The Dynamic Behavior of Wages and Prices: Cointegration Tests within a Large Macroeconomic System.

Martin B. Schmidt [*]

The dynamic relationship between wages and prices has long held a central place within the economic literature. Most macroeconomic models make assumptions as to the causal relationship between the two variables. Unfortunately, empirical investigations have produced widely divergent results. In particular, the present paper examines the results of time-series studies and argues that the lack of a consensus is due to improperly specified models. Once the wage-price relationship is embedded within a multiple vector system, identification of a wage-price cointegrating relationship is significantly improved. In addition, the increased efficiency yields evidence in favor of the dual feedback between wages and prices.

1. Introduction

A belief in a systematic and stable relationship between wage and price behavior underlies much of standard macroeconomic theory. While most models share this belief, they differ dramatically when it comes to their presumed direction of causal influence. For example, the original Phillips curve relationship hypothesized that price inflation imparts pressure on wages. Such an assumption is indicative of prices having a unidirectional impact on the wage process. In contrast, the notion that prices are set at a given percentage markup over wage costs is popular within post-Keynesian economic thought. Interestingly, such models maintain that the wage process performs an independent causal role in the inflationary process. Furthermore, the expectations-augmented Phillips curve combines these theories and would, therefore, predict joint causality between price inflation and wage inflation. Finally, in order to complete the possibilities, many monetarist models fail to recognize any systematic relationship, let alo ne any causal relationship, between the two economic variables.

Given the importance of the relationship in determining the relative merit of these opposing theories, it is not surprising that a large empirical literature exists investigating the causal responses of prices and wages. In an early study, Barth and Bennett (1975) combined consumer prices, hourly wages of production workers, and a monetary variable into a three-variable system. By examining the significance of future lags (either 4 or 8) of each variable, they determined that consumer prices were significant in explaining hourly wages, indicating that consumer prices caused hourly wages. In contrast, future hourly wages were insignificant in explaining the current behavior of consumer prices and therefore wages did not cause prices. Also, in order to highlight the importance of money in the process, money was found to cause movements in prices.

Ashenfelter and Card (1982) reported markedly different results. They modified the Barth and Bennett study to include the unemployment rate and a nominal interest rate while dropping the money variable. Using a four-variable vector autoregressive (VAR) system (with a lag of 4), the results from Granger-type equations provided strong evidence of wages Granger-causing prices. However, there was only weak support for prices Granger-causing wages. In addition, the influence of unemployment on prices and wages was rejected. Interest rates, however, did play a strong role in the behavior of prices although not for wages.

Shannon and Wallace (1986) adjusted these early studies on two levels. First, they chose unit labor costs to control for wage increases associated with productivity gains, which would not be expected to be inflationary. The second was to include a measure of output into the VAR model. Once income and money were added to the VAR equations containing the GNP deflator and unit labor costs, the results fall in line with Ashenfelter and Card, as Granger causality was estimated only from wages to prices.

Overall, these early studies indicated that the relationship between wages and prices was sensitive to the choice of additional right-hand-side variables. In addition, they provided at least marginal support for the importance of income, money, and nominal interest rate variables in altering the estimation of the wage-price relationship. However, the time-series behavior of most of these series, wages and prices included, is generally believed to be nonstationary, and therefore much of the variability in the results may have had to do with the "spurious" nature of regressing nonstationary series on each other.

An intuitive solution to control for the "spurious" effect would be to difference the nonstationary series until each series becomes stationary. Since the estimation would then involve stationary variables, the unreliability and inefficiency of the estimation would clearly be removed. However, as Hendry (1986) and Granger (1988) argue, differencing economic time-series data removes all the information about the long-run relationships. In the case where variables are cointegrated, important information would be lost. In addition, Hakkio and Rush (1989) show that taking first differences of cointegrated variables in order to obtain a stationary series may result in an omitted variable bias.

