- Email: [email protected]
On the durability of non-durable goods
Economics Letters North-Holland
39 (1992) 135~141
I35
On the durability of non-durable Some evidence
goods
from U.S. time series data
Luigi Ermini 1‘rlilwsiiy of Huwuii at Manoa, Honohtlu HI, USA Received ,\ccepted
6 January 24 March
1992 1992
lfsing both monthly and quarterly data, this paper shows that the durability rates of non-durable goods and services are significantly greater than zero, and in line with the prediction of the theory. The common practice of testing consumption and consumption-related models with the National Income and Product Account category of non-durables and services under the assumption of zero durability may thus be inappropriate.
1. Introduction This paper
seeks to reconcile
three
different
pieces of empirical
evidence:
(i) Hall’s (1978) random walk model of consumption is rejected with quarterly expenditures data of non-durable goods and services [see Myron (1986), and Campbell and Mankiw (1989) for a summary of tests results], as well as with monthly data [Ermini (1989)l. Even when suitably modified to account for durability, Hall’s model is rejected with quarterly data on durable goods [Mankiw (1982)]. ’ (ii) Contrary to what is usually assumed, quarterly data of non-durable goods used in empirical work (mainly, time series of the National Income and Product Account) do not relate to commodities that depreciate entirely within the quarter. As defined by the Bureau of Economic Analysis, the category of non-durables comprises goods with an average service life of up to three years [clothing and shoes, for example, are not included. See Byrnes et al. (1979) for details]. A similar conclusion can be reached for some expenditures on services (for example, dental services>. Thus, at quarterly frequency expenditures on non-durables and services would exhibit some degree of durability, which should then be taken into account in the model. Indeed, fitting a modified Hall’s model to Japanese panel data of quarterly consumption expenditures, Hayashi (1985) estimated very plausible durability rates for several categories of services and non-durable goods. However, he also estimated a very low durability rate for durables, thus confirming Mankiw’s rejection with a different data set. Correspondence to: Luigi Ermini, Department of Economics. University of Hawaii, 2424 Maile Way. Honolulu, HI 96822, I!SA. ’ Reversion of Mankiw’s result, however, has been obtained by introducing adjustment costs [Bernanke (19X5)] and non-separability between durables and non-durables [Startz (19X9)]. 0165.1765/92/$05.00
d 1992 - Elsevier
Science
Publishers
B.V. All rights reserved
136 Table 1 Durability Decision
I year 1 quarter 1 month 1 fortnight 1 week
L. Ermini
rates period
/ On the durability
of non-durable goods
6. Durability rate for 3-year service life ‘I
Range of durability (non-durables)
Range of durability (durables)
0.368 0.779 0.920 0.959 0.979
O&O.368 o-o.779 O-0.920 o-o.959 O-0.979
0.36X- I 0,779-l 0.920-I 0.959- 1 0.97YSl
“ Based on a 5% scrap value at the end of the three years.
(ii) Monthly expenditures on non-durables and services appear to follow an integrated, first-order moving average process with a negative MA coefficient [henceforth, negative IMA(l,l)], thus suggesting that consumption decisions may be generated by this process, rather than by a random walk or IMA(l,O) [Ermini (1989)]. Moreover, this alternative generating process of consumption is consistent with the dynamic behavior of both monthly and quarterly data even when consumption decisions are generated at intervals shorter than a month. The possibility that consumers make decisions at shorter intervals than the interval of data observation, which gives rise to the phenomenon of temporal aggregation, is gaining popularity in empirical work [see, inter alia, Breeden, Gibbons and Litzenburger (19891, Christiano, Eichenbaum and Marshal1 (1991), Ermini (1988, 1989, 1991a, b), Grossman, Melino and Shiller (1987)]. As a negative IMA(l,l) is precisely the generating mechanism of consumption implied by Hall’s model suitably modified to account for a geometrically decaying durability of goods [Mankiw (1982)], the purpose of this paper is to investigate whether this type of durability, possibly combined with the phenomenon of temporal aggregation, can explain the time series behavior of US expenditures on these two categories. The investigation compares the durability rates inferred by the data with the theoretical values of table 1, which reports for different decision periods the durability rate 6 (e.g., the fraction surviving after each decision period) of a commodity that depreciates geometrically in three years. The table also reports the ranges of durability rates corresponding to the two NIPA categories of non-durables and durables, each containing a full range of service lives under and over three years respectively. For example, for consumption decisions taken at quarterly intervals one would expect an average durability rate between 0 and 0.779 for non-durables. Note that as the decision intervals become smaller, the durability rates increase. The main result of the paper is that the estimated durability rates for non-durables and services are significantly greater than zero and in line with the ranges of table 1; furthermore, these estimates increase as the decision intervals become smaller. This evidence undermines the common practice of testing consumption models and consumption-related models with NIPA quarterly data of non-durables and services under the assumption of zero durability. For examples of models that explicitly incorporate durability in consumption, see Dunn and Singleton (1986), and Ferson and Constantinides (1991). On the other hand, regarding durable goods, the paper estimates much lower durability rates than the ranges indicated in table 1, thus confirming for this category of goods Mankiw’s and Hayashi’s rejections. This finding seems to support the conjecture that to explain expenditures on durable goods additional factors such as adjustment costs, liquidity constraints and perhaps rules of thumb should be taken into account. For examples of research along this line, see Bernanke (19851, Lam (1989), Startz (1989), Caballero (1990, 1991), and Campbell and Mankiw (1989).
