Multivariate Regime Switching Model with Flexible. Threshold Variable: With an Application to Returns. from a Portfolio of U.S.

Transcription

1 Multivariate Regime Switching Model with Flexible Threshold Variable: With an Application to Returns from a Portfolio of U.S. Stocks Daniele Massacci Einaudi Institute for Economics and Finance January 16, 2014 Abstract This paper proposes a novel multivariate regime switching model that allows the threshold variable to be a linear combination of covariates with unknown coef- cients. The paper considers least squares estimation of the model and provides a suitable algorithm; and it proposes a test for the number of regimes based on theoretical results from multivariate statistics. Finite sample results from Monte Carlo analysis strengthen the methodological contribution of the paper. An application to asset pricing in stock markets with a focus on portfolio allocation illustrates the This paper greatly bene ts from comments and suggestions from Massimo Guidolin. I am very grateful to Hashem Pesaran for letting me visit the Center for Applied Financial Economics (CAFE), USC, and for providing me with the data I used in this work. Valuable insights from a conversation with Marco Lippi are acknowledged. Errors and omissions are my own responsibility. Financial support from the Associazione Borsisti Marco Fanno and from UniCredit and Universities Foundation is gratefully acknowledged. Address correspondence to Daniele Massacci, EIEF, Via Sallustiana 62, Roma, Italy. Tel.: dm355@cantab.net. 1

2 usefulness of the proposed model for applied work. JEL classi cation: C32, C58, G12. Keywords: Multivariate Threshold Model, Flexible Threshold Variable, Stock Returns, Asset Pricing. 2

3 1 Introduction There now exists substantial empirical evidence that the data generating processes of many economic and nancial variables are well approximated by time series econometric models displaying a nite number of regimes: these models can either be univariate or multivariate, depending on the purpose of the analysis or the nature of the data (or both). A rst approach requires estimation and model selection in the presence of structural instabilities: for example, Paye and Timmermann (2006) follow Bai and Perron (1998) and document the presence of structural instabilities in models for ex post predictability of stock returns. When the underlying assumption is that "history repeats", regime switching models become a useful tool for empirical investigation (see Timmermann, 2008). Markov-Switching models as proposed in Hamilton (1989) assume that regime changes are driven by a latent variable: applications include Guidolin and Timmermann (2006a) to the term structure of risk, Ang et al. (2008) to interest rates, and Henkel et al. (2011) to stock returns. Under transition models, the prevailing regime depends on an observable variable, and the transition itself can be either smooth (see Terasvirta, 1994) or stepwise (see Tsay 1989; 1998), the latter generating threshold models: Audrino and Medeiros (2011) highlight the importance of smooth regimes in modelling interest rates; and Potter (1995) and Pesaran and Potter (1997) employ threshold models to investigate the dynamics of U.S. output. The focus of this paper is on transition models. An important component of these models is the variable that drives the dynamics between regimes. Common practice is to a priori choose a single observable variable such as the appropriately lagged dependent variable (see Tong, 1978; Terasvirta, 1994) or a more general exogenous covariate (see Hansen, 2000): this strategy would lead to model misspeci cation should the chosen 3

4 variable be di erent from the true one or should the regimes be driven by more than one covariate. Within the univariate framework, several studies have proposed exible parameterisations for the variable that determines the regime. Under smooth transition dynamics, Medeiros and Veiga (2003; 2005) and Becker and Osborn (2012) model the transition variable as an average of lagged values of the dependent variable. In threshold models, Seo and Linton (2007) propose a exible speci cation for the threshold variable based on a linear combination of exogenous covariates with unknown coe cients; and Massacci (2013a) modi es Seo and Linton s (2007) model to allow for a di erent identi cation strategy 1. The aforementioned contributions relate to univariate models and suitable extensions to multivariate counterparts are called for: this would be valuable for empirical work since "the search for an appropriate threshold variable [...] in an application needs a careful investigation," as stated in Tsay (1998) 2. This paper focuses on the class of threshold models and it extends to a multivariate framework the exible parameterisation for the threshold variable advanced in Seo and Linton (2007). The paper makes a number of methodological and empirical contributions. From a methodological standpoint, it explicitly addresses the issues of identi cation and estimation: it derives su cient conditions for the former to be achieved; and it provides an algorithm for the latter to be performed. It proposes a test for the number of regimes to be applied to the model: following Luukkonen et al. (1988), the test is speci ed in variable addition form and it is constructed by utilising results from multivariate statistics (see Anderson, 1984). Finally, in order to illustrate the potential empirical usefulness of the theoretical framework, the paper provides an empirical application to an asset pricing exercise in stock markets with a focus on portfolio allocation. 1 More discussion on this point is provided in Section See Tsay (1998), p

5 The rest of the paper is organised as follows: Section 2 introduces the model; Section 3 deals with the issues of identi cation and estimation; Section 4 develops a test to determine the number of regimes; Section 5 performs a Monte Carlo analysis; Section 6 provides an application to stock returns; nally, Section 7 concludes. 2 Model This paper introduces the model y t = (X t 1 + u 1t ) I ( 0 q t ) + (X t 2 + u 2t ) I ( 0 q t > ) ; t = 1; : : : ; T; (1) where I () is the indicator function; y t (y 1t ; : : : ; y Nt ) 0 2 Y R N is the N 1 vector of dependent variables; X t I N x 0 t is a N N (K x + 1) matrix, with x t 2 X R Kx+1 being a (K x + 1)1 vector of K x explanatory variables (which may include lagged values of y t ) and the intercept, denoting the Kronecker product, and I N being the N N identity matrix; the threshold variable 0 q t is a linear combination of the elements of the K q 1 vector of random variables q t (q 1t ; : : : ; q Nt ) 0 2 Q R Kq with coe cients 0; collected in the K q 1 vector of parameters 1 ; : : : ; Kq is the threshold value; 1 and 2 are N (K x + 1) 1 vectors de ned as j 0 j1; 0 j2; : : : ; jn 0 0 for j = 1; 2, where ji is of dimension (K x + 1) 1, for i = 1; : : : ; N; u 1t and u 2t are N 1 vectors of error terms such that E (u jt jx t ; q t ) = 0; E u jt u 0 jt jx t ; q t = j ; j = 1; 2; (2) where 1 and 2 are N N positive de nite matrices. 5