Rather than omit what may be relevant information, the focus has moved toward analyzing the cointegrating relationship between the GNP deflator and unit labor costs. [1] However, the question of Granger causality is more complicated when variables are cointegrated. A rejection of Granger causality requires not only that the lagged variables be insignificant but also that the speed of adjustment coefficient be zero. Vector error-correction equations (VEC) modify the VAR structure to incorporate the speed of adjustment variable(s). [2]

Two recent studies have incorporated both cointegration and VEC estimation. Both Mehra (1991) and Darrat (1994) reexamine the relationship between price and wage inertia accounting for the nonstationary behavior of the data. Surprisingly, the two obtain contradictory results. Using unit labor costs, the GNP deflator and an output-gap variable, Mehra offers augmented Dickey-Fuller (ADF) test results that indicate that both unit labor cost and the GNP deflator have two unit roots, that is, I(2), while the output-gap variable has one, that is, I(1). Consistent with the ADF results, cointegration to I(0) is then found between the differences of prices and the difference of unit labor costs but not between their levels. Also, the inclusion of the speed of adjustment parameters within the VEC yields support for Granger causality from prices to wages. However, the results fail to generalize as wages were not found to Granger-cause prices.

Following Lutkepohl (1982), Darrat argues that Mehra's Granger-causality tests are subject to an "omission-of-variables" bias. In addition, since cointegration and VEC models are closely related to Granger causality, these also may suffer from the same bias. Darrat suggests that the wage-price relationship would be more accurately estimated within a general inflation equation. Specifically, Darrat follows much of the earlier literature by including a money variable and an interest rate variable in addition to introducing a measure of exchange rates into the wage-price vector. Once these omitted variables are included, Darrat is unable to find a cointegrating relationship between either the levels or the differences of unit labor cost and the GNP deflator. The results support Gordon's (1988) view that prices and wages have little to offer one another and that wages "live a life of their own.

The present paper is equally concerned with the possible biases introduced when estimating improperly specified cointegrating vectors. Recent theoretical work by Phillips (1991) and Johansen (1992) demonstrates that the omission of relevant variables in an analysis of cointegration may produce biased and inefficient estimates of the number of both cointegrating relationships and cointegrating coefficients. Given the fact that most, if not all, economic variables and/ or relationships are not determined in isolation, estimating these in a single equation format may subject the results to the concerns raised by Johansen and Phillips. Recent studies by King et al. (1991), Cutler et al. (1997), and Cutler, Davies, and Schmidt (1997) have found empirical support for the efficiency gains associated with embedding a single macroeconomic equation within the framework of a larger macroeconomic system.

Therefore, unlike Darrat, who introduces additional variables into the wage-price vector itself, the present paper opts to embed the wage-price vector within a larger set of macroeconomic relations. Such an approach allows the vector to be estimated within a set of simple, stable, and theoretically consistent relationships. The system of equations incorporates many of the additional right-hand variables suggested in the previous studies but introduces them in a more systematic way. Therefore, the approach may be viewed as less arbitrary than what would be necessitated by the constant inclusion and exclusion of right-hand-side variables whose theoretical importance may be argued and whose impact may vary over time.

The macroeconomic system used here, with its multiple vectors, follows King et al. (1991), Cutler et al. (1997), and Cutler, Davies, and Schmidt (1997). King et al. examine the behavior of six macroeconomic variables (real income, consumption, investment, real money balances, the nominal interest rate, and the inflation rate). From these, they are able to estimate investment, money, and consumption functions. Cutler et al. modify the King et al. system of variables to include imports and are able to derive the additional import function. Cutler, Davies, and Schmidt incorporate nominal Ml and the GNP deflator and find stronger evidence of an Ml demand relationship than much of the previous literature. For the present purpose, unit labor costs are added to the King et al. model. Therefore, the wage-price relationship is placed in the middle of many of the processes that influence the vector's behavior.

In order to demonstrate the gains a systems approach to estimation yields, the wage-price relationship is estimated within a one-equation, two-variable system and then within the larger system of equations. Estimation of the cointegrating relationship(s) follows the method developed by Johansen and Juselius (1992). Also, in order to address the concerns raised about the sensitivity of varying sample periods, a more systematic rolling regression approach is used. Finally, the existence of Granger causality is examined through the behavior of the speeds of adjustment (error-correction) equations.

Overall, significantly more support for a cointegrating relationship between prices and wages and for the dual feedback between wages and prices is found than by other researchers, which may be due to the efficiency gains of full-system estimation. In addition, the systems approach may yield an interesting side result: A breakdown in the M2 relationship does appear to have occurred, although significantly later than had been previously thought. Interestingly, this does not seem to have affected the wage-price relationship.

