# An alternative model of U.S. clothing expenditures: application of cointegration techniques.

There are two purposes to this study. The first purpose is to analyze
the relationship between U.S. consumer expenditures on clothing and its
determinants by using an Error Correction Mechanism (ECM) model
expounded by Davidson et al. (1978). An ECM model of consumer clothing
expenditures, in its basic form, proposes modifying shortrun
proportional changes in consumer clothing expenditures with disposable
income by utilizing the feedback from past mistakes (or past
disequilibrium). This model has been applied to total consumer
expenditure data by Baltagi and Mokhtari (1990). The second purpose is
to emphasize the inherent nonstationarity of time-series data that
renders inferences from the usual t- and F-statistics invalid.(1) To
circumvent the problems associated with the use of non-stationary
variables in modeling clothing expenditures, this paper applies a simple
but powerful two-step modeling strategy recommended by Engle and Granger
(1987).

Despite the well-known properties of the ECM model (see, for example, Gilbert (1986) and references therein), this specification has never been used for modeling clothing expenditures. In this regard, the ECM model provides a richer dynamic structure than conventional models such as the Partial Adjustment Model (PAM). In fact, it takes the PAM as a special case. The ECM modeling procedure (1) is easy to implement; (2) sharpens insight into the dynamics of consumer decision making; (3) is fairly general, giving an accurate account of the observations; and (4) is an excellent pedagogical tool for modeling and studying many aspects of consumer activities over time.

This paper also emphasizes the inherent nonstationarity of time-series data that renders the usual textbook inferences from t-tests and F-tests invalid, because real world data do not satisfy the conventional (Gauss-Markov) properties assumed when reporting tables of critical values. Hence, research using old methodologies, statistics, and tables of critical values is invalid, when modeling nonstationary variables.(2) This paper, by application and example, demonstrates a methodology that is easy to implement and does not lead to estimating spurious relationships among variables.(3) Indeed, it can be seen that existence of an equilibrium relationship between nonstationary variables (say, consumption and income)(4) naturally leads to estimating an ECM model that is a proper framework for robust inference. This is due to the fact that the regressors and regressand of an ECM model are ensured to follow a stationary process.

CLOTHING EXPENDITURE STUDIES

Pioneering work by Winakor (1962, 1969) and recent studies (Bryant and Wang 1990; Courtless 1988; Nelson 1989; Norum 1989; Wagner 1986; Wagner and Hanna 1983; Winakor 1986, 1989) provide valuable contributions to the analysis of clothing expenditures in the United States. Norum (1990) includes the latest review of literature on modeling clothing expenditures and provides a synthesis of the major findings.

Norum's (1990) estimates of a modified stock-adjustment model (Bryant and Wang 1990) for the time period 1929-1987 show that consumers react inelastically to changes in clothing prices and income variables.(5) This generally supports findings of Bryant and Wang (1990) and Winakor (1986). Nonetheless, note that Bryant and Wang, treating clothing and shoes as durables, find a unit price elasticity for these goods. However, Norum finds no significant effect for price on consumers' clothing expenditures, a negative relationship between clothing expenditures and population, and emphasizes that other characteristics of the population should be used in modeling U.S. clothing expenditures. It is also of interest to note that Winakor (1962) finds a negative time trend for clothing expenditures, which may have been approximating the trend in the population variable used in Norum's study.

Applications of standard and innovative adjustment models in other studies do not include estimates of the ECM for U.S. clothing expenditures. Given the well-known theoretical and empirical properties of the ECM model (see Gilbert (1986) for relevant references), it is surprising that this specification has never been used. Note that dynamic modeling, of which ECM modeling is a procedure, allows theories and data to interact and provides a richer dynamic structure than the conventional partial adjustment model. The ECM model allows for shortrun dynamics (adjustments) to be suggested by the data and reproduces theoretical assertions (equilibrium) in the long run. The basic idea is that observations (e.g., clothing expenditure data) are the outcomes of shortrun economic activity, while theoretical assertions, such as the proportionality hypothesis (homogeneity of degree one), hold in the long run. Dynamic modeling methodology is surveyed in Hendry, Pagan, and Sargan (1984).

ESTIMATION WHEN MODEL INCLUDES NONSTATIONARY TIME SERIES

Because inclusion of nonstationary variables in the regression analysis renders the usual methods for hypothesis testing invalid, alternative procedures that circumvent this problem are desirable.(6) Two decades ago, Box and Jenkins (1970) suggested differencing as a method of achieving stationarity. However, they did not take into account that nonstationary components of variables (e.g., trends) convey the longrun information that economic theory is set to establish. This suggests that differencing could be more costly than beneficial. As a method of achieving stationarity in variables (while the relationship is in levels of variables), conventional differencing leads to the loss of a priori information (the longrun relationship) suggested by economic theory. For example, a regression that uses first differences of variables may provide shortrun evidence on the relationship of variables, but is likely to lack relevant longrun information nested in the levels. Clearly, potential for specification error, i.e., ignoring the longrun information, makes inference suspect. In this respect, extreme care must be taken. Hence, a framework that circumvents statistical difficulties associated with use of trended variables and simultaneously exploits their longrun information is needed. Engle and Granger's (1987) two-step procedure provides the relevant framework for an analysis involving nonstationary variables.

Relevant tests for determining nonstationarity of data are shown, in this paper, to be easy to implement; hence, it is important to test the null hypothesis of nonstationarity before any time-series data are used.(7) Moreover, to circumvent pitfalls of working with stochastically trended data, Engle and Granger's two-step modeling approach can be used to construct a model of clothing expenditures that, in the second step, allows the usual t- and F-statistics to be used in hypothesis testing. The suggested two-step method utilizes both the differences and the levels of variables included in the models. However, valid inferences can be made because the two-step technique ensures the stationarity of the variables in the final model. Engle and Granger also suggest that the ECM model is general enough to take other dynamic models as a special case and still use differences and levels of variables as the dependent and explanatory variables. The attractiveness of this methodology is that for testing and modeling economic variables (e.g., clothing expenditures) no sophisticated estimation technique, apart from Ordinary Least Squares (OLS), is needed.

ERROR CORRECTION MECHANISM

An ECM model of clothing expenditures may be derived by minimizing a fairly general quadratic cost function based on adjusting the natural log of actual clothing expenditures, lnC, to its desirable value, lnC*. For related discussion of dynamic cost functions and their applications see Nickell (1985). The cost function for clothing expenditures takes the following form.

ln|L.sub.t~ = ||lambda~.sub.1~(ln|C.sub.t~ - ln|C*.sub.t~).sup.2~ + ||lambda~.sub.2~(ln|C.sub.t~ - ln|C.sub.t-1~).sup.2~ - 2||lambda~.sub.3~(ln|C.sub.t~ - ln|C.sub.t-1~)(ln|C*.sub.t~ - ln|C*.sub.t-1) (1)

Where adjustment factors ||lambda~.sub.1~, ||lambda~.sub.2~, and ||lambda~.sub.3~ |is greater than or equal to~ 0, and (ln|C.sub.t~ - ln|C.sub.t-1~)(ln|C*.sub.t~ - ln|C*.sub.t-1~) reflect the attenuation of cost if actual consumers' clothing purchases move toward the desired (optimal) position. This cost function is fairly general as it nests other functions that lead to the well-known PAM. Setting ||lambda~.sub.3~ = 0 generates the cost function that, upon minimization with respect to ln|C*.sub.t~, |delta~ln|L.sub.t~/|delta~ln|C*.sub.t~ = 0, assuming that lnC = |kappa~ + |eta~lnY, reproduces the well-known PAM; i.e., ln|C*.sub.t~ = ||beta~.sub.0~ + ||beta~.sub.1~ ln|Y.sub.t~ + ||beta~.sub.3~ln|C.sub.t-1~, where ||beta~.sub.0~ = ||lambda~.sub.1~|kappa~/(||lambda~.sub.1~ + ||lambda~.sub.2~), ||beta~.sub.1~ = ||lambda~.sub.1~/(||lambda~.sub.1~ + ||lambda~.sub.2~) and ||beta~.sub.3~ = ||lambda~.sub.2~/(||lambda~.sub.1~ + ||lambda~.sub.2~).(8)

