Error correction mechanisms and short-run expectations.

Antzoulatos, Angelos A.

I. Introduction

Ever since the publication of Engle and Granger's [12] seminal paper, it has been widely known that a system of co-integrated variables has an error-correction representation in which the vector autoregression (VAR) in differenced variables contains an error correction mechanism (ECM). In Engle and Granger's paper, the ECM emerges as a statistical property of the data, a property that bodes well with the theoretical explanations which stress ECM's resemblance to a feedback control rule driven by some partial adjustment mechanism [10; 14]. Nevertheless, the nature of economic decisions suggests that the ECM may arise from forward-looking behavior and, thus, reflect expectations about future events [1; 4]. In such a case, as this paper shows, the estimated ECM coefficients can misleadingly appear to be insignificant or to have the opposite-than-expected sign if the other explanatory variables in the error-correction representation generate poor conditional forecasts for the system's endogenous variables. In turn, the erroneous inferences about the ECM coefficients can lead to misspecified econometric models in which the ECM's promise of better short-run forecasts will not materialize.

This paper explores the problem of erroneous inferences about the estimated ECM coefficients using the system of consumption and income, and demonstrates its potential magnitude with U.S. data. More specifically, section II combines the forward-looking nature of consumption with a typical partial adjustment mechanism to derive a theoretical model for consumption growth. In it, consumption growth is increasing in contemporaneous and (expected) future income growth and decreasing in the ECM, while the ECM is increasing in future income growth.

On the other hand, in the typical error-correction representation, consumption growth is a function of the ECM, and lagged values of consumption and income growth. Provided that these lagged values generate good income-growth forecasts, the estimated ECM coefficient in the error-correction representation will be negative. If not, as in the case of U.S. data, the coefficient will be positively "biased," a reflection of the ECM's positive correlation with future income growth in conjunction with the latter's positive correlation with consumption growth. More important, the bias can be so severe that the ECM coefficient can misleadingly appear to be insignificant or, even worse, positive. The bias can be reduced though with the inclusion in the error-correction representation of other stationary variables which can help predict income growth.

The empirical evidence, in section III, confirms these expectations. The ECM, the lagged residuals of the co-integrating regression of log consumption on log income and a constant, is positively correlated with future income growth, as postulated. But its coefficient in the error-correction representation is positive and insignificant. To test whether this result is due to the poor income-growth forecasts generated by lagged consumption and income growth terms, contemporaneous income growth is included in the regression (the working hypothesis is that expected values differ from realized ones by an unpredictable stochastic term [8]). In this equation, the ECM coefficient becomes negative but remains insignificant. However, the addition of future income growth makes the coefficient significantly negative at the 5% level. The addition of another variable which can help predict future income growth, the first lag of the growth rate of the Composite Index of Eleven Leading Indicators, increases the coefficient's significance to the 1% level. Overall, in accordance with the theory outlined in section II, each additional variable correlated with future income growth helps increase both the significance level and the absolute value of the estimated ECM coefficient.

Closing, the forward-looking nature of economic decisions and the difficulty of modeling expectations suggest that the conditions for erroneous inferences about the estimated ECM coefficients are likely to apply to many other settings. For this reason, section IV, which concludes the paper, recommends a re-evaluation of the evidence in cases where the ECM coefficient appears to be insignificant or with the wrong sign.

II. The Case of Consumption and Income

Reflecting the forward-looking nature of the consumer's decision problem, optimal consumption [Mathematical Expression Omitted] in equation (1) is increasing in contemporaneous and expected future income, [E.sub.t][i.sub.t+k] (k [greater than or equal to] 0). Small letters denote logs, E is the usual expectations operator, [[Epsilon].sub.t] is a stochastic term unrelated to variables known at t - 1 or before, while the coefficients [[Mu].sub.k] (k [greater than or equal to] 0) are positive. Because of some sort of adjustment costs, people cannot set actual consumption, [c.sub.t], equal to [Mathematical Expression Omitted]. Instead, [c.sub.t] adjusts towards [Mathematical Expression Omitted] as described by equation (2) (This partial adjustment mechanism has been adapted from Davidson and MacKinnon [9, 680]). The term (1 - [Xi]), 0 [less than] (1 - [Xi]) [less than] 1, measures the speed of adjustment, while [e.sub.t] is a stochastic term unrelated to variables known at t - 1 or before.

