Structural Breaks in GARCH Models.

Transcription

1 Structural Breaks in GARCH Models. Daniel R. Smith Faculty of Business Administration Simon Fraser University 8888 University Drive, Burnaby, BC, Canada, V5A 1S6 Date: July 16, 2003 The is a preliminary first draft and should not be quoted without the authors permission.

2 Structural Breaks in GARCH Models. Abstract The GARCH model is one of the most important and widely applied econometric models in finance and is used extensively in empirical asset pricing and risk management. However, GARCH models are only rarely examined with diagnostic tests and almost never tested for structural breaks. In this paper we propose a new test for model specification which uses GMM and apply the structural break tests of Andrews (1993) and Andrews and Ploberger (1994) to GARCH models fitted to a number of financial time series. We find that many of the estimated GARCH models fail the structural break test while passing diagnostic tests for autocorrelation and heteroscedasticity. To endogenously allow for time-varying parameters we fit a Markov-Switching GARCH model to exchange rates and stock indices and find that it fails to solve the structural break problem. Keywords: GARCH, Diagnostic tests, Structural breaks, Regime-switching. JEL: C22, C52. 1

3 Contents 1 Introduction 3 2 GARCH Models 6 3 Diagnostic Tests 7 4 Testing for Structural Breaks 11 5 Data 14 6 Empirical Results Diagnostic Tests Probabilistic Structural Breaks The Model Markov-Switching GARCH Model MSGARCH Diagnostic Tests Results Conclusions 32 A Filtering Algorithm for Probabilistic Structural Break Model 36 2

4 1 Introduction One of the more important innovations in finance over the past two decades has been the modelling of conditional volatility. Possibly the most influential model of conditional volatility which has been the subject of numerous empirical applications are the GARCH model of Engle (1982) and Bollerslev (1986). One important limitation of this literature is the paucity of structural break tests which are applied to GARCH models. Structural break tests are an extremely important diagnostic tools available to econometricians. Nearly all good intermediate undergraduate econometrics texts devote at least some attention to structural break tests. One of the basic assumptions used to derive useful properties of the common estimation methods (i.e. maximum likelihood, least squares, generalized method of moments, etc.) are that the model is well specified which in tern requires parameters which stay constant through time. When modelling random regression coefficients or time-varying volatility we require that the parameters which describe the data generating process of the random coefficients or volatility themselves be stable through time. Parameter instability is evidence of model misspecification and standard econometric theory no-longer applies. Given the importance of structural break tests when specifying means, it is curious that despite the literally thousands of empirical applications of the GARCH models, only a handful of tests for structural breaks have been implemented. Examples of these tests include Lundbergh and Terasvirta (2002), who develop a number of specification tests, including a test for parameter constancy against a threshold-garch alternative (which is related to structural break tests), unfortunately there were no empirical applications in their paper. In this paper we apply a well know standard test for actual structural breaks which more directly answers the question: Are there structural breaks in GARCH models? Another relevant paper is Malik (2003), who develops a test based on the itterated cumulated sums of squares algorithm and analyzes five exchange rates from January

5 to September 2000 and find a number of structural breaks in the data. The ICCS algorithm works by identifying breaks in the volatility of a time series, and assumes that the volatility between two break points is constant. Malik finds a number of breaks in the different exchange rate series that were analyzed. Although being a very interesting line of enquiry, this paper also is unable to test for structural breaks in GARCH models and it suffers from a number of logical shortcomings. In particular, after determining the break points by this ICSS algorithm, Malik fits a dummy variable to allow the unconditional volatility (the constant term in the conditional volatility formulae) to be different between these break dates. Malik s test requires constant volatility between break dates, yet these are used to fit break points in a model which explicitly recognizes the random nature of volatility. Secondly, the dummy variables are endogenously determined and subject to estimation error. This will influence the standard errors of the parameters. A key finding in Malik is that accounting for these breaks reduces the estimated persistence of volatility shocks. This inference would depend on the estimated break dates. The paper does not test if the coefficients on the dummy variables are different from zero, which would suggest the existence of structural breaks, but again failure to account for estimation error would preclude the use standard hypothesis testing procedures. This paper fills the gap by applying the structural break tests of Andrews (1993) and Andrews and Ploberger (1994) to GARCH models fitted to stock return and exchange rate data. Current practice in diagnostic testing of GARCH models is to test for serial correlation in the standardized residuals and squared standardized residuals (as does Malik (2003)). We apply diagnostic tests which are inspired by these residual diagnostic tests but which account for estimation error. One limitation of current diagnostic tests are that they assume that residuals are known. Examples of this include the size, positive and negative sign bias tests of Engle and Ng (1993), and the common practice of calculating portmanteau statistics for serial correlation in the standardized residuals and squared standardized residual. In this paper we develop and apply diagnostic tests for the standardized residuals which account for parameter uncertainty and look for autocorrelation, omitted linear dependence in volatility 4