Similarly, to derive a variant of the ECM model, ln|L.sub.t~ should be minimized with respect to ln|C.sub.t~, |Mathematical Expression Omitted~. This minimization yields:

ln|C*.sub.t~ = ||beta~.sub.1~ln|C.sub.t~ + ||beta~.sub.2~ln|C.sub.t-1~ + ||beta~.sub.3~ln|C*.sub.t-1~ (2)

where ||beta~.sub.1~ = (||lambda~.sub.1~ + ||lambda~.sub.2~)/(||lambda~.sub.1~ + ||lambda~.sub.3~), ||beta~.sub.2~ = -||lambda~.sub.2~/(||lambda~.sub.1~ + ||lambda~.sub.3~), and ||beta~.sub.3~ + ||lambda~.sub.3~/(||lambda~~.sub.1~ + ||lambda~.sub.3~). Rearranging (2) yields an unrestricted ECM model of clothing expenditures,

ln|C(*).sub.t~ - ln|C*.sub.t - 1~ = ||beta~.sub.t~ (ln|C.sub.t~ - ln|C.sub.t -1~) + (1 - ||beta~.sub.3~)(|theta~ln|C.sub.t - 1~ - ln|C*.sub.t - 1~) or

|delta~ln|C*.sub.t~ = ||beta~.sub.1~|delta~ln|C.sub.t~ + (1 - ||beta~.sub.3~)(|theta~ln|C.sub.t-1~ - ln|C*.sub.t-1), (3)

where |theta~ = (||beta~.sub.1~ + ||beta~.sub.2~)/(1 - ||beta~.sub.3~) which is either given by economic theory (e.g., homogeneity of degree one: ||beta~.sub.1~ + ||beta~.sub.2~ + ||beta~.sub.3~ = 1) or estimated directly from the data. Equivalently, (2) can be written as:

|delta~ln|C*.sub.t~ = ||beta~.sub.1~|delta~ln|C.sub.t~ + |phi~ln|C.sub.t-1~ + (1 - ||beta~.sub.3~)(ln|C.sub.t-1 - ln|C*.sub.t-1~), (4)

where |phi~ = ||beta~.sub.1~ + ||beta~.sub.2~ + ||beta~.sub.3~ - 1. Note that (|theta~ln|C.sub.t-1~ - ln|C*.sub.t-1~) in (3) and (ln|C.sub.t-1~ - ln|C*.sub.t-1~) in (4) and the so-called error correction terms, representing disequilibrium (or mistakes) experienced by consumers. In further transformation of (3) and (4), to express ln|C*.sub.t~ and ln|C*.sub.t-1~ in observable terms, reasonable presumptions or theoretical conjectures (priors), such as lnC* = lnC + |epsilon~, lnC = |kappa~ + |eta~lnY, and/or lnC* = |kappa~ + lnY, where lnY is an independent variable (income) and |epsilon~ is a transitory term, are made. Accordingly, (3) and (4) can be written as:

|delta~ln|C.sub.t~ = ||alpha~.sub.0~ + ||alpha~.sub.1~|delta~ln|Y.sub.t~ + |gamma~(|mu~ln|Y.sub.t-1~ - ln|C.sub.t-1~), and (3')

|delta~ln|C.sub.t~ = ||alpha~.sub.0~ + ||alpha~.sub.1~ |delta~ln|Y.sub.t~ + ||alpha~.sub.2~ln|Y.sub.t-1~ + |gamma~(ln|Y.sub.t-1~ - ln|C.sub.t-1~) (4')

where ||alpha~.sub.0~, ||alpha~.sub.1~, ||alpha~.sub.2~, |gamma~, and |mu~ are parameters to be estimated. It should be emphasized that the change of notation is to show moving from the theoretical discussion about unobserved values to empirical specification and observed values. Note also that unit income elasticity or homogeneity of degree one implies that |mu~ = 1 or, equivalently, |alpha.sub.2~ = 0. In Engle and Granger's (1987) two-step procedure, |mu~ (longrun income elasticity) is estimated in the first step and |alpha.sub.1~ (shortrun income elasticity) is estimated in the second step. In (3'), the |lambda~ (adjustment or loading factor) measures the responsiveness of clothing expenditures to past disequilibrium (|mu~ln|Y.sub.t-1~ -- ln|C.sub.t-1~).

In practice, to apply a model such as the one given in (3') or (4') to actual nonstationary data, researchers require a methodology that provides a general framework for modeling nonstationarity of the variables. Engle and Granger's two-step (cointegration) procedure is the appropriate method for the analysis of nonstationary time series.

TWO-STEP COINTEGRATION PROCEDURE

Empirical models of the relationship between clothing expenditures and other variables ignore the fact that these variables may follow some nonstationary (integrated) process. If nonstationarity of variables is not taken into account, regression models yield statistics that are unreliable for inference. Hence, explicit account of the non-stationary behavior of time series in any model used for inference is a vital issue. Engle and Granger's (1987) cointegration techniques account for the nonstationary properties of data and allow a valid analysis to be conducted.

Cointegration theory suggests that two variables, each being stationary in its changes, are cointegrated, if some linear combination of them is stationary in levels. The equilibrium relationship implies existence of a unique set of parameters (e.g., |mu~ in (3')) that does not allow the variables in the model (dependent and a linear combination of the independent variables) to depart from one another by a significant degree; hence, the residuals of an equilibrium relationship should be stationary. In the statistical sense, an equilibrium relationship implies that OLS estimates of the parameters of the model are consistent--as the sample size increases the bias and variance tend to zero. Engle and Granger suggest that equilibrium relationships imply cointegration among the variables of interest. Moreover, they prove that cointegrated varieables have an ECM representation and vice versa. Engle and Grange recommend a two-step procedure for modeling economic relationships; in the first step, one has to determine the order of integration, I(d), of the individual time series. An integrated series is a nonstationary series (does not have constant mean and variance) that can be made stationary by differencing the series d times. In other words, the order of integration of a variable, say clothing expenditures, |C.sub.t~, is the number of times (d) that |C.sub.t~ needs differencing in order to generate a stationary series. For example, if |C.sub.t~ needs to be differenced only once to achieve stationarity, then |C.sub.t~ is I(1) and, hence, |delta~|C.sub.t~ is I(0). After determining the order of integration of relevant variables, one has to find that linear combination of the individual series, i.e., I(1) series, which forms a stationary process. This is done by running a regression of the dependent variable, an I(1) variable, on the independent variables, which are I(1) or I(0), and then testing the estimated residuals for stationarity, i.e., being I(0). This regression is the so-called cointegrated regression, and OLS is the proper technique for estimating its parameters. Nonetheless, conventional estimates of other statistics, such as t-tests, are not proper for inference, as they do not follow standard distributions. However, if the estimated residuals are stationary then coefficient estimates are consistent.(9)

Consistency, implied by testing for the stationarity of residuals, indicates that coefficient estimates are unique and not time dependent. Conversely, invariance of the coefficient estimates reflects bounded variability (stationarity) of the residuals. Hence, dependent and independent variables cannot deviate from their joint longrun path by too much. This tight relationship is characteristic of a unique equilibrium and is captured in the residuals of the cointegrated (equilibrium) regression. In practice, rejecting nonstationarity of residuals, which is equivalent to testing for a longrun relationship between variables, is the only viable inference that prevents us from accepting a spurious relationship.

In the second step, the lagged residuals from the cointegrated regression in (1) are entered as the error correction term, |EC.sub.t-1~, in an ECM representation. If an ECM model adequately represents the shortrun dynamic and the longrun static (equilibrium) behavior of variables then the error correction term is significant (see the Granger Representation theorem in Engle and Granger (1987)). In other words, if the observations (dependent variable) are generated by an ECM model, then the left-hand side and the right-hand side variables should be cointegrated in levels.(10) Conversely, the validity of an ECM model implies equilibrium (cointegration) among the economic variables. Because every term in the ECM model is stationary (no violation of the Gauss-Markov assumption is occurring), valid inferences using t-, F-, and other statistics can be made.

In the following section, the hypothesis that logarithmic values of U.S. clothing expenditures, disposable income, relative price of clothing, the elderly/population ratio, and unemployment rate are nonstationary in levels but follow an equilibrium (cointegrated) path, is explored. Tests for the adequacy of an ECM model as the data generating process (proper model) of U.S. clothing expenditures for the period 1929-1987 are performed.