L. Ermini
/ On the durability
of non-durable
goods
137
2. The model A derivation similar to Hall’s (1978) with a utility function defined over the flows of consumption services, S,, provided by each category of commodities yields a random walk process with drift for S,, AS, = p + u,, with ~1, a zero-mean white noise process. Under durability of goods, this flow of services is affected also by past expenditures C, for each category. Letting S, = C~=o~,C,_, (6, = l), one gets a K-order autoregressive process in the first differences of expenditures, or ARIMA(k,l,O):
;~,AcT_~+u,.
AC,=p-
This is the representation chosen by Hayashi (1985) with K = 4 to fit Japanese data. Alternatively, following Mankiw (1982) one can approximate the relation between expenditures and consumption services with the more parsimonious geometric decaying Koyck representation, aj = 6’. For K + a, one gets the negative IMA(l,l) process
(2) where 6 is the (average) durability rate of each category of goods over the decision period (0 I 6 _< 1) and (Y= (1 - S)p. When 6 = 0, the goods depreciate entirely within the decision period, and Hall’s random walk obtains. When 6 = 1, consumption follows a white noise process with a linear time trend. The null hypothesis in this paper is that consumption expenditures are generated by the representative agent according to model (2) at a decision interval of unknown length shorter than or equal to one month, with 6 restricted to the values indicated in table 1 for for each category of goods. In empirical work the decision interval is usually assumed to coincide with the observation interval. Here, the possibility that the former is shorter than the latter is allowed, thus necessitating to take into account the effects of temporal aggregation on (2). Let ct be the monthly aggregation of C,, i.e. C, = Cy=;,‘C,nt_j, with the index t referring to monthly periods, and with m the sampling ratio, that is the ratio between one month and the shorter interval of decision. Under temporal aggregation, the IMA(l,l) process (2) remains an IMA(l,l) process for all values of m greater than one [see, for example, Weiss (1984)]. Thus, c, is generated as AC, =
LX,,, +
VI -
%Iv,~I
)
(3)
where cy,, and 6,, are indexed on m to indicate that in general they depend on the sampling ratio. The effect of temporal aggregation on the MA coefficient 6 can be derived from the relation between the first-lag autocorrelation of AC,, p = -a/(1 + S’), and the first-lag autocorrelation of AC,, P,n = - a,,/( 1 + 6;) [Ermini (1989)]: (m’-
Pm =
1) + 2( m2 + 2)p
2(2m2 + 1) + 8( m2 - l)p ’
(4)
for all p E (-0.5, 0.5). For p = -0.5, the process C, is a white noise, and (4) does not apply. For p = 0, Working’s (1960) result on the effects of temporal aggregation on random walks obtains. By estimating P,,~from model (3) with monthly data, and solving (4) for p with different values of m, one can infer the durability rate 6 corresponding to different decision intervals, under the
138 Table 2 Model K,
L. Ermini
= (Y+ U, - 6u,_,
fitted to monthly
of non-durable
/ On the durability
data (in parentheses:
standard
goods
errors).
“
u
6atm=l (durability over month)
6atm=2 (durability over fortnight)
6atm=4 (durability over week)
Durables
3.252 (1.075)
0.376 (0.048)
0.620
0.795
Non-durables
3.073 (0.737)
0.364 (0.049)
0.601
0.780
2.03
Services
8.141 (0.603
0.260 (0.050)
0.528
0.729
4.41
Services + non-durables
1 I .22 (1.13)
0.253 (0.051)
0.503
0.719
4.80
Tot. consumption
14.483 (1.873)
0.254 (0.050)
0.517
0.729
” National Income and Product Account (NIPA) from 1959 to 1989 (no. of observations = 366).
series
of per-capita,
monthly,
seasonally
adjusted,
SN *
constant
dollars
‘82,
maintained hypothesis that (2) is true. Of course, this procedure is acceptable only if the estimated mode1 (3) is an appropriate description of the time series behavior of monthly data. Note also that this procedure can be equally applied to quarterly data.
