In particular, we develop the asymptotic theory of the inf-t test for the null hypothesis of a unit root in a wide class of nonlinear autoregressive models with parameters identified only under the alternative of stationarity. Therefore, it is well expected that the zero asymptotics depends on the limit of the random parameter space. This dependence of the limit parameter space on the asymptotic critical values has never been properly addressed in the literature.
Therefore, critical test values can be calculated for any type of transition function and parameter space specification. We first look at the prototype version of the transient AR models that will be considered in the paper. Therefore, the usual assumption on the compactness of the parameter space is not appropriate in this case.
In the previous literature, none of the marginal null distributions were obtained under appropriate assumptions about the parameter space. In the unit root literature, it is routinely assumed that the fourth conditional moment of the innovation sequence is bounded.
Test Statistics and Asymptotic Theory
It is very important to note that the space for the parameter θ in the processes Mn and M is not restricted to be a compact space. This is crucial for our asymptotic theories, which allow for the parameter space remaining to be random in the limit. Here and elsewhere in the paper we denote by ˆσ2n(θ) the error variance estimate given by ˆσ2n(θ) = (1/n)Pn.
To distinguish them from the unnormalized original parameters, we let µn and κn be the location and scale parameters, respectively, in the normalized transition function for the TAR, LSTAR, and ESTAR models. The actual implementation of our test in the TAR, LSTAR and ESTAR models is quite simple, especially thanks to the result in Corollary 3.3. We now establish the consistency of our inf-ttest, which is given in the following statement.
Extensions
The critical values for the inf-t test with the parameter spaces in (22) and (23) are tabulated in Table 1 for the TAR, LSTAR, and ESTAR models. As a result, it provides much more believable and realistic transition dynamics for AR models. For the D-TAR and D-LSTAR models, in this paper we mainly deal with the symmetry restriction ρ1=ρ2= 1.
The motivation for the definition of the transition term in (31) and (32) is exactly the same as the one given for (6) and (7). The parameter spaces for the location parameters µi's and the scale parameters κi's can be set as in (22) and (23) for these models. For the TAR, LSTAR, ESTAR, D-TAR, and D-LSTAR models with intercept, we need to set the parameter spaces for the parametersνi's andρi's.
The critical values of the inf-test are listed in table 1 for the D-TAR and D-LSTAR models, and in table 2 for the TAR, LSTAR, ESTAR, D-TAR and D-LSTAR models with intercept. All the critical values are calculated using (22) and (23), respectively, as the parameter space for the location and scale parameters in these models. Finally, the consistency of the inf test can be established for the general models exactly as in Proposition 3.4.
In particular, we can show that our tests are valid for the models with (out) driven by a general linear process. For the order of the fitted autoregression, we write p=pn to make it explicit that it is a function of the sample size and assume. The condition in Assumption 4.4 is very mild and the same as that used in Chang and Park (2002) to derive the asymptotics of the ADF unit root test.
However, the obtained bound for the cross product of the transition function and the various delay terms is tighter in Lemma 4.3, compared to the one given in Lemma 3.1.
Monte Carlo Experiment
The result in Lemma 4.3 ensures asymptotic orthogonality between the transition term and all lagged differences, the number of which increases as the sample size is large. The tighter bound obtained in Lemma 4.3 requires the differentiability of the transition function introduced in Assumption 4.5. For some of the models used in our empirical applications, we note that the size-corrected critical values can differ moderately from the asymptotic critical values of Tables 1 and 2 .
Of the three transitional AR models we consider in our simulations, the TAR model produces the most noticeable small sample size distortions for the inf-t test. The distortions of the inf-t tests are less significant in the LSTAR model, and much less so in the ESTAR model. To evaluate the powers of the inf ttests, we set λ < 0 in the same model (36) to generate the time series.
Our simulation results for the magnitude-adjusted powers of the inf-t tests are given in Tables 4 and 5. When the stationary regime has less stable roots, including λ=−1 and −0.5, the inf-t test outperforms the ADF test essentially regardless of sample size and location parameter. When the root becomes close to unity with λ= -0.1, the power of the inf-t test in the TAR model is smaller than that of the ADF test with a small sample size.