DATA AND EMPIRICAL EVIDENCE

The data used in this study are the same as those used in Norum (1990). Annual real per capita clothing expenditures (|C.sub.t~) and disposable incomes (|Y.sub.t~) for the period of 1929-1987 are the basic inputs to this analysis. Figures 1, 2, and 3 show the relationship between the log of these two variables (ln|C.sub.t~ and ln|Y.sub.t~), their first changes |delta~ln|C.sub.t~ and |delta~ln|Y.sub.t~), and their ratios ln|C.sub.t~/ln|Y.sub.t~. Figure 1 shows a growing gap between clothing expenditures and income, with the former variable (ln|C.sub.t~) tending to stabilize. It is notable that the slope of ln|C.sub.t~ is gentler than that of ln|Y.sub.t~ implying a negative trend in the ratio of ln|C.sub.t~/ln|Y.sub.t~ as income increases. Figure 2 is more revealing. It shows the growth rate of clothing expenditures to have more variability than the growth rate of disposable income. This is opposite to what is found for aggregate consumption expenditures and disposable income. Figure 3 clearly depicts a remarkably complex dynamic relationship between ln|C.sub.t~, ln|Y.sub.t~, and ln|C.sub.t~/ln|Y.sub.t~. This figure also shows a negative trend in the ln|C.sub.t~/ln|Y.sub.t~ ratio as disposable income grows. Although one may be able to capture the longrun trend of the relationship between U.S. clothing expenditures and other variables using regressions on levels of variables, shortrun movement of the data requires a dynamic model. However, a complete model of clothing expenditures should allow for both the shortrun dynamics and the longrun trends.

To explain the complexity of U.S. clothing expenditures, one needs other relevant and influential factors. The relative price of clothing expenditures, i.e., the ratio of the Consumer Price Index for apparel and upkeep to the private consumption expenditure deflator for the same time period, is clearly an important variable for explaining U.S. clothing expenditures. Additionally, for controlling economic conditions, the unemployment rate is included in the model. Finally, to capture demographic and taste changes, the ratio of the population over 65 years of age to the general population is utilized.

Testing for Nonstationarity

To determine the number of times (d) that variables used in this study require differencing to become stationary, the Dickey and Fuller (1981) t-test (denoted by DF-test) is used.(11) This test is conducted by running each variable, |C.sub.t~, on a constant and then testing the disturbances of this regression (say, |e.sub.t~) for integration of order one I(1) (a simple random walk behavior which is a nonstationary process). Hence, the null hypothesis is |Mathematical Expression Omitted~, with |Mathematical Expression Omitted~, where ||omega~.sub.t~ is the usual disturbances. The alternative hypothesis is that |e.sub.t~ follows a stationary process, |Mathematical Expression Omitted~; that is, being integrated of order zero, I(0).

To conduct the DF-test, an OLS estimate of the following regression is obtained: |delta~|e.sub.t~ = |delta~|e.sub.t-1~ + ||omega~.sub.t~. This regression is obtained by reducing |e.sub.t-1~ from both sides of (5). Now, the DF-test consists of a significance test of |Mathematical Expression Omitted~, which is a t-test. If |e.sub.t~is stationary then |delta~ is negative |Mathematical Expression Omitted~ and significant. The critical values for the DF-test are given in Fuller (1976) and differ from the conventional ones in textbooks. Given the sample size in this study and the way the test of nonstationarity is conducted (i.e., running the variables on a constant and then testing), the relevant critical value is 2.9 (Fuller 1976, 373).

Table 1 contains the estimated DF-tests for the levels and first differences of individual time series. The DF-values indicate nonstationary I(1) behavior for the levels of all variables under consideration. Although not reported here, the findings are that the levels of prices (not their ratios) need differencing more than one time to become stationary. Clearly, this indicates the inappropriateness of including levels of prices in models that have stationary or I(1) variables as the dependent variable. However, Table 1 also shows that first differencing yields stationary series I(0) for the variables used in this study.

Testing for Cointegration

These findings and the theoretical notion that clothing expenditures are a function of income and other variables, such as relative prices, age structure, and departure of the economy from full employment, allow the formulation of a longrun equilibrium relationship (cointegration regression) in levels. Application of the OLS estimation technique yields

|Mathematical Expression Omitted~

Adjusted |R.sup.2~ = 0.930, S.E. = 0.0556, DF-test = -2.52; where S.E. is the standard error of the regression (t-tests in parentheses). While the conventional t-tests for the coefficient estimates are reported, they do not follow a standard distribution and, hence, are inappropriate for inference (see Phillips and Durlauf (1986)). This is due to the fact that the right-hand side variables are nonstationary, which violates the basic assumption of having bounded variability in the regression analysis.

In testing the estimated residuals for nonstationarity, the variable (||epsilon~.sub.t~)is not run on a constant, so a different critical value for the DF-test is required. The relevant critical value is 1.95 (Fuller 1976, 373). The DF-test shows that the estimated residuals from this cointegration regression are stationary. Stationarity of the residuals implies that estimated coefficients are consistent (see Stock 1987) and present unique equilibrium (longrun) responses of the dependent variable to the right-hand side variables.(12) It should be noted that the longrun responses, estimated coefficients in (6), appear to have their appropriate signs; i.e., apart from the coefficient of income, other estimated coefficients are negative.

ECM Model of U.S. Clothing Expenditures

Entering the lagged residuals from (6) as the error correction term (|EC.sub.t-1~ = ||epsilon~.sub.t-1~) in a standard ECM model and applying OLS yields

|Mathematical Expression Omitted~

Adjusted |R.sup.2~ = 0.783, S.E. = 0.0291, |F.sub.1~(5, 51) = 41.6, |F.sub.2~(4, 47) = 1.39, LM(1) = .81, ARCH(1) = 0.87; where |v.sub.t~ is the error term and |delta~ is the first difference operator (t-tests in parentheses). The first differences of the variables expressed in log, i.e., |delta~|lnc.sub.t~ and |delta~|lny.sub.t~ equal the percentage change in these variables (%|delta~|c.sub.t~ and %|delta~|y.sub.t~). Accordingly, the estimated coefficients of the right-hand side variables in (6) that can be expressed as the ratios of two growth rates, for instance, %|delta~|c.sub.t~/%|delta~|y.sub.t~ are known as the impact multipliers or the shortrun elasticities. For the longrun solution of the estimated model, %|delta~|c.sub.t~ and %|delta~|y.sub.t~ are set to zero, leaving only the levels of the logged variables in the model. Coefficients on the log of the levels of variables are the total multipliers or the longrun elasticities, as they reflect the longrun response of the dependent variable to the independent variable.

An important feature of (6) is that, as the reported t-ratios test the significance of the coefficients of stationary variables, valid hypothesis testing can be conducted. Note that values in the parentheses below the coefficient estimates are heteroscedasticity-consistent t-tests (White 1980). Other relevant statistics for this regression, with degrees of freedom in parentheses, are also reported. |F.sub.1~(5, 51) tests the null hypothesis that, apart from the constant, other estimated coefficients are jointly insignificant. |F.sub.2~(4, 47) tests the relevance of the restrictions that are nested in the |EC.sub.t-1~ term. This test is obtained by estimating an unrestricted version of (6), i.e., replacing |EC.sub.t-1~ by ln|C.sub.t-1~, ln|Y.sub.t-1~, ln(|Pc.sub.t-1~/|Pc.sub.t-1~), ln(|POP65.sub.t-1~/|POP.sub.t-1~), and |U.sub.t-1~ and then estimating the model. LM(1) is the Lagrange Multiplier test for first order autoregressive disturbances (see Breusch 1978; Godfrey 1978); LM(1) is distributed as chi-squared with one degree of freedom. ARCH(1) is a test for first order autoregressive conditional heteroscedasticity among the disturbances (see Engle 1982); ARCH(1) is distributed as chi-squared with one degree of freedom.

