σ may be counterbalanced by a larger

Transcription

1 Questons CHAPTER 5: TWO-VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING 5.1 (a) True. The t test s based on varables wth a normal dstrbuton. Snce the estmators of β 1 and β are lnear combnatons of the error u, whch s assumed to be normally dstrbuted under CLRM, these estmators are also normally dstrbuted. (b) True. So long as E(u ) = 0, the OLS estmators are unbased. No probablstc assumptons are requred to establsh unbasedness. (c) True. In ths case the Eq. (1) n App. 3A, Sec. 3A.1, wll be absent. Ths topc s dscussed more fully n Chap. 6, Sec (d) True. The p value s the smallest level of sgnfcance at whch the null hypothess can be rejected. The terms level of sgnfcance and sze of the test are synonymous. (e) True. Ths follows from Eq. (1) of App. 3A, Sec. 3A.1. (f) False. All we can say s that the data at hand does not permt us to reject the null hypothess. (g) False. A larger σ may be counterbalanced by a larger s only f the latter s held constant, the statement can be true. x. It (h) False. The condtonal mean of a random varable depends on the values taken by another (condtonng) varable. Only f the two varables are ndependent, that the condtonal and uncondtonal means can be the same. () True. Ths s obvous from Eq. (3.1.7). (j) True. Refer of Eq. (3.5.). If X has no nfluence on Y, ˆβ wll be zero, n whch case y = uˆ. 33

2 5. ANOVA table for the Food Expendture n Inda Source of varaton SS df MSS Due to regresson (ESS) Due to resdual (RSS) TSS F = = wth df = 1 and 53, respectvely Under the hypothess that there s no relatonshp between food expendture and total expendture, the p value of obtanng such an F value s almost zero, suggestng that one can strongly reject the null hypothess. 5.3 (a) se of the ntercept coeffcent s 6.153, so the t value under H 0 : β 1 = 0, s: =.353. Wth 3 degrees of freedom, the cutoff for the 5% level of sgnfcance s.04 (usng 30 d.f. snce 3 s not n the table n the textbook s appendx), so the ntercept IS statstcally sgnfcant. (b) se of the slope coeffcent s , so the t value under H 0 : β = 0, s: 0.00 = As noted n part a, the cutoff for the 5% level of sgnfcance s.04, so the slope IS statstcally sgnfcant. (c) The 95% confdence nterval for the true slope coeffcent would be: 0.00 ± (.04) ( ) , (d) If per capta ncome s $9000, the mean forecast value of cell phones demanded s (9000) = per 100 persons. For the predcton confdence nterval, we frst need to compute var( Ŷ 0 )= σ 1 ( n + X 0 X ) x. var( Ŷ 0 )= ( ) + = Now the 34 1, 668, 91,885 confdence nterval s gven as 34

3 Pr Ŷ 0 t α se( Ŷ 0 ) Y 0 Ŷ0 + t α se ( Ŷ 0 ) = 1 α = Pr ( ) Y ( ) = , Verbally, the hypothess states that there s no correlaton between the two varables. Therefore, f we can show that the covarance between the two varables s zero, then the correlaton must be zero. 5.5 (a) Use the t test to test the hypothess that the true slope coeffcent ˆ β s one. That s obtan: t = = = 0.81 se( ˆ β ) For 38 df ths t value s not sgnfcant even at α = 10%. The concluson s that over the sample perod, IBM was not a volatle securty (b) Snce t = =.405, whch s sgnfcant at the two percent level of sgnfcance. But t has lttle economc meanng. Lterally nterpreted, the ntercept value of about 0.73 means that even f the market portfolo has zero return, the securty's return s 0.73 percent. 5.6 Under the normalty assumpton, ˆβ s normally dstrbuted. But snce a normally dstrbuted varable s contnuous, we know from probablty theory that the probablty that a contnuous random varable takes on a specfc value s zero. Therefore, t makes no dfference f the equalty s strong or weak. 5.7 Under the hypothess that β = 0, we obtan ˆ β ˆ β x ˆ β x t = = = se( ˆ β ˆ ) σ y (1 r ) ( n ) uˆ y (1 r ) because ˆ σ = =, from Eq.(3.5.10) ( n ) ( n ) ˆ β x ( n ) = y (1 r ) 35

4 But snce r x = ˆ β y r ( n ) ˆ β Thus, t = = (1 r) ˆ σ Emprcal Exercses, then x r = ˆ β, and x β r ( n ) t = F = = ˆ 1 r ˆ σ x y, from Eq. (5.9.1), from Eq.(3.5.6). 5.8 (a) There s a postve assocaton n the LFPR n 197 and 1968, whch s not surprsng n vew of the fact snce WW II there has been a steady ncrease n the LFPR of women. (b) Use the one-tal t test t = = For 17 df, the one-taled t value at α =5% s Snce the estmated t value s sgnfcant, at ths level of sgnfcance, we can reject the hypothess that the true slope coeffcent s 1 or greater. (c) The mean LFPR s : (0.58) To establsh a 95% confdence nterval for ths forecast value, use the formula: ±.11(se of the mean forecast value), where.11 s the 5% crtcal t value for 17 df. To get the standard error of the forecast value, use Eq. (5.10.). But note that snce the authors do not gve the mean value of the LFPR of women n 1968, we cannot compute ths standard error. (d) Wthout the actual data, we wll not be able to answer ths queston because we need the values of the resduals to plot them and obtan the Normal Probablty Plot or to compute the value of the Jarque-Bera test. 36

