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Unformatted text preview: PAM 305  Introduction to Multivariate Analysis Fall 2006 Suggested Solutions Homework 2 Due date: September 24th Question 1 The data set WAGE1.dta, contains a sample of N=526 individuals drawn from the population of people in the workforce in 1976 (so wages are in dollars of 1976). Report your answers by means of a STATA log file. Here is the list of commands you will need in order to solve the homework. regress, plot, tab, count, egen, generate, summarize, correlate. Suppose the true model is wage = β o + β 1 educ + u (1) a. Using STATA, obtain the sample regression function. d wage = ˆ β o + ˆ β 1 educ See the STATA log file b. Is it true that in order to get an unbiased estimator of the ceteris paribus effect of education on wages, we must assume that E [ u  educ ] = 0? Show that this implies that E [ wage  educ ] = β o + β 1 educ . Yes, that is our assumption number 4 (the conditional expectation of the unobservables is the same and equal to 0) this is one of the assumptions needed for OLS estimators to be unbiased, if it fails OLS are biased. 1 Here is the proof for the second part of the question E [ wage  educ ] = E [ β o + β 1 educ + u  educ ] = E [ β o  educ ] + E [ β 1 educ  educ ] + E [ u  educ ] = β o + β 1 E [ educ  educ ] + E [ u  educ ] = β o + β 1 educ + 0 = β o + β 1 educ where I have used the following properties of the Expectation operator: It can be distributed over the sum, the expectation of a constant is the constant itself, the expectation of a constant times a random variable is the constant times the expectation of the random variable, and functions of random variables behave as constants when we condition on the same random variable c. If we also make the homoskedasticity assumption, then V ar [ u  educ ] = σ 2 does not depend on the level of education. Argue that this is the same as assuming V ar [ wage  educ ] = σ 2 ....
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 Fall '06
 LUCARELLI
 Regression Analysis, Variance, Probability theory, hourly wage, stata log file

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