- Email: [email protected]
The role of expectations on automobile demand
Transpn. Res:B Rnled m Great
Vol. ZIB.No.4. Brim
pp. 331-337.
1987 6
THE ROLE OF EXPECTATIONS AUTOMOBILE DEMAND
0191-2615/87 $3.00+ .oO 1987 F’ergamon loumalr Lid.
ON
HAROLD E. DYCK Southwest Missouri State University, Springfield, Missouri 65804-0095, U.S.A. (Received 30 November
1984; in
revisedform
15 September
1986)
Abstract-This study investigates the influence of adaptive expectations on the purchase of automobiles by income quintile. Through maximum likelihood estimation, it is found that the coefficients of adaptation to income exhibit a trend towards faster rates in the upper quintiles. The effects of a “truncation remainder” and of serial correlation are also noted.
1. INTRODUCTION
This study investigates the influence of adaptive expectations on the purchase of automobiles by income quintile. Through maximum likelihood estimation, it is found that the coefficients of adaptation to income exhibit a trend towards faster rates in the upper quintiles. The effects of a “truncation remainder” and of serial correlation are also noted. The main focus of this investigation is the division of income into permanent and transitory income. Unlike other studies, the present study does not insist on the uniform application of the adaptive expectations assumption across quintiles. The coefficient of adaptation is estimated by use of maximum likelihood technique described in Dhrymes (1971), Zellner and Geisel (1970), and Dyck (1982, 1984). We begin by assuming an adaptive expectations model where the change in expected or permanent income is proportional to the difference between the current period’s observed income level and the period’s previously formed expected value. The rate of adjustment, /3, is the coefficient of adaptation. If p = 0, there is no adjustment, whereas, if p = 1, the adjustment is instantaneous. There seems to be no reason, a priori that the rate of adjustment should be the same in each quintile. Thus, the alternative hypothesis that the rates of adjustment are different is tested.
2. PERMANENT
INCOME
2.1 The restricted model Permanent income, yr* , is estimated & la Friedman as the sum of two parts: at, a trend value, and a geometrically declining weighted average of deviations from past values of the trend. Thus, y:=w+p~(l-p)jD,-j J=o
t=i
,...,
T,
(1)
where D, = yI - at. This formulation is designed to allow for “predicted secular growth.” The weight parameter, p, is similar to our coefficient of adaptation. Once a series for permanent income is derived, it is used to predict purchase rates of automobiles for the five quintiles in the equation, 4, = a,
+ U,Y,* + Uz(Y, 331
TR 21:48-P
Y?) + U,r
(2)
332
H. E. DYCK
where q1 is the purchase rate of a particular quintile, and the difference (y, - y,*) represents transitory income. The restricted model is estimated by Smith (1975) for three values of l3: 0.0, 0.4, and 1.O. It is assumed that all quintiles have the same value of l.3.When p = 0.0, then the consumers are using income trend only to estimate permanent income. Thus, for p = 0 we have: y: = or which implies that q, =
a0
+
u*y, + (a, -
u*)cxt + u,.
(3)
But if p = 1.O, then there is no adjustment towards the long-run trend and permanent income. Therefore, we have YI* =
at + (y, - at) = y,
which implies that 4r =
=o + QIY, +
(4)
4.
That is, current income is used by the consumer as the best prediction for permanent income. Finally, when p = 0.4 we have a partial adjustment to long-run trend. Smith truncates the lags after five years due to the short time series data used and rounds up the latter coefficients “to compensate for earlier periods. ” The model proposed by Smith has several shortcomings: only three values of p are compared, all income quintiles are assumed to have the same value of l3, and for l3 = 0.4, the lag is truncated after five years because of the short time series. There are other problems as well. It is not clear how cx is estimated. It should be estimated simultaneously with l3 as part of the expectational hypothesis. Further, the random error term is attached to the final regression equation determining each quintile purchase frequency rather than to the equation describing the formation of expectations. To circumvent these problems, the following model employing the maximum likelihood of parameters is proposed. 2.2 The unrestricted model Rather than truncate after five years, we propose the following derivation, beginning with eqn (1). Note that it is possible to attach an error term to eqn (1) and carry it along without complicating it into a second or higher order term. From eqn (l), we have [See Dyck (1982)]. * _ 41 Yr -
- P)
P
+ (3
C(1 -
PYY,-j + 4
I=0
From the above equation, if l3 = 0, then, y: = ctt + u,, and
(3
41 = =o + o(a, - oz)t +
a2yt +
(a,
-
Q2h4,
by substitution into eqn (2). On the other hand, if l3 = 1, then, yr* = at + (y, - at)
+ u,
= y, + u,, and 4, = a0 +
@Y,
+
(6) (4
-
a2h,.