6 (ARCH effects), and skewness. This basic testing approach is also applied to a GARCH model which endogenously allows parameters to change through time (a Markov-Switching GARCH model). We find some evidence that these time series exhibit structural breaks. We analyze the relatively short post-1989 sample period and find mixed evidence of structural breaks. The stock index, Canadian and Pound exchange rates and several stocks exhibited evidence of structural breaks. In our full-sample analysis every GARCH model is rejected. This point deserves repeating. Every GARCH model is rejected as exhibiting a structural break. This is true of those series which pass the residual diagnostic tests for autocorrelation and omitted ARCH effects. The discovery that a well accepted time-series model fails a test for structural break is not without precedent in the finance literature. Ghysels (1998) demonstrates that a number of different conditional asset pricing models which were and still are widely used in the empirical literature exhibit structural breaks. These models include implicit beta methods, which do not require specifying the explicit dynamics of either asset betas or covariances; and explicit models where the asset betas are modelled as linear functions of conditioning variables. These models received wide application in the literature before Ghysels (1998) demonstrated the existence of breaks in the series. The Markov-switching GARCH (MS-GARCH hereafter) models of Gray (1996), Dueker (1997), and Klaassen (2002) are generalizations of the basic GARCH model that is similar in spirit to a structural break since they allow the underlying parameters of the GARCH model to change through time. We consider a special case of the MS-GARCH model in which there is a probabilistic structural break. In this model the break can occur at each point in time and this probability is an estimated parameter. We develop a recursive filter which updates the probability of each breakpoint using progressively more and more data. This allows us to date and draw inference regarding the date of the structural break. This probabilistic structural break model helps bridge the gap between the structural 5

7 break results and the MS-GARCH literature. One could interpret the evidence rejecting the basic single-regime GARCH models as qualitatively supporting the need to allow for discrete regime shifts as recognized by the MS-GARCH literature. Klaassen (2002) demonstrates that the MS-GARCH models (in the context of foreign exchange rates) show significant improvement in forecasting future market volatility. The two-regime constant transition probability MS-GARCH model fails the structural break tests. The remainder of the paper proceeds as follows. Section 2 briefly discusses GARCH models and current diagnostic testing procedures. Section 3 discusses and develops general diagnostic tests based on GMM and quasi-maximum lieklihood. Section 4 discusses structural break tests where the break point is not prespecified. Section 5 describes the data used in the empirical portion of the paper. Section 6 presents the basic empirical results. Section 7 presents the probabilistic structural break GARCH model and discusses the evidence on breaks it contains. Section 8 presents the MS-GARCH model and tests it for structural breaks. The papers results are summarized and conclusions are drawn in section 9. 2 GARCH Models The GARCH model was introduced by Engle (1982) and Bollerslev (1986) in which conditional volatility is a (typically linear) function of lagged squared residuals and lagged conditional volatility. The most common variant is the GARCH(1,1) model (Bollerslev, Chou, and Kroner, 1993) in which volatility is modelled as: h t = ω + αe 2 t 1 + βh t 1 (1) where e t = y t E t 1 y t is the unexpected component of y t. The original ARCH model was motivated by the observation of volatility clustering that is observable in many financial and economic time series. Specifically, conditional volatility is modelled as E t 1 (e 2 t ) = ω + αe 2 t 1. 6

8 This model assumes that the relationship between lagged surprises and conditional volatility is symmetric. However, it is a well known empirical regularity that in equity market conditional volatility rises more following negative surprises than following positive returns. This observation was initially documented by (Black 1976) and Christie (1982) and its most common incarnation in the GARCH literature are by Glosten, Jagannathan, and Runkle (1993) (the GJR model) and Nelson (1990) (the EGARCH model). Engle and Ng (1993) analyzes a number of different variants of GARCH models that capture this so-called leverage or volatility feedback effect, and found that in their daily stock index series that both the EGARCH and GJR models fit the data the best and perform comparably. The GJR(1,1,1) model is a very simple extension of the basic GARCH(1,1) model h t = ω + αe 2 t 1 + βh t 1 + δe 2 t 11(e t 1 < 0). (2) In our empirical applications we use a standardized T distribution with ν 1 degrees of freedom, with ν being estimated as a free parameter. The variance of y t is σt 2 = h t ν 1 /(ν 1 2) where h t, which we will term volatility, is a scale parameter so e t / (h t ) has unit variance. 3 Diagnostic Tests It is common practice in building GARCH models to analyze the standardized residuals for serial correlation and squared serial dependence. If the econometric specification is adequate, then it should account for linear dependence in both the conditional mean and variance. This implies that the standardized residual will be uncorrelated with past residuals, and the squared residuals will be uncorrelated with lagged squared residuals. A common practice is to construct portmanteau statistics which measure the predictability of residuals and squared residuals using past standardized residuals and squared standardized residuals. However, these statistics ignore the estimation error that arises when estimated standardized residuals 7

9 are used rather than the true standardized residuals. We present diagnostic tests which are designed to detect unmodelled linear dependence in the mean and variance and which accounts for parameter uncertainty. We also construct a comparable test for nonzero skewness. These tests are constructed using Generalized Method of Moments of Hansen (1982). If our GARCH model is correctly specified (and a symmetric distribution is assumed), the following moment conditions will be satisfied: E(e t e t j ) = 0 for i = 1,..., n E[(e 2 t σt 2 )e 2 t j] for i = 1,..., n E(e 3 t ) = 0. We can test these three moment conditions using GMM. To implement these three diagnostic tests we construct the following three moment vectors which are theoretically mean zero, corresponding to each of the three moments: m kt for k {1, 2, 3}: m 1t = [e t e t 1 e t e t 2... e t e t n ] m 2t = [(e 2 t σt 2 )e 2 t 1 (e 2 t σt 2 )e 2 t 2... (e 2 t σt 2 )e 2 t n] m 3t = e 3 t. In this notation m k refers to the test relating to the appropriately transformed k-th moment or autocomoment. As mentioned above, the first moment condition is linear independence of residuals (autocorrelation) ; the second moment condition is that the realized volatility is unforecastable beyond the GARCH-implied conditional volatility (omitted ARCH effects); and we are using symmetric distributions which implies that the third central moment is zero. The first two tests are n-dimensional and third is scalar. These dimensions determine the degrees of freedom in the test statistics chi-squared distribution. 8