3. The empirical
results
Consider first the case of monthly data. The first two columns of table 2 report the maximum likelihood estimates (standard errors in parentheses) of mode1 (3) fitted to monthly, per-capita, seasonally adjusted, constant dollars ‘82, data of all NIPA categories of services, non-durables and durables, and their aggregation. ’ This mode1 was tested against the set of all nesting univariate ARIMA( p, 1, q + 1) alternatives, with p + q I 3, that passed a white-noise Q-test for residuals [see Granger and Newbold (1986) for details], and could not be rejected at the 5% level for all categories of data. As the random walk, or IMA(l,O), model is also rejected against the IMA(l,I), one can safely conclude that model (3) is a satisfactory representation of monthly consumption data [see also Ermini (1989)]. The estimated MA coefficients in the second column of the table give directly the durability rate over the month. The remaining columns report the durability rates over shorter decision intervals (a fortnight and a week), inferred with the procedure previously described. With the exception of the category of durables, all the estimates satisfy the prediction of the theory, e.g. (i> the durability rates are in line with the theoretical values of table 1, and (ii) the durability rates increase as the decision interval becomes smaller. One problem with monthly data is their noisiness due to measurement errors. This issue is not usually raised with quarterly data, as the two-thirds reduction of the noise variance through the ’ Source: Citibank database 1990. For example, the series of per-capita non-durables was obtained (in Citibank notation) as GMCN82 x Gh4YDP8/GMYDP82. Monthly data was treated as if the deseasonalization procedure X-11 adopted by the Bureau of Economic Analysis, as well as the procedures of interpolation/extrapolation from quarterly and annual data used to construct parts of the monthly series. has no effect on the dynamic structure of consumption innovations.
L. Ermini Table 3 Model AC, = (Y + u, - 6u,_,
fitted to quarterly
of non-durable goods
/ On the durability
data (in parentheses:
standard
errors).
139
“
cr
6
6atm=3 (durability over month)
Durables
7.87 1 (2.418)
- 0.082 (0.077)
0.323
bon-durables
7.916 (2.018)
- 0.126 (0.075)
0.247
Scwices
20.261 (1.664)
-0.192 (0.076)
O.096
Services + non-durables
28.177 (3.147)
-0.211 (0.075)
0.036
Tot. consumption
36.050 (4.741)
- 0. IO9 (0.077)
0.280
“ NIPA series of per-capita, 169).
quarterly.
seasonally
adjusted.
constant
dollars
‘82, from 1947-l to 1989-l (no. of observations
=
temporal aggregation of the noise-corrupted monthly data to quarterly frequency is deemed sufficient for the problem to disappear. Some economists argue against the use of monthly data in macroeconometrics, as it is too noisy to make any sensible inference. Others ignore the noisiness issue altogether [for example, in Dunn and Singleton (1986), and Eichenbaum, Hansen and Singleton (19SS>]. On the other hand, not using monthly data is a waste of potentially relevant information, which in fact can be extracted under some plausible conjectures about the dynamic properties and the relative importance of the measurement errors [as, for example, in de Leeuw (1990>, Ermini (1989, 1991b), and Wilcox (1990)]. Regarding the effects of measurement errors on the MA coefficient of an IMA(l,l) process, it can be shown, for example, that additive uncorrelated errors reduce the value of the coefficient proportionally to the relative importance of the error [see Ermini (1989, 1991b) for details]. So, for sufficiently large error variance, monthly consumption may be observed with a negative MA coefficient, and yet uncorrupted consumption may be generated with a positive MA coefficient, in which case the durability hypothesis would be rejected. However, the durability hypothesis can still be valid (with lower inferred durability rates) if the error variance is relatively low. The last column of table 2 reports the signal-to-noise ration SN*, e.g. the ratio of the variance of consumption innovations over the variance of the measurement error, above which the durability hypothesis is not rejected. For example, for the category of non-durable goods, an error variance not greater than half the variance of consumption innovations is sufficient to corroborate the durability hypothesis. Note that different signal-to-noise ratios across categories of goods may affect comparisons of relative durability rates. Table 3 reports the results for the same procedure applied to quarterly data. ’ First, under similar tests, the IMA(l,l) model turns out to be a quite satisfactory representation also for quarterly data. However, the MA coefficient is positive, thus implying unacceptable negative durability rates. For the category of durables, the table essentially replicates Mankiw’s (1982)
