CHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS

CHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS QUESTIONS 9.1. (a) In a log-log model the dependent and all explanatory varables are n the logarthmc form. (b) In the log-ln model the dependent varable s n the logarthmc form but the explanatory varables are n the lnear form. (c) In the ln-log model the dependent varable s n the lnear form, whereas the explanatory varables are n the logarthmc form. (d) It s the percentage change n the value of one varable for a (small) percentage change n the value of another varable. For the log-log model, the slope coeffcent of an explanatory varable gves a drect estmate of the elastcty coeffcent of the dependent varable wth respect to the gven explanatory varable. (e) For the ln-ln model, elastcty = slope Y. Therefore the elastcty wll depend on the values of and Y. But f we choose and Y, the mean values of and Y, at whch to measure the elastcty, the elastcty at mean values wll be: slope. Y 9.. The slope coeffcent gves the rate of change n (mean) Y wth respect to, whereas the elastcty coeffcent s the percentage change n (mean) Y for a (small) percentage change n. The relatonshp between two s: Elastcty = slope. For the log-lnear, or log-log, model only, the elastcty and Y slope coeffcents are dentcal. 9.3. Model 1: ln Y = B + 1 B ln : If the scattergram of ln Y on ln shows a lnear relatonshp, then ths model s approprate. In practce, such models 66

are used to estmate the elastctes, for the slope coeffcent gves a drect estmate of the elastcty coeffcent. Model : ln Y = B + 1 B : Such a model s generally used f the objectve of the study s to measure the rate of growth of Y wth respect to. Often, the varable represents tme n such models. Model 3: : If the objectve s to fnd out the absolute change n Y for a relatve or percentage change n, ths model s often chosen. Model 4: Y = B + 1 B ln ) : If the relatonshp between Y and s curvlnear, as n the case of the Phllps curve, ths model generally gves a good ft. 9.4. (a) Elastcty. Y (1/ = B + B 1 (b) The absolute change n the mean value of the dependent varable for a proportonal change n the explanatory varable. (c) The growth rate. (d) dy d Y (e) The percentage change n the quantty demanded for a (small) percentage change n the prce. (f) Greater than 1; less than 1. 9.5. (a) True. d ln Y dy =, whch, by defnton, s elastcty. d ln d Y (b) True. For the two-varable lnear model, the slope equals and the elastcty = slope = B Y the log-lnear model, slope = whle the elastcty equals regresson model. B Y B B, whch vares from pont to pont. For Y, whch vares from pont to pont. Ths can be generalzed to a multple 67

(c) True. To compare two or more same. R s (d) True. The same reasonng as n (c). (e) False. The two, the dependent varable must be the r values are not drectly comparable. 9.6. The elastcty coeffcents for the varous models are: (a) B ( / Y ) (b) - (1 / Y ) (c) B B (d) - B (1 / ) (e) B (1 / Y ) (f) B (1 / ) Model (a) assumes that the ncome elastcty s dependent on the levels of both ncome and consumpton expendture. If B > 0, Models (b) and (d) gve negatve ncome elastctes. Hence, these models may be sutable for "nferor" goods. Model (c) gves constant elastcty at all levels of ncome, whch may not be realstc for all consumpton goods. Model (e) suggests that the ncome elastcty s ndependent of ncome,, but s dependent on the level of consumpton expendture, Y. Fnally, Model (f) suggests that the ncome elastcty s ndependent of consumpton expendture, Y, but s dependent on the level of ncome,. 9.7 (a) Instantaneous growth: 3.0%; 5.30%; 4.56%; 1.14%. (b) Compound growth: 3.07%; 5.44%; 4.67%; 1.15%. (c) The dfference s more apparent than real, for n one case we have annual data and n the other we have quarterly data. A quarterly growth rate of 1.14% s about equal to an annual growth rate of 4.56%. PROBLEMS 9.8. (a) MC = (b) AVC = B + + B3 3B4 B + B + B 3 4 1 (c) AC = B 1 + B + B3 + B4 By way of an example based on actual numbers, the MC, AVC, and AC from Equaton (9.33) are as follows: 68

