RDP 8612: A Weekly Model of the Floating Australian Dollar 2. Univariate Time Series Modelling

(a) Methodology

Analysis of the univariate time series properties of exchange rate data provides a useful starting point, prior to econometric analysis. Finance theory suggests that ‘speculative prices’ ought to be represented by fairly simple time series processes, if financial markets are efficient. Naturally, these simple processes have important implications for the design of structural econometric models.^[3] Accordingly, this section develops the time series model to recover the lag structure, trend effects and unusual features of the exchange rate series.

The univariate models considered had the general form:

where e_t is the logarithm of the spot exchange rate (as a deviation from its initial value) of domestic currency in terms of foreign currency, u_t is assumed to be a weakly (covariance) stationary, possibly heteroskedatic error process, ζ(t) is a polynomial function of time, and ϕ(L) and η(L) are pth and qth order polynomials in the lag operator. If the characteristic function of ϕ(L) has d unit roots, then it can be factored to give ϕ(L) = ϕ^*(L)∇^d where ∇ is a difference operator, and the order of ϕ^*(L) is p = P – d. The model would then be an ARIMA (p, d, q) incorporating a time polynomial.

On the basis of likelihood ratio testing of nested ARIMA models, the exchange rate series in common with most other economic series is of a low order in the polynomials. To begin, the following model will be considered, higher order autoregressive terms being statistically irrelevant

A key issue is the value of ϕ₁ and ϕ₂, the roots of ϕ(L). If ϕ₁, is unity, ϕ(L) factorises to (1−ϕ₂L)∇, and if ϕ₁ and ϕ₂ are unity we get ∇². Tests are undertaken for these hypotheses, and if accepted, estimation is redone using the appropriately differenced form.

If there is to be any credibility in the deduced time series properties, the selected model should be checked for the following criteria (at least).

Presence of Unit Roots

One of the critical issues in modelling the autoregressive part of univariate time series models is the test procedure for the presence of unit roots. This issue is of especial importance when examining ‘speculative price’ data. An efficient capital market will fully reflect all existing and publically available information, and one would expect the data to be consistent with a random walk perhaps with a time-dependent drift (for example, see Granger and Morgenstern (1976)). This would imply (at least) one root lying on the unit circle. However, when estimating autoregressive parameters, one normally presumes that the time series is weakly (covariance) stationary, with characteristic roots lying outside the unit circle. But under the null hypothesis of a unit root, the time series is not stationary and the asymptotic variance of the series is not finitely defined; hence classical t and F tests cannot be undertaken. Fuller (1976), Dickey and Fuller (1979) and Hasza and Fuller (1979) derive the appropriate test statistics and their distributions for the null hypothesis of one or two unit roots of an autoregressive process. The distribution of the autoregressive parameter estimates under the null is decidedly skewed to the left of unity, and one should not be surprised to obtain estimates that are significantly less than unity on classical t tests.

In the case of a single unit root, the form of the test statistic is identical to that for the studentised t, and the method is a likelihood ratio test for the unit root (and if included, a zero mean and/or linear trend). In the regressions, the statistic is reported as τ_τ, and the distribution tables are obtained from Fuller (1976, page 373). For two unit roots, the form of the test statistic is similar to the F and two forms are reported: Φ₃(2) and Φ₃(4).^[4] The first is a likelihood ratio test of the two unit roots alone, and the second is a joint one of the roots and of a zero mean and no linear trend. The tables for the distributions of these statistics can be found in Hasza and Fuller (1979, page 1116).

If the null hypothesis of a unit root can not be rejected, non-stationarity of the underlying process likewise can not be rejected. By ignoring the problem, one can generate frequently ‘significant’ but spurious regression outcomes (see Granger and Newbold (1977) for spurious regressions of one random walk on another). One procedure for dealing with this type of non-stationarity is to prefilter all the data by regressing all variables on a polynomial of time, and to use the residuals as the data. This strategy is effective for that express purpose, but dubious if one is also concerned with forecasting and structural explanation. A perceived trend in a sample may well turn out to be a transitory feature of a more complex dynamic process. One objective of multivariate and structural analysis is to explain the causes of this perceived trend – the prefiltering strategy precludes such an analysis.

