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Using EViews on Shiller16 Data

Ron Smith: [email protected]

Autumn 2018

Department of Economics, Mathematics and Statistics, Birkbeck, University of London

1. Data

This exercise is designed to teach you to use a variety of different estimators in EViews and interpret the output. There are notes for other programs, Stata and gretl, but more detail is given in

these notes. In general you should get the same answers from different programs. You may get different results on some tests, e.g. for heteroskedasticity, on different versions of EViews, and different programs, because the alternative hypothesis, the assumed form of heteroskedasticity is different. In non-linear routines you may also get different answers.because the optimisation routines are different or it converges to a different maximum. The data is in an Excel file: Shiller16.xls, on Moodle. The data is updated

from Robert J Shiller ‘Market Volatility’, MIT Press 1989 and downloaded from his webpages. This is a subset of the full data he provides. We will use Shiller16 to re-examine the hypotheses in a famous paper J Lintner ‘Distribution of Income of Corporations among Dividends, Retained Earnings and Taxes’, American Eco- nomic Review May 1956. The file contains annual US data from 1871 to 2016 on NSP: S&P composite stock price index January value. ND: nominal dividends for the year NE : nominal earnings (profits) for the year R : average short interest rate for the year RL: average long interest rate for the year

CPI Consumers Price Index RR: real interest rate RC: real consumption in 2005 dollars Some data are not provided for the whole period.

2. Initial Data Handling

2.1. Loading the data

Click on EViews icon. Click on File, Create a new EViews workfile. In the dialog box specify annual data (the default), and put 1871 2016 in the

boxes for beginning and end. You will then get a box with two variables, C for the constant and Resid for

the residuals. Notice that we renamed the variable C in the original Shiller File to RC because EViews uses C for the constant, similarly D is a reserved name you cannot use it for a variable. Choose File, Import, Import from file, then you will get a browser and find

Shiller16, click on that. A dialog box will appear, note what it is doing and click next, next, finish.

Notice that it will start reading data at B2. This is correct, column A has years, which it already knows and row 1 has names, which it will read as names You will see the variables in the workfile box. You can name and save this file and any changes to it. Notice that you have two menu bars one at the top and one in the workfile

box. You will get a third menu bar when you do operations like graph or regress. There is a white box where you can type commands.

2.2. Transforming and graphing the data

Using the top menu click Quick (six from the left) Graph and enter ND and NE in the box. OK. See the various types of graph you can do, but accept the default line graph. Notice that the trends and the change in scale dominate the data, earnings are usually above dividends but it is diffi cult to see. You can see the effect of the great recession on earnings in 2008 Close graph. It will ask Delete untitled graph? Choose Yes. There is a box to name it if you wanted to save it.

In looking at data, it is often useful to form ratios. Construct the pay out ratio, the proportion of earnings paid out in dividends PO = ND/NE which removes the common trend in the two variables and is more stationary. Generate transformations of the data to create new series Type Quick, Generate Series and type into box PO=ND/NE Press OK. You will see, PO has been added to the list of variables in the workfile. Click

on it, you will see the series. Click on View, top left of the workfile box. Notice the options. Choose graph, accept the defaults to get a line graph. Notice the years when the ratio was above one, firms were paying out more in dividends than they earned. Click on the arrow on the workfile menu next to where it says default, this gives you various transformations you can do. There is a slider at the bottom which allows you to change the sample. Close the graph. Get summary statistics on the ratio. Click on the series name, PO, choose

view, descriptive statistics, histogram and stats. This will give minimum and maximum values (check these are sensible), mean, median, skewness (which is zero for a normal distribution) and kurtosis (which is 3 for a normal distribution) and the JB test for the null hypothesis that the distribution is normal. If the p value is less than 0.05, you reject the hypothesis of normality. It is clearly not normal. Always graph the data and transformations of it and look at the descriptive

statistics before starting any empirical work. Make sure series are in comparable units before putting them on the same graph. We are going to work with the logarithms of the data. Type Quick, Generate

Series and type into box LD=LOG(ND) And OK. You will get a new series in the box LD. Similarly generate LE=LOG(NE) LSP=LOG(NSP)

3. Regression: a static model of dividends and earnings

Run a static regression Click, Quick, Estimate an Equation, Type in LD C LE You should always include C for the constant.

