Regime Switching in the Presence of Endogeneity

Transcription

1 ISSN X Department of Econometrics and Business Statistics Regime Switching in the Presence of Endogeneity Tingting Cheng, Jiti Gao and Yayi Yan May 2018 Working Paper 9/18

2 Regime Switching in the Presence of Endogeneity Tingting Cheng and Jiti Gao 1 and Yayi Yan School of Finance, Nankai University, Department of Econometrics and Business Statistics, Monash University Abstract In this paper, we propose a state-varying endogenous regime switching model the model), which includes the endogenous regime switching model by Chang et al. 2017), the model, as a special case. To estimate the unknown parameters involved in the model, we propose a maximum likelihood estimation method. Monte Carlo simulation results show that in the absence of state-varying endogeneity, the model and the model have similar performance, while in the presence of state-varying endogeneity, the model performs much better than the model. Finally, we use the model to analyze the China stock market returns and our empirical results show that there exists strongly state-varying endogeneity in volatility switching for the Shanghai Composite Index returns. Moreover, the model can indeed produce a much more realistic assessment for the regime switching process than the one obtained by the model. Keywords: Latent factor, Maximum likelihood estimation, Markov chain, Regime switching models, State-varying endogeneity JEL clasifications: C22, C32 1 Introduction Regime switching models have been widely used in economics and finance after the publication of an influential article by Hamilton 1989), which extended Markov switching models by including an autoregressive process. Asymptotic properties for regime switching models have been discussed extensively in the literature, such as Bickel et al. 1998); Jensen and Petersen 1999); Douc et al. 2004), and Cho and White 2007), among others. Moreover, regime switching models have also been applied in various other models, such as vector autoregression with regime switching Krolzig 2013), dynamic factor model with regime switching Kim and Nelson 1998) and panel data model with regime switching Chen 2007). Numerous empirical studies can also be found in Capponi 1 Corresponding author: Department of Econometrics and Business Statistics, Monash University, Caulfield East, Victoria 3145, Australia. Telephone: Fax: Jiti.Gao@monash.edu. 1

3 and Figueroa-López 2014), Fink et al. 2017) and etc. Recently, many studies suggest that the assumption of a constant transition probability for Markov regime switching models may be too restrictive for many empirical applications. For example, it is often the case where the estimated state variable has a strongly business cycle correlation Sims and Zha 2006). As a result, researchers have started to consider how to relax the exogenous switching assumption, which can roughly be classified into two streams as follows: 1) Markov-switching is driven by a set of observed variables e.g. Diebold et al. 1994; Filardo 1994; Bazzi et al. 2017); and 2) Markov regime switching models incorporate endogenous switching e.g. Kim et al. 2008; Chang et al. 2017). In particular, Chang et al. 2017) proposed an endogenous regime switching model in which regime switching is allowed to be correlated with both the innovations and the observed time series. In their model, the endogeneity of the regime switching is captured by a constant parameter. They showed that the presence of endogeneity in regime switching is indeed strong and helps to extract more information from the observed data on the latent regimes. However, the endogeneity parameter in their model is assumed to be constant, which implies that the correlation between latent factor and the observed time series is invariant during the whole time period. As illustrated above, such assumption may be restrictive for empirical applications. In particular, we find that the endogeneity of regime switching in some cases is complicated and a constant endogeneity parameter cannot fully capture the characteristics of real data. For example, in the volatility switching model for a given financial asset, the sign of endogeneity parameter would represent leverage effect or anti-leverage effect, and the value of endogeneity parameter corresponds to the degree of leverage effect. Many researchers have found that there exist both leverage effect and anti-leverage effect in financial markets at different time period, and the degree of leverage effect may change over time. Therefore, it may be better to consider a state-varying endogeneity of regime switching than the constant endogeneity. In this paper, we propose a new approach to dealing with the endogeneity issue of regime switching that allows for the endogeneity parameter to be varying in different latent regimes. For simplicity, we refer this model as state-varying endogenous regime switching model or model for short). The state-varying endogeneity means that we allow for the correlation between regime switching and observed time series to be varying in different latent regimes. Note that if the endogeneity of regime switching is invariant in different states, our model will reduce to the constant 2