3 Source: Citibank database 1990. For example, GCNX2 x GYDPCS / GYDP82.
the series of per-capita
non-durables
was obtained
(in Citibank
notation)
as
140
L. Ermini / On the durability of non-durable goods
result; for the other categories, the table confirms the rejection of the random walk model for consumption. Nonetheless, as under temporal aggregation a positive IMA(l,l) can be generated by a negative IMA(l,l), quarterly data could still be consistent with the durability hypothesis for m greater than 1. As an example, the third column of table 3 reports the durability rates over monthly intervals inferred from quarterly data with the procedure described in section 2. Although the inferred values are not systematically consistent with those directly derived from monthly data, especially for the category of services, the differences could be explained, as previously anticipated, as a consequence of noisier measurement errors in monthly data. A final comment concerns the relation between estimated durability rates at different frequencies. From table 3, for example, under the maintained model (2) consumers would face an average durability rate over the month of approximately 0.1 for services and 0.25 for non-durables. Thus, one would conclude that, as 0.25’ and 0.13 are practically zero, these monthly estimates are consistent with a zero durability over the quarter, and hence corroborate the reported rejections with quarterly data of the durability hypothesis. This conclusion, however, would be incorrect. The durability of a category of goods over a given period is the average over each single good in the category, i.e. C,~=,N,CS,, with 6, the good-j durability and (Y, the share of the good in the category. Over m periods, the average durability for the category is C;~,(Y,C?I)I. Due to the convexity of the exponential function, this value is greater than or equal to (C;“_,C,I,~~)“‘, so that the true average durability for non-durables over the quarter may well be higher than 0.25’ = 0.016. For example, consider a category with equal shares of two goods with durability over the month 6, = 0 and 6, = 0.8. The average durability over the month is thus 0.4, and the average durability over the quarter becomes 0.5 x 0.8’ = 0.255, which is much greater than what one would have inferred from 0.4” = 0.06.
4. Conclusions Using both monthly and quarterly data, and introducing the possibility of temporal aggregation, this paper shows that the durability rates of non-durable goods and services are significantly greater than zero, and in line with the prediction of the theory. This evidence undermines the common practice of testing consumption models and consumption-related models with the National Income and Product Account category of non-durables and services under the assumption of zero durability. The paper, however, estimates very low durability rates for the category of durable goods, thus corroborating the conjecture that to explain expenditures on durables additional ad hoc factors must be taken into account.
References Bernanke, B., 1985, Adjustment costs, durables, and aggregate consumption, Journal of Monetary Economics 15. 41-68. Breeden. D.T., M.R. Gibbons and R.H. Litzenberger, 1989, Empirical tests of the consumption-oriented CAPM, Journal of Finance 44, no. 2, 231-262. Byrnes, J.C., G.F. Donahoe, M.W. Hook and R.P. Parker, 1979, Monthly estimates of personal income, taxes and outlays. Survey of Current Business 59, no. 11 (U.S. Department of Commerce, Washington, DC) 1X-36. Caballero, R.J., 1990, Expenditures on durable goods: A case for slow adjustment, Quarterly Journal of Economics IV. no. 3, 727-744. Caballero, R.J., 1991, Durable goods: An explanation for their slow adjustment, Working paper no. 3748 (NBER, Cambridge, MA). Campbell, J.Y.. and N.G. Mankiw. 1989, Consumption, income and interest rate: Reinterpreting the time series evidence, Macroeconomics Annual (NBER, Cambridge, MA).
