Basic Economerics, Gujarai and Porer CHAPTER 7: DYNAMIC ECONOMETRIC MODELS: AUTOREGRESSIVE AND DISTRIBUTED-LAG MODELS 7. (a) False. Economeric models are dynamic if hey porray he ime pah of he dependen variable in relaion o is pas values. Models using cross-secional daa are no dynamic, unless one uses panel regression models wih lagged values of he regressand. (b) True. The Koyc model assumes ha all he disribued lag coefficiens have he same sign. (c) False. The esimaors are biased as well as inconsisen. (d) True. For proof, see he Johnson ex cied in foonoe # 30. (e) False. The mehod produces consisen esimaes, alhough in small samples he esimaes hus obained are biased. (f) True. In such siuaions, use he Durbin h saisic. However, he Durbin d saisic can be used in he compuaion of he h saisic. (g) False. Sricly speaing, i is valid in large samples. (h) True. The Granger es is a measure of precedence and informaion conen bu does no, by iself, indicae causaliy in he common use of he erm. 7. Mae use of Equaions (7.7.), (7.6.), and (7.5.). * * Y = β + β X + u () 0 Y Y = δ ( Y Y ) () * * * * X X = γ ( X X ) (3) Rewrie Equaion () as * Y = δy + ( δ ) Y (4) Rewrie Equaion (3) as * γ X = X (5) ( γ ) L where L is he lag operaor such ha LX = X. Subsiue Eq. () ino Eq. (4) o obain * Y = δβ0 + δβx + δ u + ( δ ) Y (6) Subsiue Eq. (5) ino Eq. (6) o obain 90
Basic Economerics, Gujarai and Porer γ Y = δβ0 + δβ[ X ] + ( δ ) Y δ u ( γ ) L + Simplifying Eq. (7), we obain Y = α + α X + α3y + α4y (7.7.) where he α's are (nonlinear) combinaions of he various parameers enering ino Eq. (7). 7.3 Y u λu E Y E Y u λu = E[( u )( u λu )] 7.4 The cov[,( )] = {[( ( )][ ]}, since E( u ) = 0. correlaion. = - λσ. (7), since [Y - -E (Y - )] = u. = - λe[( u ) ], since here is no serial * P values are 00, 05, 5, 35, and 60, respecively. 7.5 (a) The esimaed Y values, which are a linear funcion of he he nonsochasic X variables, are asympoically uncorrelaed wih he populaion error erm, v. (b) The problem of collineariy may be less serious. 7.6 (a) The median lag is he value of ime for which he fracion of adjusmen compleed is ½. To find he median lag for he Koyc scheme, solve period response β0( λ ) /( λ) = = long run response β0 /( λ) Simplifying, we ge λ =. Therefore, ln λ = ln( ) = ln. Therefore, ln =, which is he required answer. ln λ λ ln λ ln Median lag (b) 0..6094 0.693 0.4307 0.4 0.963 0.693 0.7565 0.6 0.508 0.693.3569 0.8 0.3 0.693 3.063 9
Basic Economerics, Gujarai and Porer 7.7 (a) Since β = β0 λ ;0 < λ < ; = 0,,... mean lag = β 0 0 0 β 0 β λ λ /( λ ) λ = = = β λ /( λ) λ (b) If λ is very large, he speed of adjusmen will be slow. 7.8 Use he formula β β. For he daa of Table 7., his becomes:.36 0.986 0.959.03 = 7.9 (a) Following he seps in Exercise 7., we can wrie he equaion for M as: β ( γ ) β ( γ ) M = α + Y + R + u γ L γ L which can be wrien as: M = β + β ( γ ) Y β γ ( γ ) Y + β ( γ ) R 0 - βγ ( γ ) R + ( γ + γ ) M ( γ γ ) M + +[ u ( γ + γ ) u + ( γ γ ) u ] where β0 is a combinaion of α, γ,andγ. Noe ha if γ = γ = γ, he model can be furher simplified. (b) The model jus developed is highly nonlinear in he parameers and needs o be esimaed using some nonlinear ieraive procedure as discussed in Chaper 4. 