Economics 272 Midterm Exam Solutions: Regression Analysis and Econometric Models - Prof. D, Exams of Introduction to Econometrics

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Economics 272 David Guilkey Spring 2004

MIDTERM EXAM

1. The following STATA log runs a regression of the birth weight in grams of children in Cebu the Philippines against sex of the child, whether the family lives in an urban location, the mother’s age and years of education, and the mother’s height. . regress bw sexchild urban mothage motgrd moheight

Source | SS df MS Number of obs = 3022 -------------+------------------------------ F( 5, 3016) = 35. Model | 32540996.5 5 6508199.3 Prob > F = 0. Residual | 546471499 3016 181190.815 R-squared = 0. -------------+------------------------------ Adj R-squared = 0. Total | 579012496 3021 191662.528 Root MSE = 425.

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bw | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- sexchild | 40.85196 15.52463 2.63 0.009 10.41202 71. urban | 44.26499 18.98823 2.33 0.020 7.033813 81. mothage | 8.716754 1.301816 6.70 0.000 6.164218 11. motgrd | .679521 2.474886 0.27 0.784 -4.173113 5. moheight | 16.23067 1.577784 10.29 0.000 13.13703 19. _cons | 258.8253 235.8906 1.10 0.273 -203.6974 721.

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. imtest

Cameron & Trivedi's decomposition of IM-test

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Source | chi2 df p ---------------------+----------------------------- Heteroskedasticity | 30.92 18 0. Skewness | 14.57 5 0. Kurtosis | 10.00 1 0. ---------------------+----------------------------- Total | 55.49 24 0.

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. regress bw sexchild urban mothage motgrd moheight,robust

Regression with robust standard errors Number of obs = 3022 F( 5, 3016) = 34. Prob > F = 0. R-squared = 0. Root MSE = 425.

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| Robust bw | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- sexchild | 40.85196 15.4868 2.64 0.008 10.4862 71. urban | 44.26499 18.65565 2.37 0.018 7.685904 80. mothage | 8.716754 1.369204 6.37 0.000 6.032087 11. motgrd | .679521 2.443228 0.28 0.781 -4.11104 5. moheight | 16.23067 1.545999 10.50 0.000 13.19935 19. _cons | 258.8253 233.486 1.11 0.268 -198.9826 716.

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a. Interpret the coefficient for sex of the child.

B. Explain the results of the test and the subsequent regression.

2. Given the following model:

where all terms are scalars, X is non-stochastic, and the ,’s are iid normal with mean zero and variance. You can assume

that the full Gaus-Markov assumptions hold.

a. Derive the least squares estimator for $. Show that it is unbiased.

b. Derive the method of moments estimator for $.

c. Suppose that we can no longer assume that X is non-stochastic. Furthermore. However, there exists Z’s such

that. Derive the method of moments estimator.

d. Suppose we add the following equation to the model:

1). Test for identification using the order condition. 2). Discuss the two-stage least squares method of estimation. 3). Show that the two-stage least squares estimator and the method of moments estimator from part c are identical.

3. Given the following model:

where all the standard assumptions are satisfied except:

where the :’s are independent with mean zero and variances and.

a. Derive the covariance matrix of ,.

b. Show that the OLS estimator of $ is unbiased.

c. Discuss a method of obtaining a consistent estimator for the covariance matrix of the OLS estimator for $.