6 From a statistical standpoint, the model in (1) belongs to the general class of threshold models discussed in Tong (1978): according to the threshold principle introduced in Pearson (1900), the change in regime in the data generating process of y t occurs when the threshold variable 0 q t crosses the unknown threshold value. The model in (1) extends the class of multivariate threshold models proposed in Tsay (1998) by letting the threshold variable 0 q t be a linear combination of the elements of q t : this general speci cation has originally been suggested in Seo and Linton (2007) for univariate threshold models; for identi cation purposes, it has been modi ed in Massacci (2013a), as discussed more into details in Section 3.1 below. The speci cation in (1) and (2) allows for threshold e ects induced both on the conditional mean and the conditional variance of y t : letting vech () denote the half-vectorisation operator, the former and the latter are respectively de ned as 2 1 = ( ) 0 ; : : : ; ( 2N 1N ) ; : : : ; 0 N ; (3) and vech ( 2 ) vech ( 1 ) : (4) Several contributions have proposed exible parameterisations for the transition variable in transition models: Medeiros and Veiga (2003; 2005) and Becker and Osborn (2012) focus on univariate smooth transition models; Seo and Linton (2007) and Massacci (2013a) consider univariate threshold regressions; and Galvao and Marcellino (2012) propose a multivariate model analogous to (1) and (2) where the vector of parameters is assumed to be known a priori and does not have to be estimated. This paper therefore contributes to the literature in nonlinear time series econometrics by introducing a multivariate threshold model in which the threshold variable is a linear combination of the 6

7 elements of a random vector with unknown coe cients: it addresses the methodological issues of identi cation, estimation and testing for the number of regimes; and it provides an application to asset pricing in stock markets with a focus on portfolio allocation. 3 Identi cation and Estimation This section deals with identi cation and estimation of the model in (1) and (2): the former and the latter are addressed in Sections 3.1 and 3.2, respectively. 3.1 Identi cation The model in (1) and (2) is piecewise linear: whereas the linear component within each regime does not pose any identi cation issue, the argument of the indicator function I () needs to be suitably constrained. To this purpose, let L and A be the parameter spaces of and, respectively, with generic elements labelled l and a, respectively. The following de nition is introduced: De nition 3.1 The model in (1) and (2) is said to be identi ed if 0 P [I ( 0 q t ) = I (l 0 q t a)] < 1; 8l 2 L; 8a 2 A: Let denote the Cartesian product operator. According to De nition 3.1, the model in (1) and (2) is identi ed provided that I (l 0 q t a) evaluated at l = and a = is di erent from I (l 0 q t a) in any other point in L A with positive probability. Given De nition 3.1, the vector of parameters ( 0 ; ) 0 is observationally equivalent to (l 0 ; a) 0 = v ( 0 ; ) 0 7

8 for 0 < v < 1: formally, P [I ( 0 q t ) = I (v 0 q t v)] = 1; 80 < v < 1; and the model is not identi ed. Let Kq denote the K q 1 vector of ones. Theorem 3.1 below provides su cient conditions for identi cation: Theorem 3.1 The model in (1) and (2) is identi ed according to De nition 3.1 if at least one of the following two conditions is satis ed: (a) m = c m for at least one m = 1; : : : ; K q, where c m is a known constant such that c m 6= 0; (b) 0 Kq = c, where c is a known constant such that c 6= 0. Proof of Theorem 3.1. See Appendix A. Theorem 3.1 provides su cient conditions for identi cation of the model in (1) and (2): condition (a) is more general than the one in Seo and Linton (2007), which imposes c m = 1 for one m = 1; : : : ; K q ; and condition (b) extends Theorem 3.1 in Massacci (2013a) by allowing for c 6= 0 rather than c > 0. Condition (b) will be used in the Monte Carlo analysis and in the empirical application in Sections 5 and 6, respectively. 3.2 Least Squares Estimation This section looks at estimation of the model in (1) and (2): the least squares estimator and the related proposed algorithm are illustrated in Sections and 3.2.2, respectively. 8

9 3.2.1 Least Squares Estimator The model in (1) and (2) is estimated by least squares by suitably extending the theory developed in Tsay (1998). To this purpose, de ne I 1t (l; a) I (l 0 q t a) ; I 2t (l; a) I (l 0 q t > a) ; 8l 2 L; 8a 2 A; and P T j (l; a) T I jt (l; a) ; j = 1; 2; 8l 2 L; 8a 2 A: t=1 For given l and a, the least squares estimators for 1 and 2, and for 1 and 2, are T P 1 T P ^ j (l; a) = I jt (l; a) X 0 tx t I jt (l; a) X 0 ty t ; j = 1; 2; 8l 2 L; 8a 2 A; t=1 t=1 and ^ j (l; a) = TP t=1 h i h i 0 I jt (l; a) y t X t^j (l; a) y t X t^j (l; a) ; j = 1; 2; 8l 2 L; 8a 2 A; T j (l; a) (K x + 1) respectively. De ne S (l; a) 2 P j=1 h i S j (l; a) ; S j (l; a) [T j (l; a) (K x + 1)] tr ^ j (l; a) ; j = 1; 2; 8l 2 L; 8a 2 A; where tr () denotes the trace operator; the estimator for ( 0 ; ) 0 is then de ned as ^0 ; ^ 0 arg min (l 0 ;a) 0 2(LA) S (l; a) : 9