7.0 The esimaion of Eq. (7.7.) poses he same esimaion problem as he Koyc or adapive expecaions model in ha each is auoregressive wih similar error srucure. The model is inrinsically a nonlinear regression model, requiring nonlinear esimaion echniques. 7. As explained by Griliches, since he serial correlaion model includes lagged values of he regressors and he Koyc and parial adjusmen models do no, he serial correlaion model may be appropriae in siuaions where we are ransforming a model o ge rid of (firs-order) serial correlaion, even hough i may resemble he Koyc or he PAM. 9
Basic Economerics, Gujarai and Porer 7. (a) Yes, in his case he Koyc model may be esimaed wih OLS. (b) There will be a finie sample bias due o he lagged regressand, bu he esimaes are consisen. The proof can be found in Henri Theil, Principles of Economerics, John Wiley & Sons, New Yor, 97, pp. 408-4. (c)since boh ρ and λ are assumed o lie beween 0 and, he assumpion ha hey boh are equal is plausible. 7.3 Similar o Koyc, Al, Tinbergen, and oher models, his approach is ad hoc and has lile heoreical underpinning. I assumes ha he imporance of he pas values declines coninuously from he beginning, which may a reasonable assumpion is some cases. By using he weighed average of curren and pas explanaory variables, his riangular model avoids he problems of mulicollineariy ha may be presen in oher models. 7.4 (a) On average, over he sample period, he change in employmen is posiively relaed o oupu, negaively relaed o employmen in he previous period and negaively relaed o ime. The negaive sign of he ime coefficien and he negaive sign of he ime-squared variable sugges ha over he sample period he change in employmen has been declining, bu declining a a faser rae. Noe ha he ime coefficien is no significan a he 5% level, bu he ime-squared coefficien is. (b) I is 0.97 (c)to obain he long-run demand curve, divide he shor-run demand funcion hrough by δ and drop he lagged employmen erm. This gives he long-run demand funcion as: 47.879 + 0.579Q + 0.094 + 0.00 (d) The appropriae es saisic here is he Durbin h. Given ha n = 44 and d =.37, we obain: d n h = ( ) n var( coeff of E ) = [- d ] 44 44(0.00089) =.44 Since h asympoically follows he normal disribuion, he 5% criical value is.96. Assuming he sample of 44 observaions is reasonably large, we can conclude ha here is evidence of firs-order posiive auocorrelaion in he daa. 93
Basic Economerics, Gujarai and Porer 7.5 (a) I is ( 0.0864) = 0.36. (b) The shor-run price elasiciy is 0.8, and he long-run price elasiciy is (-0.8/0.36) = -.60. (c) The shor-run ineres rae elasiciy is 0.855. The long-run elasiciy is (-0.855/0.36) = -6.87. (d) The rae of adjusmen of 0.36 is relaively low. This may be due o he naure of echnology in his mare. Remember ha racors are a durable good wih a relaively long life. 7.6 The lagged erm represens he combined influence of all he lagged values of a regressor (s) in he model, as we saw in developing he Koyc model. 7.7 The degree of he polynomial should be a leas one more han he number of urning poins in he observed ime series ploed over ime. Thus, for he firs figure in he upper lef hand corner, use a 4 h degree polynomial; for he figure in he upper righ hand corner, use a second degree polynomial; for he figure in he lower lef hand corner, use a 6 h degree polynomial, and for he figure in he boom righ hand corner, use a second degree polynomial. 