One final issue is the integrated behavior of prices and wages. Both Mehra and Darrat report ADF test results that suggest that both variables are I(2). The ADF tests reported in Table 1 also suggest that prices may be I(2). The tests, however, reject the null of a second unit root for wages. Since variables with different levels of integration cannot be cointegrated, this finding, by itself, would indicate that wages and prices cannot be cointegrated. However, there is good reason to believe that first-differencing the price deflator is sufficient to induce a stationary series. Miller (1991) argues that DF and ADF tests yield coefficients on the lagged level (the second difference of the series) that significantly exceed minus one; such results are indicative of overdifferencing. In addition, Miller cites evidence from the autocorrelation and partial autocorrelation functions, which indicate that first-differencing leaves a highly autocorrelated series with a slowly declining autocorrelation function, while the second-differencing produces only one significant spike in the autocorrelation function. [3] Similar results were found here.

In addition, much of the money demand literature implicitly estimates the behavior of P as I(1). The results in Table 1 indicate that M2 is I(1) and P is I(2). If so, then any linear combination of P and M2 must necessarily be I(2). However, it is generally accepted that real money balances are I(1). Additional cursory evidence of the price deflator being I(1) is presented by Cutler, Davies, and Schmidt (1997), where the estimation of a nominal M1 money demand function significantly improves once the level of prices is included into the money vector. Also, Miller (1991) is able to estimate a nominal M2 money demand vector with the additional price variable. Finally, Miyao (1996) introduces P as I(1) in an examination of the stability of an M2 vector. Overall, then, there exists significant evidence of the price level being I(1), and therefore this paper treats both the price and the wage variables as I(1).

The rest of this paper is organized as follows. Section 2 presents the economic approach and model used to reduce the biases and inefficiencies associated with the estimation of a partial system. Section 3 describes the empirical results and the speeds of adjustment parameters. Section 4 contains a brief conclusion.

2. Econometric Approach and Model

Econometric Approach

The Jobansen and Juselius reduced rank regression approach is one of the more popular techniques for jointly estimating a group of cointegrating relationships (Johansen 1988; Johansen and Juselius 1992). Their approach begins with a vector error correction model (VEC), such as the following:

[Delta][X.sub.t] = [mu] + [[[sum].sup.k-1].sub.i=1] + [[Gamma].sub.i][Delta][X.sub.t-1] + [prod][X.sub.t-k] + [[epsilon].sub.t]. (1)

The variable [Delta][X.sub.t] represents a p-element vector of observations on all variables in the system at time t, the [[Gamma].sub.i][Delta][X.sub.t-i] terms account for stationary variation related to the past history of system variables, and the [Pi] matrix contains the cointegrating relationships. All variables must be nonstationary in levels, and it is hypothesized that [Pi] = [alpha][beta]', where the cointegrating vectors are in the [beta] matrix and the [alpha] matrix describes the speed at which each variable changes to return the markets to long-run equilibrium. Cointegration, then, requires that the [beta] matrix contain parameters such that [Z.sub.t], where [Z.sub.t] = [beta]'[X.sub.t], is stationary. Also, the [alpha] matrix is thought to represent the speed with which each variable changes to retum the individual vectors to their respective long-run equilibrium.

In order to examine the robustness of the cointegration results to the omission of relevant variables, Johansen (1992) examines the statistical properties of two versions of Equation 1; first, where all the system's variables are included and, second, where only a subset of the variables is included in the cointegration analysis. Rather than drop the excluded variables from the system completely, the remaining variables are introduced as conditioning variables. Interestingly, the results indicate that when the number of variables retained in the subset is less than the number of true cointegrating vectors, linear combinations rather than unbiased estimates of each vector are estimated. In addition, even if the number of subset variables exceeds the number of cointegrating vectors, the estimates are still generally inefficient but not biased. Only when the excluded variables are weakly exogenous are estimates efficient and unbiased.

In a parallel investigation, Phillips (1991) argues similar points. Rather than adding the omitted variables as conditioning variables, Phillips excludes the variables completely. His results suggest that if the excluded variables contain unique common trends, biased and inefficient estimates may result. The bias is introduced through the nonzero correlation between nonstationary components of the included variables and the common trends that have been excluded from the estimation. Overall, then, whether relevant variables are included as conditioning variables or are completely excluded from the estimation, significant inefficiency and bias are possible.

