- Email: [email protected]
A comment on durable goods consumption
JOURNALOF
Monetary ELSEV1ER
Journal of Monetary Economics 37 (1996) 381 391
ECONOMICS
A comment on durable goods consumption Kiseok Hong Department of Economics, Harvard University, Boston, MA 02138, USA
(Received February 1994;final version received January 1996)
Abstract This paper examines the time series behavior of durables/nondurables and services consumption expenditure of six OECD countries, following Caballero's approach. In each country, durables expenditure exhibits greater reversion than nondurables and services, when lower frequencies of the series are examined. The results reject the frictionless PIH as a model for durable goods consumption. However, they strongly support Mankiw's hypothesis on the nature of durable goods consumption. Key words: Autocorrelation; Durable goods; Moving average process; Permanent in-
come hypothesis J E L classification: E21
1. Introduction In one of his early papers, Mankiw (1982) noticed that according to the standard permanent income hypothesis (PIH), the growth rate of consumption expenditure on durable goods should follow a first-order moving average (MA(I)) process with the MA coefficient equal to the negative of one minus the rate of depreciation of durable goods stock, while the expenditure on nondurables and services should follow a random walk. Intuitively, past purchase of durable goods should affect current expenditure on those goods, and therefore we expect durables expenditure to exhibit different time series behavior from nondurables and services expenditure. Mankiw tested this implication of the
I am grateful to N. Gregory Mankiw and Ricardo J. Caballero for their many helpful comments.All errors are mine. 0304-3932/96/$15.00 © 1996Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 3 9 3 2 ( 9 6 ) 0 1 2 4 5 - 7
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
382
PIH with postwar U.S. quarterly data and, contrary to the theory, found no significant difference between nondurables and services expenditure and durables expenditure; each expenditure series seemed to follow a random walk. Caballero (1990b) suggested that if people respond to a shock with some lags in their consumption behavior, the change in durables expenditure will follow a higher-order than a first-order MA process, and therefore the difference between the two expenditure series may not be captured by MA(1) estimation. He tested his model with U.S. quarterly/annual data and found that the two expenditure series do behave differently and the difference is consistent with the predictions of the model. In particular, the data shows a clear reversion of the impact of an initial shock in durables expenditure. This paper attempts to extend Caballero's result to six O E C D countries and finds that in each country durables expenditure exhibits the reversion implied by the theory, once lagged responses are allowed in consumption behavior. While the frictionless PIH is rejected by the results, the original hypothesis suggested by Mankiw (1982) - that durable consumption expenditures exhibit a larger degree of reversion than nondurables and services - is strongly supported.
2. Theory The accumulation of durable goods stock over time can be represented as follows: K t = (1 - 6 ) K t _ 1 + Ct,
(1)
with K t durables stock at time t, 6 the depreciation rate of durables stock which is a constant between zero and one, and Ct durables expenditure at time t. If we assume that the flow of service from durables stock is a constant fraction of the stock, the standard PIH implies AK, = u,,
(2)
where ut represents a shock at time t to the consumption of durable goods. From Eqs. (1) and (2) we can derive A C , = u, -
(1 - 6 ) u , _ 1 .
(3)
The standard PIH, combined with several simplifying assumptions, implies that the change in durables expenditure should follow an MA(1) process, with the MA coefficient equal to the negative of one minus 3. Note that nondurables and services expenditure can be thought of as a special case with 6 = 1. Suppose, for whatever reasons, that the standard PIH is wrong and the consumption process is better described by a more general MA process, A K = ut + oq u t - 1 + o~2ut 2 ~-
" ' "
~- O~nUt-n"
(4)
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
383
C o n s u m p t i o n expenditure is n o w represented by ACt=u, +(~x-(1-f))u,_l
+(~2-~1(1-3))u,
z+
"'"
+ (0~n- 0~n- 1(1 -- 6))U,-n -- ~n(l -- CS)Ut-n-1 = u, + f l l u ,
1 + f12u,-2 +
... + fl, u, n + f l n + l U , - . - 1 .
(5)
If we estimate Eq. (3) while c o n s u m p t i o n expenditure follows Eq. (5), we m a y not see the difference between the two expenditure series because Eq. (3) misspecifies the true process. With more than one M A term for the expenditure series as in Eq. (5), one succinct way to capture the difference between the two series is to c o m p a r e the sum of autocorrelation coefficients, a It can be shown that the sum of autocorrelations for Eq. (5) equals (Y~i/31 + Zi ~jfld3j)/(1 + Y~/32). Therefore, if the sum of the /~'s is close to - 1, which is the case regardless of the values of the c~'s whenever 6 is close to zero, the sum of autocorrelations is approximately - 0.5. If the sum of the /~'s is 0.35, which is Caballero's favorite estimate for nondurables and services expenditure in U.S., the sum of autocorrelations is a b o u t 0.3. A n o t h e r way to capture the difference between the two series as implied by Eq. (5) is to estimate the M A coefficients (/~'s) directly. We expect the sum of the M A coefficients to be close to - 1 for durables expenditure and typically some positive n u m b e r for nondurables and services expenditure. N o t e that Eq. (5) is derived by combining Eq. (4) with several assumptions on the nature of durable goods consumption. Therefore, it does not represent any particular c o n s u m p t i o n hypothesis unless an explicit explanation is given for why c o n s u m p t i o n exhibits lagged responses as in Eq. (4).
