1. EC220 - Introduction to econometrics (chapter 6)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 6)
Slideshow: reparameterization of a model and t test of a linear restriction
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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource]
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2. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Linear restrictions can also be tested using a t test. This involves the reparameterization of a regression model and we will start with this. 1

3. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Suppose that you have a regression model shown and that the regression model assumptions are valid. 2

4. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
We fit the model in the usual way. This enables us to obtain estimates of the parameters and their standard errors. 3

5. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Suppose, however, that we are interested in a linear combination, q, of the parameters, where the lj are weights. 4

6. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Obviously, the can obtain a point estimate of q as the corresponding linear combination of the estimates of the individual parameters. If the regression model assumptions are valid, it can easily be shown that it is unbiased and that it is the most efficient estimator of q. 5

7. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
However you do not have information on its standard error and hence you are not able to construct confidence intervals for q or to perform t tests. There are three ways that you might use to obtain such information. 6

8. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
(1) Some regression applications have a special command that produces it. For example, Stata has the lincom command. 7

9. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
(2) Given the appropriate command, most regression applications will produce the variance-covariance matrix for the estimates of the parameters. This is the complete list of the estimates of their variances and covariances, for convenience arranged in matrix form. 8

10. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
The standard errors in the ordinary regression output are the square roots of the variances. The estimate of the variance of the estimate of q is given by the expression shown, where s subscript bpbj is the estimate of the covariance between bp and bj. 9

11. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
This method is cumbersome and avoided when possible. 10

12. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
(3) The third method is to reparameterize the model, manipulating it so that q and its standard error are estimated directly as part of the regression output. 11

13. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
To do this, we rewrite the expression for q so that one of the b parameters is expressed in terms of q and the other b parameters. This will be illustrated with two simple examples, the general case being left as an additional exercise (see the study guide on the website). 12

14. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Suppose the regression model is as shown and suppose we are interested in the sum of b2 and b3. 13

15. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
We rewrite the relationship as shown, expressing one of the b parameters in terms of q and the other b parameter. 14

16. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Substituting in the original model, we reparameterize it as shown. 15

17. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Thus if we define a new variable Z = X2 – X3, and regress Y on Z and X3, the coefficient of Z will be an estimate of b2 and that of X3 will be an estimate of q. 16

18. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
The estimate of q will be exactly the same as that obtained by summing the estimates of b2 and b3 in the original model. The difference is that we obtain its standard error directly from the regression results. The estimate of b2 and its standard error will be the same as those obtained with the original model. 17

19. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Suppose instead that we were interested in the difference between b2 and b3. 18

20. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
We rewrite this as shown and substitute in the original model. Thus if we define a new variable Z = X2 + X3 and regress Y on Z and X3, the coefficient of Z will be an estimate of b2 and that of X3 will be an estimate of q. 19

21. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
The estimate of q will be exactly the same as that obtained by taking the difference of the estimates of b2 and b3 in the original model and we obtain its standard error directly from the regression results. The estimate of b2 and its standard error will be the same as those obtained with the original model. 20

22. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
An obvious application of reparameterization is its use in the testing of linear restrictions. Suppose that your hypothetical restriction is as shown, where a is a scalar. 21

23. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Define q as shown and reparameterize. q will become the coefficient of one of the variables in the model, and a t test of H0: q = 0 is effectively a t test of the restriction. 22

24. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
As an illustration. we will use the example discussed in the previous sequence. The model relates years of schooling, S, to the cognitive ability score ASVABC and years of schooling of the mother and the father, SM and SF. 23

25. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
It was hypothesized that mother’s education and father’s education are equally important for educational attainment, implying the restriction b4 = b3. 24

26. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
The restriction may be rewritten as b4 – b3 = 0. Define q as shown. 25

27. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
Hence express one of the b parameters in terms of q and the other b parameter, and substitute in the original model. 26

28. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
SP is defined as the sum of SM and SF. A t test on the coefficient of SF is a test of the restriction. 27

29. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
. reg S ASVABC SP SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SP |
SF |
_cons |
Here is the corresponding regression. We see that the coefficient of SF is not significantly different from zero, indicating that the restriction is not rejected. 28

30. REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION
. reg S ASVABC SP SF
Source | SS df MS Number of obs =
F( 3, 536) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
S | Coef. Std. Err t P>|t| [95% Conf. Interval]
ASVABC |
SP |
SF |
_cons |
It can be shown mathematically that the F test of a restriction and the corresponding t test are equivalent. The F statistic is the square of the t statistic and the critical value of F is the square of the critical value of t. 29

31. Copyright Christopher Dougherty 2011.
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The content of this slideshow comes from Section 6.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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20 Elements of Econometrics