Always look at the dialogue window and note the options. Notice the default estimationmethod isLS- LeastSquares (NLSandARMA). NLS isnon-linear least squares, arma, autoregressive moving average. We use these below. If you click the arrow on the right of LS, you will see that there are other methods you could choose: including Two stage Least Squares and GARCH, which we will use below. There is an option tag at the top, which you can use to get Heteroskedasticity and Autocorrelation Consistent (HAC) Standard Errors. You could also have entered Log(ND) C Log(NE) directly in the equation box rather than generating them. Click OK and you will get the following output

Dependent Variable: LD Method: Least Squares Date: 08/04/16 Tim e: 14:25 Sample (adjusted): 1871 2014 Included observations: 144 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C -0.428854 0.021586 -19.86709 0.0000 LE 0.875617 0.010850 80.70330 0.0000

R-squared 0.978663 Mean dependent var 0.352726 Adjusted R-squared 0.978512 S.D. dependent var 1.579280 S.E. of regression 0.231501 Akaike info criterion -0.074676 Sum squared resid 7.610149 Schwarz criterion -0.033428 Log likelihood 7.376650 Hannan-Quinn criter. -0.057915 F-statistic 6513.023 Durbin-Watson stat 1.148272 Prob(F-statistic) 0.000000

To copy this equation so that you can paste it into a word processing file, highlight what you want to copy; use the edit button on the top menu and click copy. You can save the equation in EViews by using the name box on the equation toolbar menu. Both the constant and the coeffi cient of LE are very significant, t ratios much

bigger than 2 in absolute values and p values (Prob) of zero. The P value gives you what you can loosely think of as the probability of getting that value of the test statistic if the null hypothesis (in this case that the coeffi cient is zero) were true. It is conventional to reject the null hypothesis if the p value is less than 0.05. However the Durbin Watson Statistic (which should be close to 2) of 1.15 indicates severe serial correlation that suggests dynamic misspecification. The serial correlation will also bias the standard errors upwards. To get standard

errors that are robust to serial correlation, click estimate on the equation box, options tab beside the specification tab at the top, change covariance method from ordinary to HAC (Newey-West) OK you will get the equation again with the same coeffi cients, but different standard errors, t stats and p-values. The standard error on LE increases from 0.010850 to 0.012746. From the equation box menu choose View; Actual Fitted Residual; Actual

Fitted Residual Graph and you will get a graph of the residuals in blue and the actual in red and the fitted in green. The graph shows that the residuals are not random, there are quite long runs where the actual is above or below the fitted and there are some big spikes, larger positive residuals than one would expect, where the actual is much higher than the fitted. These were cases where earnings dropped sharply, but dividends did not respond, because dividends were smoothed relative to earnings. Alwasys plot the residuals.

4. Dynamic model of earnings and dividends

Given that the serial correlation in the original regression suggested dynamic misspecification, we add lagged values, denoted by (-1) in EViews. Click, Quick, estimate equation and type in LD C LE LE(-1) LD(-1) @TREND @trend, is a variable that goes 1,2,3, etc.

Dependent Variable: LD Method: Least Squares Date: 08/04/16 Tim e: 14:29 Sample (adjusted): 1872 2014 Included observations: 143 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C -0.194563 0.042665 -4.560248 0.0000 LE 0.195017 0.025996 7.501697 0.0000

LE(-1) 0.119784 0.032892 3.641803 0.0004 LD(-1) 0.630665 0.034941 18.04949 0.0000

@TREND 0.000987 0.000643 1.534663 0.1272

R-squared 0.997148 Mean dependent var 0.364613 Adjusted R-squared 0.997065 S.D. dependent var 1.578354 S.E. of regression 0.085509 Akaike info criterion -2.046045 Sum squared resid 1.009033 Schwarz criterion -1.942449 Log likelihood 151.2922 Hannan-Quinn criter. -2.003949 F-statistic 12060.64 Durbin-Watson stat 1.795495 Prob(F-statistic) 0.000000

The standard error of regression is much smaller at 0.0855 rather than 0.2315 the Durbin Watson is much better at 1.795 and all the variables except the trend are significant. Click View on the equation box; then Actual Fitted Residual; then Actual Fitted Residual Graph. The estimates of the residuals still show some outliers, big errors.