But even in such a case, the power of the inf-ttest ultimately increases as the sample size increases. This is reasonable and expected since the transition function of the LSTAR model approaches that of the TAR model as the scale parameter becomes large. These are the values of κ for which the difference in the power properties of the inf-t test and the ADF test is most apparent.
In this sense, the advantage of the inf-t test in the ESTAR model is not as obvious as in the TAR and LSTAR models, and it can vary widely across the models with different parameter values.
Empirical Results
For this reason, it would be more informative to assess the power of the tests in each empirical application using the models with estimated parameters or the parameter values that appear to be relevant. Since the 5% asymptotic critical values for the two tests are -2.86 and -3.41 respectively, neither rejects the null hypothesis. Since the 5% asymptotic critical values for the two tests are -3.39 and -3.45, respectively, both tests significantly reject the unit root hypothesis.
In summary, using size-corrected critical values does not change the results of the ADF and inf-t tests. A comparison of the magnitudes of the adjusted powers reported in the bottom half of Table 6, however, shows a clear advantage of the inf-ttest over the ADF test, at least for this particular case. As in the case of the unemployment rate, the null hypothesis of a unit root is not rejected based on the ADF tests at the 5% significance level.
The conclusion does not change even if we use size-adjusted critical values instead of asymptotic critical values. Furthermore, a power comparison based on the estimated models shows a higher power property of the inf-t test compared to the ADF test. Our third empirical example is the application of inf-t tests based on D-TAR and D-LSTAR to a target area exchange rate model.
During the 1980s and 1990s, exchange rates within the European Monetary System (EMS) were subject to interventions by central banks based on ranges set at ± 2.25% around the central rate in most Member States. However, these results are not very surprising considering that the series is by definition limited to a band, as well as the availability of the series with a large sample size. To demonstrate such a possibility, we consider a subsample based on the series up to the time of the third realignment (in 1982) in the total of six realignments.
The higher size-adjusted powers of the inf-t tests, relative to those of the ADF tests, are also consistent with the fact that only the inf-t tests can reject the unit root in this subsample case.
Conclusion
For this series, both the ADF tests with a constant and with a trend reject the unit root hypothesis at the 5% significance level. For this subsample (n = 819), the ADF tests fail to reject the null hypothesis of a unit root. On the other hand, both the inf-t tests applied to the D-TAR and D-LSTAR models give values of -4.26 and -4.53, respectively, rejecting the unit root hypothesis quite strongly.
Such a test is motivated by the fact that parameters in transient AR models are not identified under the null hypothesis of a unit root. To illustrate, we consider several economic time series and test a unit root against several popular transient AR models. The unit root model is unequivocally rejected for all series we investigated, in favor of stationary transitory AR models.
Proof of Lemma A1 Fixθ0 ∈Rm arbitrarily and letπδ and πδ be the regular functions on Rsuch thatπδ ≤π(x, θ)≤πδ(x) for allkθ−θ0k< δ, and such that()−πδ(x) As in the proof of Lemma A1 in Park and Phillips (2001), we can write π = π+ −π−, where π+ and π− are the positive and negative parts of π, respectively, and consider them separately. To prove part (b) we look at υπδ and υπ2δ, for which the rest of the proof goes completely analogously to that of part (a).
For arbitrarily chosen θ0 ∈ Rm, let πδ and πδ be the regular functions such that πδ(x)≤π(x, θ)≤πδ(x) for allkθ−θ0k< δ, and. Proof of Lemma A4 As in the proof of Lemma A3, it suffices to show that sup. Proof of Lemma 4.3 The result can be obtained using the similar argument used in the proof of Lemma 3.1 and using the result from Lemma A4 instead of Lemma A3.
The extension for the model in (29) is simple and can be done as shown in the proof of Theorem 4.1. The desired results follow from combining the results for An(θ), Bn(θ) and ˆσn2(θ) and using the limit distribution in the proof of Theorem 3.2. Carrasco (2004a) “Tests for Unit-Root versus Threshold Specification with an Application to the Purchasing Power Parity Relationship,”.
