ANSWER KEY. Economics 3950 Spring 2004 Dr. Richard Mueller. Assignment #4

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1 Economics 3950 Spring 2004 Dr. Richard Mueller Assignmen #4 Insrucions: These quesions should be answered using a ex edior or a word processor where you can cu and pase oupu from your saisical program (where necessary). Please mark quesion numbers clearly. This assignmen is due on Monday, March 29, 2004 in class a 15: (50 poins oal) Exercise 9.13, p a. µ = ρµ + ε. 1 b. H o : ρ = 0. c. N = 34, k =1, d U = 1.514, and d L = Since d < d L, we rejec H 0 and conclude ha here is significan firs-order aucorrelaion. d. Regress PATENTS agains and consan and R&D, compue µ ˆ = PATENTS βˆ ˆ R & D 1 β and generae µ ˆ 2. Then regress µˆ agains a consan, 1 R & D, and µ ˆ, using observaions 1961 hrough e. LM = (n 1)R 2 = 33[1 (793.09/ )] = f. Under H 0, LM is disribued as a chi-squared wih 1 degree of freedom. g. LM* = h. Since LM > , we rejec H 0 and conclude ha here is significan firs-order aucorrelaion. i. OLS esimaors are unbiased and consisen bu no efficien. Also, ess are invalid. j. Cov ( µ µ ) = 0 when does no equal s. In oher words, he assumpion is ha he wo are uncorrelaed. 2. (50 poins oal) Exercise 9.18, p The following is he GRETL oupu: s MODEL 1: OLS esimaes using he 21 observaions ) cons

2 3) sales ** Mean of dep. var S.D. of dep. variable Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (1, 19) p-value for F() Durbin-Wason sa Firs-order auocorr. coeff The plo of he residuals looks like his: Regression residuals (= observed - fied profis) 20 residual Observaion I appears ha serial correlaion could be a problem. Also, we have evidence of posiive serial correlaion since d = < d L = (n = 21 and k = 1). We can also do a BG or LM es by regressing he residual agains a consan, sales, and he residual lagged. We will now have 20 observaions. MODEL 2: OLS esimaes using he 20 observaions Dependen variable: uha1 0) cons ) sales ) uha1_ * Mean of dep. var S.D. of dep. variable Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (2, 17) p-value for F() Durbin's h sa. undefined Firs-order auocorr. coeff We have (n - 1)R 2 = 20*0.177 = 3.54 which is less han he 5% criical value of Sill, he p-value of means ha we can rejec he null of no auocorrelaion a he 6% level. 2

3 Anoher way of doing his is simply o use he auocorrelaion es in he pull down menu afer he firs model was esimaed (his saves you having o do he calculaion and look up he criical value in he able). Using he CORC procedure we ge: MODEL 3: Cochrane-Orcu esimaes using he 20 observaions ) cons ** 3) sales *** Saisics based on he rho-differenced daa: R-squared is compued as he square of he correlaion beween observed and fied values of he dependen variable. Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (1, 18) p-value for F() Durbin-Wason sa Firs-order auocorr. coeff And using he HILU we ge: MODEL 4: Hildreh-Lu esimaes using he 20 observaions ) cons ** 3) sales *** Saisics based on he rho-differenced daa: R-squared is compued as he square of he correlaion beween observed and fied values of he dependen variable. Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (1, 18) p-value for F() Durbin-Wason sa Firs-order auocorr. coeff The resuls are very similar in boh cases. Noe he DW saisic is less han he criical value of (n = 20, k = 1) in boh cases, indicaing ha posiive serial correlaion sill exiss. Now we have o es for higher order serial correlaion. We can do his using a LM es (p. 399 of he ex). Regressing he residual agains a consan, sales, and he residuals lagged for 1, 2 and 3 periods, we ge ( n 3)R 2 = 18*.70 = which exceeds he criical 3

4 value of 7.81 in a chi-squared disribuion wih 3 d.f. a he 5% level. Therefore, we can rejec he null ha AR(3) does no exis. We now esimae he model and include he residuals lagged for one, wo, and hree periods (see p. 400 of your book, or jus use he Auoregression esimaion command on he Model in GRETL. This yields: MODEL 5: AR esimaes using he 18 observaions ) cons ** 3) sales *** Esimaes of he AR coefficiens: u_ u_ u_ Sum of AR coefficiens = Saisics based on he rho-differenced daa (R-squared is compued as he square of he correlaion beween observed and fied values of he dependen variable): Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (1, 16) p-value for F() Durbin-Wason sa Firs-order auocorr. coeff Noe ha he coefficien on he hird lag is insignifican, so we drop i and esimae again, his yields: MODEL 6: AR esimaes using he 19 observaions ) cons ** 3) sales *** Esimaes of he AR coefficiens: u_ u_ Sum of AR coefficiens = Saisics based on he rho-differenced daa (R-squared is compued as he square of he correlaion beween observed and fied values of he dependen variable): 4

5 Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (1, 17) p-value for F() Durbin-Wason sa Firs-order auocorr. coeff This is he final version of he model. Noe ha an increase in sales of $1.00 will increase profis by 2.55 cens, his compares he 2.65 cens in he OLS model. In hindsigh, i probably wasn worh he rouble o go hrough his exercise! 3. (50 poins oal) Exercise 10.7, p Firs, we lag he income variables for a number of periods. I have chosen six periods. Then we regress consumpion on income and all of is lags: MODEL 1: OLS esimaes using he 30 observaions Dependen variable: C 0) cons ** 2) Y *** 3) Y_ ) Y_ ) Y_ ) Y_ ) Y_ ) Y_ Mean of dep. var S.D. of dep. variable Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (7, 22) p-value for F() Durbin-Wason sa Firs-order auocorr. coeff The DW es suggess he presence of posiive AR(1) as does an LM es where we regress he residuals on heir lagged values and he oher independen variables included in Model 1. This yields: MODEL 2: OLS esimaes using he 29 observaions Dependen variable: uha1 0) cons ) uha_ *** 2) Y ) Y_ ) Y_

6 5) Y_ ) Y_ ) Y_ ) Y_ Mean of dep. var S.D. of dep. variable Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (8, 20) p-value for F() Durbin's h sa Firs-order auocorr. coeff (Using variable 10 for h sa, wih T' = 28) Here we ge (n 1)R 2 = 29*0.474 = which has a p-value of Clearly, AR(1) is a problem. We can also see his from he residual plo (no shown here). Nex, we apply he CORC-HILU procedure in GRETL. This yields: MODEL 3: Hildreh-Lu esimaes using he 29 observaions Dependen variable: C 0) cons ) Y *** 3) Y_ ) Y_ ) Y_ ) Y_ ) Y_ ) Y_ Saisics based on he rho-differenced daa: R-squared is compued as he square of he correlaion beween observed and fied values of he dependen variable. Error Sum of Sq (ESS) Sd Err of Resid. (sgmaha) Unadjused R-squared Adjused R-squared F-saisic (7, 21) p-value for F() Durbin-Wason sa Firs-order auocorr. coeff The long run muliplier is achieved by adding all he coefficiens on income ogeher. This yields a figure of Obviously, mulicollineariy is a huge problem here, which explains why he -saisics are so low. Removing insignifican variables one a a ime does lile o improve he model and he long run muliplier remains close o one. 4. (50 poins oal) Exercise 10.19, p Skipped. 6

7 Grand Toal: poins 7