MC = 63.4776 5.930 +.8188 AVC = 63.4776 1.9615 + 0.9396 1 AC = 141.7667 + 63.4776 1.9615 + 0.9396 (d) The plot wll show that they do ndeed resemble the textbook U-shaped cost curves. 1 9.9. (a) = B1 + B Y (b) Y = B + B 1 9.10. (a) In Model A, the slope coeffcent of -0.4795 suggests that f the prce of coffee per pound goes up by a dollar, the average consumpton of coffee per day goes down by about half a cup. In Model B, the slope coeffcent of -0.530 suggests that f the prce of coffee per pound goes up by 1%, the average consumpton of coffee per day goes down by about 0.5%. 1.11 (b) Elastcty = -0.4795 = -0.190.43 (c) -0.530 (d) The demand for coffee s prce nelastc, snce the absolute value of the two elastcty coeffcents s less than 1. (e) Antlog (0.7774) =.1758. In Model B, f the prce of coffee were $1, on average, people would drnk approxmately. cups of coffee per day. [Note: Keep n mnd that ln(1) = 0]. (f) We cannot compare the two r values drectly, snce the dependent varables n the two models are dfferent. 9.11. (a) Ceters parbus, f the labor nput ncreases by 1%, output, on average, ncreases by about 0.34%. The computed elastcty s dfferent from 1, for t = 0.3397 1 = -3.5557 0.1857 For 17 d.f., ths t value s statstcally sgnfcant at the 1% level of sgnfcance (two-tal test). 69

(b) Ceters parbus, f the captal nput ncreases by 1%, on average, output ncreases by about 0.85 %. Ths elastcty coeffcent s statstcally dfferent from zero, but not from 1, because under the respectve hypothess, the computed t values are about 9.06 and -1.65, respectvely. (c) The antlog of -1.654 = 0.1916. Thus, f the values of = 1, 3 = then Y = 0.1916 or (0.1916)(1,000,000) = 191,600 pesos. Of course, ths does not have much economc meanng [Note: ln(1) = 0]. (d) Usng the R varant of the F test, the computed F value s: 0.995 / F = = 1,691.50 (1 0.995) / 17 Ths F value s obvously hghly sgnfcant. So, we can reject the null hypothess that B = B 0. The crtcal F value s F = 6.11 for α = 1%. 3 = Note: The slght dfference between the calculated F value here and the one shown n the text s due to roundng. 9.1. (a) A pror, the coeffcents of ln(y / P) and lnσ BP,17 should be postve and the coeffcent of lnσ should be negatve. The results meet the pror E expectatons. (b) Each partal slope coeffcent s a partal elastcty, snce t s a log-lnear model. (c) As the 1,10 observatons are qute a large number, we can use the normal dstrbuton to test the null hypothess. At the 5% level of sgnfcance, the crtcal (standardzed normal) Z value s 1.96. Snce, n absolute value, each estmated t coeffcent exceeds 1.96, each estmated coeffcent s statstcally dfferent from zero. (d) Use the F test. The author gves the F value as 1,151, whch s hghly statstcally sgnfcant. So, reject the null hypothess. 9.13. (a) If (1 / ) goes up by a unt, the average value of Y goes up by 8.743. 8.743 (b) Under the null hypothess, t = = 3.0635, whch s statstcally.8478 sgnfcant at the 5% level. Hence reject the null hypothess. 70

(c) Under the null hypothess, F = 1,15 t15, whch s the case here, save the roundng errors. 1 (d) For ths model: slope = - 1 B = -8.743 t.5 = -3.8775. 1 (e) Elastcty = - 1 B = -8.743 = -1.117 t Y t (1.5)(4.8) Note: The slope and elastcty are evaluated at the mean values of and Y. (f) The computed F value s 9.39, whch s sgnfcant at the 1% level, snce for 1 and 15 d.f. the crtcal F value s 8.68. Hence reject the null hypothess that r = 0. 9.14. (a) The results of the four regressons are as follows: Dependent Varable Intercept Independent Varable Goodness of Ft 1 Ŷ t = 38.9690 + 0.609 t r = 0.943 t 3 t t = (10.105) (15.655) l ˆ n Y = 1.4041 + 0.5890 ln t t = (8.954) (0.090) r = 0.964 l ˆ n Y = 3.9316 + 0.008 t r = 0.984 t = (84.678) (13.950) 4 Ŷ t = -19.9661 + 54.16 ln t r = 0.9543 t = (-11.781) (17.703) (b) In Model (1), the slope coeffcent gves the absolute change n the mean value of Y per unt change n. In Model (), the slope gves the elastcty coeffcent. In Model (3) the slope gves the (nstantaneous) rate of growth n (mean) Y per unt change n. In Model (4), the slope gves the absolute change n mean Y for a relatve change n. (c) 0.609; 0.5890(Y / ); 0.008(Y); 54.16(1 / ). (d) 0.609( / Y); 0.5890; 0.008(); 54.16(1 / Y). 71