333
Expectations on automobile demand
The only difference so far in the final equation with p = 0 and p = 1 are in the error terms. Now, with 0 < p < 1, we have
YP(cyTPI
=
41 - P)
P
+ PUP, 3) + PC1- P>'Yo*+
= 41 - P) + PZh(P,Y,) + 9Yo*UP)
4
+ u,
P
(7)
1-l
where Z,, = c
j=O
(1 - WY,-,,
Gr
=
(1 - p)‘,
yo* = c j=O
(1 -
P)jY-j
and9, = (Y,, Y,-~, . . . , Y& From eqn (2), 4, = a0 +
=
a,Y?
+ %(Y, -
y3 + u,
(
i
- P) + azy, + P(a, - %P,, + Pyo*(u, - u*)Z*, +
a, + ; (4 - Ml
(where E, = (a, - a&,) or 4, = bo + b,y, +b,Z,,+b3Z,,+c,
t=
I,...,
E,
(8)
T.
Equations 7-8 show explicitly how the coefficients of the reduced form of the model (the b’s) depend joinr2y on the structural coefficients of the model and so must be estimated simultaneously. Intuitively, eqn (8) states that while the purchase rate of a particular quintile is
a function of permanent income and transitory income, the way permanent income is measured allows the purchase rate to be estimated as a function of actual income and the two Z’s. The variable Z, is the sum of weighted past sample values of actual income with exponentially declining weights. The asymptotically zero variable, Zz, is used to estimate the ‘Yruncation remainder,” y$, which represents the effect of the presample values on the sample period. Of special interest here is the effect of including Z2 in the equation when a small sample (T = 16) is used. More generally, eqn (8) can be rewritten as: q = Zb + E
where
Maximizing the log-likelihood function
L(b, p, a*jq, Z) = -5
log 2n - 5 log a* - & (q - Zb)‘(q - Zb),
with respect to b and u2 yields 6(p) = (Z’Z)-‘Z’q
and
I%*(P) = ; (q - Zb)‘(q - Zb).
334
H. E. DYCK
1. MLE of em 8
Table MLE
A Quintile
6
6,
I;’
with> = 1
DW
bl
ii2
b2
DW
aI
1
.OOl
A808
.1208
3.1
.3071
.1443
2.7
.3062
2
.2
.0410
.5656
2.1
.7691
.0703
.2296
1.3
.3271
3
.3
.9358
.8328
2.3
.6466
-.0658
.7296
1.4
.3147
4
.OOl
.0960
.4706
1.8
.1543
.0563
.3327
1.3
.1866
5
.5
A380
.3975
2.7
.6673
-.2936
.0592
1.7
.0675
-.0376
A search can then be carried out over the range of p to find the global maximum of the log likelihood function. The asymptotic standard errors of the structural coefficients are found in the manner of Dhyrmes (1971) and Appendix B of Dyck (1982). To investigate the effects of serial correlation, let Et =
Qrlrrlt+x) =
W-I
+ r(,Y
1
where In,/ < 1,
E(q,)
= 0 and
aq*fors = 0 for s _# 0
0
Then eqn (8) becomes 91 =
bo + blY,
+
b*Z,,
+
w2,
+
E,
where E, = pe,-i 9f -
P4r-1 = bo(l -
P) + MY,
-
PY,-1) +
b*(Z,,
+ -q,, and -
PZ,,-J
+
b&G,
-
pZ*,_,) + q,. (9)
The regressions can then be carried out using the transformed variables of eqn (9) without violation of the usual assumptions regarding the error term, since q, - NID(0, at). 2.3 Empirical results The data sources used in the present study are as follows. The new car purchase rates by quintile and attitudes came from Table A.2 of Smith (1975), p. 90. (Originally from the University of Michigan Survey Research Center, annual Survey of Consumer Finance). The average income per quintile in current dollars came from Table A.5 of Smith (1975), p. 92. [Originally from Historical Statistics of the U.S., series G 107-l 12, mean family personal income of 1962. After 1962, median income figures came from the Statistical Abstract of the U.S., (Dept. of Commerce, pub.). For the top quintile the median figures are adjusted by the ratio of median to mean estimated from the years prior to 19621. Maximum likelihood estimation of eqn (8) yielded the