4.1. Misspecification/Diagnostic tests

Click View on the equation box, choose Residual Diagnostics, Serial Correlation LM tests, and accept the default number of lags to include 2. You will get the LM serial correlation test. It just fails to reject the hypothesis of no serial correlation up to second order at the 5% level, p=0.0571 and the second lag of the resiudal is just significant. On diagnostic tests, the null hypothesis is that they are well specified, p values below 0.05 indicate that there is a problem. Click View, Residual tests, histogram- normality test. You will get the his-

togram and in bottom right the JB test of 100.48 and a p value of 0.0000. There is clearly a failure of normality, caused by the outliers. Click View, residual, Heteroskedasticity tests. Choose the first Breush-Pagan-

Godfrey, thepvalue is 0.0037 so there is an indicationof heteroskedasticity, in that the variances of the residuals seem to be related to the values of the regressors, so at the 5% level we would reject the null hypothesis of constant variance. Notice that thare are a long list of heteroskedasticity tests which differ in what they make the squared residuals a function of, e.g. ARCH. They all have the same null, constant variance, but different alternatives. Click View, stability diagnostics, Ramsey Reset tests, accept the default num-

ber of fitted terms at 1, This tests for the null of linearity by adding powers of the fitted values. Look at the regression below. There are other stability diagnostics. If you wanted to test for a change in the

parameters at a particular date, you would use the Chow breakpoint test, specify- ing the date at which you thought the relationship changed. The Breakpoint tests for equality of the regression coeffi cients before and after the break, assuming the variances in the two periods are constant. The Chow Forecast tests whether the estimates for the first period forecast the second period. If you do not know the breakpoint choose recursive estimates and look at the

CUSUM and CUSUM of squares graphs. The test statistics should not cross the confidence intervals if the model is stable. Alternatively the Quandt Andrews test will determine the most likely breakpoint. It identifies a significant break in 1972.

Diagnostic tests for the same null hypothesis (that the model is well speci- fied e.g. homoskedasticity or structural stability) and can give conflicting results because they are testing against different alternative hypotheses. View coeffi cient diagnostics, redundant variables tests allows us to test for

deletingvariables, e.g. the trendwhich is insignificanthereandommittedvariables allows us to test for adding variables. Click View, coeffi cient tests, Wald and type in to the box: (C(2)+C(3))/(1-

C(4))-1=0. This tests that the long-run coeffi cient on log earnings equals unity. Click OK. The hypothesis is clearly rejected with Chi-squared p value of 0.0003. Wald tests are not invariant to how you write non-linear restrictions. We could have written the same restriction: C(2)+C(3)+C(4)-1=0. This gives a Chi- squared p value of 0.0017, so we still reject. But there are cases where writing the restriction one way leads to rejection and another way to acceptance.

4.2. ARDL and ECM

We re-estimate it without a trend, first in the ARDL form above and then in the statistically identical ECM reparameterisation. Note that D(..) takes first difference of the variable. Although the trend was not significant even at the 10% level the AIC would choose the model with trend over the model without trend −2.046 < −2.043, whereas the Schwarz criterior chooses the model without trend. Although the ECM and ARDL are statistically identical with the same standard error of regression, the R2 of the ARDL at 0.997 is much larger than the R2 of the ECM at 0.526. This is because the dependent variable in the ECM is the change in log dividends (growth in dividends) not the level of log dividends. The model explains a smaller proportion of the change than of the level. This does not mean that the ARDL is better. It means that R2 can be misleading.

Dependent Variable: LD Method: Least Squares Date: 08/04/16 Tim e: 15:34 Sample (adjusted): 1872 2014 Included observations: 143 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C -0.133988 0.016276 -8.232040 0.0000 LE 0.203134 0.025576 7.942294 0.0000

LE(-1) 0.115419 0.032928 3.505239 0.0006 LD(-1) 0.651616 0.032320 20.16136 0.0000

R-squared 0.997099 Mean dependent var 0.364613 Adjusted R-squared 0.997036 S.D. dependent var 1.578354 S.E. of regression 0.085925 Akaike info criterion -2.043109 Sum squared resid 1.026254 Schwarz criterion -1.960232 Log likelihood 150.0823 Hannan-Quinn criter. -2.009431 F-statistic 15924.80 Durbin-Watson stat 1.805160 Prob(F-statistic) 0.000000

