Solutions to Odd Number Exercises in Chapter 6

Transcription

1 1 Soluions o Odd Number Exercises in 6.1 R y eˆ y 6.3 From eˆ ( T K) ˆ R 1 1 SST SST SST (1 R ) 55.36(1.7911) we have, ˆ T K ( ) 6.5 y ye ye y e 1 1 Consider he erms e and xe b b x e y e b e b x e y e e y b b x y Tb b x xe x y b bx xy b x b x The las expressions in each of hese equaions become zero from he normal equaions ha are used o solve for he leas squares esimaors. Subsiuing e = and xe = back ino he original equaion, we obain y ye. 6.7 (a) Knowledge of he relaionship beween he rae of change of wages and he unemploymen rae is very imporan for governmen policy makers and unions. Typically, governmens like o keep he lid on inflaion, bu he cos of doing so may be increasing unemploymen. Unions are always aemping o negoiae pay rises for heir employees. However, hey mus be wary of doing so if wage rises mean ha fewer of heir workers can ge jobs. An economic model for relaing he rae of change in wages o unemploymen was derived in Secion I is called he Phillips' curve and is given by 1 %w 1 u A corresponding saisical model is %w 1 1 e u

2 where some saisical assumpions are needed for he random error e. We assume he e are independen normally disribued random variables wih zero mean and consan variance. To esimae his model, we use a se of 18 observaions on w and u for he period 1949 o The esimaion mehod used is he linear leas squares mehod regressing %w on 1/u. This mehod gives he esimaed equaion as % w 1 = R =.393 (. 355) (. 838) u where sandard errors are in parenheses. We would expec 1 < and >. Thus, from he esimaed equaion, b 1 and b give he expeced signs. The sandard error of b 1 is.355 which is relaively large and leads o b 1 being saisically insignifican. The esimae for b is saisically significan, bu, as we discovered in Exercise 6.(h), is large sandard error leads o a wide confidence inerval ha conveys lile economic informaion. Some of he economic implicaions were discussed in he answers o Exercise 6.. (b) From he Economic Repor o he Presiden we can consruc he following able where u is he unemploymen rae for all workers, w is average gross hourly earnings in curren dollars for he oal privae non-agriculural secor, and % w 1( w w 1) / w 1. Year w u %w (c) The esimaed equaion for he Phillips' curve using he daa for he years 1975 o 1983 is % w 1 = R =.635 ( ) ( ) u where sandard errors are in parenheses. The esimae b 1 =.8311 suggess ha he annual wage increase will never drop below.83%, even as he unemploymen rae becomes very large. This oucome may be unrealisic; we may expec workers o accep a drop in real wages for very high unemploymen raes. If so, b 1 should be negaive. The very high sandard error of b 1, and is consequen insignificance, do no rule ou he possibiliy of 1 being negaive. The esimae b = wih se(b ) = is economically feasible and significan. However, he resuling confidence inerval is sill wide. I conveys lile informaion, alhough he large value of b suggess wage changes are very responsive o he unemploymen rae.

3 3 6.9 In he learning curve model ln( u) 1 ln( q), 1 is he logarihm of he uni cos of producion for he firs uni produced and is he elasiciy of uni cos wih respec o cumulaive producion. A 1% change in he cumulaive producion leads o a % change in he uni cos equal o.

4 The resuls are summarised in he following able. Sandard errors are in parenheses. Changing he unis of y only changes boh b 1 and b, making hem one hundred imes smaller. Changing he unis of x only influences only b, making i 1 imes bigger. When boh x and y are scaled b 1 changes bu no b. Regression b 1 b (i) y on x (.14) (35) (ii) y on x* (.14) (3.5) (iii) y* on x (.14) (35) (iv) y* on x* (.14) (35) 6.13 To es H : normally disribued errors for he preferable equaion, equaion 3, we need values of skewness and kurosis. They are S = and k = The es saisic is T ( k 3) JB S The hypohesis H is rejeced if JB () for a given significance level. Since () a he 5 level of significance, we do no rejec H. Therefore, here is insufficien evidence o conclude ha he normal disribuion assumpion is unreasonable. Differen sofware packages can use differen esimaors for skewness and kurosis; and some repor excess kurosis ( k 3) as he kurosis. For example, SAS yields S.8741 and k 3.411, leading o a es value of JB (a) For households wih 1 child wfood ln( oexp) (41) (9) se For households wih children: (5.19) (-16.7) R.33 wfood ln( oexp) (365) (8) se (6.1) (-16.16) R 6

5 5 (b) In erms of we would expec a negaive value because as he oal expendiure increases he food share should decrease. Boh esimaions give he expeced sign. The sandard errors for b 1 and b from boh esimaions are relaively small resuling in high values of raios and significan esimaes. For households wih 1 child, he average oal expendiure is and b1b ln( oexp) ln(94.848) 1 ˆ.5461 b b ln( oexp) ln(94.848) 1 For households wih children, he average oal expendiure is and b1bln( oexp) ln(11.17) 1 ˆ.6366 b b ln( oexp) ln(11.17) 1 Boh of he elasiciies are less han one; herefore, food is a necessiy.

