Statistical Papers (2008) 49:581–593 DOI 10.1007/s00362-006-0035-2 R E G U L A R A RT I C L E
A note on testing for nonstationarity in autoregressive processes with level dependent conditional heteroskedasticity Paulo M. M. Rodrigues · Antonio Rubia
Received: 28 October 2005 / Revised: 30 October 2006 / Published online: 30 November 2006 © Springer-Verlag 2006
Abstract In this paper, we investigate the empirical distribution and the statistical properties of maximum likelihood (ML) unit-root t-statistics computed from data sampled from a first-order autoregressive (AR) process with level-dependent conditional heteroskedasticity (LDCH). This issue is of particular importance for applications on interest rate time-series. Unfortunately, the extent of the technical complexity related associated to LDCH patterns does not offer a feasible theoretical analysis, and there is no formal knowledge about the finite-sample size and power behaviour or the ML test for this context. Our analysis provides valuable guidelines for applied work and directions for future work. Keywords Maximum likelihood estimation · Nonstationarity · Volatility · Interest rates 1 Introduction In this paper, we discuss finite-sample size and power behaviour of a maximum likelihood (ML) based autoregressive (AR) unit-root t-test. The distinctive feature of our study is that data is sampled from a data generating process (DGP) in which the errors exhibit level-dependent conditional heteroskedasticity
P. M. M. Rodrigues Faculty of Economics, University of Algarve, Faro, Portugal e-mail:
[email protected] A. Rubia (B) Department of Financial Economics, University of Alicante, Campus de San Vicente, CP 03080, Alicante, Spain e-mail:
[email protected]
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(henceforth LDCH). This special form of heteroskedasticity is particularly important when modelling short-term interest rates. Specifically, we consider a fairly general family of LCDH models in which the conditional variance ht of the error term, say εt = yt − Et−1 (yt ) , is a measurable function with almost-surely positive paths given by 2γ
ht = σt2 (λ) yt−1 ;
γ >0
(1)
and λ being a vector of unknown parameters. We focus our attention on AR models such that Et−1 (yt ) = c + φyt−1 , and investigate the properties of the t-statistic under the null hypothesis H0 : φ = 1 against H0 : |φ| < 1 when all the parameters in the model are jointly estimated by ML. Our analysis is relevant, though not necessarily limited to, theoretical and empirical applications on short-term interest rates. The vast majority of models in this literature assume that this variable follows a mean-reverting process with time-varying volatility which is positively correlated with the level; see, among others, the equilibrium model of Cox et al. (1985)[CIR], and its generalisation in Chan et al. (1992)[CKLS]. Empirical evidence largely supports the suitability of LCDV models over standard alternatives in financial modelling such as GARCH or stochastic volatility processes.1 However, the hypothesis of mean reversion, central for most of these models, finds little (if any) statistical support, with unit root tests typically not rejecting the null. This unappealing feature besides lacking the theoretical support in the dynamics of integrated processes in this context is incompatible with the economic properties of short-term interest-rates.2 Hence, it has been argued that evidence of nonstationary behaviour of these series is spurious and due to the statistical failure of the testing procedure. This is a sensible explanation since the properties of unit root tests are likely to be influenced by LDCH patterns. In fact, LDCH models constitute such a degree of complexity for theoretical analysis that the particular distribution of the relevant test statistics under the null of integration has yet to be derived in literature. 3 In other words, when applying a unit root test on real data one does not have a formal understanding on (i) what the correct critical values for the test are, (ii) whether the critical values are sensitive to the sample characteristics such as LDCH patterns (i.e., if the distribution is invariant), and (iii) whether the testing procedure has rea1 When using GARCH models, explosive patterns are often found, thereby suggesting model misspecification; see, for instance, Engle et al. (1987). 2 In contrast to other financial assets, market efficiency does not imply random walk behavior of
short-term interest rates. Furthermore, the random walk model induces implausible descriptions of the term structure; see Campbell et al. (1997). 3
While there exists a considerable body of literature related to unit-root testing and standard GARCH errors (see Kim and Schmidt 1993; Ling and Li 1998; Seo 1999; Li et al. 2002, among others), the results related to LDCH models are very limited. Ling (2004) shows some theoretical results for maximum likelihood estimation in an alternative specification of ht considered above related to the linear case γ = 1. Rodrigues and Rubia (2005) discuss through experimental analysis the performance of several least-squares unit root tests, such as the well-known Dickey–Fuller test.
