0.0001; eps2=0.005):
Strong convergence
Used starting values:
0.39367 -0.017166 -0.11667 0.0017844
---- Maximum likelihood estimation of ARFIMA(2,0,0) model ----
The estimation sample is: 1973(5) - 1990(2)
The dependent variable is: DLFrance/Belgium
The dataset is: new09.in7
Coefficient Std.Error t-value t-prob
AR-1 0.399155 0.07004 5.70 0.000
AR-2 -0.0633158 0.06988 -0.906 0.366
Constant 0.00178289 0.001164 1.53 0.127
log-likelihood 623.986982
no. of observations 202 no. of parameters 4
AIC.T -1239.97396 AIC -6.13848497
mean(DLFrance/Belgium) 0.00178439 var(DLFrance/Belgium) 0.000142048
sigma 0.0110163 sigma^2 0.000121359
BFGS using numerical derivatives (eps1=0.0001; eps2=0.005):
Strong convergence
Used starting values:
0.40113 -0.063965 0.0017844
---- Maximum likelihood estimation of ARFIMA(1,0,0) model ----
The estimation sample is: 1973(5) - 1990(2)
The dependent variable is: DLFrance/Belgium
The dataset is: new09.in7
Coefficient Std.Error t-value t-prob
AR-1 0.375174 0.06496 5.78 0.000
Constant 0.00178020 0.001239 1.44 0.152
log-likelihood 623.577387
no. of observations 202 no. of parameters 3
AIC.T -1241.15477 AIC -6.14433056
mean(DLFrance/Belgium) 0.00178439 var(DLFrance/Belgium) 0.000142048
sigma 0.0110389 sigma^2 0.000121857
BFGS using numerical derivatives (eps1=0.0001; eps2=0.005):
Strong convergence
Used starting values:
0.37701 0.0017844
Test for excluding: AR-1
Subset Chi^2(1) = 33.3597 [0.0000] **
Descriptive statistics for residuals:
Normality test: Chi^2(2) = 58.149 [0.0000]**
ARCH 1-1 test: F(1,198) = 7.5716 [0.0065]**
Portmanteau(36): Chi^2(35) = 47.627 [0.0755]
ARCH
ARCH coefficients:
Lag Coefficient Std.Error
1 0.14213 0.07377
2 0.15012 0.0743
3 0.16443 0.07515
4 -0.028656 0.07482
5 0.047427 0.07478
6 -0.039926 0.07482
7 -0.013674 0.07469
8 -0.011489 0.07465
9 0.029838 0.07458
10 -0.0027954 0.07309
11 -0.052836 0.07233
12 0.020861 0.07185
RSS = 1.47674e-005 sigma = 0.000289665
Testing for error ARCH from lags 1 to 12
ARCH 1-12 test: F(12,176) = 1.7613 [0.0579]
Residual [1973( 5) - 1990( 2)] saved to new09.in7
GARCH
VOL( 2) Modelling DLFrance/Belgium by restricted GARCH(1,1)
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error robust-SE t-value t-prob
Constant X 0.00137385 0.0005496 0.0006335 2.17 0.031
alpha_0 H 2.53091e-005 6.838e-006 1.225e-005 2.07 0.040
alpha_1 H 0.548753 0.1647 0.2417 2.27 0.024
beta_1 H 0.386169 0.1059 0.1239 3.12 0.002
log-likelihood 647.090499 HMSE 7.74626
mean(h_t) 0.000169302 var(h_t) 5.89695e-008
no. of observations 202 no. of parameters 4
AIC.T -1286.181 AIC -6.36723266
mean(DLFrance/Belgium) 0.00178439 var(DLFrance/Belgium) 0.000142048
alpha(1)+beta(1) 0.934922 alpha_i+beta_i>=0, alpha(1)+beta(1)<1
Initial terms of alpha(L)/[1-beta(L)]:
0.54875 0.21191 0.081834 0.031602 0.012204 0.0047127
0.0018199 0.00070278 0.00027139 0.00010480 4.0472e-005 1.5629e-005
Used sample mean of squared residuals to start recursion
Robust-SE based on analytical Information matrix and analytical OPG matrix
BFGS using analytical derivatives (eps1=0.0001; eps2=0.005):
Strong convergence
Used starting values:
0.0017844 2.1595e-005 0.70855 0.13943
Test for excluding: alpha_1
Subset Chi^2(1) = 11.0952 [0.0009] **
Using robust standard errors:
Subset Chi^2(1) = 5.15536 [0.0232] *
Portmanteau statistic for scaled residuals
Autocorrelation function (ACF) from lag 1 to 12:
0.24488 0.047178 -0.058108 -0.085325 -0.059613 -0.080555
-0.0051672 0.13997 0.19257 0.0078786 -0.13371 -0.15760
Partial autocorrelation function (PACF):
0.24488 -0.013605 -0.070740 -0.057554 -0.023454 -0.065465
0.024169 0.14067 0.12540 -0.091215 -0.13049 -0.078908
Portmanteau(12): Chi^2(12) = 38.040 [0.0002]**
Portmanteau statistic for squared scaled residuals
Autocorrelation function (ACF) from lag 1 to 12:
-0.011257 0.027938 -0.024482 0.032919 0.050857 -0.051928
-0.035495 -0.054797 0.075378 0.023350 -0.028213 0.0044665
Partial autocorrelation function (PACF):
-0.011257 0.027815 -0.023885 0.031675 0.052970 -0.053474
-0.038122 -0.051434 0.071194 0.027495 -0.027254 0.010154
Portmanteau(12): Chi^2(10) = 4.0149 [0.9467]
EGARCH
VOL( 3) Modelling DLFrance/Belgium by EGARCH(1,1)
The dataset is: new09.in7
The estimation sample is: 1973(5) - 1990(2)
Coefficient Std.Error robust-SE t-value t-prob
Constant X 0.00101843 0.0005399 0.0005579 1.83 0.069
alpha_0 H -1.41749 0.6391 0.9994 -1.42 0.158
eps[-1] H -0.143496 0.09108 0.1445 -0.993 0.322
|eps[-1]| H 0.691818 0.1431 0.2420 2.86 0.005
beta_1 H 0.833174 0.07042 0.1136 7.33 0.000
log-likelihood 649.124517 HMSE 8.68633
mean(h_t) 0.000159174 var(h_t) 4.75962e-008
no. of observations 202 no. of parameters 5
AIC.T -1288.24903 AIC -6.37747046
mean(DLFrance/Belgium) 0.00178439 var(DLFrance/Belgium) 0.000142048
Used sample mean of squared residuals to start recursion
Robust-SE based on numerical Hessian matrix and numerical OPG matrix
BFGS using numerical derivatives (eps1=0.0001; eps2=0.005):
Strong convergence
Used starting values:
0.0017844 1.0047 0.70855 0.00000 0.13943
(5) Forecast
Forecasting DLFrance/Belgium from 1990(3) to 1991(5)
Horizon Forecast (SE) Actual CondVar
1.0000 0.0010184 0.0051081 0.00013464 2.6093e-005
2.0000 0.0010184 0.0060643 -0.00097591 3.6775e-005
3.0000 0.0010184 0.0069962 0.0043737 4.8947e-005
4.0000 0.0010184 0.0078812 0.0029580 6.2113e-005
5.0000 0.0010184 0.0087034 -0.0045257 7.5749e-005
6.0000 0.0010184 0.0094536 0.0019288 8.9370e-005
7.0000 0.0010184 0.010128 -0.0019952 0.00010257
8.0000 0.0010184 0.010726 -0.0011920 0.00011505
9.0000 0.0010184 0.011251 0.0031894 0.00012660
10.000 0.0010184 0.011709 0.0056647 0.00013710
11.000 0.0010184 0.012104 .NaN 0.00014651
12.000 0.0010184 0.012443 .NaN 0.00015484
13.000 0.0010184 0.012734 .NaN 0.00016214
14.000 0.0010184 0.012980 .NaN 0.00016849
mean(Error) = -6.2387e-005 RMSE = 0.0030268
SD(Error) = 0.0030261 MAPE = 165.93
image1.png
FMBF computer lab1.pdf
FMBF: Computer Practical 1
Introduction
There are four workshops for the FMBF module this term which will be organised as
follows:
Practical 1 ARIMA modelling and unit root testing.
Practical 2 Cointegration procedures.
Practical 3 ARCH/GARCH modelling and forecasting.
1 ARIMA Modelling
1.1 Aims
In the first part of this session we will firstly load a dataset and experiment with ARIMA
modelling. Once the concept has been demonstrated you will be required to find a preferred
model for the stock price data that we will be using. The stock price series is raw data so you
will need to calculate returns yourself using the calculator function in PcGive.
We will demonstrate the procedure for using the Time Series Models package in PcGive by
estimating an AR(1) model:
1.2 Key Steps
Importing and transforming the data
1. We will be using monthly data on the price of British Airways from 1996 to 2002.
Download the file BA.xls from duo that contains these data.
2. Start PcGive.
(Start > Search PcGive12)
3. Create new database in PcGive.
(File > New… > OxMetrics (Data: *.in7) > Frequency: Monthly > Observations: 74 > OK).
Practical 4 Review.
4. Copy the price data column from Excel to PcGive. Double click on the first row and type
the column title: BA.
Note: You could also click File > Open data file... and import the Excel file directly.
However, PcGive will not automatically recognise the dates in the leftmost column. If you
chose to do it this way you could choose Edit > Change Sample to let PcGive know we are
dealing with monthly data from 1996.
5. Use Calculator to calculate the log return, i.e., generate new series Ri which equals the
dlog of BA.
Prior to constructing the model
You should plot and carefully examine the ACF and PACF. Recall that we plotted the ACF
and PACF last term, and that you can do this by going to the Graphics button in PcGive.