The error correction term, |EC.sub.t-1~, which is significant at the five percent level, supports the appropriateness of the ECM model for clothing expenditures in the United States. The high adjusted |R.sup.2~ indicates that a large portion of variation in the dependent variable is explained by the explanatory variables. The adjusted |R.sup.2~ of 78 percent is considered to be high as the dependent variable is the rate of growth rather than a trended variable. Also the dependent variable, being in the differenced form, reduces the possibility of spurious correlation that is encountered when the levels of variables are used. In addition to the high |R.sup.2~, other statistics, at the five percent level of significance, support the ECM model as an alternative model of U.S. clothing expenditures. |F.sub.1~, being larger than the critical value of 2.3, rejects the null of no relationship. On the other hand, |F.sub.2~, being smaller than the critical value of 2.5, does not reject the nested restriction in the error correction term (|EC.sub.t-1~). LM and ARCH, also being smaller than the critical value of 3.8, support (6) by not rejecting the nulls of "no first order autoregression" and "no ARCH-effect" among the disturbances, respectively.

Shortrun and Longrun Elasticities

Income

Given that the estimated coefficients of the logs of the variables are elasticities, the estimated slopes in (5) and (6) reflect the longrun and shortrun elasticities respectively. Estimated shortrun and longrun income elasticities of clothing expenditures do not appear to be too different from each other. These elasticities (0.48 and 0.50) indicate that the demand for clothing is income inelastic (see Bryant and Wang (1990)). In particular, findings in this paper indicate that a ten percent increase (decrease) in income causes clothing expenditures to increase (decrease), upon impact, by 4.8 percent. Because the short- and longrun income elasticities are similar, no substantial role for the longrun adjustment to an income change is found. Nonetheless, the reported estimates in this paper are about half of the shortrun income elasticity (i.e., 0.97) reported by Norum (1990, 77). This could be due to the fact that the ECM allows for a richer dynamic structure than models that use conventional stock adjustment procedures to represent the dynamic reactions of the variables of interest. Clearly, the lower short- and longrun income elasticities obtained in this study indicate that changes in income due to government policies, the business cycle, or other factors are not as influential in changing clothing expenditures in the United States as might be inferred from higher estimates reported in other studies.

Prices

Estimates in this paper show that, in the short run, clothing expenditures are highly price elastic (-1.9) while in the long run this elasticity settles at unity (-1.0). It is of interest to note that these findings indicate that a ten percent increase (decrease) in the relative price of clothing causes clothing expenditures to decrease (increase) in the short run by 19 percent. However in the long run, changes in the relative price of clothing and clothing expenditures are proportional. These findings differ markedly from some previous studies. For example, the longrun price elasticity of clothing expenditures in this study is about four times the value reported by Norum (1990). This can also be contrasted with the inelastic clothing expenditures reported in Winakor (1989). The finding of unitary price elasticity is in agreement with the value obtained by Bryant and Wang (1990). Overall, the short- and longrun price elasticity estimates obtained in this study show that clothing expenditures in the United States are highly responsive to price changes. These results support the idea that substantial increases in clothing expenditures could result if relative prices were to drop significantly. Obviously, reduction of relevant trade barriers, for example between the United States and Mexico that may lead to a decrease in the relative price of apparel and up-keep, is a potential driving force for increased clothing expenditures.

Population

Estimated responses of clothing expenditures to the elderly/population ratio show that, despite a statistically insignificant shortrun rise in clothing expenditures due to an increase in the elderly/population ratio, the longrun effect of a rise in this ratio is negative and reduces U.S. clothing expenditures. Basically, inclusion of the elderly/population ratio allows the researcher to control for changes in taste. As it is well-known that the elderly/population ratio is increasing in the United States, discovery that in the short run no significant change in clothing expenditures is likely to occur due to the increase in the elderly/population ratio is important. However, slow evolution of taste (and other factors) reflected in this demographic variable (the elderly/population ratio), while leaving clothing expenditures unchanged in the short run, implies a negative longrun impact on clothing expenditures.

Unemployment

Interpreting the unemployment rate as the general condition of the economy, a significant decline in clothing expenditures due to a rise in the unemployment rate in the short run is found. Similarly, the longrun impact of a rise in the unemployment rate on U.S. clothing expenditures is 1.6 times that of the shortrun impact. These findings are consistent with the observed behavior of clothing expenditures and its growth rate during the 1930s depicted in Figures 1 and 2. Norum (1990) also found a negative coefficient estimate for a dummy variable that she included to account for the depression years (1929-1933).

CONCLUSION

In this paper an error correction model of U.S. clothing expenditures is estimated using recent developments in testing for stationarity and modeling cointegrated variables using Engle and Granger's (1987) two-step procedure. On the basis of the reported statistics, it is found that logarithmic values of U.S. clothing expenditures, disposable income, relative price of clothing, elderly/population ratio, and the unemployment rate are stationary in their first changes. Moreover, the Error Correction Mechanism model and Engle and Granger's two-step procedure are argued to be an appropriate technique for estimating clothing expenditures (a nonstationary variable) in the United States.

The estimates of the shortrun and longrun elasticities of clothing expenditures with respect to income and prices indicate that demand for clothing, although income inelastic, is highly price elastic. In the long run, expenditures for clothing appear to have a unit elasticity with respect to prices. Inclusion of the number of elderly (above the age of 65) relative to the total population and the unemployment rate show that longrun expenditures on clothing in the United States decrease as these two ratios increase.

Future research may greatly benefit by taking into account properties of time-series data in modeling economic activities. Relevant tests, modeling and estimation techniques, as demonstrated, are accessible enough to suggest Engle and Granger's two-step technique for further analysis of historical time-series observations.

Manouchehr Mokhtari is Assistant Professor, Department of Family Studies, University of Maryland at College Park.

I would like to thank Janet Wagner for her generous comments and suggestions. I also would like to thank Carole Makela and two anonymous referees for helpful comments and Pam Norum for providing me with her data set. This paper is dedicated to Professor Rachel Dardis. This research was supported by a grant from HUEC/DRIF and the computing facilities of the Computer Science Center at the University of Maryland at College Park.

1 A nonstationary variable (process) does not possess constant mean and variance, while a stationary process does. Use of nonstationary variables in the regression analysis leads to the violation of Gauss-Markov assumptions that may cause inconsistency in parameter estimates.

2 If variables included in a regression model are nonstationary then distributions of the usual statistics are not the same as those derived when only stationary variables are included. See, for example, Phillips and Durlauf (1986). In particular, parameters of the regression will not converge in probability as the sample size increases. Hendry (1986) provides relevant references and historical background.

3 More sophisticated econometric approaches to the problem of nonstationary regressors and regressands exist, but for the sake of simplicity of exposition are not used here. Interested readers are referred to Johansen (1988).

4 Nelson and Plosser (1982) show that most U.S. time series follow some nonstationary process.

5 See, also, Bryant (1983) and Norum (1989). In these studies, clothing is assumed to be a semidurable or durable good. Bryant (1990, 36-37) notes that clothing and shoes, lasting for less than three years, are classified as nondurable goods by the U.S. Department of Commerce.

6 Inclusion of nonstationary variables in the regression model cause the distributions of t- and F-tests to diverge and the Durbin-Watson statistic to approach zero.

7 In recent years, proliferation of testing for nonstationarity in data (unit root testing) is indicative of ever growing concerns over the inclusion of nonstationary variables in models (Hendry 1986).

8 Many other cost functions can be postulated and variants of the PAM and the ECM can be derived. In this paper, with no significant loss, only the simple and convenient functions, estimation techniques, and tests are considered.

9 Variables included in this regression, being nonstationary, violate the Gauss-Markov assumption, making the usual standard errors of coefficients and t- and F-statistics useless for inference.

10 Cointegration implies the existence of an ECM as the data generating process and vice versa.

11 It is worth emphasizing that many other tests for nonstationarity exist. However, only the simplest one (DF-test) is reported here.

12 Restriction on the disturbances implies restriction on the parameters in the model and vice versa.

REFERENCES

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Nelson, J. (1989), "Individual Consumption within the Household: A Study of Expenditures on Clothing," The Journal of Consumer Affairs, 23: 21-44.

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Norum, P. S. (1989), "Economic Analysis of Quarterly Household Expenditures on Apparel," Home Economics Research Journal, 17: 228-240.

