Final Exam. 1. Definitions: Briefly Define each of the following terms as they relate to the material covered in class.

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1 Name Answer Key Economics 170 Spring 2003 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 (15 points) 5 (40 points) Bonus (10 points) Total (120 points) 1. Definitions: Briefly Define each of the following terms as they relate to the material covered in class Under identification In a system of simultaneous equations, the situation in which it there does not exist a solution for the structural parameters in terms of the parameters of the reduced form Consistency a n is a consistent estimator of a if the lim (probability( a-a > e)) Y 0 as n Y 4 for any e > 0 i.e., the plim(a n ) = a 1.3. Simultaneous equation bias If one of the independent variables in a regression equation is endogenous (is systematically related to other variable so that the covariance between the variable and error term is not 0) then OLS estimation of the equation will result in biased parameter estimates 1.4. Durbin-Watson statistic dw statistic is used to test for 1 st order autocorrelation of errors. It is based on an estimate of the correlation coefficient between errors at t and t-1. Yt = β1 + β2 Xt β k Xtk + µ t error specification: µ t = ρµ t 1 + εt with 1 ρ 1 Step1: estimate with OLS and compute residuals µ $ t = Yt β $ β $ 1 2 Xt2... β $ k Xtk t= n 2 Step2: compute d = t= 2 ( µ $ t µ $ t 1) µ $ 2 t Step3: test H0: ρ = 0against ρ > 0and H0: ρ = 0 against ρ < heteroskedasticity Failure of the one of the classical assumption for OLS estimates (errors have constant variance) to be BLUE. Econ 170 Final Page 1 of 18

2 Failure of the assumption: Given X t, m t has constant varaince. Var(m t X)= s 2 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 6 is called a) multicollinearity b) serial correlation c) heteroscedasticity d) endogeneity 2.2. A failure assumption 4 is called a) multicollinearity b) serial correlation c) heteroscedasticity d) endogeneity 2.3. If one of the X varaibles is endogenous, 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. Assuming that all 8 assumptions hold and that the parameter estimates of the β' i ; β $ i are made using OLS, then β $ i is distributed σ $ β $ i a) t under H o : β $ i = βi c) P 2 under H o : b) t under H o : β $ i = 0 d) P2 under H o : β $ i = βi $ β i = Assuming that all 8 assumptions hold, the nr 2 (where R 2 =1-(ESS/TSS) fro the OLS regression) is distributed a) F under H 0 : B$ $ 1 = β2 =... = βk = 0 a) P 2 under H 0 : $ $ B1 = β2 =... = 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 18

3 Econ 170 Final Page 3 of 18

4 Problems: (Points as indicated) 3. (30 points total) Consider the following regression mode to be estimated on a sample of Y and X values: Yi = α + β Xi + µ i i N 2 µ ~ 0, σ i and σi = σ + γx i α, β, γ, and σ are parameters and assume σ + γxi > 0, i 3.1. (6 points) True or False $ β (The OLS estimate of β) is unbiased if γ =0 unbiased if γ 0 efficient if γ =0 efficient if γ (12 points) Suppose that you knew the true values of F and ( but do not know the true values of " and $. Describe how you might use the available date to generate unbiased efficient estimators for " and $. Explain your answer clearly The problem here is that the error terms do not have constant variance (heteroskedasticity) though the OLS estimates will be unbiased, they will not be efficient. If we knew the values of F and ( then we would know the value of s i = F +(X i If we then transform the original regression equation by dividing both sides of the equation by s i we get Yi α Xi µ = + β + i This transformed equation now has an error term that is standard norm and so fits the σi σi σ i σ i classical assumptions for OLS regress to be BLUE. The procedure would be to regress the transformed Y against the transformed X and 1/s i with no constant term. The OLS estimate of b would be BLUE (12 points) Now suppose that you don t know the true values of F and ( but only have a data set on Y and X. Describe how you might use the available date to generate estimators for " and $ and describe what the properties of these estimates would be. Explain your answer clearly. If you did not know the true values of F and (, then you would have to estimate them. One could do an OLS regression on the original equation Yi = α + β Xi + µ i and then predict the residuals $µ i. The predicted residuals could then be regressed on X (or X and powers) of X to get estimates of F and (. If you take the specification of σ i and convert it to a variance you get σ i 2 σ 2 σγxi γ 2 Xi 2 orσi 2 2 = + 2 = = δ0 + δ1xi + δ3xi you could think of this as 2 2 an estimating equation where we proxy for σ i with $µ i. Taking the square roots of the predicted values of the estimated variance equation will provide the weights to us for transforming the variables as in 3.2. Econ 170 Final Page 4 of 18

