A new test for simultaneous estimation of unit roots and GARCH risk in the presence of stationary conditional heteroscedasticity disturbances

Abstract

According to previous research, standard unit root tests are considered robust to stationary GARCH distortions. These conclusions are in fact correct when the number of observations is extraordinarily high. However, simulation experiments in this study, using more normal sample sizes, reveal that eight of the most commonly applied unit root tests exhibit considerable bias in the size in the presence of fairly moderate GARCH distortions. As a remedy for the disturbances from GARCH, this article presents size-corrected unbiased critical values for all these examined tests. Nevertheless there is still reduced power in the presence of stationary GARCH distortions. As a solution, a completely new test is formulated which simultaneously models unit roots and the interconnected parameters of GARCH risk. For empirically relevant sample sizes, this new test exhibits superior size and power properties compared with all the traditional unit root tests in the presence of GARCH disturbances.

Acknowledgement

I would gratefully like to acknowledge the financial support provided by Sparbankernas Forskningsstiftelse.

Notes

¹ Bollerslev's GARCH model was independently proposed by Taylor (1986).

² See a simple proof in Brooks (2002), pp. 453–4.

³ There are many variants of ARCH models, e.g. ABARCH, APARCH, ARCH, ATACH, CC–MGARCH, CGARCH, EARCH, EGARCH, FIGARCH, GARCH, GARCH–M, GJR–GARCH, GQ–ARCH, IGARCH, LOG–ARCH, NARCH, NARCHK, NPARCH, NPARCHK, PARCH, PGARCH, POWER GARCH, SAARCH, SDGARCH, STAR–GARCH, STAR–STGARCH, TARCH, TGARCH, TPARCH, VC–MGARCH, VS–GARCH. All of these models are nonrobust and sensitive to whether the true mean equation process is integrated or not.

⁴ Granger and Newbold (1974) reached their conclusions by simulation. The asymptotic distribution theory valid for their experiment was worked out more than a decade later by Phillips (1986). A concise presentation of these developments can be found in Granger (2001).

⁵ In the present article eight of the most commonly applied unit root tests are studied: the Dickey–Fuller test (1979), Phillips–Perron's test (1988), Elliott, Rothenberg and Stock's Dickey–Fuller–GLS test (1996), Elliott, Rothenberg and Stock's Point-Optimal test (1996) and Ng–Perron's (2001) modified unit root tests , MSB ^D and .

⁶ ARMA–GARCH is the general form of the test. However, the reason for the notation AR(MA) is due to the fact that the true DGP in the experiments follows AR and not ARMA processes. This traditional approach should not be penalised in the evaluation. Therefore, only AR–GARCH models are estimated.

⁷ When measuring the size, critical values from MacKinnon's formula (Engle and Granger, 1991) are applied.

⁸ When measuring the power, size-corrected critical values are applied which implies that the size is adjusted in order to reach unbiasedness (5%).

⁹ This is considered the most commonly applied approach for estimating the GARCH parameters; see e.g. Enders (2003).

¹⁰ ADF–BEST is an abbreviation for the unit root test constructed by Dickey and Fuller, while best is an acronym for Bollerslev (the initiator of GARCH), Engle (the inventor of ARCH) and Sjölander test. The ‘A's in ADF–BEST represents the Augmentation part of the augmented Dickey–Fuller test (ADF test).

¹¹ AR(MA)–GARCH provides valid garch estimates only in cases when the variable is clearly stationary (not near integrated or integrated) and for large sample sizes. However, a priori we cannot know whether the true process is integrated or not since unit root estimation is misleading in the presence of GARCH. Furthermore, it is worthwhile to note that a common ARCH–LM test only detects the existence of ARCH-distortions; it does not measure the magnitudes of the economically interpretable GARCH parameters.

¹² In Pantula's article ρ = 1 is the unit root null hypothesis. α is the ARCH parameter and δ is the GARCH parameter.

¹³ The ‘A's in ADF–BEST represents the augmentation part of the ADF test.

¹⁴ Thus, this is the same augmentation property as in the ADF test. All information criteria have their pros and cons and it is not obvious which information criteria to choose since there is a trade-off between asymptotic efficiency and consistency. However, if many different information criteria support the same model, this makes the choice simpler. Therefore, it is suggested that different information criteria are applied simultaneously and that the residuals are studied carefully by diagnostic checks and by misspecification tests. Furthermore, if the true model is not among the studied ones, it is impossible to detect this problem using information criteria. Information criteria must be combined with diagnostic checks and misspecification tests to identify such problems.