Norum, P. S. (1990), "U.S. Clothing Expenditures: A Time Series Analysis, 1929-1987," in American Council on Consumer Interests--The Proceedings, Mary Carsky (ed.), Columbia, MO: ACCI: 75-81.

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Wagner, J. (1986), "Expenditure for Household Textiles and Textile Home Furnishings: An Engle Curve Analysis," Home Economics Research Journal, 15(1): 21-31.

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Winakor, G. (1989), "The Decline in Expenditures for Clothing Relative to Total Consumer Spending, 1929-1986," Home Economics Research Journal, 17: 195-215.

Despite the well-known properties of the ECM model (see, for example, Gilbert (1986) and references therein), this specification has never been used for modeling clothing expenditures. In this regard, the ECM model provides a richer dynamic structure than conventional models such as the Partial Adjustment Model (PAM). In fact, it takes the PAM as a special case. The ECM modeling procedure (1) is easy to implement; (2) sharpens insight into the dynamics of consumer decision making; (3) is fairly general, giving an accurate account of the observations; and (4) is an excellent pedagogical tool for modeling and studying many aspects of consumer activities over time.

This paper also emphasizes the inherent nonstationarity of time-series data that renders the usual textbook inferences from t-tests and F-tests invalid, because real world data do not satisfy the conventional (Gauss-Markov) properties assumed when reporting tables of critical values. Hence, research using old methodologies, statistics, and tables of critical values is invalid, when modeling nonstationary variables.(2) This paper, by application and example, demonstrates a methodology that is easy to implement and does not lead to estimating spurious relationships among variables.(3) Indeed, it can be seen that existence of an equilibrium relationship between nonstationary variables (say, consumption and income)(4) naturally leads to estimating an ECM model that is a proper framework for robust inference. This is due to the fact that the regressors and regressand of an ECM model are ensured to follow a stationary process.

CLOTHING EXPENDITURE STUDIES

Pioneering work by Winakor (1962, 1969) and recent studies (Bryant and Wang 1990; Courtless 1988; Nelson 1989; Norum 1989; Wagner 1986; Wagner and Hanna 1983; Winakor 1986, 1989) provide valuable contributions to the analysis of clothing expenditures in the United States. Norum (1990) includes the latest review of literature on modeling clothing expenditures and provides a synthesis of the major findings.

Norum's (1990) estimates of a modified stock-adjustment model (Bryant and Wang 1990) for the time period 1929-1987 show that consumers react inelastically to changes in clothing prices and income variables.(5) This generally supports findings of Bryant and Wang (1990) and Winakor (1986). Nonetheless, note that Bryant and Wang, treating clothing and shoes as durables, find a unit price elasticity for these goods. However, Norum finds no significant effect for price on consumers' clothing expenditures, a negative relationship between clothing expenditures and population, and emphasizes that other characteristics of the population should be used in modeling U.S. clothing expenditures. It is also of interest to note that Winakor (1962) finds a negative time trend for clothing expenditures, which may have been approximating the trend in the population variable used in Norum's study.

Applications of standard and innovative adjustment models in other studies do not include estimates of the ECM for U.S. clothing expenditures. Given the well-known theoretical and empirical properties of the ECM model (see Gilbert (1986) for relevant references), it is surprising that this specification has never been used. Note that dynamic modeling, of which ECM modeling is a procedure, allows theories and data to interact and provides a richer dynamic structure than the conventional partial adjustment model. The ECM model allows for shortrun dynamics (adjustments) to be suggested by the data and reproduces theoretical assertions (equilibrium) in the long run. The basic idea is that observations (e.g., clothing expenditure data) are the outcomes of shortrun economic activity, while theoretical assertions, such as the proportionality hypothesis (homogeneity of degree one), hold in the long run. Dynamic modeling methodology is surveyed in Hendry, Pagan, and Sargan (1984).

ESTIMATION WHEN MODEL INCLUDES NONSTATIONARY TIME SERIES

Because inclusion of nonstationary variables in the regression analysis renders the usual methods for hypothesis testing invalid, alternative procedures that circumvent this problem are desirable.(6) Two decades ago, Box and Jenkins (1970) suggested differencing as a method of achieving stationarity. However, they did not take into account that nonstationary components of variables (e.g., trends) convey the longrun information that economic theory is set to establish. This suggests that differencing could be more costly than beneficial. As a method of achieving stationarity in variables (while the relationship is in levels of variables), conventional differencing leads to the loss of a priori information (the longrun relationship) suggested by economic theory. For example, a regression that uses first differences of variables may provide shortrun evidence on the relationship of variables, but is likely to lack relevant longrun information nested in the levels. Clearly, potential for specification error, i.e., ignoring the longrun information, makes inference suspect. In this respect, extreme care must be taken. Hence, a framework that circumvents statistical difficulties associated with use of trended variables and simultaneously exploits their longrun information is needed. Engle and Granger's (1987) two-step procedure provides the relevant framework for an analysis involving nonstationary variables.

Relevant tests for determining nonstationarity of data are shown, in this paper, to be easy to implement; hence, it is important to test the null hypothesis of nonstationarity before any time-series data are used.(7) Moreover, to circumvent pitfalls of working with stochastically trended data, Engle and Granger's two-step modeling approach can be used to construct a model of clothing expenditures that, in the second step, allows the usual t- and F-statistics to be used in hypothesis testing. The suggested two-step method utilizes both the differences and the levels of variables included in the models. However, valid inferences can be made because the two-step technique ensures the stationarity of the variables in the final model. Engle and Granger also suggest that the ECM model is general enough to take other dynamic models as a special case and still use differences and levels of variables as the dependent and explanatory variables. The attractiveness of this methodology is that for testing and modeling economic variables (e.g., clothing expenditures) no sophisticated estimation technique, apart from Ordinary Least Squares (OLS), is needed.

ERROR CORRECTION MECHANISM

An ECM model of clothing expenditures may be derived by minimizing a fairly general quadratic cost function based on adjusting the natural log of actual clothing expenditures, lnC, to its desirable value, lnC*. For related discussion of dynamic cost functions and their applications see Nickell (1985). The cost function for clothing expenditures takes the following form.

ln|L.sub.t~ = ||lambda~.sub.1~(ln|C.sub.t~ - ln|C*.sub.t~).sup.2~ + ||lambda~.sub.2~(ln|C.sub.t~ - ln|C.sub.t-1~).sup.2~ - 2||lambda~.sub.3~(ln|C.sub.t~ - ln|C.sub.t-1~)(ln|C*.sub.t~ - ln|C*.sub.t-1) (1)

Where adjustment factors ||lambda~.sub.1~, ||lambda~.sub.2~, and ||lambda~.sub.3~ |is greater than or equal to~ 0, and (ln|C.sub.t~ - ln|C.sub.t-1~)(ln|C*.sub.t~ - ln|C*.sub.t-1~) reflect the attenuation of cost if actual consumers' clothing purchases move toward the desired (optimal) position. This cost function is fairly general as it nests other functions that lead to the well-known PAM. Setting ||lambda~.sub.3~ = 0 generates the cost function that, upon minimization with respect to ln|C*.sub.t~, |delta~ln|L.sub.t~/|delta~ln|C*.sub.t~ = 0, assuming that lnC = |kappa~ + |eta~lnY, reproduces the well-known PAM; i.e., ln|C*.sub.t~ = ||beta~.sub.0~ + ||beta~.sub.1~ ln|Y.sub.t~ + ||beta~.sub.3~ln|C.sub.t-1~, where ||beta~.sub.0~ = ||lambda~.sub.1~|kappa~/(||lambda~.sub.1~ + ||lambda~.sub.2~), ||beta~.sub.1~ = ||lambda~.sub.1~/(||lambda~.sub.1~ + ||lambda~.sub.2~) and ||beta~.sub.3~ = ||lambda~.sub.2~/(||lambda~.sub.1~ + ||lambda~.sub.2~).(8)

