1. 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
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2. 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.
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3. 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
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4. 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
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5. 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
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6. 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.
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7. 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
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8. Equation estimation results:
For the General Electric equation
For the Westinghouse equation
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9. 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
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10. 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
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11. To obtain standard error for the Westinghouse coefficient estimates, go to View / Coefficient Diagnostics / Wald – Coefficient Restrictions
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12. 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
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13. 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"
@last
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14. 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
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15. 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
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16. Estimation result:
Name the output as “table_15_11”
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17. 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
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18. Previous estimator using indicator (dummy) variables
Estimation result:
Previous estimator using indicator (dummy) variables
Estimator using Pool object
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19. 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
equation we.ls inv_we c v_we k_we
series ehat_we=resid
scalar
series ee=ehat_ge*ehat_we
scalar
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
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20. Estimation result:
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21. 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
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22.
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
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23.
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24. 15.8.5 Seemingly unrelated regressions1 2
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25. Reject the null hypothesis of equal coefficients
Reject the null hypothesis of equal coefficients
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26. Thank You !