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sonable power to reject the null of a unit root (inadmissible from an economic perspective). The main aim of this paper is to provide insight on these three key questions through Monte Carlo simulations. In absence of a theoretical framework, the results put forward in our analysis set valuable guidelines for applied work and serve as a reference for further theoretical results. Our results can be summarised as follows. First, when the innovations of an autoregressive process are driven by LDCH models with constant scale parameter, σt (λ) = σ , the distribution under the null of integration of the ML t-statistics of the estimated AR coefficient is accurately approximated by the Dickey–Fuller distribution even in small samples. This result is robust and holds for a wide range of economically plausible (and non-plausible) DGPs. Therefore, practitioners could rely on the critical values from the Dickey–Fuller distribution to conduct inference in applied work. On the other hand, the power behaviour of the test may strongly be affected by LDCH patterns. In particular, the distortion is remarkably pervasive in the near-integration context, the most likely situation for practical applications, as interest rates represent highly-persistent series. We observe power reductions due to LDCH that reach 50% in our analysis. Therefore, it is not very surprising to find it particularly difficult to reject the null of integration. Finally, if we allow for a more general class of LDCH errors by assuming time-varying dynamics in σt (λ) , such as GARCH(1,1) dynamics (as is usually done in applied work), the distribution of the test is not longer invariant and depends at least on the short- and long-run dependence generated by GARCH effects. A similar result has been discussed when considering GARCH effects alone in the context of ML estimation; see Ling and Li (1998, 2003) and Seo (1999). This aspect is crucial for applied work since inference is implicitly conducted as if the distribution is invariant, which unfortunately does not seem to be the case. Statistical inference may not be as straightforward as believed before. The remainder of the paper is organised as follows. In Sect. 2, the general time series processes analysed in this paper are described. Section 3 presents the main features of the simulation process and the main results. Finally, Sect. 4 draws on important conclusions. 2 Maximum likelihood-based unit root testing with LDCH models Let (, F, P) be a probability space and {Ft } a sequence of increasing σ -fields. Consider a first-order AR model with LDCH patterns. In particular, we begin our analysis by assuming the discrete time version of the well-known CKLS model: yt = µ + αyt−1 + εt εt = σ |yt−1 |γ ηt ; ηt ∼ iid (0, 1) ;
γ >0
(2) (3)
where µ, σ > 0, and εt being measurable with respect to Ft . This specification is one of the most popular frameworks in the empirical modelling of interest rate time-series. In the sequel, we shall mantain the assumption that the observable
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process {yt }T t=1 is strictly positive for all t (e.g., the short-term interest rate timeseries) and simply set εt = σ yt−1 γ ηt . The σ parameter is a scale factor, while γ controls the degree to which the interest rate level influences the volatility and, hence, non-zero values of this parameter leads to LDCH dynamics. Chan et al. (1992) argue that the particular value of γ is the most important feature distinguishing interest rate models. By placing restrictions on γ , a number of short-term interest rate models arise as particular cases. In a wide sense, this model simply generalises the homoskedastic AR(1) process which arises as a particular case by setting γ = 0. Due to its generality, (2), (3) will serve as starting points for our analysis. From a sample of observations (y1 , . . . , yT ) the vector of parameters θ = (µ, α, σ , γ ) can be estimated by quasi-maximum likelihood (QML) through the optimisation of a Gaussian log-likelihood function as discussed in Bollerslev and Wooldridge (1992) and Newey and Steigerwald (1997).4 Given the estimate αˆ T , the null hypothesis of integration H0 : α = 0 is usually tested against the alternative HA : α < 0 through a t-statistic, say t0α . Although it is well known that the t-statistic for a unit root in an homoskedastic AR