4 endogenous regime switching models in Chang et al. 2017). For simplicity, we refer the model proposed in Chang et al. 2017) as the model in the rest of this paper. To evaluate the performance of our proposed model, we conduct a set of simulation studies. The results can be summarized as follows. First, in the presence of the state-varying endogeneity, the model performs much better than the model. In this case if we do not take the presence of state-varying endogeneity into account, there will exist both estimation bias and significant efficiency loss. Second, when we set the endogeneity to be the same in different latent regimes i.e. constant endogeneity), then the estimates using the model are very similar to those obtained from the model, which is consistent with our expectation. Finally, we find that the likelihood ratio test for state-varying endogeneity works well if regime switching is assumed to be stationary. To demonstrate the effectiveness of the model in practice, we use this model to analyze the China stock market returns and our empirical results show that there exists strong state-varying endogeneity in volatility switching for the Shanghai Composite Index returns, and the transitional probabilities estimated based on the model are indeed much more realistic for regime switching process than the one obtained by the model. To summarize, Our proposed work has the following contributions. First, we develop a statevarying endogenous regime switching model model) which allows for the endogeneity parameter to be varying in different latent regimes. Second, we propose a maximum likelihood estimation method to estimate the unknown parameters in the model. Third, we propose a likelihood ratio test for the state-varying endogeneity in the stationary case. Finally, we conduct Monte Carlo simulation studies and an empirical study to investigate the performance of the above method we propose in this paper. The rest of the paper is organized as follows. In Section 2, we introduce the model and its estimation method. In Section 3, we present the results of our simulation studies. Section 4 reports the analysis of the China stock market returns using the volatility switching model. Section 5 concludes this paper, and an appendix collects the additional figures and proofs. 3

5 2 Models with state-varying endogenous regime switching As we mentioned in the introduction, we refer to the model we consider in this paper as the state-varying endogenous regime switching model or the model for short). We begin with a two-regime switching model for illustration, and this model can easily be extended to an N-regime switching model. 2.1 Model specification Consider the following regime switching model for a time series {y t } n t=1 : y t = x tβ st + σ st u t, u t i.i.d.n0, 1), 2.1) where y t is the dependent variable, x t is a vector of observed variables, which may include the lagged values of y t, β st and σ st are state-dependent parameters. The state variable s t determines the regimes of an economy depending on whether the value of a latent factor w t is above a specific threshold level or not. Denote the two regimes by 0 and 1, and set 0 if w t < τ s t = 1 if w t τ, 2.2) where τ refers to a threshold level and w t follows an AR1) process given by w t = αw t 1 + v t, v t N0, 1), 2.3) in which α 0, 1] is an autoregressive coefficient. In order to model the state-varying endogeneity in regime switching, u t and v t+1 are independent and identically distributed i.i.d.) as u t v t+1 N 0, 1 ρ s t, 2.4) 0 ρ st 1 where the endogeneity parameter ρ st denotes the correlation between regime switching and the observed time series. Note that ρ st in 2.4) is state dependent instead of being a constant in the 4

6 model and it is easy to see that if ρ 0 = ρ 1, then our model reduces to the model. Given 2.3) and 2.4), we may equivalently define the latent factor w t as w t = αw t 1 + ρ st 1 u t ρ 2 s t 1 v t, 2.5) where u t and v t are i.i.d. normally distributed random variables. Equation 2.5) shows that the link between latent factor w t and a shock to the observed time series can be different in different latent states. This extension is quite straightforward for volatility switching model. For example, if y t denotes returns from a financial asset, the sign of the endogeneity parameter will represent leverage effect ρ < 0) or anti-leverage effect ρ > 0), and the value of endogeneity parameter corresponds to the degree of leverage effect. In Section 4 below, we will see that there exist both leverage effect and anti leverage effect in stock index returns and these effects are state-dependent. 2.2 Maximum likelihood estimation Our model can be estimated by maximum likelihood estimation. Let F t = y t, y t 1,..., y 1 ) and Ω t = x t, x t 1,..., x 1 ) be vectors containing observed data, and θ = β 0, σ 0, β 1, σ 1, ρ 0, ρ 1, α, τ) be the vector of unknown parameters. The conditional log-likelihood function for the observed data is constructed as n ly 1,...y n ) = log fy 1 θ) + log fy t Ω t, F t 1 ; θ). 2.6) t=2 The maximum likelihood estimator ˆθ is given by ˆθ = arg max ly 1,...y n ). To obtain the conditional density function fy t Ω t, F t 1 ; θ) for each period, we extend the modified Markov switching filter in in terms of transition probabilities. Particularly, the transitional probabilities of the state process s t defined in 2.2) for several different settings of the model are given in Propositions below. Proposition 2.1. When ρ st < 1, the transitional probability ω = ωs t 1,..., s t k 1, y t 1,..., y t k 1 ) of s t to the low state conditional on the previous states and the observed data is given as follows: 5