10 Finally, the least squares estimators for 1, 2, 1 and 2 are de ned as ^ j ^ j ^; ^ ; ^ j ^ j ^; ^ ; j = 1; 2: Let a:s:! and d! denote almost sure convergence and convergence in distribution, respectively. Under conditions analogous to those imposed in Assumption 1 in Tsay (1998), as T! 1 it follows that ^0 ; ^ 0 a:s:! ( 0 ; ) 0 ; ^j a:s:! j ; ^ j a:s:! j ; j = 1; 2; and h T j ^; ^ i 1/2 ^j j d! N (0; j j ) ; j = 1; 2 : the (K x + 1) (K x + 1) matrix j is de ned as j fe [I jt (; ) x t x 0 t]g 1 ; j = 1; 2; and it is estimated as ^j h i 1 P T T j ^; ^ I jt ^; ^ x t x 0 t t=1 1 a:s:! j ; j = 1; 2: Estimation Algorithm From a computational standpoint, this paper extends Massacci (2013a) and proposes a two-step algorithm to rst estimate ( 0 ; ) 0 and then 1, 2, 1 and 2 : the algorithm rst estimates ( 0 ; ) 0 by constructing an objective function de ned on the parameter space L A of ( 0 ; ) 0 only; it then estimates 1, 2, 1 and 2 given the estimator for 10

11 ( 0 ; ) 0 obtained in the rst step. The proposed algorithm is implemented as follows: 0 1. De ne the set L L 1 L Kq with generic point l l 1 ; : : : ; l Kq 2 L such that l m 2 L m, for m = 1; : : : ; K q : L can be constructed according to economic arguments such as those put forward in Section 6. For l 2 L, de ne the set A (l) with generic element a (l) 2 A (l): as in Tong and Lim (1980), Tsay (1989) and Kapetanios (2000), the elements of A (l) are given quantiles of the empirical distribution function of l 0 q t. For all l 2 L and a (l) 2 A (l), consider T P 1 T P ^ j [l; a (l)] = I jt [l; a (l)] X 0 tx t I jt [l; a (l)] X 0 ty t ; j = 1; 2; t=1 t=1 and ^ j [l; a (l)] = TP t=1 n I jt [l; a (l)] y t o hn X t^j [l; a (l)] y t T j [l; a (l)] (K x + 1) oi 0 X t^j [l; a (l)] ; j = 1; 2; and de ne S [l; a (l)] 2 P j=1 o S j [l; a (l)] ; S j [l; a (l)] ft j [l; a (l)] (K x + 1)g tr n^ j [l; a (l)] ; j = 1; 2 : the least squares estimator for ( 0 ; ) 0 is then obtained as ^0 ; ^ 0 = arg min [l 0 ;a(l)] 0 2LA(l) S [l; a (l)] : 2. Given ^0 ; ^ 0, the estimators for 1, 2, 1 and 2 are obtained as in Section

12 4 Testing for the Number of Regimes This section deals with the problem of testing for the number of regimes in the model in (1) and (2): it provides an introduction to the problem in Section 4.1 and it proposes a formal testing procedure in Section Preliminary Discussion A question that naturally arises is whether regimes are actually present in the data generating process of the variables of interest: in order to provide an answer, a formal testing procedure has to be developed. The model in (1) and (2) reduces to y t = X t + u t ; E (u t jx t ; q t ) = 0; E (u t u 0 t jx t ; q t ) = ; t = 1; : : : ; T; under the single-regime null hypothesis H 0 : ( 1 = 2 = ) \ ( 1 = 2 = ), and the parameters vector ( 0 ; ) 0 no longer is identi ed: this is an example of the Davies problem, rst addressed in Davies (1977; 1987). As compared to the framework considered in Tsay (1998), the set of unidenti ed parameters includes the vector of parameters, as well as the threshold value. Several solutions to the Davies problem exist for threshold models: Hansen (1996) develops statistical tests based on empirical process theory; Petruccelli and Davies (1986) and Tsay (1989; 1998) transform threshold regressions into changepoint models by using arranged regressions; and Luukkonen et al. (1988) treat threshold models as limiting cases of logistic smooth transition models and construct variable addition tests by applying Taylor approximations of suitable order to the transition function. In order to construct a test for the number of regimes, this paper suitably extends the 12

13 procedure advanced in Luukkonen et al. (1988) to the model of interest. To this purpose, de ne the logistic function F (z) [1 + exp ( z)] 1 ; z 2 R; (5) and consider the model y t = (X t 1 + u 1t ) f1 F [h ( 0 q t )]g + (X t 2 + u 2t ) F [h ( 0 q t )] ; h > 0; t = 1; : : : ; T; (6) where u 1t and u 2t are de ned as in (2). The model in (5) and (6) extends the univariate set up considered in Medeiros and Veiga (2003; 2005) and Becker and Osborn (2012) by introducing a multivariate logistic smooth transition model with exible transition variable, which nests (1) as a limiting case as h! 1. The speci cation for the conditional mean of y t in (6) is equivalent to y t = X t 1 + X t F [h ( 0 q t )] + u t ; h > 0; t = 1; : : : ; T; (7) where is de ned in (3) and u t u 1t f1 F [h ( 0 q t )]g + u 2t F [h ( 0 q t )] : the single-regime null hypothesis is then stated as H 0 : h = 0. Following Luukkonen et al. (1988), a test for the single-regime null hypothesis can be constructed as a variable addition test by applying a Taylor approximation of suitable order to the transition function F (): as in Harvill and Ray (1999), this results in a multivariate extension of 13