7.8 (a) p ˆ j j+ p ˆ ˆ ˆ i = i a j + i a j ap j= 0 j< p var( β ) var( ) cov(, ) A similar expression follows, excep ha now we have an addiional erm. (b) This is no necessarily so. This is because, as seen in par (a), he variances of he esimaes of βi involves boh variances and covariances of he esimaed a coefficiens and covariances can be negaive. 7.9 Given ha β i = a0 + ai + ai If β0 = 0 a0 = 0and when β4 = 0 a0 + 4a + 6a = 0 a = 4a. Therefore, he ransformed model is: 4 Y = α + ( β X ) + u i i i= 0 = α + ( a0 + ai + ai ) X i + u = α + a [ 4 ] ix i + i X i + u 94
Basic Economerics, Gujarai and Porer 7.0 Y = α + β X + u i i i= 0 / = α + iβ X + ( i) β X + u i= 0 i i i= ( + ) = α + β[ ix + ( i) X ] + u = α + β Z + u i i 7. Here n = 9 and d =.54. Alhough he sample is no very large, jus o illusrae he h es, we find he h value as: Empirical Exercises d n h = ( ) ( n) var( coefficien of PF ).54 9 = (- ).3773 9(0.04) = This h value is no significan a he 5% level. So, here is no evidence of firs-order posiive serial correlaion, eeping in mind ha our sample may no be large enough o accep his resul. 7. Using he soc adjusmen, or parial adjusmen model (PAM), he shor-run expendiure funcion can be wrien as (see Eq. 7.6.5): Y = δβ0 + δβx + ( δ ) X + u () where Y = expendiure for new plan and equipmen and X = sales. From he given daa he regression resuls are as follows: Yˆ = 5.04 + 0.69X + 0.7Y = ( 3.94) (6.433) (.365) R F d = 0.987; = 690.056; =.59 From he coefficien of he lagged Y value we find ha δ =0.78. The long-run expendiure funcion is: ˆ* Y = 0.738 + 0.864X which is obained from () by dividing i by 0.78 and dropping he lagged Y erm. () 95
Basic Economerics, Gujarai and Porer We have o use he h saisic o find ou if here is serial correlaion in he problem. Using he formula for he h saisics, i can be shown ha in he presen example h =.364. Asympoically, his value is no significan a he 5% level. So, asympoically, here is no serial correlaion in our daa. 7.3 Using he same noaion as in Exercise 7., he shor-run expendiure funcion can be wrien as: ln Y = ln δβ0 + δβ ln X + ( δ ) Y + u () The regression resuls are: ln Yˆ =.078 + 0.905ln X + 0.60 lny = ( 5.854)(8.3) (.96) = 0.994; = 45.9; =.479 R F d From hese resuls, we find ha ˆ δ = 0.740. The long-run expendiure funcion is: ln Yˆ* = 0.376 +. ln X The h saisic for his problem is.34. Asympoically, herefore, we rejec he hypohesis ha here is firs-order posiive correlaion in he error erm. Boh models give similar resuls. The advanage of he log model is ha he esimaed slope coefficiens give direc esimaes of he elasiciy coefficiens, whereas in he linear model he slopes only measure he rae of change. 7.4 The saisical resuls are he same as in Problem 7.. However, since his is he adapive expecaions model, he inerpreaion is differen. Now he esimaed δ is inerpreed as he fracion ha expecaions abou invesmen in plan and equipmen in manufacuring are revised each period. The populaion error srucure is now differen, as noed in he ex. 7.5 Here we use he combinaion of adapive expecaions and PAM. The esimaing equaion is: Y = β δγ + β δγ X + [( δ ) + ( γ )] Y + [( δ ) + ( γ )] Y +v 0 where v = [ δ u + δ ( γ ) u ] () which, for convenience, we express as: Y = α0 + αx + αy + α3y + v Based on he daa, he regression resuls are: 96