Dependent Variable: D(LD) Method: Least Squares Date: 08/04/16 Tim e: 15:35 Sample (adjusted): 1872 2014 Included observations: 143 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C -0.133988 0.016276 -8.232040 0.0000 D(LE) 0.203134 0.025576 7.942294 0.0000 LE(-1) 0.318554 0.028591 11.14183 0.0000 LD(-1) -0.348384 0.032320 -10.77922 0.0000

R-squared 0.526412 Mean dependent var 0.035118 Adjusted R-squared 0.516191 S.D. dependent var 0.123533 S.E. of regression 0.085925 Akaike info criterion -2.043109 Sum squared resid 1.026254 Schwarz criterion -1.960232 Log likelihood 150.0823 Hannan-Quinn criter. -2.009431 F-statistic 51.50140 Durbin-Watson stat 1.805160 Prob(F-statistic) 0.000000

5. Theoretical background.

Lintner suggested that there was a target or long-run dividend pay-out ratio, say, Θ, such that D∗t = ΘEt. We will take logs of this relationship, using lower case letters for logs, e.g. dt = log(Dt), etc. Notice natural logs are almost universally

used. Taking logs we get d∗t = log(Θ) + et. This can be written in an unrestricted form as d∗t = θ0 + θ1et, where his theory suggests that θ1 = 1 and θ0 = log(Θ). To this he added a ‘Partial Adjustment Model’(PAM) and a random error

∆dt = λ(d ∗ t −dt−1) + ut

dt = λθ0 + λθ1et + (1 −λ)dt−1 + ut. dt = a0 + b0et + a1dt−1 + ut

The PAM can be justified if, for instance, firms smooth dividends, not adjusting them completely to short term variations in earnings. Our estimates above suggest that lagged earnings are significant and this can

be allowed for using the more general error correction model, ECM, with a long run equilibrium determining d∗t and an adjustment process towards it.

d∗t = θ0 + θ1et,

∆dt = λ1∆d ∗ t + λ2(d

∗ t−1 −dt−1) + ut

∆dt = λ1θ1∆et + λ2θ0 + λ2θ1et−1 −λ2dt−1 + ut ∆dt = a0 + b0∆et + b1et−1 + a1dt−1 + ut

which is statistically identical to the ARDL

dt = α0 + β0et + β1et−1 + α1dt−1 + ut;

a0 = α0; .b0 = β0; a1 = a1 − 1; b1 = β0 + β1;

θ1 = − b1 a1

= β0 + β1 1 −α1

= 0.914374(0.01228)

where the standard error of the long-run coeffi cient is got using the Wald com- mand.

6. Estimate the ECM by Non-linear Least Squares

Close the equation, you could name it and save it, and click, quick, estimate an equation again, type in D(LD)=C(1)*C(4)*D(LE)+C(2)*(C(3)+C(4)*LE(-1)-LD(-1)) The D(.. .) first differences the data on LD. This estimates the ECM giving

estimatesof the long-runparametersandspeedofadjustmentdirectly.: C(1) = λ1, C(2) = λ2, C(3) = θ0,C(4) = θ1. We get the same estimate of θ1 and its standard error as we got above.

Dependent Variable: D(LD) Method: Least Squares (Gauss-Newton / Marquardt steps) Date: 11/09/16 Tim e: 10:28 Sample (adjusted): 1872 2014 Included observations: 143 after adjustments Convergence achieved after 5 iterations Coefficient covariance computed using outer product of gradients D(LD)=C(1)*C(4)*D(LE)+C(2)*(C(3)+C(4)*LE(-1)-LD(-1))

Coefficient Std. Error t-Statistic Prob.

C(1) 0.222157 0.028335 7.840289 0.0000 C(4) 0.914374 0.012283 74.43963 0.0000 C(2) 0.348384 0.032320 10.77922 0.0000 C(3) -0.384599 0.023683 -16.23973 0.0000

Save this by giving it a name and then close it. This is a non-linear procedure, it took 5 iterations to get to the minimum sum

of squared residuals. Most programs ask you to provide starting values, EViews does not, and this can lead to problems. To provide some good starting values, if you did not get the results above type param c(1) 0.3 c(2) 0.3 c(3) 1 c(4) 1 in the command window at the top under

the toolbar. To provide some bad starting values type param c(1) 0.0 c(2) 0.0 c(3) 0.0 c(4) 0 in the command window at the top under the toolbar. This sets the starting

values for all parameters at 0.0. Now open the non-linear regression again and run it again with these starting values and see what happens. It should not converge and give silly values for the parameters. When doing non-linear estimation, try to start with sensible starting values,

using the economic interpretation, scale or preliminary OLS regressions to give you sensible values. Also experiment with different starting values to make sure

that you have not got a local minimum.