Therefore, any investigation of the wage-price cointegrating relationship must be concerned with excluding variables that are cointegrated with either wages or prices. For example, a money demand function argues that prices are cointegrated with nominal interest rates, income, and nominal money. As Darrat correctly points out, the exclusion of these while estimating the cointegrating behavior of wages and prices creates a specification bias. However, income and interest rates are cointegrated with consumption, investment, and imports. The exclusion of these may also create a specification bias.

Economic Model

Given the concerns raised by Phillips and Johansen, a possible solution would be to incorporate the additional right-hand-side variables directly into the cointegrating vector. However, as is pointed out in the introduction, this approach has met with variable success. The present paper offers an alternative solution. Rather than introduce the additional variables into a single cointegrating vector, this paper opts to embed the two-variable wage-price vector into a system of equations. The system of equations incorporates many of the additional variables utilized in the earlier literature. As this literature indicated, real income, the level of the interest rate, and the nominal money supply are important in determining the behavior of the wage-price vector. However, these variables are at the core of most macroeconomic relationships, whose behavior could further influence the wage-price vector.

In addition to these concerns exist a need and a desire to limit the size of the system and to maximize its simplicity. A simple, yet comparatively complete, explanation of the movements of the macroeconomy has been provided by King et al. (1991), who combine real income, real consumption, and real investment to describe the real side of the economy. The nominal side of the economy is introduced through the inclusion of real money balances and the nominal interest rate. These five variables are thought to produce three cointegrating relationships, two of which reflect the consumption and investment "great ratios" and a third yielding a money demand relationship.

A further justification for the inclusion of the wage-price vector within a larger system of equations is that the King et al. model is intended to describe the behavior of the economy. In particular, the behavior of the "great ratios" are intended to capture the permanent or long-run movements in output. Moreover, Cochrane (1994) maintaines that there exists a significant amount of stability in the "great ratio" for consumption and income and that consumption is believed to determine trend movements in income. Deviations from the trend, which are captured within the error-correction terms, should then reflect both the demand and the supply shocks, which are believed to influence the wage-price relationship. [4]

The present paper makes two modifications on the King et al. structure. The first is to explicitly recognize the relative importance of the real interest rate in influencing the behavior of the "great ratios." The second is the introduction of the nominal money supply rather than its real counterpart. Since the fundamental relationship in question investigates the behavior of the aggregate price level, it would seem more efficient to estimate the price coefficient than to impose the existence of a unity coefficient. [5] Overall, then, a simple way to express the macroeconomic relations is as follows:

[c.sub.t] - [a.sub.0] - [a.sub.1][y.sub.t] - [a.sub.2][r.sub.t] = [[epsilon].sub.ct], (2)

[i.sub.t] - [b.sub.0] - [b.sub.1][y.sub.t] - [b.sub.2][r.sub.t] = [[epsilon].sub.it], (3)

[M.sub.t] - [d.sub.0] - [d.sub.1][y.sub.t] - [d.sub.2][r.sub.t] - [d.sub.3][P.sub.t] = [[epsilon].sub.Mt], (4)

[W.sub.t] - [f.sub.0] - [f.sub.1][P.sub.t] = [[epsilon].sub.WPt]. (5)

where y represents the log of real output, c the log of real domestic consumption, i the log of real domestic investment, r the ex post real interest rate, M the log of nominal money balances, P the log of the aggregate price level, and W the log of the productivity-adjusted wage level, while t is a time subscript. The variables [[epsilon].sub.ct], [[epsilon].sub.it], [[epsilon].sub.Mt], and [[epsilon].sub.WPt] represent the respective disequilibrium error terms. The accompanying appendix details the specific data.

Each equation represents a long-run equilibrium relation with the error terms capturing disequilibria in each equation; therefore, each is predicted to yield a separate cointegrating relationship. As mentioned earlier, most macroeconomic models predict that P and Vi must satisfy Equation 5 over time such that [[epsilon].sub.wpt] is a stationary process. In the end, the seven variables in the system are combined into four equations whose residuals are thought to be unique stationary processes.

One additional modification is required. The real interest rate rather than the theoretically relevant nominal interest rate is incorporated into the money demand relationship. Overall, the system cannot accommodate both the real and the nominal interest rate along with the price level because perfect correlation exists between the two interest rates and the difference in the price level. Since the fundamental question at hand concerns the behavior of prices and that the real interest rate appears in two of the three other hypothesized vectors, the nominal interest rate is dropped. [6] Finally, it should be noted that more complicated formulations of this macroeconomic system failed to qualitatively alter the results. For example, the inclusion of imports directly or fiscal policy measures indirectly as a conditioning variable produced similar results.