Similarly, to derive a variant of the ECM model, ln|L.sub.t~ should be minimized with respect to ln|C.sub.t~, |Mathematical Expression Omitted~. This minimization yields:

ln|C*.sub.t~ = ||beta~.sub.1~ln|C.sub.t~ + ||beta~.sub.2~ln|C.sub.t-1~ + ||beta~.sub.3~ln|C*.sub.t-1~ (2)

where ||beta~.sub.1~ = (||lambda~.sub.1~ + ||lambda~.sub.2~)/(||lambda~.sub.1~ + ||lambda~.sub.3~), ||beta~.sub.2~ = -||lambda~.sub.2~/(||lambda~.sub.1~ + ||lambda~.sub.3~), and ||beta~.sub.3~ + ||lambda~.sub.3~/(||lambda~~.sub.1~ + ||lambda~.sub.3~). Rearranging (2) yields an unrestricted ECM model of clothing expenditures,

ln|C(*).sub.t~ - ln|C*.sub.t - 1~ = ||beta~.sub.t~ (ln|C.sub.t~ - ln|C.sub.t -1~) + (1 - ||beta~.sub.3~)(|theta~ln|C.sub.t - 1~ - ln|C*.sub.t - 1~) or

|delta~ln|C*.sub.t~ = ||beta~.sub.1~|delta~ln|C.sub.t~ + (1 - ||beta~.sub.3~)(|theta~ln|C.sub.t-1~ - ln|C*.sub.t-1), (3)

where |theta~ = (||beta~.sub.1~ + ||beta~.sub.2~)/(1 - ||beta~.sub.3~) which is either given by economic theory (e.g., homogeneity of degree one: ||beta~.sub.1~ + ||beta~.sub.2~ + ||beta~.sub.3~ = 1) or estimated directly from the data. Equivalently, (2) can be written as:

|delta~ln|C*.sub.t~ = ||beta~.sub.1~|delta~ln|C.sub.t~ + |phi~ln|C.sub.t-1~ + (1 - ||beta~.sub.3~)(ln|C.sub.t-1 - ln|C*.sub.t-1~), (4)

where |phi~ = ||beta~.sub.1~ + ||beta~.sub.2~ + ||beta~.sub.3~ - 1. Note that (|theta~ln|C.sub.t-1~ - ln|C*.sub.t-1~) in (3) and (ln|C.sub.t-1~ - ln|C*.sub.t-1~) in (4) and the so-called error correction terms, representing disequilibrium (or mistakes) experienced by consumers. In further transformation of (3) and (4), to express ln|C*.sub.t~ and ln|C*.sub.t-1~ in observable terms, reasonable presumptions or theoretical conjectures (priors), such as lnC* = lnC + |epsilon~, lnC = |kappa~ + |eta~lnY, and/or lnC* = |kappa~ + lnY, where lnY is an independent variable (income) and |epsilon~ is a transitory term, are made. Accordingly, (3) and (4) can be written as:

|delta~ln|C.sub.t~ = ||alpha~.sub.0~ + ||alpha~.sub.1~|delta~ln|Y.sub.t~ + |gamma~(|mu~ln|Y.sub.t-1~ - ln|C.sub.t-1~), and (3')

|delta~ln|C.sub.t~ = ||alpha~.sub.0~ + ||alpha~.sub.1~ |delta~ln|Y.sub.t~ + ||alpha~.sub.2~ln|Y.sub.t-1~ + |gamma~(ln|Y.sub.t-1~ - ln|C.sub.t-1~) (4')

where ||alpha~.sub.0~, ||alpha~.sub.1~, ||alpha~.sub.2~, |gamma~, and |mu~ are parameters to be estimated. It should be emphasized that the change of notation is to show moving from the theoretical discussion about unobserved values to empirical specification and observed values. Note also that unit income elasticity or homogeneity of degree one implies that |mu~ = 1 or, equivalently, |alpha.sub.2~ = 0. In Engle and Granger's (1987) two-step procedure, |mu~ (longrun income elasticity) is estimated in the first step and |alpha.sub.1~ (shortrun income elasticity) is estimated in the second step. In (3'), the |lambda~ (adjustment or loading factor) measures the responsiveness of clothing expenditures to past disequilibrium (|mu~ln|Y.sub.t-1~ -- ln|C.sub.t-1~).

In practice, to apply a model such as the one given in (3') or (4') to actual nonstationary data, researchers require a methodology that provides a general framework for modeling nonstationarity of the variables. Engle and Granger's two-step (cointegration) procedure is the appropriate method for the analysis of nonstationary time series.

TWO-STEP COINTEGRATION PROCEDURE

Empirical models of the relationship between clothing expenditures and other variables ignore the fact that these variables may follow some nonstationary (integrated) process. If nonstationarity of variables is not taken into account, regression models yield statistics that are unreliable for inference. Hence, explicit account of the non-stationary behavior of time series in any model used for inference is a vital issue. Engle and Granger's (1987) cointegration techniques account for the nonstationary properties of data and allow a valid analysis to be conducted.

Cointegration theory suggests that two variables, each being stationary in its changes, are cointegrated, if some linear combination of them is stationary in levels. The equilibrium relationship implies existence of a unique set of parameters (e.g., |mu~ in (3')) that does not allow the variables in the model (dependent and a linear combination of the independent variables) to depart from one another by a significant degree; hence, the residuals of an equilibrium relationship should be stationary. In the statistical sense, an equilibrium relationship implies that OLS estimates of the parameters of the model are consistent--as the sample size increases the bias and variance tend to zero. Engle and Granger suggest that equilibrium relationships imply cointegration among the variables of interest. Moreover, they prove that cointegrated varieables have an ECM representation and vice versa. Engle and Grange recommend a two-step procedure for modeling economic relationships; in the first step, one has to determine the order of integration, I(d), of the individual time series. An integrated series is a nonstationary series (does not have constant mean and variance) that can be made stationary by differencing the series d times. In other words, the order of integration of a variable, say clothing expenditures, |C.sub.t~, is the number of times (d) that |C.sub.t~ needs differencing in order to generate a stationary series. For example, if |C.sub.t~ needs to be differenced only once to achieve stationarity, then |C.sub.t~ is I(1) and, hence, |delta~|C.sub.t~ is I(0). After determining the order of integration of relevant variables, one has to find that linear combination of the individual series, i.e., I(1) series, which forms a stationary process. This is done by running a regression of the dependent variable, an I(1) variable, on the independent variables, which are I(1) or I(0), and then testing the estimated residuals for stationarity, i.e., being I(0). This regression is the so-called cointegrated regression, and OLS is the proper technique for estimating its parameters. Nonetheless, conventional estimates of other statistics, such as t-tests, are not proper for inference, as they do not follow standard distributions. However, if the estimated residuals are stationary then coefficient estimates are consistent.(9)

Consistency, implied by testing for the stationarity of residuals, indicates that coefficient estimates are unique and not time dependent. Conversely, invariance of the coefficient estimates reflects bounded variability (stationarity) of the residuals. Hence, dependent and independent variables cannot deviate from their joint longrun path by too much. This tight relationship is characteristic of a unique equilibrium and is captured in the residuals of the cointegrated (equilibrium) regression. In practice, rejecting nonstationarity of residuals, which is equivalent to testing for a longrun relationship between variables, is the only viable inference that prevents us from accepting a spurious relationship.

In the second step, the lagged residuals from the cointegrated regression in (1) are entered as the error correction term, |EC.sub.t-1~, in an ECM representation. If an ECM model adequately represents the shortrun dynamic and the longrun static (equilibrium) behavior of variables then the error correction term is significant (see the Granger Representation theorem in Engle and Granger (1987)). In other words, if the observations (dependent variable) are generated by an ECM model, then the left-hand side and the right-hand side variables should be cointegrated in levels.(10) Conversely, the validity of an ECM model implies equilibrium (cointegration) among the economic variables. Because every term in the ECM model is stationary (no violation of the Gauss-Markov assumption is occurring), valid inferences using t-, F-, and other statistics can be made.

In the following section, the hypothesis that logarithmic values of U.S. clothing expenditures, disposable income, relative price of clothing, the elderly/population ratio, and unemployment rate are nonstationary in levels but follow an equilibrium (cointegrated) path, is explored. Tests for the adequacy of an ECM model as the data generating process (proper model) of U.S. clothing expenditures for the period 1929-1987 are performed.