6.1. Estimate the ECM allowing for non-normality and ARCH.

Above we estimated the model on the assumption that ut ∼ IN(0,σ2). But there was evidence that the errors were heteroskedastic and non-normal and this was mainly caused by excess kurtosis. Now we are going to assume that ut ∼ It(0,ht,ν), the errors are independent with a student t distribution, expected value zero, a time varying variance E(u2t ) = ht, and degrees of freedom ν. The degrees of freedom determine how thick the tails of the distribution are. If ν is small, the tails are much fatter than the normal distribution, if v is around 30, it is very similar to the normal. The form of time varying variance we will use is GARCH(1, 1)

ht = c0 + c1û 2 t−1 + c2ht−1 + εt

Close or save any equations. Click quick, estimate an equation, enter D(LD) C D(LE) LE(-1) LD(-1) and then change method from LS to ARCH using the arrow on the right of the method box. You will now get a GARCH box. Change error distribution from Normal to Student’s t. Accept the other defaults, click OK.

Dependent Variable: D(LD) Method: ML ARCH - Student's t distribution (BFGS / Marquardt steps) Date: 08/04/16 Tim e: 16:22 Sample (adjusted): 1872 2014 Included observations: 143 after adjustments Convergence achieved after 60 iterations Coefficient covariance computed using outer product of gradients Presample variance: backcast (parameter = 0.7) GARCH = C(5) + C(6)*RESID(-1)^2 + C(7)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C -0.089663 0.010518 -8.524973 0.0000 D(LE) 0.161176 0.017711 9.100466 0.0000 LE(-1) 0.234743 0.017747 13.22749 0.0000 LD(-1) -0.254972 0.019814 -12.86813 0.0000

Variance Equation

C 0.000542 0.000538 1.008014 0.3134 RESID(-1)^2 0.255953 0.212146 1.206496 0.2276 GARCH(-1) 0.749364 0.123303 6.077413 0.0000

T-DIST. DOF 2.859034 0.785262 3.640866 0.0003

R-squared 0.494314 Mean dependent var 0.035118 Adjusted R-squared 0.483399 S.D. dependent var 0.123533 S.E. of regression 0.088789 Akaike info criterion -2.383006 Sum squared resid 1.095811 Schwarz criterion -2.217252 Log likelihood 178.3849 Hannan-Quinn criter. -2.315652 Durbin-Watson stat 1.860517

It is very non-normal v = 2.859. The t distibution has moments greater than its degrees of freedom. Since, ν > 2, a variance exists, but third and fourth moments do not. There is a very strong and significant GARCH effect c2 = 0.749 though the lagged squared residual is not significant. Both short-run adjustment, λ1 and long run adjustment λ2 are slightly slower than previous estimates and the long run coeffi cient 0.234743/0.254972=0.92 is similar to before.

7. ARIMA and unit roots

Estimate a random walk model for log stock prices, up till 2006; then an ARIMA model and use it to forecast. Use quick estimate an equation, set the sample to 1873-2006 and type in D(LSP) C.

Youshouldgetanestimateof thedrift (C)0.041586, MLL=43.96852, s=0.174939. Estimate an ARIMA(1,1,1) model for log stock prices. Click estimate on the equation equation box, check the sample is 1873 2006

and type in D(LSP) C AR(1) MA(1) You will get estimates with MLL=47.46484, s=0.172309. Notice that both

the AR (t=-2.56) and MA (t=4.30) terms are significant. Click forecast on the equation box. Set the forecast period to 2006 2016 look at the graph. It will save the forecast as LSPF. Close the equation and graph LSP and LSPF. This is clearly not a great forecast. AlthoughtheARandMAtermsare individually significant, theydonot reduce

the standard error of the regression very much relative to a random walk, and on a likelihood ratio test they are just jointly significant, LR=2(47.46-43.97)=6.98 compared to a χ2(2) at the 5% of 5.99 but not at the 1% of 9.21. There may be a common factor which cancels out. It would probably be better to use real stock prices rather than nominal ones.