L. Ermini / On the durability
of non-durable
goods
141
Christiano, L.J., M. Eichenbaum and D. Marshall, 1991, The permanent income hypothesis revisited, Econometrica 59, no. 2, 397-424. de Leeuw, F., 1990, The reliability of U.S. Gross National Product, Journal of Business and Economic Statistics 8, no. 2, 191-203. Dunn, K.B. and K.J. Singleton, 1986, Modeling the term structure of interest rates under nonseparable utility and durability of goods, Journal of Financial Economics 17, 27-55. 1988, A time series analysis of representative agent models of Eichenbaum, MS., L.P. Hansen and K.J. Singleton, consumption and leisure choice under uncertainty, Quarterly Journal of Economics CIII, no. 1, 51-78. Ermini, L., 1988, Temporal aggregation and Hall’s model of consumption behavior, Applied Economics 20, 1317-1320. Ermini, L., 1989, Some new evidence on the timing of consumption decisions and on their generating process, Review of Economics and Statistics 71, no. 4, 643-650. Ermini, L.. IYYla, Reinterpreting a temporally aggregate consumption CAP-model, Journal of Business and Economic Statistics 9, no. 3, 325-328. Ermini, L., 1991b, Effects of transitory consumption and measurement errors on tests of the permanent income hypothesis, Discussion paper no. 91-8 (Department of Economics, University of Hawaii, Honolulu, HI). Ferson, W.E. and G.M. Constantinides, 1991, Habit persistence and durability in aggregate consumption: Empirical tests, Journal of Financial Economics 29, IYY-240. Cirossman, S.J., A. Melino and R.J. Shiller, 1987. Estimating the continuous-time consumption-based asset pricing model, Journal of Business and Economic Statistics 5, no. 3, 315-327. Ilall, R., 1978, Stochastic implications of the life-cycle permanent income hypothesis: Theory and evidence, Journal of Political Economy 86, no. 6, 125881279. Iiayashi, F., 1985, The permanent income hypothesis and consumption durability: Analysis based on Japanese panel data, Quarterly Journal of Economics C. no. 4, 1083-1113. I-am, P.S., 1989, Irreversibility and consumer durables expenditures, Journal of Monetary Economics 23, 135-150. Mankiw, N.G.. 1982. Hall’s consumption hypothesis and durable goods, Journal of Monetary Economics 10, 417-425. Myron, J.A., 1986, Seasonal fluctuations and the life-cycle permanent income model of consumption, Journal of Political Economy 94, no. 6, 1258-1279. Startz. R.. lY89, The stochastic behavior of durable and non-durable consumption, Review of Economics and Statistics 71, 356-363. Weiss, A.A., 1984, Systematic sampling and temporal aggregation in the time-series models, Journal of Econometrics 26, 271-281. Wilcox, D.W., 1990, What do we know about consumption?, Working paper no. 107 (Division of Research and Statistics, Board of Governors of the Federal Reserve System, Washington, DC). Working, H., 1960, Note on the correlation of first differences of averages in a random-chain, Econometrica 28, 916-918.
39 (1992) 135~141
I35
On the durability of non-durable Some evidence
goods
from U.S. time series data
Luigi Ermini 1‘rlilwsiiy of Huwuii at Manoa, Honohtlu HI, USA Received ,\ccepted
6 January 24 March
1992 1992
lfsing both monthly and quarterly data, this paper shows that the durability rates of non-durable goods and services are significantly greater than zero, and in line with the prediction of the theory. The common practice of testing consumption and consumption-related models with the National Income and Product Account category of non-durables and services under the assumption of zero durability may thus be inappropriate.
1. Introduction This paper
seeks to reconcile
three
different
pieces of empirical
evidence:
(i) Hall’s (1978) random walk model of consumption is rejected with quarterly expenditures data of non-durable goods and services [see Myron (1986), and Campbell and Mankiw (1989) for a summary of tests results], as well as with monthly data [Ermini (1989)l. Even when suitably modified to account for durability, Hall’s model is rejected with quarterly data on durable goods [Mankiw (1982)]. ’ (ii) Contrary to what is usually assumed, quarterly data of non-durable goods used in empirical work (mainly, time series of the National Income and Product Account) do not relate to commodities that depreciate entirely within the quarter. As defined by the Bureau of Economic Analysis, the category of non-durables comprises goods with an average service life of up to three years [clothing and shoes, for example, are not included. See Byrnes et al. (1979) for details]. A similar conclusion can be reached for some expenditures on services (for example, dental services>. Thus, at quarterly frequency expenditures on non-durables and services would exhibit some degree of durability, which should then be taken into account in the model. Indeed, fitting a modified Hall’s model to Japanese panel data of quarterly consumption expenditures, Hayashi (1985) estimated very plausible durability rates for several categories of services and non-durable goods. However, he also estimated a very low durability rate for durables, thus confirming Mankiw’s rejection with a different data set. Correspondence to: Luigi Ermini, Department of Economics. University of Hawaii, 2424 Maile Way. Honolulu, HI 96822, I!SA. ’ Reversion of Mankiw’s result, however, has been obtained by introducing adjustment costs [Bernanke (19X5)] and non-separability between durables and non-durables [Startz (19X9)]. 0165.1765/92/$05.00
d 1992 - Elsevier
Science
Publishers
B.V. All rights reserved
136 Table 1 Durability Decision
I year 1 quarter 1 month 1 fortnight 1 week
L. Ermini
rates period
/ On the durability
of non-durable goods
6. Durability rate for 3-year service life ‘I
Range of durability (non-durables)
Range of durability (durables)
0.368 0.779 0.920 0.959 0.979
O&O.368 o-o.779 O-0.920 o-o.959 O-0.979
0.36X- I 0,779-l 0.920-I 0.959- 1 0.97YSl
“ Based on a 5% scrap value at the end of the three years.
