Wooldridge Introductory Econometrics 5th Edition Chapter 7 Problem 10 For a chil

Question

Wooldridge Introductory Econometrics 5th Edition Chapter 7 Problem 10

For a child i living in a particular school district, let voucheri be a dummy variable equal to one if a child is selected to participate in a school voucher program, and let scorei be that child’s score on a subsequent standardized exam. Suppose that the participation variable, voucheri , is completely randomized in the sense that it is independent of both observed and unobserved factors that can affect the test score. (i) If you run a simple regression scorei on voucheri using a random sample of size n, does the OLS estimator provide an unbiased estimator of the effect of the voucher program? (ii) Suppose you can collect additional background information, such as family income, family structure (e.g., whether the child lives with both parents), and parents’ education levels. Do you need to control for these factors to obtain an unbiased estimator of the effects of the voucher program? Explain. (iii) Why should you include the family background variables in the regression? Is there a situation in which you would not include the background variables?

Explanation / Answer

(i) Yes, simple regression does produce an unbiased estimator of the effect of the voucher program. we can write

scorei = b1 + b2voucheri + ui

where, voucher is independent of u,that is, all other factors affecting score.Therefore, the key assumption for unbiasedness of simple regression is satisfied.

Assumption : The sample outcomes on x; namely, {xi ; i = 1; :::; n} ; are not all the same value.

ii) No, we do not need to control for background variables. In the equation from part (i), these are factors in the error term, u. But voucher was assigned to be independent of all factors, including the listed background variables.

(iii) We should include the background variables to reduce the sampling error of the estimated voucher effect. By pulling background variables out of the error term, we reduce the error variance – perhaps substantially. Further, we can be sure that multicollinearity is not a problem because the key variable of interest, voucher , is uncorrelated with all of the added explanatory variables. (This zero correlation will only be approximate in any random sample, but in large samples it should be very small.) The one case where we would not add these variables – or, at least, when there is no benefit from doing so – is when the background variables themselves have no affect on the test score. Given the list of background variables, this seems unlikely in the current application.

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