Econ 120B Ramu Ramanathan Spring 2003 Answers to Exam 2 Using data for the 58 counties in California, the following model was estimated: u++++++=URBUNEMPCOLLEGEHIGHSCHLFAMSIZEMEDINC654321ββββββ MEDINC = Median family income in thousands of dollars FAMSIZE = Number of persons per household HIGHSCHL = Percent of the population (25 years and over) that had only a high school education COLLEGE = Percent of the population (25 years and over) that completed four years of college or higher UNEMP = Percent unemployment rate URB = Percent of urban population The estimated unrestricted model (U), with standard errors in parentheses, is given as follows: 837.0 Adjusted578.7298ESSURB0.029 UNEMP0.403COLLEGE 1.021HIGHSCHL 335.0FAMSIZE 382.15 41.464MEDINC22(0.029)(0.229)(0.132))126.0()033.3((15.742)===+−+++−=RR Let iβbe a typical regression coefficient, i = 1, 2, …, 6. 1. (2 points) To test whetheriβ is significantly different from zero, state the null and alternative hypotheses: H0: iβ = 0 H1: iβ ≠ 0 2. (3 points) Write down an expression for the test statistic (don’t compute it yet). Be sure to define symbols you use. tc = isiββˆˆ where is the regression coefficient and is the standard error. iβˆisβˆ 3. (2 points) State its distribution under the null and the numerical value of the d.f. Under the null, tc is distributed as t with 58 − 6 = 52 d.f. 14. (3 points) Write down the range for the critical value for a 10% level of significance. (1.671, 1.684) 5. (15 points) In the table below, enter the numerical value of the test statistic for the above test, indicate whether you would reject the null or not, and whether the variable is a candidate for omission or retention from Model U. _____________________________________________________________ Variable Test statistic Reject/ Omit/ Not reject Retain _____________________________________________________________ Constant Ignore Ignore Ignore FAMSIZE 5.07 Reject Retain HIGHSCHL 2.66 Reject Retain COLLEGE 7.73 Reject Retain UNEMP −1.76 Reject Retain URB 1.00 Not Reject Omit A second model (Model R) was estimated and its coefficients and related statistics are given next. 826.0 Adjusted.229640ESSCOLLEGE1.213HIGHSCHL498.0FAMSIZE971.1661.766MEDINC2(0.079))109.0()746.2((13.584)==+++−=R Use the first model as the unrestricted model and the second as the restricted model, and perform a relevant test at the 10 percent level by carrying out the following steps. 6. (2 points) State the null and alternative hypotheses in terms of theβs of Model U. H0: 65ββ= = 0 H1: At least one of them is not zero 7. (4 points) Compute the test statistic. Fc = )/()/()(knESSUmkESSUESSR−−− = 52/7298.5782/)7298.578229.640(− = 2.76 28. (2 points) State its distribution including the d.f. F2,52 9. (3 points) State the range for the critical value. (2.39, 2.44) 10. (2 points) Based on the above, would you reject the null hypothesis or not? Why or why not? Because Fc > F*, we reject the null hypothesis. 11. (3 points) What do you conclude in terms of the significance or not of the coefficients of the variables omitted in Model U? At least one of 5βand 6βis not zero. 12. (3 points) Is there a contradiction between this test and the one you did in (5)? If yes, point out where. There is no contradiction because we find from the t-test that 5βis significantly different from zero. 13. (3 points) Model R had UNEMP and URB omitted from Model U. Suppose this was wrong and, say, UNEMP should have been kept. Does this omission affect any of the properties of unbiasedness, consistency, BLUE, and valid of tests? If yes, in what way? If not, why not? Because 5βwas significantly different from zero, it should have been kept. The omission causes the estimates to be biased and inconsistent. Also hypothesis tests are invalid. 14. (3 points) Suppose some of the independent variables were highly correlated. Does it affect the properties of unbiasedness, consistency, BLUE, valid of tests? If yes, in what way? If not, why not? The proofs of unbiasedness, consistency, BLUE, and the valid of tests do not make any assumption about multicollinearity. Therefore these properties still hold.