For the frst, thrd and the fourth model, the elastctes at the mean values are, respectvely, 0.5959, 0.6165, and 0.563. (e) The choce among the models ultmately depends on the end use of the model. Keep n mnd that n comparng the the dependent varable must be n the same form. r values of the varous models, 9.15. (a) ˆ1 = 0.0130 + 0.0000833 Y t = (17.06) (5.683) r = 0.8015 The slope coeffcent gves the rate of change n mean (1 / Y) per unt change n. dy B (b) = d B + B ) ( 1 At the mean value of, = 38.9, ths dervatve s -0.3146. (c) Elastcty = coeffcent s -0.1915. 1 (d) Ŷ = 55.4871 + 11.1797 t = (17.409) (4.45) dy. At = 38.9 and Y = 63.9, ths elastcty d Y r = 0.695 (e) No, because the dependent varables n the two models are dfferent. (f) Unless we know what Y and stand for, t s dffcult to say whch model s better. 9.16. For the lnear model, r = 0.99879, and for the log-ln model, r = 0.99965. Followng the procedure descrbed n the problem, r = 0.99968, whch s comparable wth the r = 0.99879. 9.17. (a) Log-lnear model: The slope and elastcty coeffcents are the same. Log-ln model: The slope coeffcent gves the growth rate. Ln-log model: The slope coeffcent gves the absolute change n GNP for a percentage n the money supply. 7

Lnear-n-varable model: The slope coeffcent gves the (absolute) rate of change n mean GNP for a unt change n the money supply. (b) The elastcty coeffcents for the four models are: Log-lnear: 0.988 Log-ln (Growth): 1.0007 (at = 1,755.667) Ln-log: 0.960 (at Y =,791.473) Lnear (LIV): 0.9637 (at = 1,755.667 and Y =,791.473). (c) The r s r s of the log-lnear and log-ln models are comparable, as are the of the ln-log and lnear (LIV) models. (d) Judged by the usual crtera of the t test, all the models more or less gve smlar results. r values, and the elastctes, (e) From the log-lnear model, we observe that for a 1% ncrease n the money supply, on the average, GNP ncreases by about 1%, the coeffcent 0.988 beng statstcally equal to 1. Perhaps ths model supports the monetarst vew. Snce the elastcty coeffcents of the other models are smlar, t seems all the models support the monetarsts. 9.18. (a) Ŷ t = 8.3407 + 0.9817 0.595 t se = (1.417) (0.0193) (0.015) t = (0.0617) (50.7754) (-17.0864) 3t R = 0.9940 p value = (0.0000)* (0.0000)* (0.0000)* R = 0.9934 * Denotes a very small value. (b) Per unt change n the real GDP ndex, on average, the energy demand ndex goes up by about 0.98 ponts, ceters parbus. Per unt change n the energy prce, the energy demand ndex goes down about 0.6 ponts, agan holdng all else constant. (c) From the p values gven n the above regresson, all the partal regresson coeffcents are ndvdually hghly statstcally sgnfcant. (d) The values requred to set up the ANOVA table are: TSS = 6,746.9887; ESS = 6,706.863, and RSS = 40.704. The computed F value s 1,647.638 wth a p value of almost zero. Therefore, we can reject the null hypothess 73

that there s no relatonshp between energy demand, real GDP, and energy prces (Note: These ANOVA numbers can easly be calculated wth the regresson optons n Excel). (e) Mean value of demand = 84.370; mean value of real GDP = 89.66, and the mean value of energy prce = 13.135, all n ndex form. Therefore, at the mean values, the elastcty of demand wth respect to real GDP s 1.048 and wth respect to energy prce, t s -0.3787. (f) Ths s straghtforward. (g) The normal probablty plot wll show that the resduals from the regresson model le approxmately on a straght lne, ndcatng that the error term n the regresson model seems to be normally dstrbuted. The Anderson-Darlng normalty test gves an s about 0.188, thereby supportng the normalty assumpton. A value of 0.50, whose p value (h) The normalty plot wll show that the resduals do not le on a straght lne, suggestng that the normalty assumpton for the error term may not be tenable for the log-lnear model. The computed Anderson-Darlng A s 1.00 wth a p value of about 0.009, whch s qute low. Note: Any mnor coeffcent dfferences between ths log-lnear regresson and the log-lnear regresson (9.1) are due to roundng. Regardng the Anderson-Darlng test, t s avalable n MINITAB. If you do not have access to MINITAB, you can use the normal probablty plots n EVews and Excel for a vsual nspecton, as descrbed above. EVews also has the Jarque-Bera normalty test, but you should avod usng t here because t s a large sample asymptotc test and the present data set has only 3 observatons. In fact, the Jarque-Bera test wll show that the resduals of both the lnear and the log-lnear regressons satsfy the normalty assumpton, whch s not the case based on the Anderson-Darlng A and the normal probablty plots. () Snce the lnear model seems to satsfy the normalty assumpton, ths model may be preferable to the log-lnear model. 9.19. (a) Ths wll make the model lnear n the parameters. 74