process converges to the well-known Dickey–Fuller (1979) distribution under mild assumptions (see Phillips and Perron 1988), it is not evident which distribution rules the critical values for LDCH errors when γ > 0. Furthermore, even the simplest case of constant-scale LDCH considered above introduces a remarkable degree of complexity which makes the theoretical analysis extremely difficult. Note that, in addition to the degree of nonlinearity strong in the variance, H0 : α = 0 leads directly to E εt2 = ∞, thereby violating one of the most common assumptions in parametric modelling and testing, namely that of prediction errors with a well-defined variance. Furthermore, yt finds an absorbing barrier at zero which is asymptotically attainable, so the process degenerates at zero asymptotically. Finally, yt may be nonstationary even under the alternative, if γ > 1, as shown in Broze et al. (1995). These features largely interfere with the asymptotic analysis whereby the derivation of the limit distribution of t0α , in a general context, is a challenging and unsolved problem. In order to shed light on this issue we present experimental evidence on a general class of LDCH errors in the following section. 3 Experimental analysis 3.1 Experiment design Our analysis assumes a conditional mean given by a first-order AR process as in (2) and LDCH patterns as in (3) with two different specifications for σ . In 4 Note that most empirical studies which assume time-varying conditional volatility use the QML
procedure to estimate the unkown parameters. The QML estimator is known to be consistent under correct specification of both the conditional mean and the conditional variance (see, among others, Bollerslev and Wooldridge 1992; Lee and Hansen 1994; Lumsdaine 1996; Newey and Steigerwald 1997).
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particular, we assume that the scale parameter (a) is constant and (b) follows stationary GARCH(1,1) dynamics. Note that it has been argued that allowing the scale to be a measurable function on the information set gives rise to a superior characterisation of interest rate models, and as such often estimated (see for instance, Koedijk et al. 1997; Anderson and Lund 1997; Bali 2003). In particular, the general DGP we consider is, yt = µ + αyt−1 + εt γ εt = σt yt−1 ηt ; ηt ∼ iid (0, 1) ; γ > 0 γ 2 2 σt2 = ω0 + ω1 εt−1 /yt−2 + ω2 σt−1 ;
(4) (5) (6)
with ω0 > 0; ω1 , ω2 ≥ 0; ω1 + ω2 < 1. In case (a), we set ω1 = ω2 = 0 and thus analyse the CKLS model as the most simple specification. In the more general case, we allow for clustering as well as level-dependence effects (the so-called mixed models).5 The vectors of unknown coefficients in cases (a) and (b) are respectively θa = (µ, α, γ , σ ) and θb = (µ, α, γ , ω0 , ω1 , ω2 ) . These can be estimated by maximising a Gaussian log-likelihood function under the QML approach, given the assumption of conditional normality and some set of starting values, say y˜ 0 , leading to:
LT yt |θ , y˜ 0
T εt2 1 2 2γ =− log σt yt−1 + 2γ 2 σt2 yt−1 t=1
(7)
for the different specifications of σt2 considered. Our main interest lies in the distribution of t0α under the null hypothesis, i.e., the distribution of the ratio t0α = αˆ T / ωˆ α
(8)
where αˆ T is the ML estimate of α, and ωˆ α is the 2 × 2 element of the estimated asymptotic covariance matrix of either θa or θb . The random shocks ηt in our simulations are generated from a standard normal distribution. The estimated models considered in (a) and (b) given (7) are therefore correctly specified and hence the parameter estimates are in fact Gaussian maximum-likelihood optimisers. This simplifies the issue of estimating the asymptotic covariance matrix since the Hessian-based estimator and the outer product of the score are equivalent. In a more general setting, the covariance matrix is estimated through White’s method in a quasi-maximum likelihood setting. We conduct the following experiments. First, we analyse the empirical distribution of t0α as a function of different values of γ and for different values of a constant scale parameter σ . Second, using the critical values obtained from these distributions, we further analyse the finite sample size-adjusted power of 2 5 The normalisation of ε t−1 found in σt is necessary to allow for separate identification of distinct
heteroskedasticity effects (i.e., clustering and level-dependence).
