15.5 The Hausman test For the random effects estimator to be unbiased in large samples, the effects must be uncorrelated with the explanatory variables. The Hausman test is a test of the significance of the difference between the fixed effects estimates and the random effects estimates. Correlation between the random effects and the explanatory variables will cause these estimates to diverge. So if the difference is not significant, there is no evidence of the offending correlation. Go to View / Fixed / Random Effects Testing / Correlated Random Effects - Hausman Test. 1 2 3 24/48

corresponding p-value Suggest the null hypothesis of no correlation between the explanatory variables and the random effects should be rejected p-values for separate tests on the difference between each pair of coefficients   At a 5% significance level, the null hypothesis is rejected for TENURE2, SOUTH and UNION, but not for EXPER, EXPER1 and TENURE. The variables BLACK, EDUC, and the constant are not included because we do not have coefficient estimates for them in the fixed effects model. 25/48

15.7 Sets of regression equations Open involves T = 20 time series observations on just N = 2 cross sectional units: the firms General Electric (GE) and Westinghouse (WE) INV - investment V - market value of stock K - capital stock 26/48

15.7.1 Equal coefficients, equal error variances When the two equations are assumed to have the same coefficients and same variances, we can simply regress INV on V and K without distinguishing between what values belong to Westinghouses and what values belong to General Electric. Command: ls inv c v k 27/48

15.7.2 Different coefficients, equal error variances To relax the assumption of identical coefficients for the two firms, we introduce an indicator (dummy) variables DUM that is equal to 1 for the Westinghouses observations and 0 for the General Electric observations. Command: series dum = (firm=2). The do equation estimation. Command: ls inv c dum v dum*v k dum*k 28/48

Testing the equality of the coefficients, go to View / Coefficient Diagnostics / Wald – Coefficient Restrictions Conclusion: reject the null hypothesis, the coefficients for the two firms are not equal. 29/48

15.7.3 Different coefficients, different error variances If we assume the equations have both different coefficients and different error variances, then estimation is equivalent to running two separate regressions, one for each firm. For the General Electric equation For the Westinghouse equation General Electric Westinghouse 30/48

Equation estimation results: For the General Electric equation For the Westinghouse equation 31/48

Estimation after restructuring the workfile To restructure the workfile, go to Proc / Structure / Resize Current Page 1 2 3 4 5 6 Panel data structure 32/48

The coefficients for General Electric The alternative way of estimating the two equations with different coefficients and different error variances, go to Quick / Estimate Equation 1 2 3 The coefficients of C, V, and K for Westinghouse are obtained by adding the General Electric coefficients to the corresponding dummy variable coefficients.   The coefficients for General Electric 33/48

To obtain standard error for the Westinghouse coefficient estimates, go to View / Coefficient Diagnostics / Wald – Coefficient Restrictions   34/48

15.7.4 Seemingly unrelated regressions The seemingly unrelated regression is the equation estimation with relaxing the assumption that the General Electric and Westinghouse errors for the same year are uncorrelated. Go to Quick / Estimate Equation. 1 2 3 35/48

15.8 Using the pool object To get ride of using indicator (dummy) variables to model coefficients, we use Pool object to obtain the coefficients estimates and their standard error directly. First, to create a series FIRMID with the labels GE and WE, use command: smpl if firm =1 alpha firmid = "ge" smpl if firm=2 alpha firmid = "we" smpl @first @last 36/48

15.8.1 Unstacking the data To distinguish the data for each firm, go to Proc / Reshape Current Page / Unstack in New Page. 6 Name the new page 1 2 Retain the current name for INV, V and K, but adding a suffix _GE or _WE to indicating the firm 4 The series to unstack 5 3 37/48

15.8.2 Equal coefficients, equal error variances Open Pool FIRMID, go to Estimate 2 1 Open series names with identifier replaced by ? 3 4 We assume the General Electric coefficients are identical to the Westinghouse coefficients If we assume the their coefficients are different We assume equal variances 38/48

Estimation result: Name the output as “table_15_11” 39/48

15.8.3 Different coefficients, equal error variances Relax the assumption that the coefficients are equal, use Pool FIRMID, go to Estimate We still assume equal variances This time, we assume their coefficients are different 1 2 40/48

Previous estimator using indicator (dummy) variables Estimation result: Previous estimator using indicator (dummy) variables Estimator using Pool object 41/48

15.8.4 Different coefficients, different error variances Way 1: use the usual single equation commands: equation ge.ls inv_ge c v_ge k_ge series ehat_ge=resid scalar sig_ge=@se equation we.ls inv_we c v_we k_we series ehat_we=resid scalar sig_we=@se series ee=ehat_ge*ehat_we scalar cov=@sum(ee)/17 scalar r=cov/(sig_ge*sig_we) Equation estimation for GE Save the residuals Save estimates of the error standard deviations Equation estimation for WE Estimate the covariance between the two errors Estimate of the contemporaneous correlation between the errors 42/48

Estimation result:   43/48

15.8.4 Different coefficients, different error variances Way 2: use the Pool object, use Pool FIRMID, go to Estimate We assume variances are not equal Again, we assume their coefficients are different 1 2 44/48

www.themegallery.com To find the covariance and correlation matrices for the residuals, go to View / Residual Diagnostic / Covariance Matrix and View / Residual Diagnostic / Correlation Matrix Same Different   45/48

www.themegallery.com   46/48

15.8.5 Seemingly unrelated regressions   1 2 47/48

Reject the null hypothesis of equal coefficients   www.themegallery.com Reject the null hypothesis of equal coefficients 48/48

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