(b) The slope coeffcents n the two models are, respectvely: dy dt B 1 dy = and = B ( A + Bt) Y dt (c) In models (1) and () the slope coeffcents are negatve and are statstcally sgnfcant, snce the t values are so hgh. In both models the recprocal of the loan amount has been decreasng over tme. From the slope coeffcents already gven, we can compute the rate of change of loans over tme. (d) Dvde the estmated coeffcents by ther t values to obtan the standard errors. (e) Suppose for Model 1 we postulate that the true B coeffcent s -0.14. Then, usng the t test, we obtan: t = 0.0 ( 0.14) = -7.3171 0.008 Ths t value s statstcally sgnfcant at the 1% level. Hence, t seems there s a dfference n the loan actvty of New York and non-new York banks. [Note: s.e. = (-0.0)/(-4.5) = 0.008]. 9.0. (a) For the recprocal model, as Table 9-11 shows, the slope coeffcent (.e., the rate of change of Y wth respect to s B(1 / ). In the present nstance B = 0.0549. Therefore, the value of the slope wll depend on the value taken by the varable. (b) For ths model the elastcty coeffcent s B (1 / Y). Obvously, ths elastcty wll depend on the chosen values of and Y. Now, = 8.375 and Y = 0.433. Evaluatng the elastcty at these means, we fnd t to be equal to -0.0045. 9.1. We have the followng varable defntons: TOTAL PCE () = Total personal consumpton expendture; EP SERVICES ( Y 1 EP DURABLES ( ) = Expendture on servces; Y EP NONDURABLES ( ) = Expendture on durable goods; Y 3 ) = Expendture on nondurable goods. 75

Plottng the data, we obtan the followng scatter graphs: 3000 900 EP SERVICES (Y1) 800 700 600 500 400 400 4400 4600 4800 5000 500 TOTAL PCE () 750 EP DURABLES (Y) 700 650 600 550 500 400 4400 4600 4800 5000 500 TOTAL PCE () 76

1600 EP NONDURABLES (Y3) 1550 1500 1450 1400 1350 1300 400 4400 4600 4800 5000 500 TOTAL PCE () It seems that the relatonshp between the varous expendture categores and total personal consumpton expendture s approxmately lnear. Hence, as a frst step one could apply the lnear (n varables) model to the varous categores. The regresson results are as follows: (the ndependent varable s TOTAL PCE and fgures n the parentheses are the estmated t values). Dependent varable EP SERVICES EP DURABLES EP NONDURABLES ( Y 1 ) ( Y ) ( Y 3 ) Intercept.5759 (11.981) -554.5943 (-16.8744) 335.764 (4.7647) Slope (TOTAL PCE) 0.5164 (19.8600) 0.484 (35.468) 0.345 (81.1599) R 0.9988 0.9836 0.9968 Judged by the usual crtera, the results seem satsfactory. In each case the slope coeffcent represents the margnal propensty of expendture (MPE) that s the addtonal expendture for an addtonal dollar of TOTAL PCE. 77

Ths s hghest for servces, followed by durable and nondurable goods expendtures. By fttng a double-log model one can obtan the varous elastcty coeffcents. 9.. The EVews results for the frst model are as follows: Dependent Varable: Y Sample: 1971 1980 Varable Coeffcent Std. Error t-statstc Prob. C 1.79719 7.688560 0.166445 0.8719 1.069084 0.38315 4.486004 0.000 R-squared 0.715548 The output for the regresson-through-the-orgn model s: Dependent Varable: Y Sample: 1971 1980 Varable Coeffcent Std. Error t-statstc Prob. 1.08991 0.191551 5.6899 0.0003 R-squared 0.714563 Ths R may not be relable. Snce the ntercept n the frst model s not statstcally sgnfcant, we can choose the second model. 9.3. Usng the raw raw r formula, we obtan : = ( ) Y (11,344.8) r = = 0.785 Y (10,408.44)(15,801.41) You can compare ths wth the ntercept-present R value of 0.7155. 9.4. Computatons wll show that the raw r s 0.7318. The one n Equaton (9.40) s 0.7353. There s not much dfference between the two values. Any mnor dfferences between regressons (9.39) and (9.40) n the text and the same regressons based on Table 6-1 are due to roundng. 78