Name Economics 170 Spring 2004 Honor pledge: I have neither given nor received aid on this exam including the preparation of my one page formula list and the preparation of the Stata assignment for the exam Final Exam John F. Stewart Instructions: Complete each part of this exam in the space provided. If you need additional space, use the backs of pages but clear indicate where your answers are. Neatness and clarity of exposition count. You must attach your formula list and the output from the Stata assignment to this exam. Question 1 (20 points) 2 (25 points) 3 (30 points) 4 (25 points) 5 (10 points) 6 (40 points) Total (150 points) Bonus question (10) 1. Definitions: Briefly Define each of the following terms as they relate to the material covered in class. 1.1. Over identification 1.2. Durbin-Watson statistic 1.3. Serial correlation 1.4. adjusted R 2 1.5 exact multicollinearity Econ 170 Final Page 1 of 10
2. Short Answer, multiple choice, (5 points each) Use the following list of Ramanathan s assumptions for the linear regression model. 1. The regression model is linear in unknown coefficients a and b i. Y t = a + b 1 X 1t +...+ b k X kt + m t for t=1,2,...,n 2. Not all observations on X are the same, i.e. Var(X) > 0 3. The error term m t is a random variable with E( m t X t ) = 0 4. X t is given and nonrandom, implying that it is uncorrelated with ms that is Cov(X t, m s ) = 0 5. Given X t, m t has constant varaince. Var(m t X)= s 2 6. Given X t, m t and m s are independently distributed for all t s so Cov(m t, m s X) = 0 7. The number of observation (n) must be greater than the number of regression coefficients estimate 8. For a given X, m t is normally distributed. m t X ~ N(0, s 2 ) 2.1. A failure of assumption 5 is called a) multicollinearity b) serial correlation c) heteroscedasticity d) both b) and c) 2.2. A failure assumption 6 is called a) multicollinearity b) serial correlation c) heteroscedasticity d) both b) and c) 2.3. If error terms are heteroscedastic, OLS (ordinary least squares) estimates of the parameter of the regression equation will be (check all that apply) a) biased b) unbiased c) efficient d) not efficient e) consistent 2.4. If one independent variable is omitted from the regression and that variable is correlated with one or more independent variables that are included in the regression then assumption number will be violated and the estimated regression will a) still be unbiased but will be inefficient or b) will be biased. 2.5. Assuming that all 8 assumptions hold, the nr 2 (where R 2 =1-(ESS/TSS) from the OLS regression) is distributed a) F under H 0 : B$ $ 1 = β2 =... = β k = 0 a) P 2 under H 0 : B$ β$... 1 = 2 = = β k = 0 b) F under H 0 : B$ 1 = β1,......, β $ k = β k b) P 2 under H 0 : B$ 1 = β1,......, β $ k = βk Econ 170 Final Page 2 of 10
Problems: (points as indicated) 3. (30 points) Consider a simple regression model where Y = α +βx + µ where Y is college GPA X is high school GPA µ is a random error You are given the following information about the data Term Value n, number of observations 427 Y Sample mean of Y 2.79 3.56 X Sample mean of Y ( Yi Y ) 1= 1... n 2 = TSS (total sum of squares) 124.60 ( X i X ) 1= 1... n ( Yi Y )( X i X ) i = 1... n ( $ ) µ i i=1... n 2 OLS regression) 2 = ESS (the error sum of squares from the 74.00 39.31 103.99 Show your work. 3.1. Calculate the sample standard deviation of X. 3.2. Calculate the value of the OLS estimate of β, β $ = 3.3. Calculate the value of the OLS estimate of a, α $ = 3.4. Calculate the R 2 for the OLS regression R 2 = Econ 170 Final Page 3 of 10
3.5. Calculate the estimated error variance of the OLS model s 2 = 3.6. My daughter had a high school GPA=4.0. Given the additional information that the standard error of prediction for her observation is.04 and that the 95% critical value from the relevant normal distribution is 1.65, calculate a 95% confidence interval on her predicted college GPA. 4. Consider the following time series model: Y t = α +βx t + µ t. We suspect that the model is characterize by a second order auto regressive process. µ t = ρ 1 µ t-1 + ρ 2 µ t-2 + ε t where ε t is white noise i.e it is iid. 4.1. (10 points) Describe step by step how you could test the hypothesis H0: µ t = ε τ against the alternative H1: µ t = ρ 1 µ t-1 + ρ 2 µ t-2 + ε t 4.2. (5 points) Suppose that after doing the test proposed in 4.1., that you cannot reject the null hypothesis of autocorrelation, what are the consequences of estimating the original model with OLS. (What properties will OLS estimates have in this situation?) 4.3. (10 points) Again assuming that we cannot reject the null hypothesis of autocorrelation,, describe step by step the procedure that would result in consistent and asysmtotically efficient estimators of the parameters of this model. Econ 170 Final Page 4 of 10