14 Tsay s (1986) univariate test. 4.2 Testing Procedure The rst-order Taylor approximation of the function F (z) in (5) about z = 0 is F (z) ' z; (8) 4 and application of (8) to (7) leads to the auxiliary model 1 y t = X t 1 + X t h (0 q t ) + t = X t h hx t 0 q t + t : (9) Notice that x 0 t i 0 q t = vec (x 0 t i 0 q t ) = (q 0 t x 0 t) vec i 0 ; i = 1; : : : ; N; where vec () denotes the vectorisation operator and de ne 1 = h h0 1 ; : : : ; 0 1N N 4 h0 N ( 0 11; : : : ; 0 1N) 0 0 ; (10) h vec 1 0 ; : : : ; vec N 0 0 ( 0 21; : : : ; 0 2N) 0 ; (11) and Z t I N (q 0 t x 0 t) ; 14

15 where 1 is an N (K x + 1) 1 vector, 2 is a N [K q (K x + 1)] 1 vector, Z t is a N N [K q (K x + 1)] matrix, and i and i are de ned in (3) for i = 1; : : : ; N: the model in (9) is then equivalent to y t = X t 1 + Z t 2 + t ; (12) and the test for the number of regimes is obtained by testing the null hypothesis H 0 : 2 = 0 against the alternative H 1 : 2 6= 0. The test for the number of regimes can be equivalently implemented as follows. De ne the N 1 vector of residuals under the null hypothesis of linearity as ~ t y t X t ~ 1 ; t = 1; : : : ; T; where ~ 1 is the least squares estimator of 1 from (12) under H 0 : 2 = 0. Construct the auxiliary regression ~ t X t 1 + Z t 2 + w t ; t = 1; : : : ; T; with N 1 vector of residuals ^w t. De ne the total sum of squares SST, the errors sum of squares SSE, and the Wilks lambda statistic as SST T P t=1 ~ t ~ 0 t; SSE T P t=1 ^w t ^w 0 t; det (SSE) det (SST) ; respectively, where det () denotes the determinant operator. As T! 1, the test statistic LM = gd b 1 1/2 F [NK NK q (K x + 1) 1/2 q (K x + 1) ; gd b] (13) 15

16 under the single-regime null hypothesis 3, where b NK q (K x + 1) 2 1; d " N 2 K 2 q (K x + 1) 2 4 N 2 + K 2 q (K x + 1) 2 5 # 1/2 ; g T (K x + 1) 1 2 [N + K q (K x + 1) + 1] : When the dimensionality of the model is N = 2, the LM test statistic de ned in (13) reduces to 4 LM = T (K x + 1) K q (K x + 1) 1 K q (K x + 1) 1 1/2 1/2 F f2k q (K x + 1) ; 2 [T (K x + 1) K q (K x + 1) 1]g : (14) The power of the proposed test deserves some discussion. The asymptotic distribution of the LM statistic in (13) is derived under the null hypothesis of no threshold e ect in the conditional mean and variance of y t : the test has then power when the threshold e ect is induced only on the conditional second moment, namely when = 0 and 6= 0, where and are de ned in (3) and (4), respectively. Following Luukkonen et al. (1988), the test deserves special attention when 6= 0 and = 0: as proved in Appendix B, when P (q t = Sx t ) = 1 where S is a K q (K x + 1) matrix of coe cients (i.e., when each element of q t is a linear combination of the elements of x t ), the test does not have power against the alternative that the threshold e ect is induced only on the intercept in all equations in the system or in a subset of them; since this case is not relevant for the empirical analysis in Section 6, this paper does not address the problem and leaves it to future research 5. 3 See Anderson (1984), Section 8:5:4. 4 See Anderson (1984), p. 305, Theorem 8:4:6. 5 As in Luukkonen et al. (1988), the problem is solved by using higher order expansions of the function 16

17 5 Monte Carlo Analysis This section assesses the nite sample properties of the least squares estimator and of the test for the number of regimes in relation to the model proposed in this paper: the Monte Carlo design and the results are illustrated in Sections 5.1 and 5.2, respectively. 5.1 Design The Monte Carlo experiment addresses the issues of estimation and inference on the number of regimes, and the speci c designs are presented in Sections and 5.1.2, respectively Estimation The Data Generating Process (DGP) is y r it = ( 1i1 + x r t 1i2 + u r 1it) I [q r 1t + (1 ) q r 2t ] + [ 2i1 + x r t 2i2 + u r 2it] I [q r 1t + (1 ) q r 2t > ] ; i = 1; 2; t = 1; : : : ; T; (15) where r = 1; : : : ; R, r refers to the replication, T is the sample size, and R is the total number of replications. The experiment is run in Ox 7:01 (see Doornik, 2012). The sample sizes T are equal to T = 125; 250; 500; The number of replications is equal to R = The seed of the random number generator is equal to 1. The restrictions 111 = 112 = 121 = 122 = 1 and 211 = 212 = 221 = 222 = 2 are imposed: the same threshold e ect is induced on the intercept and on the slope coe cient in both equations and it is equal to ( 2 1 ). The scalar parameters 1, 2 and are xed throughout the replications. F (z) in (5). 17

