Bayesian Inference for a Threshold Autoregression with a Unit Root

Transcription

1 Bayesian Inference for a Threshold Autoregression with a Unit Root Penelope Smith Melbourne Institute The University of Melbourne Parkville, VIC 3010 September 1, 2005 Abstract A Bayesian approach to distinguishing between non-linear and unit root behaviour o ers several practical advantages over equivalent frequentist procedures. Foremost among these advantages is the simplicity of the approach. This paper compares the small sample power and size properties of a simple Bayesian test for unit roots in the presence of a possible threshold e ect with Caner and Hansen s (2001) frequentist strategy. The results from Monte Carlo experiments indicate that the simpler Bayesian test performs at least as well as Caner and Hansen s procedure. 1 Introduction Unit root tests are usually speci ed so that the time series under consideration has a constant dynamic structure. However, it is by now well known that the manner in which alternative models are speci ed can matter a great deal when testing for unit roots. Perron (1989) rst noted that standard tests, such as Dickey and Fuller s (1981) augmented Dickey Fuller (ADF) or the Phillips and Perron (1988) (PP) test, have reduced power against stationary alternatives in the presence of structural breaks in either the growth rate or the level of a time series. Findings of unit roots in bounded series, particularly those measured in rates, also cast doubt on how evidence in favour of the unit root hypothesis is assessed. There is a steadily growing interest in testing for unit roots outside the standard linear framework. This is motivated by a growing interest in the idea that asymmetry may be important for describing the behavior of macroeconomic time series, particularly output and unemployment. 1 However, because such pasmit@unimelb.edu.au 1 See, for example, Ramsey and Rothman (1996) who found evidence of both steepness and deepness in several of Nelson and Plosser s variables. 1

2 testing problems are non-standard, only limited progress in this area has been made. A simple and popular method of introducing nonlinearity into a time series model is via a threshold e ect. Threshold models are, in their simplest form, piece-wise regression models in which the parameters of the model can switch according to the value of a threshold variable. The threshold variable may be exogenous, or some function of the dependent variable, as is the case in Tong s (1978) the self exciting threshold autoregression (SET AR). Pippenger and Goering (1993) considered the problem of testing for unit roots when the data generating process (DGP) is believed a priori to be subject to threshold e ects. Through a series of Monte Carlo experiments, they found that the power of Dickey-Fuller type tests to be reduced for threshold processes. Caner and Hansen (2001), developed an asymptotic distribution theory for threshold models when the underlying process may contain a unit root. They noted that the Wald test for a threshold e ect has a non-standard asymptotic null distribution. This is partly due to Davies (1977) problem (the presence of a nuisance parameter which is not identi ed under the null hypothesis) and partly due to discontinuities in the likelihood functions associated with unit root processes (see, for example Sims and Uhlig (1991)). Making use of the asymptotic theory they established, Caner and Hansen developed a method for bootstrapping the critical values of their tests. They proposed a sequential testing strategy that involved testing for a threshold e ect when a unit root may be present and then, if a threshold e ect is found, testing for a unit root against stationary and partial unit root alternatives. This testing strategy has been adopted in applied work by Aretis, Cipollini and Fattouh (2002), Bec, Salem and MacDonald (2002) and Henry and Shields (2004). An equivalent test is relatively simple within the Bayesian framework. As described by Koop and Potter (1999a), the Bayesian approach circumvents Davies problem by adopting an informative prior for the threshold parameter and averaging over all observable values of the threshold to nd and compare the marginal likelihoods associated with linear and nonlinear speci cations. Furthermore, as the Bayesian approach allows the practitioner to jointly test for unit roots and threshold e ects pre-testing biases are avoided. This paper presents a simple Bayesian procedure to jointly test for threshold and unit root behavior in a univariate time series. It is based upon Bayesian tests for nonlinearity outlined by Koop and Potter (1999a) and is closely related to Bayesian tests for unit roots in the presence of structural change developed by Phillips and Zivot (1994) and Marriot and Newbold (2000). The performance of the test is benchmarked against the Caner and Hansen s suite of tests via a set of Monte Carlo experiments and by application to the male unemployment rate in the United States. Because analytical results are available due to the use of natural conjugate priors, the advantage of this approach lies in its simplicity. 2

3 2 Nonlinearity and the Unit Root Hypothesis The standard linear relationship most often used for evaluating the evidence in favour of a unit root in a given time series Y e T equation = fy t g T t= k 1 is the ADF test kx y t = + t + y t 1 + j y t j + e t (1) where e t i:i:d:n 0; 2. When testing for a unit root one generally proceeds by obtaining an estimate of and then using the appropriate critical values to determine whether or not can be said to lie within the stationary region of the parameter space, 2 < < 0, with some pre-speci ed degree of con dence. ADF tests have been applied to a large array of macroeconomic time series. However, the suitability of this constant-parameter speci cation has been questioned by a number of authors. Perron (1989) rst noted that the power of standard unit root tests, such as the ADF or PP tests are reduced in the presence of structural breaks in the level or growth rate of a series. Perron s solution to this problem was to modify the standard ADF test to incorporate a dummy variable to account for a structural break at some point in the sample which is known. Perron applied this modi ed test to U.S. real GNP and found that if anomalous events such as the Great Depression and the 1973 oil price shock were treated as points of structural change, then the preferred model of U.S. real GNP is the trend stationary model rather than the di erence stationary model preferred by Nelson and Plosser (1982). 2 Perron (1994) considered the case of a time series that undergoes a structural change in its mean level and reaches similar conclusions to Perron (1989): standard unit root tests are biased towards non-rejection of the unit root hypothesis. Implementing a sequential test similar to that of Zivot and Andrews (1992) Perron concluded that three time series which were commonly found to be nonstationary, the interest rate, unemployment rate and terms of trade index, were in fact stationary around a changing mean. Parallel to the discussion of unit roots tests and structural breaks is the growing literature concerned with asymmetry and nonlinearity in macroeconomic time series, particularly in real output and unemployment series. 3 Pippenger and Goering (1993) present an alternative model to structural change for explaining the apparent failure of exchange rates to return to some long run equilibrium, even though this data is bounded and although the theory of purchasing power parity (PPP) suggests that they should be stationary. They argue that if price di erentials across spatially segregated markets are smaller 2 Perron s results have been criticized by Christiano (1992), Banerjee, Lumsdaine and Stock (1992) and Zivot and Andrews (1992). These authors argue that because Perron s test is dependent on the timing of the structural break being known a priori, break points will be correlated with the data and the test will be biased against the unit root hypothesis. Zivot and Andrews, adopting a sequential Dickey-Fuller test for the analysis of the Nelson-Plosser data, and found less evidence against the unit root hypothesis than Perron. 3 See, for example Koop and Potter (1999b). j=1 3

