RDP 2003-03: Australia's Medium-Run Exchange Rate: A Macroeconomic Balance Approach Appendix C: Time Series Properties and Model Selection

Unit Root Tests

We tested for the presence of a unit root in all series. Both the augmented Dickey-Fuller test (ADF) and the Phillips-Perron (PP) tests suggest that all the data are I(1) with one exception. We can reject empirically that world output has a unit root over our sample period. However, we assume that the series is I(1) a priori on economic grounds.

Model Selection

As our data are I(1) we have to test for cointegration between the volume components and possible explanatory variables. Our chosen procedure is the one suggested by Johansen (1991, 1995). However, this procedure may fail to detect genuine cointegration for small sample sizes. We therefore tested whether the deviations from the Johansen long-run relationships were stationary using ADF and PP tests. We also tested whether the residuals from each equation in the VECM are white noise. Finally, we compared our results from the VECM estimation with ECM and OLS estimates to ensure the signs and magnitudes were similar.

The number of cointegrating equations

Assuming that we are satisfied that all of the relevant variables are I(1), then we require that there be some cointegrating vector that will combine them in such a way as to make them stationary. If we fail to find a cointegrating relationship, there is little we can do in modelling these variables. Hence, our choice of lag length and explanatory variables was partly driven by the need to find a cointegrating vector, rather than traditional model selection criteria such as the AIC or BIC.

Moreover, it is unclear how to treat a relationship where there may exist more than one cointegrating vector. This consideration also influenced our choice of lag length and explanatory variables. We present the results of the Johansen procedure for various lag lengths in Table C2.

Before we apply the Johansen procedure we have to make some assumptions regarding the deterministic trends in the VAR. A general form of the VECM is given below.

where y is a vector of endogenous variables, i is a vector of ones, t is a time trend, and π = αβ'. The terms in brackets represent the long-run relationship or cointegration space. In this general specification it is possible to have a constant and/or a time trend in both cointegration space and in the error correction system. For our purposes we assume that δ₁ = δ₂ = 0. That is, we allow only for a constant but no time trend. Because the dependent variables are expressed in differences, using a time trend implies that this variable changes at an increasing rate with time, that is, a quadratic trend. On economic grounds, we find it difficult to justify using a quadratic trend to model our variables.

We use the Trace test proposed by Johansen (1995) to test for the number of non-zero characteristic roots of π (which is equivalent to the number of cointegrating relationships). For the hypothesis that there are at most r distinct cointegrating vectors, the Trace test statistic is given by:

where are the eigenvalues of π sorted from highest to lowest. The critical values used in these tests are those from Osterwald-Lenum (1992).

Robustness to lag length

While our choice of lag length was influenced by the results of the Johansen procedure, we took great care not to impose a lag structure to which our point estimates were not robust. It may have been the case that we chose a lag structure because it allowed for a single cointegrating vector over a lag structure that was preferred on the basis of the AIC and BIC. However, in these cases we checked that the point estimates under our chosen lag structure were robust to this lag structure. The results of these tests are presented in Table C3.

Are deviations from the estimated long-run stationary?

In one particular case (service import volumes) we found no evidence of a cointegrating vector using the Johansen methodology. However, when we combined the variables according to the ‘candidate cointegrating vector’ (i.e., eigenvector corresponding to the largest eigenvalue), we found the deviations from the long-run relationship to be stationary. This, in spite of the fact that the ADF and PP tests have low power and thus have difficulty in rejecting the null of a unit root in favour of stationarity. In this case we went against the results of the Johansen procedure and imposed the ‘candidate cointegrating vector’ as a cointegrating relationship. A possible justification is that the sample size in this particular case may have been too low for the Johansen procedure to give reliable results.

In general, we tested whether each of our cointegrating relationships gave stationary deviations from the long run they implied. The results are presented in Table C4.

Are the VECM residuals white noise?

Another consideration when choosing our lag structure was to ensure that the residuals from each equation in the VECM were white noise. We tried to choose a lag structure that satisfied this requirement. In order to test this we visually inspected the ACF and PACF of each residual series. The results of these tests are presented in Table C5.

Consistency with ECM and OLS

For some choices of explanatory variables and lag length we found that the results given by the VECM were inconsistent with those given by the ECM and OLS. We were particularly concerned about sign changes, although these were rare. However, we sought to have a VECM specification that had the same sign as the ECM and OLS and a similar order of magnitude and significance test results to the ECM. In Table C6 we present each VECM estimate and the corresponding ECM estimate.^[23]

Footnote

The estimates with (static) OLS are not reported here, but the results are in line with those from the VECM and the ECM (results available on request from the authors). [23]