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ps6_Spring2018.pdf

ps6_Spring2018.pdf - Department of Economics Columbia...

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Department of Economics UN3412 Columbia University Spring 2018 Problem Set 6 Introduction to Econometrics Seyhan Erden and Simon Lee Part I Question 1 True, False, Uncertain with Explanation: (a) If the key explanatory variable is constant over time, we cannot use fixed effects to estimate its effect on y (the dependent variable). (b) Using fixed effects is mechanically the same as allowing a different intercept for each cross- sectional unit. (c) In the fixed-effects regression model, you should exclude one of the binary variables for the entities when an intercept is present in the equation. (d) Time fixed effects regressions are useful in dealing with omitted variables if these omitted variables are constant over time but not across entities. Question 2 A researcher investigating the determinants of crime in the United Kingdom has data for 42 police regions over 22 years. She estimates by OLS the following regression 𝐿?𝑔(𝑐𝑟??) 𝑖𝑡 = ? 𝑖 + 𝜑 𝑡 + ? 1 ??𝑟?? 𝑖𝑡 + ? 2 ?𝑟?𝑦?ℎ 𝑖𝑡 + ? 3 log(??) 𝑖𝑡 + ? 𝑖𝑡 where cmrt is the crime rate per head of population, unrtm is the unemployment rate of males, proyth is the proportion of youths, and pp is the probability of punishment measured as (number of convictions)/(number of crimes reported). ? and 𝜑 are area and year fixed effects, coefficient 𝜑 1 is not included. (a) What is the purpose of excluding 𝜑 1 ? What are the terms ? and 𝜑 likely to pick up? Discuss the advantages of using panel data for this type of investigation. (b) Estimation by OLS using heteroskedasticity-robust standard errors results in the following output, where the coeffcients of the fixed effects are not reported: 𝐿?𝑔(𝑐𝑟??) 𝑖𝑡 ̂ = 0.063??𝑟?? 𝑖𝑡 + 3.739?𝑟?𝑦?ℎ 𝑖𝑡 − 0.558log(??) 𝑖𝑡 ; 𝑅 2 = 0.904 (0.109) (0.179) (0.024) Comment on the results. In particular, what is the effect of a ten percent increase in the probability of punishment? (c) To test for the relevance of the area fixed effects, you restrict the regression by dropping all regional fixed effects and adding a single constant. The relevant F-statistic is 135.28. What are the degrees of freedom? What is the critical value from your F table ?
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(d) Although the test rejects the hypothesis of eliminating the fixed effectsfrom the regression, you want to analyze what happens to the coefficients and their standard errors when the equation is re-estimated without fixed effects. In the resulting regression, ? ̂ 2 and ? ̂ 3 do not change by much, although their standard errors roughly double. However, ? ̂ 1 is now 1.340 with a standard error of 0.234. Why do you think that is? Question 3: (a) Explain the differences between panel data and repeated cross-sections. (b) In panel data, it is recommended that empirical researchers use clustered standard errors.
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