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Subtracting [c.sub.t - 1] from both sides of equation (1), re-arranging terms and multiplying both sides of the resulting equation with (1 - [Xi]) give the following expression for equation (2):

[Delta][c.sub.t] = [c.sub.t] - [c.sub.t - 1] = (1 - [Xi])[Mu] - (1 - [Xi])([c.sub.t - 1] - [Lambda][i.sub.t - 1]) + [[Psi].sub.0]([i.sub.t] - [i.sub.t - 1]) + [[Psi].sub.1] ([E.sub.t][i.sub.t - 1] - [i.sub.t]) + [[Psi].sub.2]([E.sub.t][i.sub.t + 2] - [E.sub.t][i.sub.t + 1]) + ... + [[Psi].sub.p]([E.sub.t][i.sub.t + p] - [E.sub.t][i.sub.t + p - 1]) + [(1 - [Xi])[[Epsilon].sub.t] + [e.sub.t]]

= (1 - [Xi])[Mu] - (1 - [Xi])([c.sub.t - 1] - [Lambda][i.sub.t - 1]) + [summation of] [[Psi].sub.k][E.sub.t][Delta][i.sub.t + k] where k = 0 to p + [u.sub.t] (3)

where [E.sub.t][i.sub.t] = [i.sub.t], [[Psi].sub.p] = (1 - [Xi])[[Mu].sub.p], [[Psi].sub.k] = (1 - [Xi]) [summation of] [[Mu].sub.p - j] (k = 0, 1, 2, ..., p - 1) where j = 0 to p - k and [Lambda](1 - [Xi]) = [[Psi].sub.0] The income-growth coefficients ([[Psi].sub.k], 0 [less than or equal to] k [less than or equal to] p) and [Lambda] are positive, while -(1 - [Xi]), the coefficient of the ECM (equation (4)), is negative. Further, since equation (3) contains a constant, the income-growth terms can be expressed as deviations from their means and, thus, reflect short-run income expectations.

[ECM.sub.t - 1] = [c.sub.t - 1] - [Lambda][i.sub.t - 1] (4)

Also reflecting the forward-looking nature of the consumer's problem, [ECM.sub.t - 1] is positively correlated with future income. [ECM.sub.t - 1] is increasing in [c.sub.t - 1] which, in turn, is increasing in [E.sub.t - 1][i.sub.t + k](k [greater than or equal to] 0). Since [E.sub.t - 1][i.sub.t + k] differs from [E.sub.t][i.sub.t + k](k [greater than or equal to] 0) by a stochastic term arising from revisions of expectations between t - 1 and t (so, this term is unrelated to variables known at t - 1), [c.sub.t - 1] and [ECM.sub.t - 1] are positively correlated with [E.sub.t][i.sub.t + k](k [greater than or equal to] 0).

The derivation of equation (3) illustrates the analytical foundations of the error-correction representation. Even though it departs from the strict stochastic implications of the permanent income hypothesis,(1) equation (3) is consistent with the forward-looking nature of consumption and the empirical regularities found in the U.S. consumption data [3]. For completeness, the appendix discusses another partial adjustment mechanism in the spirit of rational expectations models which culminates in an equation for consumption growth similar to (3).

In equation (3), the omission of [ECM.sub.t - 1] will induce a negative bias in the regression estimates of [[Psi].sub.k](k [greater than or equal to] 0). This bias reflects the positive correlation between future income and [ECM.sub.t - 1] in conjunction with [ECM.sub.t - 1]'s negative sign, as in the typical case of omitted variables in a regression equation analyzed in Johnston [15, 260]. Similarly, the omission of [E.sub.t - 1][Delta][i.sub.t + k](k [greater than or equal to] 0) will induce a positive bias in [ECM.sub.t - 1]'s coefficient estimate which, as a result, can appear insignificant or even positive.

By extension, the [ECM.sub.t - 1] coefficient will be positively "biased" when the other right-hand-side variables in the typical error-correction representation for consumption growth, equation (5), do not generate good income-growth forecasts.

[Delta][c.sub.t] = [Alpha] + [[Phi].sub.1][Delta][i.sub.t - 1] + ... + [[Phi].sub.p][Delta][i.sub.t - p + 1] + [[Theta].sub.1][Delta][c.sub.t - 1] + ... + [[Theta].sub.p][Delta][c.sub.t - p + 1] + [Beta][ECM.sub.t - 1] + [u.sub.t] (5)

The argument proceeds as follows. The coefficient [Beta] in equation (5) corresponds to -(1 - [Xi]) in (3). Also, the terms [Delta][i.sub.t - m] and [Delta][c.sub.t - m](m = 1, ... p) in (5) can be thought of as proxies of [E.sub.t][Delta][i.sub.t + k](k [greater than or equal to] 0) in (3). If these proxies do not generate good forecasts for [Delta][i.sub.t + k](k [greater than or equal to] 0), [Beta] will be a positively biased estimate of -(1 - [Xi]), a reflection of the [ECM.sub.t - 1]'s positive correlation with [E.sub.t][Delta][i.sub.t + k](k [greater than or equal to] 0) in conjunction with the latter's positive correlation with [Delta][c.sub.t]. More important, the "bias" can be so strong as to render [Beta] insignificant or even positive. However, the inclusion of other stationary variables which are correlated with expected income growth will help reduce [Beta]'s "bias". Such variables, labeled here as exogenous, allow a richer specification of the system's short-run dynamics and more efficient estimation without undermining the analytical foundations of the error-correction representation. By the way, if [Beta] in equation (5) is positive, the system of consumption and income would be unstable as [ECM.sub.t - 1] is positively correlated with expected income growth.

