COMPREHENSIVE WRITTEN EXAMINATION, PAPER III FRIDAY AUGUST 18, 2006, 9:00 A.M. 1:00 P.M. STATISTICS 174 QUESTIONS

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1 COMPREHENSIVE WRITTEN EXAMINATION, PAPER III FRIDAY AUGUST 18, 2006, 9:00 A.M. 1:00 P.M. STATISTICS 174 QUESTIONS Answer all parts. Closed book, calculators allowed. It is important to show all working, especially with numerical calculations. Statistical tables are provided. You may freely quote results from the course notes or text without proof, but to the extent that it is feasible to do so, state precisely the result you are quoting. Consider the following bivariate linear regression model BLRM: y i,1 = y i,2 = p x i,j β j,1 + ǫ i,1, 1 j=1 p x i,j β j,2 + ǫ i,2, 2 j=1 where i = 1,2,...,n, ǫ i = ǫ i,1,ǫ i,2 has i.i.d. bivariate normal distribution with mean zero, Varǫ i,k = σ 2 k, and Covǫ i,1,ǫ i,2 = σ pts Let β k = β 1,k,...,β p,k, k = 1,2, β = β 1,β 2, x i = x i,1,...,x i,p, and X = x 1,...,x n. Write down the BLRM in matrix form pts Suppose σ 2 1,σ2 2,σ 12 are known, derive the generalized least squares GLS estimator β GLS. An economist wants to fit the above BLRM to his data, but he only knows how to run ordinary LS OLS regression. So he estimated β 1 and β 2 seperately by applying OLS procedures to 1 and 2 respectively. Will his estimate in general be the same as β GLS? If your answer is yes, prove your claim. If your answer is No, give a counter example pts Assuming that X is full rank, give unbiased estimators of σ 2 1 and σ 12, prove that they are indeed unbiased, and give the distribution of σ pts Derive the variance-covariance matrix for β GLS. 1

2 The price and consumption per capita of beef and pork annually from 1925 to 1941 together with food consumption per capita index are given below. 1. PBE = Price of beef cents/lb 2. CBE = Consumption of beef per capita lbs 3. PPO = Price of pork cents/lb 4. CPO = Consumption of pork per capita lbs 5. CFO = Food consumption per capita index = 100 YEAR PBE CBE PPO CPO CFO The Pork Producers Association PPA would like to understand the relationship between pork/beef price and their per capita consumption. A BLRM is fitted to the data, and following are the R outputs: > bp.lm <- lmcbindcbe,cpo~year+pbe+ppo+cfo > summarybp.lm Response CBE : Call: lmformula = CBE ~ YEAR + PBE + PPO + CFO 2

3 Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr> t Intercept *** YEAR *** PBE e-06 *** PPO *** CFO * --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 12 degrees of freedom Multiple R-Squared: ,Adjusted R-squared: F-statistic: on 4 and 12 DF, p-value: 2.795e-06 Response CPO : Call: lmformula = CPO ~ YEAR + PBE + PPO + CFO Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr> t Intercept e-09 *** YEAR e-08 *** PBE ** PPO e-10 *** CFO ** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 12 degrees of freedom Multiple R-Squared: ,Adjusted R-squared: F-statistic: on 4 and 12 DF, p-value: 4.548e-10 3

4 In addtion, we know that i e i,1e i,2 = , where e i,k are the residuals from the BLRM, and X X 1 is given by Intercept YEAR PBE PPO CFO Intercept YEAR PBE PPO CFO pts Write a report of no more than 200 words explaining the fitted model to someone from PPA who don t know much about statistics pts If for the year 1942 the CFO remains the same as 1941, and the PBE is expected to decrease by 6 cents/lb. In order to maintain the pork per capita consumption at the same level as in 1941, how much should the PPO decrease? Give a point estimator and its 95% confidence interval pts Suppose for the year 1942, CFO=95, PBE=50, and PPO=40. Estimate the difference between per capita consumption of beef and pork for 1942, and compute its standard error pts Outline in no more than 200 words any further analysis you can to do before writing a final report for PPA on relationship between PBE, CBE, PPO, and CPO. 4

5 Solutions, 2006 CWE Let Y = Y 1,Y 2, ǫ = ǫ 1,ǫ 2, and Z = diagx,x, the BLRM can be written as Y = Zβ + ǫ. 2. βgls = Z V 1 Z 1 Z V 1 Y, where V = VarY = σ 2 1 I σ 12I σ 12 I σ 2 2 I. Matrix computation leads to β X X 1 X Y 1 GLS = X X 1 X, i.e., the GLS and OLS esitmater Y 2 is the same, and for the rest we drop the subscript GLS. 3. Let e = Y Z β = e 1,e 2, σ 1 2 = e 1 e 1/n p χ 2 n p, and σ 12 = e 1 e 2/n p. σ Var β = X X 1 σ 12 X X 1 σ 12 X X 1 σ2 2X X Interpretation of the coefficients. 6. Let x 0 be the price decrease for PPO, then β 22 6β 32 x 0 β 42 = 0. Point estimator: x 0 = / = CI can be obtained using Feiller s method: solve for x 0 the following inequality β 22 6β 32 x 0 β 42 2 /Varβ 22 6β 32 x 0 β 42 t 2 12,0.975, and we have x , Let x h = 1,1942,50,40,95, the difference can be written as x h, x h β. Point estimator: x h, x h β = Variance: x h, x h Var βx h, x h = σ1 2 + σ2 2 2σ 12x h X X 1 x h = Diagnostics, check for temporal correlation, colinearity, etc. 5