Testing for Cointegrating Relationships with Near-Integrated Data
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MacKinnon, 1990, Critical Values for Cointegration Tests
Hendry David , Neale Adrian , and Ericsson Neil . 1990. “PC-NAIVE: An Interactive Program for Monte Carlo Experimentation in Econometrics,” Version 6.01.
Elliott Graham , and Stock James . 1992. “Inference in Time Series Regression when the Order of Integration of a Regressor is Unknown.” National Bureau of Economic Research Technical Working Paper No. 122.
De Boef, 1999, Testing for Cointegrating Relationships when Data are Near-Integrated
O refers to the order of probability. A sequence of random variables is said to be Op (T–½) if for every v > 0 there exists an M > 0 such that P{|Xt | > (M/)} < v for all T (Hamilton 1994).
s may be thought of as a “nuisance parameter.”
All integrals range from 0 to 1.
Assumptions about exogeneity become particularly important in the single-equation setup. For much of the literature in political science that relies on error correction representations, the necessary assumption of weak exogeneity is reasonable.
The ECM is estimated with an extra lagged independent variable to break homogeneity.
It may be that the absence of cointegration will be caught in the second step, but this is not known and we can provide no evidence in this regard.
For q to equal 0, β must equal 1. (In all other cases β is set to 0.5.) In this case, we chose unit variances so that the variance ratio, s, equals 1. Analytical results indicate that the statistics are invariant to s in this case. See the Appendix.
The results below should apply equally for larger samples and values of ρ closer to 1. For ρ = 0.95, we would falsely accept the null 50% of the time in samples as large as 130.
The derivation of s is given by De Boef and Granato (1999, Appendix, Part C).
Under the alternative hypothesis of cointegration, or near-cointegration, wt is a stationary process so standard asymptotic results apply. The distribution of both tests under the null hypothesis have been derived elsewhere and relevant results are presented in the Appendix.
To see the common factor restriction, express yt in terms of wt as yt = xt + wt . Nowgiven Eq. (11), rewrite wt as Wt = (1 + b)wt –1 + ψ. Substituting into the equation for yt , we have yt = xt + (1 + b)wt –1 + ψ. Finally, substitute for wt – lagged one period and collect terms: yt [1 – (1 + b)L] = [1 – (1 + b)L]xt + ψ, where L is a lag operator. Both yt and xt share a common factor in this representation, where the common factor is given by [1 – (1 + b)L] (Hendry and Mizon 1978). This restriction is not made in the ECM t test. In other words, the ECM t test does not impose a common factor restriction.
We assume that α = 0 without loss of generality.
The pretest analyses proceed in the same manner as follows below for η.
Rejection frequencies are unaffected by the variance of the error term.
We might prefer to think of tests for cointegration in the context of near-integrated processes as tests to distinguish significant long-run relationships from spurious relationships rather than as tests for cointegration, but given our definition of near-cointegration, they are the same in finite samples.
The ECM representation may be applicable when the data are stationary as well. Hendry (1995) elaborates on the versatility of ECM representations of stationary data.
The ADL(1,1) model can be written as yt = α + ϕyt– 1 + π0 xt + π 1 xt –1 + εt , where λ = φ – 1, β = π0, and π0 + π1 + φ – 1 = α from the ECM representation. The DGP can thus be thought of as a process whose levels are caused both by its own past values and the current and past values of other variables. Alternatively, we can think of the DGP as a process whose changes are caused by past changes and the distance the series is from its long-run relationship (i.e., whether approval is too high or too low for current levels of economic conditions.).
The assumption of weak exogeneity is critical. It is also a reasonable assumption in many cases in which this type of DGP has been proposed in political science. Beck (1992) notes that in many cases the “symmetric treatment” of political time series does not make sense and that assuming weak exogeneity is reasonable. However, one should always test for weak exogeneity. Weak exogeneity cannot be tested directly, but tests for parameter constancy are the usual approach (Engle et al. 1983).
Even longer time series may be needed to distinguish near-integrated prcesses from unit root processes when the series is estimated, as in the case of tests on the residuals from a cointegrating regression.
We use PC-NAIVE (Hendry et al. 1990) to conduct the Monte Carlo simulations.
The form of the test on the residuals from the cointegrating regression is the same as that used in the pretest phase; however, different critical values are used to reflect the fact that the residual series is estimated.
