Error correction mechanisms and short-run expectations.

• Author: Antzoulatos, Angelos A.

1. 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.

2. 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...