4 than transaction costs, then the equilibrating force of arbitrage will not come into e ect. Instead there will be a small equilibrium band, rather than a point, within which prices can be expected to uctuate randomly. This is equivalent to modeling the price di erential process as being subject to a threshold e ect with a unit root in one regime. In a similar vein, Gonzalez and Gonzalo (1997) present a class of models which describe a process as having either a unit or stable root depending on the value of a threshold variable. They nd that while under certain conditions, such a process would be both stationary and geometrically ergodic, it would also be able to mimic the behavior of variables measured in rates, such as unemployment or in ation rates. Pippenger and Goering (1993) consider the case of testing for unit roots when the DGP is believed a priori to be a threshold autoregressive process. In this model the adjustment process towards the long-run equilibrium of y t depends on the location of the previous value of the series y t 1 relative to some threshold value : 1 y y t = t 1 + e t if y t 1 < (2) 2 y t 1 + e t if y t 1 = where e t i:i:d:n 0; 2. By Monte Carlo experiment Pippenger and Goering nd that the power of Dickey-Fuller type tests is reduced for this DGP. They contend that this result supports arbitrage based economic theories, such as purchasing power parity, although they do not develop a unit root test for the special case of a threshold process. Nelson, Piger and Zivot (2001) note increasing support for the idea that the number and nature of regime changes in a given economic time series may be better modeled as a probabilistic process, such as in the Markov switching model of Hamilton (1989). They ask what the e ects of Markov switching in the DGP of a time series might have on the result of several standard unit root tests, including the ADF, Zivot and Andrew s (1992) and Perron s (1994) tests. They nd that the size distortions found in standard unit root tests when there is a break in the variance or growth rate of the DGP do not appear to generalize to Markov switching in the mean or variance. However, Nelson Zivot and Piger do nd that the presence of a switching variance can lead to over-rejection in tests that allow for only a single break in the growth rate or the trend. Nelson Zivot and Piger do not develop a procedure for testing for unit roots in the presence of Markov switching. 3 The Testing Problem Despite the apparent importance of developing joint tests for asymmetric and nonstationary behavior in economic time series, research in this area has thus far been limited. This is perhaps attributable to the non-standard nature of the testing problem. Enders and Granger (1998) consider the problem of testing for a unit root in a series y t that follows a threshold autoregressive process of a form similar to 4

5 equation (2). They assume that the value of the threshold is known, but do not consider the possibility of testing for a threshold process when there may be a unit root in y t. Caner and Hansen (2001) developed a set of procedures for testing for a unit root when there may be a threshold e ect DGP and the reciprocal case of testing for threshold nonlinearity when there may be a unit root in the DGP. In particular, Caner and Hansen consider the following self exciting threshold autoregression of order k (SET AR(k)): 8 P >< t + 1 y t 1 + k 1j y t j + e t if Z t 1 < j=1 y t = P >: t + 2 y t 1 + k (3) 2j y t j + e t if Z t 1 = : j=1 Here Z t 1 is some threshold variable constructed from lagged values of y t and the threshold,, is unknown. Caner and Hansen specify the threshold variable to be Z t = y t y t m for some m = 1 but note that the de nition of Z t is not central to the analysis and that their results extend to any strictly stationary and ergodic Z t which is pre-determined and has a continuous distribution. De ne i = [ i ; i ; i ; i ] to be the 1 n vector of the parameters for alternative threshold values i 2 f1; 2 g and x t = [1; t; y t 1 ; y t 1 ; : : : ; y t k ] 0. Also de ne = [ 1 ; 2 ; ]. Assuming independently, identically distributed Gaussian disturbances, e t i:i:d:n 0; 2, the likelihood function associated with (3) for a given parameter point t is f t y t jyt k 1; ; 1 = p exp (y! t 1 x t I fzt 1<g 2 x t I fzt 1=g) : Here yt k 1 = y t 1 ; : : : ; y t k 1 and I fg is a binary indicator function taking on the value 1 if the identity fg is true and 0 otherwise. For data series Y e T the likelihood function associated with (3) is f T ( e Y T ; ; ) = TY f t y t jyt k 1; t=1 The series Y e T follows a threshold process if 1 6= 2 and may have a unit root in either regime if 1 = 0 or 2 = 0. When testing for a threshold e ect, the relevant null hypothesis is that of a linear autoregression with the alternative : H 0 : 1 = 2 (4) H 1 : 1 6= 2 : (5) Let 0 denote the parameter vector under the null hypothesis. It is easy to see that the likelihood function under the null hypothesis is independent 5