information in Table 1. The last column gives the coefficients obtained by Smith (Table Cl, p. 102, (1975)) of a, when p = 1. The coefficient a, represents the coefficient on permanent income (actual income when l3 = 1) and a2 the coefficient on transitory income (actual income when p = 0). The pattern of l3 coefficients strongly suggests that adjustments to changes in permanent income take place much more rapidly the higher the level of income. For quintile 4, many of the results did not seem to follow the pattern set by the other quintiles. This phenomenon can be partially explained by observing the data (see data sources above). For that quintile the percentage of families purchasing a car shifted dramatically from 10% to 16% in 1964, and remained at the higher level. Other quintiles did not shift as dramatically or remain at the higher level. Smith’s results are also distorted, as he points out (1975, p. 81). Nonlinear estimation of eqn (8) suggests that the estimated value of l3 is significant (at
Expectations
on automobile
335
demand
least asymptotically) for quintiles 2, 3, and 5. An observable pattern does not seem to be a characteristic exhibited by the asymptotic standard errors. Table 2 gives the estimated structural coefficients (with r-values in parenthesis) and negative log-likelihoods for each beta of interest (p = 0, MLE p and p = I) by quintile. Likelihood ratio tests show the MLE p to be significantly different (at the .05 level) from either 0 or 1 in all but the first (lowest) income quintile. In Fig. 1 there is a set of three curves for each quintile showing how S,(p) varies with p: 1) with 2, and Z2 included in eqn (8), 2) with Z, only included (i.e. Z2 omitted) and 3) with the autocorrelation parameter, p, added. The third curve uses the envelope of curves S,(p, p), with the p which minimizes S,(p). Note that different scales are used for each quintile. Curves of type 2) and 3) are discussed below. It is clear that with ZI and Z2 included in eqn (8), a definite trend in betas minimizing S, is observed. 2.4 Asymptotic considerations The contribution of the variable, Z2, = (1 - p)’ becomes negligible as I -+ m. Furthermore, Z2, * 0 faster as t -+ m the closer p is to 1 .O. Therefore it is of interest to observe trends in the contribution of Z2, across quintiles, as measured by the significance of the estimated truncation remainder yt , and the stability of fi in each quintile, especially since we are dealing with a relatively short time series (T = 16). Also of interest is the effect of omitting Zzl from, and the inclusion of the autocorrelation parameter into, eqn (8). Nonlinear estimation of the structural coefficients by quintile shows the truncation remainder decreases for the first four quintiles but is statistically insignificant in each case. Only in the fifth quintile where fi is the largest does y$ become significant (with a t-statistic of about 5). This fact is contrary to what one would expect, since the faster a quintile reacts to changes in permanent income, the less weight is placed on older values of permanent income. Among other possible explanations, it is suspected that the rapidly declining series of Z2, has led to ill-conditioned matrices in the nonlinear program. Nevertheless, its significance points out the desirability of including the term in short time series studies. Next, the effect of omitting Z2, from eqn (8) can be observed by referring to Fig. 1 and Table 3. Without ZZ,, the