3. Empirical results To test the model, I used annual data on durables / nondurables and services expenditure of six O E C D countries as reported by the OECD National Accounts. The estimation period is from 1951 to 1990 for Italy and N o r w a y and from 1950 to 1990 for Canada, D e n m a r k , France, and U.K. The selection of countries was solely determined by the availability of the data. All the series are in logarithms. 2
1A finite-order AR approximation of Eq. (5) will still misspecifythe model, because the importance of MA terms does not diminish by the approximation. I follow Caballero's (1990b) approach in comparing autocorrelation coefficients and MA coefficients of the two expenditure series. 2Even though the model in Section 2 refers to the levels, not the logarithms, of consumption expenditure, taking logarithms does not change the basic nature of the model. See the Appendix for details. Caballero (1993) also takes logarithms of durable goods expenditure, noting that the difference in levels and logarithms have similar time series properties and that exponential detrending and heteroskedasticity adjustment are more convenient with logarithms.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
384
Canada
2.5 o
2.0 o
1.5
e8
2.0: 1.5
1.0 0.5
Denmark
2.5
g o
s
0.5
0 -o.5
f J
Y
0 --=
-oz
_
-1.0
-1.0 2
3
4
5
6
7
8
9
34
10
89
Year 4 - durables
--*-- nondurables and services
-,,- durables
+
nondurables and services
France
Italy
2.5
.£ "U
8
2.5
2.0
o
1.5 1.0
S
2.0
'~
1.5
b
1.0
o
0.5
~
0.5
o -0.5
~ E
-1.0
o -o.5 -1.0
2
3
4
5
6
7
8
9
10
2
3
4
Year -4'- durables
--,-- nondurables and services
-,,- durables
~
o
1.0
o
=
7
8
9
10
U.K.
1.5
2.0 1.5
6
nondurables and services
Norway
"~ .-g
5
Year
2.5 o
io
Year
o
1.0
o
0.5
o
05
0
o E # -0.s
~= -0.5 ce
-1.0
-1.0 2
3
4
5
6
7
8
9
10
2
Year -~- durables
3
4
5
6
7
8
9
10
Year
- ~ nondurables and services
--- durables
Fig. 1. Sum of autocorrelations.
--*- nondurables and services
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
385
Table 1 Sum of MA coefficients (D = durables, ND = nondurables, S = services) MA(1) Canada
Denmark
France
Italy
Norway
U.K.
MA(2)
MA(3)
MA(4)
MA(5)
D
0.248 (0.158)
0.269 (0.274)
- 0.910 (0.117)
- 0.918 (0.121)
- 1.381 (0.484)
ND + S
0.629 (0.135)
1.082 (0.253)
1.160 (0.469)
1.002 (0.765)
3.785 (0.749)
D
0.170 (0.166)
0.147 (0.263)
- 0.126 (0.321)
- 0.293 (0.360)
- 0.888 (0.213)
ND + S
0.419 (0.153)
0.326 (0.302)
1.086 (0.293)
1.952 (0.376)
1.699 (0.643)
D
0.162 (0.164)
0.253 (0.257)
0.270 (0.323)
0.595 (0.361)
0.610 (0.466)
ND + S
0.757 (0.117)
1.196 (0.300)
1.863 (0.428)
1.956 (0.722)
1.682 (0.814)
D
0.370 (0.161)
0.040 (0.278)
- 0.145 (0.323)
- 0.195 (0.361)
- 0.394 (0.370)
ND + S
0.519 (0.146)
0.504 (0.303)
1.737 (0.229)
2.106 (0.626)
1.663 (0.898)
D
0.152 (0.167)
0.140 (0.260)
0.022 (0.336)
0.034 (0.428)
0.205 (0.562)
ND + S
0.471 (0.151)
0.682 (0.290)
0.711 (0.413)
0.315 (0.515)
0.538 (0.990)
D
0.248 (0.164)
0.928 (0.065)
- 1.245 (0.184)
- 1.268 (0.241)
-0.932 (0.082)
ND + S
0.956 (0.046)
0.693 (0.330)
0.576 (0.442)
0.311 (0.565)
- 0.683 (0.313)
Note: Each expenditure series is log differenced. Standard errors are in parentheses.
B e f o r e t h e e s t i m a t i o n , I e x a m i n e d t h e t i m e series p l o t o f t h e c h a n g e s in e a c h c o n s u m p t i o n e x p e n d i t u r e . I t w a s c l e a r t h a t in e a c h c o u n t r y d u r a b l e s e x p e n d i t u r e f l u c t u a t e d m o r e in t h e s e n s e t h a t a p o s i t i v e d e v i a t i o n f r o m t h e m e a n t e n d e d to be followed by a negative one, 3 suggesting the reversion implied by the theory. I first e s t i m a t e d t h e a u t o c o r r e l a t i o n c o e f f i c i e n t s o f t h e c h a n g e s i n e a c h c o n s u m p t i o n e x p e n d i t u r e f o r e a c h c o u n t r y u p t o t e n lags. T h e c u m u l a t e d s u m o f
3For the countries whose log differenced expenditure series have a time trend, deviations from the trend exhibit this pattern.
K. Hong /Journal of Monetary Economics 37 (1996) 381 391
386
Canada .o
1.0
1.01 0.5
0.5
~
0
0
' t ~ -0.5 -1.0
, 1
, 2
, 3
~ -0.5
1.0
oV:
0.5
5
6
7
8
9
10
2
6 7 8 9 10 Year --.- nondurables and services
3
4
~ durables
5
Italy
1.5 o
8o
1.0 0.5
0
-0.5 -1.0
~ ~ ~ ~ ~ ~ ~ ~;o
~ -0.5 m -1.0
2 3 4 5 6 7 8 9 1 0 ~ar --- durables --,- nondurables and services
Year --~ nondurables and services
-~- durables
Norway
1.5 .~
1
France
1.5 o
~
-1.0 ~ 4
Year --*- nondurables and services
-4,- durables
o ,--t
Denmark
1.5
U.K.
