Time Irreversibility in Consumers’ Expenditure: An Analysis of Disaggregated Data

Abstract

The literature on testing for the presence of cyclical asymmetry in consumers’ expenditure is extended via the application of tests for time irreversibility to UK data subject to a higher degree of disaggregation than considered in previous studies. The empirical findings reported provide support for a positive relationship between the durability of goods and the asymmetric, and specifically time irreversible, behaviour they exhibit at a fine level of disaggregation. Further investigation of the underlying causes of such time irreversibility exhibit pronounced difference according to the degree of durability, in that nonlinearity in the underlying data generating process is a prevalent feature of highly disaggregated durable good expenditures, but is a less marked feature of semi‐durable and non‐durable expenditures. Prominent among the durable and semi‐durable good expenditure categories exhibiting such nonlinearities are expenditures relating to housing fittings and communication equipment. These findings are consistent with threshold effects in inventory control as well as the effects of credit rationing, such that these expenditures are more likely to be made at times when those constraints are eased by income windfalls or by the easing of credit availability, possibly associated with mortgage (re‐) financing.

Key Words:

• Time reversibility
• time irreversibility
• nonlinearity
• consumers’ expenditure
• disaggregated data

Jel Classifications:

• C12
• E21
• E32

Notes

1. See Psaradakis (2000) for discussion of a number of commonly employed tests of asymmetry and nonlinearity.

2. For an illustration of such consequences, see Arden et al. (2000).

3. Relatedly, Cook (2000b) reports evidence of significant inverse steepness in Australian total consumers’ expenditure, while Cook (2000c) reports the results of asymmetry tests applied to total consumers’ expenditure for seven major OECD economies, with mixed results. Cook (1999) confirms significant deepness and steepness asymmetry in US consumer expenditure on durable goods, and insignificant test results for non‐durable goods expenditure.

4. Hodrick & Prescott (1997). For criticisms of and alternatives to the HP filter see, for example, Nelson & Kang (1981), Harvey & Jaeger (1993), Cogley & Nason (1995), Kaiser & Maravall (2000), Psaradakis & Sola (2003) and Christiano & FitzGerald (2003).

5. In comparison, Speight & McMillan (1997) examine 13 categories of consumers’ expenditure; Cook (2000a) examines 2 categories, while Holly & Stannett (1995) consider the total consumers’ expenditure series alone.

6. While the choice of K is essentially arbitrary, we select K=5 following Ramsey & Rothman (1996), who argue that this is an appropriate value given the degrees of freedom typically available for economic time series. In the results of robustness exercises (not reported here), we found that adopting alternative values of K made no qualitative and little quantitative difference to the results reported in Section 4.

7. Note that it is a requirement of the TR test statistic that the data possess a finite sixth moment. Chen et al. (2000) have proposed an alternative to the TR test statistic that does not have any moment restrictions. However, as Chen et al. note, their test is not directly applicable to model residuals because it is a test of unconditional symmetry, and cannot therefore be used in order to discriminate between Type I and Type II time irreversibility.

8. Under this latter approach an estimate of the variance of the variance of is calculated by fitting a linear autoregressive (AR) model to the data, obtaining an estimate of the innovations variance, and then simulating a series using the estimated AR coefficients and generating a Gaussian error process with zero mean and variance equal to that estimated in the preceding stage. Values of are calculated for each such replication for N replications, where N=100, permitting straightforward computation of the estimated variance using the replicated values for . If the process is truly linear Gaussian, and time reversible, this is an exact simulation procedure. If the series is truly nonlinear (Type I time irreversible), the linear model constitutes a local approximation to the unknown nonlinear model, but the procedure should nonetheless provide asymptotically unbiased estimates of the variance of in the presence of uncorrelated innovations.

9. Note that while Type I time irreversibility implies nonlinearity, the converse is not necessarily true, because there exist stationary nonlinear processes that are time reversible (e.g. Lewis et al., 1989). The TR test cannot therefore be considered as equivalent to a test for nonlinearity of unknown form. For more detailed discussion of this test procedure and its rationale, see Ramsey & Rothman (1996). On the uses and possible limitations of the TR test statistic as a guide to model specification tool in application to nonlinear conditional mean and conditional variance model residuals, see Rothman (1999) and Belaire‐Franch & Contreras (2002, 2003).

10. The two aggregate series considered are ‘national’ and ‘domestic’ measures. The series differ as a consequence of the latter’s inclusion of net tourist expenditure.