10 To construct the GMM-based test for the hypothesis, collect all model parameters (both mean and variance) into the p-dimensional vector θ. The moment conditions consistent with (quasi) maximum likelihood (QML hereafter) are that the score l t = log ft(yt Y t 1,θ) θ has zero expectation E(l t ) = 0 where Y t 1 is the sigma-field generated by past observations on y t. The QML parameter estimates θ set the sample moments identically to zero. The covariance matrix based on GMM using scores as moments equals the QML standard errors of White (1982) and Bollerslev, Engle, and Wooldridge (1988). Our tests are constructed We construct the moment conditions. g jt = [m kt l t ] (3) We use the generalized method of moments (GMM hereafter) of Hansen (1982) to construct the test statistic. The model defines a set if r moment conditions of the form E(g t ) = 0 r 1. The moments depend on the data and a p-vector of parameters θ. In our case r = p + n; the p parameters estimated by QML and the K diagnostic test. Normally in GMM we need at least as many moments as parameters for identification. The parameters are estimated so as to set the sample moments close to their theoretical counterparts (i.e. zero) as possible. Formally, the parameter estimation problem is to minimize the quadratic form J T (θ) = g T W g T using the sample moments g T = 1 T T t=1 g t and the symmetric positive semi-definite weighting matrix W. If there are exactly as many parameters as moments, all moment conditions can be set to zero and the system is said to be exactly identified. A model is overidentified when there are more moments than parameters. In this case not all moments can be set to zero (typically none are set identically to zero) and the weighting matrix summarizes the relative importance of different moments. 1 1 The efficient weighting matrix which has the lowest sampling variability is W = S 1, in inverse of the moment variance-covariance matrix. This places higher weight on moments which have lower variance and are consequently measured most precisely by sample estimates. 9

11 Given these assumptions, we can construct a test statistic H T is constructed as: H T = T g T [(I D T (D T W D T ) 1 D T W )S T (I W D T (D T W D T ) 1 D T )] + g T (4) where + denotes the pseudo-inverse which is required since the covariance matrix of g T singular, having rank n = r p, D T = g T θ, and S T is a consistent estimate of the variancecovariance matrix of the moment conditions and can be constructed using heteroscedasticity and autocorrelation consistent estimates of Newey and West (1987) and Andrews (1991). In maximum likelihood the moment conditions are obtained by setting the expectation of the score equal to zero. In this setting there are as many moments as parameters. The maximum likelihood parameter estimates are obtained by setting the first-order conditions equal to zero. This results in all sample moments, the sample score and first-order conditions, being set equal to zero. When the score is the chosen moment conditions the objective function is set to zero for all weighting matrices. To construct our model specification tests we will augment the p dimensional score vector with n extra moments satisfied when the model is correctly specified, as discussed above. By defining the r r weighting matrix with a p dimensional identity matrix with zeros elsewhere: 0 n p W = 0 n n. (5) 0 p n I p The resulting parameter estimates will be the QML estimates since the minimum of the quadratic objective function will obtain by setting each of the p-elements of the sample score equal to zero as does the QML estimates. The H T statistic for each diagnostic test turns out to be a Wald-type test in terms of h T since the only non-zero sample moments are those related to the diagnostic test. Denote by is V = (I D T (D T W D T ) 1 D T W )S T (I W D T (D T W D T ) 1 D T )/T 10

12 the variance covariance matrix of the augmented moment condition vector g T. partition this covariance matrix as V = V hh V sh V hs V ss, We will with V sh = Vhs. We can now express the test statistic as H T = h T (V hh V sh V 1 ss V hs ) 1 h T. The covariance matrix of the moment conditions h T includes two terms: the standard variance of the moment conditions, and the adjustment V sh Vss 1 V hs which corrects for estimation error. Ignoring estimation error overstates the variance of the moment conditions, reducing test statistics which will result in too few rejections of the null hypothesis (Breunig and Pagan forthcoming). These diagnostic tests are similar in spirit to the approach of Breunig and Pagan (forthcoming). An advantage of this diagnostic test is its generality. It can in principal be applied to any function of the residuals, so the portmanteau tests with standardized residuals and squared standardized residuals are nested in this approach. 2 4 Testing for Structural Breaks Imagine an econometric model parameterized by θ t which is fit to time series data y t for t = 1,..., T. We are particularly interested in testing for a one time change in the parameters. The change point occurs at observation πt with π (0, 1) being the proportion of the sample occurring before the break. The model parameters before the break are θ 1 and after 2 We favor testing with the residuals and the difference between the squared residual and its conditional expectation to avoid standardizing, which is a nonlinear function of the conditional volatility, which can be confounded by estimation error. 11