5 5.9 (a) PAY SPEND (b) Pay = Spend se = ( ) (0.3117) r = ; RSS =.65E+08 (c) If the spendng per pupl ncreases by a dollar, the average pay ncreases by about $3.31. The ntercept term has no vable economc meanng. (d) The 95% CI for β s: ± (0.3117) = (.684,3.931) Based on ths CI you wll not reject the null hypothess that the true slope coeffcent s 3. (e)the mean and ndvdual forecast values are the same, namely, (5000) 8,667. The standard error of the mean forecast value, usng eq.(5.10.), s (dollars) and the standard error of the ndvdual forecast, usng Eq.(5.10.6), s The confdence ntervals are: Mean Predcton: 8,667 ± ( ), that s, ( $7,66, $9,708) Indvdual Predcton: 8667 ± (38.337), that s, ($ 3,90, $33,43) As expected, the latter nterval s wder than the former. (f) Seres: Resduals Sample 1 51 Observatons 51 Mean 9.13E-1 Medan Maxmum Mnmum Std. Dev Skewness Kurtoss Jarque-Bera Probablty The hstogram of the resduals can be approxmated by a normal curve. The Jarque-Bera statstc s.197 and ts p value s about So, we do not reject the 37

6 normalty assumpton on the bass of ths test, assumng the sample sze of 51 observatons s reasonably large The ANOVA table for the busness sector s as follows: Source of Varaton SS df MSS Due to Regresson(ESS) Due to resdual (RSS) Total(TSS) The F value s = Under the null hypothess that there s no relatonshp between wages and productvty n the busness sector, ths F value follows the F dstrbuton wth 1 and 44 df n the numerator and denomnator, respectvely. The probablty of obtanng such an F value s , that s, practcally zero. Thus, we can reject the null hypothess, whch should come as no surprse. (b) For the non-farm busness sector, the ANOVA table s as follows: Source of Varaton SS df MSS Due to regresson (ESS) Due to resdual (RSS) Total Under the null hypothess that the true slope coeffcent s s zero, the computed F value s: F = If the null hypothess were true, the probablty of obtanng such an F value s practcally zero, thus leadng to the rejecton of the the null hypothess. 38

7 5.11 (a) The plot shown below ndcates that the relatonshp between Impressons AdExp 00 the two varables s nonlnear. Intally, as advertsng expendture ncreases, the number of mpressons retaned ncreases, but gradually they taper off. (b) As a result, t would be napproprate to ft a bvarate lnear regresson model to the data. At present we do not have the tools to ft an approprate model. As we wll show later, a model of the type: Y = β1 + β X + β3x + u may be approprate, where Y = mpressons retaned and X s advertsng expendture. Ths s an example of a quadratc regresson model. But note that ths model s stll lnear n the parameters. (c) The results of blndly usng a lnear model are as follows: Y = X se (7.089) (0.0971) r =

8 5.1 (a) U.S. CPI vs Canada CPI Candan CPI The plot shows that the nflaton rates n the two countres generally move together. (b)& (c) The followng output s obtaned from EVews 3 statstcal package. Sample: Included observatons: 6 Varable Coeffcent Std. Error t-statstc Prob. C ICAN R-squared F-statstc Adjusted R-squared Prob(F-statstc) As ths output shows, the relatonshp between the two varables s postve. One can easly reject the null hypothess that there s no relatonshp between the two varables, as the t value obtaned under 40

9 that hypothess s , and the p value of obtanng such a t value s practcally zero. Although the two nflaton rates are postvely related, we cannot nfer causalty from ths fndng, for t must be nferred from some underlyng economc theory. Remember that regresson does not necessarly mply causaton (a) The two regressons are as follows: Goldprce t = CPI t se = ( ) (0.4038) t = (3.955) (.5718) r = NYSEIndex t = CPI t se = ( ) (3.9584) t = ( ) (1.7066) r = (b) The Jarqu-Bera statstc for the gold prce equaton s wth a p value The JB statstc for the NYSEIndex equaton s wth a p value At the 5% level of sgnfcance, n both cases we do not reject the normalty assumpton. (c) Usng the usual t test procedure, we obtan: t = = Snce ths t value does not exceed the crtcal t value of.04, we cannot reject the null hypothess. The true coeffcent s not statstcally dfferent from 1. (d) & (e) Usng the usual t test procedure, we obtan: t = = Snce ths t value exceeds the crtcal t value of.04, we reject the null hypothess. The estmated coeffcent s actually greater than 1. For ths sample perod, nvestment n the stock market probably was a hedge aganst nflaton. It certanly was a much better hedge aganst nflaton that nvestment n gold (a) None appears to be better than the others. All statstcal results are very smlar. Each slope coeffcent s statstcally sgnfcant at the 99% level of confdence. (b) The consstently hgh r s cannot be used n decdng whch monetary aggregate s best. However, ths does not suggest that t makes no dfference whch equaton to use. 41