1.5 o
1.o
1.0
ol
0.5
0.5 o
~
0
~ -0.5 -1.0
~ -0.5 -1.0
2
-~- durables
3
4
5
6
7
8
9
10
Year --,- nondurables and services
, 1
2
3
-~- durables
4
5
6
7
8
9
1
0
Year --,- nondurables and services
Fig. 2. Sum of autocorrelations: Detrended series.
K. Hong /Journal of Monetary Economics 37 (1996) 381-391
387
Table 2 Sum of MA coefficients (D = durables, N D = n o n d u r a b l e s , S = services): Detrended series MA(1)
Canada
Denmark
France
Italy
Norway
U.K.
MA(2)
MA(3)
MA(4)
MA(5)
D
0.236 (0.158)
0.246 (0.270)
- 0.886 (0.116)
- 0.898 (0.110)
- 0.920 (0.125)
ND + S
0.490 (0.146)
0.357 (0.330)
0.890 (0.448)
0.347 (0.576)
0.817 (0.699)
D
0.117 (0.166)
- 0.956 (0.060)
- 0.958 (0.081)
- 0.878 (0.104)
- 0.868 (0.181)
ND + S
0.336 (0.158)
0.201 (0.286)
1.035 (0.300)
1.893 (0.385)
0.317 (0.567)
- 0.187 (0.168)
- 0.320 (0.223)
- 0.902 (0.111)
- 0.879 (0.133)
- 0.846 (0.156)
ND + S
0.683 (0.129)
1.021 (0.308)
1.108 (0.472)
1.152 (0.606)
0.976 (0.723)
D
0.363 (0.161)
- 0.025 (0.270)
- 0.317 (0.298)
- 0.407 (0.311)
- 0.864 (0.169)
ND + S
0.441 (0.153)
0.338 (0.291)
0.065 (0.422)
0.915 (0.503)
0.056 (0.611)
D
0.069 (0.166)
- 0.170 (0.235)
- 0.472 (0.241)
- 0.881 (0.121)
- 0.833 (0.124)
ND + S
0.453 (0.151)
0.610 (0.289)
0.366 (0.374)
- 0.267 (0.380)
- 0.745 (0.285)
D
0.258 (0.163)
- 0.916 (0.067)
- 0.954 (0.076)
- 1.350 (0.308)
- 0.965 (0.038)
ND + S
0.951 (0.048)
0.673 (0.327)
0.541 (0.433)
0.387 (0.538)
- 0.656 (0.375)
D
Note: Each expenditure series is log differenced. Standard errors are in parentheses.
the autocorrelations is graphed in Fig. 1. The sum is consistently lower for durables expenditure and, for most countries, as low as -0.5 at its minimum as implied by the theory. For France and Norway, however, the sum does not seem to be fully consistent with the theory. Next, I directly estimated MA(1) through MA(5) processes for each expenditure series. For space limitations, only the sums of the MA coefficients are reported in Table 1, and they are consistent with the results in Fig. 1. In most countries, the sum is close to - 1 for durables, if we assume an MA(4) or MA(5) process, and consistently positive for nondurables and services. Again, France and Norway are exceptions; the sum is not significantly negative for durables in these countries.
388
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
The high values of the sum of autocorrelations observed in some countries suggest a possible time trend in the expenditure series. In fact, I did find a significantly negative time trend for some countries, including France. Autocorrelation coefficients are not well defined for a nonstationary series. In particular, the existence of a time trend will bias upward the estimated autocorrelations. Therefore, I detrended each series for each country, and Fig. 2 and Table 2 report the estimation results from the detrended series. The changes in the estimates are particularly remarkable for France. The estimates do not change much for the countries without significant time trends. Overall, the detrended series behave more consistently with the theory. Although I have not reported the individual MA coefficients here, upon examining them, I found that the implied ~'s tend to be larger for durables consumption. ¢ This suggests that people respond to shocks with more lags in their durables than in nondurables and services consumption. One reason for this may be the presence of transactions costs. Purchasing durable goods often requires a considerable amount of time and information, and since the secondary markets for durable goods are far from perfect, reselling is costly as well. Bertola and Caballero (1990) and Grossman and Laroque (1990) show that threshold behavior can be derived as an optimal rule in the presence of transactions costs. Under the (S, s) rule, durables are allowed to deviate from their optimal level until some threshold is reached; only those close to the trigger point will respond to a shock immediately.
4. Conclusion I examined the time series behavior of durables/nondurables and services expenditure in six O E C D countries following Caballero's approach, and found that the durables expenditure exhibits clear reversion as implied by the theory when lower frequencies of the data are examined. Mankiw tested the joint hypothesis of the standard P I H and several assumptions on durable goods consumption and rejected it. The results of this paper suggest that Mankiw's puzzle was not due to the restrictions of his hypothesis on the nature of durable goods but rather to the failure of the standard PIH. Once the standard P I H is relaxed to allow for lagged responses in consumption, the data becomes consistent with Mankiw's hypothesis on durable goods, s
CThe MA coefficientsthat I estimated are the fl's in Eq. (5). The estimates of the ~'s can be derived from the estimates of the fl's. 5Caballero (1993) interprets his results as being consistent with the basic implications of the PIH. However, Section 2 of this paper shows that the results do not support any particular consumption hypothesis.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
389
The existence of lagged responses in consumption has long been recognized (e.g., Flavin, 1981). Christiano et a1.(1991) argue that once time aggregation is taken into account, there is surprisingly little evidence against the PIH. However, even with time aggregation, durables expenditure is still expected to follow an MA(1)process with the MA coefficient close t o - 1 (Heaton, 1993). Therefore, this paper provides some evidence against the time aggregation argument, at least in durables consumption. Some researchers have introduced more complicated preferences such as habit formation (Constantinides and Ferson, 1991; Muellbauer, 1988) and precautionary-saving motive (Caballero, 1990a; Deaton, 1992). These approaches have the potential to explain the lagged responses in consumption, but have not been very successful empirically. Campbell and Mankiw (1989) suggest that consumption exhibits lagged responses because it is too sensitive to current income. I have applied their hypothesis to the durables expenditure series of the six O E C D countries and the U.S., and obtained somewhat mixed results. 6 The importance of fixed adjustment costs may be responsible for the lags in durables consumption. Caballero (1993) finds that the (S, s) model is consistent with aggregate durables expenditure series. Eberly (1994) provides some micro evidence for the model from her analysis of household automobile purchases. Deciding which hypothesis better explains the lags requires further research.