3. Empirical Results

Testing for the Number of Cointegrating Vectors and Parameter Values

As described earlier, the unrestricted matrix of cointegrating vectors was estimated using Johansen's method. Two levels of hypothesis testing were then performed on these results. First, the maximum eigenvalue test, the trace test, as well as the Hannan-Quinn criteria were used to determine the number of cointegrating vectors within the system of variables. Second, Johansen and Juselius's (1992) [H.sub.6] and [H.sub.5] tests were employed to determine whether the restrictions derived from Equations 2 to 5 are valid individually.

The first test is concerned with whether the estimated number of cointegrating vectors is consistent with the proposed model. The seven variables are predicted to yield money, consumption, and investment vectors in addition to the wage-price vector. Therefore, four significant eigenvalues and associated eigenvectors are expected. However, as Johansen and Juselius (1992) point out, these unconstrained vectors may be linear combinations of the true vectors and therefore may not have an economic interpretation. The second stage of hypothesis testing should then be to determine whether estimated unconstrained vectors reduce to the following form:

([y.sub.t][c.sub.t][i.sub.t][M.sub.t][P.sub.t][r.sub.t][W.sub.t]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = ([[epsilon].sub.ct][[epsilon].sub.it][[epsilon].sub.Mt][[epsilon].sub .WPt]). (8)

The last column contains the cointegrating coefficients for the wage-price relationship, while the first three columns include appropriate values for the consumption, investment, and money demand equations, respectively.

Johansen and Juselius's (1992) [H.sub.5] test allows for the restriction of all coefficients within a specified number of vectors. Under the null hypothesis, the [H.sub.5] test statistic is [[chi].sup.2] with (p - r)[r.sub.1] degrees of freedom, where p represents the number of variables, r is the total number of vectors, and [r.sub.1] is the number of restricted vectors. Restrictions may be made on a single vector, so that one vector is restricted while other vectors are freely estimated. [7]

One remaining issue concerns the lag order of the VEC model. The latter part of Table 1 reports the results from adjusted log-likelihood ratio, AIC, and SBC tests in order to determine the appropriate lag length. Unfortunately, each test defines a separate lag structure. In particular, the adjusted log-likelihood ratio test indicates that an order of 6 is appropriate, while the AIC and SBC estimate a lag structure of 3 and 2, respectively. In order to examine the robustness of the empirical results to the choice of lag, the following empirical sections were estimated with lags of 2 to 6. Interestingly, the results for all lag specifications were qualitatively similar, especially relative to the wage-price results. In the end, a lag of 4 was chosen. While the choice is somewhat arbitrary, the choice reflects the fact that a lag of 2 would seem to be too short given the literature, while the choice of 6 would seem inappropriate given the MG and SBC results. [8]

The Estimated Number of Cointegrating Vectors

In order to assess the number of cointegrating relationships the seven variables are estimated to yield, the latter part of Table 2 reports the results of the maximum eigenvalue and trace tests of the stochastic matrix produced by the seven variables for the entire sample period, 1960.2-l994.4. [9] The results provide fairly strong support for the system of equations with the trace test estimating the predicted four cointegrating vectors and the maximum eigenvalue estimating three. In addition, although not reported, the Hannan-Quinn criterion also confirms four cointegrating vectors. As will become apparent in the next section, further tests indicate that the more restrictive maximum eigenvalue test may pick up the loss of the M2 cointegrating vector during the 1991-1992 period, while the other two tests are unable. However, this failure does not seem to have impacted the other three vectors.

As a point of comparison, the standard single-equation representation for the wage-price vector was also estimated and is reported in the initial portion of Table 2. [10] The estimation entailed including only the price and wage variable and, again, adopting the Johansen procedure to estimate the unrestricted stochastic matrix. In which case, the estimation is similar, in spirit, to Mehra and Darrat's estimation. [11] The results seem to reinforce Darrat's findings since the trace test rejects the existence of a cointegrating relationship all together. However, the maximum eigenvalue test did estimate the predicted cointegrating vector at the 10% level.