10 (c) One cannot tell from the regresson results. But lately the Fed seems to be targetng the M measure Wrte the ndfference curve model as: 1 Y = β1( ) + β + u X Note that now β1 becomes the slope parameter and β the ntercept. But ths s stll a lnear regresson model, as the parameters are lnear (more on ths n Ch.6). The regresson results are as follows: Y ˆ = 3.87( 1 X ) se =(1.599) (0.6817) r = The "slope" coeffcent s statstcally sgnfcant at the 9% confdence coeffcent. The margnal rate of substtuton (MRS) Y 1 of Y for X s: = X X 5.16 (a) Let the model be: Y = β1 + β X + u where Y s the actual exchange rate and X the mpled PPP. If the PPP holds, one would expect the ntercept to be zero and the slope to be one. (b) The regresson results are as follows: Y ˆ = X se = (6.9878) (0.074) t = (-1.6) (66.137) r = To test the hypothess that β = 1, we use the t test, whch gves t = = Ths t value s hghly sgnfcant, leadng to the rejecton of the null hypothess. Actually, the slope coeffcent s s greater than 1. From the gven regresson, the reader can easly verfy that the ntercept coeffcent s not dfferent from zero, as the t value under the hypothess that the true ntercept s zero, s only Note: Actually, we should be testng the (jont) hypothess that the ntercept s zero and the slope s 1 smultaneously. In Ch. 8, we wll show how ths s done. (c) Snce the Bg Max Index s "crude and hlarous" to begn wth, t probably doesn't matter. However, for the sample data, the 4

11 results do not support the theory (a) Lettng Y represent the male math score and X the female math score, we obtan the followng regresson: Yˆ = X se = (1.875) (0.065) t = (15.435) (5.33) r = (b) The Jarque-Bera statstc s wth a p value of Therefore, asymptotcally we cannot reject the normalty assumpton. (c) t = = Therefore, wth 99% confdence we can reject the hypothess that β = 1. (d) The ANOVA table s: Source of Varaton SS df MSS ESS RSS TSS Under the null hypothess that β = 0, the F value s , The p value of obtanng such an F value s almost zero, leadng to the rejecton of the null hypothess (a) The regresson results are as follows: Y ˆ = X se = (33.74) (0.067) t = (3.937) (11.187) r = (b) The Jarque-Bera statstcs s 1.1 wth a p value of Therefore we can reject the null hypothess of non-normalty. (c) Under the null hypothess, we obtan: t = = The crtcal t value at the 5% level s.04 (or -.04). Therefore, we can reject the null hypothess that the true slope coeffcent s 1. (d) The ESS, RSS, and TSS values are, respectvely, (l df), 73. (34 df), and (35 df). Under the usual null 43

12 hypothess the F value s The p value of such an F value s almost zero. Therefore, we can reject the null hypothess that there s no relatonshp between the two varables (a) CPI vs PPI (WPI) PPI (WPI) The scattergram as well s shown n the above fgure. (b) Treat CPI as the regressand and WPI as the regressor. The CPI represents the prces pad by the consumers, whereas the WPI represents the prces pad by the producers. The former are usually a markup on the latter. (c) & (d) The followng output obtaned from Evews6 gves the necessary data. 44

13 Dependent Varable: CPI Method: Least Squares Sample: Included observatons: 7 CPI=C(1)+C()*PPI Coeffcent Std. Error t-statstc Prob. C(1) C() R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood Durbn-Watson stat The estmated t value of the slope coeffcent s under the null hypothess that there s no relatonshp between the two ndexes. The p value of obtanng such a t value s almost zero, suggestng the rejecton of the null hypothess. The hstogram and Jarque-Bera test based on the resduals from the precedng regresson are gven n the followng dagram. 45

14 Hstogram Frequency More Bn The Jarqe-Bera statstc s wth a p value Therefore, we cannot reject the normalty assumpton. The hstogram also shows that the resduals are slghtly left-skewed, but not too far from symmetrc. 46

15 5.0 (a) There seems to be a general postve relatonshp between Smokng and Mortalty. Mortalty vs Smokng Smokng Index (b) Y ˆ = X se = (3.0337) (0.09) t = (-0.153) (4.9) r = (c) The slope coeffcent has a t statstc of 4.9, whch ndcates a p value of almost 0. Therefore, we can reject the null hypothess and conclude that Smokng s related to Mortalty at the 5% level of sgnfcance. (d) The rskest occupatons seem to be Furnace forge foundry workers, Constructon workers, and Panters and decorators. One reason for why these occupatons are more rsky could be that they all work around toxc fumes and/or chemcals and therefore breathe n dangerous toxns frequently. (e) Unless there s a way to categorze the occupatons nto fewer groups, we cannot nclude them n the regresson analyss (ths wll be addressed later n the dscusson of dummy, or ndcator, varables n chapter 9). 47