6 of the threshold parameter. That is, f T ( Y e T ; 0; ) = f T ( Y e T ; 0). This is known as Davies problem after Davies (1977) who noted that when a nuisance parameter is unidenti ed under the null hypotheses a consistent estimator of the nuisance parameter will not exist and the distributions of the usual test statistics, such as the Wald (W T ) Lagrange multiplier (LM T ) and likelihood ratio (LR T ) test statistics will be non-standard. Furthermore, as described by Andrews and Ploberger (1994), Davies problem places these test statistics outside the domain of standard asymptotic optimality results. Andrews and Ploberger consider the problem of optimal testing when a nuisance parameter is unidenti ed under the null hypothesis against local alternatives of the form f T 0 + B 1 T h for some h 2 R and some nonrandom diagonal matrix B: They show that for particular integrable weighting functions Q (h) on values of h and a chosen weight function J () on values of a nuisance parameter, optimal tests of will be of the form Z 1 Exp-M T = (1 + c) n=2 c exp c M() dj () : Here M() is one of the standard W T, LM T or LR T test statistics given, and c > 0 is a scalar constant depending on the weight functions Q (). Andrews and Ploberger show that tests of this form will be optimal in the sense that they have the greatest average weighted power over all tests, ' T of asymptotic level,. That is, they are the class of tests that maximizes Z Pr ' T rejects H B 1 T h ; dq (h)dj () ; lim T!1 where lim is the average lim. T!1 T!1 Despite Andrews and Ploberger s optimality results, their Exp-M T test is not the dominant approach to testing for nonlinearity. This is possibly because Andrews and Ploberger do not discuss how critical values for the Exp- M T test statistic should be obtained. 4 This dominant approach to testing for non-linearity is via Hansen s (1996) Sup-Wald test. Hansen developed a solution to Davies problem by approximating asymptotically correct critical values for the null hypothesis by bootstrapping the data under the assumption that (3) is a stationary process. For the testing problem described by (4) and (5), Hansen s Sup-Wald statistic is 0 1 sup W T () = T b2 0 1A : (6) b 2 b Here b 2 ( b ) is an unrestricted estimate of the residual variance from (3) conditional upon the estimated value of the threshold b and b 2 0 is an estimate of the 4 Hansen (1996) demonstrates that the Sup-Wald test has near optimal power against distant alternatives. 6

7 residual variance under the null hypothesis. b is the least squares estimate of the threshold variable, obtained by nding the value of that minimizes 2 (). In turn, 2 () is a vector of estimates of variance for alternative vales of the threshold variable, obtained by sequential least squares over the observed values of the threshold variable. If a unit root is present in (3), the asymptotic distribution of (6) will be discontinuous in the parameters at the boundary = 0 and the bootstrap distribution of (6) will no longer be consistent. 5 To address this problem Caner and Hansen augmented Hansen s procedure to allow for a unit root in (3). They do so by imposing a unit root on the bootstrap in order to locate the correct asymptotic distribution for this case. They show that the constrained bootstrap will be rst order correct under the null hypothesis of a linear model (4) if the DGP has a unit root, but incorrect otherwise. As the order of integration of the DGP is usually unknown, Caner and Hansen advise that prudence be applied in the implementation of their test. They suggest that the p-values be bootstrapped both with and without a unit root, with inference based on the larger, more conservative p-value. Caner and Hansen (2001) also to develop a theory of inference for the complementary case of testing for a unit root when a threshold e ect may be present. The relevant null hypothesis in this case is H 0 : 1 = 2 = 0: (7) How the alternative hypothesis of a stationary process should be formulated is less obvious. This is because the region of the parameter space for which SET AR models will be stationary and ergodic is not well understood. Chan and Tong (1985) show that when k = 1; the stationarity of the process depends upon the i parameters. In particular they demonstrate that the necessary and su cient conditions for the stationarity of (3) are 1 < 0; 2 < 0 and (1 + 1 )(1 + 2 ) < 1: However, Chan and Tong note that there are no general conditions for the geometrical ergodicity of higher-order models. 6 Based on this analysis the rst alternative hypothesis proposed by Caner and Hansen (2001) is that Y e T is stationary and ergodic: H 1 : 1 < 0 and 2 < 0: (8) and the second alternative is the partial unit root hypothesis: 8 < 1 < 0 and 2 = 0: H 2 : or : 1 = 0 and 2 < 0: 5 Caner and Hansen (2001), p Chen and Tsay (1991) show that the dynamic properties of (3) are also dependent upon m. In addition to the conditions of Chan and Tong (1985), (3) must also satisfy (1 + 1 ) s(m) (1 + 2 ) t(m) < 1 and (1 + 1 ) t(m) (1 + 2 ) s(m) < 1 where t(m) and s(m) are nonnegative integers depending on m and odd and even numbers. (9) 7

