Solutions to Exercises in Chapter 12

Transcription

1 Chaper in Chaper. (a) The leas-squares esimaed equaion is given by (b)!i = Y 0.8 R R = 0.86 (.5) (0.07) (0.6) Boh b and b 3 have he expeced signs; income is expeced o have a posiive effec on invesmen whereas an increase in he ineres rae should reduce invesmen. The sandard errors for b and b are relaively small suggesing ha he corresponding coefficiens are significanly differen from zero. However, he sandard error of b 3 is large, yielding a raio ha is less han wo. Based on his sandard error, we can quesion wheher we should include R in he equaion, alhough economic heory suggess R should have a srong influence on I. The plo of he leas squares residuals in Figure. reveals a few long runs of negaive and posiive residuals, suggesing he exisence of auocorrelaion. e Figure. Residuals for Invesmen Equaion (c) (d) In his conex, he Durbin-Wason es is a es for H 0 : ρ = 0 agains H : ρ > 0 in he firsorder auoregressive model e = ρe + v. We find he compued value for he Durbin- Wason saisic is d = 0.85 and he p-value of he es if P(d < 0.85) = Because he p-value is less han 0.05, we rejec H 0 and conclude ha auocorrelaion does exis. The esimaor for ρ inroduced on p. 8 of he ex is no he one uilized by all sofware. Differen programs ofen employ slighly differen esimaors ha yield differen esimaes. Consequenly, for his quesion we repor hree esimaes for ρ and he hree corresponding generalized leas-squares esimaed equaions. The resuls from using SHAZAM, EViews and SAS are:

2 Chaper (e)!i = Y 0.85 R ˆρ = (SHAZAM) (.86) (0.) (0.079) Iˆ = Y 0.96R ρ= ˆ 0.66 (EViews) (3.73) (0.7) (0.080)!I = Y 0.85 R!ρ = (SAS) (.90) (0.5) (0.08) Comparing hese resuls wih hose form par (a), we find ha here has been lile change in he coefficien esimaes, bu a considerable change in he sandard errors. The sandard error on he coefficien of Y has increased, suggesing ha, if we did no correc for auocorrelaion, our confidence inerval for β would be oo narrow, giving us a false sense of he reliabiliy of our esimae. The opposie has occurred for β 3. Here he sandard error has dropped afer correcing for auocorrelaion. From he resuls in par (a) we migh be misled ino hinking ha ineres rae has no impac on invesmen (he esimaed coefficien is no significan). Afer correcing for auocorrelaion we have a relaively narrow confidence inerval ha does no include zero. Given he nex year values of Y and R are Y T+ = 36 and R T+ =, he appropriae forecas using he SHAZAM resuls is I! = β! + β! Y + β! R + ρ!~ e = (36) 0.85() (.53) = T+ T+ 3 T+ T Wih EViews and SAS, he resul urns ou o be 3.66 and 3.36, respecively. If auocorrelaion is ignored, our predicion is!it+ = b + byt+ + b3rt+ = (36) 0.8() = There is no a large difference beween he wo predicions.. From he equaion for he AR() error model we have from which we ge σ ρ σ σ To find Eee ( ) ( e) = ρ ( e ) + ( v) + ρ ( e v) var var var cov, = + +, or e( ) e e v 0 σ e σv = ρ σ ρ = σ, and hence we noe ha ee = ρ e + e v. Taking expecaions, ( ) ρee ( ) Eee = + 0 = ρσ e Similarly, ee = ρ e e + e v, and ( ) ( ) 0 ( ) Eee = ρee e + = ρeee = ρ σ e v

3 3 Chaper.3 (a) The leas-squares esimaed equaion is (b) (c) ln( JV ) = ln(u ) R = (0.89) (0.555) Using he value c =.07, a 95% confidence inerval for β is b ± c se(b ) = (.93,.890) The value of he Durbin-Wason saisic is d =.09. In erms of is p-value, we find ha P(d <.09) = Since his p-value is less han 0.05, we rejec H 0 and conclude ha posiive auocorrelaion exiss. The exisence of auocorrelaion means he original assumpion for e, ha he e are independen, is no correc. This problem also causes he confidence inerval for β in par (a) o be incorrec, meaning we will have a false sense of he reliabiliy of he coefficien esimae. Afer correcing for auocorrelaion, he esimaed equaion is ln( JV ) = ln(u )!ρ = 0.7 (SHAZAM) (0.377) (0.3) ^ ln( JV) = ln( U) ρ= ˆ 0.86 (EViews) (0.87) (0.3) ln( JV ) = ln(u )!ρ = 0.38 (SAS) (0.37) (0.7) The resuls for SAS, EViews and SHAZAM differ slighly because hey use differen esimaors for ρ. The 95% confidence inervals for β from SHAZAM, EViews and SAS are, respecively, (.87,.360), (.875,.35) and (.879,.353). These confidence inervals are slighly narrower han ha given in par (a). A direc comparison wih he inerval in par (a) is difficul because he leas squares sandard errors are incorrec in he presence of AR() errors. However, given he change in sandard errors is no grea, and ha we know leas squares is less efficien, one could conjecure ha he leas squares confidence inerval is narrower han i should be, implying unjusified reliabiliy.. (a) The marginal funcions are obained by differeniaing he oal funcions. Tha is, dr ( ) dc ( ) mr = = β + βq mc = = α + α3q dq dq (b) If mc = mr, hen β + β = α + α soluion q α = q 3q, and q ( 3) = β ( β α3) β α α β, leading o he