The cointegrating regression may contain multiple independent variables as well as trends or events. In cases where the relationship is symmetric, the choice of which variable to treat as (in)dependent is irrelevant.
This is a special case of cointegration and the one we consider.
Alternative representations of cointegrating relationships are often used in economics. These include the Johansen vector error correction model (1988) and the Engle and Yoo three-step estimator (1991), but these techniques have not been widely used in political science. But see, for example, Granato and West (1994) or Clarke and Stewart (1994, 1995) Testing for Cointegrating Relationships
For an exchange on this debate see Beck (1992), Williams (1992), and Smith (1992).
The results of Kremers et al. (1992), with T = 20, show a little less power—91.6 and 94.3% rejection frequencies for p = 0.05 for MacKinnon and t critical values, respectively.
Krause, 1998, Rivalry, Reciprocity, and the Dynamics of Presidential-Congressional Institution Building, 54, 325
The true DGP imposes homogeneity, that is, yt = xt in equilibrium. In practice, this assumption may be unreasonable. However, the results generalize in the absence of homogeneity. Note that the estimates of λ and β are unaffected whether we estimate Δyt = α + βΔxt + λ(yt– 1 – xt– 1) + γxt –1 + εt or Δyt = α + βΔxt + λ(yt –1 – θxt–1) + εt . DGPs without homogeneity can thus be similarly estimated by including an additional xt –1 term on the right (Banerjee et al. 1993).
Banerjee et al. (1986) developed and Kremers et al. (1992) corrected the distributional results for the ECM t ratio.
Macroideology is measured as the percentage of respondents claiming to be liberal from the total number of respondents claiming to be either liberal or conservative. It may be thought of as the balance of liberal, relative to conservative, ideological self-placement.
Dickey, 1979, Distribution of the Estimators for Autoregressive Time Series with a Unit Root, Journal of the American Statistical Association, 84, 427
Engle, 1991, Cointegrated Economic Time Series: An Overview with New Results, 237
The mean and variance of aunit root process depend on time, so that the mean does not converge and the series’ variance tends toward infinity. All references to integrated series in this paper refer more specifically to unit root processes.
Other tests for cointegration include tests in the context of the Engle-Yoo three-step procedure (1991) and the Johansen (1988) procedure. If weak exogeneity appears to be an unreasonable assumption, then these tests take simultaneity into account. See Granato and West (1994) and Krause (1998) for applications of the Johansen procedure and Clarke and Stewart (1994, 1995) for use of the Engle-Yoo three-step procedure.
We fix the variance of σε to 1 and vary σμ to simulate different values of q.
Adding additional lagged values of changes in the residuals to pick up any autocorrelation in the DF t test, using the augmented Dickey–Fuller test (ADF), does not provide any significant improvement on the inferences drawn from the DF t test. This follows from the fact that the omitted dynamics in the DF t test are from xt in the cointegrating regression itself, and not from lagged changes in the residuals. Thus the residuals from the cointegrating regression can be white noise even if the common factor restriction is not valid. As a result, adding lagged changes in the residuals—using the augmented Dickey–Fuller test—will not help to solve the problem. We did repeat the experiments using the ADF t test to verify this claim. In none of the six sets of experiments does the ADF perform significantly better than the DF t test. At most, the true null hypothesis is rejected with a 2% lower frequency. This occurs with the smallest ρ and the larger sample—as intuition would suggest. When ρ is near 1, the rejection frequencies (using either set of critical values) differ by an average of less than 0.20. It is also the case that the test is more conservative whether the null hypothesis is true or false so that inferences will be neither clearly better nor worse using the ADF t test. Clearly, however, if the DGP is more general than that proposed here, the analyst should check the residuals from the DF t test to see if additional lags need to be included in the test.
De Boef, 1999, Modeling Near-Integrated Data
It is important to note that the critical values for these test statistics vary given the presence of a constant or a trend in the cointegrating regression and the sample size. Further, these values are distinct from the critical values used when testing an individual series for a unit root. There is considerable controversy on unit root pretesting (Stock 1994; Elliott and Stock 1992). The critical values reported by MacKinnon (1990) are considered superior to those reported by Dickey and Fuller (1979) and Engle and Granger (1987).