8 In order to distinguish between these hypotheses, Caner and Hansen (2001) propose two test statistics. The rst is the two-sided Wald statistic from the least squares estimates of (3), which is the standard test of the null of a stationary process in both regimes against the unrestricted alternative 1 = 0 or 2 = 0: R 2T = t t 2 2: Here t 1 and t 2 are the t ratios of the OLS estimates of 1 and 2. Caner and Hansen note that as the two alternative hypotheses are one-sided, this twosided version of the test may have less power than the one sided test. Given that it is unclear how to form an optimal one-sided test when unit roots may be present 7, Caner and Hansen suggest that a one-sided Wald test focusing on negative values of the OLS estimates of 1 and 2 should also be considered: R 1T = t 2 11 fb1 <0g + t 2 21 fb2 <0g: Rejection of (7) gives no indication of whether, H 1 or H 2 ; is more likely. To overcome this di cultly Caner and Hansen suggest that the individual t statistics, t 1 and t 2, should be examined as tests of the stationary alternative H 1. The signi cance of only one of the t statistics would be consistent with the partial unit root hypothesis, H 2. In order to conduct these tests, sampling distributions of the tests under the null hypothesis H 0 are required. Caner and Hansen show that the asymptotic distributions of the tests of H 0 are di erent depending on whether or not the there is a threshold e ect in the DGP. This is because the null hypothesis of a unit root is consistent with both 1 = 2 and 1 6= 2 : Separate bootstrap procedures are therefore required for each case. Caner and Hansen nd the unidenti ed threshold model to be slightly preferred in terms of size. Both the one sided and two sided tests are found to have good power when compared with the standard ADF test. The one sided test, R 1T, performs slightly better than the two sided test R 2T. Caner and Hansen s approach provides some valuable advances in asymptotic theory. However there are several disadvantages associated with this approach. First, as already stated, the test for a threshold e ect is only nearly optimal, even when the DGP is stationary. From a practical perspective, the procedure is both analytically complex and computationally expensive. In addition, the approach s sequential structure subjects it to a pretesting bias. Finally, Caner and Hansen s testing procedure is not easily generalizable to alternative nonlinear speci cations, such as Markov switching. To do so would require a complete re-working of the asymptotic theory. Tong and Lim (1980, pp ) list ve criteria for the employment of nonlinear speci cations to be useful. These are in Tong and Lim s order of preference: 1. The estimation and identi cation of the model should not entail excessive computation. 7 See Elliot, Rothenberg and Stock (1996) 8

9 2. The model should be generalizable to capture a wide range of nonlinear e ects. 3. The predictive performance of the nonlinear model should be an improvement on the linear model. 4. There should be some theoretical basis for the adoption of a nonlinear model. 5. They should produce a degree of generality, not just in theory, but also in practice. Caner and Hansen s test arguably fails on the rst and second of these criteria. 4 Bayesian Analysis There are several practical advantages associated with the Bayesian approach to inference in the context of nonlinear time series models. 8 In the present context, the ability to circumvent Davies problem by adopting a proper prior for the nuisance parameter and averaging over all possible values to nd and compare marginal likelihoods of competing models via Bayes factors is foremost among them. This makes it feasible to jointly test for unit roots and threshold e ects. There is a clear correspondence between the Bayesian approach to inference when a nuisance parameter is unidenti ed under the null hypothesis and that of Andrews and Ploberger (1994). In fact, Andrews and Ploberger a rm that their test statistic may be given a Bayesian interpretation if the weight functions, J() and Q, are interpreted as priors. However, they argue that their approach is preferred because it avoids the necessity of placing a prior on the nuisance parameter under the null hypothesis and because their approach is computationally more e cient. It is true that Bayesian methods can be computationally complex when a closed-form solutions to the posterior distributions of the model parameters does not exist or when the posterior distributions are not known. This is usually the case for more complex nonlinear speci cations, such as Markov switching models, which must be estimated using numerical methods of integration. However, these methods are arguably no more complex than the bootstrapping techniques proposed by Hansen (1996) and Caner and Hansen (2001). In addition, as will be outlined in the following section of this paper, analytical solutions to the posterior distribution can be found for model (3) when normal, natural conjugate priors are adopted. Under these priors, the Bayesian approach to testing for nonlinearity will be asymptotically equivalent to Andrews and Ploberger s optimal test statistic. Andrews and Ploberger also note that Bayesian posterior odds ratios are asymptotically equivalent to the Exp-LM T test, but do not describe why, or 8 See Koop and Potter (1999) for a detailed discussion. 9

10 elucidate the conditions under which this will be true. This result follows from Andrews (1994). Adopting Andrews s (1994) notation, let 2 (0; 1) denote the prior probability assigned to the null hypothesis, and let Q denote the prior distribution function for the data. De ne () to be a probability distribution on R + = fr 2 R : r 0g that depends on the prior Q and and let g () be a function for 2 R. The following strictly increasing function of the Wald, Lagrange multiplier or likelihood ratio statistic can be de ned for M = W T ; LM T or LR T : P O (M; ) = 1 Z r 2 exp g Mr 2 d r 2 2 The posterior odds statistic associated with the null and alternative hypotheses will be P O T (Q ) = 1 Z f T 0 + T 1=2 h d Q (h) f T ( 0) Andrews (1994) shows that if the priors and the data satisfy certain regularity conditions, then under both the null (4) and the alternative (5) P O T (Q ) P O (M T ; ) p! 0: (10) Andrews notes that one prior which satis es these conditions is the multivariate normal distribution with a variance proportional to a scalar, c: Q N (0; c). Andrews shows that if this is the case then c = p c 2 n and P O (M; ) = 1 (1 + c) n=2 1 c exp 2 (1 + c) M for the two sided testing case where n 1:This is Andrews and Ploberger s Exp-M T statistic for J () equal to a point mass at 0 and = 1 2. More generally, if we de ne P O T (Q ; ) = 1 Z Z f T 0 + T 1=2 dq (h) dj () h; f T ( 0) and P O (M; ; ) = 1 Z (1 + c) n=2 1 exp 2 c (1 + c) M () dj () then because (10) will hold for all points 2, it follows that under the regularity conditions outlined Andrews (1994) and for = 1 2 P O T (Q ; ) p! P O (M; ; ) or P O T p! Exp-LMT : 10

