CHAPTER 7 THE TWOVARIABLE REGRESSION MODEL: HYPOTHESIS TESTING


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1 CHAPTER 7 THE TWOVARIABLE REGRESSION MODEL: HYPOTHESIS TESTING QUESTIONS 7.1. (a) In the regresson contet, the method of least squares estmates the regresson parameters n such a way that the sum of the squared dfference between the actual Y values (.e., the values of the dependent varable) and the estmated Y values s as small as possble. (b) The estmators of the regresson parameters obtaned by the method of least squares. (c) An estmator beng a random varable, ts varance, lke the varance of any random varable, measures the spread of the estmated values around the mean value of the estmator. (d) The (postve) square root value of the varance of an estmator. (e) Equal varance. (f) Unequal varance. (g) Correlaton between successve values of a random varable. (h) In the regresson contet, TSS s the sum of squared dfference between the ndvdual and the mean value of the dependent varable Y, namely, ( Y Y ). () ESS s the part of the TSS that s eplaned by the eplanatory varable(s). (j) RSS s the part of the TSS that s not eplaned by the eplanatory varable(s), the X varable(s). (k) It measures the proporton of the total varaton n Y eplaned by the eplanatory varables. In short, t s the rato of ESS to TSS. (l) It s the standard devaton of the Y values about the estmated regresson lne. 44
2 (m) BLUE means best lnear unbased estmator, that s, a lnear estmator that s unbased and has the least varance n the class of all such lnear unbased estmators. (n) A statstcal procedure of testng statstcal hypotheses. (o) A test of sgnfcance based on the t dstrbuton. (p) In a onetaled test, the alternatve hypothess s onesded. For eample: H 0 : μ μ 0 aganst H 1 : μ > μ0 or μ < μ 0, where μ s the mean value. (q) In a twotaled test, the alternatve hypothess s twosded. (r) It s a shorthand for the statement: reject the null hypothess. 7.. (a) False. It mnmzes the sum of resduals squared, that s, t mnmzes e. (b) True. (c) True. (d) False. The OLS does not requre any probablstc assumpton about the error term n estmatng the parameters. (e) True. The OLS estmators are lnear functons of u and wll follow the normal dstrbuton f t s assumed that u are normally dstrbuted. Recall that any lnear functon of a normally dstrbuted varable s tself normally dstrbuted. (f) False. It s ESS / TSS. (g) False. We should reject the null hypothess. (h) True. The numerator of both coeffcents nvolves the covarance between Y and X, whch can be postve or negatve. () Uncertan. The p value s the eact level of sgnfcance of a computed test statstc, whch may be dfferent from an arbtrarly chosen level of sgnfcance, α (a) t (b) se( b ) (c) 0 and 1 (d) 1 and +1 (e) ESS (f) ESS (g) the standard error of the estmate (h)( Y ) b + e Y () 45
3 7.4. The answers to the mssng numbers are n boes: Ŷ X r se ( ) ( ) n 0 t ( ) (18.73) The crtcal t value at the 5% level for 18 d.f. s.101 (twotaled) and (onetaled). Snce the estmated t value of far eceeds ether of these crtcal values, we reject the null hypothess. A twotaled test s approprate because no a pror theoretcal consderatons are known regardng the sgn of the coeffcent r y e ) / y ( ŷ / y b / y, followng Equatons (7.34) and (7.35) In provng the last equalty, note that follows by substtuton. ŷ b y y. Then the result 7.6. e n Y n Y b X ) n b X 0. See also Problem 6.. ( PROBLEMS 7.7. (a) The d.f. here are 14. Therefore, the 5% crtcal t value s.145. So, the 95% confdence nterval s: 3.4 ±.145(1.634) (0.649, ) (b) The precedng nterval does nclude B. Therefore, do not reject the null hypothess. (c) t 3.4 / Snce car sales are epected to be postvely related to real dsposable ncome, the null and alternatve hypotheses should be: : B 0 and : B 0. Therefore, an onetaled t test s H0 H1 > approprate n ths case. The 5% onetaled t value for 14 d.f. s Snce the computed t value of eceeds the crtcal value, reject the null hypothess (one and two tal tests sometmes gve dfferent results). 46
