Advanced Statistics with R code 90% doen just check it and make a summary

CSC425 – Time series analysis and forecasting

Homework 5 – not be submitted

The goal of this assignment is to provide students with some practical training on

GARCH/EGARCH/TGARCH models for the analysis of the volatility of a stock return.

Solutions will be posted on Thursday November 7th 2013

Reading assignment:

1. Read Chapter 4 in short book and Chapter 3 in long book on volatility models 2. Review course documents posted under week 7 and 8.

Problem

Use the datafile nordstrom_w_00_13.csv that contains the Nordstrom (JWN ) stock weekly prices

from January 2000 to October 2013. The data file contains dates (date), daily prices (price). You

can also use the code and the analysis of the S&P500 returns used in week 7 and 8 lectures as

your reference for the analysis of this data. Analyze the Nordstrom stock log returns following

the steps below.

1. Compute log returns, and analyze their time plot.

Moments

N 693 Sum Weights 693

Mean 0.00268299 Sum Observations 1.85931229

Std Deviation 0.06068019 Variance 0.00368209

Skewness -0.2804891 Kurtosis 7.55351876

Uncorrected SS 2.55299156 Corrected SS 2.54800304

Coeff Variation 2261.66262 Std Error Mean 0.00230505

Time plot of returns show that Nordstrom stock returns shows periods of high volatility at

various times, and consistently high volatility from 2007 to 2010. Most returns are

between +/- 10%. Sample moments show that distribution of log returns is somewhat

symmetric with very fat tails (kurtosis = 7.55).

return

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date

01/01/2000 01/01/2002 01/01/2004 01/01/2006 01/01/2008 01/01/2010 01/01/2012 01/01/2014 01/01/2016

2. Is there evidence of serial correlations in the log returns? Use autocorrelations and 5% significance level to answer the question.

Log returns are not serially correlated as shown by the Ljung-Box test with p-values >

0.05 and the autocorrelation plot. .

Name of Variable = return

Mean of Working Series 0.002683

Standard Deviation 0.060636

Number of Observations 693

Autocorrelation Check for White Noise

To Chi- Pr >

Lag Square DF ChiSq --------------------Autocorrelations--------------------

6 6.16 6 0.4050 -0.009 0.011 -0.048 -0.009 0.078 -0.006

12 9.08 12 0.6957 -0.042 -0.038 -0.019 0.011 -0.015 0.016

18 26.17 18 0.0960 0.033 -0.026 0.088 -0.032 0.101 -0.056

24 31.58 24 0.1377 0.029 -0.037 0.025 -0.042 -0.026 -0.048

30 33.84 30 0.2871 -0.029 -0.003 -0.037 0.023 0.011 -0.016

3. Is there evidence of ARCH effects in the log returns? Use appropriate tests at 5% significance level to answer this question.

The analysis below shows a strong ARCH effect. The squared returns are strongly

correlated. The Ljung-Box tests on squared residuals are highly significant with p-values

< 0.001, and autocorrelation plots shows large autocorrelations for the first 10 lags.

Name of Variable = returnsq

Mean of Working Series 0.003684

Standard Deviation 0.011302

Number of Observations 693

Autocorrelation Check for White Noise

To Chi- Pr >

Lag Square DF ChiSq --------------------Autocorrelations--------------------

6 331.56 6 <.0001 0.534 0.281 0.123 0.101 0.167 0.240

12 407.05 12 <.0001 0.246 0.187 0.053 0.036 0.032 0.082

18 469.70 18 <.0001 0.069 0.072 0.128 0.127 0.156 0.145

24 494.90 24 <.0001 0.149 0.090 0.050 0.035 0.014 0.031

30 501.01 30 <.0001 0.033 0.064 0.051 0.023 0.010 -0.004

4. Fit an GARCH(1,1) model for the log returns using a t- distribution for the error terms. Perform model checking (analyze if residuals are white noise, if squared residuals are

white noise, and check if t-distribution is a good fit for the data) and write down the fitted

model.

The GARCH model with t-distribution is an adequate model for the volatility of the

returns. All parameters are significant and the squared residuals are white noise (LB

tests on squared residuals are non significant).

Fitted GARCH model can be expressed as follows:

rt=0.005 +at

at=σtet with σt 2 =0.1483 a

2 t-1+0.836 σ

2 t-1

(Intercept ARCH0 can be considered equal to zero)

Where error term et has t-distribution with 1/0.185=5 degrees of freedom.

The AUTOREG Procedure

GARCH Estimates

SSE 2.55069233 Observations 693

MSE 0.00368 Uncond Var 0.00531549

Log Likelihood 1102.73026 Total R-Square .