Appendix This appendix shows that the use of logarithms in Section 3 does not change the basic nature of the model. Suppose Eq. (4) describes the movement of the log difference of durable goods stock. Homoskedastic process in logarithms is commonly assumed in the estimation of aggregate time series (e.g., Campbell and Deaton, 1989; MaCurdy, 1982) and implies conditional heteroskedasticity in levels. Now, Eq. (4) can be approximated by +~2ut
AlnKt=ut+~lut_l Kt
-- Kt-
Kt
1
1
2 + ""
KI -- Kt
Kt
1
'
6When applied to durables expenditure, Campbell and Mankiw's hypothesis implies AC,-).(AYt
(I--6)AY, 1)+(1--2)(u,--(1
6)u, 1).
The equation can be estimated consistently by lagging instruments more than one period. For U.S., I could not reject the hypothesis that the coefficients on AYf and AYt 1 are of the same size and opposite sign. The residuals were almost white noise suggesting that the fraction of permanent income consumers may be negligible. For the other six countries, the results were less favorable to the model. Detailed results are available from the author upon request.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
390
for small changes in K. N o t e that Eq. (5) was derived from the following relationship on levels: K, - Kt_ 1 =
(]
--
6)(K,_,
-
Kt-2) + ( C t - C,- 1).
By dividing both sides by Kt-1, we obtain Kt - Kt 1
- (1 -- 6)
Kt-
Kt- 1 - -
1
Kt-2
+
Ct - Ct-1 Kt
K t -- K t - 1 -
Kt-1
Ct- 1
A In C, ~
--
Kt- 1
Ct - C t - 1 C t - 1 Ct
1
Kt (1
Kt
-
1 -- K t - 2
6)
1
,
1 ,
Kt-1
.~ A In K t - (1 - 6) A In K t - 1.
If we a s s u m e K t_ 1/Ct 1 = K -k Ih - 1, where K is a constant close to the inverse of the depreciation and ~/,_ 1 is a mean zero random variable, the autocorrelations of the log difference of consumption expenditure depend in part on the relative magnitude of K and ~/,_ 1. In particular, it can be shown that the sum of autocorrelations is still close to - 0.5 if var(~/)/K 2 is small, which is likely to be the case in practice for the aggregate consumption series. The etimation of MA coefficients is also based on the autocorrelation coefficients and is therefore not much affected by the use of logarithms.
References Bertola, G. and R.J. Caballero, 1990, Kinked adjustment costs and aggregate dynamics, in: O.J. Blanchard and S. Fisher, eds., NBER macroeconomics annual 1990 (MIT Press, Cambridge, MA). Caballero, R.J., 1990a, Consumption puzzles and precautionary savings, Journal of Monetary Economics 25, 113 136. Caballero, R.J., 1990b, Expenditure on durable goods: A case for slow adjustment, Quarterly Journal of Economics 105, 727 743. Caballero, R.J., 1993, Durable goods: An explanation for their slow adjustment, Journal of Political Economy 101,351-384. Campbell, J.Y. and A. Deaton, 1989, Why is consumption so smooth?, Review of Economic Studies 56, 357 373. Campbell, J.Y. and N.G. Mankiw, 1989, Consumption, income, and interest rates: Reinterpreting t h e time series evidence, in: O.J. Blanchard and S. Fisher, eds., NBER macroeconomics annual 1989 (MIT Press, Cambridge, MA). Christiano, L.J., M. Eichenbaum, and D. Marshall, 1991, The permanent income hypothesis revisited, Econometrica 59, 397 423. Constantinides, G.M. and W.E. Ferson, 1991, Habit persistence and durability in aggregate consumption: Empirical tests, NBER working paper no. 3631. Deaton, A., 1992, Understanding consumption (Clarendon Press, Oxford). Eberly, J.C., 1994, Adjustment of consumer's durables stocks: Evidence from automobile purchases, Journal of Political Economy 102, 403-436.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
391
Flavin, M.A., 1981, The adjustment of consumption to changing expectations about future income, Journal of Political Economy 89, 974-1009. Friedman, M., 1957, A theory of the consumption function (Princeton University Press, Princeton, N J). Grossman, S.J. and G. Laroque, 1990, Asset pricing and optimal portfolio choice in the presence of illiquid durable consumption goods, Econometrica 58, 25-51. Heaton, J., 1993, The interaction between time-nonseparable preferences and time aggregation, Econometrica 61,353 385. Mankiw, N.G., 1982, Hall's consumption hypothesis and durable goods, Journal of Monetary Economics 10, 417 425. MaCurdy, T.E., 1982, The use of time-series processes to model the error structure of earnings in longitudinal data analysis, Journal of Econometrics 18, 83 114. M uellbauer, J., 1988, Habits, rationality and myopia in the life cycle consumption function, Annales d'Economie et de Statistique, 47-70.