11 Recall that the Exp-LM T was shown to be asymptotically optimal when a nuisance parameter is unidenti ed under the null by Andrews and Ploberger (1994). The signi cance of this result is that under certain regularity conditions and a for particular class of priors, the Bayesian P O T test will be asymptotically equivalent to the optimal Exp-LM T test. The Bayesian approach to testing for a unit root in the presence of nonlinearity has received relatively little attention. However, as discussed by Phillips and Zivot (1994) for the case of unknown break points, Bayesian methods avoid the pretesting bias which arises from the fact that the delay lag of the threshold parameter, m, is unknown and must rst be estimated before tests can be conducted. Phillips and Zivot also emphasize that the asymptotic distribution theory associated with these test statistics is complex and often at odds with the corresponding nite sample distributions. Furthermore, Bayesian techniques allow joint testing for nonlinearity and unit roots through the comparison of posterior model probabilities. The correspondence between Bayesian and frequentist procedures for inference breaks down when unit roots are present in the DGP. This was initially noted by Sims (1988), who argued that a at prior Bayesian approach to inference was simpler than the frequentist approach, avoiding complications such as disjoint con dence regions or uncertainty regarding trend speci cation. Di erences between the classical and Bayesian approaches have been emphasized at an applied level by DeJong and Whiteman (1991). These authors adopted a Bayesian at prior methodology to revisit Nelson and Plosser s data set, and found far less support for unit roots. In particularly, they rejected the unit root hypothesis for stock-price data, dividend data and U.S. real GNP. Phillips (1991) also noted that Bayesian unit root tests are able to avoid the complexity of the frequentist approach, but warns against the use of at priors. This is because at priors are informative in the sense that they down-weight the nonstationary region of the posterior distribution. In order to develop an objective, or non-informative Bayesian unit test Phillips formulated a framework for the application of Je reys prior on the autoregressive parameters. Phillips found that results from Bayesian unit root tests accord far more closely with the frequentist alternative. Sims argument was largely based on the assertion that inference based on the likelihood principle will be unchanged, even if the DGP is a random walk. The likelihood principle, expressed simply, is that only observed sample outcomes are relevant to statistical inference. As such, all information relevant for inference is contained in the likelihood. Sims observed that the likelihood for the AR(1) model is normal, whether or not the data are stationary. This point has been powerfully illustrated by Sims and Uhlig (1991) and shown to be more generally applicable by Kim (1994, 1998). Kim demonstrated that subject to a fairly general set of conditions, the shape of the likelihood will be asymptotically normal. 9 9 It is useful to note that one prior for which asymptotic normality will not hold is Phillips (1991) version of Je reys prior. As discussed by Kim (1998), Phillips prior is dependent on sample size and diverges to in nity as the sample size becomes large, resulting in a very 11

12 Kim s (1998) result of asymptotic posterior normality is useful in the present context because it carries the implication that the posterior probabilities involved in testing for nonlinearity will be unchanged whether or not there is a unit root in the DGP. In contrast, the asymptotic properties of the alternative Exp-LM T and Sup-Wald statistics under similar conditions are unknown. When a unit root is present, the second derivative of the log of the likelihood function, L 00 () will be discontinuous. Furthermore, L 00 () will no longer converge to a constant. These properties result in non-standard asymptotic distributions of both the Exp-LM T and the Sup-Wald statistics. It is therefore unclear whether or not the Exp-LM T test for a threshold e ect will retain its optimality properties under the null hypothesis of a unit root. In addition, a unit root in the DGP will violate several of the regularity conditions required for the P O T and Exp- LM T to be asymptotically equivalent. The Bayesian test, however will retain its form when a nuisance parameter is unidenti ed under the null hypothesis, with no discontinuities, even when the DGP may follow a unit root. These results form a strong motivation for the adoption of a Bayesian testing strategy with priors that conform to Q N (0; c) whenever a nuisance parameter is unidenti ed under the null hypothesis, particularly when the DGP may be nonstationary. The remainder of this paper is devoted to the development of such a strategy for Caner and Hansen s testing problem. 4.1 A Bayesian Test First assume that the threshold is known. De ne x t = (1; t; y t 1 ; y t 1 ; :::; y t k ) and let x 1 t = x 0 ti t and x 2 t = x 0 t(1 I t ) where 1 if Zt I t = 1 < 0 otherwise. De ne X 1 and X 2 to be the vectors x 1 = [x 1 1; x 1 2; :::; x 1 T ]0 and x 2 = [x 2 1; x 2 2; :::; x 2 T ]0 respectively, with elements re-ordered according to the observed value of the threshold variable Z t associated with each x t. This implies that if there are g observations for which Z t 1, then the last g observations of X 1 will be zero and the rst T g observations of X 2 will be zero. Finally, de ne X () to be the T (4 + 2k) matrix [X 1 X 2 ]. Using this notation, (3) can be expressed as Y () = X () + e where = ( 0 1; 0 2) and Y () and e are the vectors (y 1 ; y 2 ; :::; y T ) 0 and (e 1 ; e 2 ; :::; e T ) 0 respectively, also re-ordered by the observed value of the threshold variable Z t at time t. Based on the hypotheses described in section 3, there are seven competing models of interest de ned by restrictions on and 1 ; 2 2 : M 1 : 1 = 2 and 1 = 2 = 0: a linear autoregression with a unit root non-normal asymptotic posterior distribution. 12

