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Economics 513

Spring 2021

Midterm

Be concise. You can cite results and formulas in the lecture notes if you make the attribution clear. For instance: ... In lecture 5, p .. it is shown that (cite formula) is the test statistic for the null hypothesis ... that under the null has a t-distribution with n-1 degrees of freedom. All subquestions are 10 points. Good luck.

Problem 1 Are the following statements true or false? Give a concise argument for your answer.

a. We can obtain the sampling distribution of the OLS estimator of regression coefficients only if we have multiple data sets with observations on the dependent and independent variables.

b. The t-statistic for the hypothesis that a single regression coefficient has some particular value, e.g. 0, is only a valid test statistic if the errors of the regression model have a normal distribution.

c. If the sample size is large we can derive the sampling distribution of the OLS estimator as if the errors have a normal distribution.

d. The bootstrap estimator of the variance of the OLS estimator of a regres-sion coefficient is close to the exact sampling variance of the OLS estimator even in small samples.

e. If one omits an independent variable in a linear regression model,that variable does not affect the OLS estimator of the regression coefficients of the remaining independent variables.

f. In the linear regression model the parametric bootstrap estimator of the variance of the OLS estimator of a regression coefficient is a consistent estimator of the population sampling variance, even if the errors are het-eroskedastic.

Problem 2 Your task is to estimate a demand equation for automobiles. The dependent variable is the number of cars sold for a car model in a year. You are particularly interested in the effect of price on the demand for cars. It is assumed that prices for car models are set by the producers on the basis of the attributes of the car models and that they supply the quantity that the consumers want to buy at that price. Therefore we do not have to consider a supply equation for cars. The data are sales and prices of all car models on the market in a particular year. This is a cross-sectional data set.

a. If the dependent variable is the logarithm of the number of cars sold for a model and the independent variable the logarithm of its price, what is the interpretation of the coefficient on the log price?

b. Are all cars of the same quality? Would this bias the OLS estimator of the coefficient on log price in a linear regression of log quantity on log price (and a constant) and what is the likely sign of the bias? Why?

From a consumer magazine we obtain the test score for each model on a scale from 1 to 100. We want to use this variable to reduce or possibly eliminate the bias in the OLS estimate of the coefficient on log price.

c. Under what assumptions can you use this variable to eliminate the bias in the OLS estimator?

d. Describe the estimation procedure that will give a consistent estimator of the regression coefficient on log price under the assumptions in c.

From a car yearbook we obtain data on the attributes of the various car mod-els (think of engine size, interior space, fuel economy etc.). It is argued that including these attributes in the regression model will reduce the bias in the estimated coefficient of log price.

e. Under what condition is the OLS estimate unbiased after inclusion of these attributes? You may assume that the list of observed attributes is incomplete, i.e., there is at least one unobserved attribute.

f. Given that car manufacturers use all attributes in setting the price, is this condition likely to be met? Why (not)?

Problem 3 Consider a linear regression model

yi = β1 + β2xi2 + β3xi3 + β4xi4 + εi

a. We have 40 observations and R2 = .4. Test the hypothesis that all re-gression coefficients except the intercept are 0 using the F-test at the 5% significance level.

b. We want to construct a 95% confidence interval for β2 + β3. Suggest a regression model with transformed independent variables that allows us to use the standard error on a single coefficient to obtain the desired confidence interval.

c. The relevant coefficient in the regression in b. is .85 with a standard error of .068. Give the 95%confidence interval and test (at the 5% significance level) the hypothesis that β2 + β3 = 1.

d. In a. what assumption on the distribution of the random errors did you implicitly make?