4 Chaper (c) The leas squares esimaed models (wih sandard errors in parenheses) are r i = 7.8 qi 0.50 q i (.5) (0.035) r i = qi q i (77.) (9.5) (0.0889) The saisical models ha are appropriae for he above esimaion are r i = β q i + β q i + e i c i = α + α q i + α 3 q i + e i where e i and e i are boh independen random variables wih zero means and consan variances σ and σ, respecively. The profi maximizing level of oupu suggesed from he leas squares esimaes is or q 0. q = ( ) = 0.3 (d) For q = 0, oal cos and oal revenue predicions in each of he nex hree monhs are given by r = 7.8(0) T + h 0.50(0) = 3679 h =,,3 (e) c = (0) + T + h 0.768(0) = 555 h =,,3 Thus, a predicion for profi is!π = = 85. The Durbin-Wason saisics and p-values for each of he equaions are Equaion DW-value p-value revenue cos (f) Boh p-values are subsanially below a nominal significance level of We herefore conclude ha he errors in boh equaions are auocorrelaed. The answers o he remaining pars of his quesion could depend on he compuer sofware package ha is being used. Differen sofware packages ofen use differen esimaors for ρ and varying esimaes of ρ flow hrough o he generalized leas squares esimaes and heir sandard errors, as well as predicions of fuure values. From his par on we repor SHAZAM, EViews and SAS esimaes. Generalized leas squares esimaes of he coefficiens appear in he able below. Sandard errors are in parenheses.

5 5 Chaper Equaion r c Variable SHAZAM EViews SAS SHAZAM EViews SAS consan 37.7 (608.) 35.7 (66.0) 38.6 (633.) qi 7.9 (5.90) 7.58 (8.0) 75.8 (.60) (7.90) (8.00) 5.9 (8.63) qi 0.53 (0.088) (0.08) (0.055) 0.0 (0.0) 0.5 (0.07) (0.05) ρ (g) The profi maximizing level of oupu suggesed by he SHAZAM resuls in par (f) is q = = 8. ( ) or, q 8. Using he EViews and SAS resuls in par (f) we obain q 8 and q 9, respecively. (h) In his case, because he errors are assumed auocorrelaed, he oal revenue and oal cos errors for monh 8 have a bearing on he predicions, and he predicions will be differen in each of he fuure hree monhs. For he oal revenue funcion he SHAZAM esimaed error for monh 8 is ~e,8 = 835 [7.9(83) 0.533(83) ] = Therefore, for q = 8, oal revenue predicions for he nex 3 monhs (using he SHAZAM resuls) are given by r T + = r 9 = 7.9(8) 0.533(8) + (0.8887) ( ) = 50 r T + = r 50 = 7.9(8) 0.533(8) + (0.8887) ( ) = 387 r T +3 = r 5 = 7.9(8) 0.533(8) + (0.8887) 3 ( ) = 598 Using he EViews resuls he corresponding predicions are r 9 = 0979 r 50 = 090 r 5 = 95 Using he SAS resuls he corresponding predicions are r 9 = 87 r 50 = 899 r 5 = 6 For he oal cos funcion, he SHAZAM error for he las sample monh is ~ e 8, = (83) 0.0(83) = 77.5