Basic Economerics, Gujarai and Porer Yˆ = 9.770+ 0.75X + 0.565Y 0.409Y = ( 4.467) (8.33) (4.50) ( 3.460) R = 0.99; F = 5653.34; d =.367 The esimaed coefficiens are all saisically significan. Bu since he esimaed coefficiens are nonlinear combinaions of he original coefficiens, i is no easy o ge heir direc esimaes. In principle, we should esimae his model using he nonlinear mehods discussed in Chaper 4. Tha will give direc esimaes of he various parameers, which can hen be compared wih hose obained from Problems 7.,7.3 and 7.4. 7.6 Null hypohesis H 0 : sales do no Granger cause invesmen in plan and equipmen. The resuls of he Granger es are as follows: Number of lags F saisic p value Conclusion 7.394 0.000 rejec H 0 3 5.687 0.07 rejec H 0 4 3.309 0.068 do no rejec H 0 5.379 0.606 do no rejec H 0 6.307 0.4463 do no rejec H 0 H 0 : Invesmen in plan and expendiure does no Granger cause sales: Number of lags F saisic p value Conclusion.865 0.000 rejec H 0 3 3.009 0.0004 rejec H 0 4 7.346 0.0065 rejec H 0 5 5.867 0.06 rejec H 0 6 3.053 0.939 do no rejec H 0 As you can see from hese resuls, he Granger causaliy es is sensiive o he number of lagged erms inroduced in he model. Up o 3 lags, here is bilaeral causaliy, up o 5 lags here is causaliy from invesmen o sales. A six lags, neiher variable causes he oher variable. 7.7 One illusraive model fied here is a second degree polynomial model wih 4 lags. Using he forma of Eq. (7.3.5) and leing Y represen invesmen and X sales, he regression resuls are: Y ˆ = -35.493 + 0.890X +0.355X - -0.03X - 0.79X -3 = (-4.33) (5.04) (3.676) (-0.530) (-.09) -0.83X -4 (-0.656) The reader is urged o ry oher combinaions of lags and he degree 97
Basic Economerics, Gujarai and Porer of he polynomial. You may use he Aaie informaion crierion o choose among he compeing models. 7.8 Using EViews, we obained he following resuls. Coefficien NER FER BER Inercep -3.3844-36.0936-5.9303 ( -.3578) (-4.6740) (-0.8799) X 0.388 0.87 0.5 (3.579) (5.505) (9.943) X - 0.444 0.355 0.945 (3.954) (0.4464) (9.943) X - 0.3677 0.0045 0.88 (5.43) (0.0993) (9.943) X -3 0.0976-0.697 0.945 (.948) (-.065) (9.943) X -4-0.3686-0.7 0.5 (-.6678) (-.7730) (9.943) Noes: NER, FER, and BER denoe near-end, far-end, and bohend resricions. Figures in he parenheses are he raios. As you can see, puing resricions on he coefficiens of he models produce vasly differen resuls. Noe he ineresing finding ha imposing boh-end resricions give idenical sandard errors and he raios. Unless here is srong a priori expecaion, i is is beer no o impose any resricions. Of course, sill he number of lagged erms o be inroduced and he degree of he polynomial are he quesions ha need o answered in each case. 7.9 (a) Direcion of causaliy # of lags F Probabiliy Y X 0.0695 0.939 X Y.877 0.0705 * Y X 3 0.338 0.939 X Y 3.489 0.0793* Y X 4 0.407 0.9655 X Y 4.839 0.533 * Significan a he 0% level. In each case he arrow indicaes he direcion of causaliy. The null hypohesis in each case is ha he variable on he 98