7 a) If α < 1, ω = [1 s t 1 ) τ 1 α 2 ] +s t 1 τ τ ρ st 1 u ) t αx/ 1 α 1 α Φ 2 2 ϕx)dx 1 ρ 2 st 1 1 s t 1 )Φτ 1 α 2 ) + s t 1 [1 Φτ 1 α 2 )]. b) If α = 1, for t = 1, ωs 0 ) = Φτ) and for t 2, ω = [ 1 s t 1 ) τ/ t 1 ) ] +s t 1 τ/ τ ρ st 1 u t x/ t 1 t 1 Φ ϕx)dx 1 ρ 2 st 1 1 s t 1 )Φτ/ t 1) + s t 1 [1 Φτ/ t 1)], where ϕ ) and Φ ) denote the density and distribution functions of the standard normal random variable, respectively. Note that Theorem 3.1 in Chang et al. 2017) is a special case of Proposition 2.1 when ρ 0 = ρ 1. Proposition 2.2. When ρ st = 1, the transitional probability ω = ωs t 1,..., s t k 1, y t 1,..., y t k 1 ) of s t to the low state conditional on the previous states and the observed data is given as follows: a) If α = 0, ω = I { ρ st 1 < τ }, where I { } is an indicator function. b) If 0 < α < 1, ω = 1 s t 1 ) min 1, + s t 1 max 0, Φ τ ρ 0 u t 1 ) Φ τ ρ 1 u t 1 ) 1 α 2 α ) Φτ 1 α 2 ) ) 1 α 2 α Φτ 1 α 2 ) 1 Φτ. 1 α 2 ) c) If 1 < α < 0, ω = 1 s t 1 ) max 0, + s t 1 min 1, Φ τ ) 1 α 2 Φ τ ρ 0 u t 1 ) 1 Φ τ ρ 1 u t 1 ) 1 Φτ 1 α 2 ) Φτ 1 α 2 ) ) 1 α 2 α. 1 α 2 α ) 6

8 d) If α = 1, for t = 1, ωs 0 ) = Φτ), and for t 2, 1 s t 1, if ρ st 1 u t 1 > 0 ω = 1 Φ τ ρ st 1 ) t 1 ) s t 1 Φτ/. t 1) 1 s t 1 )Φτ/ t 1)+s t 1 [1 Φτ/ t 1)], otherwise Note that Corollary 3.2 in Chang et al. 2017) is a special case of Proposition 2.2 when ρ 0 = ρ 1. Using the modified Markov switching filter in, we can also easily extract the latent autoregressive factor w t. The algorithms are similar to that of except that the transition density of w t conditional on the previous states and the past values of observed time series for the model, which yields the inferred factor Ew t F t ), for all t = 1, 2,..., n, can be replaced by those in Proposition 2.3. Proposition 2.3. The transitional density of w t conditional on the previous states and the past values of observed time series for the model is given as follows: a) When α < 1 and ρ st < 1, and pw t s t 1 = 1, s t 2,..., s t k 1, F t 1 ) 1 Φ = 1 ρ 2 1 +α2 ρ ρ Φτ 1 α 2 ) τ αwt ρ 1u t 1 ) 1 ρ 2 1 +α2 ρ 2 1 ) )) N ρ 1 u t 1, 1 ρ2 1 + α2 ρ 2 ) 1 1 ρ 2 1 pw t s t 1 = 0, s t 2,..., s t k 1, F t 1 ) ) ) 1 ρ Φ 2 0 +α2 ρ 2 0 τ αwt ρ 0u t 1 ) 1 ρ ρ 2 0 = +α2 ρ Φτ 1 α 2 ) N ρ 0 u t 1, 1 ρ2 0 + α2 ρ 2 ) 0 1 ρ 2. 0 where Na, b) also signifies the density of normal distribution with mean a and variance b. b) When α < 1 and ρ st = 1, pw t s t 1 = 1, s t 2,..., s t k 1, F t 1 ) = 1 α 2 α ϕ wt ρ1 u t 1 α 1 α 2) 1 Φτ 1 α 2 ) I {w t ατ + ρu t 1 } 7