18 Three features of the model are controlled for: (i) the value of that assigns the weights to q1t r and q2t r in (15); (ii) the magnitude of the threshold e ect ; (iii) the proportion of observations of q1t r + (1 ) q2t r below. The variables x r t, q1t r and q2t r in (15) are generated as x r t = x (1 x ) + x x r t 1 + (1 2 x) 1/2 r xt; t = 49; : : : ; 0; : : : ; T; x 50 = x ; and q r mt = q 1 q + q q r m;t q 1/2 r qmt; t = 49; : : : ; 0; : : : ; T; q r m; 50 = q ; m = 1; 2; where x, x, q and q are xed in repeated samples, with x N (1; 1), x U (0:05; 0:95) and q U (0:05; 0:95), while the choice of q is described below. The shocks r xt, r q 1 t and r q 2 t are generated as r xt = xqf r t + r xt 2 xq + 1 1/2 ; r q mt = xqf r t + r q mt 2 xq + 1 1/2 t = 49; : : : ; 0; : : : ; T; m = 1; 2; where f r t IIDN (0; 1), r xt IIDN (0; 1), r q 1 t IIDN (0; 1), r q 2 t IIDN (0; 1) and xq = 1: in this way, E (x r t ) = x ; E (q r 1t) = E (q r 2t) = q ; Var (x r t) = Var (q r 1t) = Var (q r 2t) = 1; and Corr (x r t ; q r 1t) = Corr (x r t ; q r 2t) = Var (q r 1t; q r 2t) = 0:5: In order to reduce the e ect induced by the initial values x 50 = x, q r 1; 50 = q and 18

19 q r 2; 50 = q, the rst 50 observations in the DGP for x r t, q r 1t and q r 2t are discarded. The shocks u r 11t, u r 21t, u r 12t and u r 22t in (15) are generated as u r jit = uf r t + r u i t ( 2 u + 1) 1/2 j; i; j = 1; 2; t = 1; : : : ; T; where r u i t IIDN (0; 1), and u, 2 1 and 2 2 are xed in repeated samples with u = 1, 2 1 = 1 and 2 2 = 2: in this way, Var u r j1t = Var u r j2t = 2 j ; Corr u r j1t; u r j2t = 0:5; j = 1; 2; and the threshold e ect induced on the conditional second moment de ned in (4) is equal to = (1; 0:5; 1) 0. Two values for in (15) are considered, namely = 0:10; 0:50. In order to control for the threshold e ect on the slope coe cients, ( 2 1 ) is set equal to = 0:25; 1:00; 1:75: given 1 = 1, 2 is assigned the values 2 = 1:25; 2:00; 2:75, respectively. In order to control for the proportion of observations of q r 1t + (1 ) q r 2t lying below, the latter is set equal to = 2. It follows that P [q1t r + (1 ) q2t r ] 8 q1t r + (1 ) q2t r E (q1t) r + (1 ) E (q2t) r >< 2 Var (q = P 1t) r + (1 ) 2 Var (q2t) r + 2 (1 ) Cov (q1t; r q2t) r 1/2 E (q r >: 1t) + (1 ) E (q2t) r 2 Var (q1t) r + (1 ) 2 Var (q2t) r + 2 (1 ) Cov (q1t; r q2t) r 1/2 " # = x ; 1/2 9 >= >; where () denotes the cumulative distribution function of a random variable with a standard normal distribution; the analytical expression for x is then obtained in closed 19

20 form as x = 1 () /2 : Five values for are considered, namely = 0:15; 0:30; 0:50; 0:70; 0:85. The parameters of the model in (15) are estimated by means of the procedure discussed in Section In order to estimate, the set of points L is a priori de ned as L f0:00; 0:10; 0:20; 0:30; 0:40; 0:50; 0:60; 0:70; 0:80; 0:90; 1:00g ; for l 2 L, the set A (l) is made of 19 equally spaced quantiles of the empirical distribution function of lq r 1t + (1 l) q r 2t, namely 5%; 10%; 15%; : : : ; 85%; 90%; 95%, and the true value = 2. The focus is on estimation of 111, 112, and in (15). The estimators are assessed by computing bias and RMSE. In the case of 111, these are de ned as bias = 1 R RP r=1 r 1 2; ^r ; RMSE = ^r 111 R 111 respectively, where ^ r 111 denotes the estimate of 111 obtained from the r th replication; analogous de nitions hold in the case of 112, and. In the case of 111 and 112, the size of the tests for the null hypotheses H 0 : 111 = 1 and H 0 : 112 = 1 obtained from the relevant t-statistics at 5% signi cance level are also reported Inference on the Number of Regimes In order to assess the size of the proposed test for the number of regimes, the DGP is y r it = i1 + x r t i2 + u r it; i = 1; 2; t = 1; : : : ; T; 20

21 where 11 = 12 = 21 = 22 = = 1; the variable x r t is generated as described in Section 5.1.1; the shocks u r 1t and u r 2t are generated as u r it = uft r + r u i t ; i = 1; 2; t = 1; : : : ; T; ( 2 1/2 u + 1) where r u 1 t IIDN (0; 1), r u 2 t IIDN (0; 1), and u and 2 are xed in repeated samples and set equal to u = 1 and 2 = 1. The power of the test is assessed by using the DGP described in Section Under this set up, the dimensionality of the system is N = 2 and the test statistic is constructed according to (14). 5.2 Results Results from model estimation are shown in Tables 1 4, whereas size and power properties of the test for the number of regimes are summarised in Table 5. Table 1 about here Table 2 about here Table 3 about here Table 4 about here Table 5 about here The bias of the least squares estimators for 111 and 112 (see Tables 1 and 2, respectively) is always of limited magnitude, with the exception of the case in which the threshold e ect is small (i.e., = 0:25), the relevant regime occurs rarely (i.e., = 0:15) and 21