(Graphics > Choose the variables and click All plot types > Time-series properties > In
Dynamic Properties tick ACF and PACF and click OK)
Is a particular model suggested by this graphical analysis?
Constructing the model
These steps show the construction of an AR(1) model for demonstration.
1 In PcGive select Model > Category: Models for time series data > Model class: ARFIMA
models using PcGive.
2 Select Formulate…
3 Select Ri as the dependent variable (Y), add a constant term (PcGive should do this
automatically), and press OK.
4 Set AR order as 1 and fix the Fractional parameter d at 0. Treatment of the mean should
automatically be set to None (or using constant as regressor).
5 Press OK, and select Maximum Likelihood.
6 Press OK and the Estimation Results will appear.
7 Examine the results. Is this model satisfactory?
Exercice 1: Identifying your preferred model
Carefully examine the results from your model. You should:
Check the residuals to see if they conform to white noise using ACF and PACF and the Q-
statistic (Ljung Box statistic).
Re-estimate the model, overfitting in an effort to isolate the preferred specification.
You may then want to utilise the AIC when choosing between competing models.
If you are not prompted to do so by the steps above, you should at the least estimate an MA(1)
and an ARIMA(1,0,1) model to be clear on the procedure of specifying such models in
PcGive.
2 Unit root testing
2.1 Aims
In the second part of this session we will examine stationarity testing using PcGive.
Price/Earnings data for the US will be used to demonstrate how the software can easily test
for the presence of a unit root and display results in a fashion that easily allows us to choose a
preferred model cf. the number of lags to include. We will then test for stationarity in
monthly data for the FTSE 100 and FTSE All Share.
Example 1: Annual Price/Earnings Ratio
This example is based on the illustration of the annual price/earnings ratio proposed in
Verbeek (2004, p.274). The literature has focussed on whether the price/earnings ratio is
mean reverting (why might this be interesting?).
1 Load the data PE.xls into PcGive (File > Open… > PE.xls). This is annual data on the
ratio of the S&P Composite Stock Price Index and S&P Composite Earnings. The sample
(annual data) runs from 1871 to 2002.
2 Plot the log of the series (LOGPE) using GiveWin graphics (Graphics > Choose LOGPE >
Actual series) . Does it look stationary?
3 We will test for stationarity with the standard Dickey-Fuller regression, i.e. if we denote
the log P/E ratio as Yt
1 1t t tY Y (1)
Note that you will need to use Calculator to diff the LOGPE series.
Modelling this using Single equation dynamic modelling... gives
10.335 0.125t t tY Y e (2)
(Model > Category: Model for time-series data > Model class: Single-equation Dynamic
Modelling using PcGive > Formulate… > Select DLOGPE as Y and choose lagged
LOGPE > Click OK)
The full output from PcGive is
We can calculate the DF test statistic as -2.57, the 5% critical value being -2.88. Thus we
cannot reject the null of a unit root. However, for this result to hold we must have included
lags to the extent that the error term is white noise.
2.2 Automated unit root testing
Fortunately, PcGive makes it easy to perform unit root testing automatically, saving us
having to run a regression each time. It also produces a neat summary table allowing easy
selection of the appropriate number of lag terms.
We will first replicate the results previously obtained:
1 Select Model > Category: Other models > Model class: Descriptive Statistics using
PcGive)
2 Select Formulate
3 Add LOGPE to the model and click OK
4 In Descriptive Statistics select Unit-root tests, and then edit Unit-root test settings so that
Lag length for differences is 0 and Constant) is selected
5 Click OK and then OK in the Estimate Model dialog
6 The following results should appear in the Results area in GiveWin
These are the results we obtained previously. Note that the software automatically calculates
the correct test statistic and critical value.
2.3 ADF testing
We will now add lags and look at the results from augmented Dickey-Fuller tests:
1 Formulate a new model (Descriptive Statistics)
2 Edit Unit-root test settings so that Lag length for differences is 6 and Constant) is selected.
Make sure that Report summary table only is selected.
The following results are presented
There is a rejection of the null of a unit root at the 5% level with one lag. However, none of
the other ADF tests reject the null of nonstationarity. Does this concur with the graphical plot
of the P/E series that you produced above?
Exercice 2: Annual Price/Earnings Ratio cont...
Test for the presence of a second unit root in annual price/earnings data (Hint: you will need
to difference the data once more). Can you reject the null of nonstationarity? What
conclusion does this lead you towards?
Exercice 3: Stationarity in the FTSE 100 and ALL SHARE
1. Load the data FTSEDATA.xls that is on duo. This contains monthly data for the FTSE 100
and ALL SHARE from 1985:1.
2. Create logarithms of the two indices, naming them LFTSE100 and LFTALLSH.
3. Plot the series then test for stationarity adding an appropriate number of lags.
4. Create the first difference of LFTSE100 and LFTALLSH and test for stationarity after
plotting the differenced series.
5. Come to conclusions about the presence of a unit root in the two series.
2.4 Points to note
Make sure you are clear on what the columns in the tables of unit root test output mean;
see Figure 1 and Figure 2.
Perron procedure (not Perron's test) may be used to test for unit root in ammore systematic
way (see Harris and Sollis p. 47 and your lecture notes). This will not be considered here.
References
M. Verbeek. A Guide to Modern Econometrics. John Wiley & Sons, Inc., 2004.
Figure 1: Interpreting unit root test output: summary table option
Figure 2: Interpreting unit root test output: summary table vs. standard output
Guide_to_ACF_PACF_plots(computer lab1).pdf
Guide to ACF/PACF Plots
The plots shown here are those of pure or theoretical ARIMA processes. Here are some general guidelines for identifying the process:
Nonstationary series have an ACF that remains significant for half a dozen or more lags, rather than quickly declining to zero. You must difference such a series until it is stationary before you can identify the process.
Autoregressive processes have an exponentially declining ACF and spikes in the first one or more lags of the PACF. The number of spikes indicates the order of the autoregression.
Moving average processes have spikes in the first one or more lags of the ACF and an exponentially declining PACF. The number of spikes indicates the order of the moving average.
Mixed (ARMA) processes typically show exponential declines in both the ACF and the PACF.
At the identification stage, you do not need to worry about the sign of the ACF or PACF, or about the speed with which an exponentially declining ACF or PACF approaches zero. These depend upon the sign and actual value of the AR and MA coefficients. In some instances, an exponentially declining ACF alternates between positive and negative values.
ACF and PACF plots from real data are never as clean as the plots shown here. You must learn to pick out what is essential in any given plot. Always check the ACF and PACF of the residuals, in case your identification is wrong. Bear in mind that:
Seasonal processes show these patterns at the seasonal lags (the multiples of the seasonal period).
1
2
You are entitled to treat nonsignificant values as zero. That is, you can ignore values that lie within the confidence intervals on the plots. You do not have to ignore them, however, particularly if they continue the pattern of the statistically significant values.
An occasional autocorrelation will be statistically significant by chance alone. You can ignore a statistically significant autocorrelation if it is isolated, preferably at a high lag, and if it does not occur at a seasonal lag.
Consult any text on ARIMA analysis for a more complete discussion of ACF and PACF plots.
ARIMA(0,0,1), θ>0
ACF PACF
3
Guide to ACF/PACF Plots
ARIMA(0,0,1), θ<0
ACF PACF
ARIMA(0,0,2), θ1θ2>0
ACF PACF
4
ARIMA(1,0,0), φ>0
ACF PACF
ARIMA(1,0,0), φ<0
ACF PACF
5
Guide to ACF/PACF Plots
ARIMA(1,0,1), φ<0, θ>0
ACF PACF
ARIMA(2,0,0), φ1φ2>0
ACF PACF
6
ARIMA(0,1,0) (integrated series)
ACF
- Table of Contents
- 1. Guide to ACF/PACF Plots
- Index
FMBF computer lab2.pdf
FMBF: Computer Practical 2
Introduction
This workshop session covers cointegration, using the Engle-Granger and Johansen
approaches. You should be aware of the benefits and drawbacks of each approach.
1 Engle-Granger
Exercise 1: Cointegration between the S&P and FTSE All-Share
1. Load the data file FMBF Prac2.xls from duo. This contains monthly data on the
S&P 500 and FTSE All Share from January 1 1965 to January 1 2004.
2. Log both series (Calculator) and then use the unit root testing facility in
Descriptive Statistics to assess the degree of integration of the series (Model >
Category: Other models; Model class: Descriptive statistics using PcGive).
Notes: most financial variables are I(1) series. To conduct the EG procedure, we
should firstly check whether the two time series are I(1).
3. Regress LS&P on LFTSE and a constant using OLS (Model > Category:
Models for time-series data; Model class: Single-equation dynamic
Modelling using PcGive).
4. Save the cointegration regression residual in the database (Test > Test Menu:
Store Residuals etc. in Database… > Store in database: Residuals).
5. Test for cointegration by performing a unit-root test on the saved residuals (do
not include deterministic components here).
6. Evaluate the results and establish whether or not the series cointegrate (note:
make sure you use the correct critical values).
7. If appropriate, build an ECM. Do this by regressing DLS&P on a constant,
DLFTSE and the one-period lagged residuals that were previously stored in the
database. Interpret your findings.
Notes: according to the unit root test of the residuals, since the residuals are not
stationary, it is not appropriate to put the non-stationary residuals into the ECM.
In other words, the residuals are I(1). Therefore, we should estimate a model
containing only first differences.
2 Johansen
Example 1: Long-run PPP
This is a PcGive implementation of the long-run purchasing power parity example
presented in Verbeek (2004, p.331). We begin by loading the data file ppp.xls from
duo, which contains monthly observations from 1981:1 to 1996:6 on price indices and
exchange rates from France and Italy. The variables contained in the file are as
follows:
This example investigates the concept of PPP, where exchange rate equals the ratio of
price levels. In logarithms, we represent PPP as:
*
ttt pps (1)
where ts is the log of the spot exchange rate, tp
is the log of domestic prices and
*
tp is the log of foreign prices.