Monetary ELSEV1ER
Journal of Monetary Economics 37 (1996) 381 391
ECONOMICS
A comment on durable goods consumption Kiseok Hong Department of Economics, Harvard University, Boston, MA 02138, USA
(Received February 1994;final version received January 1996)
Abstract This paper examines the time series behavior of durables/nondurables and services consumption expenditure of six OECD countries, following Caballero's approach. In each country, durables expenditure exhibits greater reversion than nondurables and services, when lower frequencies of the series are examined. The results reject the frictionless PIH as a model for durable goods consumption. However, they strongly support Mankiw's hypothesis on the nature of durable goods consumption. Key words: Autocorrelation; Durable goods; Moving average process; Permanent in-
come hypothesis J E L classification: E21
1. Introduction In one of his early papers, Mankiw (1982) noticed that according to the standard permanent income hypothesis (PIH), the growth rate of consumption expenditure on durable goods should follow a first-order moving average (MA(I)) process with the MA coefficient equal to the negative of one minus the rate of depreciation of durable goods stock, while the expenditure on nondurables and services should follow a random walk. Intuitively, past purchase of durable goods should affect current expenditure on those goods, and therefore we expect durables expenditure to exhibit different time series behavior from nondurables and services expenditure. Mankiw tested this implication of the
I am grateful to N. Gregory Mankiw and Ricardo J. Caballero for their many helpful comments.All errors are mine. 0304-3932/96/$15.00 © 1996Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 3 9 3 2 ( 9 6 ) 0 1 2 4 5 - 7
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
382
PIH with postwar U.S. quarterly data and, contrary to the theory, found no significant difference between nondurables and services expenditure and durables expenditure; each expenditure series seemed to follow a random walk. Caballero (1990b) suggested that if people respond to a shock with some lags in their consumption behavior, the change in durables expenditure will follow a higher-order than a first-order MA process, and therefore the difference between the two expenditure series may not be captured by MA(1) estimation. He tested his model with U.S. quarterly/annual data and found that the two expenditure series do behave differently and the difference is consistent with the predictions of the model. In particular, the data shows a clear reversion of the impact of an initial shock in durables expenditure. This paper attempts to extend Caballero's result to six O E C D countries and finds that in each country durables expenditure exhibits the reversion implied by the theory, once lagged responses are allowed in consumption behavior. While the frictionless PIH is rejected by the results, the original hypothesis suggested by Mankiw (1982) - that durable consumption expenditures exhibit a larger degree of reversion than nondurables and services - is strongly supported.
2. Theory The accumulation of durable goods stock over time can be represented as follows: K t = (1 - 6 ) K t _ 1 + Ct,
(1)
with K t durables stock at time t, 6 the depreciation rate of durables stock which is a constant between zero and one, and Ct durables expenditure at time t. If we assume that the flow of service from durables stock is a constant fraction of the stock, the standard PIH implies AK, = u,,
(2)
where ut represents a shock at time t to the consumption of durable goods. From Eqs. (1) and (2) we can derive A C , = u, -
(1 - 6 ) u , _ 1 .
(3)
The standard PIH, combined with several simplifying assumptions, implies that the change in durables expenditure should follow an MA(1) process, with the MA coefficient equal to the negative of one minus 3. Note that nondurables and services expenditure can be thought of as a special case with 6 = 1. Suppose, for whatever reasons, that the standard PIH is wrong and the consumption process is better described by a more general MA process, A K = ut + oq u t - 1 + o~2ut 2 ~-
" ' "
~- O~nUt-n"
(4)
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
383
C o n s u m p t i o n expenditure is n o w represented by ACt=u, +(~x-(1-f))u,_l
+(~2-~1(1-3))u,
z+
"'"
+ (0~n- 0~n- 1(1 -- 6))U,-n -- ~n(l -- CS)Ut-n-1 = u, + f l l u ,
1 + f12u,-2 +
... + fl, u, n + f l n + l U , - . - 1 .
(5)
If we estimate Eq. (3) while c o n s u m p t i o n expenditure follows Eq. (5), we m a y not see the difference between the two expenditure series because Eq. (3) misspecifies the true process. With more than one M A term for the expenditure series as in Eq. (5), one succinct way to capture the difference between the two series is to c o m p a r e the sum of autocorrelation coefficients, a It can be shown that the sum of autocorrelations for Eq. (5) equals (Y~i/31 + Zi ~jfld3j)/(1 + Y~/32). Therefore, if the sum of the /~'s is close to - 1, which is the case regardless of the values of the c~'s whenever 6 is close to zero, the sum of autocorrelations is approximately - 0.5. If the sum of the /~'s is 0.35, which is Caballero's favorite estimate for nondurables and services expenditure in U.S., the sum of autocorrelations is a b o u t 0.3. A n o t h e r way to capture the difference between the two series as implied by Eq. (5) is to estimate the M A coefficients (/~'s) directly. We expect the sum of the M A coefficients to be close to - 1 for durables expenditure and typically some positive n u m b e r for nondurables and services expenditure. N o t e that Eq. (5) is derived by combining Eq. (4) with several assumptions on the nature of durable goods consumption. Therefore, it does not represent any particular c o n s u m p t i o n hypothesis unless an explicit explanation is given for why c o n s u m p t i o n exhibits lagged responses as in Eq. (4).