An Example

A more concrete example will help illustrate the nature of the bias, the importance of good income proxies, and the efficiency gains afforded by exogenous variables which can help improve income-growth forecasts. For simplicity, but without loss of generality, let [Delta][c.sub.t] in equation (3) be a function of [ECM.sub.t - 1], [E.sub.t][Delta][i.sub.t] and [E.sub.t][Delta][i.sub.t + 1] only. Let also [Delta]c, ECM, E[Delta][i.sub.t] and E[Delta][i.sub.t + 1] denote the vectors of realizations of these variables; X be the matrix with columns ECM, E[Delta][i.sub.t] and E[Delta][i.sub.t + 1]; and [Psi] be the coefficient vector [Psi][prime] = (-(1 - [Xi]), [[Psi].sub.0], [[Psi].sub.1]).

The theoretical model for consumption growth - the equivalent of (3) - is given by

[Delta]c = X[Psi] + u. (6)

Next, let [z.sub.0,t - 1] and [z.sub.1,t - 1] be two proxies of expected future income, as in equation (5). These proxies can be lagged values of income or consumption growth, or exogenous stationary variables. Let also the matrix Z of the regressors in (5) have as columns ECM, [z.sub.0] and [z.sub.1]. The OLS coefficients [Zeta][prime] = ([Beta], [[Zeta].sub.0], [[Zeta].sub.1]) are:

[Zeta] = [(Z[prime]Z).sup.-1]Z[prime][Delta]c. (7)

Substituting (6) into (7) gives (under the assumption Z[prime]u = 0):

[Zeta] = [(Z[prime]Z).sup.-1]Z[prime][Delta]c

= [(Z[prime]Z).sup.-1]Z[prime](X[Psi] + u)

= [(Z[prime]Z).sup.-1]Z[prime]X[Psi] + [(Z[prime]Z).sup.-1]Z[prime]u

= [(Z[prime]Z).sup.-1]Z[prime]X[Psi]

= [Theta][Psi]. (8)

The (3, 3) matrix [Theta] = [(Z[prime]Z).sup.-1]Z[prime]X has as columns the regression coefficients of ECM, E[Delta][i.sub.t] and E[Delta][i.sub.t + 1] (matrix X) on ECM, [z.sub.0] and [z.sub.1]. Let [Theta] be

[Mathematical Expression Omitted]

which implies the following regression equations:

[ECM.sub.t - 1] = 1[ECM.sub.t - 1] + 0[z.sub.0,t + 1] + 0[z.sub.1,t - 1] (9)

[E.sub.t][Delta][i.sub.t] = [Pi][ECM.sub.t - 1] + [[Pi].sub.0][z.sub.0,t - 1] + [[Pi].sub.1][z.sub.1,t - 1] (10)

[E.sub.t][Delta][i.sub.t + 1] = [Omega][ECM.sub.t - 1] + [[Omega].sub.0][z.sub.0,t - 1] + [[Omega].sub.1][z.sub.1,t - 1]. (11)

So, the coefficient [Beta] in (8), the equivalent of the ECM coefficient in the error-correction representation (equation (5)), will be:

[Beta] = -(1 - [Xi]) + [Pi][[Psi].sub.0] + [Omega][[Psi].sub.1]. (12)

Since [ECM.sub.t - 1] and [Delta][i.sub.t + k](k = 0, 1) are positively correlated, one can reasonably expect that the coefficients [Pi] and [Omega] in equations (10) and (11) will be positive.(2) Taking into account that [[Psi].sub.0] and [[Psi].sub.1] are positive, [Beta]'s bias and the possibility that [Beta] is insignificant or positive are increasing in the value of [Pi] and [Omega]. Thus, the bias will be highest when [z.sub.i,t - 1](i = 0, 1) are unrelated to [E.sub.t][Delta][i.sub.t + k](k = 0, 1) in which case [[Pi].sub.i] and [[Omega].sub.i](i = 0, 1) are equal to zero while [Pi] and [Omega] attain their highest values. In general, as the explanatory value of [z.sub.i,t - 1](i = 0, 1) for [E.sub.t][Delta][i.sub.t + k](k = 0, 1) increases, the value of [Pi] and [Omega] (and the bias) will decrease. The bias will be totally eliminated only when [z.sub.0,t - 1] = [E.sub.t][Delta][i.sub.t] and [z.sub.1,t-1] = [E.sub.t][Delta][i.sub.t + 1]; in this case, [[Pi].sub.0] = [[Omega].sub.i] = 1 and [Pi] = [Omega] = [[Pi].sub.1] = [[Omega].sub.0] = 0.

