Time-varying lag cointegration

This paper proposes an alternative estimation method for cointegration, which allows for variation in the leads and lags in the cointegration relation. The method is more powerful than a standard method. Illustrations to annual inflation rates for Japan and the USA and to seasonal cointegration for quarterly consumption and income in Japan shows its ease of use and empirical merits. © 2021 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
This paper introduces an alternative estimation method for cointegration. This method makes use of the fact that if two integrated variables y t and x t are cointegrated, that then y t and x t+1 and y t and x t−1 are also cointegrated. This fact can be used to estimate an error correction variable y t − θx t−lagt where the variable lag t can take values −1, 0, or 1, throughout the sample t = 1, 2, 3 . . . , T .
Commonly applied methods to test for cointegration analyze the variables y t and x t at the same time, that is, these methods consider y t − θx t . There seems however no a priori reason to only allow for a contemporaneous cointegration relation. In fact, it may be restrictive to have shocks ε t to hit the two variables always exactly at the same moment. Also, it may be that sometimes y t leads and that sometimes x t leads. A visual impression of this feature is presented in Fig. 1, which depicts annual CPI (Consumer Price Index) based inflation for Japan and the USA, for the sample 1960 to 2015. For some years, these two series seem to move together, like around 1975 and 1980, but in other periods, the USA inflation rates increase or decrease one year earlier than inflation in Japan does, while in other periods it is the other way around. 1 The outline of this paper is as follows. Section 2 shows that cointegration is not dependent on the specific lead and lag structure across the nonstationary variables. In fact, the same cointegration relation is found across various cases. Section 2 further provides a detailed illustration using annual CPI based inflation for Japan and the USA. Section 3 presents results of a few simulation experiments where it is examined what happens if the Data Generating Process (DGP) includes an error correction term like y t−1 − θx t−1−lagt , where lag t can take values −1, 0, or 1, and when the standard Engle and Granger [1] test method is used. It is found that the standard statistical method loses power. Section 4 addresses seasonal cointegration [2], where the new estimation method may be even more useful, because the seasonal random walk implies that ''summer can become winter''. This feature makes it even more unlikely that exactly the same error process makes these changes to happen at the same time. Section 5 concludes with various avenues for further research.

Cointegration and lags
This section considers an exemplary case where two variables are cointegrated and can be captured by a single equation Error Correction Model (ECM).
Consider two variables y t and x t , for t = 1, 2, 3., . . . , T , and suppose that these two variables are connected via the Autoregressive Distributed Lag (ADL) model where ε t is a standard zero mean white noise process with common variance σ 2 ε . To save notation, an intercept is excluded.

Error correction model
It is assumed that the two variables are integrated of order 1, I(1). That is, y t − y t−1 and x t − x t−1 are stationary, I(0).
It is further assumed that there is a single cointegration relationship between the two variables. In that case, ADL model in (1) can be written as an ECM like where the cointegration relation is (2) is the adjustment parameter.
An alternative way of writing an ECM from the ADL model in (1) is which shows that the parameter in the cointegration relation does not change when the long-run relation holds between y t and x t+1 . A third way of writing the ADL model in (1) as an ECM is (4) and this shows that the very same cointegration relation holds for y t and x t−1 .

Two inflation series
One method to test for cointegration is the Engle and Granger [1] two-step method. The first step amounts to a regression of y t on a constant and x t (or x t−1 or x t+1 ). The second step is to run a Dickey-Fuller (DF) unit root test on the residuals. One can also look at the Durbin Watson (CRDW) test statistic as recommended in Sargan and Bhargava [3]. 2 The first regression for the inflation series (for the effective sample 1961-2015 for comparison purposes) corresponding to (2) results in with a CRDW value of 0.449, and where the DF test (no lags of the first differences needed) obtains the value −2.775. The 5% critical value for this test is −3.37, and hence the null hypothesis of a unit root is not rejected. There seems to be no evidence of cointegration. The second regression corresponding with (3) results in with a CRDW value of 0.716, and a DF test value equal to −3.384. This suggests the rejection of the unit root null hypothesis at the 5% level, and hence now there is evidence for cointegration. The third regression corresponding with (4) results in with a CRDW value equal to 0.621 and a DF test value of −3.185, which now suggests the absence of cointegration.
Clearly, in this empirical case, the results give mixed evidence for cointegration.

An alternative estimation method
An alternative estimation method resorts to a time-varying lag cointegration model, that is, to consider the cointegration relation as where lag t ∈ {−1, 0, 1}. One way to determine the lag t variable is to consider the three regressions above, compute for each regression the residuals, compare the absolute values of these residuals, and set the lag at each time t at the value that corresponds with the smallest absolute residual.
An application of this simple method to two inflation series, results in the lag t variable, depicted in Fig. 2. The associated cointegration regression is with a CRDW value equal to 0.804 and a DF test value of −3.739, which provides strong evidence for the presence of cointegration amongst the two variables. 3 A suitable error correction model turns out to be with estimated standard errors 0.289, 0.107 and 0.171, respectively. Comparing the parameters with their standard errors suggests that the long-run and short-run relations between the inflation series are about equal to 1.

