Midterm10 - Department of Economics ECONOMETRICS I Fall...

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ECONOMETRICS I Fall 2010 – Tuesday, Thursday, 11:50 – 1:10 Professor William Greene Phone: 212.998.0876 Office: KMC 7-78 Home page:www.stern.nyu.edu/~wgreene Office Hours: Open Email: [email protected] URL for course web page: www.stern.nyu.edu/~wgreene/Econometrics/Econometrics.htm Midterm 1 . In the linear regression model, y i = x i ′ β + ε i the least squares estimator, b LS = ( X X ) -1 X y is unbiased and consistent. The least absolute deviations estimator, b LAD = argmin Σ i | y i - x i ′ β | is consistent, but biased and inefficient (compared to b LS ). On the other hand, b LAD appears to have desirable small sample properties – e.g., a small mean squared error and a tolerable small sample bias. [5 points] a. Explain the terms unbiased and consistent. Does unbiased imply consistent? Does consistent imply unbiased? Explain. [5 points] b. Consider an estimator b MIXED that is designed to take advantage of the good properties of both estimators. We compute b MIXED as follows: (1) toss a fair coin. (Probability of HEAD exactly = probability of TAIL = 0.5.) (2) If HEADS, b MIXED = b LS . If TAILS, b MIXED = b LAD . Is b MIXED unbiased? Prove your answer. Is b MIXED consistent? Prove your answer. [5 points] c. The estimator of the asymptotic covariance matrix of the least squares estimator is s 2 ( X ̒ X ) -1 . There is no comparable result for the LAD estimator, so researchers usually use bootstrapping to estimate the covariance matrix for b LAD . Show how the technique of bootstrapping is used to obtain a covariance matrix for the LAD estimator. Department of Economics
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2 . The regression results below are based on a sample of 2,500 observations on 500 American banks observed from 1996-2000. The dependent variable is C, the log of costs. The independent variables are a constant, W1,W2,W3,W4 = logs of prices of 4 inputs (C and W1-W4 are all divided by the price of a 5 th input, W5, before taking logs, so that the cost function is homogeneous of degree one in prices.) Q1, Q2, Q3,Q4, Q5 are the logs of 5 outputs. T is a time trend that takes the values 1,2,3,4,5, since there are 5 years of data. In the third regression result, T2 is T 2 /2 = 1/2 times the square of T. First Regression, Unrestricted +----------------------------------------------------+ | Ordinary least squares regression | | LHS=C Mean = 11.46039 | | Standard deviation = 1.174110 | | WTS=none Number of observs. = 2500 | | Model size Parameters = 11 | | Degrees of freedom = 2489 | | Residuals Sum of squares = 152.5817 | | Standard error of e = .2475933 | | Fit R-squared = .9557087 | | Adjusted R-squared = .9555307 | | Model test F[ 10, 2489] (prob) =5370.71 (.0000) | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant .60183154
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Midterm10 - Department of Economics ECONOMETRICS I Fall...

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