III. Empirical Evidence

All series are constructed from CITIBASE data. Real per capita consumption, non-durables and services, and personal disposable income are measured in 1982 dollars. The sample is restricted to 1953:1-1988:4, to hedge against the distortive impact on the income series of the Korean War period, and the revisions of the original consumption and income series [5]. Nevertheless, the results are qualitatively the same for the whole sample period, 1947 through 1990. The critical values for the ADF tests are taken from Tables 2 and 3 in Charemza and Deadman [7]. Throughout this section, the numbers in parentheses below the estimated coefficients correspond to t-statistics, while one, two and three asterisks denote significance at the 10%, 5% and 1% levels, respectively.

Table I summarizes the correlation coefficients between income growth and some variables of interest. The first row indicates that [Delta][i.sub.t] exhibits very little autocorrelation: with the exception of the third lead, all the autocorrelation coefficients are well below 0.10. The second row confirms the expectation that consumption growth should be positively correlated with contemporaneous and future income growth. Taken together, the first two rows indicate that lagged values of [Delta][i.sub.t] and [Delta][c.sub.t] will generate poor forecasts of future income growth, [Delta][i.sub.t + k](k [greater than or equal to] 1). The third row confirms the expectation that [ECM.sub.t] should be positively correlated with [Delta][i.sub.t + k](k [greater than] 1). Finally, the last row shows that the growth rate of the Composite Index of Eleven Leading Indicators, denoted as [DL3.sub.t], is positively correlated with [Delta][i.sub.t + k](k [greater than] 1). As such, [DL3.sub.t] is expected to help reduce the bias of the [ECM.sub.t] coefficient in the error-correction representation for consumption growth. [TABULAR DATA FOR TABLE I OMITTED] Further, the correlation coefficients between [ECM.sub.t] and [Delta][c.sub.t - k], [Delta][i.sub.t - k], [DL3.sub.t - k], (k = 0, 1, 2, 3, ...) are below 0.2 and frequently below 0.1. These low coefficients imply that [Pi] and [Omega] in equations (10) and (11) will likely be positive and, consequently, [Beta] will be a positively biased estimate of -(1 - [Xi]) (see the discussion in the previous footnote).

A series of ADF tests established that [c.sub.t] and [i.sub.t] are I(1). The lower critical values at the 1% level for 100 and 150 observations are -2.70 and -2.68 (Table 2, m = 0) [7] without intercept, and -2.90 and -2.79 (Table 3, m = 0) [7] with intercept. The number of available observations is approximately 140. Regressing consumption growth, [Delta][c.sub.t], on [c.sub.t - 1] and [Delta][c.sub.t - k](k = 1, 2) gave a t-statistic for [c.sub.t - 1] of 5.13 which, obviously, is not significantly negative. The inclusion of an intercept in the regression gave a t-statistic of -0.07 which is not significantly negative either. Next, regressing the second difference, [Delta][Delta][c.sub.t], on [Delta][c.sub.t - 1] and [Delta][Delta][c.sub.t - k](k = 1, 2) resulted in a t - statistic for [Delta][c.sub.t - 1] of -3.07 without and -5.20 with an intercept. Both are below the lower critical values at the 1% level. Similarly, in the regression of [Delta][i.sub.t] on [i.sub.t - 1] and [Delta][i.sub.t - k], k = 1, 2, the t-statistics of [i.sub.t - 1] were 4.79 without and -0.29 with intercept. In the [Delta][Delta][i.sub.t] regression, the appropriate t-statistics were -3.99 and -5.64, respectively, which are below the lower critical values at the 1% level. The stationarity tests were also conducted with three and four lags of [Delta][c.sub.t] and [Delta][i.sub.t]. Since the conclusions regarding the order of integration for [c.sub.t] and [i.sub.t] are the same, only the results for k = 1, 2 are reported to save space.

The results of the co-integrating regression, the equivalent of equation (4) above, at the first step of Engle and Granger's [12] two-step estimator, are summarized below.

[c.sub.t] = -0.041 + 0.912[i.sub.t] + [v.sub.t] (-4.48) (209.6)

[R.sup.2] = 0.997, D.W. = 0.378

In general, the OLS estimate of [i.sub.t]'s coefficient is biased. However, as Davidson and McKinnon [9, 724] remark, the bias seems to be least severe when the [R.sup.2] is close to 1, as is the case here.

Regressing the change in the residuals, [Delta][v.sub.t], on [v.sub.t - 1] and [Delta][v.sub.t - 1] established that the null of non-co-integration between [c.sub.t] and [i.sub.t] can be rejected at the 5% level. In more detail, the t-statistic of [v.sub.t - 1] was -3.30, while the lower critical value for 100 and 150 observations at the 5% level with one estimated parameter (Table 2, m = 1) [7] is -2.87.