1. Following Verbeek's reasoning we will first run a test with p = 3, excluding a
time trend (Model > Category: Models for time-series data; Model class:
Multiple-equation dynamic Modelling using PcGive). Note that PcGive
automatically restricts the Constant term (shown by the U to its left. We remove
this to restrict the constant following Verbeek's example. Select U Constant in
Selection > Change Use default status to Clear status and click Set, then the
U is removed. Note that this now corresponds to Model 2 in the Pantula
Principle). Click OK and choose Unrestricted system.
2. Press Test > Test Menu: Dynamic Analysis and Cointegration Tests... >
Dynamic Analysis: I(1) cointegration analysis > OK. You are presented with
results, the firrst part of which are the eigenvalues:
We can see from the results that there are two small eigenvalues that are significant at
the 1 percent level. In this case we reject H0 : r = 0 and also H0 : r = 1, but we cannot
reject H0 : r = 2 against the alternative of H1 : r = 3. Therefore using Johansen we
conclude that there are two cointegrating relationships. P-values are based on Doornik
(1998) (reprinted in McAleer and Oxley (1999)). Verbeek (2004, p.332) reminds us
that in this particular example, Engle-Granger finds that the null of no cointegration
could not be rejected. (Note: you can follow the E-G procedure that was used above to
verify this). One possible explanation is that the number of lags is too small.
Therefore we formulate a model with p = 12, supported by the use of monthly data:
These results are clearly weaker than with p = 3. We do, however, have a rejection of
H0 : r = 1. We can now move on to build a cointegrated VAR model. We will assume
r = 1.
3. Formulate the model used for estimating the cointegration test for p = 12.
Remember that we have a restricted constant (Model > Category: Models for
time-series data; Model class: Multiple-equation dynamic Modelling using
PcGive).
4. Click OK and then select Cointegrated VAR and press OK once more.
5. In the Cointegrated VAR Settings dialog box make sure the Cointegrating rank is
set to 1. Click OK and OK to estimate a Reduced Rank Regression
You should be presented with the following output in the Results:
The most interesting part of these results, relating to estimating the cointegrating
vector, β, are shown under Reduced form beta. The normalized cointegrating
therefore corresponds to:
*756.14346.6 ttt pps (2)
As Verbeek, p.332 points out, this “does not seem to correspond to an economically
interpretable long-run relationship.”
3 The Pantula Principle
Following Johansen (1992) we can use the so-called Pantula Principle to determine
the choice for deterministic components in the cointegration space and/or the
short-run, and also establish the order of the cointegration rank r. Recall that we
estimate all three models and present the results from the r = 0 (model 2, the most
restrictive) to r = n-1 (model 4, the least restrictive). Moving through these models
and examining them in turn, we look to the trace statistic and stop once the null
hypothesis cannot be rejected. In the case of the example we are concerned with
identifying the deterministic components. One further point to consider are the correct
critical values for models 1 to 4.
Exercice 2: The Pantula Principle
In the above worked example we estimated a long-run PPP model and tested for
cointegration. Effectively, what we did corresponds with Model 2 in the Pantula
Principle.
You should now follow this principle, estimating Models 2, 3 and 4 in sequence.
You should examine your results and establish which specification is preferred.
To assist you, Table 1 contains pointers on setting up each model in PcGive. It is
helpful to construct a table similar to Table 5.5 in Harris and Sollis (2003).
In the example, we estimated with both 3 lags and 12 lags. You should come to
your own conclusion about this by first looking at appropriate graphic analysis.
Specifically you can go to Test > Graphic Analysis and look at Actual and
fitted values, Cross plot of actual and fitted, Residuals (scaled), Residual
density and histogram (kernel estimate) and Residual correlogram (ACF).
System reduction is important in our goal to find the preferred model. You should
examine F-tests on the retained regression to see if it is possible to delete all the
lags of the same length (i.e. those that are not significant) whilst keeping the
sample period unchanged. You can use Test > Exclusion Restrictions... to
evaluate.
Estimate your preferred model as a cointegrated VAR.
As usual, diagnostic tests are important. See Test > Test Summary for
equation-by-equation and system-wide tests, which you should examine.
3.1 Imposing Restrictions
We can test for restrictions on α and β with PcGive. For example, to test the
restriction
0,0, . (3)
We would go to Model > Category: Models for time-series data; Model class:
Multiple-equation dynamic Modelling using PcGive and select Cointegrated VAR.
Press OK and enter a Cointegrating rank of 1. We then select General restrictions
and press OK.
The General Restrictions dialog box opens, where we specify the restriction in the
form &1=0;&2=0;&3=0 - see Figure 2 for a screenshot. We are testing a null of
weakly exogenous - you may wish to try this for the PPP model estimated above.
We can also use PcGive's ability to impose general restrictions to test for unique
cointegrating vectors and in addition jointly test restrictions on α and β. See Harris
and Sollis (2003, p.135-163) for full details, examples and references to imposing
restrictions in PcGive.
4 Points to note
Make sure you are clear about the differences between the E-G and Johansen
approaches to cointegration.
In the example above we used the trace test. This evaluates whether the smallest k
- r0 eigenvalues significantly differ from 0. However, we can also use the
maximum eigenvalue test. This tests H0 : r ≤ r0 against H1 : r = r0 + 1. PcGive
gives the eigenvalues so it is possible to calculate these. For example, in the PPP
example the first eigenvalue is 0.30091 so the axm statistic can be calculated
as 183*LN(1-0.30091), i.e. 65.509 which can be set against the correct critical
value, in this case 22.04.
References
J. A. Doornik. Approximations to the asymptotic distribution of cointegration tests.
Journal of Economic Surveys, 12:573{593, 1998.
R. Harris and R. Sollis. Apple Time Series Modelling and Forecasting. John Wiley &
Sons Ltd., Chichester, 2003.
D. F. Hendry and J. A. Doornik. Empirical Econometric Modelling Using Pc-Give,
volume 1. Timberlake Consultants Ltd., 3 edition, 2001.
S. Johansen. Cointegration in partial systems and the effciency of single equation
analysis. Journal of Econometrics, 52:389{402, 1992.
M. McAleer and L. Oxley. Practical Issues in Cointegration Analysis. Blackwell
Publishers, Oxford, 1999.
M. Verbeek. A Guide to Modern Econometrics. John Wiley & Sons, Inc., 2004.
Johansen Test by PcGive(computer lab2).pdf
— Appendix ————-—
Cointegration Analysis Using the Johansen Technique: A Practitioner's
__ Guide to PcGive 10.1
This appendix provides a basic introduction on how to implement the Johan- sen technique using the PcGive 10.1 econometric program (see Doornik and Hendry, 2001 for full details). Using the same data set as underlies much of the analysis in Chapters 5 and 6, we show the user how to work through Chapter 5 up to the point of undertaking joint tests involving restrictions on a and p.
This latest version of PcGive brings together the old PcGive (single equa- tion) and PcFiml (multivariate) stand-alone routines into a single integrated software program (that in fact is much more than the sum of the previous versions, since it is built on the Ox programming language and allows various bolt-on Ox programs to be added—such as dynamic panel data analysis (DPD), time series models and generalized autoregressive conditional heteroscedastic (GARCH) models—see Chapter 8). It is very flexible to operate, providing drop-down menus and (for the present analysis) an extensive range of modelling features for 7(1) and 7(0) systems (and limited analysis of the 7(2) system).1 Cointegration facilities are embedded in an overall modelling strategy leading through to structural vector autoregression (VAR) modelling.
After the data have been read-in to GiveWin2 (the data management and graphing platform that underpins PcGive and the other programs that can operate in what has been termed the Oxmetrics suite of programs), it is first necessary to (i) start the PcGive module, (ii) select 'Multiple-equation Dynamic Modelling' and then (iii) 'Formulate' a model. This allows the user to define the model in (log) levels, fix which deterministic variables should enter the co- integration space, determine the lag length of the VAR and decide whether
1 PcGive also allows users to run batch jobs where previous jobs can be edited and rerun. 2 The program accepts data files based on spreadsheets and unformatted files.
260 APPENDIX
Figure A.I. Formulating a model in PcGive 10.1: Step (1) choosing the correct model option.
7(0) variables, particularly dummies, need to be specified to enter the model in the short-run dynamics but not in the cointegration spaces (see Figures A. 1 and A.2).
When the 'Formulate' option is chosen, the right-hand area under 'Data- base' shows the variables available for modelling. Introducing dummies and transformations of existing variables can be undertaken using the 'Calculator' or 'Algebra Editor' under Tools' in GiveWin, and these new variables when created will also appear in the 'Database'. In this instance, we will model the demand for real money (rm) as a function of real output (y), inflation (dp) and the interest rate (rstar), with all the variables already transformed into log levels. The lag length (k) is set equal to 4 (see lower right-hand option in Figure A.2); if we want to use an information criterion (1C) to set the lag length, then k can be set at different values, and when the model is estimated it will produce the Akaike, Hannan—Quinn and Schwarz 1C for use in deter- mining which model is appropriate.3 (However, it is also necessary to ensure that the model passes diagnostic tests with regard to the properties of the residuals of the equations in the model—see below—and therefore use of an 1C needs to be done carefully.)
Each variable to be included is highlighted in the 'Database' (either one at a time, allowing the user to determine the order in which these variables enter, or all variables can be simultaneously highlighted). This will bring up an '<<Add' option, and, once this is clicked on, then the model selected appears on the left-hand side under 'Model'. The 'Y' next to each variable indicates
3 Make sure you have this option turned on as it is not the default. To do this in PcGive. choose 'Model', then 'Options', 'Additional output' and put a cross in the information criterion box.