3. Empirical results To test the model, I used annual data on durables / nondurables and services expenditure of six O E C D countries as reported by the OECD National Accounts. The estimation period is from 1951 to 1990 for Italy and N o r w a y and from 1950 to 1990 for Canada, D e n m a r k , France, and U.K. The selection of countries was solely determined by the availability of the data. All the series are in logarithms. 2
1A finite-order AR approximation of Eq. (5) will still misspecifythe model, because the importance of MA terms does not diminish by the approximation. I follow Caballero's (1990b) approach in comparing autocorrelation coefficients and MA coefficients of the two expenditure series. 2Even though the model in Section 2 refers to the levels, not the logarithms, of consumption expenditure, taking logarithms does not change the basic nature of the model. See the Appendix for details. Caballero (1993) also takes logarithms of durable goods expenditure, noting that the difference in levels and logarithms have similar time series properties and that exponential detrending and heteroskedasticity adjustment are more convenient with logarithms.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
384
Canada
2.5 o
2.0 o
1.5
e8
2.0: 1.5
1.0 0.5
Denmark
2.5
g o
s
0.5
0 -o.5
f J
Y
0 --=
-oz
_
-1.0
-1.0 2
3
4
5
6
7
8
9
34
10
89
Year 4 - durables
--*-- nondurables and services
-,,- durables
+
nondurables and services
France
Italy
2.5
.£ "U
8
2.5
2.0
o
1.5 1.0
S
2.0
'~
1.5
b
1.0
o
0.5
~
0.5
o -0.5
~ E
-1.0
o -o.5 -1.0
2
3
4
5
6
7
8
9
10
2
3
4
Year -4'- durables
--,-- nondurables and services
-,,- durables
~
o
1.0
o
=
7
8
9
10
U.K.
1.5
2.0 1.5
6
nondurables and services
Norway
"~ .-g
5
Year
2.5 o
io
Year
o
1.0
o
0.5
o
05
0
o E # -0.s
~= -0.5 ce
-1.0
-1.0 2
3
4
5
6
7
8
9
10
2
Year -~- durables
3
4
5
6
7
8
9
10
Year
- ~ nondurables and services
--- durables
Fig. 1. Sum of autocorrelations.
--*- nondurables and services
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
385
Table 1 Sum of MA coefficients (D = durables, ND = nondurables, S = services) MA(1) Canada
Denmark
France
Italy
Norway
U.K.
MA(2)
MA(3)
MA(4)
MA(5)
D
0.248 (0.158)
0.269 (0.274)
- 0.910 (0.117)
- 0.918 (0.121)
- 1.381 (0.484)
ND + S
0.629 (0.135)
1.082 (0.253)
1.160 (0.469)
1.002 (0.765)
3.785 (0.749)
D
0.170 (0.166)
0.147 (0.263)
- 0.126 (0.321)
- 0.293 (0.360)
- 0.888 (0.213)
ND + S
0.419 (0.153)
0.326 (0.302)
1.086 (0.293)
1.952 (0.376)
1.699 (0.643)
D
0.162 (0.164)
0.253 (0.257)
0.270 (0.323)
0.595 (0.361)
0.610 (0.466)
ND + S
0.757 (0.117)
1.196 (0.300)
1.863 (0.428)
1.956 (0.722)
1.682 (0.814)
D
0.370 (0.161)
0.040 (0.278)
- 0.145 (0.323)
- 0.195 (0.361)
- 0.394 (0.370)
ND + S
0.519 (0.146)
0.504 (0.303)
1.737 (0.229)
2.106 (0.626)
1.663 (0.898)
D
0.152 (0.167)
0.140 (0.260)
0.022 (0.336)
0.034 (0.428)
0.205 (0.562)
ND + S
0.471 (0.151)
0.682 (0.290)
0.711 (0.413)
0.315 (0.515)
0.538 (0.990)
D
0.248 (0.164)
0.928 (0.065)
- 1.245 (0.184)
- 1.268 (0.241)
-0.932 (0.082)
ND + S
0.956 (0.046)
0.693 (0.330)
0.576 (0.442)
0.311 (0.565)
- 0.683 (0.313)
Note: Each expenditure series is log differenced. Standard errors are in parentheses.
B e f o r e t h e e s t i m a t i o n , I e x a m i n e d t h e t i m e series p l o t o f t h e c h a n g e s in e a c h c o n s u m p t i o n e x p e n d i t u r e . I t w a s c l e a r t h a t in e a c h c o u n t r y d u r a b l e s e x p e n d i t u r e f l u c t u a t e d m o r e in t h e s e n s e t h a t a p o s i t i v e d e v i a t i o n f r o m t h e m e a n t e n d e d to be followed by a negative one, 3 suggesting the reversion implied by the theory. I first e s t i m a t e d t h e a u t o c o r r e l a t i o n c o e f f i c i e n t s o f t h e c h a n g e s i n e a c h c o n s u m p t i o n e x p e n d i t u r e f o r e a c h c o u n t r y u p t o t e n lags. T h e c u m u l a t e d s u m o f
3For the countries whose log differenced expenditure series have a time trend, deviations from the trend exhibit this pattern.
K. Hong /Journal of Monetary Economics 37 (1996) 381 391
386
Canada .o
1.0
1.01 0.5
0.5
~
0
0
' t ~ -0.5 -1.0
, 1
, 2
, 3
~ -0.5
1.0
oV:
0.5
5
6
7
8
9
10
2
6 7 8 9 10 Year --.- nondurables and services
3
4
~ durables
5
Italy
1.5 o
8o
1.0 0.5
0
-0.5 -1.0
~ ~ ~ ~ ~ ~ ~ ~;o
~ -0.5 m -1.0
2 3 4 5 6 7 8 9 1 0 ~ar --- durables --,- nondurables and services
Year --~ nondurables and services
-~- durables
Norway
1.5 .~
1
France
1.5 o
~
-1.0 ~ 4
Year --*- nondurables and services
-4,- durables
o ,--t
Denmark
1.5
U.K.