13 the break θ 2, in particular θ 1 (π) θ t = θ 2 (π) for t = 1,..., πt for t = πt + 1,..., T (6) The null hypothesis is that H 0 : θ t = θ 0 t. When the break point π is known, testing for structural breaks is a standard and relatively trivial exercise. The Chow test is a very well know textbook structural break test and is constructed as an F -test for the null hypothesis that θ 1 = θ 2 for known break point π. However, when the break point is unknown, standard tests no longer have standard distributions. To that end, Andrews (1993) and Andrews and Ploberger (1994) develop tests for structural breaks which are valid for unknown break points, and tabulate asymptotic critical values for these test statistics. The Lagrange Multiplier (LM hereafter) test is often used in constructing specification tests since it only requires parameter estimates obtained under the null hypothesis. This is important because it is typically much easier to estimate the null model, and specification tests are applied to an already estimated model which is correct under the null to examine the alternative of incorrect model specification. Under the null of no structural break there is one common parameter vector which holds for the whole sample and this is estimated using maximum likelihood. If one knew the break point was π the Lagrange Multiplier test for the structural break alternative would be constructed as LM T (π) = T ( θ, π) S 1 π(1 π)ḡ1t T D T (D T S 1 T D T ) 1 D T S 1 T ḡ1t ( θ, π). Here ḡ 1T ( θ, π) = 1 T ḡ T ( θ, π) = 1 T T t=1 g(y t; θ) D T = 1 T T t=1 πt t=1 g(y t; θ), S T = 1 T T t=1 (g(y t; θ) ḡ T ( θ))(g(y t ; θ) ḡ T ( θ)) with g(y t; θ) θ. This LM statistic is asymptotically distributed as a chi-squared random variable with degrees of freedom equal to the dimensionality of the parameter vector. Maximum likelihood sets the score, or partial derivative of the log-density with respect to the parameter vector: g(y t ; θ) = log f(yt; θ) θ equal to zero, so D T is the hessian-based estimate 12

14 of the information matrix, and S T is the outer-product estimate. The sample mean of the moment condition is set set to zero by construction, so the two partial means must add to zero. If the null hypothesis is true, then each sub-sample score will be close to zero since the model is correctly specified. However, if there is a structural break in the parameter, the estimate which sets the whole sample score will do a very poor job in each sub-sample. In other words, ḡ 1T need not equal zero and in fact m 1T ( θ, π) = ḡ 2T ( θ, π) which can take any value. This last result illustrates why the LM test only uses the first partial mean. Unfortunately the break point is typically unknown and standard distributional theory no-longer applies since the information matrix is singular under the null hypothesis of no structural break the log-likelihood takes the same value for all possible break points when θ 1 = θ 2. Andrews (1993) considers the sup-lm test which is inspired by the work of Davies (1977, 1987): sup LM T (π). (7) π Π The null hypothesis is rejected for large values of this test statistic. Andrews (1993) notes that the sup-lm statistic is poorly behaved when Π = [0, 1] since it diverge as the sample size increases: lim T sup π [0,1] LM(π) =. However, when π is bounded away from zero and one, the test statistic has a well defined asymptotic distribution which is dependent on the number of parameters and the proportion of the sample excluded by this bounding exercise. Critical values of the test statistic are tabulated in Andrews (1993). One limitation of the sup-lm test is that it depends on only one break point. The power of the tests can be improved by considering test statistics over a range of different possible break points. Andrews and Ploberger (1994) develop optimal tests which are (weighted) averages of the individual LM T (π) tests over different possible break points π. Assuming that π is uniformly distributed over [π 1, π 2 ] (π 1 π 2 and bounded away from zero and one), 13

15 the exponential Lagrange multiplier statistic (exp-lm hereafter) is given by 1 exp LM T = log( T (1 2π 0 ) [ T [π 0 T ] 1 t=[π 0 T ]+1 exp(lm T (t/t )/2) + ([π 0 T ] + 1 π 0 T ){exp(lm T ([π 0 T ]/T )/2) + exp(lm T ((T [π 0 T ])/T )/2)}]), (8) and the average Lagrange multiplier statistic (avg-lm hereafter) avglm T = T [π 0 T ] 1 1 T (1 2π 0 ) [ LM T (t/t ) + t=[π 0 T ]+1 ([π 0 T ] + 1 π 0 T ){LM T ([π 0 T ]/T ) +LM T ((T [π 0 T ])/T )}]. (9) The avg-lm and exp-lm consider the value of the structural break test at all possible break points and has more power than the sup-lm test. The distribution of the test depends on the amount of the sample dropped and on the number of parameters. Although the test statistics don t have a closed form distribution, critical values for the sup-lm test are tabulated in Andrews (1993), and the avg-lm and exp-lm tests are tabulated in Andrews and Ploberger (1994). The tests are relatively straightforward to apply and because they are formulated in a GMM context, they are robust to distributional assumptions and heteroscedasticity, and can be constructed to account for autocorrelation all the standard issues that plague financial time series data. 5 Data We consider GARCH models for 13 common time series which are observed daily: 14