Basic Economerics, Gujarai and Porer lef of he arrow causes he variable on he righ side of he arrow. In each case i seems ha invesmen in informaion processing equipmen does no Granger cause sales. Bu here is some wea evidence ha sales cause invesmen. Try oher lags and see if his conclusion changes. (b) The resuls of causaliy beween invesmen and ineres rae are ineresing in ha up o 5 lags, here is no causal relaionship beween he wo, bu a lag 6 ineres rae causes invesmen bu no vice versa. A lags 7 and 8 again here is no causal connecion beween he wo. I is hard o jusify hese resuls inuiively. (c)in he linear form here was no discernible disribued lag effec of sales on invesmen. In he log-linear model wih 4 lags and second degree polynomial and imposing near end resricion, we ge he following resuls: ln Yˆ = 5.508 + 0.008ln X + 0.388ln X + 0.384ln X ( ) ( ) = ( 73.85)(3.896) (5.0794) (9.6896) 0.3665ln X ( 3) + 0.76 ln X ( 4) + (5.078) (.9) If you plo he coefficiens of he various ln X erms, you will find ha he coefficiens increase up o lag and hen decline, showing an invered U-shaped curve. 7.30 (a) & (b) Applying he Granger causaliy es, i can be shown ha up o 4 lags here is bilaeral causaliy beween he wo variables, bu beyond 4 lags here is no unilaeral or bilaeral causaliy. For example, a lag 3 we find ha Produciviy compensaion F = 3.84 (p value 0.034) Compensaion produciviy F = 3.97 (p value 0.084) A lag 4 we find ha Produciviy compensaion F =.7 (p value 0.0888) Compensaion produciviy F = 3.6 (p value 0.065) (c)for example, we could regress compensaion on produciviy and he unemploymen rae o see he (parial) effec of unemploymen ne of he produciviy effec. The resuls are as follows: Yˆ = 6.7834 + 0.6907X + 0.6680X 3 = (.8468) (33.34) (.7053) R = 0.9694; d = 0.47 where Y = compensaion, X = produciviy and X 3 = he unemploymen rae. All he esimaed coefficiens seem o be saisically 99
Basic Economerics, Gujarai and Porer significan. The posiive sign of he unemploymen rae variable may be couner-inuiive, unless one can mae an argumen ha higher unemploymen booss produciviy which hen leads o higher compensaion. Since he d saisic in he presen insance is quie low, i is possible ha his model eiher suffers from auocorrelaion or specificaion bias, or boh. 7.3 To perform he Sim's es, we ran Y (invesmen in plan and equipmen) on X (sales) wih four lead erms of X and obained he following resuls for regression (): Dependen Variable: Y Sample (adjused): 3 Included observaions: 0 afer adjusmens Variable Coefficien Sd. Error -Saisic Prob. C -9.770 4.45609-4.46707 0.0004 Y_LAG 0.564880 0.395 4.499 0.0006 Y_LAG -0.408687 0.85-3.4590 0.003 X 0.74564 0.085857 8.374 0.0000 R-squared 0.9990 Mean dependen var.9975 Adjused R-squared 0.990383 S.D. dependen var 50.06889 S.E. of regression 4.9008 Aaie info crierion 6.9793 Sum squared resid 385.7340 Schwarz crierion 6.396439 Log lielihood -57.9793 F-saisic 653.34 Durbin-Wason sa.367385 Prob(F-saisic) 0.000000 Dependen Variable: Y Sample (adjused): 3 8 Included observaions: 6 afer adjusmens Variable Coefficien Sd. Error -Saisic Prob. C -0.8799 6.58770-0.3475 0.8979 Y_LAG 0.454705 0.3078 4.066 0.0038 Y_LAG -0.3087 0.094089 -.64730 0.0533 X.3303 0.6007 7.65838 0.000 X_LEAD -0.60796 0.03959 -.78667 0.369 X_LEAD -0.33990 0.9479 -.668 0.447 X_LEAD3-0.08393 0.84305-0.54055 0.884 X_LEAD4-0.03588 0.4096-0.66003 0.873 R-squared 0.996843 Mean dependen var 96.08000 Adjused R-squared 0.994080 S.D. dependen var 40.3379 S.E. of regression 3.095684 Aaie info crierion 5.404748 Sum squared resid 76.66609 Schwarz crierion 5.7904 00