5 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 + ε t where ε t is white noise i.e it is iid (5 points) Describe step by step how you could test the hypothesis H0: µ t = ε t against the alternative H1: µ t = ρ 1 µ t-1 + ε t 1. estimate model Y t = α +βx t + µ t. with OLS and predict residuals e t 2. do auxiliary regression: e t = α +βx t +ρ 1 e t-1 3. calculate L = (n-m)r 2 where n is the number of observations, m is number of lags in the auxiliary regression and R 2 is its unadjusted R 2 4. L is distributed χ 2 m under null hypothesis of all lagged error coefficients in auxiliary regression are 0, If we cannot reject, have serial correlation Because this is a first order autregressive procoss, you could also use the Durbin-Watson test (5 points) Suppose that after doing the test proposed in 4.1., that you cannot reject the null hypothesis of no autocorrelation, what are the consequences of estimating the original model with OLS. Estimates from the OLS model will still be unbiased, but they will not be efficient (5 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. 1. estimate model Y t = α +βx t + µ t. with OLS and predict residuals e t 2. do auxiliary regression: e t =ρ 1 e t-1 +ρ 2 e t-2 3. use r the OLS estimate of ρ to transform the original data as follows Y * t =Y t - ry t-1 X* t =X t - rx t-1 4. estimate original model on transformed data and repeat until the estimates of the rhos no longer change (iterative procedure) or search over rho for the value producing the best fit (search procedure).. Econ 170 Final Page 5 of 18

6 5. (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. 5.1.a. (5 points) First consider you 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? If you look at the two sets of results, you find that Model 1 has a negative insignificant coefficient on spend01 and very low R 2. However when pr_02 is added the sign on spend01 changes and becomes significant. These are classic symptoms of omitted variable bias.. Model 1: reg sat_tot spend F( 1, 48) = 0.14 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = _cons Model 2: reg sat_tot spend01 pr_ F( 2, 47) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = pr_ _cons Econ 170 Final Page 6 of 18

7 5.1.b. (5 points) Though you were not asked to do it on the assignment, if, after running Model 2, you had run Whites general test you would have gotten the following output.. whitetst White's general test statistic : Chi-sq( 5) P-value =.0551 Model Yt = β1 + β2 X2 + β 3X3 + µ t assumed error structure: 2 σt = α1 + α2 X2 + α3x3 + α4x2 2 + α5x3 2 + α6 X2 X3 µ $ 2 1. Estimate model and calculate 2. Estimate auxiliary regression 2 3. Compute nr for auxiliary regression 2 4. nr ~ χ5 2 under the hypothesisα2 = α3 = α4 = α5 = α6 = 0 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. The White test is a test for Heteroskedasticity (error terms whose variance is not constant). The test that is performed is as follows. The test statistic in the output above is nr 2 (where n=50 and R2 is from the auxiliary regression on the estimated residuals from Model 2. A Chi-sq of has a P-value of.0551 so at not quite the 95% level of confidence we can reject the hypothesis of no heteroskedasticity. If the residuals are heteroskedastic, then the OLS coefficient estimates, while still unbiased are inefficient (so hypotheses tests are invalid) 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 5.2.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) Econ 170 Final Page 7 of 18

8 δsat _ tot Note that if we take Model 4, = β1 + β4pov _ rate using the coefficients from the regression we get the δspend 01 numbers above. Also note from the summary statistics that you produced, that participation rate is measured in full number e.g. 15% = 15 (not.15) in the actual data. Econ 170 Final Page 8 of 18

9 5.2.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. Restriction between Model 4 and Model 2 is that β 3 =β 4 =0 The output of the test is. test pov_rate pr_spend ( 1) pov_rate = 0.0 ( 2) pr_spend = 0.0 F( 2, 45) = Prob > F = With 99% + confidence we can reject the hypothesis that β 3 =β 4 =0, that is we can reject Model 2 in favor of Model 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. Here the implied restriction is that β 3 =0 We could rely on the t statistic on β 3 or, better we could remember that we can construct the F for Wald s test using the ESS of the restricted and unrestricted regressions. For a linear restriction of coefficients the following test statistic is distributed as F under the null hypothesis that the restriction is valid. 2 2 ( U R ) R R / ( k m) is distributed as F 1 R / ( n k ) 2 1, k m U Where the R 2 s are the R 2 from the restricted and unrestricted regressions, k-m is the number of restriction and k is the number of parameters estimated in the unrestricted model. From the two regression run on models 2 and 4 in the exercise, you can get the two R 2 measures. 5.2.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 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: 2) pr_02 = γ + γ 1 spend01 + γ 2 sat_tot + γ 3 fam_y + β 4 col_grad + υ 5.3.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 5.3.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 9 of 18