DATA AND EMPIRICAL EVIDENCE

The data used in this study are the same as those used in Norum (1990). Annual real per capita clothing expenditures (|C.sub.t~) and disposable incomes (|Y.sub.t~) for the period of 1929-1987 are the basic inputs to this analysis. Figures 1, 2, and 3 show the relationship between the log of these two variables (ln|C.sub.t~ and ln|Y.sub.t~), their first changes |delta~ln|C.sub.t~ and |delta~ln|Y.sub.t~), and their ratios ln|C.sub.t~/ln|Y.sub.t~. Figure 1 shows a growing gap between clothing expenditures and income, with the former variable (ln|C.sub.t~) tending to stabilize. It is notable that the slope of ln|C.sub.t~ is gentler than that of ln|Y.sub.t~ implying a negative trend in the ratio of ln|C.sub.t~/ln|Y.sub.t~ as income increases. Figure 2 is more revealing. It shows the growth rate of clothing expenditures to have more variability than the growth rate of disposable income. This is opposite to what is found for aggregate consumption expenditures and disposable income. Figure 3 clearly depicts a remarkably complex dynamic relationship between ln|C.sub.t~, ln|Y.sub.t~, and ln|C.sub.t~/ln|Y.sub.t~. This figure also shows a negative trend in the ln|C.sub.t~/ln|Y.sub.t~ ratio as disposable income grows. Although one may be able to capture the longrun trend of the relationship between U.S. clothing expenditures and other variables using regressions on levels of variables, shortrun movement of the data requires a dynamic model. However, a complete model of clothing expenditures should allow for both the shortrun dynamics and the longrun trends.

To explain the complexity of U.S. clothing expenditures, one needs other relevant and influential factors. The relative price of clothing expenditures, i.e., the ratio of the Consumer Price Index for apparel and upkeep to the private consumption expenditure deflator for the same time period, is clearly an important variable for explaining U.S. clothing expenditures. Additionally, for controlling economic conditions, the unemployment rate is included in the model. Finally, to capture demographic and taste changes, the ratio of the population over 65 years of age to the general population is utilized.

Testing for Nonstationarity

To determine the number of times (d) that variables used in this study require differencing to become stationary, the Dickey and Fuller (1981) t-test (denoted by DF-test) is used.(11) This test is conducted by running each variable, |C.sub.t~, on a constant and then testing the disturbances of this regression (say, |e.sub.t~) for integration of order one I(1) (a simple random walk behavior which is a nonstationary process). Hence, the null hypothesis is |Mathematical Expression Omitted~, with |Mathematical Expression Omitted~, where ||omega~.sub.t~ is the usual disturbances. The alternative hypothesis is that |e.sub.t~ follows a stationary process, |Mathematical Expression Omitted~; that is, being integrated of order zero, I(0).

To conduct the DF-test, an OLS estimate of the following regression is obtained: |delta~|e.sub.t~ = |delta~|e.sub.t-1~ + ||omega~.sub.t~. This regression is obtained by reducing |e.sub.t-1~ from both sides of (5). Now, the DF-test consists of a significance test of |Mathematical Expression Omitted~, which is a t-test. If |e.sub.t~is stationary then |delta~ is negative |Mathematical Expression Omitted~ and significant. The critical values for the DF-test are given in Fuller (1976) and differ from the conventional ones in textbooks. Given the sample size in this study and the way the test of nonstationarity is conducted (i.e., running the variables on a constant and then testing), the relevant critical value is 2.9 (Fuller 1976, 373).

Table 1 contains the estimated DF-tests for the levels and first differences of individual time series. The DF-values indicate nonstationary I(1) behavior for the levels of all variables under consideration. Although not reported here, the findings are that the levels of prices (not their ratios) need differencing more than one time to become stationary. Clearly, this indicates the inappropriateness of including levels of prices in models that have stationary or I(1) variables as the dependent variable. However, Table 1 also shows that first differencing yields stationary series I(0) for the variables used in this study.

TABLE 1 Testing for Nonstationarity in Clothing Expenditure Variables DF-test Variable Levels First Difference ln|C.sub.t~ -1.00 -5.39 (Log of Clothing Expenditure) ln|Y.sub.t~ 0.28 -5.06 (Log of Disposable Income) ln(P|c.sub.t~/|P.sub.t~) -1.90 -5.46 (Log of Relative Price) ln(|POP65.sub.t~/|POP.sub.t~ -0.93 -2.32 (Log of Elderly/Population Ratio) |U.sub.t~ -1.71 -4.66 (Unemployment Rate)

Testing for Cointegration

These findings and the theoretical notion that clothing expenditures are a function of income and other variables, such as relative prices, age structure, and departure of the economy from full employment, allow the formulation of a longrun equilibrium relationship (cointegration regression) in levels. Application of the OLS estimation technique yields

|Mathematical Expression Omitted~

Adjusted |R.sup.2~ = 0.930, S.E. = 0.0556, DF-test = -2.52; where S.E. is the standard error of the regression (t-tests in parentheses). While the conventional t-tests for the coefficient estimates are reported, they do not follow a standard distribution and, hence, are inappropriate for inference (see Phillips and Durlauf (1986)). This is due to the fact that the right-hand side variables are nonstationary, which violates the basic assumption of having bounded variability in the regression analysis.

In testing the estimated residuals for nonstationarity, the variable (||epsilon~.sub.t~)is not run on a constant, so a different critical value for the DF-test is required. The relevant critical value is 1.95 (Fuller 1976, 373). The DF-test shows that the estimated residuals from this cointegration regression are stationary. Stationarity of the residuals implies that estimated coefficients are consistent (see Stock 1987) and present unique equilibrium (longrun) responses of the dependent variable to the right-hand side variables.(12) It should be noted that the longrun responses, estimated coefficients in (6), appear to have their appropriate signs; i.e., apart from the coefficient of income, other estimated coefficients are negative.

ECM Model of U.S. Clothing Expenditures

Entering the lagged residuals from (6) as the error correction term (|EC.sub.t-1~ = ||epsilon~.sub.t-1~) in a standard ECM model and applying OLS yields

|Mathematical Expression Omitted~

Adjusted |R.sup.2~ = 0.783, S.E. = 0.0291, |F.sub.1~(5, 51) = 41.6, |F.sub.2~(4, 47) = 1.39, LM(1) = .81, ARCH(1) = 0.87; where |v.sub.t~ is the error term and |delta~ is the first difference operator (t-tests in parentheses). The first differences of the variables expressed in log, i.e., |delta~|lnc.sub.t~ and |delta~|lny.sub.t~ equal the percentage change in these variables (%|delta~|c.sub.t~ and %|delta~|y.sub.t~). Accordingly, the estimated coefficients of the right-hand side variables in (6) that can be expressed as the ratios of two growth rates, for instance, %|delta~|c.sub.t~/%|delta~|y.sub.t~ are known as the impact multipliers or the shortrun elasticities. For the longrun solution of the estimated model, %|delta~|c.sub.t~ and %|delta~|y.sub.t~ are set to zero, leaving only the levels of the logged variables in the model. Coefficients on the log of the levels of variables are the total multipliers or the longrun elasticities, as they reflect the longrun response of the dependent variable to the independent variable.

An important feature of (6) is that, as the reported t-ratios test the significance of the coefficients of stationary variables, valid hypothesis testing can be conducted. Note that values in the parentheses below the coefficient estimates are heteroscedasticity-consistent t-tests (White 1980). Other relevant statistics for this regression, with degrees of freedom in parentheses, are also reported. |F.sub.1~(5, 51) tests the null hypothesis that, apart from the constant, other estimated coefficients are jointly insignificant. |F.sub.2~(4, 47) tests the relevance of the restrictions that are nested in the |EC.sub.t-1~ term. This test is obtained by estimating an unrestricted version of (6), i.e., replacing |EC.sub.t-1~ by ln|C.sub.t-1~, ln|Y.sub.t-1~, ln(|Pc.sub.t-1~/|Pc.sub.t-1~), ln(|POP65.sub.t-1~/|POP.sub.t-1~), and |U.sub.t-1~ and then estimating the model. LM(1) is the Lagrange Multiplier test for first order autoregressive disturbances (see Breusch 1978; Godfrey 1978); LM(1) is distributed as chi-squared with one degree of freedom. ARCH(1) is a test for first order autoregressive conditional heteroscedasticity among the disturbances (see Engle 1982); ARCH(1) is distributed as chi-squared with one degree of freedom.