(ii) Monthly expenditures on non-durables and services appear to follow an integrated, first-order moving average process with a negative MA coefficient [henceforth, negative IMA(l,l)], thus suggesting that consumption decisions may be generated by this process, rather than by a random walk or IMA(l,O) [Ermini (1989)]. Moreover, this alternative generating process of consumption is consistent with the dynamic behavior of both monthly and quarterly data even when consumption decisions are generated at intervals shorter than a month. The possibility that consumers make decisions at shorter intervals than the interval of data observation, which gives rise to the phenomenon of temporal aggregation, is gaining popularity in empirical work [see, inter alia, Breeden, Gibbons and Litzenburger (19891, Christiano, Eichenbaum and Marshal1 (1991), Ermini (1988, 1989, 1991a, b), Grossman, Melino and Shiller (1987)]. As a negative IMA(l,l) is precisely the generating mechanism of consumption implied by Hall’s model suitably modified to account for a geometrically decaying durability of goods [Mankiw (1982)], the purpose of this paper is to investigate whether this type of durability, possibly combined with the phenomenon of temporal aggregation, can explain the time series behavior of US expenditures on these two categories. The investigation compares the durability rates inferred by the data with the theoretical values of table 1, which reports for different decision periods the durability rate 6 (e.g., the fraction surviving after each decision period) of a commodity that depreciates geometrically in three years. The table also reports the ranges of durability rates corresponding to the two NIPA categories of non-durables and durables, each containing a full range of service lives under and over three years respectively. For example, for consumption decisions taken at quarterly intervals one would expect an average durability rate between 0 and 0.779 for non-durables. Note that as the decision intervals become smaller, the durability rates increase. The main result of the paper is that the estimated durability rates for non-durables and services are significantly greater than zero and in line with the ranges of table 1; furthermore, these estimates increase as the decision intervals become smaller. This evidence undermines the common practice of testing consumption models and consumption-related models with NIPA quarterly data of non-durables and services under the assumption of zero durability. For examples of models that explicitly incorporate durability in consumption, see Dunn and Singleton (1986), and Ferson and Constantinides (1991). On the other hand, regarding durable goods, the paper estimates much lower durability rates than the ranges indicated in table 1, thus confirming for this category of goods Mankiw’s and Hayashi’s rejections. This finding seems to support the conjecture that to explain expenditures on durable goods additional factors such as adjustment costs, liquidity constraints and perhaps rules of thumb should be taken into account. For examples of research along this line, see Bernanke (19851, Lam (1989), Startz (1989), Caballero (1990, 1991), and Campbell and Mankiw (1989).
L. Ermini
/ On the durability
of non-durable
goods
137
2. The model A derivation similar to Hall’s (1978) with a utility function defined over the flows of consumption services, S,, provided by each category of commodities yields a random walk process with drift for S,, AS, = p + u,, with ~1, a zero-mean white noise process. Under durability of goods, this flow of services is affected also by past expenditures C, for each category. Letting S, = C~=o~,C,_, (6, = l), one gets a K-order autoregressive process in the first differences of expenditures, or ARIMA(k,l,O):
;~,AcT_~+u,.