6 6 Chaper Therefore, using he SHAZAM resuls, oal cos predicions for q = 8 are c T + = c 9 = (8) + 0.0(8) + (0.39) (77.5) = 56 c T + = c 50 = (8) + 0.0(8) + (0.39) (77.5) = 5306 c T +3 = c 5 = (8) + 0.0(8) + (0.39) 3 (77.5) = 566 The corresponding predicions from EViews are c 9 = 5630 c 50 = 53 c 5 = 56 The corresponding predicions from SAS are c 9 = 5569 c 50 = 57 c 5 = 56 Profis for he monhs of 9, 50 and 5 (obained by subracing oal cos from oal revenue) are (SHAZAM)!π 9 = 558!π 50 = 608!π 5 = 63 (EViews)!π 9 = 539!π 50 = 5779!π 5 = 603 (SAS)!π 9 = 598!π 50 = 667!π 5 = 706 Because ~ e T is negaive, and is impac declines as we predic furher ino he fuure, he oal revenue predicions become larger he furher ino he fuure we predic. The opposie happens wih oal cos; i declines because ~ e T is posiive. Combining hese wo influences means ha he predicions for profi increase over ime. These predicions are, however, much lower han 85, he predicion for profi ha was obained when auocorrelaion was ignored. Thus, even alhough auocorrelaion has lile impac on he opimal seing for q, i has considerable impac on he predicions of profi. This impac is caused by a relaively large negaive residual for revenue, and a relaively large posiive residual for cos, in monh 8..5 (a) For he AR() error model e = ρe + v, we are esing H 0 : ρ = 0 agains H : ρ > 0. The compued value of he Durbin-Wason saisic is and is p-value is Since < 0.0, we rejec H 0 and conclude ha he errors are auocorrelaed. (b) The leas squares resuls yield b = and se(b ) = The generalized leas squares resuls from SHAZAM are β! = and se( β! ) = Using EViews, we obain β ˆ = and se( β ˆ ) = 0.9. Using SAS, we obain β! = and se( β! ) = The differen resuls arise from using differen esimaors for ρ, and because EViews auomaic command drops he firs observaion; he oher wo sofware packages do no. In small samples dropping he firs observaion can make a subsanial difference. This is one of hose occasions. The differen esimaes for ρ are!ρ =

7 7 Chaper (c) (SHAZAM), ρ= ˆ 0.68 (EViews) and!ρ = 0.69 (SAS). Using c =.5 ( degrees of freedom), he wo confidence inervals for β are: LS = ( 0.63, 0.308) GLS = ( 0.88, 0.6) (SHAZAM) GLS = ( 0.77, 0.86) (EViews) GLS = ( 0.85, 0.7) (SAS) The wider inerval obained using GLS esimaes suggess ha incorrec sandard errors have made he leas squares inerval oo narrow. The leas squares inerval suggess our informaion abou β is more reliable han i acually is. To answer his quesion we need o predic uni cos for a cumulaive producion value of Our predicion will be differen depending on wheher or no we assume he exisence of auocorrelaed errors. (i) Wihou auocorrelaed errors our predicion is compued from he leas squares resuls ln (!u 97 ) = b + b ln(3800) = (8.76) =.83989!u 97 = exp(.83989) = 7.39 (ii) When he exisence of auocorrelaed errors is recognized, he relevan predicion using he SHAZAM esimaed value for ρ is ( u ) ln! = β! + β! ln( 3800) + ρ!~ e = (8.76) ( ) =.79659!u 97 = exp(.79659) = Using he EViews esimaed value for ρ, he resuls are ln (!u 97 ) =.763 and!u 97 = Using he SAS esimaed value for ρ, he resuls are ln (!u 97 ) = and!u 97 = Since uni cos is 6.63 in 970, he leas squares predicion suggess cos will be greaer in 97, whereas he generalized leas squares predicion suggess cos will be less. Inuiively, we would expec cos o decline wih he increase in cumulaive producion. The leas squares resuls do no predic a decline because he residual for 970 is negaive; he acual value for 970 is below he value ha would be prediced for ha year. The generalized leas squares resuls recognize he negaive residual in 970 and make allowance for he fac ha anoher negaive residual is likely for (a) The leas squares esimaed equaion is (b)!q = P I F R = 0.79 (0.70) (0.83) (0.00) (0.000) The calculaed -value for he hypohesis H 0 : β = 0 is.5, indicaing ha b is no significanly differen from zero. From his resul one migh be emped o conclude ha price is no a relevan variable for explaining he demand. However, a close look a he daa shows ha here has been lile variaion in price. Thus, i is more likely ha he