9 and pw t s t 1 = 0, s t 2,..., s t k 1, F t 1 ) = 1 α 2 α ϕ wt ρ0 u t 1 α 1 α 2) Φτ 1 α 2 ) I {w t < ατ + ρu t 1 }. c) When α = 1 and ρ st < 1, we have and pw t s t 1 = 1, s t 2,..., s t k 1, F t 1 ) 1 Φ = t tρ 2 1 +ρ2 1 1 ρ 2 1 τ wt ρ 1u t 1 t tρ 2 1 +ρ2 1 1 Φτ/ t 1) ) )) N ρ 1 u t 1, t tρ2 1 + ) ρ2 1 t 1 pw t s t 1 = 0, s t 2,..., s t k 1, F t 1 ) ) ) t tρ Φ 2 0 +ρ2 0 τ wt ρ 0u t 1 1 ρ 2 0 t tρ 2 0 = +ρ2 0 Φτ/ t 1) d) When α = 1 and ρ st = 1, pw t s t 1 = 1, s t 2,..., s t k 1, F t 1 ) = N ρ 0 u t 1, t tρ2 0 + ) ρ2 0. t 1 ) 1 t 1 ϕ wt ρ t 1 1 u t 1 1 Φτ/ t 1) I {w t τ + ρ 1 u t 1 } and pw t s t 1 = 0, s t 2,..., s t k 1, F t 1 ) = ) 1 t 1 ϕ wt ρ t 1 0 u t 1 1 Φτ/ t 1) I {w t < τ + ρ 0 u t 1 }. Note that Corollary 3.3 in Chang et al. 2017) is a special case of Proposition 2.3 when ρ 0 = ρ 1. In what follows, we will examine the performance of the model by Monte Carlo simulation and empirical analysis. 3 Simulation studies In this section, we conduct a set of simulations to evaluate the performance of our model and the estimation method discussed in Section 2. 8

10 3.1 Simulation models In our simulations, we consider both mean and volatility switching models. Our volatility switching model is given by y t = σ st u t, u t N0, 1). 3.1) In this study, we set σ 0 = 0.02 and σ 1 = 0.05, which are roughly the same as our estimates for the weekly China stock returns using volatility switching model in our empirical study. Meanwhile, our mean model is specified as y t = µ st + γy t 1 µ st 1 ) + σu t, u t N0, 1). 3.2) Following, we set the parameter values as µ 0 = 0.6, µ 1 = 3, γ = 0.5 and σ = 0.8. For both mean and volatility switching models, u t and v t are generated as specified in 2.2)-2.4) with sample sizes T = 100, 200 and 500. For each sample size, we conduct 500 replications. We consider three different pairs for ρ st, corresponding to highly state-varying endogeneity ρ 0 = 0.9, ρ 1 = 0.9), moderate state-varying endogeneity ρ 0 = 0.5, ρ 1 = 0.5), and constant endogeneity ρ 0 = ρ 1 = 0). As discussed earlier, if ρ 0 = ρ 1, our model reduces to the constant endogenous regime switching model considered by. Following, we consider three types of the autoregressive coefficient α and threshold τ given by α, τ) = 0.4, 0.5), 0.8, 0.7), 1, 9.63), corresponding to different persistence of regime switching. In addition, we measure the accuracy of the estimates by computing the average of the 500 maximum likelihood estimates and the root mean squared error RMSE) of the 500 maximum likelihood ML) estimates: averageˆθ) = i=1 θ i and RMSEˆθ) = θ i θ 0 ) 2, where θ i is the estimate of parameter θ 0 for the i-th replication, and θ 0 is the true value of parameters which is θ 0 = µ 0, µ 1, σ 0, σ 1 ) = 0.6, 3.0, 0.02, 0.05). i=1 9

11 3.2 Simulation results The simulation results for both the mean and volatility switching models are presented in Tables below. For the parameter σ 0, σ 1, µ 0 and µ 1, each table shows the average of the 500 ML estimates, as well as the RMSE of the 500 ML estimates in the parentheses. Tables reveal the following findings: In all cases, the average of the 500 maximum likelihood estimates of our model are very close to their true values in both mean and volatility switching models and the RMSE decreases when sample size increases. In the presence of state-varying endogeneity, our model performs much better than the model. In the case of constant endogeneity ρ 0 = ρ 1 = 0), the estimates of our model and the model are almost identical for both mean and volatility switching models. As discussed earlier, in this case, the model reduces to the model. Our simulation results just demonstrate this point. In the nonstationary case α = 1), the estimates of our model and the model are both close to their true values for both mean and volatility switching models. The effect of state-varying endogeneity on the parameter estimates in both mean and volatility switching models are insignificant when α = 1. 10

12 Table 3.1: Monte Carlo results when ρ 0 = 0.9, ρ 1 = 0.9 for the volatility model 3.1). T σ 0 = 0.02 σ 0 = α = 0.4, τ = ) 0.002) 0.001) 0.039) 0.020) 0.004) ) 0.006) 0.008) 0.041) 0.066) 0.152) α = 0.8, τ = ) 0.003) 0.002) 0.038) 0.027) 0.012) ) 0.005) 0.007) 0.038) 0.044) 0.220) α = 1, τ = ) 0.003) 0.003) 0.028) 0.006) 0.005) ) 0.003) 0.004) 0.023) 0.008) 0.007) Notes: Each cell contains the average of the 500 ML estimates for the parameter listed in the column heading, and the RMSE of the estimates in parentheses. 11