SBC -2172.7554 AIC -2195.4605

MAE 0.04086373 AICC -2195.3732

MAPE 113.334007 HQC -2186.6796

Normality Test 126.0152

Pr > ChiSq <.0001

Parameter Estimates

Standard Approx

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.004653 0.001570 2.96 0.0030

ARCH0 1 0.0000848 0.0000429 1.98 0.0482

ARCH1 1 0.1483 0.0369 4.01 <.0001

GARCH1 1 0.8358 0.0359 23.30 <.0001

TDFI 1 0.1851 0.0448 4.14 <.0001

Inverse of t DF

R output Robust Standard Errors: Estimate Std. Error t value Pr(>|t|)

mu 0.004729 0.001465 3.2281 0.001246 omega 0.000086 0.000050 1.7302 0.083590 alpha1 0.145749 0.052318 2.7858 0.005340 beta1 0.836471 0.051967 16.0962 0.000000 shape 5.345569 1.009241 5.2966 0.000000

5. Fit and EGARCH(1,1) model for the NDX log returns using a normal distribution for the

error terms. Perform model checking and write down the fitted model.

The EGARCH model is adequate although the Gaussian distribution on the error terms is

not sufficient to describe most extreme events. No ARCH effects are shown in residuals.

The leverage parameter “theta” is significant showing that volatility of Nordstrom stock

returns is affected more heavily by negative shocks.

The fitted model can be written as follows:

rt=0.0012 +at

at=σtet with ln σt 2 =-0.194+0.213 g(et-1) +0.996 ln σ

2 t-1

g(et-1)= -0.47et-1+[|et-1|-E(et-1)]

The AUTOREG Procedure

Exponential GARCH Estimates

SSE 2.54945444 Observations 693

MSE 0.00368 Uncond Var .

Log Likelihood 1093.55049 Total R-Square .

stres

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quantiles

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t-distribution QQplot for residuals of GARCH model

SBC -2154.3958 AIC -2177.101

MAE 0.04095052 AICC -2177.0136

MAPE 101.222881 HQC -2168.32

Normality Test 71.9939

Pr > ChiSq <.0001

Parameter Estimates

Standard Approx

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.001236 0.001752 0.71 0.4806

EARCH0 1 -0.1940 0.0636 -3.05 0.0023

EARCH1 1 0.2133 0.0316 6.75 <.0001

EGARCH1 1 0.9662 0.0106 91.41 <.0001

THETA 1 -0.4748 0.1121 -4.23 <.0001

R output

Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.001271 0.001774 0.71655 0.473654 omega -0.169047 0.110169 -1.53443 0.124924 alpha1 -0.095670 0.035615 -2.68620 0.007227 beta1 0.970507 0.018820 51.56682 0.000000 gamma1 0.194542 0.059579 3.26526 0.001094

In R model is written as

rt=0.0012 +at

at=σtet with ln σt 2 =-0.169-0.095( et-1 +0.194(|et-1|-E(et-1))+0.970 ln σ

2 t-1

where error term has t-distribution with 6 degrees of freedom.

Note that the leverage coefficients of et-1 in SAS and R are equivalent.

6. Fit and EGARCH(1,1) model for the Nordstrom log returns using a t- distribution for the error terms. Perform model checking and write down the fitted model.

The EGARCH model t-distributed errors is a good model for the data. No ARCH effects

are shown in residuals. The leverage parameter “theta” is significant and negative

showing that volatility of Nordstrom stock returns is affected more heavily by negative

shocks.

The fitted model can be written as follows:

rt=0.0032 +at

at=σtet with ln σt 2 =-0.191+0.158 g(et-1) +0.974 ln σ

2 t-1

g(et-1)= -0.55et-1+[|et-1|-E(et-1)]

et has t-distribution with 6 degrees of freedom.

The MODEL Procedure

Nonlinear Liklhood Summary of Residual Errors

DF DF Adj

Equation Model Error SSE MSE Root MSE R-Square R-Sq

7 686 2.5482 0.00371 0.0609 -0.0001 -0.0088

nresid.y 686 1028.4 1.4991 1.2244

Nonlinear Liklhood Parameter Estimates

Approx Approx

Parameter Estimate Std Err t Value Pr > |t|

intercept 0.003273 0.00160 2.05 0.0410

earch0 -0.19123 0.0932 -2.05 0.0405

earch1 0.158378 0.0490 3.23 0.0013

egarch1 0.974881 0.0139 70.01 <.0001

theta -0.55013 0.1969 -2.79 0.0053

df 5.993367 1.3083 4.58 <.0001 R output

Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.003374 0.001552 2.1743 0.029681 omega -0.137932 0.098212 -1.4044 0.160191 alpha1 -0.104435 0.034086 -3.0638 0.002185 beta1 0.977603 0.016446 59.4433 0.000000 gamma1 0.180592 0.075474 2.3928 0.016722 shape 5.783774 1.169335 4.9462 0.000001

In R model is written as

rt=0.0033 +at

at=σtet with ln σt 2 =-0.138-0.104( et-1 +0.180(|et-1|-E(et-1))+0.978 ln σ

2 t-1

where error term has t-distribution with 6 degrees of freedom.

