Bootstrap-based unit root tests for higher order autoregressive models with GARCH(1, 1) errors

ABSTRACT

Bootstrap-based unit root tests are a viable alternative to asymptotic distribution-based procedures and, in some cases, are preferable because of the serious size distortions associated with the latter tests under certain situations. While several bootstrap-based unit root tests exist for autoregressive moving average processes with homoskedastic errors, only one such test is available when the innovations are conditionally heteroskedastic. The details for the exact implementation of this procedure are currently available only for the first order autoregressive processes. Monte-Carlo results are also published only for this limited case. In this paper we demonstrate how this procedure can be extended to higher order autoregressive processes through a transformed series used in augmented Dickey–Fuller unit root tests. We also investigate the finite sample properties for higher order processes through a Monte-Carlo study. Results show that the proposed tests have reasonable power and size properties.     

The sensitivity of robust unit root tests

The power properties of the rank-based Dickey–Fuller (DF) unit root test of Granger and Hallman [C. Granger and J. Hallman, Nonlinear transformations of integrated time series, J. Time Ser. Anal. 12 (1991), pp. 207–218] and the range unit root tests of Aparicio et al. [F. Aparicio, A. Escribano, and A. Siplos, Range unit root (RUR) tests: Robust against non-linearities, error distributions, structural breaks and outliers, J. Time Ser. Anal. 27 (2006), pp. 545–576] are considered when applied to near-integrated time series processes with differing initial conditions. The results obtained show the empirical powers of the tests to be generally robust to smaller deviations of the initial condition of the time series from its underlying deterministic component, particularly for more highly stationary processes. However, dramatic decreases in power are observed when either the mean or variance of the deviation of the initial condition is increased. The robustness of the rank- and range-based unit root tests and their higher power results relative to the seminal DF test have both been noted previously in the econometrics literature. These results are questioned by the findings of the present paper.     

Double unit root tests for cross-sectionally dependent panel data

This paper proposes various double unit root tests for cross-sectionally dependent panel data. The cross-sectional correlation is handled by the projection method [P.C.B. Phillips and D. Sul, Dynamic panel estimation and homogeneity testing under cross section dependence, Econom. J. 6 (2003), pp. 217–259; H.R. Moon and B. Perron, Testing for a unit root in panels with dynamic factors, J. Econom. 122 (2004), pp. 81–126] or the subtraction method [J. Bai and S. Ng, A PANIC attack on unit roots and cointegration, Econometrica 72 (2004), pp. 1127–1177]. Pooling or averaging is applied to combine results from different panel units. Also, to estimate autoregressive parameters the ordinary least squares estimation [D.P. Hasza and W.A. Fuller, Estimation for autoregressive processes with unit roots, Ann. Stat. 7 (1979), pp. 1106–1120] or the symmetric estimation [D.L. Sen and D.A. Dickey, Symmetric test for second differencing in univariate time series, J. Bus. Econ. Stat. 5 (1987), pp. 463–473] are used, and to adjust mean functions the ordinary mean adjustment or the recursive mean adjustment are used. Combinations of different methods in defactoring to eliminate the cross-sectional dependency, integrating results from panel units, estimating the parameters, and adjusting mean functions yields various available tests for double unit roots in panel data. Simple asymptotic distributions of the proposed test statistics are derived, which can be used to find critical values of the test statistics.

We perform a Monte Carlo experiment to compare the performance of these tests and to suggest optimal tests for a given panel data. Application of the proposed tests to a real data, the yearly export panel data sets of several Latin–American countries for the past 50 years, illustrates the usefulness of the proposed tests for panel data, in that they reveal stronger evidence of double unit roots than the componentwise double unit root tests of Hasza and Fuller [Estimation for autoregressive processes with unit roots, Ann. Stat. 7 (1979), pp. 1106–1120] or Sen and Dickey [Symmetric test for second differencing in univariate time series, J. Bus. Econ. Stat. 5 (1987), pp. 463–473].     