13 Table 3.2: Monte Carlo results when ρ 0 = 0.5, ρ 1 = 0.5 for the volatility switching model 3.1). T σ 0 = 0.02 σ 0 = α = 0.4, τ = ) 0.002) 0.003) 0.025) 0.007) 0.005) ) 0.006) 0.007) 0.034) 0.243) 0.562) α = 0.8, τ = ) 0.003) 0.002) 0.010) 0.008) 0.004) ) 0.005) 0.012) 0.012) 0.024) 0.456) α = 1, τ = ) 0.003) 0.003) 0.011) 0.007) 0.006) ) 0.004) 0.005) 0.012) 0.008) 0.013) Notes: Each cell contains the average of the 500 ML estimates for the parameter listed in the column heading, and the RMSE of the estimates in parentheses. 12

14 Table 3.3: Monte Carlo results when ρ 0 = ρ 1 = 0 for the volatility switching model 3.1). T σ 0 = 0.02 σ 0 = α = 0.4, τ = ) 0.003) 0.003) 0.018) 0.008) 0.006) ) 0.023) 0.005) 0.018) 0.061) 0.009) α = 0.8, τ = ) 0.003) 0.003) 0.011) 0.008) 0.005) ) 0.004) 0.004) 0.019) 0.026) 0.009) α = 1, τ = ) 0.003) 0.001) 0.013) 0.008) 0.006) ) 0.004) 0.003) 0.012) 0.008) 0.007) Notes: Each cell contains the average of the 500 ML estimates for the parameter listed in the column heading, and the RMSE of the estimates in parentheses. 13

15 Table 3.4: Monte Carlo results when ρ 0 = 0.9, ρ 1 = 0.9 for the mean switching model 3.2). T µ 0 = 0.6 µ 1 = α = 0.4, τ = ) 0.120) 0.084) 0.191) 0.142) 0.091) ) 0.349) 0.354) 0.284) 0.215) 0.157) α = 0.8, τ = ) 0.137) 0.091) 0.247) 0.178) 0.107) ) 0.351) 0.331) 0.402) 0.324) 0.257) α = 1, τ = ) 0.184) 0.105) 0.414) 0.347) 0.362) ) 0.185) 0.116) 1.227) 0.935) 0.954) Notes: Each cell contains the average of the 500 ML estimates for the parameter listed in the column heading, and the RMSE of the estimates in parentheses. 14

16 Table 3.5: Monte Carlo results when ρ 0 = 0.5, ρ 1 = 0.5 for the mean switching model 3.2). T µ 0 = 0.6 µ 1 = α = 0.4, τ = ) 0.146) 0.080) 0.261) 0.181) 0.100) ) 0.238) 0.223) 0.305) 0.214) 0.162) α = 0.8, τ = ) 0.157) 0.087) 0.274) 0.207) 0.115) ) 0.241) 0.155) 0.336) 0.279) 0.153) α = 1, τ = ) 0.165) 0.130) 0.441) 0.431) 0.305) ) 0.206) 0.132) 0.812) 0.675) 0.470) Notes: Each cell contains the average of the 500 ML estimates for the parameter listed in the column heading, and the RMSE of the estimates in parentheses. 15

17 Table 3.6: Monte Carlo results when ρ 0 = ρ 1 = 0 for the mean switching model 3.2). T µ 0 = 0.6 µ 1 = α = 0.4, τ = ) 0.152) 0.096) 0.315) 0.176) 0.111) ) 0.150) 0.093) 0.311) 0.178) 0.109) α = 0.8, τ = ) 0.168) 0.089) 0.353) 0.233) 0.127) ) 0.165) 0.088) 0.363) 0.223) 0.125) α = 1, τ = ) 0.169) 0.105) 0.502) 0.352) 0.320) ) 0.171) 0.103) 0.644) 0.512) 0.377) Notes: Each cell contains the average of the 500 ML estimates for the parameter listed in the column heading, and the RMSE of the estimates in parentheses. Finally, we consider testing for the presence of state-varying endogeneity in regime switching models on the basis of the likelihood ratio LR) test given by 2lˆθ) l θ)), 3.3) where l stands for the log-likelihood function and θ and ˆθ denote their maximum likelihood 16