6 WFOOD X1 X1 Figure 6.4a Figure 6.4b (c) Figures 6.4a and 6.4b are he fied curve and he residual plos for households wih 1 child. The funcion linear in wfood and ln(oexp) seems o be an appropriae one. However, he observaions vary considerably around he fied line, consisen wih he low R value. Also, he absolue magniude of he residuals appears o decline as ln(oexp) increases. In Chaper 11 we discover ha such behavior suggess he exisence of heeroskedasiciy. Figures 6.5a and 6.5b are plos of he fied equaion and he residuals for households wih children. They are similar o he ones wih 1 child, indicaing ha he funcional form is a reasonable one. The values of JB for esing H : errors are normally disribued are and for households wih 1 child and children, respecively. Since boh values are greaer han he criical () 5.991, we rejec H. The p-values obained are 5 and 41, respecively, confirming ha H is rejeced. We conclude ha for boh cases he errors are no normally disribued. (Using SAS esimaes for skewness and excess kurosis, he JB values are 11.6 and 6.65, respecively.) WFOOD X Figure 6.5a Figure 6.5b X

7 (a) For households wih 1 child wfuel ln( oexp) (17) (48) For households wih children: (14.3) (-15) R.1458 se (b) wfuel ln( oexp) (198) (43) se (15.) (-1.71) R.115 We expec fuel o be a necessary good and as oal expendiure increases he fuel share should decrease. Tha is, we expec he sign of o be negaive. Boh esimaions give he expeced sign. The sandard errors for b 1 and b from boh esimaions are relaively small resuling in high values of raios. For households wih 1 child, he average oal expendiure is and b1bln( oexp) ln(94.848) 1 ˆ.4494 b b ln( oexp) ln(94.848) 1 For households wih children, he average oal expendiure is and b1bln( oexp) ln(11.17) 1 ˆ.4647 b b ln( oexp) ln(11.17) 1 Boh of he elasiciies are less han one; herefore, fuel is a necessiy WFUEL X1 Figure 6.6a Figure 6.6b (c) Figures 6.6a and 6.6b are he fied curve and he residual plos for households wih 1 child. The plo of he acual observaions and he fied equaon is in Figure 6.6a. The funcional form is reasonable in he sense ha i is difficul o sugges an alernaive funcion which will be a beer fi o he scaer of poins. On he oher hand, he funcion provides a poor explanaion of variaion in he budge share of fuel. Consisen wih he low R, he observaions vary widely around he fied line. Also he variaion of he observaions around he fied line decreases as he oal X1

8 8 expendiure increases. The posiive residuals vary in magniude from zero o.3 whereas he range of he negaive residuals is zero o.1, suggesing a highly skewed error disribuion. Figures 6.7a and 6.7b are he fied curve and he residual plos for households wih children. They are similar o he ones wih 1 child. While i is difficul o sugges a funcional form which would fi beer han he linear-log one, he funcion is no a good one for explaining variaion in he fuel budge share. Perhaps a muliple regression model wih addiional explanaory variables would be an improvemen WFUEL X Figure 6.7a Figure 6.7b X The values of JB for esing H : errors are normally disribued are and for households wih 1 child and children, respecively. Since boh values are greaer han he criical () 5.991, we rejec H. The p-values are for boh cases confirming ha H is rejeced. We conclude ha for boh cases he errors are no normally disribued (a) Figures 6.1a, 6.1b, 6.1c and 6.1d are he plos of observaions of y agains x, ln(y) agains ln(x), y agains ln(x) and ln(y) agains x, respecively. The funcional forms o choose from are specified in par (b). They are all linear funcions of y and x or he logarihms of hem. The preferable form is he one where he plo of he observaions is bes represened by a linear line. This is he plo of y and x in Figure 6.1a. In all he oher figures a line wih some curvaure would be a beer fi. Thus, he chosen funcional form is he equaion y 1x e. (b) (i) (ii) yˆ x (3) (.691) se (4.5) (.667) R.9753

9 9 ln( y) ln( x) (.3356) (55) se (.5114) (6.6418) R.774 (iii) (iv) yˆ ln( x) (.76) (454) se (5.644) (4.13).5663 ln( y) x (.1556) (3.5589) se (-11.33) (6.5488) R.7674 To check wheher he level of arsenic in he waer influences he level of arsenic in he oenails we es each equaion for wheher,, and are zero. We rejec an H if a raio is greaer han he absolue criical value. A level of significance 5, and 13 degrees of freedom, The calculaed raios for he hypoheses from each equaion are all higher han he criical value. We herefore rejec H and conclude ha here is evidence o sugges ha he level of arsenic in waer does influence he level of arsenic in he oenails. (c) The plos of he residuals from equaion (i) o (iv) are presened in Figures The residuals in Figures 6.1b and 6.13b display a rough U-shape, while hose in Figure 6.14b exhibi a rough invered U-shape. Thus, in hese cases, he residuals for small and large values of x or ln(x) have signs differen o he residuals ha correspond o middle values of x or ln(x). Such paerns sugges inappropriae funcional forms. Since he residuals for he linear funcion do no display such a paern, he plos suppor our choice of funcion in par (a). R Y X Figure 6.11a Figure 6.11b X

10 LNY LNX LNX Figure 6.1a Figure 6.1b Y LNX LNX Figure 6.13a Figure 6.13b LNY X Figure 6.14a Figure 6.14b X

11 (a) The esimaed equaions for dishwasher shipmens as a funcion of durable goods expendiure and as a funcion of privae residenial invesmen are below. DISH DISH DUR (95.5) (.548) R RES (534.7) (1.95) R.7465 The equaion ha gives he beer predicor is he equaion ha has he higher R. In his case i is he equaion wih DUR as he explanaory variable. Hence, DUR is he beer predicor. (b) (c) DISH DISH DUR (8.) RES (67.) 4119 Since he acual DISH , for his fuure observaion RES is a beer predicor han DUR. The residual plos for he wo equaions using DUR and RES as he explanaory variables are in Figures 6.15a and 6.15b, respecively. The residuals appear o be correlaed. There is a endency for posiive residuals o follow posiive residuals and negaive residuals o follow negaive residuals DISH Residuals Figure 6.15a using DUR DISH Residuals Figure 6.15b using RES