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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 variancecovariance 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
 öcal
 Econometrics

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