To establish that the joint distribution of [c.sub.t] and [i.sub.t] is an error correction system, in the second step of Engle and Granger's estimator, consumption growth is regressed on [v.sub.t - 1] and five lags of [Delta][c.sub.t] and [Delta][i.sub.t]. An F test indicated that, with the exception of [Delta][c.sub.t - 1], all the lagged terms were jointly insignificant. More specifically, the F test for the joint hypothesis [Delta][c.sub.t - k] = [Delta][i.sub.t - m] = 0(k = 2, 3, 4, 5 and m = 1, 2, 3, 4, 5) was F(9, 139) = 1.29, far below the critical values at all conventional significance levels. Proceeding with a "general to specific" modeling approach, the more parsimonious equation shown below is estimated.

[Mathematical Expression Omitted]

In this equation, the estimated ECM coefficient, [Beta] = 0.023, is not only insignificant, but also has the opposite - than-expected sign. Moreover, there is strong evidence of high order serial correlation in the residuals. Regarding the positive coefficient of [Delta][c.sub.t - 1], it probably reflects [Delta][c.sub.t - 1]'s positive correlation with [Delta][i.sub.t + k](k [greater than or equal to] 0).

To test whether the positive sign and the statistical insignificance of [Beta] are due to the fact that lagged values of [Delta][c.sub.t] and [Delta][i.sub.t] are poor proxies for future income growth, [Delta][i.sub.t] is included in the set of the regressors (In the spirit of equation (12), this will render [Pi] = 0 and reduce [Beta]'s positive bias). The working hypothesis is that, under rational expectations, expected income growth differs from realized one by an unpredictable stochastic term [8]. As the equation below indicates, the inclusion of [Delta][i.sub.t] renders lagged consumption growth insignificant and makes the ECM coefficient negative. Still, however, [Beta] is not significant at any conventional level. On the positive side, this model - as well as the next two - passes diagnostic tests for serial correlation and autoregressive conditional heteroskedasticity (ARCH) in the residuals.

[Mathematical Expression Omitted]

More importantly, the inclusion of future income growth (which is equivalent to setting [Omega] = 0 in (12)) makes [Beta] significant at the 5% level. In addition, all the estimated coefficients have the theoretically predicted sign.(3)

[Mathematical Expression Omitted]

Further, the inclusion of [DL3.sub.t - 1] increases both the t-statistic of [Beta] and its absolute value. This provides further evidence about the hypothesis that "exogenous" variables which can help predict income growth may help reduce the bias of the ECM coefficient.

In this step-by-step procedure, every time a new variable correlated with future income growth is added to the error-correction representation, both the significance level and the absolute value of the estimated ECM coefficient increase. Thus, in accordance with the theory, such variables help reduce the coefficient's bias.

The results were similar for the whole sample, 1947 to 1990. The ECM coefficient was marginally significant at the 10% level (t-statistic -1.68) in the unrestricted equation which included lagged values of [Delta][c.sub.t] and [Delta][i.sub.t]. The inclusion of [DL3.sub.t - 1] in the set of the regressors raised ECM's significance to the 5% level (t-statistic -2.1). The addition of [Delta][i.sub.t] further raised the significance level to 1% (t-statistic -3.25). Also, in accordance with the theory, the ECM coefficient kept increasing in absolute value as more regressors correlated with future income were added.

Finally, an error-correction model was estimated for income. The original model included six lags of consumption and income growth, as well as the error-correction mechanism and the first lag of the growth rate of the Composite Index of Eleven Leading Indicators. A series of F tests indicated that only the first three lags of consumption growth and the error-correction mechanism had significant predictive power for income growth. As one might expect from the theoretical analysis and the correlation coefficients in Table I, the coefficient of [ECM.sub.t - 1] was positive and significant at the 1% level. The [ECM.sub.t - 1]'s coefficient positive sign provides further support for the argument that a positive [Beta] in equation (5) would indicate an unstable system.

IV. Concluding Remarks

This paper argues that biased error-correction coefficients may be pervasive in error-correction models that are meant to capture behavior that is influenced in an essential way by expectations. Combining a typical partial adjustment mechanism with the forward-looking nature of the consumer's decision problem, it shows that the ECM coefficient in the error-correction representation of consumption will be biased upwards if the other right-hand-side variables are poor proxies of future income expectations. The bias will be reduced though when better income-growth proxies are included as regressors. The paper also demonstrates that the theoretical predictions are borne out by U.S. data. Lagged growth rates of consumption and income are poor proxies of future income growth and, presumably, of consumers' income expectations. In the typical error-correction representation which includes only these (poor) proxies in the right-hand side, the ECM coefficient is positive but insignificant. However, when better predictors of future income growth are added to the right-hand-side, the estimated ECM coefficient eventually turns negative and significant, as predicted by the theory.