16 The excess returns on value-weighted NYSE stock index from CRSP (VWNYSE) which spans July 1963 to December Four US dollar denominated exchange rate series: the Canadian Dollar (CAD), the Japenese Yen, the German Deutsche Mark (GDM), and the British Pound (GBP). All series start from January 1974 to avoid overlapping with the fixed exchange rate system which ended in We follow Calvet and Fisher (2003) and start the CAD series later (in July 1974) since it was held in parity with the dollar for several months after Bretton Woods ended, and finish GDM in December 1998 which it was replaced with the Euro. Daily returns on eight stocks from July 1963 to December 1998 are used: ATT, Bethlehem Steel (BS), Dupont (DU), General Electric (GE), General Motors (GM), IBM (IBM), Philip Morris (PM), Texaco (TX). The data is plotted in figures 1 and 2. Summary statistics are reported in Table 1. The data present the standard set of well-known stylized facts about financial time series; nonnormality, limited evidence of short term predictability and strong evidence of predictability in volatility. All series are presented in daily percentage growth rates/returns. Individual stocks have the highest unconditional variance, while the exchange rate series have the lowest. The conditional mean return on the individual stock is significantly higher because they are measured as raw returns, which the index is measured in excess returns. The Bera-Jarque test for normality is convincingly rejected for all series: the 5 percent critical value os 5.99 the lowest test statistic is around 20 times that. Most of the nonnormality comes from excess kurtosis rather than skewness. The stock index is negatively skewed and has fat tails. Roughly one half of the exchange rates and individual stock returns are negatively skewed and the others are positively skewed, though the negatively skewed series are significantly larger in magnitude. The asymptotic standard error of the skewness statistic under the null of normal is by 6/T, and the standard error of the kurtosis statistic is 24/T. Approximately two-thirds of the skewness statistics are statistically different from 15

17 zero (using the asymptotic Z-score), while all 13 series exhibit kurtosis which is statistically distinguishable from 3. Thus one would argue that fat tails is a more significant problem. For this reason we follow Bollerslev (1987) and model the standardized residual using a T random variable, and use robust standard errors to account for possible skewness. We use the Box-Pierce portmanteau, or Q statistic with 5 lags to test for serial correlation in the data, and we adjust the Q statistic for autocorrelation following Diebold (1986). The standard Q test indicates autocorrelation in all but GDM, however after adjusting for possible ARCH effects the evidence becomes weaker. When using only the post-1989 data the evidence is weak at best. The evidence of linear dependence in the squared demeaned returns, which is an indication of ARCH effects in the data, is much stronger. These results are fairly standard: much stronger evidence of predictability in volatility than for return predictability. A note of caution is in order: these statistics are a useful summary statistic and are widely used, but their distribution relies on distributional assumptions not likely to apply in all cases. When choosing the model specification below we will use a combination of robust Lagrange Multiplier and Wald tests whose distributions are more reliable than the Q tests. 6 Empirical Results It is well known that the basic GARCH(1,1) with an AR(1) model for the mean is able to capture much of the dynamics in the conditional mean and variance, and that the GJR model is able to capture asymmetry in the mean. There is also much evidence that there is excess kurtosis in returns and some evidence of skewness. These stylized facts are present in our data also. Because of these stylized facts we focus on models which are special cases of the AR(1)-GJR(1,1,1)-T model, which allows for a first order AR(1) process to capture serial dependence in the data, a GARCH(1,1) model augmented by an asymmetry parameter following Glosten, Jagannathan, and Runkle (1993), and we follow Bollerslev (1987) and 16

18 model the standardized innovation as a T random variable to capture fat tails. We model the inverse of the degrees of freedom parameter ν 2 to facilitate testing the nested null of gaussian disturbances which obtains as lim ν 1 0. We start out with the parsimoneous AR(0)-GJR(1,1,0) model and test whether the autocorrelation and asymmetry parameters are equal to zero. We also conduct diagnostic tests for autocorrelation, uncaptured ARCH, and skewness of the residuals. Even when the null hypothesis that the AR parameter is zero is not rejected but there is evidence of serial correlation in our short sample period we add the AR parameter. When estimating parameters we impose the constraint of covariance stationarity using the sufficient conditions ω, α, β, δ > 0 and α+β +.5δ < 1 (assuming symmetry), and φ < 1. In particular we constrain α + β +.5δ For three of the exchange rate series, the constraint that α + β.9999 binds. In this case we drop the redundant parameter β when calculating standard errors. (Since β =.9999 α the standard error of β equals the standard error of α which is the only of the two which is a freely estimated parameter.) Table 2 reports the parameter estimates for the four exchange rate series, the stock index and the 8 individual stocks for the post-1989 sample, while table 4 reports the same for the entire sample. We follow the above procedure for both sample periods, though we are less concerned by rejections of the diagnostic tests in the full sample. As is common with daily GARCH models are quite persistent with α + β close to unity, and for most of the model α is approximately 0.05 for both sample periods for the index and exchange rate data. The distribution of α s and persistence are more disbursed for the individual stock data. There is a strong leverage effect in the VWNYSE in both subperiods, and for most individual stocks in the full-sample period but much less in the post-1989 sample. We assume that all series are distributed as T random variables and estimate the degree of freedom. In fact it is the inverse of the degrees of freedom which is estimated so that the normal distribution is a special case as ν 0 to facilitate hypothesis testing. It is clear that there are fat tails in all time series. The point estimates are all many times their standard 17