22 the sample size is not large enough to let asymptotic results hold (i.e., T = 125; 250; 500); other conditions being equal, the bias tends to decrease in, T and, while it shows no dependence upon. The RMSE tends to decrease in (with the exception of the case T = 125 and = 0:25), T and, and it is not dependent on. The empirical size of the tests approaches the theoretical value as, T and increase, and it is independent of. Bias and RMSE of the estimators for and (see Tables 3 and 4, respectively) have similar behaviour to those of the estimators for 111 and 112 previously discussed. Finally, the test for the number of regimes (see Table 5) is always correctly sized; as expected, the power of the test monotonically increases in T and. In conclusion, the least squares estimators for 111, 112, and, and the proposed test for the number of regimes have desirable nite sample properties: this paper then provides a valid tool for applied work. 6 Empirical Application This section provides an empirical application of the theory developed in this paper to an asset pricing problem in stock markets with a focus on portfolio allocation: the aim is to show how the proposed model may be used to characterise the degree of crosssectional dependence between returns from portfolios of assets, which plays a key role in diversi cation strategies. In what follows, Section 6.1 provides a description of the data and model speci cation, and Section 6.2 presents the results. 22

23 6.1 Data and Model Speci cation Following Perez-Quiros and Timmermann (2000; 2001), this paper looks at nonlinear dynamics in the joint distribution of U.S. stock returns as related to capitalisation e ects. The vector of dependent variables consists of monthly excess returns (in percentage terms) on size-sorted decile portfolios over the benchmark T -bill rate collected from the Center for Research in Security Prices (CRSP): the sample period begins in January 1957 and ends in December 2012, a total of 672 observations. Descriptive statistics and correlation matrix for the dependent variables are reported in Table 6. Table 6 about here Two features are worth highlighting: in line with standard asset pricing models, mean and standard deviations decline almost monotonically as one moves from smallest rms to largest rms portfolios; and the correlation between portfolios is higher the closer they are ranked in terms of degree of market capitalisation. Joint analysis of nonlinear dynamics of random variables requires a multivariate system: following Guidolin and Timmermann (2006b), a two-regime system is employed as only stock returns are considered. For i = 1; : : : ; 10, the model for the excess return it at time t is it = ( 1i1 + 1i2 I t 1 + 1i3 Def t 1 + 1i4 Y ield t 1 + u 1it ) I Cli t 2 (1 ) V ol t 1 + ( 2i1 + 2i2 I t 1 + 2i3 Def t 1 + 2i4 Y ield t 1 + u 2it ) I Cli t 2 (1 ) V ol t 1 > : The right-hand side variables consist of monthly observations from Welch and Goyal s 23

24 (2008) updated dataset 6, augmented by the composite leading indicator. The predictors I t 1, Def t 1 and Y ield t 1 are one-month lagged values of the 3 month Treasury Bill Secondary Market rate, the default premium measured as the di erence between the yields on BAA and AAA-rated corporate bonds, and the dividend yield, respectively, all multiplied by 100: these are all consistent with the model employed in Perez-Quiroz and Timmermann (2001). The variables Cli t 2 and V ol t 1 are standardised values for the year-on-year log-di erence in the composite leading indicator and for the stock variance (computed as sum of squared daily returns on the S&P 500), respectively 7 : the choice of the former and the latter as variables driving the regimes follows arguments put forward in Perez-Quiroz and Timmermann (2000) and Massacci (2013b), respectively; and the linear combination Cli t 2 (1 ) V ol t 1 is motivated by the fact that volatility increases during recessions (see Schwert, 1989). The event Cli t 2 (1 ) V ol t 1 (Cli t 2 (1 ) V ol t 1 > ) then identi es a regime with low (high) growth rate in the composite leading indicator and high (low) volatility in the stock market. In order to empirically assess the contribution of the paper, the model is estimated for = 0, = 1 and 2 [0; 1]: the threshold variable is then equal to V ol t 1, Cli t 2, and a convex linear combination of the two, respectively. The analysis is conducted in Ox 7:01 (see Doornik, 2012). For 2 [0; 1], the algorithm suggested in Section is used: and are estimated by searching over L f0:00; 0:01; 0:02; : : : ; 0:98; 0:99; 1:00g and over A (l) made of the f15%; 16%; 17%; : : : ; 83%; 84%; 85%g quantiles of the empirical distribution function of lcli t 2 (1 l) V ol t 1, for l 2 L. For = 0 and = 1, the search for is made over the f15%; 16%; 17%; : : : ; 83%; 84%; 85%g quantiles of (1 l) V ol t 1 6 The dataset is kindly made available online at 7 The variable Cli t 2 is lagged by two periods because macroeconomic indicators are not available in real time. Both Cli t 2 and V ol t 1 are standardised variables for two reasons: to prevent any of them dominating the indicator function I (); and to ensure that Cli t 2 (1 ) V ol t 1 is a pure number. 24

25 and Cli t 2, respectively. It is important to stress that a model for the joint distribution of returns from all decile-sorted portfolios has not been previously constructed and estimated: this therefore provides a further contribution of the paper; and it allows to characterise the risk of the whole portfolio in terms of the conditional covariance matrix. 6.2 Results The test proposed in Section 4.2 as applied to the model described in Section 6.1 strongly rejects the null hypothesis of one regime in the joint distribution of excess stock returns from size-sorted decile portfolios and the multivariate threshold system is estimated 8. Results from model estimation are shown in Tables 7 and 8. Table 7 about here Table 8 about here Given the contribution of the paper, it is important to understand how the model with exible threshold variable di ers from those in which either Cli t 2 or V ol t 1 are a priori chosen. To this purpose, Table 7 shows results from estimation of the regime indicator I Cli t 2 (1 ) V ol t 1 and from characterisation of the degree of cross-sectional dependence when is equal to the least squares estimator ^ and under the restrictions = 0 and = 1. The degree of cross-sectional dependence is measured by computing the maximum eigenvalue of the conditional correlation matrix in each regime 9 : this provides valuable information that can be used for asset allocation purposes. 8 The p-value of the test for the number of regimes is equal to 0: See Bailey et al. (2013) and the references therein for an overview of the literature on measuring cross-sectional dependence. 25

