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Unformatted text preview: Chapter 6 Specification of regression variables: a preliminary skirmish Overview This chapter treats a variety of topics relating to the specification of the variables in a regression model. First there are the consequences for the regression coefficients, their standard errors, and R 2 of failing to include a relevant variable, and of including an irrelevant one. This leads to a discussion of the use of proxy variables to alleviate a problem of omitted variable bias. Next come F and t tests of the validity of a restriction, the use of which was advocated in Chapter 3 as a means of improving efficiency and perhaps mitigating a problem of multicollinearity. The chapter concludes by outlining the potential benefit to be derived from examining observations with large residuals after fitting a regression model. Further material This section contains some new material on the following topics: 1. The reparameterization of a regression model 2. The application of reparameterization to t tests of linear restrictions 3. The testing of multiple restrictions 4. Tests of zero restrictions The reparameterization of a regression model Suppose that you have fitted the regression model (1) u X Y k j j j + + = ∑ = 2 1 β β and that the regression model assumptions are valid. Let the fitted model be (2) ∑ = + = k j j j X b b Y 2 1 ˆ as usual. Suppose that, as well as the individual parameter estimates, you are interested in some linear combination: (3) ∑ = = k j j j 1 β λ θ To obtain a point estimate of θ , it is natural to construct the statistic , and indeed, given that the regression model assumptions are valid, it can easily be shown that this is unbiased and the most efficient estimator of θ . However you do not have information on its standard error and hence you are not able to construct confidence intervals or to perform t tests. There are three ways that you might use to obtain such information: ∑ = = k j j j b 1 ˆ λ θ (1) Some regression applications have a special command that produces it. For example, Stata has the lincom command. (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 October 2007 2 covariances, for convenience arranged in matrix form. The standard errors in the ordinary regression output are the square roots of the variances. The estimate of the variance of the estimate of θ is given by (4) ∑ ∑ ≠ = + = j p b b j p k j b j j p j s s s λ λ λ θ 2 1 2 2 2 ˆ where is the estimate of the covariance between b j p b b s p and b j . (3) The third method is to reparameterize the model, manipulating it so that θ and its standard error are estimated directly as part of the regression output. To do this, we rewrite (3) so that one of the b parameters is expressed in terms of θ and the other b parameters. This will be illustrated with two simple examples, the general case being left as an additional exercise. examples, the general case being left as an additional exercise....
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.
- Spring '10