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Econ 3120 1st Edition Exam # 2 Study GuideIn three or fewer sentences, describe each of the following concepts. Do not use or simply translate any formulas. (a) Gauss-Markov Theorem Answer. The Gauss-Markov Theorem states that under assumptions MLR.1-MLR.5, the OLS estimators are the best (minimum variance) linear unbiased estimators. (b) Collinearity Answer. In the linear regression context, collinearity occurs when one independent variable can be expressed as a linear combination of the other independent variables. When this occurs, the OLS estimates for the collinear variables cannot be computed. (c) Slope parameterAnswer. The slope parameter tells us how the dependent variable changes when there is a change in the independent variable. It may be interpreted differently based on the functional forms of both the dependent and independent variables. (d) Explained sum of squares (SSE) Answer. The explained sum of squares (SSE) is the sample variation in the predicted values that result from a regression. It captures thevariation in the dependent variable that can be explained by the independent variable. 2. Suppose you estimate the following model of weight on height for a random sample of 20-year-old men: Weight = α0 +α1 ·Height +u, where Weight is measured in pounds and Height is measured in inches. (a) Give an interpretation of the estimated slope coefficient. 1 Answer. The slope coefficient, α1, is the change in Weight (in pounds) for a one-inch change in Height. (b) Suppose instead that you estimate: log(Weight) =β0 +β1 · Height +ε. Now what is the interpretation of the estimated slope coefficient? Answer. The slope coefficient, β1, is the percentage change in Weight for a one-inch change in Height. (c) Finally, suppose you estimate: log(Weight) = γ0 +γ1 · log(Height) +ψ. What is the interpretation of the estimated slope coefficient in this case? Answer. The slope coefficient, γ1, is the percentage change in Weight for a one-percent change in Height. 3. Consider the wage regression model: log(wage) = β0 + β1educ + β2exper + β3exper2 + u, where exper represents the worker’s experience in the labor market (usually calculated as age-18) (d) Explain what the coefficient β1 represents. Show how to derive this by taking the derivative of both sides of the equation with respect to education, holding experience fixed. Solution: taking the derivative of both sides of the equation with respect to educ, we have β1 = dlog(wage) deduc = dwage wage deduc = ∆wage wage ∆educ , By rearranging the above equation, we have ∆wage wage ×100% = β1×100% ×∆educ. Thus, based on these math derivation, we claim that additional one year of education will predict β1 ×100% increase in salary. (e) Perform the following hypothesis tests. For each tests, state the null hypothesis, the alternative, the test statistic, the distribution of the test statistic under the null, the rejection rule, and the outcome of your test. The Appendix contains Stata output of OLS regressions that may be useful. Note that we have supressed some of the statistics in the output. i. Test whether a year of education has no impact on wages, at the 1% level. Find the p-value for this test. Solution: • The null hypothesis and alternative hypothesis: 2 H0 : β1 = 0 H1 : β1 6= 0 • We construct the t-statistics t = ˆ √ β1−β1 var( ˆβ1) = 0.077986−0 0.0066242 = 11.773 • The distribution of the test statistics under the nullfollows t-distribution t932 with degrees of freedom equal 932. Because the sample size n = 935 > 30, under the Central Limit Theorem, the test statistics follows the standard normal distribution N(0,1). • The rejection rule is that we reject the null hypothesis if p-value is smaller than α = 1% or the t-statistics is greater than the 1% critical value under the standard normal distribution, which is 2.5758. • We reject the null hypothesis that education has no impact on wages because the pvalueP({|Z| > |t|) = 0 < α = 0.01 by reading Table G.1 in the appendix . Or because the t-statistics 11.773 is greater than 2.5758. (f) Using the regression output from the full model, show how you can calculate the r-squared from the other information given. Solution: Using the first estimation regression result in appendix, from the left top table, we know that total sum of squares (SST)=165.656283, explained sum of squares (model sum of squares, SSE)=21.688779 residual sum of squares (SSR)=143.967504. Recall the definition of r-squared, which is r 2 = SSE SSR = SST−SSR SST , thus plugging the corresponding value into either one ratio on the righ hand side, we have r 2 = 21.688779 143.967504 = 1 − 143.967504 165.65628 = 0.1309. 4. The following question is based on a data set of domestic airline flights in the United States. Let passen be the average number of passengers per day on a domestic flight. Let f are denote the average one-way fare in dollars for a flight, and let dist denote the one-way distance (in thousands of miles) covered by the flight. Our interest is to estimate a demand function, where log(passen) is the dependent variable, and log(f are), dist, and its square, dist2 are the independent variables. The 3 population model is specified as follows: log(passen) = β0 +β1log(f are) +β2dist +β3dist2 +u (0.1) Using 2000 data, the estimated regression equation is: log\(passen) = 8.899 (0.173) − 0.555 (0.0366) log(f are)−0.246 (0.084) dist +0.141 (0.031) dist2 n = 1149, R 2 = 0.063 Assume that assumptions MLR.1 - MLR.5 hold. (a) Is there strong evidence, at α = 1%, that dist should be included in the model? Be sure to state the null hypothesis in terms of the regression parameters, the test statistic, its distribution under the null hypothesis, the rejection rule for your test, and the outcome of your test. (You are only testing whether dist should be included, not dist2 .) Answer. This involves a t-test of the coefficient on dist. The null hypothesis is H0 : β2 = 0 and the alternative hypothesis is H1 : β2 6= 0. The test statistic is β2−0 se(β2) = −0.246 0.084 = −2.93, which has an approximately standard normal distribution under the null. The critical value for this test is 2.57 (two-sided) and we reject if |t| >tcrit. We reject the null since | −2.93| >2.57. (b) What percentage of the variation in log(passen) is explained by the model? Answer. The R 2 = ESS T SS = 0.063 = 6.3%. (c) Name an omitted variable that might bias the

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