10 Simple answer here (see log material) is that they don t differ much at all coefficient sign, magnitudes, and implied significance are virtually identical. A case where doing it right, gets you essentially the same answer as doing it wrong Bonus Question (10 points) Along with your answer to 4.3., 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 F( 5, 44) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = pr_ spend pov_rate col_grad _cons What can you add to your answer to 5.3 with this additional information? This is the Durbin-Wu-Hausman exogeneity test. A reduced form equation for pr_02 is run and the residuals are calculated. These residuals are then included in the estimation of structural equation 1) as an additional variable. If the coefficient on the residuals does not differ from zero (as in this case) then we cannot reject exogeneity. The reason we get the same results on equation 1 with OLS and 2SLS is that pr_02 is statistically exogenous. Full Log of STATA Exercise: use "D:\work\courses\econ170\Econ 170 Exams\final_hw_s03_a.dta", clear. do "D:\work\courses\econ170\Econ 170 Exams\final_s03_a.do". /* Key for final Exam Homework*/. /*Part A: Using final_hw_s03_a.dta*/. pause on. sum Variable Obs Mean Std. Dev. Min Max Econ 170 Final Page 10 of 18

11 state 0 pov_rate pr_ ver_ math_ sat_tot spend pr_spend fam_y col_grad pr_02bak corr pov_rate pr_02 ver_02 math_02 sat_tot spend01 pr_spend fam_y col_grad (obs=50) pov_rate pr_02 ver_02 math_02 sat_tot spend01 pr_spend pov_rate pr_ ver_ math_ sat_tot spend pr_spend fam_y col_grad fam_y col_grad fam_y col_grad reg sat_tot spend F( 1, 48) = 0.14 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = spend _cons reg sat_tot spend01 pr_02 Econ 170 Final Page 11 of 18

12 F( 2, 47) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = spend pr_ _cons whitetst White's general test statistic : Chi-sq( 5) P-value = reg sat_tot spend01 pr_02 pov_rate F( 3, 46) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = spend pr_ pov_rate _cons reg sat_tot spend01 pr_02 pov_rate pr_spend F( 4, 45) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = spend pr_ pov_rate Econ 170 Final Page 12 of 18

13 pr_spend _cons /*Wald Test, two ways to the same answer*/. test pov_rate pr_spend ( 1) pov_rate = 0.0 ( 2) pr_spend = 0.0 F( 2, 45) = Prob > F =. test pov_rate=0 ( 1) pov_rate = 0.0 F( 1, 45) = 0.93 Prob > F = test pr_spend=0, accum ( 1) pov_rate = 0.0 ( 2) pr_spend = 0.0 F( 2, 45) = 9.82 Prob > F = display ( *5) display ( *15) /*Making predictions if all states had participation at same level*/. replace pr_02=37.4 (50 real changes made). /* Setting all observation at average participation*/. predict sat_hat (option xb assumed; fitted values). sort sat_tot. list state sat_tot state sat_tot Econ 170 Final Page 13 of 18

14 1. GA SC TX FL PA NY IN DE ME RI HI NJ CA VA CT MD VT NV MA AL NH WV AR OR WA WY OH ID MT CO NM KY MS AZ TN AK LA UT OK MI NE MO KS SD MN IL WI IA ND 1207 Econ 170 Final Page 14 of 18

15 . sort sat_hat. list state sat_tot sat_hat state sat_tot sat_hat 1. LA WV MS NM AL AR KY UT AZ TX CA OK MT TN SC ID FL NY GA SD NV MO WA HI CO OH IL OR VA ME RI AK PA ND MI KS WY IA NE MA MD NH IN Econ 170 Final Page 15 of 18

16 45. VT DE WI MN CT NJ /*Restoring original values to pr_02*/. replace pr_02=pr_02bak (50 real changes made). /*Model 5*/. /* OLS estimation of the SAT equation of model 5*/. reg sat_tot pr_02 spend01 pov_rate col_grad F( 4, 45) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = pr_ spend pov_rate col_grad _cons /* 2sls estimation of the SAT equation of model 5*/. ivreg sat_tot (pr_02=fam_y) spend01 pov_rate col_grad Instrumental variables (2SLS) regression F( 4, 45) = 9.93 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = pr_ spend pov_rate col_grad _cons Econ 170 Final Page 16 of 18

17 Instrumented: pr_02 Instruments: spend01 pov_rate col_grad fam_y. /* Bonus question endogeneity test for model 5*/. reg pr_02 spend01 col_grad pov_rate fam_y F( 4, 45) = 4.46 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = pr_02 Coef. Std. Err. t P> t [95% Conf. Interval] spend col_grad pov_rate fam_y _cons predict pr_hat (option xb assumed; fitted values). predict er_pr,resid. reg sat_tot pr_02 er_pr spend01 pov_rate col_grad F( 5, 44) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = pr_ er_pr spend pov_rate col_grad _cons Econ 170 Final Page 17 of 18

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