19 errors, indicating that they are different from the normal; and all are less than 1/2 which indicates that they have a variance (i.e. degrees of freedom greater than 2). The typical degree of freedom parameter is roughly between 5 and 10. Leverage effects are not normally found in exchange rate models (see, e.g. Calvet and Fisher (2003), Malik (2003), Klaassen (2002) for recent references). We however find a different result for the CAD series which has an non-zero asymmetry parameter, while all other series appear to be symmetric. Although it is clear from a Wald test that δ is significantly negative (grater than 10), and a likelihood ratio statistic is greater than 12, the robust Lagrange Multiplier test is It appears that a quadratic approximation to the likelihood function which is required for all three tests to be identical is patently false: in the neighborhood of the GARCH null hypothesis the likelihood surface is flat in δ, while the function increases rapidly towards the alternative and the curvature is quite steep around the alternative point estimate. It is rather interesting that in both sample periods there is evidence of an asymmetric effect in the CAD series, but that positive shocks increase future volatility than a negative shock of comparable magnitude. For example, if volatility were at its unconditional value, a positive two standard deviation shock will increase volatility by about 24 percent, while a negative two standard deviation shock will increase volatility by only about 8 percent. 3 This could be interpreted as evidence of policy uncertainty following a positive shock to the Canadian dollar. A devaluation is typically perceived as good news for exporters. The sharp increase in the Canadian dollar in early 2003 resulted in calls from exporters for the Bank of Canada to decrease interest rates and narrow the gap between the US and thereby devalue the Loonie to aid export competitiveness. This tradeoff between inflation and export potential though somewhat unusual would create no conflict were the roles reversed and the dollar had received a negative shock. This can be viewed in line with the increase in inflation uncertainty following positive shocks to inflation due to policy uncertainty as discussed in 3 Assuming symmetry of the innovations, the unconditional variance is h = /( ), and the proportional increase is 0.24 = ( h h)/ h 1 for a positive shock, and 0.08 = ( ( ) 2 2 h h)/ h 1 18

20 Ball (1992). 6.1 Diagnostic Tests The specification of the GARCH models are analyzed using structural break tests for autocorrelation, omitted ARCH and skewness using a GMM procedure which corrects for estimation error. The results of these tests are reported in tables 3 and 5. There is some evidence of skewness in the series which is not captured by the symmetric T distribution. The H test has a 5 percent critical value of The index, half of the exchange rates, and five of the stocks have statistically significant skewness in the full sample, although the VWNYSE index and CAD and JPY are skewed in the shorter sample, the evidence of skewness in individual stocks vanishes in all but one case (TX). The T distribution was chosen to capture fat tails but because it is symmetric, it cannot handle skewness. For this reason we report robust standard errors which correct for misspecification of the distribution while maintaining consistency and asymptotic normality. It is reasonable to expect a series which contains a structural break in its mean dynamics to exhibit serial correlation, and that a break in volatility will result in serial correlation in squared residuals (after correcting for conditional volatility). Take the simple case of a constant mean or constant variance model where an unrecognized break occurs to increase the relevant moment. A positive shock to the true model will result in a sample estimate that will likely be between the two true means. This in turn results in positive residuals on average before the break, since the initial sample mean is too high, and negative residuals on average after the break, since the latter sample mean is too low. The autocovariance is the summation of the products of the residuals e t e t 1 /T which will tend to be positive since both before and after the break the products of sequential residuals will tend to be positive since they will have the same sign both before and after the omitted break. A similar story will hold for conditional volatility and squared residuals. 19

21 This result will not hold always since the break may occur in the degrees of freedom, the AR coefficient etc which are less likely to be detected using structural break tests. So one would not automatically expect a series which exhibits a structural break to fail other diagnostic tests. In point of fact this happens for a number of series in the post-1989 sample period. They passed the diagnostic tests but failed the structural break tests (most notably VWNYSE and CAD, and a number of stocks). Conversely, it is well known that structural breaks in volatility can give rise to GARCH-type effects. 4 When we move to the full sample period a number of points stand out. Firstly, the constraint on α + β binds in three of the four exchange rates which it did not on the initial sample period. This suggests that there is something amiss with the model over the longer sample period. Secondly, There is much more evidence of autocorrelation and asymmetry in the individual stocks. More importantly is the universal failure to pass the structural break test. To get a feeling for the severity of the problem, the highest number of parameters in any model is 7 (AR(1) with asymmetry and estimated degrees of freedom) and the critical values increase monotonically in the number of parameters. The 1 percent critical value for the sup-lm test for this model is Every statistic is greater than this critical value, and in most cases this is true several times over. 7 Probabilistic Structural Breaks On balance, the evidence presented in the previous section suggests the existence of structural breaks in GARCH models commonly used in finance. In section 8 we will estimate a Regime-Switching GARCH model which explicitly models periodic discrete shifts in model parameters. Before we report this we develop and analyze a probabilistic structural break model which serves a number of purposes. It bridges the gap from our discussion of structural breaks to the regime switching model since it can be thought of as a special case. 4 For a discussion of this in the context of foreign exchange markets and the reduction in volatility persistence that follows after adjusting for breaks in unconditional volatility see Malik (2003). 20