Note that the leverage coefficients of et-1 in SAS and R are equivalent.

Name of Variable = resid2

Mean of Working Series 1.483942

Standard Deviation 2.819517

Number of Observations 693

Autocorrelation Check for White Noise

To Chi- Pr >

Lag Square DF ChiSq --------------------Autocorrelations--------------------

6 5.11 6 0.5299 0.026 0.020 -0.044 -0.061 0.024 -0.004

12 13.61 12 0.3261 0.025 0.088 0.014 -0.041 0.017 -0.039

18 19.28 18 0.3748 0.043 -0.032 -0.038 -0.028 -0.049 0.020

24 23.55 24 0.4873 0.004 0.028 -0.025 -0.051 -0.039 -0.021

7. Find a GJK-TGARCH(1,1) model for the Nordstrom log returns using a t-distribution for the innovations. Perform model checking and write down the fitted model.

Similar to the AR(0)- EGARCH(1,1) model, the AR(0)-GJR(1,1) model is also a good

model for the data. No ARCH effects are shown in residuals. The leverage parameter is

significant and positive showing that volatility of Nordstrom stock returns is affected

more heavily by negative shocks.

The fitted model can be written as follows:

rt=0.0036 +at

at=σtet with σt 2 =(0.034 +0.13Nt-1)a

2 t-1 +0.800 σ

2 t-1

with Nt-1 = 1 if at-1<0, and Nt-1 = 0 if at-1>0

et has t-distribution with 5 degrees of freedom.

The MODEL Procedure

Nonlinear Liklhood Parameter Estimates

Approx Approx

Parameter Estimate Std Err t Value Pr > |t|

intercept 0.003589 0.00157 2.28 0.0226

df 5.028263 0.9762 5.15 <.0001

arch0 0.000095 0.000044 2.18 0.0297

arch1 0.034154 0.0259 1.32 0.1869

garch1 0.800444 0.0584 13.71 <.0001

gamma 0.129939 0.0496 2.62 0.0090

R output

Robust Standard Errors: Estimate Std. Error t value Pr(>|t|) mu 0.003880 0.001519 2.5535 0.010665 omega 0.000083 0.000053 1.5798 0.114152 alpha1 0.039211 0.032406 1.2100 0.226283 beta1 0.858372 0.058370 14.7058 0.000000 gamma1 0.153341 0.074560 2.0566 0.039723 shape 5.767186 1.177356 4.8984 0.000001

The fitted model can be written as follows:

rt=0.0039 +at

at=σtet with σt 2 =(0.039 +0.15Nt-1)a

2 t-1 +0.858 σ

2 t-1

with Nt-1 = 1 if at-1<0, and Nt-1 = 0 if at-1>0

et has t-distribution with 6 degrees of freedom.

8. Is the leverage effect significant at the 5% level?

The leverage parameters in both the EGARCH and the GJR models are significant, and

have values that indicate that volatility react more heavily to negative shocks.

9. What model provides the best fit for the data? Explain. Since the leverage parameters in the AR(0)-EGARCH(1,1) and AR(0) – GJR(1,1) models

are significant, and residuals are white noise with no ARCH effect, we can conclude that

both models provide a good explanation of the volatility behavior for Nordstrom stock

returns. Either models can be chosen, there is no clear winner. (Note that in SAS, PROC

MODEL does not compute BIC criterion – backtesting can be used to compare models.)

In R the egarch(1,1) and the GJR-GARCH(1,1) models have very similar BIC values-

garch11.t egarch11 gjrgarch11 Akaike -3.172122 -3.186143 -3.183976 Bayes -3.139358 -3.146827 -3.144660

10. Use the selected model to compute up to 5 step-ahead forecasts of the simple returns and its volatility.

The following predictions are based on the AR(0)-EGARCH(1,1) model. sigma is the

predicted conditional standard deviation or volatility, and series is the predicted return

(note that predicted returns are the sample mean returns since returns are white noise)

R output for forecasts sigma series 2013-10-29 0.02732 0.003374 2013-10-30 0.02764 0.003374 2013-10-31 0.02796 0.003374 2013-11-01 0.02827 0.003374 2013-11-04 0.02858 0.003374

SAS code

*import data from file;

proc import datafile='nordstrom_w_00_13.csv' out=myd replace;

resid

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t-distribution QQplot of residuals for GJRGARCH model