5. (10 points) Consider a sample of 60 individuals who applied for medical school. The variable Y measures whether or not the individual was accepted to medical school (accept = 1 if the student was accepted and accept = 0 if the student was not accepted). The variable mcat measure the individual s score on the med school entrance exam, and the variable gpa measures the students college grade point average. 5.1. (4 points) Consider estimating the model accept = a + b 1 mcat = b 2 gpa + m using OLS methods. a) Which assumption(s) required for OLS estimators to be unbiased and efficient is (are) necessarily violated (use the list of assumptions from Question 2)? b) Standard hypothesis test of the OLS coefficient estimates (t tests and F tests) are or are not valid. Explain your answer. 5.2. (6 points) The table below show the model estimated using probit. Probit estimates Number of obs = 60 LR chi2(2) = 27.25 Prob > chi2 = 0.0000 Log likelihood = -27.962445 Pseudo R2 = 0.3276 ------------------------------------------------------------------------------ accept Coef. Std. Err. z P> z [95% Conf. Interval] -------------+---------------------------------------------------------------- gpa -.0300694.4246595-0.07 0.944 -.8623868.802248 mcat.0997133.0289851 3.44 0.001.0429036.156523 _cons -4.823301 1.435672-3.36 0.001-7.637167-2.009435 ------------------------------------------------------------------------------ Sam has a 4.0 college gpa and a mcat score of 66, what is the based on the probit model, what is the predicted probability that Sam will get accepted into medical school. Explain your answer. Cumulative standard normal probability f(x).65.70.75.80.85.90.95.99 x.385.524.674.841 1.04 1.28 1.65 2.33 Econ 170 Final Page 5 of 10
6. (40 points) This question uses Part A of the Final Exam Homework Stata assignment. Note: I am trying to cover a lot of ground with one data set. So please treat the questions as sequential and only use the information that is specifically requested for each part. 6.1.a. (5 points) First consider your OLS estimation results for Model 1 and Model 2 from the Final Exam Homework assignment. The economic model is the cross state variation in average performance on the SAT score, depends on how much the state spends per pupil on education and possibly other factors. Compare the results you obtained from these two model (particularly concentrating on the differences). What explanation would you offer as to why the two models differ? 6.1.b. (5 points) Though you were not asked to do it on the assignment, if, after running Model 2, you had run White s general test you would have gotten the following output.. whitetst White's general test statistic : 10.81928 Chi-sq( 5) P-value =.0551 What have we tested for with this test? How was the test actually done? And, how do you interpret the above results? How do these test results change your interpretation of the OLS estimators you obtained for Model 2, if at all. 6.2. Now consider Model 2, Model 3, and Model 4. For this section we have added have added the states poverty rate as another determinant of SAT scores. In Model 3 it is added separate entering linearly variable; in Model 4 poverty rate enters both linearly and interacted with spending. Model 2: sat_tot = α + β 1 spend01 + β 2 pr_02 + µ Model 3: sat_tot = α + β 1 spend01 + β 2 pr_02 + β 3 pov_rate + µ Model 4: sat_tot = α + β 1 spend01 + β 2 pr_02 + β 3 pov_rate + β4(pov_rate spend01) + µ 6.2.a. (5 points) Using your estimated results for Model 4, State A has a poverty rate of 5% and State B has a poverty rate of 15%. An additional $1 of spending per pupil in will result in how many additional points on the SAT scores in State A?, in State B? (show you work, and note in the data a 10% poverty rate Econ 170 Final Page 6 of 10
appears as 10", not.1) 6.2.b. (5 points) Using the Wald Test from your STATA assignment, can you reject the specification in Model 2 (as H 0 when the alternative hypothesis (H 1 ) is Model 4? Explain. 6.2.c. (5 points) Suppose our interest was in rejecting Model 2 (as H 0 ) when the alternative hypothesis (H 1 ) is Model 3, Do you have enough information on your print outs to do this test? Explain. 6.2.d. (5 points) Using the information you generate in part 5 of the STATA assignment, where does North Carolina rank compared to the other states in average SAT scores? Rank, Where do you predict North Carolina would rank in average SAT scores if all states had participation in the exams at the same level? Rank. 6.3. Now consider a model of SAT scores in which we also consider some additional factors and we consider the determinants of the participation rate pr_02. Model 5: 1) sat_tot = α + β 1 spend01 + β 2 pr_02 + β 3 pov_rate + β 4 col_grad + µ 2) pr_02 = γ + γ 1 spend01 + γ 2 sat_tot + γ 3 fam_y + β 4 col_grad + υ 6.3.a. (5 points) In theory, using OLS to estimate equation 1) of model 5 will result in parameter estimates that are (check those that apply) biased efficient unbiased inefficient asymptotically efficient not asymptotically efficient consistent not consistent 6.3.b. (5 points) Using your STATA output, compare the estimates you obtained to OLS and 2SLS procedures. How do they differ? Econ 170 Final Page 7 of 10