22 Secondly, we can use the data to conduct inference on the date of the structural break. To conserve space, we focus the remainder of the paper on the VWNYSE stock index and the four foreign exchange rates. The various LM structural break tests discussed and implemented above have one thing in common: they work when the break point is unknown. This is very important since we rarely know when the break occurs. Even when the break is due to a known event, such as the Volker period in interest rates, the exact date is unknown. One may be interested in identifying the break date empirically. One complication of this is the number of estimations involved. Take the GARCH(1,1) model on daily stock return data from July 1963 to December 2002, a series of nearly 10,000 observations. To estimate the structural break one must perform a grid search over the possibly break points, calculating the parameter estimates for each break point τ (1,..., T ), θ(τ). This involves over 9,000 optimizations very time consuming. One problem with the structural break model is that estimation of the break point requires that a computationally intensive search be undertaken. For each breakpoint τ with 1 < τ < T the parameters for both before and after the break have to be estimated by numerical maximum likelihood. The breakpoint corresponding to the highest likelihood is then our estimate. However it is clear that the likelihood function is non-differentiable with respect to this parameter. This section presents an alternative approach in which the break is probabilistic. An alternative approach allows easier estimation, and allows us to construct confidence intervals around the estimated break point. The model is a special case of the Markovswitching GARCH models considered by Gray (1996), Dueker (1997), and Klaassen (2002) and which are discussed below in section 8. We estimate a Markov-switching model that is in the spirit of the structural break model. In a structural break model the parameters take one value in the early part of the sample, and after the break take another value. In the Markov-break model the break point is random. This is parameterized by having two regimes, denoted by regime 1 and 2. The transitions between these states occur at discrete 21

23 points in time. The setup is somewhat nonstandard to the extent that the probability of being in regime 1 at the start of the sample is set to one. We model breaks as the switch from regime 1 to regime 2 which occurs with probability p: P (S t = 2 S t 1 = 1) = p. State 2 is an absorbing state once the break occurs the new parameter values take effect forever. Another advantage of this model is that inference regarding the break can be performed taking account of uncertainty regarding the break date. Performing an extensive grid search and choosing the date which results in the highest loglikelihood ignores the uncertainty about this date and the impact this will have on the other parameters standard errors. We are also able to construct confidence intervals for the date of the structural break. 7.1 The Model We present the algorithm for estimating the probabilistic structural break model in appendix A. At each time point t in the algorithm we must keep track of t distinct possible sequences of states: one for a break occurring at each of the t 1 possible breaks from the second period to the current (τ = 2,..., t), and the possibility that no break has occurred. The density of y t depends on the entire sequence of past states since conditional volatility is regime dependent and is defined recursively. 5 The filtering algorithm uses the definition of conditional volatility to update the probability of each of these t possible sequences as new information, i.e. new observations on y, become available, i.e. updating P (τ = s Ψ t 1 ) to P (τ = s Ψ t ). We run through this filter using the entire sample of data and finally determine the probability that the unobserved break occurred at each point in time, i.e. P (τ = s Ψ T ). This set of T probabilities can be used to infer the break date, or to calculate the mean T +1 ˆτ = tp (τ = t Ψ T ) (10) t=2 5 In a GARCH model, volatility today depends recursively on past conditional volatility. To the extent that current volatility model depends on the current regime, lagged volatility will depend on the state in the last period and so on. Since volatility is also dependent on past volatility, the conditional volatility today will also depend on its entire past. This is discussed in more detail in the appendix and following section. 22

24 and variance T +1 var(ˆτ) = (t ˆτ) 2 P (τ = t Ψ T ) (11) t=2 of the estimated break point. A byproduct of this algorithm are the unconditional densities f(y t Ψ t 1 ) which are used to construct the log-likelihood for numerical estimation of the model parameters. We estimate this model are report the parameter estimates for the full model in table 6 along with the estimated mean break date and the standard deviation of this estimate. We only report estimates using the full sample period. For brevity we only report the analysis of the probabilistic structural break model for the daily VWNYSE and the four exchange rate series. These are also more interesting given the vast volume of empirical work that has focused on these series (and the paucity of work done on individual stocks). The first point to note is the dramatic improvement in loglikelihoods. We cannot test if the GARCH dynamics before and after the break are different since the transition probability is an unidentified parameter, but the improvement is sufficiently dramatic to render extensive empirical analysis irrelevant. The smallest likelihood ratio statistic (LR) is for the Canadian foreign exchange rate is is 35. All other series have LR statistics between 109 and 191. As a rule of thumb, if we consider the maximum number of common parameters is 7 and count the nuisance parameter as a restriction we could conjecture that a chi-squared distribution with 8 degrees of freedom should dominate the actual distribution, which would suggest a 5 percent critical value of 15. Our test statistics are between 2 to 12 times that value! The dynamics of the VWNYSE series is rather interesting. In the initial sample, which concludes relatively early in the sample, around May 1966, the leverage effect dominates. The leverage effect refers to the empirical observation that volatility increases more following negative shocks than positive shocks. In the initial period the positivity constraint on α binds and the only influence on volatility is following negative shocks. The initial sample is also rather close to normal since v 1 is only a little more than 2 standard errors from zero. The unconditional volatility in the latter sample is much larger (0.76 vs 0.12). 23