18 estimates with and without the constant endogeneity restriction, ρ 0 = ρ 1, respectively. The likelihood ratio test has a Chi-square limit distribution with one degree of freedom. Tables 3.7 and 3.8 report the size and power of the LR test at the 5% significance level for both the mean and volatility switching models, respectively. From Tables 3.7 and 3.8, we have the following findings: For the mean switching model 3.2), under the cases of autoregressive coefficient α = 0.4, 0.8, when the null hypothesis is true, the LR test has roughly correct size with T = 200 and 500. Power increases rapidly as the degree of state-varying endogeneity increases and sample size becomes large. For the volatility switching model 3.1), when the null hypothesis is true, the LR test is oversized with T = 100, 200 and 500. Power increases rapidly as the degree of state-varying endogeneity increases and sample size becomes large under the cases of autoregressive coefficient α = 0.4, 0.8. In contrast, the LR test does not work well when α = 1, and the power increases very slowly as the degree of state-varying endogeneity increases for both mean and volatility models. To summarize, the Monte Carlo experiments suggest that ML estimates using our model are quite accurate, while the constant endogenous estimator produces biased estimates when the true process has state-varying endogenous switching and the latent factor is stationary. Moreover, the likelihood ratio test appears to be a reliable test for the state-varying endogenous switching under the stationary case. 17

19 Table 3.7: Power of LR test for state-varying endogeneity in mean switching model. Size Power Power ρ 0 = ρ 1 = 0 ρ 0 = 0.5, ρ 1 = 0.5 ρ 0 = 0.9, ρ 1 = 0.9 T = 100 α = 0.4, τ = α = 0.8, τ = α = 1, τ = T = 200 α = 0.4, τ = α = 0.8, τ = α = 1, τ = T = 500 α = 0.4, τ = α = 0.8, τ = α = 1, τ = Notes: Each cell of the table contains the percentage of 500 Monte Carlo simulations for which the likelihood ratio test rejected the null hypothesis that ρ 0 = ρ 1 at the 5% significance level. For both columns labeled size and power, the critical values are based on a Chi-square limit distribution with one degree of freedom. 18

20 Table 3.8: Power of LR test for state-varying endogeneity in volatility switching model. Size Power Power ρ 0 = ρ 1 = 0 ρ 0 = 0.5, ρ 1 = 0.5 ρ 0 = 0.9, ρ 1 = 0.9 T = 100 α = 0.4, τ = α = 0.8, τ = α = 1, τ = T = 200 α = 0.4, τ = α = 0.8, τ = α = 1, τ = T = 500 α = 0.4, τ = α = 0.8, τ = α = 1, τ = Notes: Each cell of the table contains the percentage of 500 Monte Carlo simulations for which the likelihood ratio test rejected the null hypothesis that ρ 0 = ρ 1 at the 5% significance level. For both columns labeled size and power, the critical values are based on a Chi-square limit distribution with one degree of freedom. 4 Empirical results To empirically illustrate our approach, we analyse the China stock market returns using the volatility switching model in 3.1) with state-varying endogenous regime switching and the constant endogenous regime switching model considered by Chang et al. 2017). Define stock market returns as the weekly Shanghai Composite Index returns which are obtained from Wind Economic Database for the period of August 2011 to April Figure 4.1 presents the time series plots of weekly 19

21 returns of the Shanghai Composite Index which shows the typical volatility clustering, that is, sometimes the returns appear quite clam and at other times highly volatile and this transition is strongly persistent. Figure 4.1: The weekly returns of the Shanghai Composite Index from August 2012 to April In order to estimate the volatility switching model by the ML method via the modified Markov switching model, we use numerical optimization procedures, including the Broyden-Fletcher-Goldfarb- Shanno BFGS) algorithm. The standard errors of estimates can be constructed from an estimate of the inverse of the information matrix, which is the negative of the second derivative of the log-likelihood function. Our estimates are reported in Table 4.1. The columns labelled exogenous and collect the estimated parameters from the conventional Markov switching model and the model, respectively. The column labelled collects the estimated parameters from our model. The estimates for the endogeneity parameter ρ are respectively, ρ = by the model and ρ 0 = , ρ 1 = by our model. The result from the model means weak endogeneity in volatility switching and few anti-leverage effect in the stock market. By contrast, the estimates using our model show a completely different result that in the low volatility state, stock index returns exists moderate anti-leverage effect while in the high volatility state, there exists substantial leverage effect in stock index returns. These results show that the endogeneity in regime switching for market volatility is quite subtle. Our model can extract more information through the correlation of the lagged value of innovation and the current state. In this case, the estimates of α and τ using our model are ) and ) 20