trend in p is now decreasing, rather than increasing, with income, and the standard error of the equation is now smaller in quintiles 2 and 3. There was no pattern to the significance pf the s’s and only in quintile two was fi significant (at the .05 level). The effect on p of including an autocorrelation parameter, p, into the equation is dramatic in the first four quintiles compared to the p’s when Z, and Zz are included (again referring to Fig. 1 and Table 2). The Hildreth-Lu procedure was used to estimate p. The residual sum of squares decreases in quintiles 1, 3, and 4. In the fifth quintile the p of .5 with Z, and Z2 is confirmed. The effect in the first quintile is rather peculiar, fi jumping from .OOl to .999. However the relatively small variation observed in that quintile’s purchase
Table 2. Estimated
structural
coefficients MLE 6
6=0
6=1 -1ogL
a1
-1ogL
.3071 (~562)
14.251
.3070 (1.62)
15.523
1.120 (1.23)
.7691 (1.79)
21.046
.3265 (2.60)
26.360
a2
-1ogL
.3137 LO331
.2876 (.744)
15.521
2 -.06541 C.0397)
.4252 (.961)
26.325
r6419 (1.16)
.7455 (2.96)
22.226
(6.6;
.4274 (3.06)
.6466 (1.66)
19.340
.3147 (6.72)
23.695
-.3191 C.176)
.2628 t.940)
34.765
.OOl LOOi?
56.43 C.007)
.1543 C.466)
32.240
.1665 (3.15)
34.633
5 -.05611 C.767)
.32’77 (2.35)
39.346
.OEOl (1.39)
.6673 (2.66)
37.066
.0675 (1.734)
41.370
(16.;
a1 1
.¶I
by quintile
6 .OOl C.021) (5.0:
a1 144.6 t.0227)
a2
;I
2”
g 4
Note: al = coeff. on permanent income a = coeff on transitory income Studant t statistics in parentheses
H. E. DYCK
336
2.x)
4
2.20
i
2 .lO
1.10
-I
4 ;
3
1.00
G
3.40
1.30 2 1.20
1.10
1
1i
Zl&Z2 -.ZlOdY Zl&Z2WIthP,in----
Key:
Fig. I. S.(p) vs. p by income quintile.
Table 3. Effect on 0 of omitting Zz and of including Quintile ^s Zl EJz2
2
3
4
5
.OOl
.2
.3
.OOl
.s
S,“s
.048
.040
.035
.142
.021
5.563
13.011
10.508
52.703
96.618
.3
.3
.3
.2
.l
S,i
.789
.130
.212
.356
.I16
RSS( 8)
6.312
13.269
10.590
64.524
122.554
^s Zl& z2 with P
1
RSSCc1
i Zl OflY
fi
Sip RSS( s, 3
.999
.4
.4
.2
.5
.Ol
.Ol
.Ol
.09
.Ol
4.475
12.299
9.463
45.036
97.026
Expectations on automobile demand
rates makes it relatively unimportant. Also the comparatively alternatives makes the choice of fi of little consequence.
337
level curves under each of the
3. CONCLUSION
The preceding results suggest that the asymptotically zero variable Zz, should be included, particularly for studies incorporating short time series. The inclusion of the autocorrelation term p, however, led to insignificant values of 0 = .OOl in most cases, dramatic changes in fi, and inconclusive results. Additional work was done to test the role of attitudinal variables [see Dyck (1984)]. Basically the inclusion of such variables rendered the income variable insignificant for most of the quintiles. This was not unexpected, however, for the attitudinal index should depend largely on income.
REFERENCES Dhrymes P. J. (1971) Disrribured Lags: Problems of Estimationand Formulation. Holden-Day, Inc., San Francisco. Dyck H. E. (1982) Econometric Models of AutomobileDemand. Ph.D. Thesis, Purdue University. Dyck H. E. and Kadiyala K. R. (1984) Alternative Measures of Expectation and their Effect on Automobile Demand by Income Quintile, Krannert Institute Paper, Purdue University. Smith, R. P. (1975) Consumer Demandfor Curs in the USA. Cambridge University Press, Cambridge. Zellner A. and Geisel M. S. (1970) Analysis of distributed lag models with applications to consumption function estimation. Econometrico 36, 865-888.