25 The exchange rate series are also very interesting. Unlike Malik (2003) we find that accounting for a structural break does very little to reduce the persistence of shocks to volatility. This is likely because our model has one break over a relatively long period, while Malik allows for multiple breaks over a much shorter time period. We also observe that the constraint that α + β binds in three of the four series in the initial sample, with α being relatively large. The post-break sample also suggests persistent shocks, with point estimates around The anti-leverage effect in the Canadian exchange rate is still present. In the Yen, Mark and Pound series the tails become less fat after the structural break. The standard deviations of the structural break date estimate are relatively small. Recall that we are working with daily data, so the standard deviations of 30 for VWNYSE suggest an approximate 95 percent confidence band of a little over two months on either side of the break date (assuming 22 trading days per month), while we have nearly a year either size of the GJR(1,1,1) model for the Canadian dollar. This conclusion can be seen visually in the plots of the cumulative probability of a break by each date presented in figure 3 which suggest a similar conclusion since the jump from zero to one is very quick. 8 Markov-Switching GARCH Model Consider a model where the data generating process depends on the state or regime in which the world is at each point in time. Denote by S t the random state of the world at time t which can take two value s t = {1, 2}. In the Markov-regime switching literature the regime evolves through time as a first-order Markov process. It becomes quite simple to parameterize. We will denote by e t it = y t µ t St=i t and by h t t 1,it,i t 1 = E(e 2 t Ψ t 1, S t = i t, S t 1 = i t 1 ) the conditional variance conditional when the current period is in state i and the last periods state was j. One major problem with Markov-switching GARCH models is their inherent path dependence. In ARCH models the conditional volatility today is a function of lagged 24

26 squared residuals and the current regime. The current literature focuses almost exclusively on two-regime models where the regimes of interest are labelled as high volatility and low volatility. This illustrates the importance of allowing the parameters driving the GARCH process to depend on the current regime. For example, Cai (1994) models an ARCH process where the unconditional volatility (the intercept, loosely speaking) switches between regimes. Because the residuals in previous periods depend on the regime in those past periods, conditional volatility becomes a function of lagged regimes also. To clarify, consider a MS-ARCH(1) model where all parameters depend on the regime and denote by i t the regime that occurred at time t (i.e. S t = i t ) and i t 1 as the regime in time t 1 (i.e. S t 1 = i t 1 ): h t it,i t 1 = E(e 2 t Ψ t 1, S t = i t, S t 1 = i t 1 ) = ω it + α it e 2 t 1 i t 1 where e t 1 it 1 = y t 1 µ t 1 j. To model a MS-ARCH(p) model with K regimes (typically K = 2), one needs to keep track of the probabilities of each of the K p+1 possible (unobserved) regime sequences occurring. As p becomes large this becomes computationally burdensome but not infeasible. In the MS-GARCH model this path dependence issue becomes more severe in fact is becomes technically infeasible since the current conditional volatility will depend on the entire past sequence of states. The volatility today depends on the current regime and the volatility last period; while the volatility last period depends on the regime in that lagged period and the regime before that, and so on: h 2 i2,i 1 = ω i2 + α i2 e 2 1 i 1 + β i2 h 1 i1 h 3 i3,i 2,i 1 = ω i3 + α i3 e 2 2 i 2 + β i3 h 2 i2,i 1. The number of possible sequences grows exponentially since at time t one needs to keep track of K t+1 possible sequences of regimes for 1 t T. 25

27 Note that this problem only holds because the regimes are recurrent. When we have a regime which is an absorbing state as in the probabilistic shifting model, then the path dependence is manageable as it only grows linearly with time and not exponentially. Early research in allowing for multiple regimes such as Cai (1994) and? only allowed for ARCH terms because of the path dependence problem. The first paper to overcome this path dependence problem and estimate a MS-GARCH model was Gray (1996). Other MS-GARCH models include Dueker (1997), Klaassen (2002), and Calvet and Fisher (2003). In this paper we follow Dueker (1997) and avoid the path dependence problem by building conditional volatility recursively using the formulae h t it,i t 1,t 1 = ω it + α it e 2 t 1 i t 1 + β it ĥ t 1 it 1. (12) Note that for any time point t, the conditional volatility depends only on the regime in the current period and in the previous period. The dependence of lagged volatility on states in previous periods is integrated out by substituting the entire path dependent h t 1 it 1,i t 2,... with ĥt 1 i t 1 = E(e 2 t Ψ t 1, S t 1 = i t 1 ). We truncate the dependence of volatility on lagged states so the dimensionality does not get away from us. 6 The path dependence which is inherent in regime-switching GARCH models is integrated out following (Dueker 1997): K ĥ t it = P (S t 1 = i t 1 S t = i t, Ψ t 1 )h t iti t 1 (13) i t 1 =1 where P (S t 1 = i t 1 S t = i t, Ψ t 1 ) = P (S t = i t, S t 1 = i t 1 Ψ t 1 ) K i t 1=1 P (S t = i t, S t 1 = i t 1 Ψ t 1 ). (14) 6 Kim (1994) applied this approach to Kalman filters and showed that the approximation performs well, Dueker (1997) applied it to solving the path dependence problem in MS-GARCH models, Smith (2002) applied this to estimating Markov-Switching Stochastic Volatility models using quasi-maximum likelihood, Smith (2003) applies the approach to constructing path independent Lagrange multiplier specification tests. 26