1.5 o
1.o
1.0
ol
0.5
0.5 o
~
0
~ -0.5 -1.0
~ -0.5 -1.0
2
-~- durables
3
4
5
6
7
8
9
10
Year --,- nondurables and services
, 1
2
3
-~- durables
4
5
6
7
8
9
1
0
Year --,- nondurables and services
Fig. 2. Sum of autocorrelations: Detrended series.
K. Hong /Journal of Monetary Economics 37 (1996) 381-391
387
Table 2 Sum of MA coefficients (D = durables, N D = n o n d u r a b l e s , S = services): Detrended series MA(1)
Canada
Denmark
France
Italy
Norway
U.K.
MA(2)
MA(3)
MA(4)
MA(5)
D
0.236 (0.158)
0.246 (0.270)
- 0.886 (0.116)
- 0.898 (0.110)
- 0.920 (0.125)
ND + S
0.490 (0.146)
0.357 (0.330)
0.890 (0.448)
0.347 (0.576)
0.817 (0.699)
D
0.117 (0.166)
- 0.956 (0.060)
- 0.958 (0.081)
- 0.878 (0.104)
- 0.868 (0.181)
ND + S
0.336 (0.158)
0.201 (0.286)
1.035 (0.300)
1.893 (0.385)
0.317 (0.567)
- 0.187 (0.168)
- 0.320 (0.223)
- 0.902 (0.111)
- 0.879 (0.133)
- 0.846 (0.156)
ND + S
0.683 (0.129)
1.021 (0.308)
1.108 (0.472)
1.152 (0.606)
0.976 (0.723)
D
0.363 (0.161)
- 0.025 (0.270)
- 0.317 (0.298)
- 0.407 (0.311)
- 0.864 (0.169)
ND + S
0.441 (0.153)
0.338 (0.291)
0.065 (0.422)
0.915 (0.503)
0.056 (0.611)
D
0.069 (0.166)
- 0.170 (0.235)
- 0.472 (0.241)
- 0.881 (0.121)
- 0.833 (0.124)
ND + S
0.453 (0.151)
0.610 (0.289)
0.366 (0.374)
- 0.267 (0.380)
- 0.745 (0.285)
D
0.258 (0.163)
- 0.916 (0.067)
- 0.954 (0.076)
- 1.350 (0.308)
- 0.965 (0.038)
ND + S
0.951 (0.048)
0.673 (0.327)
0.541 (0.433)
0.387 (0.538)
- 0.656 (0.375)
D
Note: Each expenditure series is log differenced. Standard errors are in parentheses.
the autocorrelations is graphed in Fig. 1. The sum is consistently lower for durables expenditure and, for most countries, as low as -0.5 at its minimum as implied by the theory. For France and Norway, however, the sum does not seem to be fully consistent with the theory. Next, I directly estimated MA(1) through MA(5) processes for each expenditure series. For space limitations, only the sums of the MA coefficients are reported in Table 1, and they are consistent with the results in Fig. 1. In most countries, the sum is close to - 1 for durables, if we assume an MA(4) or MA(5) process, and consistently positive for nondurables and services. Again, France and Norway are exceptions; the sum is not significantly negative for durables in these countries.
388
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
The high values of the sum of autocorrelations observed in some countries suggest a possible time trend in the expenditure series. In fact, I did find a significantly negative time trend for some countries, including France. Autocorrelation coefficients are not well defined for a nonstationary series. In particular, the existence of a time trend will bias upward the estimated autocorrelations. Therefore, I detrended each series for each country, and Fig. 2 and Table 2 report the estimation results from the detrended series. The changes in the estimates are particularly remarkable for France. The estimates do not change much for the countries without significant time trends. Overall, the detrended series behave more consistently with the theory. Although I have not reported the individual MA coefficients here, upon examining them, I found that the implied ~'s tend to be larger for durables consumption. ¢ This suggests that people respond to shocks with more lags in their durables than in nondurables and services consumption. One reason for this may be the presence of transactions costs. Purchasing durable goods often requires a considerable amount of time and information, and since the secondary markets for durable goods are far from perfect, reselling is costly as well. Bertola and Caballero (1990) and Grossman and Laroque (1990) show that threshold behavior can be derived as an optimal rule in the presence of transactions costs. Under the (S, s) rule, durables are allowed to deviate from their optimal level until some threshold is reached; only those close to the trigger point will respond to a shock immediately.
4. Conclusion I examined the time series behavior of durables/nondurables and services expenditure in six O E C D countries following Caballero's approach, and found that the durables expenditure exhibits clear reversion as implied by the theory when lower frequencies of the data are examined. Mankiw tested the joint hypothesis of the standard P I H and several assumptions on durable goods consumption and rejected it. The results of this paper suggest that Mankiw's puzzle was not due to the restrictions of his hypothesis on the nature of durable goods but rather to the failure of the standard PIH. Once the standard P I H is relaxed to allow for lagged responses in consumption, the data becomes consistent with Mankiw's hypothesis on durable goods, s
CThe MA coefficientsthat I estimated are the fl's in Eq. (5). The estimates of the ~'s can be derived from the estimates of the fl's. 5Caballero (1993) interprets his results as being consistent with the basic implications of the PIH. However, Section 2 of this paper shows that the results do not support any particular consumption hypothesis.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
389
The existence of lagged responses in consumption has long been recognized (e.g., Flavin, 1981). Christiano et a1.(1991) argue that once time aggregation is taken into account, there is surprisingly little evidence against the PIH. However, even with time aggregation, durables expenditure is still expected to follow an MA(1)process with the MA coefficient close t o - 1 (Heaton, 1993). Therefore, this paper provides some evidence against the time aggregation argument, at least in durables consumption. Some researchers have introduced more complicated preferences such as habit formation (Constantinides and Ferson, 1991; Muellbauer, 1988) and precautionary-saving motive (Caballero, 1990a; Deaton, 1992). These approaches have the potential to explain the lagged responses in consumption, but have not been very successful empirically. Campbell and Mankiw (1989) suggest that consumption exhibits lagged responses because it is too sensitive to current income. I have applied their hypothesis to the durables expenditure series of the six O E C D countries and the U.S., and obtained somewhat mixed results. 6 The importance of fixed adjustment costs may be responsible for the lags in durables consumption. Caballero (1993) finds that the (S, s) model is consistent with aggregate durables expenditure series. Eberly (1994) provides some micro evidence for the model from her analysis of household automobile purchases. Deciding which hypothesis better explains the lags requires further research.