Vol. ZIB.No.4. Brim
pp. 331-337.
1987 6
THE ROLE OF EXPECTATIONS AUTOMOBILE DEMAND
0191-2615/87 $3.00+ .oO 1987 F’ergamon loumalr Lid.
ON
HAROLD E. DYCK Southwest Missouri State University, Springfield, Missouri 65804-0095, U.S.A. (Received 30 November
1984; in
revisedform
15 September
1986)
Abstract-This study investigates the influence of adaptive expectations on the purchase of automobiles by income quintile. Through maximum likelihood estimation, it is found that the coefficients of adaptation to income exhibit a trend towards faster rates in the upper quintiles. The effects of a “truncation remainder” and of serial correlation are also noted.
1. INTRODUCTION
This study investigates the influence of adaptive expectations on the purchase of automobiles by income quintile. Through maximum likelihood estimation, it is found that the coefficients of adaptation to income exhibit a trend towards faster rates in the upper quintiles. The effects of a “truncation remainder” and of serial correlation are also noted. The main focus of this investigation is the division of income into permanent and transitory income. Unlike other studies, the present study does not insist on the uniform application of the adaptive expectations assumption across quintiles. The coefficient of adaptation is estimated by use of maximum likelihood technique described in Dhrymes (1971), Zellner and Geisel (1970), and Dyck (1982, 1984). We begin by assuming an adaptive expectations model where the change in expected or permanent income is proportional to the difference between the current period’s observed income level and the period’s previously formed expected value. The rate of adjustment, /3, is the coefficient of adaptation. If p = 0, there is no adjustment, whereas, if p = 1, the adjustment is instantaneous. There seems to be no reason, a priori that the rate of adjustment should be the same in each quintile. Thus, the alternative hypothesis that the rates of adjustment are different is tested.
2. PERMANENT
INCOME
2.1 The restricted model Permanent income, yr* , is estimated & la Friedman as the sum of two parts: at, a trend value, and a geometrically declining weighted average of deviations from past values of the trend. Thus, y:=w+p~(l-p)jD,-j J=o
t=i
,...,
T,
(1)
where D, = yI - at. This formulation is designed to allow for “predicted secular growth.” The weight parameter, p, is similar to our coefficient of adaptation. Once a series for permanent income is derived, it is used to predict purchase rates of automobiles for the five quintiles in the equation, 4, = a,
+ U,Y,* + Uz(Y, 331
TR 21:48-P
Y?) + U,r
(2)
332
H. E. DYCK
where q1 is the purchase rate of a particular quintile, and the difference (y, - y,*) represents transitory income. The restricted model is estimated by Smith (1975) for three values of l3: 0.0, 0.4, and 1.O. It is assumed that all quintiles have the same value of l.3.When p = 0.0, then the consumers are using income trend only to estimate permanent income. Thus, for p = 0 we have: y: = or which implies that q, =
a0
+
u*y, + (a, -
u*)cxt + u,.
(3)
But if p = 1.O, then there is no adjustment towards the long-run trend and permanent income. Therefore, we have YI* =
at + (y, - at) = y,
which implies that 4r =
=o + QIY, +
(4)
4.
That is, current income is used by the consumer as the best prediction for permanent income. Finally, when p = 0.4 we have a partial adjustment to long-run trend. Smith truncates the lags after five years due to the short time series data used and rounds up the latter coefficients “to compensate for earlier periods. ” The model proposed by Smith has several shortcomings: only three values of p are compared, all income quintiles are assumed to have the same value of l3, and for l3 = 0.4, the lag is truncated after five years because of the short time series. There are other problems as well. It is not clear how cx is estimated. It should be estimated simultaneously with l3 as part of the expectational hypothesis. Further, the random error term is attached to the final regression equation determining each quintile purchase frequency rather than to the equation describing the formation of expectations. To circumvent these problems, the following model employing the maximum likelihood of parameters is proposed. 2.2 The unrestricted model Rather than truncate after five years, we propose the following derivation, beginning with eqn (1). Note that it is possible to attach an error term to eqn (1) and carry it along without complicating it into a second or higher order term. From eqn (l), we have [See Dyck (1982)]. * _ 41 Yr -
- P)
P
+ (3
C(1 -
PYY,-j + 4
I=0
From the above equation, if l3 = 0, then, y: = ctt + u,, and
(3
41 = =o + o(a, - oz)t +
a2yt +
(a,
-
Q2h4,
by substitution into eqn (2). On the other hand, if l3 = 1, then, yr* = at + (y, - at)
+ u,
= y, + u,, and 4, = a0 +
@Y,
+
(6) (4
-
a2h,.