Uncorrelated Innovations

The error process should be a pure innovation with respect to information available just prior to the derivation of its elements. An implication of this is that the error process must pass a test of the null hypothesis of no autocorrelation implying that the error process cannot be predicted from its own past. The Box-Pierce statistic, distributed where q is the maximal lag, is used in this regard and is reported as BP(q). Of course, one does not rule out the possibility of influence from the current and lagged values of the error processes derived from other variables.

Homoscedastic Innovations

The error process should be checked for homoscedasticity. If the null is rejected, the estimate of the variance-covariance matrix of estimates is inconsistent. Heteroscedasticity in asset price equations is an important and distinct possibility because an influential element in asset choices is relative risk. Risk premia are difficult to specify and, generally, time varying. Misspecification of risk premia would be expected to be detected as heteroscedasticity in the error process.

Two tests are used: the Engle (1982) ARCH test for the particular autoregressive form of heteroscedasticity, which involves regressing the T squared residuals on r of their lags, with TR² being distributed ; and the less powerful White (1980) test for non-specific forms, which involves regressing the squared residuals on the products and cross-products of the k explanatory variables, with TR² being distributed . The test statistic is reported as WH(k(k+1)/2). The ARCH tests are undertaken because, if rejected, they may help to explain the existence of fat tails in the error distribution [see Engle (1982, p.992)]. The White correction for the variance-covariance matrix enables one to conduct valid inferences, provided the errors are serially uncorrelated.

Normality

The error process should be tested to see that it represents a random sample from a normal distribution. Otherwise, one could improve upon classical least squares methods. Typically, this is a difficult test to pass, and in many cases the test or the results are ignored. For samples less than fifty-one, the Shapiro-Wilk (1965) w statistic is computed, and for larger samples, the Kolmogorov D statistic is reported as KD (see Stephens (1974)).

Balanced Sample

The data sample used must be balanced in the sense that small subsets of observations must not have a substantial influence on the parameter estimates. Cook's (1979) D statistic is computed for each observation measuring the change in estimates resulting from the deletion of the observation. The statistic is distributed as an F(K,T-K). Such a test is invaluable for getting to know if there are any peculiar features in one's dataset which would require a deeper search into the causes. Such a test may indicate the need for the inclusion of dummy variables. The maximum D statistic across the sample is recorded in the tables.

Parameter Constancy

Tests for temporal stability of parameter estimates over the sample are essential if one is to accept the validity of constant parameter hypotheses. The Chow test provides the appropriate information on the assumption that the errors are homoscedatic. If heteroscedasticity is evident, the standard Chow test can seriously understate the Type 1 error. In that case, one can consult Schmidt and Sickles (1977) to get an approximate idea of the degree of the understatement.

(b) Empirical Results

Three samples were constructed for use in the regressions:

“1984–85” covered the period 14 December 1983 to 13 November 1985;
“1984” for 14 December 1983 to 21 November 1984; and
“1985” for 28 November 1984 to 13 November 1985.

The reason for the sub-sample breakdown was that at the beginning of 1985 monetary targetting was abandoned in favour of a ‘checklist’ approach. The announcement of the change came in February 1985, but the de facto switch probably occurred in the preceding months as it became apparent that monetary targets had become increasingly elusive.

The first set of regressions for the logarithm of the spot rate as in (1) are shown in Table 1.