AC,=p-
This is the representation chosen by Hayashi (1985) with K = 4 to fit Japanese data. Alternatively, following Mankiw (1982) one can approximate the relation between expenditures and consumption services with the more parsimonious geometric decaying Koyck representation, aj = 6’. For K + a, one gets the negative IMA(l,l) process
(2) where 6 is the (average) durability rate of each category of goods over the decision period (0 I 6 _< 1) and (Y= (1 - S)p. When 6 = 0, the goods depreciate entirely within the decision period, and Hall’s random walk obtains. When 6 = 1, consumption follows a white noise process with a linear time trend. The null hypothesis in this paper is that consumption expenditures are generated by the representative agent according to model (2) at a decision interval of unknown length shorter than or equal to one month, with 6 restricted to the values indicated in table 1 for for each category of goods. In empirical work the decision interval is usually assumed to coincide with the observation interval. Here, the possibility that the former is shorter than the latter is allowed, thus necessitating to take into account the effects of temporal aggregation on (2). Let ct be the monthly aggregation of C,, i.e. C, = Cy=;,‘C,nt_j, with the index t referring to monthly periods, and with m the sampling ratio, that is the ratio between one month and the shorter interval of decision. Under temporal aggregation, the IMA(l,l) process (2) remains an IMA(l,l) process for all values of m greater than one [see, for example, Weiss (1984)]. Thus, c, is generated as AC, =
LX,,, +
VI -
%Iv,~I
)
(3)
where cy,, and 6,, are indexed on m to indicate that in general they depend on the sampling ratio. The effect of temporal aggregation on the MA coefficient 6 can be derived from the relation between the first-lag autocorrelation of AC,, p = -a/(1 + S’), and the first-lag autocorrelation of AC,, P,n = - a,,/( 1 + 6;) [Ermini (1989)]: (m’-
Pm =
1) + 2( m2 + 2)p
2(2m2 + 1) + 8( m2 - l)p ’
(4)
for all p E (-0.5, 0.5). For p = -0.5, the process C, is a white noise, and (4) does not apply. For p = 0, Working’s (1960) result on the effects of temporal aggregation on random walks obtains. By estimating P,,~from model (3) with monthly data, and solving (4) for p with different values of m, one can infer the durability rate 6 corresponding to different decision intervals, under the
138 Table 2 Model K,
L. Ermini
= (Y+ U, - 6u,_,
fitted to monthly
of non-durable
/ On the durability
data (in parentheses:
standard
goods
errors).
“
u
6atm=l (durability over month)
6atm=2 (durability over fortnight)
6atm=4 (durability over week)
Durables
3.252 (1.075)
0.376 (0.048)
0.620
0.795
Non-durables
3.073 (0.737)
0.364 (0.049)
0.601
0.780
2.03
Services
8.141 (0.603
0.260 (0.050)
0.528
0.729
4.41
Services + non-durables
1 I .22 (1.13)
0.253 (0.051)
0.503
0.719
4.80
Tot. consumption
14.483 (1.873)
0.254 (0.050)
0.517
0.729
” National Income and Product Account (NIPA) from 1959 to 1989 (no. of observations = 366).
series
of per-capita,
monthly,
seasonally
adjusted,
SN *
constant
dollars
‘82,
maintained hypothesis that (2) is true. Of course, this procedure is acceptable only if the estimated mode1 (3) is an appropriate description of the time series behavior of monthly data. Note also that this procedure can be equally applied to quarterly data.
3. The empirical
results
Consider first the case of monthly data. The first two columns of table 2 report the maximum likelihood estimates (standard errors in parentheses) of mode1 (3) fitted to monthly, per-capita, seasonally adjusted, constant dollars ‘82, data of all NIPA categories of services, non-durables and durables, and their aggregation. ’ This mode1 was tested against the set of all nesting univariate ARIMA( p, 1, q + 1) alternatives, with p + q I 3, that passed a white-noise Q-test for residuals [see Granger and Newbold (1986) for details], and could not be rejected at the 5% level for all categories of data. As the random walk, or IMA(l,O), model is also rejected against the IMA(l,I), one can safely conclude that model (3) is a satisfactory representation of monthly consumption data [see also Ermini (1989)]. The estimated MA coefficients in the second column of the table give directly the durability rate over the month. The remaining columns report the durability rates over shorter decision intervals (a fortnight and a week), inferred with the procedure previously described. With the exception of the category of durables, all the estimates satisfy the prediction of the theory, e.g. (i> the durability rates are in line with the theoretical values of table 1, and (ii) the durability rates increase as the decision interval becomes smaller. One problem with monthly data is their noisiness due to measurement errors. This issue is not usually raised with quarterly data, as the two-thirds reduction of the noise variance through the ’ Source: Citibank database 1990. For example, the series of per-capita non-durables was obtained (in Citibank notation) as GMCN82 x Gh4YDP8/GMYDP82. Monthly data was treated as if the deseasonalization procedure X-11 adopted by the Bureau of Economic Analysis, as well as the procedures of interpolation/extrapolation from quarterly and annual data used to construct parts of the monthly series. has no effect on the dynamic structure of consumption innovations.
L. Ermini Table 3 Model AC, = (Y + u, - 6u,_,
fitted to quarterly
of non-durable goods
/ On the durability
data (in parentheses:
standard
errors).