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t0α when the DGP includes a near-integrated AR root. This is the most relevant case for empirical applications since interest rates are always strongly persistent time series. The ability to identify whether the process is in fact stationary (although slowly reverting) is especially important. Therefore, we focus on the cases in which the data are generated from highly persistent processes with α = {−0.01, −0.05}. Finally, the third experiment explores the effects on the empirical distribution of t0α given the existence of several types of GARCH structures in the scale parameter σt . All experiments are based on 50, 000 replications. We report the results for a sample of T = 500 observations which corresponds to a sample size of empirical relevance (see, for instance, the analysis in Chan et al. 1992), although we also conducted the analysis for samples with 1, 000 observations. Since we obtained no qualitative differences from the results commented in the next section, these are not presented in order to save space, though are available upon request. To minimise the effects of initial values in the simulations, 400 additional observations were generated and removed from the simulated paths. The values of γ used in our simulations are always in the empirical range (0, 1.5], although special attention is given to γ = 1/2, corresponding to the theoretical restriction implied in the CIR model. We set the scale values in a range typically found in the empirical framework to initiate σ . As in Rodrigues and Rubia (2005), we initially set σ 2 = 0.0001 in (a); and for comparative purposes, in (b), we consider an unconditional scale parameter σ 2 = 0.0001 and in (b.i) a high degree of persistence together with low short-run GARCH effects (ω1 = 0.05, ω2 = 0.90); and in (b.ii) high persistence together with a relatively high ARCH effect (ω1 = 0.20, ω2 = 0.70). We repeated the simulations with other values of the GARCH parameters but similar findings were drawn. Finally, since the simulated observations may take negative values under the null of integration, two alternative schemes were considered to prevent this γ problem when simulating {yt } . We generated εt = σt yt−1 ηt and recorded {yt }T t=1 as long as yt > 0, discarding the whole path otherwise. The process was repeated until 50,000 simulated paths were obtained. Alternatively, our simulations were based on εt = σt |yt−1 |γ ηt . Since both mechanisms yield similar results, we only report the ones obtained from the first scheme.
3.2 Experimental results 3.2.1 Finite-sample distribution of t0α with constant-scale parameter In this section, we first discuss results related to a pure form of level-dependent conditional heteroskedasticity with a constant scale parameter σ such as, the CKLS model. The empirical critical values for the ML t-statistic t0α are briefly summarised in Table 1. It can be observed that the distribution of this statistic gravitates around the Dickey–Fuller (DF) distribution for the wide range of values of the elasticity parameter γ analysed. Although it is usually required that E εt2 < ∞ to apply a Functional Central Limit Theorem in order to show
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Table 1 Empirical percentiles for the ML based t test of the null hypothesis H0 : α = 0 computed γ from the model yt = αyt−1 + σ yt−1 ηt ; ηt |It−1 ∼ iidN (0, 1) Probability of a smaller value
DF-T γ 0.00 0.10 0.25 0.40 0.50 0.65 0.75 0.90 0.95 1.00 1.25 1.50
0.01 −3.44
0.025 −3.13
0.05 −2.87
0.10 −2.57
0.90 −0.43
0.95 −0.07
0.975 0.24
0.99 0.63
−3.52 −3.48 −3.48 −3.44 −3.45 −3.44 −3.44 −3.43 −3.45 −3.44 −3.43 −3.40