Simulation experiments
In this section a simulation experiment is considered to examine what happens if there is a time-varying lag cointegration structure, while one relies on the standard Engle-Granger single equation test. This latter test assumes that cointegration occurs at time t for both variables. Assume that the Data Generating Process (DGP) is with u t ∼ N(0, 1) and w t ∼ N(0, 1), y 0 = 0, x 0 = 0, t = 1, 2, . . . ., 120, and where lag t ∈ {−1, 0, 1} Per observation t, the value of the lag is drawn from a multinomial distribution with equal probabilities for the outcomes −1, 0 and 1. Next, consider the regression where the parameters are estimated using Ordinary least Squares (OLS). The estimated residuals are stored, and next the test regression ε t −ε t−1 = δε t−1 + ϑ t is considered. The t ratio for δ is computed and compared against the 5% critical value −3.37. Table 1 gives the number of cases (out of 10000 replications) that the t test value indicates rejection of the null hypothesis. The first column in Table 1 concerns the case where there is no variation in the lag, and this provides the statistical power of the standard Engle-Granger test method. The next column gives the rejection frequencies in case the DGP has a lag structure with lag t ∈ {−1, 0, 1}, each with probability one-third. Obviously, there is a loss of power. One would expect the loss of power to increase when there is more variation in the lags, like for example lag t ∈ {−1, 0, 1, 2}, each with probability one-fourth. And indeed, the final column of Table 1 indicates a further loss of power.

Seasonal cointegration
When analyzing quarterly nonstationary data, each of which obeys a seasonal random walk, that is, y t − y t−4 and x t − x t−4 are stationary, one may want to examine whether there is cointegration across y t + y t−1 + y t−2 + y t−3 and x t + x t−1 + x t−2 + x t−3 , see Engle et al. [2] and Hylleberg et al. [5]. As a seasonal random walk implies that seasons may switch places, that is ''summer can become winter'', it may even be more unlikely in practice that exactly Table 1 The DGP is y t − y t−1 = (ρ − 1) ( y t−1 − x t−1−lagt ) + u t and x t − x t−1 = w t with u t ∼ N (0, 1) and w t ∼ N (0, 1), y 0 = 0, x 0 = 0, t = 1, 2, . . . ., 120. The simulations are based on 10000 replications of the Engle-Granger two-step method. is stationary. An alternative cointegration relation can now be where lag t ∈ {−3, −2, −1, 0, 1, 2, 3}. The same method as above can now be considered, where various auxiliary regressions are run, and the size of the absolute residuals can be compared across the regression model, to select the lag variables.

Consumption and income in Japan
To illustrate the usefulness of time-varying lag cointegration, consider the quarterly data for consumption and income in Japan, for 1980Q1 to 2001Q2 in Fig. 3. A trend and a seasonal pattern are clearly visible for both series. The data are analyzed after taking natural logs, and y t is the log of consumption and x t is the log of income.
An Engle-Granger type regression assuming a contemporaneous cointegration relation results in The CRDW value is 0.060, which indicates the absence of cointegration. This is confirmed by an Augmented Dickey-Fuller test (one additional lag of the first differences) on the residuals of the test regression, which obtains the value −2.571. This is not significant according to the critical values reported in Engle et al. [2].
When a time-varying lead and lag structure is allowed, the obtained time-varying lag structure is depicted in Fig. 4. This lag structure shows much variation. The alternating lag regression now gives A suitable error correction model 4 for these two series turns out to be where the parameters are estimated using Nonlinear Least Squares, and the associated standard errors are 0.065, 0.078, 0.081, 0.105, 0.013 and 0.033, respectively. The t test value on the adjustment parameter −0.043 is −3.443, which is significant at the 5% level.

Conclusion
This paper has proposed an alternative estimation method for cointegration, which allows for variation in the leads and lags in the cointegration relation. The method is more powerful than a standard method. Illustrations to cointegration amongst annual inflation rates for Japan and the USA and to seasonal cointegration for quarterly consumption and income in Japan shows its ease of use and empirical merits. It is also demonstrated that the standard test method for cointegration does not find evidence of cointegration in both cases.
Various extensions of the new estimation method seem obvious. One could want to allow for more than two variables and consider the application of the Johansen [4] method. Further, one can consider nonlinear and time-varying cointegration. Also, one may want to allow for structural breaks and shifts in trend, which may not need to happen at the same moment in time for each series.