APPENDIX , 261
Figure A.2. Formulating a model in PcGive 10.1: Step (2) choosing the 'Formulate' option.
that it is endogenous and therefore will be modelled, a 'IT indicates the vari- able (e.g., the Constant, which enters automatically) is unrestricted and will only enter the short-run part of the vector error correction model (VECM), and the variables with '_k' next to them denote the lags of the variable (e.g., rm t–1).
We also need to enter some dummies into the short-run model to take account of'outliers' in the data (of course we identify these only after estimating the model, checking its adequacy, and then creating deterministic dummies to try to overcome problems; however, we shall assume we have already done this,4
4 In practice, if the model diagnostics—see Figure A.3—indicates, say, a problem of non-normality in the equation determining a variable, plot the residuals using the graphing procedures (select, in PcGive, 'Test' and 'Graphic analysis' and then choose 'Residuals' by putting a cross in the relevant box). Visually locate outliers in terms of when they occur, then again under 'Test' choose 'Store residuals in database', click on residuals and accept the default names (or choose others) and store these residuals in the spreadsheet. Then go to the 'Window' drop-down option in Give Win and select the database, locate the residuals just stored, locate the outlier residuals by scrolling down the spreadsheet (using the information gleaned from the graphical analysis) and then decide how you will 'dummy out' the outlier (probably just by creating a dummy variable using the 'Calculator' option in Give Win, with the dummy being 0 before and after the outlier date and 1 for the actual date of the outlier).
262 APPENDIX
or that ex ante we know such impacts have occurred and need to be included). Hence, highlight these (dumrst, dumdp, dumdpl), set the lag length option at the bottom right-hand side of the window to 0 and then click on '<Add'. Scroll down the 'Model' window, and you will see that these dummies have 'Y' next to them, which indicates they will be modelled as additional variables. Since we only want them to enter unrestrictedly in the short-run model, select/highlight the dummies and then in the 'Status' options on the left-hand side (the buttons under 'Status' become available once a variable in the model is highlighted) click on 'Unrestricted', so that each dummy now has a 'U' next to it in the model.
Finally, on the right-hand side of the 'Data selection' window is a box headed 'Special'. These are the deterministic components that can be selected and added to the model. In this instance, we select 'CSeasonal (centred seasonal dummies), as the data are seasonally unadjusted, and add the seasonal dummies to the model. They automatically enter as unrest- ricted. Note that if the time 'Trend' is added, it will not have a 'U' next to it in the model, indicating it is restricted to enter the cointegration space (Model 4 in Chapter 5—see equation (5.6)). If we wanted to select Model 2 then we would not enter the time trend (delete it from the model if it is already included), but would instead click on 'Constant' in the 'Model' box and click on 'Clear' under the 'Status' options. Removing the unrest- ricted status of the constant will restrict it to enter the cointegration space. Thus, we can select Models 2–4, one at a time, and then decide which deterministic components should enter II, following the Pantula principle (see Chapter 5).
Having entered the model required, click OK, bringing up the 'Model settings' window, accept the default of 'Unrestricted system' (by clicking OK again) and accept ordinary least squares (OLS) as the estimation method (again by clicking OK). The results of estimating the model will be available in Give Win (the 'Results' window—accessed by clicking on the Give Win toolbar on your Windows status bar). Return to the PcGive window (click on its toolbar), choose the 'Test' option to activate the drop-down options, and click on 'Test summary'. This produces the output in GiveWin as shown in Figure A.3. The model passes the various tests equation by equation and by using system-wide tests.
Several iterations of the above steps are likely to be needed in practice to obtain the lag length (k) for the VAR, which deterministic components should enter the model (i.e., any dummies or other 7(0) variables that are needed in the short-run part of the VECM to ensure the model passes the diagnostic tests on the residuals) and which deterministic components should enter the cointegration space (i.e., should the constant or trend be restricted to be included in II). To carry out the last part presumes you have already tested for the rank of II, so we turn to this next.
To undertake cointegration analysis of the I(1) system in PcGive, choose Test', then 'Dynamic Analysis and Cointegration tests' and check the "7(1)
APPENDIX
rm Y dp rstar rm Y dp rstar rm Y dp rstar rm y dp rstar rm y dp rstar
263
Portmanteau(11): Portmanteau(11) : Portmanteau(11): Portmanteau(11) : AR 1-5 test: AR 1-5 test: AR 1-5 test: AR 1-5 test: Normality test: Normality test: Normality test: Normality test: ARCH 1-4 test: ARCH 1-4 test: ARCH 1-4 test: ARCH 1-4 test: hetero test: hetero test: hetero test: hetero test:
11.2164 6.76376 3.66633 11.6639 F(5, F(5, F(5, F(5, Chi' Chi- Chi' Chi' F(4, F(4, F(4, F(4, F(35 F(35 F(35 F(35
73) 73) 73) 73) 2(2) 2(2) 2(2) 2(2) 70) 70) 70) 70) ,42) ,42) ,42) ,42)
1 1 1 1 2 5 4 o
— 1
= 1 -I
- 0. = 0. = 0. = 0. - 0.
.4131
.8569
.0269
.8359
.8297
.0521
.6973
.7019
.5882
.1669
.1133 68023 38980 72070 67314 88183
[0. [0. [0. [0. [0. [0. [0. [0. [0. [0. [0. [0. [0. [0. [0. [0.
2297] 1125] 4083] 1164] 2430] 0800] 0955] 2590] 1871] 3329] 3573] 6080] 9974] 8385] 8838] 6463]
Vector Portmanteau(ll): 148.108 Vector AR 1-5 test: F(80,219)= 1.0155 [0.4555] Vector Normality test: Chi'2(8) = 15.358 [0.0525] Vector hetero test: F(350,346)= 0.47850 [1.0000] Not enough observations for hetero-X test
Figure A.3. Estimating the unrestricted VAR in PcGive 10.1; model diagnostics.
cointegration analysis' box.5 The results are produced in Figure A.4,6 provid- ing the eigenvalues of the system (and log-likelihoods for each cointegration rank), standard reduced rank test statistics and those adjusted for degrees of freedom (plus the significance levels for rejecting the various null hypotheses) and full-rank estimates of a, p and IT (the P are automatically normalized along the principal diagonal). Graphical analysis of the (J-vectors (unadjusted and adjusted for short-run dynamics) are available to provide a visual test of which vectors are stationary,7 and graphs of the recursive eigenvalues associated with each eigenvector can be plotted to consider the stability of the cointegration vectors.8
5 Note that the default output only produces the trace test. To obtain the A-max test as well as the default (and tests adjusted for degrees of freedom), in PcGive choose 'Model', then 'Options', 'Further options' and put a cross in the box for cointegration test with Max test. 6 Note that these differ from Box 5.5 and Table 5.5, since the latter are based on a model without the outlier dummies included in the unrestricted short-run model. 7 The companion matrix that helps to verify the number of unit roots at or close to unity, corresponding to the 7(1) common trends, is available when choosing the 'Dynamic analysis' option in the 'Test' model menu in PcGive. 8 Note that, to obtain recursive options, the 'recursive estimation' option needs to be selected when choosing OLS at the 'Estimation Model' window when formulating the model for estimation.
264 APPENDIX 1(1) cointegration analysis, 1964 (2) to 1989 (2)
eigenvalue
0.57076 0.11102 0.063096 0.0020654
loglik for rank 1235.302 0 1278.012 1 1283.955 2 1287.246 3 1287.350 4
rank Trace test [ Prob] Max test [ Prob] Trace test [T-nm] Max test [T-nm] 0 104.10 [0.000]** 85.42 [0.000]** 87.61 [0.000]** 71.89 [0.000]' 1 18.68 [0.527] 11.89 [0.571] 15.72 [0.737] 10.00 [0.747] 2 6.79 [0.608] 6.58 [0.547] 5.72 [0.731] 5.54 [0.676] 3 0.21 [0.648] 0.21 [0.648] 0.18 [0.675] 0.18 [0.675]
Asymptotic p-values based on: Unrestricted constant Unrestricted variables: [0] = Constant [1] = CSeasonal [2] = CSeasonal_l [3] = CSeasonal_2 [4] = dumrst [5] = dumdp [6] = dumdp1 Number of lags used in the analysis: 4
beta (scaled on diagonal; cointegrating vectors in columns) rm y dp rstar
1.0000 -1.0337 6.4188 6.7976
15.719 1.0000 -207.49 131.02
-0.046843 0.064882 1.0000
-0.039555
1.6502 -0.13051 8.6574 1.0000
alpha rm -0.18373 y -0.0081691 dp 0.022631 rstar 0.0046324
0.00073499 -0.0010447 0.00023031 -0.0011461
0.0012372 -0.16551
-0.042258 -0.0022919
-0.0010530 -0.00080063 0.0010822 0.0018533
long-run matrix, rank 4 rm y
rm -0.17397 0.19088 y -0.018159 -0.0032342 dp 0.030017 -0.026047 rstar -0.010218 -0.0063252
dp -1.3397
-0.0081182 0.064590 0.28129
rstar -1.1537 -0.18666 0.18677 -0.11673
Figure A.4. 7(1) cointegration analysis in PcGive 10.1.
After deciding on the value of r < n, it is necessary to select a reduced rank system. In PcGive, under 'Model', choose 'Model settings' (not 'Formulate'), select the option 'Cointegrated VAR' and in the window that appears set the cointegration rank (here we change '3' to '1', as the test statistics indicate that r — 1). Leave the 'No additional restrictions' option unchanged as the default, click OK in this window and the next, and the output (an estimate of the new value of II together with the reduced-form cointegration vectors) will be written to the results window in GiveWin.