22 respectively, while for the model, we have ) and ) for α and τ. This shows that the estimates of α are all close to 1 and the transition is highly persistent. As discussed in Section 3, the LR test does not work well in this case due to nonstationarity of latent factors. Table 4.1: Estimation results for volatility switching model. Models Exogenous Parameters θ1 θ2 θ3 σ ) ) ) σ ) ) ) ρ ) ρ 0 ρ ) ) log-likelihood Notes: This table reports the maximum likelihood estimates of the volatility switching model for the Shanghai Composite Index returns. The sample period is August 2011 to April The columns labelled exogenous and collect the estimated parameters from the conventional Markov switching model and the model. The column labelled collects the estimated parameters from our model. Standard errors, reported in parentheses, are based on the inverse of the information matrix in all cases. Figure 4.2 shows the striking difference in the transitional probabilities estimated from our model and the model. The left hand side graph plots the transition probability from low to high volatility state: the dot line refers to P s t = 1 s t 1 = 0, u t 1 ) estimated by our model, 21

23 while the solid line refers to P s t = 1 s t 1 = 0, u t 1 ) estimated by the model. Similarly, the right hand shows the probabilities of staying at high volatility state. As discussed earlier, our model allows for state-varying endogeneity in regime switching and the corresponding transitional probabilities can extract more information from the realized values of the lagged market returns compared to the model. This statement can be clearly seen from Figure 4.2. We can see that the transitional probabilities estimated by the model have been changing more drastically than the transition probabilities estimated by the model. Figure 4.2: Estimated transitional probability from the volatility switching model. Notes: the left panel is the transitional probability of low to high volatility sate and the right panel is show the transitional probability of staying at high volatility state. The dot line and solid line refer to the probabilities estimated by our model and the model, respectively. The state-varying endogeneity in regime switching plays an important role when the underlying regime is known. Figure 4.3 presents the weekly returns and the transitional probabilities from high volatility state to low volatility state during the stock market crash from June 2015 to July 2015, where the solid line refers to the weekly returns of the Shanghai Composite Index and the dot line indicates the transition probabilities estimated by our model, and dashed line indicates the transitional probabilities estimated by the model. Figure 4.3 illustrates that the transitional probabilities obtained by the model are indeed much more realistic for the likelihood of moving into a low volatility regime from a known high volatility regime. The shadow area indicates that the Chinese stock market is in a period of stock market crash from June 2015 to July 2015, and it is reasonable to assume that the stock market is in high volatility at this time. The estimated transition probability from high to low volatility should be very small, and when the stock market 22

24 resumes from a stock market disaster, it will have a larger transitional probability from high volatility to low volatility. Our model shows this feature very well, while the estimation results of the model are not consistent with reality. Figure 4.3: Transitional probability during stock market crash. Notes: the weekly returns and the transitional probabilities from high volatility state to low volatility state during the stock market crash from June 2015 to July 2015, where the solid line refers to the weekly returns of the Shanghai Composite Index and the dot line indicates the transitional probabilities estimated by our model, and dashed line indicates transitional probabilities estimated by the model. We also compute the filter probability of being in the high volatility regime, as well as the inferred latent factor, using the model and the model. They are presented in Figures B.1-B.2 in the Appendix. Overall the time series of the high volatility probabilities, as well as the inferred latent factor, computed from the model and the model are quite similar. However, those results obtained by the model tend to fluctuate more during the high volatility state. This is because that the state-varying endogeneity extracts more information from the observed time series on regime switching. 5 Conclusion In this paper, we have proposed a state-varying endogenous regime switching model model) which allows endogeneity parameter to be varying in different latent regimes. Our simulation studies 23

25 have shown that in the presence of state-varying endogeneity, our model performs much better than the model. The empirical evidence shows that there exists strongly state-varying endogeneity in volatility switching for the Shanghai Composite Index returns and the transitional probabilities by the model are indeed much more realistic for regime switching process than that by the model. In the near future, we will continue our work on several aspects. First, we will generalize our model by allowing for observable covariates to affect the regime switching process. We could set the effect of observable covariates to be varying in different latent states, and accordingly the state variable s t can be presented as 0 if w t < τ + τ s t 1 z t s t =, 1 if w t τ + τ s t 1 z t where τ st is state-dependent parameter, z t is a vector of observable covariates including strictly exogenous explanatory variables and lagged values of the dependent variable. Second, we will explore the forecasting performance of the model and propose a proper test for the state-varying endogeneity of regime switching in the nonstationary case. References M. Bazzi, F. Blasques, S.J. Koopman, and A. Lucas. Time-varying transition probabilities for markov regime switching models. Journal of Time Series Analysis, 383): , P.J. Bickel, Y. Ritov, and T. Ryden. Asymptotic normality of the maximum-likelihood estimator for general hidden markov models. Annals of Statistics, 264): , A. Capponi and J.E. Figueroa-López. Dynamic portfolio optimization with a defaultable security and regime-switching. Mathematical Finance, 242): , Y. Chang, Y. Choi, and J.Y. Park. A new approach to model regime switching. Journal of Econometrics, 1961): , S.W. Chen. Measuring business cycle turning points in japan with the markov switching panel model. Mathematics and Computers in Simulation, 764): , J.S. Cho and H. White. Testing for regime switching. Econometrica, 756): , F.X. Diebold, J.H. Lee, and G.C. Weinbach. Regime switching with time-varying transition probabilities. Business Cycles: Durations, Dynamics, and Forecasting, 1: , R. Douc, E. Moulines, and T. Ryden. Asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime. Annals of Statistics, 325): ,