11. Consumer Trends four character identifiers for these series, in the order listed in the results tables, are as follows: Table 1: ABJR, ZAKW, ZAKY, ZALA, ZAVO, ZAVW, ZAWC, ZAWM, ZAWW, ZAXA, ZAXS, ZAYG, ZWUN, ZWUT; Table 2: UTIH, UTID, UTIT, UTIL, UTIP; Table 3: ATQX, ATRD, ATRR, ATRV, ATRZ, ATSD, LLKX, LLKY, LLKZ, LLLA, LLLB, LLLC, TMMI, TMML, TMMZ, TMNB, TMNO, UWIC, XYJP, XYJR, XYJT, ZAYM; Table 4: ATQV, ATRF, ATRJ, ATSH, ATSL, ATSX, AWUW, CDZQ, LLLZ, LLMA, LLMB, LLMC, LLMD, XYJN, XYJO, XYJQ, XYJS, XYJU, XYJX, ZAVK; Table 5: ATSP, ATUA, AWUX, CCTK, CCTL, CCTM, CCTN, CCTO, CCTT, CCTU, CCTY, CCUA, CDZY, JRBA, LLLL, LLLM, LLLN, LLLO, LLLP, LLLQ, LTZA, LTZC, TTAB, UTHW, UTXP, UTZN, UUIS, UUVG, UWBK, UWBL, UWFD, UWFX, UWGH, UWGI, UWHO, UWIB, UWKQ, XYJW, ZAKY, ZWUN, ZWUP, ZWUR. It should be noted that although some components, for example ‘Transport’, appear under durable, semi‐durable and non‐durable expenditure, in each case expenditure is defined as that part of the component relating to the classification (durable, semi‐durable and non‐durable) being considered. It may nonetheless be the case that such expenditure are not separable across these categories as a practical matter, or in their influence on utility.

12. As a consequence of the stationarity requirement of the time reversibility tests, augmented Dickey–Fuller tests were applied to all of the above series to determine their orders of integration. Full details of the unit root test results are omitted here but are available from the authors upon request. In summary, it was found that the unit root null hypothesis was rejected only for three series (clothing materials; newspapers, books, stationery; electricity, gas and other fuels) at the 5% level of significance. However, as noted by an anonymous referee, this proportionate incidence of null hypothesis rejection may not be unexpected given the 5% significance level adopted in hypothesis testing and the associated likelihood of committing the Type I error of rejecting the null hypothesis when it is true. Therefore, the first differences of the logarithmic values are primarily employed for the time reversibility tests for all series, but the logarithmic values of the series in question are additionally reported, but not discussed, in those three cases where the unit root null hypothesis is rejected. Further to the discussion in Section 2, note that the results reported employ an estimate of the variance of calculated by Monte Carlo simulation using a linear autoregressive AR(p) model fitted to the data of order p determined by reference to the Akaike Information Criterion (AIC). Alternative results based on application of the Schwarz (Bayesian) Information Criterion (BIC), which provide qualitatively equivalent results are omitted here in the interest of conserving space, but are available from the authors on request.

13. These summary listings omit reference to the anomalous result for health, where Type I time irreversibility is indicated by the TR2 test result despite no indication of significant time irreversibility on the basis of the TR1 test result. Such anomalous test results are rare in the totality of results reported here, occurring in only five other cases, and including cases where the logarithm rather than the difference of logarithms is considered (motor vehicle spares, and transport under semi‐durables in Table 4; tobacco, newspapers, etc., and electricity, gas and other fuels under non‐durables in Table 5). As noted by an anonymous referee, the incidence of six out of 101 anomalous cases might not be unexpected given the associated likelihood of statistical errors in hypothesis testing under the significance level adopted, and we therefore do not discuss these anomalous cases further in the text.

14. Specifically: furniture and furnishings, carpets, furnishings and household equipment; photographic and optical equipment, information processing equipment, major tools; musical instruments and indoor recreation, outdoor recreation, recreation and culture; jewellery; miscellaneous goods and services.

15. Specifically: communication, and communication, telephone and telefax equipment; transport, vehicles, motor cars, motor cycles, bicycles; and major household appliances.

16. Specifically: household textiles, furnishings and household equipment; recording media, small electrical household appliances; miscellaneous goods and services.

17. We note that Rothman (1999) demonstrates by Monte Carlo simulation that where Type I irreversibility is identified, the appropriate class of non‐linear time series model to represent the underlying data generating process can be identified from the pattern of statistics generated by the TR2 test. Specifically, where the underlying nonlinear model is of threshold form, as may be the prior expectation as discussed in Section 1, and of the self exciting threshold autoregressive (SETAR) form in particular, TR2 tests results will produce a spike in the test value for k = 1, being large in magnitude and significant, but with small insignificant values for k > 1. While we focus here on reporting the portmanteau versions of the TR tests in the interests of conserving space, test results for values of k=1,2,3,4,5 (further details of which are available from the authors) strongly indicate the appropriateness of a threshold autoregressive time series model for the vast majority of those expenditure categories exhibiting Type I time irreversibility in Tables 2–4, but only for around half of the significant TR2 instances in Table 5.

18. Specifically: materials for repair and maintenance of the dwelling; electricity, newspapers, books and stationery; gas and other fuels, water supply; meat, off‐trade spirits and beer; other medical products; pets and related products; electricity, gas and other fuels.

19. Specifically: Table 1—recreation and culture, restaurants and hotels, miscellaneous goods and services; Table 2—services; Table 3—AV equipment, health, therapeutic appliances; Table 4—glassware, other personal effects, small tools and miscellaneous accessories; Table 5—gardens, plants and flowers, fish, milk, cheese, eggs, fruit, vehicle fuels and lubricants, newspapers and periodicals, housing fuels, furnishings, health, transport, gas, liquid fuels, solid fuels, off‐trade wine, bread, cereals, other food, non‐alcoholic beverages, non‐durable household goods, stationery and drawing materials.