−3.20 −3.17 −3.17 −3.14 −3.13 −3.13 −3.12 −3.12 −3.13 −3.13 −3.12 −3.10
−2.94 −2.91 −2.91 −2.89 −2.87 −2.86 −2.87 −2.86 −2.87 −2.87 −2.85 −2.84
−2.65 −2.62 −2.66 −2.60 −2.54 −2.56 −2.57 −2.57 −2.57 −2.57 −2.56 −2.54
−0.60 −0.55 −0.56 −0.50 −0.50 −0.41 −0.46 −0.44 −0.42 −0.41 −0.41 −0.40
−0.25 −0.20 −0.19 −0.12 −0.12 −0.04 −0.08 −0.07 −0.04 −0.04 −0.03 −0.03
0.06 0.13 0.14 0.23 0.22 0.29 0.25 0.27 0.29 0.30 0.29 0.29
0.42 0.51 0.54 0.63 0.64 0.68 0.66 0.68 0.68 0.68 0.68 0.67
DF-T represents the percentiles of the conventional Dickey–Fuller distribution which are presented for comparative purposes. Empirical percentiles are computed based on a sample of T = 500 observations and 50,000 replications
weak convergence in the unit root context, it is important to note that this is in fact a sufficient condition. Even though the second-order moment is not welldefined in this context, the DF distribution applies. The t-statistic converges quickly to the DF distribution and only minor deviations are observed. Further insight on this result can be obtained from Fig. 1 where the empirical cumulative distributions of the test for different values of γ are presented. Regardless of the different values of γ , there are no meaningful differences, and the relevant critical values obtained are roughly the same for all γ . Since the differences across γ do not seem to have a particular effect on the distribution, we verify the robustness of these results against different values of the constant scale parameter. For the purpose of this analysis, we consider γ = 1/2 as a benchmark,6 and a sequence of scale parameters given by σξ2 = ξ σ02 , where ξ = {0.10, 1, 10, 100} , and again σ0 = 0.0001. It should be noted that some of the resulting values are far beyond the relevant range for empirical applications on interest rate time series but it is important to verify the extent of generality of the results obtained earlier. The results from this experiment are summarised in Table 2. We can observe that the distribution of t0α is virtually invariant to the magnitude of σ and, in any case, the changes do not seem to affect the lower tail, thus providing the relevant critical values for a test of the alternative of stationarity. In many papers, the constant-scale LDCH model related to the CKLS model has been estimated and inference on αˆ T used to test whether the series are 6 Simulations were also conducted for several values of γ with no qualitative difference from the results reported in the main text. We do not present these results in order to save space but these are available from the authors upon request.
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F(x)
0.6 0.5 0.4 0.3 0.2 0.1 0 -6
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Fig. 1 Empirical cumulative density function of the t statistic for values of γ reported in Table 1 Table 2 Empirical percentiles for the ML based t test of the null hypothesis H0 : α = 0 computed 1/2
from the model yt = αyt−1 + σ yt−1 ηt ; ηt |It−1 ∼ iidN (0, 1) Probability of a smaller value
DF-T σ2 0.00001 0.0001 0.001 0.01
0.01 −3.43
0.025 −3.12
0.05 −2.86
0.10 −2.57
0.90 −0.44
0.95 −0.07
0.975 0.23
0.99 0.60
−3.42 −3.45 −3.45 −3.42
−3.11 −3.13 −3.13 −3.11
−2.84 −2.87 −2.87 −2.86
−2.54 −2.54 −2.54 −2.56
−0.42 −0.50 −0.40 −0.43
−0.06 −0.12 −0.05 −0.03
0.26 0.22 0.30 0.35
0.64 0.64 0.74 0.77
The scale parameter σ 2 takes the values presented in the first column. DF-T represents the percentiles of the conventional Dickey–Fuller distribution which are presented for comparative purposes. Empirical percentiles are computed based on a sample of T = 500 observations and 50,000 replications