26 A.J. Filardo. Business-cycle phases and their transitional dynamics. Journal of Business & Economic Statistics, 123): , H. Fink, Y. Klimova, C. Czado, and J. Stöber. Regime switching vine copula models for global equity and volatility indices. Econometrics, 51):3, J.D Hamilton. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 572): , J. L. Jensen and N.V. Petersen. Asymptotic normality of the maximum likelihood estimator in state space models. Annals of Statistics, 272): , C.J. Kim and C.R. Nelson. Business cycle turning points, a new coincident index, and tests of duration dependence based on a dynamic factor model with regime switching. Review of Economics and Statistics, 802): , C.J. Kim, J. Piger, and R. Startz. Estimation of markov regime-switching regression models with endogenous switching. Journal of Econometrics, 1432): , H.M. Krolzig. Markov-switching vector autoregressions: Modelling, statistical inference, and application to business cycle analysis, volume 454. Springer Science & Business Media, C.A. Sims and T. Zha. Were there regime switches in us monetary policy? American Economic Review, 961):54 81, A Mathematical proofs The proofs of Propositions are similar to each other. For brevity, we only show how to prove Proposition 2.1 as follows. Proof of Proposition 2.1. We now give the proof of the transition probability ω ρ in our model for the case of α < 1. Let z st = w t αw t 1 ρ u st 1 t 1, 1 ρ 2 s t 1 1 ρ 2 s t 1 we can easily deduce that z st w t 1,..., w t k 1, y t 1,..., y t k 1 N0, 1). It follows that p {w t < τ w t 1,..., w t k 1, y t 1,..., y t k 1 } = p z s t < τ αw t 1 ρ u st 1 t 1 w t 1,..., w t k 1, y t 1,..., y t k 1 1 ρ 2 s t 1 1 ρ 2 s t 1 = Φ τ αw t 1 ρ st 1 u t 1 1 ρ 2 s t 1. 25

27 Note that pw t w t 1,..., w t k 1, y t 1,..., y t k 1 ) = pw t w t 1, u t 1 ), and that w t 1 is in dependent of u t 1. Consequently, we have and p {w t < τ w t 1 < τ,..., w t k 1, y t 1,..., y t k 1 } = p {w t < τ w t 1 1 α 2 < τ } 1 α 2, w t 2,..., w t k 1, y t 1,..., y t k 1 = p {s t = 0 s t 1 = 0, s t 2,..., s t k 1, y t 1,..., y t k 1 } ) τ 1 α 2 τ αw Φ t 1 ρ 0 u t 1 ϕx)dx 1 ρ 2 0 = Φτ 1 α 2 ) p {w t < τ w t 1 τ,..., w t k 1, y t 1,..., y t k 1 } = p {w t < τ w t 1 1 α 2 τ } 1 α 2, w t 2,..., w t k 1, y t 1,..., y t k 1 = p {s t = 0 s t 1 = 1, s t 2,..., s t k 1, y t 1,..., y t k 1 } ) τ τ αw 1 α Φ t 1 ρ 1 u t 1 ϕx)dx 2 1 ρ 2 1 = 1 Φτ. 1 α 2 ) Therefore, it follows that ω ρ = [1 s t 1 ) τ 1 α 2 ] +s t 1 τ 1 α Φ 2 1 s t 1 )Φ τ 1 α 2 ) + s t 1 [ 1 Φ τ ρ st 1 u t αx/ 1 α 2 1 ρ 2 s t 1 ) ϕx)dx τ 1 α 2 )]. The proof for the case of α = 1 is virtually identical except that w t N0, t) so ignored here. 26

28 B Additional Figures Figure B.1: Filter probabilities of high volatility state. Notes: Figure B.1 plots the time series of the filter probabilities of being in high volatility state. Top panel refers to the high volatility probability series estimated by the model, while the bottom panel plots those from the model. 27

29 Figure B.2: Extracted latent factor and the threshold level. Notes: the top panel in Figure B.2 is the extracted latent factor series solid line) and the threshold level dashed line) using the model, and the bottom panel plots those from the model. 28