Normal tests for unit roots based on instrumental variable estimators

For estimating unit roots of autoregressive processes, we introduce a new instrumental variable (IV) method which discounts large values of regressors corresponding to the unit roots. Based on the IV estimator, we propose new unit root tests whose limiting null distributions are standard normal. Observation at time t is adjusted for mean recursively by the sample mean of observations up to the time t. The powers of the proposed tests are better than those of the Dickey–Fuller tests and are comparable to those of the tests based on the weighted symmetric estimator, which are known to have the best power against stationary alternatives.     

Unit-Root Tests and Asymmetric Adjustment With an Example Using the Term Structure of Interest Rates

This article develops critical values to test the null hypothesis of a unit root against the alternative of stationarity with asymmetric adjustment. Specific attention is paid to threshold and momentum threshold autoregressive processes. The standard Dickey–Fuller tests emerge as a special case. Within a reasonable range of adjustment parameters, the power of the new tests is shown to be greater than that of the corresponding Dickey–Fuller test. The use of the tests is illustrated using the term structure of interest rates. It is shown that the movements toward the long-run equilibrium relationship are best estimated as an asymmetric process.     

On the asymptotic distribution of the Dickey Fuller-GLS test statistic

In a very influential paper, Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836] show that no uniformly most powerful test for the unit root testing problem exits, derive the relevant power envelope and characterize a family of point-optimal tests. As a by-product, they also propose a ‘generalized least squares (GLS) detrended’ version of the conventional Dickey–Fuller test, denoted DF-GLS, that has since then become very popular among practitioners, much more so than the point-optimal tests. In view of this, it is quite strange to find that, while conjectured in Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836], so far there seems to be no formal proof of the asymptotic distribution of the DF-GLS test statistic. By providing three separate proofs, the current paper not only substantiates the required result, but also provides insight regarding the pros and cons of different methods of proof.     

Unit root tests: the autoregressive approach in comparison with the periodogram regression

In Monte Carlo sudies we investigate unit root tests in line with Dickey/Fuller (1979). In case of positively autocorrelated MA(1) residuals their experimental power is extremely poor. Next we compare different versions of periodogram regression suggested in the literature. Their experimental behaviour is investigated with fractionally integrated processes. It is demonstrated how unit root tests may be based on periodogram regression. There is simulation evidence that those tests may do better in terms of power than the autoregressive tests, especially when testing ARMA(1,1) series against a linear time trend.     

Performance of unit-root tests for non linear unit-root and partial unit-root processes

ABSTRACT

This paper investigates the finite-sample performance of the augmented Dickey–Fuller (ADF), Phillips–Perron (PP), momentum threshold autoregressive (M-TAR), Kapetanios–Shin–Snell (KSS), and the inf-t unit-root tests. Simulation results show that the ADF and KSS tests have better size, whereas other tests generate severe size distortions when the date-generating processes are non linear unit-root processes. In general, with regard to the combination of test powers with test sizes, the ADF and KSS tests are comparatively better than the PP, M-TAR, and inf-t tests; moreover, the inf-t test exhibits the poorest performance even for larger sample sizes.     

傅里叶型平滑结构突变下Dickey-Fuller检验的功效——兼论Shibor的平稳性

通过推导Dickey-Fuller检验功效函数,研究表明:即使中小型的傅里叶型结构突变,都会严重影响Dickey-Fuller检验的功效,从而使得含傅里叶型平滑结构突变的平稳过程被误判为单位根过程。使用3、6、9个月期和一年期Shibor日度数据发现:传统的ADF、PP、DF-GLS和KPSS几乎都指出Shibor是单位根过程;考虑平滑结构突变的单位根检验则在1%的显著性水平下拒绝了单位根的原假设,这表明Shibor是含结构突变的平稳过程。因此,预测Shibor和理解其动态行为必须考虑其结构突变特征。     

A Strongly Consistent Criterion to Decide Between I(1) and I(0) Processes Based on Different Convergence Rates

The usual procedure to determine whether a univariate time series is stationary or first-difference stationary is to perform some unit root test. In this article, an alternative methodology is presented that leads to a strongly consistent two-step criterion to estimate the number of unit roots. The criterion is based on estimating some autoregressive polynomials using regression procedures and exploiting the fact that the nonstationary roots converge at a faster rate than the stationary ones. The proposed procedure requires at most four regressions and is easy to implement. A simulation study demonstrates that it can perform significantly better in practice than the Dickey–Fuller and the generalized least squares (GLS)-detrended Dickey–Fuller tests.     