More generally, the paper demonstrates the risks inherent in treating the ECM as a mere statistical property of the data, without regard to the economic forces behind it. It also holds the promise of helping alleviate the problems of inefficient estimation, misspecified econometric models and poor short-run forecasts which arise from erroneous inferences about the estimated ECM coefficients. Owing to the forward-looking nature of economic decisions and the difficulty of modeling expectations, such problems are expected to be pervasive. Therefore, a re-examination of the evidence in cases where the ECM appears to be insignificant or to have the opposite-than-expected sign seems a worthwhile endeavor.

Appendix. An Alternative Model

This appendix derives a theoretical equation for consumption growth similar to that in (3). The analysis is the spirit of rational expectations models, while the partial adjustment of consumption to income expectations is driven by binding borrowing constraints.

Equations (13) through (15) summarize the decision problem of a forward-looking individual who maximizes the expected value of a time-separable utility function subject to a borrowing constraint.

Max [E.sub.t]{[summation of [(1 + [Delta]).sup.-k]] u([C.sub.t + k]) where k = 0 to T} ([Delta] [greater than] 0) (13)

subject to

[C.sub.t] [less than or equal to] [X.sub.t] (14)

[X.sub.t] = [RS.sub.t - 1] + [I.sub.t] = R([X.sub.t - 1] - [C.sub.t - 1]) + [I.sub.t] (R [greater than or equal to] 1). (15)

The timing and the nature of his decisions are as follows: At the beginning of period t, he decides how much to consume, [C.sub.t], in order to maximize his expected utility, equation (13), over the remaining planning horizon. Due to the borrowing constraint, equation (14), consumption cannot exceed the individual's total assets, [X.sub.t]. [X.sub.t] is equal to the savings carried over from t - 1, [RS.sub.t - 1] = R([X.sub.t - 1] - [C.sub.t - 1]), plus the realized income, [I.sub.t], as described by equation (15). For simplicity, and without loss of generality, the real interest rate (R is equal to one plus the real interest rate) and the length of the planning horizon, T, are taken as exogenous. Since the analysis is based on the stochastic implications of the permanent income hypothesis (PIH), no particular assumptions are needed for the income process [13]. Let also capital and small letters denote levels and logs, respectively.

There is no closed-form solution for optimal consumption under uncertainty and borrowing constraints. However, by using the first-order condition of the individual's optimization problem, one can show that optimal consumption is increasing in contemporaneous assets and expected future income [2; 11; 16]. In mathematical terms, [C.sub.t - 1] in equation (16) is increasing in [X.sub.t - 1] and [E.sub.t - 1][I.sub.t + k](k [greater than or equal to] 0).

[C.sub.t - 1] = [C.sub.t - 1] ([X.sub.t - 1]; [E.sub.t - 1][I.sub.t]; [E.sub.t - 1][I.sub.t + 1]; [E.sub.t - 1][I.sub.t + 2]; ...) (16)

Similarly, [E.sub.t - 1][C.sub.t] is increasing in [E.sub.t - 1][I.sub.t + k](k [greater than or equal to] 0).

When the individual is at an interior solution at t - 1 (non-binding constraint), expected consumption change, [E.sub.t - 1][Delta][C.sub.t], will be unrelated to [ECM.sub.t - 1] and any other economic variable observed at t - 1 or earlier, as the permanent income hypothesis postulates [13]. This reflects the fact that both [c.sub.t - 1] and [E.sub.t - 1][C.sub.t] are increasing in expected income. The error-correction term, however,

[ECM.sub.t - 1] = [C.sub.t - 1] - [I.sub.t - 1] (17)

will be increasing in [E.sub.t - 1][I.sub.t + k], k [greater than or equal to] 0 (equation (16)).

On the other hand, when the individual faces a binding constraint at t - 1 (corner solution), expected consumption change will be increasing in [E.sub.t - 1][I.sub.t + k](k [greater than or equal to] 0) and decreasing in [ECM.sub.t - 1]. At a corner solution, [C.sub.t - 1] = [X.sub.t - 1] = [RS.sub.t - 2] + [I.sub.t - 1]. Thus,

[ECM.sub.t - 1] = ([RS.sub.t - 2] + [I.sub.t - 1]) - [I.sub.t - 1] = [RS.sub.t - 2].