Finally, we test for restrictions on a and p (recall that these should usually be conducted together). To illustrate the issue, the model estimated in Chapter 6
APPENDIX , , 265
Figure A.5. Testing restrictions on a and B using 'General Restrictions' in PcGive 10.1.
is chosen (instead of the one above) with a time trend restricted into the cointegration space and r — 2. Thus, we test the following restrictions:
, r-i i * * o [ 0 — 1 * * *
,_ r* o * o ~~ L* o * o
using the option 'General restrictions'. To do this in PcGive, under 'Model', choose 'Model settings', select the option 'Cointegrated VAR' and in the window that appears set the cointegration rank (here we change '3' to '2', since we have chosen r = 2). Click the 'General restrictions' option, type the relevant restrictions into the window (note that in the 'Model' the parameters are identified by '&' and a number—see Figure A.5), click OK in this window (and the next) and the results will be written into Give Win (Figure A.6—see also the top half of Box 6.1).
CONCLUSION
For the applied economist wishing to estimate cointegration relations and then to test for linear restrictions, PcGive 10.1 is a flexible option. But there are others. Harris (1995) compared three of the most popular options available in the 1990s (Microfit 3.0, Cats (in Rats) and PcFiml—the latter the predecessor to the current PcGive). The Cats program9 has seen little development since its 9 Cointegration Analysis of Times Series (Cats in Rats), version 1.0, by Henrik Hansen and Katrina Juselius, distributed by Estima.
266 APPENDIX
Cointegrated VAR (4) in: [0] - rm tl] = y [2] = dp [3] = rstar Unrestricted variables: [0] = dumrst [1] = dumdp [2] = dumdp 1 [3] = Constant [4] = CSeasonal [5] = CSeasonal_l [6] = CSeasonal_2 Restricted variables: [0] = Trend Number of lags used in the analysis: 4
General cointegration restrictions: &8=-l;&9=l;&12=0; &13=0;&14=-1; &2=0;&3=0;&6=0;&7=0;
beta rm y dp rstar Trend
-1.0000 1.0000 -6.5414 -6.6572 0.00000
0.00000 -1.0000 2.8091 -1.1360
0.0066731
Standard errors of beta rm y dp rstar Trend
alpha rm y dp rstar
0.00000 0.00000 0.88785 0.33893 0.00000
0.17900 0.00000
-0.011637 0.00000
0.00000 0.00000 0.46671 0.19346
0.00020557
0.083003 0.00000 -0.15246 0.00000
Standard errors of alpha rm 0.018588 0.074038 y 0.00000 0.00000 dp 0.0078017 0.031076 rstar 0.00000 0.00000
log—likelihood 1290.6274 -T/2log|Omega| 1863.87857 no. of observations 101 rank of long-run matrix 2 beta is identified AIC -35.2253 HQ -34.3344
no. of parameters 85 no. long-run restrictions 5
SC FPE
-33.0245 1.08703e-015
LR test of restrictions: Chi"2(5) = 3.6020 [0.6080]
Figure A6. Output from testing restrictions on a and 3 using 'General restrictions' in PcGive 10.1.
APPENDIX 267
inception (although there is an 1(2) version available as a free download for users of the standard 7(1) version of Cats). Microfit 4.010 offers a modelling strategy based closely on the approach used in, for example, Garratt, Lee, Pesaran and Shin (1999), whereby the user moves toward estimating the con- ditional 7(1) model with exogenous variables. All three packages have their strengths and limitations (in comparison with each other), and therefore it is likely that different users will have different views on which they prefer.
10 Microfit 4.0, An Interactive Econometric Analysis, developed by Hashem Pesaran and Bahram Pesaran and distributed by Oxford University Press.
- Applied Time Series Modelling and Forecasting
- Contents
- Preface
- 1 Introduction and Overview
- Some Initial Concepts
- Data-generating Processes
- Role of the Error Term ut and Statistical Inference
- Forecasting
- Outline of the Book
- 2 Short- and Long-run Models
- Long-run Models
- Stationary and Non-stationary Time Series
- Spurious Regressions
- Cointegration
- Short-run Models
- Conclusion
- 3 Testing for Unit Roots
- The Dickey Fuller Test
- Perron's Procedure
- Augmented Dickey-Fuller Test
- Power and Level of Unit Root Tests
- Structural Breaks and Unit Root Test
- Seasonal Unit Roots
- Structural Breaks and Seasonal Unit Root Tests
- Periodic Integration and Unit Root-testing
- Conclusion on Unit Root Tests
- 4 Cointegration in Single Equations
- The Engle–Granger (EG) Approach
- Testing for Cointegration with a Structural Break
- Alternative Approaches
- Dynamic Models
- Fully Modified Estimators
- Problems with the Single Equation Approach
- Estimating the Short-run Dynamic Model
- Seasonal Cointegration
- Periodic Cointegration
- Asymmetric Tests for Cointegration
- Conclusions
- 5 Cointegration in Multivariate Systems
- The Johansen Approach
- Testing the Order of Integration of the Variables
- Formulation of the Dynamic Model
- Testing for Reduced Rank
- Deterministic Components in the Multivariate Model
- Testing of Weak Exogeneity and VECM with Exogenous I(1) Variables
- Testing for Linear Hypotheses on Cointegration Relations
- Testing for Unique Cointegration Vectors
- Joint Tests of Restrictions on alpha and beta
- Seasonal Unit Roots
- Seasonal Cointegration
- Conclusions
- Appendix 1 Programming in SHAZAM
- 6 Modelling the Short-run Multivariate System
- Introduction
- Estimating the Long-run Cointegration Relationships
- Parsimonious VECM
- Conditional PVECM
- Structural Modelling
- Structural Macroeconomic Modelling
- 7 Panel Data Models and Cointegration
- Introduction
- Panel Data and Modelling Techniques
- Panel Unit Root Tests
- Testing for Cointegration in Panels
- Estimating Panel Cointegration Models
- Conclusion on Testing for Unit Roots and Cointegration in Panel Data
FMBF computer lab3.pdf
FMBF: Computer Practical 3
Introduction
This workshop covers ARCH and GARCH models. You will see how to estimate the
various models and perform appropriate diagnostic testing to assist in choosing a
preferred model.
We will initially work with a dataset comprising exchange rate data for the period 2
January 1980 to 21 May 1987. The contents of this dataset are listed in Table 1.
Table 1: Data contained in GARCH.XLS
DAY Day of week (1 is Monday)
BP US$ - British Pound
CD US$ - Canadian Dollar
DM US$ - Deutsche Mark
JY US$ - Japanese Yen
SF US$ - Swiss Franc
Exercise 1: Volatility in daily exchange rates - ARCH Models
1. Download the data from duo and load into PcGive.
2. dlog all of the exchange rate series, multiplying by 100 so we have daily
percentage changes in each exchange rate, i.e for the DM it would be 100 * ln(DM)
- ln(DMt-1).
3. Plot the daily change data in PcGive; what conclusions do you come to?
4. Identify an appropriate AR model for each of the exchange rate series.
Hint: You will want to use ACF/PACF to help you here, as well as employing
‘overfitting’ to check the specification. You may find that there is no evidence of
serial correlation and thus changes in the exchange rate do not have persistence
and can be regarded as a random walk.
5. Test for ARCH effects by going to Test > Test... and selecting ARCH test with
order 1. At a minimum you should test an order of 1 and 6. Are ARCH effects
present (The null hypothesis is that there is no ARCH effect)?
Notes: you may save, square and plot the residuals (Test > Test Menu: Store in
Database… > Residuals, and use Calculator to calculate the squared error
terms), and then you can examine the ACF and PACF of these squared residuals.
6. Estimate an appropriate ARCH model based on your results from the previous
step (Model > Category: Models for financial data; Model class: GARCH
Models using PcGive > Formulate).
7. Select the variables from your AR model and click OK. In the Model Settings
dialogue box select the appropriate specification; for example, an ARCH(6) model
will be p=0 and q=6. Click OK and estimate using maximum likelihood.
8. Analyse the results from your model. In particular, you will want to check if any
of the parameters are insignificant, and if so consider a more parsimonious
specification. It may be that you want to perform a Wald test if you have more
than one insignificant parameter (Test > Test Menu: Exclusion Restrictions).
Notes: Review Brooks textbook on Page 234. The null hypothesis of Wald test is
α = 0. The example shows the results are significant, which means we reject the
null hypothesis that α1 = 0, and we do not exclude α1.
9. Perform further tests on your model from Test > Test.... Recall the Portmanteau is
the Ljung Box test. Can you improve the model?
Notes: You may wish to overfit by adding an extra AR and ARCH parameter.
Ljung Box test: Brooks textbook on Page 234-235.
We will now see if it is advantageous to formulate a more parsimonious GARCH(1,1)
model and examine different the types of GARCH models that we can estimate with
PcGive.
Notes: Brooks textbook on Page 232-444.
Exercice 2: Volatility in daily exchange rates - GARCH Models
1. Using the same AR specification identified above, go to the Model Settings
dialogue box, but this time specify p=1 and q=1 so we have a GARCH(1,1) model
(q/β captures lags of squared error terms, and p/α captures lags of conditional
variance).
2. Interpret the results. Consider if you can improve the model with a change in the
AR structure. As above, check the diagnostics.
Notes: GARCH(1,1) shows effect of lagged shocks dies out very slowly.
α+β=0.978, so the estimated process is close to being nonstationary. You may also
try a GARCH (2,1) model, although GARCH(2,2) will be too far.
3. Estimate an EGARCH(1,1) model using the AR specification you identified above.
To do this you will need to check the EGARCH box in Model Settings.
Notes: EGARCH model:
The coefficient of eps[-1] is γ. If γ<0, positive shocks generate less volatility than
negative shocks (bad news).
Brooks textbook on Page 406.
4. Estimate a GJR-GARCH(1,1) model (this type of GARCH specification.is from
Glosten et al. (1993). To do this, open Model Settings and expand the GARCH
variations section. Select Threshold GARCH. Brook Page 404.
5. Estimate a GARCH(1,1)-in-Mean. To do this, open Model Settings and expand
the GARCH variations section. Select h_t in mean. Brooks Page 409.