13 M 2 : 1 = 2 and 1 = 2 < 0: a stationary linear autoregression M 3 : 1 6= 2 and 1 = 2 = 0: a nonlinear autoregression with a unit root M 4 : 1 6= 2 and 1 = 2 < 0: a stationary nonlinear autoregression M 5 : 1 6= 2 and 1 6= 2 < 0: a stationary nonlinear autoregression M 6 : 1 6= 2 and 1 = 0 and 2 < 0: a unit root in the lower regime M 7 : 1 6= 2 and 1 < 0 and 2 = 0: a unit root in the upper regime The likelihood function for (3) conditional upon each of these hypotheses is f T ( Y e 1 (Y () X () ) 0 T ; ; jm j ) = ( p 2) exp (Y () X () ) T 2 2 (11) evaluated under M j and the conditional marginal likelihood f T ( Y e T jm j ) is obtained by integrating the joint density p(; ; ) over j, j and. ZZZ f T ( Y e T jm j ) = p(; ; )d j d j d (12) ZZZ = f T (; ; )p(; j)p()d j d j d Z = f T ( Y e T jm j ; )d The analytical evaluation of integrals like (12) is often not possible for non-linear time series models and must therefore be approximated using some numerical integration technique. However analytic solutions are available for (12), conditional on, if independent natural conjugate priors are adopted for 1, 2 and. Convention dictates that the prior densities of the parameters of the model be proper and not too di use. This is because Bayesian estimation requires the integration of f T ( Y e T jm j ; ) over the entire domain of the nuisance parameter. When non-informative or improper priors are adopted the Bayes factors will tend to favour the linear model. 10 Here, a standard normal-gamma prior is adopted for and : 10 For more information and references see Kass and Raftery (1994) section 5 and Koop and Potter (1999a). 13

14 p(j) = (2) n=2 n jaj 1=2 1 exp A vs 2 v=2 1 vs 2 p() = 2 (v=2) p(; ) = p(j)p() 2 = C n v 1 exp exp v : h vs A i where is the prior mean of, cov[j] = 2 A 1 and E[ 2 ] = vs 2 =(v 2) and v=2. C = (2) T=2 jaj 1=2 2 vs 2 (v=2) 2 This prior ful lls the conditions described by Andrews (1994) necessary for P O T Exp-LM T p! 0. Under these priors the conditional marginal likelihood will be 11 : Z f T ( Y e T jm j ) = where z! 1 2 ja j j A j + X j () 0 X j () (vs 2 + vb 2 j + Q j ) (v+t )=2 d (13) Q j = ( b j j ) 0 [A 1 j + (Xj 0 () X j () 1 )] v = T n z = [(v + T ) =2] vs2 v=2 (v=2) T=2 : 1 ( b j j ) A di culty arises here as we only have information about at discrete and unevenly spaced intervals as determined by observations on the threshold variable Z t. However, Koop and Potter (2000) note that while is continuous, its e ect on the likelihood (11) is the same as if it was discrete. This is because the possible values of can be restricted to the observed data points on Z t. Thus the likelihood (11), marginal likelihood function, (13) and the posterior distribution of will be at between the observed data points for, in this case each observation on Z t. This allows (13) to be calculated by obtaining the conditional marginal likelihood f T ( Y e T jm j ; ) for each observation on Z t and nding the weighted sum across the T observations, with weights determined by the relative size of the interval between each Z t. Then one simply has to divide these by the height of the integrating constant, which, in this case, is the sum of the value of the marginal likelihoods, calculated for each observed Z t. Following Koop and Potter a continuous uniform prior is adopted for the threshold: 1 p () = : (14) u l 11 See Judge et al (1985), p.129. for more setails 14

15 Here u is the upper allowable value for the prior and l is the lower allowable value for the prior. It is necessary to choose u and l to be within the range of observed threshold variables Z t to ensure that neither X 1 or X 2 are null vectors and that analytical solutions for (13) will exist. This restriction is similar to Caner and Hansen s (2001) restriction of to the interval [ l ; u ] such that Pr(Z t l ) = 1 and Pr(Z t u ) = 2 where 0 > 1 ; 2 > 1. Caner and Hansen s (2001) and Koop and Potter s (1999a) recommendations are followed in treating 1 and 2 symmetrically so that 2 = 1 1. This equivalent to restricting so that neither regime accounts for more than 1 % of observations on Z t. Caner and Hansen recommend that 1 be selected carefully so as to ensure that there will enough observations to identify the parameters. For the Bayesian test 1 and 2 need to be selected so that there are enough observations in each regime for the data to be reasonable informative about the posterior distributions of regime dependent parameters. Let be the vector (Z 1 ; Z 2 ; :::; Z T ) 0 sorted into ascending order and let i denote the i th element of. Under (14) the marginal conditional likelihood associated with (3) can be calculated as f T ( e Y T jm j ) = X 2T i= 1T ( ) ( i+1 i ) P 2T i= ( f T ( Y e T jm j ; = i ) (15) 1T i+1 i ) This is similar to Phillips and Zivot s (1994) Bayesian approach to inference on unit roots when break points in the DGP are unknown. Zivot and Phillips place a uniform prior on the position of the break point and produce posterior distributions for making unconditional inferences about the parameters of the model by averaging the marginal likelihood of the data over all possible break points. 12 As X 1 and X 2 are linearly independent with a prior covariance of zero, the posterior distributions of 1 and 2 will be distributed as independent, univariate-t random variables. This property can be used to nd weights for the conditional marginal likelihoods of model M 2 and models M 4 through to M 7 so that only the non-explosive regions of 1 and 2 are considered. 5 Performance of the Test This section compares the performance of the Bayesian test to Caner and Hansen s bootstrap procedure. To do so, the size and power of the bootstrap tests are compared with Bayesian size and power equivalents. The Bayesian power equivalent is de ned to be the percentage of Monte Carlo draws for which the posterior model probability associated with the null hypothesis is smaller than some constant c, (Pr(H 0 jy e T ) c). The size equivalent is de ned analogously. As the Bayesian paradigm places emphasis on the strength of evidence in favour of the null, many Bayesian statisticians, such as Gelman, Carlin, Stern 12 Marriot and Newbold (2000) adopt a similar strategy. 15