7.1. Testing for Unit Roots

Click on LSP, then view, then unit root tests. You will get a dialogue box. Leave the test as Augmented Dickey Fuller (there are lots of other alternatives), choose level, choose trend and intercept, choose Akaike, set maximum lags at 12. Choose OK. You will get the ADF test results. The ADF statistic is -1.413932, much greater than the 5% critical value of -3.446 (given on the program output). The p-value is 0.8529, so we do not reject a unit root. Below is given the regression that was run to get the results. Notice that the lag length is 5 and that the test statistic is just the t ratio on LSP(-1) in the regression. Close the equation box and repeat the process (choose view, unit root test)

set if for first difference rather than level, choose just intercept. The lag length chosen is 3. The ADF is -6.839723 which is much smaller than the 5% critical value of -2.88. Note that the critical values are different depending whether or not you have a trend. We cannot reject a unit root for LSP but we can for the first difference of

LSP, therefore LSP is clearly I(1). In practice, unit root tests are not always as clear-cut as this and can be sensitive to lag length, treatment of the deterministic elements and to choice of test.

8. VAR, cointegration and VECM

Use Quick, estimate VAR and you will get a dialogue box. Use the default, unrestricted VAR. Enter LE LD as endogenous variables. In the list of exogenous variables add @trend to C. Accept the defaults for everything else. This will give you a second order unrestricted VAR with intercept and trend. Notice: the trend is significant in both equations; the second lags of both variables are insignificant; LD(-1) is insignificant in the LE equation. ChooseView, lagstructure, lag lengthcriteriaandaccept thedefaultmaximum

lag of 8. You will get a table which shows that everything except the LR indicates that one lag is optimal. The optimal value has a star beside it. Choose View, lag structure, Granger Causality test. At the 5% level there is

Granger causality in both directions, though the p value for LD on LE at 0.0319 is larger than that for LE on LD which is 0.0000. Choose, View, lag structure, AR roots, graph.

It shows four inverse roots. One close to the unit circle, three within it, two matching complex roots. From the AR Roots table option the largest root is 0.92, which may not be significantly different from one. In the complex pair i is the square root of minus one. This suggests that there may be one stochastic trend (root on the unit circle) and one cointegrating vector. Choose estimate from the equation box and replace 1 2 by 1 1 in the lag

intervals box, keep C and @trend in the exogenous variables box. Look at the

new estimates, note that lagged dividends still do not quite significantly influence earnings. Choose View, residuals, correlation matrix, and note that the contemporane-

ous correlation between the residuals is 0.538, quite high. Choose View, impulse responses, click the impulse definition tab at the top,

choose generalised impulses. These graphs of the impulse response functions, IRFs, show the effect of a shock to each variable on itself and on the other variable. LD shows a humped shaped response to a shock to LE, which remains significantly positive for 10 years. LE shows an immediate response to a shock to LD, through the contemporaneous covariance matrix, but it declines to zero. The generalised IRFs assume that the shocks have the estimated correlation in the sample. The Choleski IRFs assume a recursive causal structure for the shocks specified by the ordering of the variables. In this case it is plausible that LE influences LD, but not vice versa. So the ordering of the variables entered is correct and we can calculate the Cholesky IRFs. Notice that here there is no significant effect of LD on LE, in period zero by construction, and subsequently because lagged dividends do not influence earnings. There is a hump shaped response of LD to LE. Choose estimate, and click Vector Error Correction, rather than unrestricted

VAR; reverse the order of the variables to LD LE, remove the @trend from the exogenous variable box; then click cointegration box at the top and choose option 4 rather than the default option 3. Notice that what EViews calls a VECM(1 1) corresponds to a VAR(1 2) and the lagged change terms are insignificant, which is what we would expect given that the second lag terms were insignificant. Choose, View, cointegration tests, and click the bottom button, option 6, summarise all 5 sets of assumptions. All the tests except quadratic intercept trend say 1 cointegrating vector. The

information criteria are given below. From the stars, you can see that both Akaike choose one cointegrating vector (equation) and option 3 (no trend in the cointe- grating equation). Go back to estimate, set the lag length at 0 0, under cointegration choose

option 3, note it is set at one cointegrating vector which is what we want and OK. There is significant feedback from the cointegrating equation on both variables. The long-run coeffi cient is 0.91, very similar to what we got with the ECM. Note you have to change the sign. You would clearly reject the hypothesis that the long- run elasticity was unity, t = (0.911845−1)/0.01120 = −7.87. View, Cointegration Graph, will give you a plot of the cointegrating relation, a measure of the deviation from equilibrium: dt −θ0 −θ1et.