Estimated Parameter Values using Johansen and Juselius 's [H.sub.5] Test

The latter part of Table 3 presents the normalized parameter values and probability levels for the [H.sub.5] tests on the full system. The choice of parameter values reflects a two-step approach. The initial parameter values were obtained from Johansen and Juselius's [H.sub.6] test. The [H.sub.6] test allows the researcher the ability to impose zero restrictions (see Eqn. 6) on the individual vector and estimates the optimal unnormalized coefficients for the remaining variables. The second step then uses the [H.sub.5] test to examine the probability of the estimated normalized coefficients. In addition, the [H.sub.5] test was used to determine the acceptable range for the individual coefficients. Finally, the sensitivity of the results to changes in the sample period is assessed by using a rolling regression approach. Impressively, reasonable parameter values at acceptable probability levels were found for the wage-price relationship for all periods. In addition, consumption and investment show similar stren gth during the sample's subperiods. Each vector is briefly discussed in the following.

The wage-price trade-off yields the expected one-to-one trade-off between the movements in wages and prices, which is consistent with long-run expectations; that is, a 1% increase in productivity-adjusted wage level is estimated to increase the overall price level by the same 1%. It should be mentioned that the range of acceptable coefficient values was quite tight, between 0.93 and 1.07. This range corresponds quite nicely to the values of 0.9 and 1.1 estimated by Mehra and they are only slightly lower than those estimated by Darrat, whose estimates were around 1.2.

In contrast, the two-variable system could not accept the expected one-to-one trade-off. The estimates were closer to the outside limit provided by Mehra and much closer to the results provided by Darrat. The two-variable results are reported in the initial part of Table 3. The single-equation, two-variable system rejected the one-to-one trade-off at every interval. However, during most periods, the smaller system was able to accept values on the upper end of the estimate of Darrat and Mehra. These results, however, were not as strong as the sample period was moved up.

As for the behavior of the three remaining markets, the results were mostly as expected, especially prior to 1992.4. Each of the three remaining markets contain both real output and the real interest rate as arguments. The three vectors yield income elasticities of -1.0 for the money vector, -0.8 for the consumption vector, and -1.7 for the investment vector. The unitary coefficient for the money vector is at the expected level, but it appears that the value for the consumption vector incorporates a tax effect. A more proper interpretation of the [a.sub.1] in the original specification of Equation 1 is [a.sub.1] = b(l - [tau]), where b is the MPC and [tau] is the tax rate. Therefore, an estimate of [a.sub.1] at -0.8 would be consistent with an MPC of 0.95 and an overall tax rate of 0.15. [12]

The analysis for the investment market is a bit more awkward with a higher-than-expected income elasticity. However, the large elasticity does appear plausible since investment is far more volatile than output and accounts for most business cycle episodes, so reactions of investment to changes in income should be elastic. The results are also consistent with previous estimates from Cutler, Davies, and Schmidt (1997).

In addition to the income variable, each of the four vectors contains the real interest rate. Interest rate elasticities for the money, consumption, and investment are 0.01, -0.01, and 0.001, respectively. While each vector yields coefficients that have the correct sign, the coefficients for all three vectors are quite low. However, they are consistent with the literature dealing with the relative insensitivity of macroeconomic variables to changes in the interest rate.

Before examining the speeds of adjustment behavior, the behavior of the money market deserves additional discussion for several reasons. First, the results seem to reinforce the findings of Miyao (1996), who argues that the M2 relationship seems to have broken down in the early 1990s. Miyao uses, among other evidence, an inability to reduce real income forecast errors as evidence of the loss of the M2 relationship. The present results may provide further evidence of such a breakdown. Furthermore, the results provide a possible explanation for the difficulty with the maximum eigenvalue test. The finding of only three cointegrating vectors may reflect a breakdown in the M2 demand relationship. Finally, the loss of the M2 relationship does not seem to have affected the wage-price transmission; that is, all periods find consistent and stable coefficients.

Speed of Adjustment Coefficients

While the results reported here suggest the existence of a stable and systematic relationship between the wage level and the price level, they fail to provide information about the dynamic responses of wages and prices to one another. However, the VEC's so-called speed of adjustment coefficients provide such information. The behavior of the wage-price vector illustrates the usefulness of this part of cointegration analysis. The error-correction equation for the wage-price vector is represented by the restricted coefficients (estimated in the previous section) of the cointegrating vector, shown in the following equation:

1.0.[W.sub.t] - 1.0.[P.sub.t] = [[epsilon].sub.wpt]. (7)

Given a positive value for [[epsilon].sub.wpt], the vector is above its equilibrium level. The adjustment back to equilibrium requires that the price level rise and/or the wage level fall. Theoretically, any combination of the two will clear the market. Therefore, the price variable should respond positively to [[epsilon].sub.wpt], and the wage variable should move negatively to eliminate positive values of [[epsilon].sub.wpt]. The logic for the other markets follows similar lines.