4 7.8. (a) The slope coeffcent of means that durng the perod a percentage pont ncrease n the market rate of return lead to about 1.06 percent ponts ncrease n the mean return on the IBM stock. In the same perod, f the market rate of return were zero, the average rate of return on the stock would have been about 0.73 percent, whch may not make economc sense. (b) About 47 percent of the varaton n the mean return on the IBM stock was eplaned by the (varaton) n the market return. (c) : B 1, : B 1. Hence: H0 H1 > ( ) t For 38 d.f, ths t value s not statstcally sgnfcant at the 5% level on the bass of the onetaled t test. Thus, durng the study perod, the beta coeffcent of IBM was not statstcally dfferent from unty, suggestng that the IBM stock was not volatle or aggressve (a) b 1 1.; b (b) se( b 1 ) ; se( b ) (c) r (d) 95% CI for B 1 : to % CI for B : 0.48 to (e) Reject H 0, snce the precedng CI does not nclude B (a) The answers to the mssng numbers are n boes: GNP t M 1 t r se ( ) (0.197) t ( ) ( ) (b) : B 0, : B 0. The null hypothess can be rejected. H0 H1 > (c) No partcular economc meanng can be attached to t. (d) Gˆ NP (55) 3,676 bllon. 47
5 7.11. (a) Negatve. (b) Yes. Here, n 14 (14 presdental electons startng n 198 and endng n 1980) and therefore d.f. 1. The computed t value of .67 s statstcally sgnfcant at the fve percent level (onetaled test). (c) Probably. But n the 1984 electons the personal popularty of Ronald Reagan was an mportant factor. (d) Snce t b / se( b ) under the null hypothess that the true B s zero, b se( b ). In the present eample these standard errors are and t , respectvely (a) It could be negatve or postve. As more output s produced as a result of ncreased capacty, prce ncreases (.e., nflaton) wll slow down. However, f capacty utlzaton s at ts optmal value, and f demand pressures contnue, nflaton may actually rse. (b) The output n EVews format s as follows: Dependent Varable: INFLATION Sample: Varable Coeffcent Std. Error tstatstc Prob. C CAPACITY Rsquared (c) The estmated slope coeffcent s negatve but also statstcally nsgnfcant, for the estmated p value s qute hgh. (d) Yes t s, for under the null hypothess that the true slope coeffcent s 1, the estmated t value s (1) t The probablty of obtanng such a t value s practcally zero. (e) To get ths, solve C 0, whch gves C 19.14, whch may be called the natural rate of capacty utlzaton. 48
6 Note: The results of the above regresson are vrtually nsgnfcant. Plus, the natural rate of capacty utlzaton that was found to be may be problematc because the measure of capacty utlzaton does not eceed 100. The reason for the regresson breakdown s the fact that the data nclude the decade of the 1970s wth ts hgh rates of nflaton and the md 1970s stagflaton. Runnng the regresson over a perod that ecludes the 1970s, say , wll produce more reasonable and statstcally sgnfcant results. In fact, f the regresson covers the perod, the reader can easly verfy that the natural rate of capacty utlzaton s appromately (a) The EVews regresson results are as follows: Dependent Varable: CAPACITY Sample: Varable Coeffcent Std. Error tstatstc Prob. C INFLATION Rsquared Note: Ths regresson s also nsgnfcant for the reason dscussed above. (b) Multplyng the two slope coeffcents, we obtan the value of whch s equal to the R value obtaned from ether equaton. Ths result s not surprsng n vew of Problem 6.1. (c) By way of another eample, let Y salary and X qualfcatons for a group of men and women. As Maddala notes, the drect regresson wll answer the queston whether men and women wth the same X value get the same Y value. The reverse regresson wll answer the questons whether men and women wth the same Y value wll have the same X value. Reverse regresson s advocated for wage dscrmnaton cases. (d) No (a) Postve. (b) and (c) The scattergram wll show that the relatonshp between the two s generally postve, although there are a few outlers. 49
7 (d) The regresson results are as follows: Ŷ t se ( ) (0.1154) (e) 99% CI: B t (0.039) (3.6406) X t r Snce the precedng nterval does not nclude zero, we can reject the null hypothess (a) The regresson results are: MA T ˆ HMt MATHFM t se (0.6353) (0.0455) t (8.579) ( ) r (b) Reject the null hypothess, snce the computed t value of far eceeds the crtcal value even at the level of sgnfcance. (c) MAˆ THM (d) CI: ( , ) (a) The regresson results are as follows: VE R ˆ BMt VERBFM t se (11.658) (0.068) t (1.713) (5.1018) r (b) Reject the null hypothess, snce the computed t value s very hgh. (c) VEˆ RBM (d) CI: ( , ) (a) There s a postve relatonshp between real return on the stock prce nde ths year and the dvdend prce rato last year: Per unt ncrease n the latter, the mean real return goes up by 5.6 percentage ponts. The ntercept has no vable economc meanng. (b) If the precedng results are accepted, t has serous mplcatons for the effcent market hypothess of modern fnance. 50