333
Expectations on automobile demand
The only difference so far in the final equation with p = 0 and p = 1 are in the error terms. Now, with 0 < p < 1, we have
YP(cyTPI
=
41 - P)
P
+ PUP, 3) + PC1- P>'Yo*+
= 41 - P) + PZh(P,Y,) + 9Yo*UP)
4
+ u,
P
(7)
1-l
where Z,, = c
j=O
(1 - WY,-,,
Gr
=
(1 - p)‘,
yo* = c j=O
(1 -
P)jY-j
and9, = (Y,, Y,-~, . . . , Y& From eqn (2), 4, = a0 +
=
a,Y?
+ %(Y, -
y3 + u,
(
i
- P) + azy, + P(a, - %P,, + Pyo*(u, - u*)Z*, +
a, + ; (4 - Ml
(where E, = (a, - a&,) or 4, = bo + b,y, +b,Z,,+b3Z,,+c,
t=
I,...,
E,
(8)
T.
Equations 7-8 show explicitly how the coefficients of the reduced form of the model (the b’s) depend joinr2y on the structural coefficients of the model and so must be estimated simultaneously. Intuitively, eqn (8) states that while the purchase rate of a particular quintile is
a function of permanent income and transitory income, the way permanent income is measured allows the purchase rate to be estimated as a function of actual income and the two Z’s. The variable Z, is the sum of weighted past sample values of actual income with exponentially declining weights. The asymptotically zero variable, Zz, is used to estimate the ‘Yruncation remainder,” y$, which represents the effect of the presample values on the sample period. Of special interest here is the effect of including Z2 in the equation when a small sample (T = 16) is used. More generally, eqn (8) can be rewritten as: q = Zb + E
where
Maximizing the log-likelihood function
L(b, p, a*jq, Z) = -5
log 2n - 5 log a* - & (q - Zb)‘(q - Zb),
with respect to b and u2 yields 6(p) = (Z’Z)-‘Z’q
and
I%*(P) = ; (q - Zb)‘(q - Zb).
334
H. E. DYCK
1. MLE of em 8
Table MLE
A Quintile
6
6,
I;’
with> = 1
DW
bl
ii2
b2
DW
aI
1
.OOl
A808
.1208
3.1
.3071
.1443
2.7
.3062
2
.2
.0410
.5656
2.1
.7691
.0703
.2296
1.3
.3271
3
.3
.9358
.8328
2.3
.6466
-.0658
.7296
1.4
.3147
4
.OOl
.0960
.4706
1.8
.1543
.0563
.3327
1.3
.1866
5
.5
A380
.3975
2.7
.6673
-.2936
.0592
1.7
.0675
-.0376
A search can then be carried out over the range of p to find the global maximum of the log likelihood function. The asymptotic standard errors of the structural coefficients are found in the manner of Dhyrmes (1971) and Appendix B of Dyck (1982). To investigate the effects of serial correlation, let Et =
Qrlrrlt+x) =
W-I
+ r(,Y
1
where In,/ < 1,
E(q,)
= 0 and
aq*fors = 0 for s _# 0
0
Then eqn (8) becomes 91 =
bo + blY,
+
b*Z,,
+
w2,
+
E,
where E, = pe,-i 9f -
P4r-1 = bo(l -
P) + MY,
-
PY,-1) +
b*(Z,,
+ -q,, and -
PZ,,-J
+
b&G,
-
pZ*,_,) + q,. (9)
The regressions can then be carried out using the transformed variables of eqn (9) without violation of the usual assumptions regarding the error term, since q, - NID(0, at). 2.3 Empirical results The data sources used in the present study are as follows. The new car purchase rates by quintile and attitudes came from Table A.2 of Smith (1975), p. 90. (Originally from the University of Michigan Survey Research Center, annual Survey of Consumer Finance). The average income per quintile in current dollars came from Table A.5 of Smith (1975), p. 92. [Originally from Historical Statistics of the U.S., series G 107-l 12, mean family personal income of 1962. After 1962, median income figures came from the Statistical Abstract of the U.S., (Dept. of Commerce, pub.). For the top quintile the median figures are adjusted by the ratio of median to mean estimated from the years prior to 19621. Maximum likelihood estimation of eqn (8) yielded the information in Table 1. The last column gives the coefficients obtained by Smith (Table Cl, p. 102, (1975)) of a, when p = 1. The coefficient a, represents the coefficient on permanent income (actual income when l3 = 1) and a2 the coefficient on transitory income (actual income when p = 0). The pattern of l3 coefficients strongly suggests that adjustments to changes in permanent income take place much more rapidly the higher the level of income. For quintile 4, many of the results did not seem to follow the pattern set by the other quintiles. This phenomenon can be partially explained by observing the data (see data sources above). For that quintile the percentage of families purchasing a car shifted dramatically from 10% to 16% in 1964, and remained at the higher level. Other quintiles did not shift as dramatically or remain at the higher level. Smith’s results are also distorted, as he points out (1975, p. 81). Nonlinear estimation of eqn (8) suggests that the estimated value of l3 is significant (at