Econ 170 Final Page 8 of 10
6.4. Bonus Question (10 points) Along with your answer to 5.3., consider the additional output that was obtained from an OLS regression 1') sat_tot = α + β 1 spend01 + β 2 pr_02 + g 0 er_pr + β 3 pov_rate + β 4 col_grad + µ where er_pr are the predicted residuals for an OLS estimation of the reduced form equation for pr_02.. reg sat_tot pr_02 er_pr spend01 pov_rate col_grad Source SS df MS Number of obs = 50 ---------+------------------------------ F( 5, 44) = 64.70 Model 182202.063 5 36440.4126 Prob > F = 0.0000 Residual 24781.717 44 563.220842 R-squared = 0.8803 ---------+------------------------------ Adj R-squared = 0.8667 Total 206983.78 49 4224.15878 Root MSE = 23.732 ------------------------------------------------------------------------------ sat_tot Coef. Std. Err. t P> t [95% Conf. Interval] ---------+-------------------------------------------------------------------- pr_02-2.67249.4787374-5.582 0.000-3.637321-1.707658 er_pr.2941339.4993292 0.589 0.559 -.7121979 1.300466 spend01.0108647.004135 2.628 0.012.0025313.0191982 pov_rate -2.539621 1.437782-1.766 0.084-5.43728.3580377 col_grad 3.747338 1.557108 2.407 0.020.6091938 6.885482 _cons 1028.728 55.9383 18.390 0.000 915.9915 1141.464 ------------------------------------------------------------------------------ What can you add to your answer to 6.3 with this additional information? Econ 170 Final Page 9 of 10
Economics 170 Spring 2004 Final Exam Assignment John F Stewart This sheet describes STATA exercises to be completed prior to the final exam. You will bring this sheet and a printed copy of the STATA output you generated in doing these this exercise to the final exam. Some questions on the exam will require information you will generate in this assignment. Your output will be turned in with your final exam. This work is an Honor Code assignment. You are not allowed to get help from any individual in completing this assignment. You may use your notes (including do files posted from lectures), the book, STATA Help and manuals, and your previous homework assignments (including keys from the web site). Data Set: final_hw_s04.dta. This is a cross sectional data set of the SAT scores and other information for 50 states in 2002. The variable definitions are as follows. state state abbreviation pov_rate percent of state population living in poverty pr_02 SAT participation rate (% of the students who took the SAT exam in 2002) sat_tot average total SAT score in state in 2002 spend01 per pupil spending on public education in the state in 2001 pr_spend (pov_rate) x (spend01) interaction of spending and poverty rate fam_y median family income in state col_grad percent of population over age 25 with college degree pr_02bak an extra copy of the values in pr_02 (see instructions) 1. Use sum to get a table of summary statistics for all the variables in the data set. 2. Generate the OLS (reg) regression estimates for the following models Model Dependent variable Independent variables 1 sat_tot spend01 2 sat_tot spend01 pr_02 3 sat_tot spend01 pr_02 pov_rate 4 sat_tot spend01 pr_02 pov_rate pr_spend 3. For Model 4: do a Wald test for the hypothesis that both the coefficient on pov_rate equals and the coeficient on pr_spend equal zero. 4. Using Model 4, create predicted SAT scores for all 50 states under the assumption that participation rate (pr_02) in each state is equal to the average participation rate for the entire sample of fifty states (pr_02 for each state is equal to the mean for all states.) Call the predicted SAT scores sat_hat. After generating your predictions sort sat_ tot and list state stat_tot and then sort sat_hat and list sat_tot state sat_hat IMPORTANT: When you have finished with question 4, make sure you restore the original values before you continue to the next part. You can do this easily by replace pr_02=pr_02bak 5. Consider the following model two equation model were participation rates are considered to be endogenous. Model 5: 1) sat_tot = α + β 1 pr_02 + β 2 spend01 + β 3 col_grad + β 4 pov_rate + µ 2) pr_02 = γ + γ 1 sat_tot + γ 2 spend01 + γ 3 col_grad + β 4 fam_y + υ Generate parameter estimates for equation 1) of the system using OLS (reg). Generate parameter estimates for equation 1) of the system using two stage least squares. (use the ivreg procedure in STATA.) Econ 170 Final Page 10 of 10