The asymptotic size and power of the augmented Dickey–Fuller test for a unit root

It is shown that the limiting distribution of the augmented Dickey–Fuller (ADF) test under the null hypothesis of a unit root is valid under a very general set of assumptions that goes far beyond the linear AR(∞) process assumption typically imposed. In essence, all that is required is that the error process driving the random walk possesses a continuous spectral density that is strictly positive. Furthermore, under the same weak assumptions, the limiting distribution of the ADF test is derived under the alternative of stationarity, and a theoretical explanation is given for the well-known empirical fact that the test's power is a decreasing function of the chosen autoregressive order p. The intuitive reason for the reduced power of the ADF test is that, as p tends to infinity, the p regressors become asymptotically collinear.     

Performance of seasonal unit root tests for monthly data

This paper uses Monte Carlo simulations to analyze the performance of several seasonal unit root tests for monthly time series. The tests are those of Dickey, Hasza and Fuller (DHF), Hylleberg, Engle, Granger and Yoo (HEGY), and Osborn, Chui, Smith and Birchenhall (OCSB). The unit root test of Dickey and Fuller (DF) is also considered. The results indicate that users have to be particularly cautious when applying the monthly version of the HEGY test. In general, the DHF and OCSB tests are preferable in terms of size and power, but these procedures may impose invalid restrictions. An empirical illustration is undertaken for UK two-digit industrial production indicators.     

Wavelet Improvement of the Over-Rejection of Unit Root Test Under GARCH Errors: An Application to Swedish Immigration Data

In this article, we use the wavelet technique to improve the over-rejection problem of the traditional Dickey–Fuller tests for unit root when the data is associated with volatility like the GARCH(1, 1) effect. The logic of this technique is based on the idea that the wavelet spectrum decomposition can separate out information of different frequencies in the data series. We prove that the asymptotic distribution of the test in the wavelet environment is still the same as the traditional Dickey–Fuller type of tests. The finite sample property is improved when the data suffers from GARCH error. The investigation of the size property and the finite sample distribution of the test is carried out by Monte Carlo experiment. An empirical example with data on the net immigration to Sweden during the period 1950–2000 is used to illustrate the performance of the wavelet improved test under GARCH errors. The results reveal that using the traditional Dickey–Fuller type of tests, the unit root hypothesis is rejected while our wavelet improved test do not reject as it is more robust to GARCH errors in finite samples.     

The Local Power of the CADF and CIPS Panel Unit Root Tests

Very little is known about the local power of second generation panel unit root tests that are robust to cross-section dependence. This article derives the local asymptotic power functions of the cross-section argumented Dickey–Fuller Cross-section Augmented Dickey-Fuller (CADF) and CIPS tests of Pesaran (2007), which are among the most popular tests around.     

Simulating Properties of the Likelihood Ratio Test for a Unit Root in an Explosive Second-Order Autoregression

This paper provides a means of accurately simulating explosive autoregressive processes and uses this method to analyze the distribution of the likelihood ratio test statistic for an explosive second-order autoregressive process of a unit root. While the standard Dickey–Fuller distribution is known to apply in this case, simulations of statistics in the explosive region are beset by the magnitude of the numbers involved, which cause numerical inaccuracies. This has previously constituted a bar on supporting asymptotic results by means of simulation, and analyzing the finite sample properties of tests in the explosive region.     

UNIT ROOT TESTING IN THE PRESENCE OF HEAVY‐TAILED GARCH ERRORS

We derive the asymptotic distributions of the Dickey–Fuller (DF) and augmented DF (ADF) tests for unit root processes with Generalized Autoregressive Conditional Heteroscedastic (GARCH) errors under fairly mild conditions. We show that the asymptotic distributions of the DF tests and ADF t‐type test are the same as those obtained in the independent and identically distributed Gaussian cases, regardless of whether the fourth moment of the underlying GARCH process is finite or not. Our results go beyond earlier ones by showing that the fourth moment condition on the scaled conditional errors is totally unnecessary. Some Monte Carlo simulations are provided to illustrate the finite‐sample‐size properties of the tests.     