Expectations
on automobile
335
demand
least asymptotically) for quintiles 2, 3, and 5. An observable pattern does not seem to be a characteristic exhibited by the asymptotic standard errors. Table 2 gives the estimated structural coefficients (with r-values in parenthesis) and negative log-likelihoods for each beta of interest (p = 0, MLE p and p = I) by quintile. Likelihood ratio tests show the MLE p to be significantly different (at the .05 level) from either 0 or 1 in all but the first (lowest) income quintile. In Fig. 1 there is a set of three curves for each quintile showing how S,(p) varies with p: 1) with 2, and Z2 included in eqn (8), 2) with Z, only included (i.e. Z2 omitted) and 3) with the autocorrelation parameter, p, added. The third curve uses the envelope of curves S,(p, p), with the p which minimizes S,(p). Note that different scales are used for each quintile. Curves of type 2) and 3) are discussed below. It is clear that with ZI and Z2 included in eqn (8), a definite trend in betas minimizing S, is observed. 2.4 Asymptotic considerations The contribution of the variable, Z2, = (1 - p)’ becomes negligible as I -+ m. Furthermore, Z2, * 0 faster as t -+ m the closer p is to 1 .O. Therefore it is of interest to observe trends in the contribution of Z2, across quintiles, as measured by the significance of the estimated truncation remainder yt , and the stability of fi in each quintile, especially since we are dealing with a relatively short time series (T = 16). Also of interest is the effect of omitting Zzl from, and the inclusion of the autocorrelation parameter into, eqn (8). Nonlinear estimation of the structural coefficients by quintile shows the truncation remainder decreases for the first four quintiles but is statistically insignificant in each case. Only in the fifth quintile where fi is the largest does y$ become significant (with a t-statistic of about 5). This fact is contrary to what one would expect, since the faster a quintile reacts to changes in permanent income, the less weight is placed on older values of permanent income. Among other possible explanations, it is suspected that the rapidly declining series of Z2, has led to ill-conditioned matrices in the nonlinear program. Nevertheless, its significance points out the desirability of including the term in short time series studies. Next, the effect of omitting Z2, from eqn (8) can be observed by referring to Fig. 1 and Table 3. Without ZZ,, the trend in p is now decreasing, rather than increasing, with income, and the standard error of the equation is now smaller in quintiles 2 and 3. There was no pattern to the significance pf the s’s and only in quintile two was fi significant (at the .05 level). The effect on p of including an autocorrelation parameter, p, into the equation is dramatic in the first four quintiles compared to the p’s when Z, and Zz are included (again referring to Fig. 1 and Table 2). The Hildreth-Lu procedure was used to estimate p. The residual sum of squares decreases in quintiles 1, 3, and 4. In the fifth quintile the p of .5 with Z, and Z2 is confirmed. The effect in the first quintile is rather peculiar, fi jumping from .OOl to .999. However the relatively small variation observed in that quintile’s purchase
Table 2. Estimated
structural
coefficients MLE 6
6=0
6=1 -1ogL
a1
-1ogL
.3071 (~562)
14.251
.3070 (1.62)
15.523
1.120 (1.23)
.7691 (1.79)
21.046
.3265 (2.60)
26.360
a2
-1ogL
.3137 LO331
.2876 (.744)
15.521
2 -.06541 C.0397)
.4252 (.961)
26.325
r6419 (1.16)
.7455 (2.96)
22.226
(6.6;
.4274 (3.06)
.6466 (1.66)
19.340
.3147 (6.72)
23.695
-.3191 C.176)
.2628 t.940)
34.765
.OOl LOOi?