Appendix This appendix shows that the use of logarithms in Section 3 does not change the basic nature of the model. Suppose Eq. (4) describes the movement of the log difference of durable goods stock. Homoskedastic process in logarithms is commonly assumed in the estimation of aggregate time series (e.g., Campbell and Deaton, 1989; MaCurdy, 1982) and implies conditional heteroskedasticity in levels. Now, Eq. (4) can be approximated by +~2ut
AlnKt=ut+~lut_l Kt
-- Kt-
Kt
1
1
2 + ""
KI -- Kt
Kt
1
'
6When applied to durables expenditure, Campbell and Mankiw's hypothesis implies AC,-).(AYt
(I--6)AY, 1)+(1--2)(u,--(1
6)u, 1).
The equation can be estimated consistently by lagging instruments more than one period. For U.S., I could not reject the hypothesis that the coefficients on AYf and AYt 1 are of the same size and opposite sign. The residuals were almost white noise suggesting that the fraction of permanent income consumers may be negligible. For the other six countries, the results were less favorable to the model. Detailed results are available from the author upon request.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
390
for small changes in K. N o t e that Eq. (5) was derived from the following relationship on levels: K, - Kt_ 1 =
(]
--
6)(K,_,
-
Kt-2) + ( C t - C,- 1).
By dividing both sides by Kt-1, we obtain Kt - Kt 1
- (1 -- 6)
Kt-
Kt- 1 - -
1
Kt-2
+
Ct - Ct-1 Kt
K t -- K t - 1 -
Kt-1
Ct- 1
A In C, ~
--
Kt- 1
Ct - C t - 1 C t - 1 Ct
1
Kt (1
Kt
-
1 -- K t - 2
6)
1
,
1 ,
Kt-1
.~ A In K t - (1 - 6) A In K t - 1.
If we a s s u m e K t_ 1/Ct 1 = K -k Ih - 1, where K is a constant close to the inverse of the depreciation and ~/,_ 1 is a mean zero random variable, the autocorrelations of the log difference of consumption expenditure depend in part on the relative magnitude of K and ~/,_ 1. In particular, it can be shown that the sum of autocorrelations is still close to - 0.5 if var(~/)/K 2 is small, which is likely to be the case in practice for the aggregate consumption series. The etimation of MA coefficients is also based on the autocorrelation coefficients and is therefore not much affected by the use of logarithms.
References Bertola, G. and R.J. Caballero, 1990, Kinked adjustment costs and aggregate dynamics, in: O.J. Blanchard and S. Fisher, eds., NBER macroeconomics annual 1990 (MIT Press, Cambridge, MA). Caballero, R.J., 1990a, Consumption puzzles and precautionary savings, Journal of Monetary Economics 25, 113 136. Caballero, R.J., 1990b, Expenditure on durable goods: A case for slow adjustment, Quarterly Journal of Economics 105, 727 743. Caballero, R.J., 1993, Durable goods: An explanation for their slow adjustment, Journal of Political Economy 101,351-384. Campbell, J.Y. and A. Deaton, 1989, Why is consumption so smooth?, Review of Economic Studies 56, 357 373. Campbell, J.Y. and N.G. Mankiw, 1989, Consumption, income, and interest rates: Reinterpreting t h e time series evidence, in: O.J. Blanchard and S. Fisher, eds., NBER macroeconomics annual 1989 (MIT Press, Cambridge, MA). Christiano, L.J., M. Eichenbaum, and D. Marshall, 1991, The permanent income hypothesis revisited, Econometrica 59, 397 423. Constantinides, G.M. and W.E. Ferson, 1991, Habit persistence and durability in aggregate consumption: Empirical tests, NBER working paper no. 3631. Deaton, A., 1992, Understanding consumption (Clarendon Press, Oxford). Eberly, J.C., 1994, Adjustment of consumer's durables stocks: Evidence from automobile purchases, Journal of Political Economy 102, 403-436.
K. Hong / Journal of Monetary Economics 37 (1996) 381-391
391
Flavin, M.A., 1981, The adjustment of consumption to changing expectations about future income, Journal of Political Economy 89, 974-1009. Friedman, M., 1957, A theory of the consumption function (Princeton University Press, Princeton, N J). Grossman, S.J. and G. Laroque, 1990, Asset pricing and optimal portfolio choice in the presence of illiquid durable consumption goods, Econometrica 58, 25-51. Heaton, J., 1993, The interaction between time-nonseparable preferences and time aggregation, Econometrica 61,353 385. Mankiw, N.G., 1982, Hall's consumption hypothesis and durable goods, Journal of Monetary Economics 10, 417 425. MaCurdy, T.E., 1982, The use of time-series processes to model the error structure of earnings in longitudinal data analysis, Journal of Econometrics 18, 83 114. M uellbauer, J., 1988, Habits, rationality and myopia in the life cycle consumption function, Annales d'Economie et de Statistique, 47-70.