Also, [S.sub.t - 1] = 0, [X.sub.t] = [I.sub.t] and

[E.sub.t - 1][Delta][C.sub.t] = [E.sub.t - 1][C.sub.t] - [C.sub.t - 1] = [E.sub.t - 1][C.sub.t]([E.sub.t - 1][X.sub.t]; [E.sub.t - 1][E.sub.t - 1][I.sub.t + 1]; [E.sub.t - 1][I.sub.t + 2]; ...) - [C.sub.t - 1]

= [E.sub.t - 1][C.sub.t]([E.sub.t - 1][I.sub.t]; [E.sub.t - 1][I.sub.t + 1]; [E.sub.t - 1][I.sub.t + 2]; ...) - ([RS.sub.t - 2] + [I.sub.t - 1]). (18)

Since [E.sub.t - 1][C.sub.t] is increasing in [E.sub.t - 1][I.sub.t + k](k [greater than or equal to] 0), expected consumption change will be increasing in [E.sub.t - 1][I.sub.t + k], k [greater than or equal to] 0 as well. There is a caveat though; if the individual is expected to be at a corner solution at t, [E.sub.t - 1][C.sub.t] = [E.sub.t - 1][X.sub.t] = [E.sub.t - 1][I.sub.t] and thus consumption change will be increasing in contemporaneous income only. Further, [E.sub.t - 1][Delta][C.sub.t] will be decreasing in [RS.sub.t - 2] and consequently in [ECM.sub.t - 1].

The above analysis postulates the following function for expected aggregate consumption change, the sum of individual [E.sub.t - 1][Delta][C.sub.t]'s:

[E.sub.t - 1][Delta][C.sub.t] = f([ECM.sub.t - 1], [E.sub.t - 1], [E.sub.t - 1][I.sub.t + 1], [E.sub.t - 1][I.sub.t + 2], ...) (19)

Because of the individuals who are at a corner solution at t - 1, [E.sub.t - 1][Delta][C.sub.t] will tend to be decreasing in [ECM.sub.t - 1] and increasing in [E.sub.t - 1][I.sub.t + k](k [greater than or equal to] 0). Thus, in the log-linear version of (19), given by equation (20) below, [[Beta].sub.0] is expected to be negative, while [[Gamma].sub.k](k [greater than or equal to] 0) are expected to be positive (the ECM's log version is given by equation (21)). Also, under the assumption that realized consumption change differs from expected one by an unpredictable stochastic term, [u.sub.t] should be unrelated to any variables observed at t - 1 or before.

[Delta][c.sub.t] = constant + [[Beta].sub.0][ECM.sub.t - 1] + [[Gamma].sub.0][E.sub.t - 1][Delta][i.sub.t] + [[Gamma].sub.1][E.sub.t - 1][Delta][i.sub.t + 1] + [[Gamma].sub.2][E.sub.t - 1][Delta][i.sub.t + 2] + ... + [u.sub.t] (20)

[ECM.sub.t - 1] = [c.sub.t - 1] - [i.sub.t - 1]. (21)

Further, because of the individuals who are at an interior solution at t - 1, [ECM.sub.t - 1] will tend to be increasing in [E.sub.t - 1][Delta][i.sub.t + k](k [greater than or equal to] 0). As a result, the omission of [ECM.sub.t - 1] in equation (20) will induce a negative bias in the regression estimates of [[Gamma].sub.k](k [greater than or equal to] 0). This bias reflects the positive relationship between future income and [ECM.sub.t - 1] in conjunction with [[Beta].sub.0]'s negative sign, as in the typical case of omitted variables in a regression equation analyzed in Johnston [15, 260]. Similarly, the omission of [E.sub.t - 1][Delta][i.sub.t + k](k [greater than or equal to] 0) will induce a positive bias in [[Beta].sub.0]'s estimate which, as a result, can be insignificant or even positive.

Finally, the comparison of equations (20) and (5) suggests that biased ECM coefficients can be pervasive even in rational expectations models when the right-hand-side variables are poor proxies of future income growth.

I thank seminar participants at the Federal Reserve Bank of New York, and especially Juann Hung, Carol Osler and Michael Boldin, as well as Edward Feasel of the George Washington University for many insightful comments and suggestions. Of course, they are not responsible for any remaining errors.

The views expressed here are those of the author and do not necessarily reflect the views of the Federal Reserve Bank of New York or the Federal Reserve System.

1. Roughly, the PIH postulates that [Mathematical Expression Omitted] and [c.sub.t] - [c.sub.t - 1] should be unpredictable. Under the PIH, [Mathematical Expression Omitted], [Mathematical Expression Omitted], while [Mathematical Expression Omitted] should - at a first approximation - be unpredictable. [v.sub.t] is an stochastic term arising from revisions of expectations between t - 1 and t.

2. By analogy to Johnston's [15, 81-86] analysis, the value of [Pi] and [Omega] is increasing in the correlation coefficient between [ECM.sub.t - 1] and [E.sub.t][Delta][i.sub.t + k](k = 0, 1), and decreasing in the correlation coefficients between [ECM.sub.t - 1] and [z.sub.i,t - 1] (i = 0, 1) and between [E.sub.t][Delta][i.sub.t + k](k = 0, 1) and [z.sub.i,t - 1](i = 0, 1).