As mentioned in the introduction, the absence of Granger causality between cointegrated variables requires not only that the lagged values are insignificant but also that the dependent variable fail to respond correctly and significantly to the error-correction information. However, as wages and prices have been found to be cointegrated, some level of Granger causality must exist. The question remains whether the causality runs from prices to wages, wages to prices, or in both directions. The latter part of Table 4 reports the results of the VEC model analysis for the seven variables in response to past disturbances to the cointegrating vectors. [13] The disequilibrium residuals ([[epsilon].sub.ct], [[epsilon].sub.it], [[epsilon].sub.Mt], and [[epsilon].sub.WPt]) and were calculated by multiplying the estimated vectors from Table 3 for each market by the relevant variable. Therefore, the results reflect the more restrictive condition that the data emulate the model's restrictions rather than any single equati on's restrictions. [14] The results from the period 1960.2-1994.4 are reported for a couple of reasons. First, the period represents an overview of the sample, and, second, the results were qualitatively similar to those obtained for the other periods.

The results for the wage-price relationship (the last column of the latter part of Table 4) provide strong support for a bidirectional feedback between wages and prices; that is, each Granger-causes the other. Both wages and prices respond significantly to disequilibrium in the wage-price vector. The price level rises in response to a positive disturbance, while the wage level falls in response to a similar disturbance. Although both variables respond correctly and significantly, the size of their coefficient differs significantly; the response of the wage variable is almost three times as strong as the response of the price variable. An interesting interpretation of this result is that the short-run adjustment of the price level is significantly more rigid than wages. [15] In addition, many of the results obtained in earlier studies carry over. In line with the results of Ashenfelter and Card as well as Darrat, the interest rate is important in determining the behavior of wages and prices. [16] Also, the re sponse of real income follows the results of Shannon and Wallace.

As before, the two-variable counterpart was also estimated, and the results are reported in the first part of Table 4. The results are in line with those of Mehra, as wages respond correctly and significantly to the disequilibrium term. Again, the results indicate that prices Granger-cause wages. However, the response of prices was in the incorrect direction and, therefore, created greater disequilibrium despite being significant. Overall, then, the additional structural equations appear to have a larger impact on the estimation of the aggregate price level than on the behavior of costs.

As for the other vectors, both the money and the investment vector have one variable that moves to clear their respective disequilibrium. Somewhat surprisingly, given the earlier tests, the money relationship is cleared by movements in the real interest rate. The interest rate, however, failed to move appropriately for any of the other relationships. Investment is brought back to equilibrium by movements in investment. The investment variable has a positive sign in its cointegrating vector, which implies that the speed of adjustment should be negative. Both the real interest rate and the real income variable, however, fail to clear the investment market.

Real income moves to clear only the consumption relationship. Income responds positively to disequilibrium in the consumption market and therefore moves to equilibrate the vector. As mentioned earlier, the connection between consumption and real income has been recently highlighted by Cochrane and Sbordone (1988), Harvey and Stock (1988), and Cochrane (1994). In particular, Cochrane (1994) maintained that a significant stability existed within the consumption-to-income ratio and that consumption determines the trend movements in income. In this case, one would predict that the relationship of real income and consumption should be more closely associated than real income and other variables. These results provide further evidence of the association.

4. Conclusion

Phillips (1991) showed that excluding variables that are cointegrated with included variables results in biased estimates of the cointegrating vectors. For example, since prices are cointegrated with money, income, and the interest rate (a money demand relationship), omitting the behavior of these while analyzing the behavior of wage and prices creates a specification problem. In addition, income is cointegrated with consumption and investment. Leaving these out also creates a specification bias. This suggests that most single-equation estimates are biased and points to an advantage of using a full-system approach.

In the present paper, the ability to identify a wage-price cointegrating vector during all sample periods within the full-system may reflect the greater precision of full system estimation. As the results of Tables 2 and 3 indicated, the single-equation estimation of the wage-price transmission was only marginally successful. In contrast, the full system estimated the correct number of vectors and stable and appropriate coefficients for all subperiods. Finally, the full-system estimation yielded error-correction results consistent with bidirectional feedback between prices and wages.