26 The features of the estimated regime indicator I Cli t 2 (1 ) V ol t 1 markedly depend on the assumption made about the threshold variable (see Panel A). Under the exible parameterisation proposed in this paper, the least squares estimators for and are equal to ^ = 0:20 and ^ = 0:1878, respectively: the event ^Cli t 2 1 ^ V ol t 1 ^ occurs with a sample frequency equal to 0:5506 and displays a correlation with the NBER recession indicator equal to 0:3716. The estimation results are di erent under the restrictions = 0 and = 1: the estimated threshold values are equal to 0:2720 and 0:1493, respectively; the sample means of the regime indicators are equal to 0:6101 and 0:4196, respectively; and the correlations between the estimated regime indicator and the NBER recession indicator are equal to 0:2337 and 0:4862, respectively. Therefore, consistently with the speci cation of the threshold variable Cli t 2 (1 ) V ol t 1, the numerical results from the exible parameterisation always lie between those obtained by a priori imposing the restrictions : this highlights the relevance of the exible speci cation proposed in this paper, as a priori restrictions on may lead to misleading empirical ndings. In order to get a sense of the economic relevance of the proposed exible model, it is instructive to consider the degree of cross-sectional dependence as measured by the maximum eigenvalue of the conditional correlation matrix in each regime (see Panel B): the higher such a dependence, the less e ective portfolio diversi cation involving only stocks will be, and the more investors will have to search for alternative asset classes to invest in (e.g., bonds). Under the exible model, the maximum eigenvalues under the regimes Cli t 2 (1 ) V ol t 1 and Cli t 2 (1 ) V ol t 1 > are equal to 374:479 and 212:982, respectively: the maximum eigenvalue decreases when shifting from the rst to the second regime, which means that conditional cross-sectional 26

27 dependence is higher in the former than in the latter. The same monotonic relationship is found when the restrictions = 0 and = 1 are a priori imposed: under the regime Cli t 2 (1 ) V ol t 1, the maximum eigenvalues are equal to 396:911 and 348:032, respectively; and to 153:985 and 268:280, respectively, when the event Cli t 2 (1 ) V ol t 1 > occurs. In both regimes, the eigenvalues from the exible model lie between those from the speci cations under the restrictions = 0 and = 1: therefore, a priori restrictions imposed on the indicator function may bias the estimate of portfolio risk (as captured by the conditional second moment), with potential consequences for allocation strategies. Table 8 presents results for the estimated conditional means, standard deviations and correlations from the exible threshold model (see Panels A, B and C, respectively). Given the contribution of the paper in relation to the empirical analysis, the focus of the discussion is on the conditional correlations (see Panel C): consistently with the monotonic relationship found in the maximum eigenvalues, correlations are generally (although not always) higher in the regime Cli t 2 (1 ) V ol t 1 than when the event Cli t 2 (1 ) V ol t 1 > occurs. 7 Conclusions This paper has generalised the class of multivariate threshold regression models considered in Tsay (1998) to let the threshold variable be a linear combination of variables with unknown coe cients: in this way, the potential problem of model misspeci cation is tackled as the threshold variable no longer needs to be a priori selected. The paper has made a number of methodological and empirical contributions. On the methodological side, it has provided su cient conditions for identi cation of the model, 27

28 it has suggested an algorithm to obtain the least squares estimator for the parameters, and it has developed a test for the number of regimes based on results from multivariate statistics: all methodological contributions have been supported by a comprehensive Monte Carlo analysis. The theoretical framework has then been applied to an asset pricing exercise in stock markets with a focus on portfolio allocation: this has illustrated how the proposed model may be used to characterise the degree of cross-sectional dependence between returns from stock portfolios. This work can be extended along several dimensions. One in particular is worth discussing, namely the application of the proposed model to an asset allocation problem: to this purpose, the contribution by Song et al. (2012) would be a useful starting point. Asset allocation problems have been considered within the framework of Markov-Switching (see Guidolin and Timmermann, 2008) and structural break models (see Pettenuzzo and Timmermann, 2011): the multivariate threshold model proposed in this paper would then provide a useful complementary tool. A Proof of Theorem 3.1 In order to prove part (a), without loss of generality assume that 1 = c 1 so that 0 q t = c 1 q 1t + PK q m=2 m q mt : For c 1 = 0 it follows that P " I PK q m=2 m q mt! = I PK q m=2 v m q mt v!# = 1; 80 < v < 1; 28