Notes: If h_t is positive and significant, it means higher risk given higher conditional
variance, yields higher expected returns.
For all of the above, you should carefully evaluate the adequacy of your model
with particular reference to the diagnostic tests.
We will now use our GARCH models to produce forecasts:
Exercice 3: Forecasting
1. Using your preferred AR(p)-GARCH(1,1) model, in the Estimate Model dialogue
box hold back a number of observations with the Less forecasts setting. Initially,
you may wish to try 14.
2. After estimating the model, go to Test > Forecast.... Select the number of forecasts
you would like. If you look under Options you are able to control the number of
pre-forecast observations that are graphed and there is also a check box for Write
results instead of graphing if you desire a print-out of the forecasts rather than a
graph.
3. Examine the forecasts from your model. Note that you can use Test > Store in
Database to commit the Forecasts and Forecasts standard errors to the database.
Notes: The conditional variance is increasing and is converging to the unconditional
variance. Several useful measures concerning the forecast errors are displayed,
including mean(Error), SD(Error), RMSE and MAPE. RMSE is root mean squared
error, and MAPE is mean absolute percent error. The smaller these errors are, the
better for the forecasting, which means the model forecasts are able to account for
much of the variability of the out-of-sample part of the data.
The procedure for forecasting with ARMA models is similar to the above (Brooks
Page 257-258):
Exercice 4: Forecasting with ARMA models
1. Estimate an appropriate ARMA model, holding observations back for forecasting.
2. Use Test > Forecast... to produce a forecast from your model.
3. If you employ Write results instead of graphing you will see that PcGive displays
the mean(Error) (Heteroscedasticity-ajusted mean square error). This is the loss
function used by Bollerslev & Ghysels (1996) for model comparison. You will also
see RMSE (root mean square error) and MAPE (mean absolute error).
4. Evaluate the forecasting ability of your model. You may wish to compare it with
another suitable specification.
Points to note
More details about PcGive's implementation of volatility models, together with a
tutorial, may be found in Hendry and Doornik (2001).
It is often useful to use Model > Progress… to produce a summary table that can
aid analysis of competing models.
References
L. Glosten, R. Jagannathan, and D. Runkle. On the relation between the expected
value and the volatility of the nominal excess return on stocks. Journal of Finance,
48(5):1779{1801, 1993.
D. F. Hendry and J. A. Doornik. Empirical Econometric Modelling Using PcGive,
volume 1. Timberlake Consultants Ltd., 3 edition, 2001.
M. Verbeek. A Guide to Modern Econometrics. John Wiley & Sons, Inc., 2004.
FMBF computer lab4.pdf
Practice for FMBF computer LAB session 4
In this session, you are aiming to review the contents in the last three sessions we had through practicing the questions below with Givewin and PcGive.
Key words: AR, MA, ARMA, ARIMA Modelling; Cointegration; Volatility models: ARCH, GARCH, EGARCH, etc.. and forecasts
AR Modelling (Dataset: BA)
1. Can Graphics Function suggest a preferred model for the price of British Airways?
2. Constructing the model for price of BA.
3. Process the diagnostic steps to test the fitness of the model built up in question 2.
Unit root test (Dataset: FTSE)
4. Manual unit root test on the variable FTSE100 and explain the result.
5. Automated unit root testing on FTSE100 and explain the result.
6. If appropriate, test for the presence of I(2) for FTSE100 series.
Cointegration-EG test (Dataset: FTSE)
7. Does cointegration exist between FTSE100 and FTSEALL?
Cointegration-Johansen test (Dataset: Johansen)
8. Detect cointegration and find out the relationship among Yb Yc Yd.
ARCH and GARCH modelling (Dataset: GARCH)
9. Build up a proper model for the series of exchange rate of Canadian Dollar (variable CD in the dataset) and explain why the model you obtain should be a preferred model.
10. Use your preferred model to produce forecasts.
additional topic.pdf
LIMITED DEPENDENT VARIABLE
MODELS
(Additional topic)
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008 1
Some Examples of when Limited Dependent Variables may
be used
• There are numerous examples of instances where this may arise, for example where we want to model:
• Why firms choose to list their shares on the NASDAQ rather than the NYSE
• Why some stocks pay dividends while others do not
• What factors affect whether countries default on their sovereign debt
• Why some firms choose to issue new stock to finance an expansion while others issue bonds
• Why some firms choose to engage in stock splits while others do not.
• It is fairly easy to see in all these cases that the appropriate form for the dependent variable would be a 0-1 dummy variable since there are only two possible outcomes. There are, of course, also situations where it would be more useful to allow the dependent variable to take on other values, but these will be considered later.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Linear Probability Model
• We will first examine a simple and obvious, but unfortunately flawed, method for dealing with binary dependent variables, known as the linear probability model.
• it is based on an assumption that the probability of an event occurring, Pi, is linearly related to a set of explanatory variables
• The actual probabilities cannot be observed, so we would estimate a model where the outcomes, yi (the series of zeros and ones), would be the dependent variable.
• This is then a linear regression model and would be estimated by OLS.
• The set of explanatory variables could include either quantitative variables or dummies or both.
• The fitted values from this regression are the estimated probabilities for yi =1 for each observation i.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
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The Linear Probability Model
• The slope estimates for the linear probability model can be interpreted as the change in the probability that the dependent variable will equal 1 for a one- unit change in a given explanatory variable, holding the effect of all other explanatory variables fixed.
• Suppose, for example, that we wanted to model the probability that a firm i will pay a dividend p(yi = 1) as a function of its market capitalisation (x2i, measured in millions of US dollars), and we fit the following line:
where denotes the fitted or estimated probability for firm i.
• This model suggests that for every $1m increase in size, the probability that the firm will pay a dividend increases by 0.012 (or 1.2%).
• A firm whose stock is valued at $50m will have a -0.3+0.01250=0.3 (or 30%) probability of making a dividend payment.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
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The Fatal Flaw of the Linear Probability Model
• Graphically, the situation we have is
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Disadvantages of the Linear Probability Model
• While the linear probability model is simple to estimate and intuitive to interpret, the diagram on the previous slide should immediately signal a problem with this setup.
• For any firm whose value is less than $25m, the model-predicted probability of dividend payment is negative, while for any firm worth more than $88m, the probability is greater than one.
• Clearly, such predictions cannot be allowed to stand, since the probabilities should lie within the range (0,1).
• An obvious solution is to truncate the probabilities at 0 or 1, so that a probability of -0.3, say, would be set to zero, and a probability of, say, 1.2, would be set to 1.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Disadvantages of the Linear Probability Model 2
• However, there are at least two reasons why this is still not adequate.
• The process of truncation will result in too many observations for which the estimated probabilities are exactly zero or one.
• More importantly, it is simply not plausible to suggest that the firm's probability of paying a dividend is either exactly zero or exactly one. Are we really certain that very small firms will definitely never pay a dividend and that large firms will always make a payout?
• Probably not, and so a different kind of model is usually used for binary dependent variables either a logit or a probit specification.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Disadvantages of the Linear Probability Model 3
• The LPM also suffers from a couple of more standard econometric problems that we have examined in previous chapters.
• Since the dependent variable only takes one or two values, for given (fixed in repeated samples) values of the explanatory variables, the disturbance term will also only take on one of two values.
• Hence the error term cannot plausibly be assumed to be normally distributed.
• Since the disturbance term changes systematically with the explanatory variables, the former will also be heteroscedastic.
• It is therefore essential that heteroscedasticity-robust standard errors are always used in the context of limited dependent variable models.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Logit and Probit: Better Approaches
• Both the logit and probit model approaches are able to overcome the limitation of the LPM that it can produce estimated probabilities that are negative or greater than one.
• They do this by using a function that effectively transforms the regression model so that the fitted values are bounded within the (0,1) interval.
• Visually, the fitted regression model will appear as an S-shape rather than a straight line, as was the case for the LPM.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Logit Model
• The logit model is so-called because it uses a the cumulative logistic distribution to transform the model so that the probabilities follow the S- shape given on the previous slide.
• With the logistic model, 0 and 1 are asymptotes to the function and thus the probabilities will never actually fall to exactly zero or rise to one, although they may come infinitesimally close.
• The logit model is not linear (and cannot be made linear by a transformation) and thus is not estimable using OLS.
• Instead, maximum likelihood is usually used to estimate the parameters of the model.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Using a Logit to Test the Pecking Order Hypothesis
• The theory of firm financing suggests that corporations should use the cheapest methods of financing their activities first (i.e. the sources of funds that require payment of the lowest rates of return to investors) and then only switch to more expensive methods when the cheaper sources have been exhausted.
• This is known as the “pecking order hypothesis”.
• Differences in the relative cost of the various sources of funds are argued to arise largely from information asymmetries since the firm's senior managers will know the true riskiness of the business, whereas potential outside investors will not.
• Hence, all else equal, firms will prefer internal finance and then, if further (external) funding is necessary, the firm's riskiness will determine the type of funding sought.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Data
• Helwege and Liang (1996) examine the pecking order hypothesis in the context of a set of US firms that had been newly listed on the stock market in 1983, with their additional funding decisions being tracked over the 1984 - 1992 period.
• Such newly listed firms are argued to experience higher rates of growth, and are more likely to require additional external funding than firms which have been stock market listed for many years.
• They are also more likely to exhibit information asymmetries due to their lack of a track record.
• The list of initial public offerings (IPOs) was obtained from the Securities Data Corporation and the Securities and Exchange Commission with data obtained from Compustat.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Aims of the Study and the Model
• A core objective of the paper is to determine the factors that affect the probability of raising external financing.
• As such, the dependent variable will be binary -- that is, a column of 1's (firm raises funds externally) and 0's (firm does not raise any external funds).
• Thus OLS would not be appropriate and hence a logit model is used.