(*.) Department of Economics, Portland State University, P.O. Box 751, Portland, OR 97207, USA; E-mail [e-mail].

The author has benefited from helpful comments from the editor and an anonymous referee. However, all remaining errors remain the sole responsibility of the author.

Received August 1998; accepted October 1999.

(1.) The literature on cointegration is ever expanding; see, for example, Engle and Granger (1987), Johansen (1988), and Phillips (1991).

(2.) Miller and Russek (1990) provide an early example of the possible biases introduced when Granger-causality tests omit the temporal error-correction effects. In investigating the causal relationship between government taxes and spending, the authors show that the Granger-causality tests suggest bidirectional feedback within the error-correction equations, while excluding the error-correction information leads to unidirectional causality. I thank an anonymous referee for providing the reference.

(3.) Enders (1995) shows that a series that has been overdifferenced generates a significant spike of around -0.5 in the autocorrelation function. In the present context, the second difference produces a significant spike of -0.44 which is indicative of a series which overdifferenced. I thank a referee for providing the citation.

(4.) See, for example, Mehra (1991) and Darrat (1994).

(5.) In addition, Cutler, Davies, and Schmidt (1997) have found that allowing the price coefficient to fluctuate yields efficiency gains in the estimation of a money cointegrating vector.

(6.) Furthermore, since the difference in the price level enters the model as part of the [Delta][X.sub.t]- 1, the stationary impact of inflation has been taken out in the first part of the Johansen technique (see Cutler, Davies, and Schmidt 1997). Also, the empirical section was completed with, first, the real interest rate and, second, with the nominal rate. As one might suspect, the inclusion of the nominal interest rate, rather than the real interest rate, does not qualitatively alter the results of this paper. These are available from the author on request.

(7.) A concern exists as to the proper acceptance level for the [H.sub.5] test. Johansen and Juselius (1992) use probability values ranging from .15 and above to accept hypotheses concerning purchasing power parity and interest rate parity. Since the equations here are also structural in nature and therefore Type II errors arc of concern, larger probability values are used to accept a restriction.

(8.) The results for lags of 2, 3, 5, and 6 are available from the author on request.

(9.) As will become apparent shortly, both the two- and the seven-variable results are, generally, robust to the choice of sample period.

(10.) In addition, the other markets were also estimated individually. The results were consistent with Cutler et al. (1997) and Cutler, Davies, and Schmidt (1997), where the existence of the single-equation cointegrating vectors was generally rejected.

(11.) Mehra's sample period was from 1961.3 to 1989.3; Darrat extended the sample to include 1959.1 to 1991.4.

(12.) The consumption estimation is equally comfortable with a coefficient of -0.7.

(13.) The results of Table 4 were estimated with all lagged first differences included. Excluding the insignificant lags (i.e., the Hendry general-to-specific approach) failed to qualitatively alter the results. Because of concerns of brevity, the estimates of the other VEC components are not reported. These are, however, available from the author on request.

(14.) Results from incorporating the single-equation restrictions failed to alter the results.

(15.) A counterexample is Spencer (1998). Using a three-variable structural VAR model, Spencer argues that nominal wage rate is less responsive to an aggregate demand shock than is the aggregate price level. However, one difficulty with the Spencer study is the ommission of the error-correction term between the nominal wage rate and the aggregate price level. In addition, the Spencer study used non-productivity-adjusted nominal wage rates.

(16.) The analysis is not exactly the same since the variable is the real rate rather than the nominal rate.

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Appendix

The data were acquired from the Citibase data tape and transformed as follows (the original Citibase labels are provided in boldface):

M2 = log(money -- nominal M2): FM2

P = log(price level -- GNP implicit price deflator): GD

y = log(real GNP): GNFQ

i = log(real gross private investment): GINQ

c = log(real domestic consumption): GCQ-GIMQ

r = ex post real interest rate: r = R - [400.In(Pt/Pt - 1)] (see King et al. 1991)

where

R = nominal interest rate -- U.S. Treasury 10-year rate (% per annum): FYGL

The wage variable was obtained from the Bureau of Labor Statistics:

w = log(productivity-adjusted wages): Unit Labor Costs Index for the Non-Farm Business Sector