Econ 3120 1st Edition Exam 2 Study Guide In 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 parameter Answer 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 the variation 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 0 1 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 1 is the change in Weight in pounds for a one inch change in Height b Suppose instead that you estimate log Weight 0 1 Height Now what is the interpretation of the estimated slope coefficient Answer The slope coefficient 1 is the percentage change in Weight for a one inch change in Height c Finally suppose you estimate log Weight 0 1 log Height What is the interpretation of the estimated slope coefficient in this case Answer The slope coefficient 1 is the percentage change in Weight for a onepercent change in Height 3 Consider the wage regression model log wage 0 1educ 2exper 3exper2 u where exper represents the worker s experience in the labor market usually calculated as age 18 d Explain what the coefficient 1 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 1 dlog wage deduc dwage wage deduc wage wage educ By rearranging the above equation we have wage wage 100 1 100 educ Thus based on these math derivation we claim that additional one year of education will predict 1 100 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 1 0 H1 1 6 0 We construct the t statistics t 1 1 var 1 0 077986 0 0 0066242 11 773 The distribution of the test statistics under the null follows 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 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 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 0 1log f are 2dist 3dist2 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 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 2 0 and the alternative hypothesis is H1 2 6 0 The test statistic is 2 0 se 2 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 estimate of 1 in the full model 0 1 Propose a direction for the bias and justify your answer based on the expected effect of the omitted variable on log passen and on the correlation between your omitted variable and log f are Hint you can use the bivariate omitted variable bias formula as an approximation Answer An omitted variable that might bias the estimate of 1 in the full model is the city size on the endpoints of the route Larger cities imply that there will be more passengers and