139
“
cr
6
6atm=3 (durability over month)
Durables
7.87 1 (2.418)
- 0.082 (0.077)
0.323
bon-durables
7.916 (2.018)
- 0.126 (0.075)
0.247
Scwices
20.261 (1.664)
-0.192 (0.076)
O.096
Services + non-durables
28.177 (3.147)
-0.211 (0.075)
0.036
Tot. consumption
36.050 (4.741)
- 0. IO9 (0.077)
0.280
“ NIPA series of per-capita, 169).
quarterly.
seasonally
adjusted.
constant
dollars
‘82, from 1947-l to 1989-l (no. of observations
=
temporal aggregation of the noise-corrupted monthly data to quarterly frequency is deemed sufficient for the problem to disappear. Some economists argue against the use of monthly data in macroeconometrics, as it is too noisy to make any sensible inference. Others ignore the noisiness issue altogether [for example, in Dunn and Singleton (1986), and Eichenbaum, Hansen and Singleton (19SS>]. On the other hand, not using monthly data is a waste of potentially relevant information, which in fact can be extracted under some plausible conjectures about the dynamic properties and the relative importance of the measurement errors [as, for example, in de Leeuw (1990>, Ermini (1989, 1991b), and Wilcox (1990)]. Regarding the effects of measurement errors on the MA coefficient of an IMA(l,l) process, it can be shown, for example, that additive uncorrelated errors reduce the value of the coefficient proportionally to the relative importance of the error [see Ermini (1989, 1991b) for details]. So, for sufficiently large error variance, monthly consumption may be observed with a negative MA coefficient, and yet uncorrupted consumption may be generated with a positive MA coefficient, in which case the durability hypothesis would be rejected. However, the durability hypothesis can still be valid (with lower inferred durability rates) if the error variance is relatively low. The last column of table 2 reports the signal-to-noise ration SN*, e.g. the ratio of the variance of consumption innovations over the variance of the measurement error, above which the durability hypothesis is not rejected. For example, for the category of non-durable goods, an error variance not greater than half the variance of consumption innovations is sufficient to corroborate the durability hypothesis. Note that different signal-to-noise ratios across categories of goods may affect comparisons of relative durability rates. Table 3 reports the results for the same procedure applied to quarterly data. ’ First, under similar tests, the IMA(l,l) model turns out to be a quite satisfactory representation also for quarterly data. However, the MA coefficient is positive, thus implying unacceptable negative durability rates. For the category of durables, the table essentially replicates Mankiw’s (1982)
3 Source: Citibank database 1990. For example, GCNX2 x GYDPCS / GYDP82.
the series of per-capita
non-durables
was obtained
(in Citibank
notation)
as
140
L. Ermini / On the durability of non-durable goods
result; for the other categories, the table confirms the rejection of the random walk model for consumption. Nonetheless, as under temporal aggregation a positive IMA(l,l) can be generated by a negative IMA(l,l), quarterly data could still be consistent with the durability hypothesis for m greater than 1. As an example, the third column of table 3 reports the durability rates over monthly intervals inferred from quarterly data with the procedure described in section 2. Although the inferred values are not systematically consistent with those directly derived from monthly data, especially for the category of services, the differences could be explained, as previously anticipated, as a consequence of noisier measurement errors in monthly data. A final comment concerns the relation between estimated durability rates at different frequencies. From table 3, for example, under the maintained model (2) consumers would face an average durability rate over the month of approximately 0.1 for services and 0.25 for non-durables. Thus, one would conclude that, as 0.25’ and 0.13 are practically zero, these monthly estimates are consistent with a zero durability over the quarter, and hence corroborate the reported rejections with quarterly data of the durability hypothesis. This conclusion, however, would be incorrect. The durability of a category of goods over a given period is the average over each single good in the category, i.e. C,~=,N,CS,, with 6, the good-j durability and (Y, the share of the good in the category. Over m periods, the average durability for the category is C;~,(Y,C?I)I. Due to the convexity of the exponential function, this value is greater than or equal to (C;“_,C,I,~~)“‘, so that the true average durability for non-durables over the quarter may well be higher than 0.25’ = 0.016. For example, consider a category with equal shares of two goods with durability over the month 6, = 0 and 6, = 0.8. The average durability over the month is thus 0.4, and the average durability over the quarter becomes 0.5 x 0.8’ = 0.255, which is much greater than what one would have inferred from 0.4” = 0.06.
4. Conclusions Using both monthly and quarterly data, and introducing the possibility of temporal aggregation, this paper shows that the durability rates of non-durable goods and services are significantly greater than zero, and in line with the prediction of the theory. This evidence undermines the common practice of testing consumption models and consumption-related models with the National Income and Product Account category of non-durables and services under the assumption of zero durability. The paper, however, estimates very low durability rates for the category of durable goods, thus corroborating the conjecture that to explain expenditures on durables additional ad hoc factors must be taken into account.
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