8 7.18 (a) The EVews regresson output s as follows: Dependent Varable: AVGHWAGE Sample: 1 13 Varable Coeffcent Std. Error tstatstc Prob. C YEARSSCH Rsquared (b) On the bass of the t test ths hypothess can be easly rejected, for the computed t value s hghly sgnfcant; ts p value s practcally zero (c) Here t Ths t value s also hghly sgnfcant, leadng to the concluson that the educaton coeffcent s statstcally dfferent from 1. The p value of obtanng the computed t value s (twotal test) Note: Ths Problem s an etenson of Problem (a) Based on the regresson, we need to calculate new varables based on the real GDP (RGDP) and the unemployment rate (UNRATE). These calculatons, based on the data n Table 61, are as follows: CHUNRATE Change n UNRATE UNRATE UNRATE(1) PCTCRGDP % Change n RGDP [RGDP / RGDP(1)]* Usng EVews, the regresson results are: Dependent Varable: PCTCRGDP Sample (adjusted): Varable Coeffcent Std. Error tstatstc Prob. C CHUNRATE Rsquared Note: The sample s adjusted to start n 1971 nstead of the ntal observaton of 1970 because we are calculatng percentage changes (RGDP) and changes (UNRATE): Ths causes the loss of the frst observaton. 51
9 (b) Yes, for the estmated slope coeffcent has a t value of whose p value s practcally zero. (c) The ntercept term s also statstcally sgnfcant. The nterpretaton here s that f the change n the unemployment rate were zero, the growth n the real GDP wll be about 3.%, whch may be called the longterm, or steadystate, rate of growth of GDP The regresson results, usng EVews, are: Dependent Varable: SP500 Sample: Varable Coeffcent Std. Error tstatstc Prob. C /MTB Rsquared As these results show, the slope coeffcent s statstcally sgnfcant at about the % level, but the ntercept s not. Any mnor dfferences wth the regresson shown n the tet are solely due to roundng The EVews regresson results for (6.7) are as follows: Dependent Varable: PRICE Sample: 1 3 Varable Coeffcent Std. Error tstatstc Prob. C AGE Rsquared The estmated slope coeffcent s hghly statstcally sgnfcant, for the p value of obtanng a t statstc of or greater under the null hypothess of a zero true populaton slope coeffcent s practcally zero. In contrast, the estmated ntercept coeffcent s statstcally nsgnfcant snce ts p value s relatvely hgh. Lkewse, the EVews results of regresson (6.8) are: (Regresson output s shown n the followng page) 5
10 Dependent Varable: PRICE Sample: 1 3 Varable Coeffcent Std. Error tstatstc Prob. C NOBIDDERS Rsquared Here both the coeffcents are ndvdually statstcally sgnfcant, as ther p values are qute low. 7.. Note: The regresson results presented here are dentcal to those of Problem (a) The results, usng EVews, are as follows: Dependent Varable: ASP Sample: 1 47 Varable Coeffcent Std. Error tstatstc Prob. C GPA Rsquared As these results suggest, GPA has a postve mpact on ASP, and t s statstcally very sgnfcant, as the p value of the estmated coeffcent s very small. (b) The results for GMAT are as follows: Dependent Varable: ASP Sample: 1 47 Varable Coeffcent Std. Error tstatstc Prob. C GMAT Rsquared These results show that GMAT has a postve and statstcally sgnfcant mpact on ASP. (c) The results for annual tuton are as follows: (Regresson output s shown n the followng page) 53
11 Dependent Varable: ASP Sample: 1 47 Varable Coeffcent Std. Error tstatstc Prob. C TUITION Rsquared Tuton (perhaps reflectng the qualty of educaton) has a postve and statstcally sgnfcant mpact on ASP. Incdentally, t can also be shown that the mpact of recruter ratng has a postve and hghly sgnfcant mpact on ASP, as t can be seen from the followng EVews output: Dependent Varable: ASP Sample: 1 47 Varable Coeffcent Std. Error tstatstc Prob. C RATING Rsquared The regresson results of ependture on mported goods (Y) and personal dsposable ncome (X), usng EVews, are as follows: Dependent Varable: Y Sample: Varable Coeffcent Std. Error tstatstc Prob. C X Rsquared These results suggest that personal dsposable ncome has a very sgnfcant postve mpact on ependture on mported goods, an unsurprsng fndng. The p value for the slope s vrtually zero, and the null hypothess s therefore rejected If we let w, we can wrte b w Y, that s, b s a lnear estmator,.e., a lnear functon of the Y values. Note that we are treatng 54
12 55 X as nonstochastc. Follow smlar steps to show that 1 b s also a lnear functon of the Y values. Now: ) ( u X B B Y y b u X B B + u B Ths s n vew of the fact that 0 ( X ) X and 1 X. Therefore, ) ( B u B E b E + Note: ) ( 1 u E u E, snce s a constant and snce X and u are uncorrelated by OLS assumpton. Follow smlar steps to prove that 1 b s also unbased Squarng Equaton (7.33) and summng, we obtan: + + e b e b y + e b snce 0 e.
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