Economics 421Homework #2
Due June 5
1) Find the expected value and variance of the estimator of the intercept term from the model Y = % + %X + µ. (Recall a = y- - bx- where y- and x- are the sample means of Y and X.)
2) Suppose you have the true model Y = % + µ and µ satisfies all of our assumptions. Find the ordinary least squares estimator of %.
3) Find the ordinary least squares estimators of the slope and intercept of : Y = % + %(X - x-) + µ where µ satisfies all our assumptions and x- is the sample mean of X. Compare these estimators to those of % and % in the usual model.
4) Find the ordinary least squares estimator of the slope if the true model is Y = %X + µ. Suppose this model is estimated as Y = %(X -x-) + µ. Will the estimator of % be the same as that of %?
5) Suppose you have the model Y = % + %X + µ. Suppose also that you hypothesize that %=1. Your estimate of the model is
Y = 6.73 + 1.35X.
Your estimate of the variance of the model is %^2 = .8, there are 50 observations (N=50), %(Xi-x-)2 = 75, and %Xi2 = 150. Find the variance of the estimates and test the hypothesis.
6) Suppose you have the model Y = % + %X + µ. Suppose also that you hypothesize that %=-1. Your estimate of the model is
Y = 6.73 + .35X.
The true variance of the model is %2 = .3, and the true variance of the slope estimate is %2b = .01. There are 50 observations (N=50). Test the hypothesis.
7) Suppose you have the model Y = % + %X + µ. Suppose also that you hypothesize that %>0. Your estimate of the model is
Y = 6.73 + .35X.
The estimated variance of the model is %^2 = .3, and of the variance of the slope estimate is %^2b = .01. There are 50 observations (N=50). Test the hypothesis.
8) Using the data set for homework 1, estimate the models in questions 1 through 4. Test the hypothesis that the slope parameter is equal to 1 in each case.
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