• The explanatory variables are a set that aims to capture the relative degree of information asymmetry and degree of riskiness of the firm.
• If the pecking order hypothesis is supported by the data, then firms should be more likely to raise external funding the less internal cash they hold.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Variables used in the Model
• The variable deficit measures (capital expenditures + acquisitions + dividends - earnings).
• Positive deficit is a variable identical to deficit but with any negative deficits (i.e. surpluses) set to zero
• Surplus is equal to the negative of deficit for firms where deficit is negative
• Positive deficit operating income is an interaction term where the two variables are multiplied together to capture cases where firms have strong investment opportunities but limited access to internal funds
• Assets is used as a measure of firm size
• Industry asset growth is the average rate of growth of assets in that firm's industry over the 1983-1992 period
• Firm's growth of sales is the growth rate of sales averaged over the previous 5 years
• Previous financing is a dummy variable equal to one for firms that obtained external financing in the previous year.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Results from Logit Estimation
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Source: Helwege and Liang (1996)
Analysis of Results
• The key variable, deficit has a parameter that is not statistically significant and hence the probability of obtaining external financing does not depend on the size of a firm's cash deficit.
• Or an alternative explanation, as with a similar result in the context of a standard regression model, is that the probability varies widely across firms with the size of the cash deficit so that the standard errors are large relative to the point estimate.
• The parameter on the surplus variable has the correct negative sign, indicating that the larger a firm's surplus, the less likely it is to seek external financing, which provides some limited support for the pecking order hypothesis.
• Larger firms (with larger total assets) are more likely to use the capital markets, as are firms that have already obtained external financing during the previous year.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Probit Model
• Instead of using the cumulative logistic function to transform the model, the cumulative normal distribution is sometimes used instead.
• This gives rise to the probit model.
• As for the logistic approach, this function provides a transformation to ensure that the fitted probabilities will lie between zero and one.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Logit or Probit?
• For the majority of the applications, the logit and probit models will give very similar characterisations of the data because the densities are very similar.
• That is, the fitted regression plots will be virtually indistinguishable, and the implied relationships between the explanatory variables and the probability that yi =1 will also be very similar.
• Both approaches are much preferred to the linear probability model. The only instance where the models may give non-negligibility different results occurs when the split of the yi between 0 and 1 is very unbalanced - for example, when yi =1 occurs only 10% of the time.
• Stock and Watson (2006) suggest that the logistic approach was traditionally preferred since the function does not require the evaluation of an integral and thus the model parameters could be estimated faster.
• However, this argument is no longer relevant given the computational speeds now achievable and the choice of one specification rather than the other is now usually arbitrary.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Parameter Interpretation for Logit and Probit Models
• Standard errors and t-ratios will automatically be calculated by the econometric software package used, and hypothesis tests can be conducted in the usual fashion.
• However, interpretation of the coefficients needs slight care.
• It is tempting, but incorrect, to state that a 1-unit increase in x2i, for example, causes a 2 % increase in the probability that the outcome corresponding to yi =1 will be realised.
• This would have been the correct interpretation for the linear probability model.
• However, for logit or probit models, this interpretation would be incorrect because the form of the function is Pi = 1 + 2 x2i + ui, for example, but rather Pi = F(x2i) where F represents the (non-linear) logistic or cumulative normal function.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Parameter Interpretation for Logit and Probit Models
• To obtain the required relationship between changes in x2i and Pi, we would need to differentiate F with respect to x2i and it turns out that this derivative is 2F(x2i) .
• So in fact, a 1-unit increase in x2i will cause a 2F(x2i) increase in probability.
• Usually, these impacts of incremental changes in an explanatory variable are evaluated by setting each of them to their mean values.
• These estimates are sometimes known as the marginal effects.
• There is also another way of interpreting discrete choice models known as the random utility model.
• The idea is that we can view the value of y that is chosen by individual i (either 0 or 1) as giving that person a particular level of utility, and the choice that is made will obviously be the one that generates the highest level of utility.
• This interpretation is particularly useful in the situation where the person faces a choice between more than 2 possibilities – see a later slide.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Goodness of Fit for Probit and Logit Models
• While it would be possible to calculate the values of the standard goodness of fit measures such as RSS, R2 these cease to have any real meaning.
• R2, if calculated in the usual fashion, will be misleading because the fitted values from the model can take on any value but the actual values will only be either 0 and 1.
• Thus if yi =1 and = 0.8, the model has effectively made the correct prediction, whereas R2 and will not give it full credit for this.
• Two goodness of fit measures that are commonly reported for limited dependent variable models are
• The percentage of yi values correctly predicted
• A measure known as „pseudo-R2‟ (also known as McFadden's R2), defined as one minus the ratio of the LLF for the logit or probit model to the LLF for a model with only an intercept.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
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Multinomial Linear Dependent Variables
• There are many instances where investors or financial agents are faced with more alternatives than a simple binary choice.
• For example:
• A company may be considering listing on the NYSE, the NASDAQ or the AMEX markets.
• A firm that is intending to take over another may choose to pay by cash, with shares, or with a mixture of both.
• A retail investor may be choosing between 5 different mutual funds.
• A credit ratings agency could assign 1 of 16 (AAA to B3/B-) different ratings classifications to a firm's debt.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Multinomial Linear Dependent Variables (Cont’d)
• Notice that the first three of these examples are different from the last one.
• In the first three cases, there is no natural ordering of the alternatives: the choice is simply made between them.
• In the final case, there is an obvious ordering, because a score of 1, denoting a AAA-rated bond, is better than a score of 2, denoting a AA1/AA+-rated bond, and so on.
• These two situations need to be distinguished and a different approach used in each case. In the first (when there is no natural ordering), a multinomial logit or probit would be used, while in the second (where there is an ordering), an ordered logit or probit would be used.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Discrete Choice Problems
• When the alternatives are unordered, this is sometimes called a discrete choice or multiple choice problem.
• The models used are derived from the principles of utility maximisation - that is, the agent chooses the alternative that maximises his utility relative to the others.
• Econometrically, this is captured using a simple generalisation of the binary setup discussed earlier. Thus the multinomial logit and probit are direct extensions of their binary counterparts.
• When there were only 2 choices (0, 1), we required just one equation to capture the
• probability that one or the other would be chosen.
• If there are now three alternatives, we would need two equations; for four alternatives, we would need three equations. In general, if there are m possible alternative choices, we need m-1 equations.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Modelling the Travel to Work Choice
• The multiple choice example most commonly used is that of the selection of the mode of transport for travel to work.
• Suppose that the journey may be made by car, bus, or bicycle (3 alternatives), and suppose that the explanatory variables are the person's income (I), total hours worked (H), their gender (G) and the distance travelled (D).
• We could set up 2 equations (e.g., for bus and car) and then travel by bicycle becomes a sort of reference point.
• While the fitted probabilities will always sum to unity by construction, as with the binomial case, there is no guarantee that they will all lie between zero and one.
• In order to make a prediction about which mode of transport a particular individual will use, given that the parameters in, the largest fitted probability would be set to one and the others set to zero.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Ordered Response Models
• Some limited dependent variables can be assigned numerical values that have a natural ordering.
• The most common example in finance is that of credit ratings, as discussed previously, but a further application is to modelling a security's bid-ask spread.
• In such cases, it would not be appropriate to use multinomial logit or probit since these techniques cannot take into account any ordering in the dependent variables.
• Using the credit rating example, the model is set up so that a particular bond falls in the AA+ category (using Standard and Poor's terminology) if its unobserved (latent) creditworthiness falls within a certain range that is too low to classify it as AAA and too high to classify it as AA.
• The boundary values between each rating are then estimated along with the model parameters.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Are Unsolicited Credit Ratings Biased Downwards?
• The main credit ratings agencies construct solicited ratings, which are those where the issuer of the debt contacts the agency and pays them a fee for producing the rating.
• Many firms globally do not seek a rating (because, for example, the firm believes that the ratings agencies are not well placed to evaluate the riskiness of debt in their country or because they do not plan to issue any debt or because they believe that they would be awarded a low rating).
• But the agency may produce a rating anyway. Such „unwarranted and unwelcome‟ ratings are known as unsolicited ratings.
• All of the major ratings agencies produce unsolicited ratings as well as solicited ones, and they argue that there is a market demand for this information even if the issuer would prefer not to be rated.
• Companies in receipt of unsolicited ratings argue that these are biased downwards relative to solicited ratings, and that they cannot be justified without the level of detail of information that can only be provided by the rated company itself.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Data and Methodology
• A study by Poon (2003) seeks to test the conjecture that unsolicited ratings are biased after controlling for the rated company's characteristics that pertain to its risk.
• The data employed comprise a pooled sample of all companies that appeared on the annual „issuer list‟ of S&P during the 1998-2000 years.
• This list contains both solicited and unsolicited ratings covering 295 firms over 15 countries and totaling 595 observations.
• As expected, the financial characteristics of the firms with unsolicited ratings are significantly weaker than those for firms that requested ratings.
• The core methodology employs an ordered probit model with explanatory variables comprising firm characteristics and a dummy variable for whether the firm's credit rating was solicited or not:
with
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
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where
• Ri are the observed ratings scores that are given numerical values as follows:
AA or above = 6, A = 5, BBB = 4, BB = 3, B = 2 and CCC or below = 1
• Ri * is the unobservable „true rating‟ (or „an unobserved continuous variable
representing S&P‟s assessment of the creditworthiness of issuer i‟)
• Xi is a vector of variables that explain the variation in ratings
• is a vector of coefficients; i are the threshold parameters to be estimated
• i is a disturbance term that is assumed normally distributed.
• The explanatory variables attempt to capture the creditworthiness using publicly
available information.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Definitions of Variables
• Two specifications are estimated: the first includes the variables listed below, while the second additionally incorporates an interaction of the main financial variables with a dummy variable for whether the firm's rating was solicited (SOL) and separately with a dummy for whether the firm is based in Japan.