mean-reverting or not without making any reference that allows the reader to identify which distribution is actually being used. From the previous analysis, we conclude that the critical values from the Dickey–Fuller distribution seem to provide correct confidence levels when the true DGP follows the CKLS model and as such this distribution may be unambiguously used in applied work. 3.2.2 Power of the t-statistics with constant-scale parameter The analysis on the power properties of the QML testing procedure allows us to gain insight on why it is so difficult to reject the null hypothesis. In addition, it is interesting to compare this performance to that from applying a least-
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Table 3 Rejection probabilities of the ML based unit root t-test at 1 and 5% nominal levels γ
α = −0.01 1%
0.10 0.25 0.40 0.50 0.65 0.75 0.90 0.95 1.00 1.25 1.50
α = −0.05 5%
1%
5%
QML
OLS
QML
OLS
QML
OLS
QML
OLS
0.049 0.030 0.028 0.026 0.026 0.026 0.026 0.024 0.025 0.024 0.024
0.041 0.028 0.024 0.017 0.006 0.006 0.015 0.017 0.018 0.022 0.026
0.189 0.138 0.120 0.122 0.119 0.118 0.120 0.118 0.117 0.120 0.118
0.169 0.127 0.119 0.105 0.087 0.090 0.103 0.106 0.109 0.117 0.121
0.743 0.727 0.740 0.728 0.728 0.726 0.730 0.718 0.723 0.726 0.740
0.718 0.719 0.683 0.606 0.396 0.485 0.581 0.608 0.624 0.680 0.710
0.971 0.965 0.964 0.965 0.963 0.963 0.963 0.964 0.962 0.965 0.967
0.962 0.963 0.960 0.947 0.927 0.930 0.947 0.950 0.952 0.961 0.965
γ
The estimated model is rt = µ + αrt−1 + σ rt−1 ηt , ηt |It−1 ∼ iidN (0, 1). The true DGP sets either α = {−0.01, −0.05} and a value of µ set as µ0 (1 + α), with µ0 = 0.15. Power is computed based on 50,000 replications of samples of 500 observations. QML and OLS denote the rejection frequencies when using quasi maximum-likelihood and ordinary least-squares estimates, respectively
squares (LS) procedure. Note, for instance, that LS estimation is the basis for DF-type tests, often applied on interest rate time series. Even though the basic assumptions of that test are not fulfilled in this context (e.g., errors with finite variance), it is shown in Rodrigues and Rubia (2005) that the OLS t statistic displays approximately correct size regardless of the values of γ , with the relevant critical values being the same for both tests. However, since the OLS inference neglects the time-varying nature of volatility, inference based on inefficient estimates is expected to yield a less powerful testing procedure. Therefore, it is important to evaluate the extent of any power enhancement in this context coming from ML estimates.7 Findings from the power analysis are presented in Table 3. First, when we compare the performance of the ML t-statistics with respect to the LS-based test, the former is not only more powerful, as would be expected, but displays robustness to departures related to non-linear persistence in volatility due to LDCH. Note, for instance, that when the true DGP is generated with α = −0.05 and the asymptotic nominal size of the test is 1%, there are strong power distortions in the LS unit-root test given the values of γ , whereas such distortions are not as significant in the ML testing procedure. The ML test is more likely to detect a slow mean-reversion process in this context than the Dickey–Fuller test, as the latter may be extremely sensitive towards LDCH patterns. However, we also note that the extent of the gain in power of the ML test is little or
7 We perform the power analysis for both tests by using the relevant critical values obtained from
the Dickey–Fuller distribution as these provide an accurate approximation.