56.43 C.007)
.1543 C.466)
32.240
.1665 (3.15)
34.633
5 -.05611 C.767)
.32’77 (2.35)
39.346
.OEOl (1.39)
.6673 (2.66)
37.066
.0675 (1.734)
41.370
(16.;
a1 1
.¶I
by quintile
6 .OOl C.021) (5.0:
a1 144.6 t.0227)
a2
;I
2”
g 4
Note: al = coeff. on permanent income a = coeff on transitory income Studant t statistics in parentheses
H. E. DYCK
336
2.x)
4
2.20
i
2 .lO
1.10
-I
4 ;
3
1.00
G
3.40
1.30 2 1.20
1.10
1
1i
Zl&Z2 -.ZlOdY Zl&Z2WIthP,in----
Key:
Fig. I. S.(p) vs. p by income quintile.
Table 3. Effect on 0 of omitting Zz and of including Quintile ^s Zl EJz2
2
3
4
5
.OOl
.2
.3
.OOl
.s
S,“s
.048
.040
.035
.142
.021
5.563
13.011
10.508
52.703
96.618
.3
.3
.3
.2
.l
S,i
.789
.130
.212
.356
.I16
RSS( 8)
6.312
13.269
10.590
64.524
122.554
^s Zl& z2 with P
1
RSSCc1
i Zl OflY
fi
Sip RSS( s, 3
.999
.4
.4
.2
.5
.Ol
.Ol
.Ol
.09
.Ol
4.475
12.299
9.463
45.036
97.026
Expectations on automobile demand
rates makes it relatively unimportant. Also the comparatively alternatives makes the choice of fi of little consequence.
337
level curves under each of the
3. CONCLUSION
The preceding results suggest that the asymptotically zero variable Zz, should be included, particularly for studies incorporating short time series. The inclusion of the autocorrelation term p, however, led to insignificant values of 0 = .OOl in most cases, dramatic changes in fi, and inconclusive results. Additional work was done to test the role of attitudinal variables [see Dyck (1984)]. Basically the inclusion of such variables rendered the income variable insignificant for most of the quintiles. This was not unexpected, however, for the attitudinal index should depend largely on income.
REFERENCES Dhrymes P. J. (1971) Disrribured Lags: Problems of Estimationand Formulation. Holden-Day, Inc., San Francisco. Dyck H. E. (1982) Econometric Models of AutomobileDemand. Ph.D. Thesis, Purdue University. Dyck H. E. and Kadiyala K. R. (1984) Alternative Measures of Expectation and their Effect on Automobile Demand by Income Quintile, Krannert Institute Paper, Purdue University. Smith, R. P. (1975) Consumer Demandfor Curs in the USA. Cambridge University Press, Cambridge. Zellner A. and Geisel M. S. (1970) Analysis of distributed lag models with applications to consumption function estimation. Econometrico 36, 865-888.