Basic Economerics, Gujarai and Porer Log lielihood -35.3798 F-saisic 360.88 Durbin-Wason sa.900403 Prob(F-saisic) 0.000000 ( ) m ( ). Applying equaion (8.7.9) o he firs model, we have F = RSS R RSS UR RSS UR n ( 385.734 76.666) 4 F = = 309.068 76.666 0 8 6.389 = 48.3763. ( ) Now for model (), we ran X (sales) on Y (invesmen in plan and equipmen) four lead erms of Y and obained he following resuls for regression (): Dependen Variable: X Sample (adjused): 3 Included observaions: 0 afer adjusmens Variable Coefficien Sd. Error -Saisic Prob. C.9458 6.45435 3.565994 0.006 X_LAG 0.343 0.44739 0.78753 0.4458 X_LAG 0.396 0.7547 0.785485 0.4436 Y 0.5436 0.46088.0893 0.0530 R-squared 0.99006 Mean dependen var 58.83 Adjused R-squared 0.98843 S.D. dependen var 57.5969 S.E. of regression 6.34870 Aaie info crierion 6.675049 Sum squared resid 6.9776 Schwarz crierion 6.87495 Log lielihood -6.75049 F-saisic 58.839 Durbin-Wason sa.6507 Prob(F-saisic) 0.000000 Dependen Variable: X Sample (adjused): 3 8 Included observaions: 6 afer adjusmens Variable Coefficien Sd. Error -Saisic Prob. C 4.65.83434 5.0359 0.000 X_LAG -0.300083 0.69853 -.07 0.984 X_LAG 0.65805 0.85.4696 0.800 Y 0.630487 0.30073.740378 0.054 Y_LEAD 0.347435 0.66734.08377 0.0707 Y_LEAD 0.630 0.96837 0.6470 0.5390 Y_LEAD3 0.3000 0.8064.030735 0.338 Y_LEAD4 0.3347 0.085047.568869 0.553 R-squared 0.998838 Mean dependen var 39.6748 0
Basic Economerics, Gujarai and Porer Adjused R-squared 0.9978 S.D. dependen var 47.9886 S.E. of regression.3694 Aaie info crierion 4.754949 Sum squared resid 40.037 Schwarz crierion 5.444 Log lielihood -30.03959 F-saisic 98.597 Durbin-Wason sa.764596 Prob(F-saisic) 0.000000 ( ) m ( ). Applying equaion (8.7.9) o he second model, we have F = RSS R RSS UR RSS UR n ( 6.978 40.033) 4 F = = 45.487 40.033 0 8 3.336 = 43.6. ( ) These resuls are relaively inconclusive because each of he F ess is very saisically significan. The reader should ry oher lead-lag srucures o see if his conclusion holds. 7.3 (a) EViews resuls are: Dependen Variable: LN_PC Sample: 960 995 Included observaions: 36 Variable Coefficien Sd. Error -Saisic Prob. C.5593 0.75579 8.838807 0.0000 LN_PDI 0.85593 0.0489 57.75899 0.0000 LTI 0.004883 0.00096 5.33454 0.0000 R-squared 0.997495 Mean dependen var.4654 Adjused R-squared 0.997344 S.D. dependen var 0.43076 S.E. of regression 0.00 Aaie info crierion -4.697689 Sum squared resid 0.0665 Schwarz crierion -4.56579 Log lielihood 87.55840 F-saisic 657.65 Durbin-Wason sa.8645 Prob(F-saisic) 0.000000 (b) An issue wih esimaion of he above model is ha here could be a spurious causaliy in effec. For example, he ineres rae, alhough a facor in privae consumpion expendiure, could also be affeced by consumpion. This is basically an issue of Granger-causaliy. One approach o address his would be o creae a muliple-equaion sysem and loo a he model from muliple perspecives. This will be addressed more in Chaper 0. 7.33 The model developmen here is lef o he reader. 0