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marginal with respect to the LS method as α approaches zero and, interestingly, the power performance of both tests tends to be equal when γ > 1. We now turn our attention to the performance of the ML t-stastistics. Even though the model is correctly specified in our analysis, and therefore the ML estimates are fully efficient, the statistic t0α has a surprisingly very poor performance in rejecting the null as the AR root approaches unity. More importantly, LDCH may strongly affect the power of the test in the near-integrated region. Although this procedure is always more powerful than the OLS-based test, their power reductions can reach up to 50% even for moderate values of γ when α = −0.01. It is clear that the ML unit root test may be biased towards non rejection even in the most favourable case in which the model specification is completely correct. In empirical applications based on the QML setting in which the normality assumption is not realistic, we can expect further power reductions as the procedure is not efficient. From this analysis, the LDCH patterns lead to the usual problems in identifying slow mean reverting series in a near-integration context and, hence, yield testing procedures with extremely poor performance. It is not very surprising, therefore, that most applied papers cannot find conclusive evidence for meanreversion through ML-based t-statistics, since the estimated autoregressive root is always in the neighborhood of unit, and the estimated γ parameter is always highly significant. 3.2.3 Time-varying scale parameter The analysis in this section is motivated by the fact that when estimating mixed models in the ML setting, the significance of α is addressed as if there were no further complication related to the more sophisticated LDCH model. There is no special distinction in the analysis suggesting that the critical values may be different in this context. To the best of our knowledge, there is no applied paper in the literature in which the inference stage is conducted as if the distribution of the resulting t0α were not invariant. We believe this way of proceeding is likely not to be correct. For instance, in the most simple case γ = 0, the mixed model reduces to an AR(1) process with (stationary) GARCH errors. Ling and Li (1998, 2003), Seo (1999), Li et al. (2002) have shown that the distribution of the unit root test based on ML estimation is a mixture of the DF t-distribution and the standard normal provided some conditions, with the relative weight depending on the magnitude of the GARCH effect and the moments of the standardized errors. Therefore, in the more complex case in which γ > 0, the distribution of t0α may behave like this mixture of distributions or not, but in any case it is not clear whether invariance holds, as it is implicitly assumed in applied work. Given the considerable computational effort related to estimate mixed models in the Monte Carlo setting, we initially focused on γ = 1/2, since the inferred value of γ when estimating mixed models tends to be in the neighborhood of this value (i.e., the CIR model). We replicate our analysis by setting γ to other values and reach the same qualitative conclusions. From these simulations, we
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observe that the GARCH parameters governing the scale dynamics are clearly nuisance parameters that strongly affect the limiting distribution of the t-statistic (see Fig. 2 below). For example, when γ = 1/2 and GARCH dynamics include low short-run effects, the (1, 5, 10%) and [90, 95, 99%] percentiles of the empirical distributions are (−3.33, −2.77, −2.50) and [0.04, 0.40, 0.90], respectively. However, when higher short-run dynamics is allowed for, the corresponding percentiles are (−3.33, −2.76, −2.44) and [0.25, 0.58, 1], respectively. Note that the computed critical values, when only level-dependence drives the conditional variance are (−3.44, −2.89, −2.60) and [−0.11, 0.23, 0.61], respectively. For these values of GARCH parameters, the clustering effect induced in the volatility pushes the distribution of the t-statistic to the left. Note that this result follows similarly as those results evidenced by Seo (1999), in which more persistent GARCH effects leading to the standard normal distribution are observed. Although it is tempting to stress the similitudes that arise from our analysis and the results in Ling and Li (1998, 2003) and Seo (1999), it should be noted that Monte Carlo simulations offer limited help in this context. The number of nuisance parameters, the precise interaction among them, and their overall effect should only be addressed theoretically, though not possible given in the extraordinarily complex context of LDCH models. Nevertheless, the statistical
γ =1/2 0.7 DF GARCH-L GARCH-H
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Fig. 2 Kernel densities of t statistics under level and mixed errors. Kernel densities of the empirical distributions of the t-stastistic with and without a time-varying scale when γ = 1/2. The
conditional variance dynamics is given by: (i) level model (Level): ω = σ02 , 0, 0 , σ02 = 0.003. (ii) (GARCH-H) mixed model with strong persistence and low ARCH coefficients, ω1 = 0.05, ω2 = 0.80. (iii) (GARCH-L) mixed model with high strong persistence and high ARCH coefficient, ω1 = 0.20, ω2 = 0.70