8 8 Chaper variaion in price is oo small o ge an accurae esimae of is effec; no ha price is unimporan. (c) The Durbin-Wason saisic is.0. Wih k = and T = 30 we have d L =. and d U =.650. Since d =.0 < dl, on he basis of his es we conclude ha auocorrelaion exiss. This decision is confirmed by a p-value of The Lagrange muliplier es gives =.077 wih a p-value of Thus, a a 5% significance level, he Lagrange muliplier es does no rejec a null hypohesis of no auocorrelaion. However, i is close o rejecion. Gahering furher evidence from he plo of leas squares residuals in Figure., we see ha hese residuals end o exhibi runs of posiive and negaive values. Thus, overall, here is evidence of auocorrelaed errors. e E-0 8E-0 E-0 0E+00 -E E-0 Figure. Residuals for Ice Cream Equaion (d) Using SHAZAM for he auocorrelaion correcion we find ha!ρ = and he esimaed equaion is!q = P I F (0.867) (0.835) (0.005) (0.0006) Wih EViews,!ρ = 0.007, and he esimaed equaion is Qˆ = P I F (0.998) (0.895) (0.006) (0.0006) Wih SAS,!ρ = , and he esimaed equaion is!q = P I F (0.895) (0.856) (0.005) (0.0005) The reason ha he esimaed equaions from EViews and SHAZAM are subsanially differen, despie he fac ha hey use virually idenical esimaes of ρ, is ha EViews drops he firs observaion when esimaing he β s; SHAZAM does no.

9 9 Chaper (e) In his case we find ha, in addiion o! β,! β 3 is also no significanly differen from zero. Thus, some doub is cas on he relevance of income in he demand for ice cream. Again, i may be ha he variaion in income is inadequae o capure is effec..7 (a) The Durbin-Wason p-value compued from he residuals for he ransformed model (from SHAZAM) is Thus, if we were esing H 0 : θ = 0 agains H 0 : θ > 0 a a 5% level of significance, where θ is he auocorrelaion parameer in he model v = θv + u, we would rejec H 0. The correcion for auocorrelaion in Exercise.6 does no seem o have cured he problem. A similar resul is obained wih SAS. For EViews he Durbin-Wason saisic of.55 falls in he inconclusive region, suggesing here is sill a poenial problem. (b) If we combine he model in par (a) wih he radiional AR() error specificaion, we have e = ρ e + v v =θ v + u where he u are independen random errors wih u ~(, ). Noe ha 0 σ u and hence e = ρe + v θv = θe θρe from which we obain ( ) e = ρe + θe θρe + u = ρ + θ e θρe + u We call his model an AR() error process. Under his assumed process our SHAZAMesimaed model is!q = P I F (0.95) (0.756) (0.00) (0.0007) The EViews esimaed model is Qˆ = P I F (0.30) (0.85) (0.006) (0.0007) The SAS-esimaed model is!q = P I F (0.905) (0.8637) (0.00) (0.0006) The esimaed model appears very sensiive o he way in which he AR() parameers are esimaed. We again find ha β! and β! 3 are no significanly differen from zero. Also, he SHAZAM resuls give β! 3 < 0; a negaive effec of income on he demand for ice cream does no seem plausible.

10 0 Chaper.8 (a) The esimae for he AR() parameer ρ is T ee ˆˆ 5553 = ρ= ˆ = = T 6336 eˆ = The approximae Durbin-Wason saisic is ( ˆ) 0.8 * d = ρ =. (b) Based on T = 90 and K = 5, dl =.566 and du =.75. Since d * is less han d L we conclude ha posiive auocorrelaion is presen. The esimae for he AR() parameer of ρ is T ee ˆˆ 6 = ρ= ˆ = = T 9 eˆ = The approximae Durbin-Wason saisic is ( ˆ).8990 * d = ρ =. Based on T = 90 and K = 6, dl =.5 and du =.776. Since d * > d U, we canno rejec a null hypohesis of no posiive auocorrelaion..9 (a) From he residual plos he residuals end o exhibi runs of posiive and negaive values, suggesing auocorrelaed errors. The Durbin-Wason saisic is.. Wih T = 6 and K = we obain dl =.30 and du =.6. Since he value of he Durbin-Wason saisic is less han d L, we conclude ha here is evidence of posiive auocorrelaion. (b) The esimaes, heir sandard errors and he confidence inervals obained from leas squares and generalised leas squares (GLS) are presened in he able below. For he leas squares mehod T = 6, K = and he criical value is 0.05 =.06. For GLS, T = 5 and K =, so 0.05 =.069. The esimaes obained from leas squares and GLS are very similar. However, he sandard errors from GLS are much higher han from leas squares, resuling in GLS confidence inervals which are much wider han hose obained from leas squares. Hence, ignoring auocorrelaion means he esimaes are less reliable han hey appear. Esimae (se) β (.66) β.766 (0.975) Leas squares Confidence inerval Esimae (se) (-60.9, -55.5) (9.7) (.759, 6.770).388 (.57) GLS Confidence inerval (-7.5, 53.75) (.3, 7.65)