29 and the condition stated in De nition 3.1 fails to hold. If 0 < c 1 < 1 then " P I q 1t PK q c 1 m=2 m c 1 q mt! = I q 1t v c 1 PK q m=2 v m q mt c 1!# = 1, v = 1; and the condition stated in De nition 3.1 holds: the case 1 < c 1 < 0 is analogous and omitted. This concludes the proof of part (a). In order to prove part (b), since 0 Kq = c then 1 = c P Kq m=2 m and 0 q t = 1 q 1t + PK q m=2 m q mt = cq 1t + PK q m=2 m (q mt q 1t ) : For c = 0 it follows that P ( I " PK q m=2 m (q mt q 1t ) # = I " PK q m=2 v m (q mt q 1t ) v #) = 1; 80 < v < 1; and the condition stated in De nition 3.1 fails to hold. If 0 < c < 1 then ( " P I q 1t c PK q m=2 m c (q mt q 1t ) # = I " q 1t v c PK q m=2 v m c (q mt q 1t ) #) = 1, v = 1; and the condition stated in De nition 3.1 holds: the case 1 < c < 0 is analogous and omitted. This concludes the proof of part (b) and of the theorem. B Power of the Test for the Number of Regimes This appendix proves that under the condition P (q t = Sx t ) = 1 imposed on the system in (1) and (2), the test for the number of regimes proposed in Section 4.2 does not have power against the alternative that the threshold e ect is induced only on the intercept in 29

30 all equations in the system or in a subset of them. To this purpose, write (12) as y it = x 0 t 1i + (q 0 t x 0 t) 2i + it ; i = 1; : : : ; N; t = 1; : : : ; T; (16) where 1i and 2i are de ned in (10) and (11), respectively. Let the K q (K x + 1) matrix S be de ned as S s 1 ; : : : ; s Kq 0 ; sm (s m1 ; _s 0 m) 0 ; m = 1; : : : ; K q ; where s m is a (K x + 1) 1 vector, for m = 1; : : : ; K q ; decompose x t, 1i and 2i as x t (1; _x 0 t) 0 ; and 1i 1i1 ; _ 1i ; 2i 0 2i1; : : : ; 0 2iK q ; 0 2im 2im1 ; _ 2im 0 0 ; m = 1; : : : ; Kq ; where 2im is a (K x + 1) 1 vector. Under the condition 2 0 P (q t = Sx t ) = 1, P q t = 6 B s C A x t = 1, P q t = 7 6 B 5 s 11 ; _s C A 1 _x t 3 1 C A = s 0 K q s Kq1; _s 0 K q 2 0, P q t = 6 B s 11 + _s 0 1 _x t. s Kq1 + _s 0 K q _x t 13 = 1 C7 A5 30

31 the model in (16) becomes y it = x 0 t 1i + (q 0 t x 0 t) 2i + it = (1; _x 0 t) 1i1 ; _ 0 0 i 0 1i + hs 11 + _s 01 _x t ; : : : ; s Kq1 + _s 0Kq _x t (1; _x 0 t) 0 2i1; : : : ; 0 2iK q + it = (1; _x 0 t) 1i1 ; _ 0 0 1i + s 11 + _s 0 1 _x t ; s 11 _x 0 t + _s 0 1 _x t _x 0 t; : : : ; s Kq1 + _s 0 K q _x t ; s Kq1 _x 0 t + _s 0 K q _x t _x 0 t 2i11 ; _ 02i1; : : : ; 2iKq1; _ 0 02iKq + it = 1i1 + _x 0 t _ 1i + PK q m=1 (s m1 + _s 0 m _x t ) 2im1 + PK q m=1 (s m1 + _s 0 m _x t ) _x 0 t _ 2im + it = 1i1 + PK q m=1 s m1 2im1! + _x 0 t " _ 1i + PK q m=1 _s m 2im1 + s m1 _ 2im # + _x 0 t PK q m=1! 0 _s m _ 2im _x t + it : for i = 1; : : : ; N and m = 1; : : : ; K q, 2im1 cannot be separately identi ed from 1i1 and _ 1i, and the test does not have power against the alternative that the threshold e ect is induced only on the intercept in all equations in the system or in a subset of them. References [1] Anderson, T. W. (1984), An Introduction to Multivariate Statistical Analysis, 2nd ed., New York: John Wiley. [2] Ang, A., G. Bekaert, and M.Wei (2008), "The Term Structure of Real Rates and Expected In ation," Journal of Finance, 63 (2),

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37 Table 1: Bias, RMSE and Size in the case of the Estimator for 111 = 1 This table presents results from the Monte Carlo analysis of the estimator for 111 = 1. The DGP is y r it = ( 1i1 + x r t 1i2 + u r 1it ) I [qr 1t + (1 ) q r 2t ] + [ 2i1 + x r t 2i2 + u r 2it ] I [qr 1t + (1 ) q r 2t > ] ; i = 1; 2; t = 1; : : : ; T; r = 1; : : : ; 2000; with = 2, 111 = 112 = 121 = 122 = 1 = 1 and 211 = 212 = 221 = 222 = 2 = 1 +. x r t, q r 1t and q r 2t are generated as x r t = x (1 x ) + x x r t x 1/2 r xt ; x 50 = x ; q r mt = q 1 q + q q r m;t q 1/2 r qmt ; q r m; 50 = q ; t = 49; : : : ; 0; : : : ; T; m = 1; 2; with r xt = (f t r + r xt) p p 2, r qmt = f t r + qmt r 2 for m = 1; 2, with x N (1; 1), x U (0:05; 0:95), x = 1 () /2, q U (0:05; 0:95), f t r IIDN (0; 1), r xt IIDN (0; 1), r qmt IIDN (0; 1) for m = 1; 2, and the rst 50 observations discarded. u r 11t, u r 21t, u r 12t and u r 22t are generated as u r jit = f p t r + uit r j 2 for i; j = 1; 2, with r uit IIDN (0; 1), 2 1 = 1 and 2 2 = 2. = 0:10 = 0:50 = 0:25 = 1:00 = 1:75 = 0:25 = 1:00 = 1:75 T Bias RMSE Size Bias RMSE Size Bias RMSE Size Bias RMSE Size Bias RMSE Size Bias RMSE Size