The first regression (la) for the two years of the float explains 98 per cent of the variance of the spot rate. On the basis of simple t tests, the parameters (apart from the constant) are significantly different from zero. Similarly, t tests for ϕ₁ and ϕ₂ being unity significantly reject that hypothesis. The estimate of ϕ₁, 0.904, is substantially below that of Meese and Singleton (1984)'s estimates for the US dollar against Swiss francs, Canadian dollars and Deutschemarks (respectively 0.999, 0.982, 1.008). Such a result may appear to suggest that the Australian experience has been somewhat different and does not lend support to the Mussa (1979) contention that the logarithms of spot rates approximately obey a random walk. Indeed it may seem to lend support for the speculative activities of financial traders based upon univariate “technical” analysis. However, such conclusions are spurious and invalid. The Meese and Singleton results were obtained using 285 observations, compared to 101 in la. Even if the true value of ϕ₁ were not unity, it is well known that ordinary least squares estimates of positive autoregressive parameters are biased downwards in small samples – White (1961) shows the bias in a first order autoregression to be −2ϕ₁/T which under the null would explain about 0.03 of the difference. But, as discussed above, the appropriate test for a unit root involves a distribution of the parameter estimate that is seriously skewed to the left of unity. Fuller (1976, p.370) shows that the probability of < 1 given ϕ = 1 asymptotically approaches 0.6826. Applying the Fuller test, the statistic τ_t is seen to be unable to reject the unit root even at 10 per cent marginal significance (for 100 observations, the critical value of τ_τ at 10 per cent is −3.15 and at 90 per cent is −1.22). The classical t test acceptance of stationarity is evidently spurious and we can accept the null hypothesis of a single unit root conditional on the assumption of no heteroscedasticity.

Two tests involving two unit roots are undertaken. Φ₃(2) is an F-type statistic that jointly tests for ϕ₁ = ϕ₂ = 1, while Φ₃(4) jointly tests for ϕ₁ = ϕ₂ = 1, ζ₀ = ζ₁ = 0. The empirical percentiles for these two statistics for 100 observations at 5 per cent (1 per cent) are 9.58, (12.31) and 5.36, (6.74) respectively. Evidently, both null hypotheses are rejected.

From the Box-Pierce statistic, the marginal significance of 0.31 indicates that we can accept the null of no autocorrelation. This means that, if heteroscedasticity is present, White's (1979) correction for the variance-covariance matrix is appropriate for making inferences. On the ARCH test, heteroscedasticity is definitely present and may be consistent with an autoregressive form. The White test indicates heteroscedasticity at the 5 per cent significance level.

The application of the White correction, shown in line 1b, has one interesting effect. The single unit root test is unaffected, but the standard error of the second autoregressive parameter is increased by nearly 40 per cent. The implication is that, after heteroscedastic correction, the logarithm of the spot rate is, in fact, closely approximated by a random walk with drift.

Before accepting this conclusion, one needs to check the balance of the data. Cook's D statistics for each observation indicates a three week period (20 February 1985 – 6 March 1985) of unusual influence. This, of course, was the so called “MX Missile Crisis”. However, the F test indicates that the crisis did not significantly “imbalance” the data set. Further tests (based on Belsley, Kuh and Welsch (1980)'s DFBETA statistics) indicate that ζ₁, ϕ₁ and ϕ₂ were the parameters most affected, but none were significantly influenced. Nevertheless, a dununy variable for the “MX Missile Crisis” was introduced, and it turned out to have a value −.019 with standard error 0.010. Other estimates were marginally reduced and all inferences remained intact. This may allow the conclusion that exchange rate crises, such as this, are merely crises of confidence that can be represented as statistical noise.

The test for normality of residuals unfortunately fails based on Kolmogorov's D statistic. This does suggest that least squares estimates could be improved upon by robust techniques. The residuals were also leptokurtic (a kurtosis coefficient of 1.48 being registered), which is often practically consistent with a distribution having a sharper peak and higher tails than the normal. We already know that the ARCH statistic was significant, and an autoregressive form of heteroscedasticity, will generally be associated with fat tails. Further the skewness coefficient had a value of 0.94 implying a positive overhang in spot rate innovations. This suggests that there may be unspecified exogenous variables which would have imparted forces for appreciation in the model. While the non-normality of the residuals is a cause for concern, it would be very surprising if robust estimates lead to a rejection of the single unit root hypothesis. This conclusion is supported by the results of the sub-sample regressions.