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analysis shown in the paper is by itself sufficient to provide important insight on the pitfalls of testing for a unit root through ML estimates in the LDCH context. The message which emerges is clear: unlike the simplest case of a constant-scale LDCH model, there does not seem to be a unique distribution for the t-stastics when complex interactions of mixed-type are allowed. As in Seo (1999), the correct critical values are sample-dependent even in the sample size is allowed to diverge and, therefore, in absence of a formal theoretical analysis, the statistical analysis in applied settings cannot be conclusive. The general analysis from a theoretical viewpoint is necessary, although it seems to be a task of extraordinary complexity whose resolution may demand techniques which are not yet available. 4 Conclusion Little attention has been given to level-dependent conditional heterokedasticity in the non-stationary literature. Most results available, which consider conditional heteroscedasticity in the errors, have focused mainly on GARCH processes. Level-dependent conditional heterokedastic models, however, have been found to be well-suited for the analysis of interest rate dynamics and raise several interesting issues of empirical relevance. The analysis of this paper aims to provide some insight on the limit distribution of the ML-based unit root test. The results provided represent an important extension of the basic GARCH framework to potentially more complex error structures. The results provided show that under the conventional form of leveldependence considered in literature, the ML-based unit root test converges to the conventional DF distribution. This distribution can be used to approach the small-sample sizes used in empirical applications. However, the test is seen to exhibit very low power in the near-integrated region, power which is strongly affected by level-dependent volatility. If mixed effects are considered (assuming both level-dependence and cluster effects), at least the parameters controlling the clustering in volatility become nuisance parameters and there is no convergence to a unique distribution. Although the Dickey–Fuller distribution might be applicable, the departures from this distribution could generate distortions in the required size. Acknowledgments We are grateful for the helpful comments and suggestions from Uwe Hassler and an anonymous referee. Any remaining error is our own responsibility. Financial support from POCTI/ FEDER (grant ref. POCTI/ECO/ 49266/2002), IVIE, and the SEJ2005-09372/ECON project is gratefully acknowledged.
References Anderson TG, Lund J (1997) Estimating continous-time stochastic volatility models of the short-term interest rates. J Econometrics 77:343–377 Bali TG (2003) Modeling the stochastic behavior of short-term interest rates: pricing implications for discount bonds. J Bank Finan 27:201–228
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Bollerslev T, Wooldridge J (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Economet Rev 11:143–172 Broze L, Scaillet O, Zaköian JM (1995) Testing for continuous-time models of the short-term interest rate. J Empirical Finan 2:199–223 Campbell J, Lo AW, Mackinlay AC (1997) The econometrics of financial markets. Princeton University Press, Princeton Chan KC, Karolyi GA, Longstaff LA, Sanders SA (1992) An empirical comparison of alternative models of the short-term interest rate. J Finan 47:1209–1227 Cox J, Ingersoll J, Ross S (1985) A theory of the term structure of interest rates. Econometrica 53:385–407 Dickey DA, Fuller WA (1979) Distribution of the estimators for autoregressive time series with a unit root. J Am Stat Asso 74:427–431 Engle RF, Lilien D, Robins R (1987) Estimating time varying risk premia in the term structure: the ARCH-M model. Econometrica 55:391–407 Kim K, Schmidt P (1997) Unit root tests with conditional heteroskedasticity. J Econometrics 59:287–300 Koedijk KG, Nissen FGJ, Schotman PC, Wolff C (1997) The dynamics of short-term interest rate volatility reconsidered. Euro Finan Rev 1:105–130 Lee S, Hansen B (1994) Asymtotic theory for the maximum likelihood estimator. Econometric Theory 10:29–52 Li WK, Ling S, McAleer M (2002) Recent theoretical results for time series models with GARCH errors. J Econ Surv 16(3):245–269 Ling S (2004) Estimation and testing stationarity for double-autoregressive models. J R Stat Soc Ser B 66:63–78 Ling S, Li WK (1998) Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroscedastic errors. Ann Stat 26:84–125 Ling S, Li WK (2003) Asymptotic inference for unit root processes with GARCH(1,1) errors. Econometric Theory 19:541–564 Lumsdaine R (1996) Consistency and asymptotic normality maximum likelihood estimator in IGARCH(1,1) and GARCH(1,1) models. J Econometrics 64:575–596 Newey W, Steigerwald D (1997) Asymptotic bias for quasi maximum likelihood estimators in conditional heteroskedasticity models. Econometrica 65:587–599 Phillips PCB, Perron P (1988) Testing for a unit root in time series regression, Biometrica 75:335–346 Rodrigues PMM, Rubia A (2005) The performance of unit root tests under level-dependent heteroskedasticity. Econ Lett 89:262–268 Seo B (1999) Distribution theory for unit root tests with conditional heteroskedasticity. J Econometrics 91:113–144