• The Japanese dummy is used since a disproportionate number of firms in the sample are from this country.
• The financial variables are ICOV - interest coverage (i.e. earnings \ interest); ROA - return on assets; DTC - total debt to capital; and SDTD - short term debt to total debt.
• Three variables SOVAA, SOVA, and SOVBBB are dummy variables that capture the debt issuer's sovereign credit rating (AA; A; BBB or below)
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Ordered Probit Results for the Determinants of Credit
Ratings
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Source: Poon (2003)
Analysis of Ordered Probit Results
• The key finding is that the SOL variable is positive and statistically significant in Model 1 (and it is positive but insignificant in Model 2).
• This indicates that even after accounting for the financial characteristics of the firms, unsolicited firms receive ratings on average 0.359 units lower than an otherwise identical firm that had requested a rating.
• The parameter estimate for the interaction term between the solicitation and Japanese dummies (SOL*JP) is positive and significant in both specifications, indicating strong evidence that Japanese firms soliciting ratings receive higher scores.
• On average, firms with stronger financial characteristics (higher interest coverage, higher return on assets, lower debt to total capital, or a lower ratio of short term debt to long term debt) have higher ratings.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Heckman 2-Step Procedure
• A major flaw that potentially exists within the above analysis is the self-selection bias or sample selection bias that may have arisen if firms that would have received lower credit ratings (because they have weak financials) elect not to solicit a rating.
• If the probit equation for the determinants of ratings is estimated ignoring this potential problem and it exists, the coefficients will be inconsistent.
• To get around this problem and to control for the sample selection bias, Heckman (1979) proposed a 2-step procedure.
• In this case would involve first estimating a 0-1 probit model for whether the firm chooses to solicit a rating and second estimating the ordered probit model for the determinants of the rating. The first stage probit model is
• where Yi = 1 if the firm has solicited a rating and 0 otherwise, and Yi * denotes the
latent propensity of issuer i to solicit a rating, Zi are the variables that explain the choice to be rated or not, and are the parameters to be estimated.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
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The Heckman 2-Step Procedure
• When this equation has been estimated, the rating Ri as defined above in will only be observed if Yi = 1.
• The error terms from the two equations, i and i follow a bivariate standard normal distribution with correlation .
• The table on the following page shows the results from the two-step estimation procedure, with the estimates from the binary probit model for the decision concerning whether to solicit a rating in panel A and the determinants of ratings for rated firms in panel B.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Heckman 2-Step Procedure: Results
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Source: Poon (2003)
The Heckman 2-Step Procedure: Analysis
• A positive parameter value in panel A indicates that higher values of the associated variable increases the probability that a firm will elect to be rated.
• Of the four financial variables, only the return on assets and the short term debt as a proportion of total debt have correctly signed and significant (positive and negative respectively) impacts on the decision to be rated.
• The parameters on the sovereign credit rating dummy variables (SOVAA, SOVA and SOVB) are all significant and negative in sign, indicating that any debt issuer in a country with a high sovereign rating is less likely to solicit its own rating from S&P, other things equal.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Heckman 2-Step Procedure: Analysis (Cont’d)
• These sovereign rating dummy variables have the opposite sign in the ratings determinant equation (panel B) as expected, so that firms in countries where government debt is highly rated are themselves more likely to receive a higher rating.
• Of the four financial variables, only ROA has a significant (and positive) effect on the rating awarded.
• The dummy for Japanese firms is also positive and significant, and so are three of the four financial variables when interacted with the Japan dummy, indicating that S&P appears to attach different weights to the financial variables when assigning ratings to Japanese firms compared with comparable firms in other countries.
• Finally, the estimated correlation between the error terms in the decision to be rated equation and the ratings determinant equation, , is significant and negative (-0.836), indicating that the results in table 11.3 above would have been subject to self-selection bias and hence the results of the two-stage model are to be preferred.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Censored and Truncated Variables
• Censored or truncated variables occur when the range of values observable for the dependent variables is limited for some reason.
• Unlike the types of limited dependent variables examined so far, censored or truncated variables may not necessarily be dummies.
• A standard example is that of charitable donations by individuals.
• It is likely that some people would actually prefer to make negative donations (that is, to receive from the charity rather than to donate it), but since this is not possible, there will be many observations at exactly zero.
• So suppose, for example that we wished to model the relationship between donations to charity and peoples' annual incomes, in pounds.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Censored and Truncated Variables (Cont’d)
• Given the observed data, with many observations on the dependent variable stuck at zero, OLS would yield biased and inconsistent parameter estimates.
• An obvious, but flawed, way to get around this would be just to remove all of the zero observations altogether, since we do not know whether they should be truly zero or negative.
• However, as well as being inefficient (since information would be discarded), this would still yield biased and inconsistent estimates.
• This arises because the error term in such a regression would not have an expected value of zero, and it would also be correlated with the explanatory variable(s).
• For both censored and truncated data, OLS will not be appropriate, and an approach based on maximum likelihood must be used, although the model in each case would be slightly different.
• We can work out the marginal effects given the estimated parameters, but these are now more complex than in the logit or probit cases.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Differences between Censored and Truncated Variables
• When the terms are used in econometrics, censored and truncated data are different
• Censored data occur when the dependent variable has been „censored‟ at certain point so that values above (or below) this cannot be observed.
• Even though the dependent variable is censored, the corresponding values of the independent variables are still observable.
• As an example, suppose that a privatisation IPO is heavily oversubscribed, and you were trying to model the demand for the shares using household income, age, education, and region of residence as explanatory variables. The number of shares allocated to each investor may have been capped at, say 250, resulting in a truncated distribution.
• In this example, even though we are likely to have many share allocations at 250 and none above this figure, all of the observations on the independent variables are present and hence the dependent variable is censored, not truncated.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Truncated Variables
• A truncated dependent variable, on the other hand, occurs when the observations for both the dependent and the independent variables are missing when the dependent variable is above (or below) a certain threshold.
• Thus the key difference from censored data is that we cannot observe the xIs either, and so some observations are completely cut out or „truncated‟ from the sample.
• For example, suppose that a bank were interested in determining the factors (such as age, occupation and income) that affected a customer's decision as to whether to undertake a transaction in a branch or on-line. Suppose also that the bank tried to achieve this by encouraging clients to fill in an on-line questionnaire when they log on. There would be no data at all for those who opted to transact in person since they probably would not have even logged on to the bank's web-based system and so would not have the opportunity to complete the questionnaire.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Truncated Variables (Cont’d)
• Thus, dealing with truncated data is really a sample selection problem because the sample of data that can be observed is not representative of the population of interest - the sample is biased, very likely resulting in biased and inconsistent parameter estimates.
• This is a common problem, which will result whenever data for buyers or users only can be observed while data for non-buyers or non-users cannot.
• Of course, it is possible, although unlikely, that the population of interest is focused only on those who use the internet for banking transactions, in which case there would be no problem.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
The Tobit Model
• The approach usually used to estimate models with censored dependent variables is known as tobit analysis, named after Tobin (1958).
• To illustrate, suppose that we wanted to model the demand for privatisation IPO shares, as discussed above, as a function of income (x2i), age (x3i), education (x4i), and region of residence (x5i). The model would be
• yi * represents the true demand for shares (i.e. the number of shares requested)
and this will only be observable for demand less than 250.
• It is important to note in this model that 2, 3, etc., represent the impact on the number of shares demanded (of a unit change in x2i, x3i, etc.) and not the impact on the actual number of shares that will be bought (allocated).
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
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Limitations of the Tobit Model
• Before moving on, two important limitations of tobit modelling should be noted.
• First, such models are much more seriously affected by non-normality and heteroscedasticity than are standard regression models, and biased and inconsistent estimation will result.
• Second, the tobit model requires it to be plausible that the dependent variable can have values close to the limit.
• There is no problem with the privatisation IPO example discussed above since the demand could be for 249 shares.
• However, it would not be appropriate to use the tobit model in situations where this is not the case, such as the number of shares issued by each firm in a particular month.
• For most companies, this figure will be exactly zero, but for those where it is not, the number will be much higher and thus it would not be feasible to issue, say, 1 or 3 or 15 shares.
• In this case, an alternative approach should be used.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Models for Truncated Dependent Variables
• For truncated data, a more general model is employed that contains two equations - one for whether a particular data point will fall into the observed or constrained categories and another for modelling the resulting variable.
• The second equation is equivalent to the tobit approach.
• This two-equation methodology allows for a different set of factors to affect the sample selection (for example the decision to set up internet access to a bank account) from the equation to be estimated (for example, to model the factors that affect whether a particular transaction will be conducted on-line or in a branch).
• If it is thought that the two sets of factors will be the same, then a single equation can be used and the tobit approach is sufficient.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
Models for Truncated Dependent Variables (Cont’d)
• In many cases, however, the researcher may believe that the variables in the sample selection and estimation equations should be different.
• Thus the equations could be
where yi = yi * for ai
* > 0 and yi is unobserved for ai * 0.
• ai * denotes the relative „advantage‟ of being in the observed sample relative to
the unobserved sample.
• The first equation determines whether the particular data point i will be observed or not, by regressing a proxy for the latent (unobserved) variable, ai
*, on a set of factors, zi.
• The second equation is similar to the tobit model.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
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Models for Truncated Dependent Variables (Cont’d)
• Ideally, the two equations will be fitted jointly by maximum likelihood.
• This is usually based on the assumption that the error terms, are multivariate normally distributed and allowing for any possible correlations between them.
• However, while joint estimation of the equations is more efficient, it is computationally more complex and hence a two-stage procedure popularised by Heckman (1976) is often used.
• The Heckman procedure allows for possible correlations between the error terms while estimating the equations separately in a clever way.
‘Introductory Econometrics for Finance’ ©
Chris Brooks 2008