11 Chaper (c) Because of he evidence of auocorrelaion he forecass are based on he GLS resuls. DISP =β ˆ +β ˆ DUR +ρˆ e" = (90) ( 77.) = DISP =β ˆ +β ˆ DUR +ρˆ e" = (95) 0.86 ( 77.) = DISP =β ˆ +β ˆ DUR +ρˆ e" = (9) 0.86 ( 77.) = (a) The LS esimaed equaion is ln( POW ) = ln( PRO ) (0.7) (0.0005) ( ) (0.0899) (b) The posiive sign for b and he negaive sign for b 3, and heir relaive magniudes, sugges ha he rend for ln(pow) is increasing a a decreasing rae. A posiive b implies he elasiciy of power use wih respec o produciviy is posiive. The value of he Durbin-Wason saisic is 0.39 wih a very small corresponding p-value of This suggess a problem of auocorrelaion in he errors. The able below compares he leas squares resuls wih hose generalised leas squares ones obained using SHAZAM, EViews and SAS. Apar from β, he leas squares and GLS esimaes are similar, alhough he EViews esimaor ha drops he firs observaion has led o some discrepancies. The mos noiceable change is he increase in sandard errors when GLS is used. Ignoring auocorrelaion conveys a false sense of reliabiliy. GLS LS EViews SAS SHAZAM β (0.7) (0.90) (0.89) (0.805) β (0.0005) (0.005) (0.00) (0.00) β ( ) (0.0000) ( ) ( ) β (0.0899) 0.96 (0.055) (0.0) (0.0)

12 Chaper (c) The p-values (from SHAZAM) for esing he hypohesis H0 : β = are 0.87 and 0.79 before and afer correcing for auocorrelaion, respecively. We do no rejec H0 in boh cases. Correcing for auocorrelaion has led o a large change in he p-value, bu he es decision is sill he same.. The model wih he resricion β = imposed is Or i can be wrien as ln( POW) = β + β + β 3 + ln( PRO) + e POW ln = β + β + β + PRO 3 e The esimaed equaion wih he resricion β = imposed is POW ln = PRO (0.07) (0.00) ( ) (SHAZAM) ln POW = PRO (0.003) (0.005) (0.0000) (EViews) ln POW = PRO (0.063) (0.00) ( ) (SAS) We can use he exponenial of he wihin sample predicion from his equaion o plo he rend for ( POW PRO ) as in he graph below. I appears ha he rend is increasing a a decreasing rae over ime...0 POW_PRO T

13 3 Chaper. (a) The model where β =α +δd and β =α +δ D can be wrien as ln( POW ) =α +δ D +β + β +α ln( PRO ) +δ D ln( PRO ) + e 3 The dummy variable D is inroduced o capure he srucural change in power use (if any) afer he year 985. The LS esimaion resul is ^ ln( POW) = D ln( PRO) (0.65) (.050) (0.0007) ( ) (0.3) 0.05D ln( PRO ) (0.) Boh of he dummy variable parameers esimaes δˆ ˆ and δ are no significanly differen from zero. To es for auocorrelaion we obain he Durbin-Wason saisic as 0.83 wih a very small p-value of , suggesing he presence of auocorrelaion. Afer allowing for auocorrelaion he GLS esimaed equaion is ^ ln( POW) = D ln( PRO) (0.576) (.30) (0.00) ( ) (0.53) 0.93D ln( PRO ) (0.36) ^ ln( POW) = D ln( PRO) (0.590) (.333) (0.007) (0.0000) (0.79) 0.66D ln( ) (0.08) PRO (SHAZAM) (EViews) ^ ln( POW) = D ln( PRO) (0.5793) (.6) (0.00) ( ) (0.60) 0.78D ln( PRO ) (0.373) (SAS) The esimaes for δ and δ change subsanially afer allowing for auocorrelaion. The ohers have changed slighly. (b) For esing H0 : δ =δ = 0 he es oucomes appear in he able below. χ Tes F Tes χ Value p-value F value p-value LS GLS (EViews) GLS (SAS) GLS (Shazam)

14 Chaper Allowing for auocorrelaion changes he es values considerably. Before correcing for auocorrelaion, he null hypohesis is bordering on rejecion a a 5% significance level. Afer correcing for auocorrelaion here is no suppor for he null hypohesis. (c) If 0 : 0 H δ =δ = is rejeced, we can conclude here is a srucural change in he paern of power use from he year 985.