¹⁵ Actually, γ = −2 is also a unit root. However, in economics, −1 ≤ γ ≤ 0 is the relevant nonoscillating area. Therefore, only H ₀: γ = 0 against H ₁: γ < 0 is tested, since this one-sided test approach leads to higher statistical power.

¹⁶ Franses et al . (2004).

¹⁷ See proof in Appendix A2.

¹⁸ See proof in Appendix A2.

¹⁹ Under the null hypothesis, this is considered an unrealistic process in economics (Elder and Kennedy, 2001).

²⁰ See Chan and Mcaleer (2003).

²¹ This article is not demonstrating the model selection properties of ADF–BEST. Therefore, an approach similar to what is found in Perron (1988), Dolado et al . (1990), Holden and Perman (1994), Ayat and Burridge (2000) and Enders (2003) will not be illustrated here.

²² Quasi-maximum likelihood covariances and standard errors is described by Bollerslev and Wooldridge (1992). However, note that QML only affects the inference, so that the estimated covariance matrix will be altered, not the parameter estimates.

²³ See e.g. Elder and Kennedy (2001) who argues that it is unrealistic that an economic time series exhibits ever increasing (or decreasing) rate of change, as is postulated by a unit root with a deterministic time trend.

²⁴ See e.g. Murray and Papell (2002), Amara and Papell (2006) or Enders (2003).

²⁵ More support for Elder and Kennedy's view can be found in Harris and Sollis (2003) and in Davison and Mackinnon (2004).

²⁶ The purpose of this article is to illustrate the problem of GARCH distortions for unit root tests. Therefore, the test specification includes a constant and no trend since this is a common starting point when testing economic theories such as the long-run PPP.

²⁷ This is an evaluation approach applied in, for instance, Edgerton and Shukur (1999).

²⁸ The aforementioned discussion in this paragraph is based on Brooks (2002).

²⁹ Due to the fact that the mean equation of the DGP follows an AR process, it is natural to choose the AR–GARCH test instead of ARMA–GARCH.

³⁰ These parameter estimates were chosen arbitrarily and this is definitely not a special case. Actually, the gains from applying ADF–BEST would have been even higher if the chosen α parameter had been higher in magnitude, or if the entire conditional heteroscedasticity process was closer to being integrated.

³¹ Inspect whether any of the estimated coefficients are negative, since models with negative coefficients are irrelevant for the study. In such cases the examined model is excluded from further analysis because negative variance makes no sense.

³² See the section ‘Analysis of all the examined GARCH tests: simulation Evidence of Propositions 3 and 6's.

³³ Due to the fact that this ranked variable by pure definition is measured on an ordinal scale, the accurate central tendency measure is the median.

³⁴ See the section ‘Analysis of all the examined GARCH tests: simulation Evidence of Propositions 3 and 6's.

³⁵ However, in this article QML is applied in order to obtain the robust SEs by Bollerslev and Wooldridge (1992). This method is easily applied since it is available in many econometric software packages.

³⁶ This approach requires pre-estimation of the GARCH magnitudes using ADF–BEST.

³⁷ ADF–BEST is superior compared with the traditional tests regarding size estimation. However, when applying the new size-adjusted critical values from this article (Appendix A1), all examined tests are unbiased.

³⁸ This is done since it resembles the real-world situation where this mean equation process is unknown.

³⁹ If the purpose is solely to estimate reliable GARCH risk parameters, we stop after Step 1. However, if the purpose is testing for unit roots we move on to step 2. Apply QLM SEs if this is supported by the applied program package. QML covariances and SEs is described by Bollerslev and Wooldridge (1992). However, note that QML only affects the inference, so that the estimated covariance matrix will be altered, not the parameter estimates.

⁴⁰ The evaluation of the size follows the performance criteria set up by Edgerton and Shukur (1999).

⁴¹ Calculating the average size bias and ranking the grand means would hardly be a good solution since this approach would not compare the number of times a size is better in relation to other tests. Outliers would also affect this mean substantially. Thus, the median handles this problem better. However, no matter what central tendency measure is calculated, mean or median, a very similar pattern emerges, so the choice of performance measure does not affect the overall conclusions in any decisive way.