16 and Rubin (1995), would argue that such a strict decision rule imposes an arti cial dichotomy between the null and alternative hypotheses. However if a comparison between the performance of Bayesian and frequentist inferential procedures is to be made, a decision rule needs to be put in place. It is well known that the posterior model probability associated with a null hypothesis can be very di erent from the p-value of a frequentist test. 13 As discussed by Berger and Sellke (1987), the prior model probability attached to a point null hypothesis must be relatively small in order to produce a posterior model probability Pr(H 0 jy e T ) 5 0: Put another way, posterior odds ratios are often more conservative than p-values in their propensity to reject the null hypothesis. As is it unclear precisely what decision rule should be employed when comparing the small sample properties of Bayesian and frequentist inferential procedures, the results of the experiments for both c = 0:05 and c = 0:5 are reported. Pr(H 0 jy e T ) = 0:05 is equivalent to a Bayes factor of 19, which represents strong evidence against the null on Kass and Raftery s scale. Alternatively Pr(H 0 jy e T ) = 0:5 corresponds to a Bayes factor of 1 and represents only weak evidence against the null. Kass and Raftery (1994) stress that the results of posterior odds are quite sensitive to how the priors are speci ed for the parameters of interest. As emphasized by Koop and Potter (1999a), the speci cation of di use priors for any parameter in will tend to bias inference towards the linear model. Similarly, a very di use prior on the i parameters will tend to bias inference towards non-rejection of the unit root hypothesis. The prior means of each i are set to zero, which is equivalent to a prior of a linear process with a unit root. The prior variance of the i and i parameters are set to 1, while the priors for the intercepts, i are more di use, with prior variances of 1/4. The prior probability assigned to each model M 1 ; :::; M 7 is distributed evenly. So that there is enough information contained in the data to make reasonably precise probability statements about any threshold e ect. The priors on the upper and lower bound of the threshold parameter, l and u are speci ed such that must lie within the middle 70% of the observed Z t. Monte Carlo experiments identical to those described in Caner and Hansen (2001) are conducted. Selected results of Caner and Hansen s Monte Carlo experiments are provided for comparison. Due to the computational expense incurred with Caner and Hansen s methods, size and power results for Caner and Hansen s procedures have been taken directly from their article and reproduced here. In each experiment the delay lag of the threshold parameter, m is set to equal 1, = 0 and k = See, for example, Berger and Sellke (1987), Edwards, Lindman and Savage (1963) and Dickey (1977) 14 Although Casella and Berger (1987) and DeGroot (1973) have shown that when comparing a one-sided null hypothesis on a scalar parameter with a one-sided alternative, p-values and posterior model probabilities may be directly comparable. 16

17 5.1 Power of the Threshold Tests Caner and Hansen (2001) examine the power of their threshold test by investigating thresholds in each of the coe cients of (3) individually. They measure the empirical power of their test by the empirical rejection rates from 2000 replications of each experiment at the nominal 5% level. Following Caner and Hansen the sample size for the data generated in each experiment is T = 100 and 2000 replications of each experiment are performed. The rst Monte Carlo experiment allows for a switching intercept only, with data simulated from the DGP y t = 1 + y t 1 + y t 1 + e t if Z t 1 < 2 + y t 1 + y t 1 + e t if Z t 1 = e t i:i:d:n (0; 1) (16) with = 0:5, = 0 and taking on values f0; 0:05g. The magnitude of the threshold e ect is controlled by the di erence = 2 1 and allowing the intercepts in each regime to vary symmetrically around zero ( 1 = 2 ). The second experiment allows for a threshold e ect in the coe cient on y t 1 : y t = + 1 y t 1 + y t 1 + e t if Z t 1 < + 2 y t 1 + y t 1 + e t if Z t 1 = e t i:i:d:n (0; 1) (17) with = 1, = 0 and = 0:5. Here the magnitude of the threshold e ect is controlled by = 2 1. In the third experiment there is a threshold e ect in the autoregressive parameters, 1 and 1 : y t = + yt y t 1 + e t if Z t 1 < + y t y t 1 + e t if Z t 1 = e t i:i:d:n (0; 1) : (18) The magnitude of the threshold e ect is de ned as = 2 1 and 1 = =2. = 1, = 0 and takes on the values {0,-0.05}. The results of these experiments are presented in table 1. The power of the Bayesian test to reject the linear model in the presence of a threshold e ect appears to compare quite favorably with Caner and Hansen s procedure. In each experiment, the power of test is monotonically increasing in the size of the threshold e ect and reasonably large, even against quite conservative alternatives for the Bayesian 5% test. The Bayesian 50% test dominates Caner and Hansen s test for all experiments except the one for which = 0 and = 2. Although, a 51% posterior model probability in favour of the threshold model represents only weak evidence against the null. 17