Bootstrapping the augmented Dickey–Fuller test for unit root using the MDIC

In this paper, we consider the bootstrap procedure for the augmented Dickey–Fuller (ADF) unit root test by implementing the modified divergence information criterion (MDIC, Mantalos et al. [An improved divergence information criterion for the determination of the order of an AR process, Commun. Statist. Comput. Simul. 39(5) (2010a), pp. 865–879; Forecasting ARMA models: A comparative study of information criteria focusing on MDIC, J. Statist. Comput. Simul. 80(1) (2010b), pp. 61–73]) for the selection of the optimum number of lags in the estimated model. The asymptotic distribution of the resulting bootstrap ADF/MDIC test is established and its finite sample performance is investigated through Monte-Carlo simulations. The proposed bootstrap tests are found to have finite sample sizes that are generally much closer to their nominal values, than those tests that rely on other information criteria, like the Akaike information criterion [H. Akaike, Information theory and an extension of the maximum likelihood principle, in Proceedings of the 2nd International Symposium on Information Theory, B.N. Petrov and F. Csáki, eds., Akademiai Kaido, Budapest, 1973, pp. 267–281]. The simulations reveal that the proposed procedure is quite satisfactory even for models with large negative moving average coefficients.     

Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

A number of recent papers have focused on the problem of testing for a unit root in the case where the driving shocks may be unconditionally heteroskedastic. These papers have, however, taken the lag length in the unit root test regression to be a deterministic function of the sample size, rather than data-determined, the latter being standard empirical practice. We investigate the finite sample impact of unconditional heteroskedasticity on conventional data-dependent lag selection methods in augmented Dickey–Fuller type regressions and propose new lag selection criteria which allow for unconditional heteroskedasticity. Standard lag selection methods are shown to have a tendency to over-fit the lag order under heteroskedasticity, resulting in significant power losses in the (wild bootstrap implementation of the) augmented Dickey–Fuller tests under the alternative. The proposed new lag selection criteria are shown to avoid this problem yet deliver unit root tests with almost identical finite sample properties as the corresponding tests based on conventional lag selection when the shocks are homoskedastic.     

Unit roots and structural breaks: The case of India 1900-1988

SUMMARY This paper tests the hypothesis of difference stationarity of macro-economic time series against the alternative of trend stationarity, with and without allowing for possible structural breaks. The methodologies used are that of Dickey and Fuller familiarized by Nelson and Plosser, and that of dummy variables familiarized by Perron, including the Zivot and Andrews extension of Perron's tests. We have chosen 12 macro-economic variables in the Indian economy during the period 1900-1988 for this study. A study of this nature has not previously been undertaken for the Indian economy. The conventional Dickey-Fuller methodology without allowing for structural breaks cannot reject the unit root hypothesis (URH) for any series. Allowing for exogenous breaks in level and rate of growth in the years 1914, 1939 and 1951, Perron's tests reject the URH for three series after 1951, i.e. the year of introduction of economic planning in India. The Zivot and Andrews tests for endogenous breaks confirm the Perron tests and lead to the rejection of the URH for three more series.     

New innovational outlier unit root test with a break at an unknown time

The Perron test which is based on a Dickey–Fuller test regression is a commonly employed approach to test for a unit root in the presence of a structural break of unknown timing. In the case of an innovational outlier (IO), the Perron test tends to exhibit spurious rejections in finite samples when the break occurs under the null hypothesis. In the present paper, a new Perron-type IO unit root test is developed. It is shown in Monte Carlo experiments that the new test does not over-reject the null hypothesis. Even for the case of a level and slope break for trending data, the empirical size is near its nominal level. The test distribution equals the case of a known break date. Furthermore, the test is able to identify the true break date very accurately even for small breaks. As an application serves the Nelson–Plosser data set.