You can impose restrictions on the cointegrating vectors and adjustment co- effi cients using the tab at the top marked VEC restrictions. Click impose restric- tions and then type B(1,1)=1,B(1,2)=-1. This imposes a long-run unit coeffi cient of unity on earnings. Click OK. You will get the restricted estimates and a Like- lihood ratio test that indicates that the restriction is rejected, as we determined above.

9. Endogeneity

Above we estimated an ARDL(1,1) model by regressing log dividends on current and lagged log earnings and lagged log dividends. The evidence of the VAR suggests that earnings may be treated as exogenous, since there was little feedback from lagged dividends to earnings. However, if E(utεt) 6= 0, this may cause et to be correlated with ut. We now investigate this. First re-estimate the ARDL by OLS, i.e. run LD C LE LE(-1) LD(-1) using

LS over the period, it will use 1873-2014, since one observation is lost for the lag. The coeffi cient on LE is 0.202599 with a standard error of 0.025582. We are now going to estimate it by IV/2SLS using the second lags and a trend as instruments. Click estimate on the equation-box toolbar, change the method from LS to

TSLS, you will get a new dialogue box with two windows. Leave the upper equation one the same and in the lower one for instrument list enter: C LE(-1) LD(-1) LE(-2) LD(-2) @TREND. Click OK and you will get the TSLS estimates. Thecoeffi cientofLEisnowlargerwitha larger standarderror. ItgivesaJstatistic (Sargan test) with a p value of 0.16, so the over-identifying restrictions are not rejected. We might have anticipated this because the trend and lagged earnings were not significant in explaining dividends. The OLS and TSLS estimates do not look significantly different, the TSLS estimate ±2 standard errors covers the LS estimate.

Dependent Variable: LD Method: Two-Stage Least Squares Date: 09/06/16 Tim e: 15:39 Sample (adjusted): 1873 2014 Included observations: 142 after adjustments Instrument specification: C LE(-1) LD(-1) LE(-2) LD(-2) @TREND

Variable Coefficient Std. Error t-Statistic Prob.

C -0.153071 0.025172 -6.080924 0.0000 LE 0.304909 0.103416 2.948389 0.0038

LE(-1) 0.043038 0.078348 0.549316 0.5837 LD(-1) 0.618579 0.047998 12.88767 0.0000

R-squared 0.996763 Mean dependent var 0.375659 Adjusted R-squared 0.996693 S.D. dependent var 1.578384 S.E. of regression 0.090770 Sum squared resid 1.136999 F-statistic 14154.51 Durbin-Watson stat 1.829364 Prob(F-statistic) 0.000000 Second-Stage SSR 1.410362 J-statistic 3.680687 Instrument rank 6 Prob(J-statistic) 0.158763

We can test this formally with a Wu-Hausman test. Estimate by OLS: LE C LE(-1) LE(-2) LD(-1) LD(-2) @TREND. This gives us the same estimates as we got for the LE equation from the VAR 2 with intercept and trend. The VAR is the reduced form. This is the first stage of two stage least squares. You should always check this first stage, to see whether the instruments explain the endogenous variable, in this case LE(-1) and @TREND are very significant and the F statistic is much bigger than 10, the rule of thumb. Close the equation, use Quick, Generate and define ULE=RESID. This saves the residuals from the first stage (reduced form equation for LE) as ULE. Then use OLS to estimate LD C LE LE(-1) LD(-1) ULE. The coeffi cient on ULE has a t statistic of -1.083368, so we do not reject the hypothesis that we can treat LE as exogenous. We could also use two stage least squares to estimate a rational expectations

model, in which dividends are determined by expected earnings in the next period, the expectations based on information in the current period. Click estimate, choose TSLS, and type into the equation box: LD C LE(1) LD(-1) and into the instrument box: C LE LE(-1) LE(-2) LD(-1) LD(-2) @trend. You will get a coeffi cient on future earnings of 0.29. Notice we have lost one observation at the end of the period, because of the future variable on the right hand side.