The error correction term, |EC.sub.t-1~, which is significant at the five percent level, supports the appropriateness of the ECM model for clothing expenditures in the United States. The high adjusted |R.sup.2~ indicates that a large portion of variation in the dependent variable is explained by the explanatory variables. The adjusted |R.sup.2~ of 78 percent is considered to be high as the dependent variable is the rate of growth rather than a trended variable. Also the dependent variable, being in the differenced form, reduces the possibility of spurious correlation that is encountered when the levels of variables are used. In addition to the high |R.sup.2~, other statistics, at the five percent level of significance, support the ECM model as an alternative model of U.S. clothing expenditures. |F.sub.1~, being larger than the critical value of 2.3, rejects the null of no relationship. On the other hand, |F.sub.2~, being smaller than the critical value of 2.5, does not reject the nested restriction in the error correction term (|EC.sub.t-1~). LM and ARCH, also being smaller than the critical value of 3.8, support (6) by not rejecting the nulls of "no first order autoregression" and "no ARCH-effect" among the disturbances, respectively.

Shortrun and Longrun Elasticities

Income

Given that the estimated coefficients of the logs of the variables are elasticities, the estimated slopes in (5) and (6) reflect the longrun and shortrun elasticities respectively. Estimated shortrun and longrun income elasticities of clothing expenditures do not appear to be too different from each other. These elasticities (0.48 and 0.50) indicate that the demand for clothing is income inelastic (see Bryant and Wang (1990)). In particular, findings in this paper indicate that a ten percent increase (decrease) in income causes clothing expenditures to increase (decrease), upon impact, by 4.8 percent. Because the short- and longrun income elasticities are similar, no substantial role for the longrun adjustment to an income change is found. Nonetheless, the reported estimates in this paper are about half of the shortrun income elasticity (i.e., 0.97) reported by Norum (1990, 77). This could be due to the fact that the ECM allows for a richer dynamic structure than models that use conventional stock adjustment procedures to represent the dynamic reactions of the variables of interest. Clearly, the lower short- and longrun income elasticities obtained in this study indicate that changes in income due to government policies, the business cycle, or other factors are not as influential in changing clothing expenditures in the United States as might be inferred from higher estimates reported in other studies.

Prices

Estimates in this paper show that, in the short run, clothing expenditures are highly price elastic (-1.9) while in the long run this elasticity settles at unity (-1.0). It is of interest to note that these findings indicate that a ten percent increase (decrease) in the relative price of clothing causes clothing expenditures to decrease (increase) in the short run by 19 percent. However in the long run, changes in the relative price of clothing and clothing expenditures are proportional. These findings differ markedly from some previous studies. For example, the longrun price elasticity of clothing expenditures in this study is about four times the value reported by Norum (1990). This can also be contrasted with the inelastic clothing expenditures reported in Winakor (1989). The finding of unitary price elasticity is in agreement with the value obtained by Bryant and Wang (1990). Overall, the short- and longrun price elasticity estimates obtained in this study show that clothing expenditures in the United States are highly responsive to price changes. These results support the idea that substantial increases in clothing expenditures could result if relative prices were to drop significantly. Obviously, reduction of relevant trade barriers, for example between the United States and Mexico that may lead to a decrease in the relative price of apparel and up-keep, is a potential driving force for increased clothing expenditures.

Population

Estimated responses of clothing expenditures to the elderly/population ratio show that, despite a statistically insignificant shortrun rise in clothing expenditures due to an increase in the elderly/population ratio, the longrun effect of a rise in this ratio is negative and reduces U.S. clothing expenditures. Basically, inclusion of the elderly/population ratio allows the researcher to control for changes in taste. As it is well-known that the elderly/population ratio is increasing in the United States, discovery that in the short run no significant change in clothing expenditures is likely to occur due to the increase in the elderly/population ratio is important. However, slow evolution of taste (and other factors) reflected in this demographic variable (the elderly/population ratio), while leaving clothing expenditures unchanged in the short run, implies a negative longrun impact on clothing expenditures.

Unemployment

Interpreting the unemployment rate as the general condition of the economy, a significant decline in clothing expenditures due to a rise in the unemployment rate in the short run is found. Similarly, the longrun impact of a rise in the unemployment rate on U.S. clothing expenditures is 1.6 times that of the shortrun impact. These findings are consistent with the observed behavior of clothing expenditures and its growth rate during the 1930s depicted in Figures 1 and 2. Norum (1990) also found a negative coefficient estimate for a dummy variable that she included to account for the depression years (1929-1933).

CONCLUSION

In this paper an error correction model of U.S. clothing expenditures is estimated using recent developments in testing for stationarity and modeling cointegrated variables using Engle and Granger's (1987) two-step procedure. On the basis of the reported statistics, it is found that logarithmic values of U.S. clothing expenditures, disposable income, relative price of clothing, elderly/population ratio, and the unemployment rate are stationary in their first changes. Moreover, the Error Correction Mechanism model and Engle and Granger's two-step procedure are argued to be an appropriate technique for estimating clothing expenditures (a nonstationary variable) in the United States.

The estimates of the shortrun and longrun elasticities of clothing expenditures with respect to income and prices indicate that demand for clothing, although income inelastic, is highly price elastic. In the long run, expenditures for clothing appear to have a unit elasticity with respect to prices. Inclusion of the number of elderly (above the age of 65) relative to the total population and the unemployment rate show that longrun expenditures on clothing in the United States decrease as these two ratios increase.

Future research may greatly benefit by taking into account properties of time-series data in modeling economic activities. Relevant tests, modeling and estimation techniques, as demonstrated, are accessible enough to suggest Engle and Granger's two-step technique for further analysis of historical time-series observations.

Manouchehr Mokhtari is Assistant Professor, Department of Family Studies, University of Maryland at College Park.

I would like to thank Janet Wagner for her generous comments and suggestions. I also would like to thank Carole Makela and two anonymous referees for helpful comments and Pam Norum for providing me with her data set. This paper is dedicated to Professor Rachel Dardis. This research was supported by a grant from HUEC/DRIF and the computing facilities of the Computer Science Center at the University of Maryland at College Park.

1 A nonstationary variable (process) does not possess constant mean and variance, while a stationary process does. Use of nonstationary variables in the regression analysis leads to the violation of Gauss-Markov assumptions that may cause inconsistency in parameter estimates.

2 If variables included in a regression model are nonstationary then distributions of the usual statistics are not the same as those derived when only stationary variables are included. See, for example, Phillips and Durlauf (1986). In particular, parameters of the regression will not converge in probability as the sample size increases. Hendry (1986) provides relevant references and historical background.

3 More sophisticated econometric approaches to the problem of nonstationary regressors and regressands exist, but for the sake of simplicity of exposition are not used here. Interested readers are referred to Johansen (1988).

4 Nelson and Plosser (1982) show that most U.S. time series follow some nonstationary process.

5 See, also, Bryant (1983) and Norum (1989). In these studies, clothing is assumed to be a semidurable or durable good. Bryant (1990, 36-37) notes that clothing and shoes, lasting for less than three years, are classified as nondurable goods by the U.S. Department of Commerce.

6 Inclusion of nonstationary variables in the regression model cause the distributions of t- and F-tests to diverge and the Durbin-Watson statistic to approach zero.

7 In recent years, proliferation of testing for nonstationarity in data (unit root testing) is indicative of ever growing concerns over the inclusion of nonstationary variables in models (Hendry 1986).

8 Many other cost functions can be postulated and variants of the PAM and the ECM can be derived. In this paper, with no significant loss, only the simple and convenient functions, estimation techniques, and tests are considered.

9 Variables included in this regression, being nonstationary, violate the Gauss-Markov assumption, making the usual standard errors of coefficients and t- and F-statistics useless for inference.

10 Cointegration implies the existence of an ECM as the data generating process and vice versa.

11 It is worth emphasizing that many other tests for nonstationarity exist. However, only the simplest one (DF-test) is reported here.

12 Restriction on the disturbances implies restriction on the parameters in the model and vice versa.

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Author: | Mokhtari, Manouchehr |
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Publication: | Journal of Consumer Affairs |

Date: | Dec 22, 1992 |

Words: | 6478 |

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