As a reminder, Johnston analyzes the case with three interrelated variables Y, [X.sub.2] and [X.sub.3] in the model

Y = [[Beta].sub.1] + [[Beta].sub.2][X.sub.2] + [[Beta].sub.3][X.sub.3] + [u.sub.t].

Let [r.sub.12], [r.sub.13] and [r.sub.23] denote their simple correlation coefficients, and [s.sub.i], (i = 1, 2, 3) denote their standard errors, with the subscript 1 referring to the Y variable.

He shows that the estimates of [[Beta].sub.2] and [[Beta].sub.3], denoted as [b.sub.2] and [b.sub.3], satisfy the relationships:

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

Focusing on the first relationship, the value of [b.sub.2] is increasing in the correlation coefficient between Y and [X.sub.1] ([r.sub.12]), and decreasing in the correlation coefficient between Y and [X.sub.3] and between [X.sub.2] and [X.sub.3] ([r.sub.13] and [r.sub.23]).

Johnston's analysis further implies that the value of [Pi] and [Omega] will decrease as the explanatory value of [z.sub.0,t - 1] and [z.sub.1,t - 1] for [E.sub.t][Delta][i.sub.t + k](k = 0, 1) increases.

3. To hedge against the simultaneous equation bias, this and the previous equation are re-estimated with instrumental variables. Since the time aggregation of consumption may induce an MA(1) term in the error term, all the instruments are lagged at least two periods [5; 6]. The instruments are a constant, [c.sub.t - 2], [i.sub.t - 2], [r.sub.t - k](k = 2, 4), [Mathematical Expression Omitted] and [Delta][i.sub.t - k](k = 2, 5). [r.sub.t] denotes the quarterly average of the three-month Treasury Bill rate (secondary market), while [Mathematical Expression Omitted] denotes the growth rate of total consumption. Also, [c.sub.t - 1] and [i.sub.t - 1] are used instead of the residuals of the co-integrating regression. In the first equation, [c.sub.t - 1] and [i.sub.t - 1] have the expected sign but are insignificant. In the second equation, they are both significant at the 5% level. Also, the income growth terms have the expected sign.

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

References

1. Alogoskoufis, George and Ron Smith, "On Error Correction Models: Specification, Interpretation, Estimation." Journal of Economic Surveys, 1991, 97-128.

2. Antzoulatos, Angelos A., "Credit Rationing and Rational Behavior." Journal of Money, Credit, and Banking, May 1994, 182-202.

3. -----, "Borrowing Constraints, Income Expectations and the Euler Equation: Theory and Evidence." Economics Letters, July 1994, 323-27.

4. Campbell, John Y., "Does Saving Anticipate Declining Labor Income? An Alternative Test of the Permanent Income Hypothesis." Econometrica, November 1987, 1249-73.

5. ----- and N. Gregory Mankiw, "Permanent Income, Current Income, and Consumption." Journal of Business and Economic Statistics, July 1990, 265-79.

6. ----- and -----, "The Response of Consumption to Income: A Cross-Country Investigation." European Economic Review, May 1991, 723-56.

7. Charemza, Wolciech W. and Derek F. Deadman. New Directions in Econometric Practice. Aldershott, U.K.: Edward Elgar Publishing Limited, 1992.

8. Cumby, Robert E., John Huizinga and Maurice Obstfeld, "Two-Step Two-Stage Least Squares Estimation In Models With Rational Expectations." Journal of Econometrics, 1983, 333-55.

9. Davidson, Russell and James G. Mackinnon. Estimation and Inference in Econometrics. New York: Oxford University Press, 1993.

10. Davidson, J. E. H., D. F. Hendry, F. Srba and S. Yeo, "Econometric Modelling of the Aggregate Time-Series Relationship between Consumers' Expenditure and Income in the United Kingdom." Economic Journal, 1978, 661-92.

11. Deaton, Angus, "Saving and Liquidity Constraints." Econometrica, September 1991, 1221-48.

12. Engle, Robert F. and C. W. J. Granger, "Co-Integration And Error Correction: Representation, Estimation, and Testing." Econometrica, March 1987, 251-76.

13. Hall, Robert E., "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence." Journal of Political Economy, 1978, 501-17.

14. Hendry, D. F., J. H. Muellbauer and A. Murphy. "The Econometrics of DHSY," in A Century of Economics: 100 Years of the Royal Economic Society and the Economic Journal, edited by J. D. Hey and D. Winch. Oxford: Basil Blackwell, 1990.

15. J. Johnston. Econometric Methods, third edition. New York: McGraw-Hill, Inc., 1984.

16. Schechtman, Jack and Vera L. S. Escudero, "Some Results on "An Income Fluctuation Problem"." Journal of Economic Theory, 1977, 151-66.