The usual parameter constancy test of Chow accepts the null that 1984 and 1985 data produced insignificantly dissimilar parameter estimates. Given the existence of heteroscedasticity, one needs to be sure that the inaccuracy of the assigned significance level (say, 5 per cent) is not too great. From the tables in Schmidt and Sickles (1977) (with equal sample sizes of about 50, and a ratio of estimated variances of (.025/.005)² = .25) the understatement will not be serious. However, there are some interesting differences arising in the sub-sample analysis – viz regressions 2a, 2b, 3a and 3b in Table 1. The unit root tests give identical results, but the ARCH tests and the normality tests are quite different. Admittedly, these differences may arise because of the power loss in decreased sample size. Nevertheless, it is worth noting that heteroscedasticity is rejected in 1984, but not in 1985. Similarly the residuals are acceptably normal in 1984, but not in 1985. This coincidence of effects is consistent with the notion of heteroscedasticity being associated with fat tails. Hence, even though parameter estimates are not significantly different between 1984 and 1985, the nature of the error process, the second moment in particular, was significantly different. The 1985 characteristics are also seen to predominate in the aggregate sample.

Before considering more fundamental reasons for this substantial difference between 1984 and 1985, it may be reasonable to think that it is the proven non-stationarity of the exchange rate process that is the source of the increasing variance. It is therefore instructive to consult Table 2 where a similar exercise is undertaken with first differences of the exchange rate as the regressand. First differences eliminate the time trend and imply a first order model. The following conclusions are obtained – now no unit root, heteroscedasticity in the combined sample (probably coming from 1985 rather 1984), non-normality in the combined sample only, no excessively influential observations, no autocorrelation and last but not least virtually no explanatory power coming from the model especially after the White adjustment for heteroscedasticity. Indeed, only the constant term shows any hint of significance; this reflects the linear time trend in Table 1. But note that the dependent variable is measured as a deviation from its initial observation (−7.74–3 for samples 1984–85, 1984 and −4.65–3 for 1985). Hence we can conclude that there is a significantly non-zero drift in the random walk model. For the full sample, the random walk model with drift is reported in line lc where the implied drift is −2.8–3.

All in all, while first differencing certainly eliminates the unit root source of non-stationarity, it does not eradicate the heteroscedasticity and non-normality problem. Further, it appears to reduce the regressand to almost complete noise. Differencing is not necessarily the best solution to the non-stationarity issue. When one cannot reject the null of a unit root, it is not appropriate on classical principles to conclude that the value of the root must be fixed at unity in subsequent testing. The appropriate procedure is to undertake inference based on distributions conditional upon the non-stationarity induced by the existence of unit roots. Unfortunately, the appropriate distributions are pathologically dependent on the particular model and the theoretical developments are still few and far between. Given this limitation, differencing is a second-best strategy. If the purpose of the exercise is forecasting, and the true model involves a unit root, then the failure to difference will result in unwarranted (classical) confidence in the forecasts. If the true root is not unity, then differencing will give rise to forecasts that are too conservative. Overdifferencing is generally less dangerous than underdifferencing.

A final point on this issue, relevant to the next section, is that if the true model does not have a unit root, but one close by, then first differencing should produce a model with a first order moving average, the parameter of which should reflect the difference of the root from unity. A first order moving average was estimated in the equations for the change in the spot rate. For the two year sample, moving average parameter was estimated as −0.54 (0.39). The other parameters were only marginally affected, and so one is left with the conclusion that the analysis in the next section should allow for the possibility of a multivariate ARIMA (1, 1, 1) model.

Footnotes

Zellner and Palm (1974) demonstrate the representational equivalence of a structural dynamic model, a multivariate ARIMA model and a set of univariate ARIMA equations. The univariate time series processes necessarily imply restrictions for multivariate and structural analysis, the presumption being that the general economic theory model encompasses the time series model. [3]

For example, Φ₃(4) = where S² is the estimate of the error variance. [4]