18 Table 1: Empirical power of the threshold tests Bayesian test (5%) Bayesian test (50%) Sup-Wald test = 0: = = 0: = = 0: = Size of the Threshold Tests Caner and Hansen examine the size of their threshold test by simulating the data under the null hypothesis of a linear AR(1) process: y t = + y t 1 + y t 1 + e (19) e t i:i:d:n (0; 1) : Here takes on the values f0; 0:05; 0:15; 0:25g and alternates between f0:5g and f 0:5g. Following Caner and Hansen the sample size of each experiment is set to T = 100 and replications of each experiment are performed. Results of this experiment, reported as the percentage of p-values which are smaller than the nominal size for the Sup-Wald test and the percentage of simulations for which posterior model probability of a linear model is less than 0.05 and 0.5 are presented in table 2. Caner and Hansen s found their constrained bootstrap procedure to perform slightly better in terms of size than the unconstrained bootstrap. Therefore, Caner and Hansen s results for the constrained bootstrap are presented in table 2. The results presented in table 2 demonstrate the more conservative nature of the Bayesian test, with the percentage of rejections of the true null hypothesis being much smaller for the 5% Bayesian test than for Caner and Hansen s procedure. 18

19 Table 2: Emprical size of the threshold tests Bayesian test 5 % = 0:25 = 0:15 = 0:05 = 0 = 0: = 0: Bayesian test 50 % = 0: = 0: Sup-Wald test = 0: = 0: Power of the Unit Root Tests Caner and Hansen examine the power of nominal 5% tests with data simulated from the DGP 1 + y t = 1 y t 1 + e t if Z t 1 < (20) y t 1 + e t if Z t 1 = e t i:i:d:n (0; 1) Following Caner and Hansen the sample size of each experiment is set to T = 100 and 1000 replications of each experiment are performed. In the rst experiment, the DGP is stationary and linear in, but subject to a threshold e ect in the intercept. Speci cally 1 = 2 are varied among {-0.05,-0.10,-0.15} and is varied among {0,1,2,3}. The second Monte Carlo experiment examines the case of a partial unit root. This time 1 = 0 and 2 takes on the values {-0.05,-0.10,-0.15}. In the nal experiment the experimental data is simulated under the condition that 1 = 0:05. Results are presented in table 3. The Bayesian 50% test outperforms Caner and Hansen s test for all experiments and, strikingly, the more conservative Bayesian 5% test outperforms Caner and Hansen s test for half of the experiments. The advantage of the Bayesian test over the R 1t test increases in the magnitude of threshold e ect. 5.4 Size of the Unit Root Tests Caner and Hansen examine the size of their unit root test by simulating the data under the null hypothesis of a unit root with a threshold e ect in the intercept only: y t = 1 + y t 1 + e t if Z t 1 < 2 + y t 1 + e t if Z t 1 = e t i:i:d:n (0; 1) : (21) 19

20 Table 3: Empirical power of the unit root tests Bayesian 5% test Bayesian 50% test R 1t test 2 : = : = : = 0: : is varied among f 0:5; 0:2; 0; 0:2; 0:5g and = 2 1 is varied among f0; 1; 2; 3g with 1 = 2. Following Caner and Hansen, the sample size for the data generated in each experiment is T = 100 and 1000 replications of each experiment are performed. Results of experiment (21) are reported as the percentage of posterior model probabilities smaller than c, and presented in table 4. Also presented in this table are the percentage of p-values which are smaller than the nominal size reported in table IV of Caner and Hansen (2001, p. 1574). Caner and Hansen nd that while the t 1 test and ADF test have reasonable size, the R 1T and R 2T tests tend to over-reject the (true) null hypothesis. The R 1T test is found to slightly outperform the R 2T test. For this reason results from both the R 1T test and the t 1 test are reported. As expected, the more conservative Bayesian 5% procedure seems to do as well as the t 1 test in terms of size for all experiments. The 50% Bayesian test also appears to perform quite well. As would be expected, the number of false rejections by the Bayesian test appears to be decreasing with the magnitude of the threshold e ect. Interestingly, the degree by which Caner and Hansen s R 1T test is oversized, seems to be increasing with the magnitude of the threshold e ect. 5.5 Prior Sensitivity Analysis As discussed in Koop and Potter (1999a) the posterior model probabilities in favour of nonlinear speci cations is reduced when the prior variance on the 20

21 Table 4: Empirical size of the unit root tests Bayesian 5 % test Bayesian 50 % test : : t 1 test R 1T test : : coe cients become large. In order to investigate the sensitivity of the results of the Monte Carlo experiments to such e ects, these experiments are repeated for two alternative prior speci cations. 15 In the rst alternative prior speci cation, the prior standard deviations of the regression coe cients are doubled. The results of the Monte Carlo experiments described above are for this prior speci cation are presented in tables 6 to 9 of Appendix A. For convenience, the results of Caner and Hansen s Monte Carlo experiments are also replicated in these tables. Obviously they are identical to those presented above. As expected, the power of the test for a threshold e ect is generally reduced for both the 5% and 50% Bayesian tests. However, the reduction in the power of the test does not substantially a ect the conclusions drawn in that power of the test remains comparable with Caner and Hansen s (2001) Sup-Wald test. Correspondingly, the size of the threshold tests are reduced under this prior for both the 5% and 50% tests. The size of the 5% test is very small, with very few false rejections. The size of the 50% test is also improved, with this test now performing better in this regard than Caner and Hansen s Sup-Wald test for = 0 and = 0:05. When the prior standard deviations of the coe cients are doubled, the power of the Bayesian unit root tests are generally reduced. However the empirical power of the Bayesian 50% test remains comparable with Caner and Hansen s R 1T test. The performance of the Bayesian test in this regard is generally superior to the R 1T test for values of which are close to 0 and smaller threshold e ects. Meanwhile the performance of the R 1T test is superior to the Bayesian 50% test for larger threshold e ects. Under this prior, the size of the 15 These alternative speci cations are similar to those adopted by Koop and Potter (1999b) for a similar investigation into the sensitivity of Bayesian inference for threshold models to prior speci cations. 21