Estimation of Markov regime-switching regression models with endogenous switching

Following Hamilton [1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384], estimation of Markov regime-switching regressions typically relies on the assumption that the latent state variable controlling regime change is exogenous. We relax this assumption and develop a parsimonious model of endogenous Markov regime-switching. Inference via maximum likelihood estimation is possible with relatively minor modifications to existing recursive filters. The model nests the exogenous switching model, yielding straightforward tests for endogeneity. In Monte Carlo experiments, maximum likelihood estimates of the endogenous switching model parameters were quite accurate, even in the presence of certain model misspecifications. As an application, we extend the volatility feedback model of equity returns given in Turner et al. [1989. A Markov model of heteroskedasticity, risk, and learning in the stock market. Journal of Financial Economics 25, 3–22] to allow for endogenous switching.     

Local GMM estimation of time series models with conditional moment restrictions

This paper investigates statistical properties of the local generalized method of moments (LGMM) estimator for some time series models defined by conditional moment restrictions. First, we consider Markov processes with possible conditional heteroskedasticity of unknown forms and establish the consistency, asymptotic normality, and semi-parametric efficiency of the LGMM estimator. Second, we undertake a higher-order asymptotic expansion and demonstrate that the LGMM estimator possesses some appealing bias reduction properties for positively autocorrelated processes. Our analysis of the asymptotic expansion of the LGMM estimator reveals an interesting contrast with the OLS estimator that helps to shed light on the nature of the bias correction performed by the LGMM estimator. The practical importance of these findings is evaluated in terms of a bond and option pricing exercise based on a diffusion model for spot interest rate.     

Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese Yuan

This paper provides closed-form likelihood approximations for multivariate jump-diffusion processes widely used in finance. For a fixed order of approximation, the maximum-likelihood estimator (MLE) computed from this approximate likelihood achieves the asymptotic efficiency of the true yet uncomputable MLE as the sampling interval shrinks. This method is used to uncover the realignment probability of the Chinese Yuan. Since February 2002, the market-implied realignment intensity has increased fivefold. The term structure of the forward realignment rate, which completely characterizes future realignment probabilities, is hump-shaped and peaks at mid-2004. The realignment probability responds quickly to economic news releases and government interventions.     

Semiparametric efficient adaptive estimation of asymmetric GARCH models

In this paper we derive a semiparametric efficient adaptive estimator of an asymmetric GARCH model. Applying some general results from Drost et al. [1997. The Annals of Statistics 25, 786–818], we first estimate the unknown density function of the disturbances by kernel methods, then apply a one-step Newton–Raphson method to obtain a more efficient estimator than the quasi-maximum likelihood estimator. The proposed semiparametric estimator is adaptive for parameters appearing in the conditional standard deviation model with respect to the unknown distribution of the disturbances.     

Sequential estimation of shape parameters in multivariate dynamic models

Sequential maximum likelihood and GMM estimators of distributional parameters obtained from the standardised innovations of multivariate conditionally heteroskedastic dynamic regression models evaluated at Gaussian PML estimators preserve the consistency of mean and variance parameters while allowing for realistic distributions. We assess their efficiency, and obtain moment conditions leading to sequential estimators as efficient as their joint ML counterparts. We also obtain standard errors for VaR and CoVaR, and analyse the effects on these measures of distributional misspecification. Finally, we illustrate the small sample performance of these procedures through simulations and apply them to analyse the risk of large eurozone banks.     

Dynamic quantile models

This paper introduces the Dynamic Additive Quantile (DAQ) model that ensures the monotonicity of conditional quantile estimates. The DAQ model is easily estimable and can be used for computation and updating of the Value-at-Risk. An asymptotically efficient estimator of the DAQ is obtained by maximizing an objective function based on the inverse KLIC measure. An alternative estimator proposed in the paper is the Method of L-Moments estimator (MLM). The MLM estimator is consistent, but generally not fully efficient. Goodness-of-fit tests and diagnostic tools for the assessment of the model are also provided. For illustration, the DAQ model is estimated from a series of returns on the Toronto Stock Exchange (TSX) market index.     

Linear and nonlinear regression with stable errors

In this paper we describe methods and evaluate programs for linear regression by maximum likelihood when the errors have a heavy tailed stable distribution. The asymptotic Fisher information matrix for both the regression coefficients and the error distribution parameters are derived, giving large sample confidence intervals for all parameters. Simulated examples are shown where the errors are stably distributed and also where the errors are heavy tailed but are not stable, as well as a real example using financial data. The results are then extended to nonlinear models and to non-homogeneous error terms.     

Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations

ARCH and GARCH models are widely used to model financial market volatilities in risk management applications. Considering a GARCH model with heavy-tailed innovations, we characterize the limiting distribution of an estimator of the conditional value-at-risk (VaR), which corresponds to the extremal quantile of the conditional distribution of the GARCH process. We propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. Finally, we apply the proposed approach to an energy market data set.     

Cointegrating rank selection in models with time-varying variance

Reduced rank regression (RRR) models with time varying heterogeneity are considered. Standard information criteria for selecting cointegrating rank are shown to be weakly consistent in semiparametric RRR models in which the errors have general nonparametric short memory components and shifting volatility provided the penalty coefficient _Cn→∞$C_n\to \infty$ and _Cn/n→0$C_n/n\to 0$ as n→∞$n\to \infty$. The AIC criterion is inconsistent and its limit distribution is given. The results extend those in Cheng and Phillips (2009a) and are useful in empirical work where structural breaks or time evolution in the error variances is present. An empirical application to exchange rate data is provided.     

ARCH/GARCH with persistent covariate: Asymptotic theory of MLE

The paper considers a volatility model which introduces a persistent, integrated or near-integrated, covariate to the standard GARCH(1, 1) model. For such a model, we derive the asymptotic theory of the quasi-maximum likelihood estimator. In particular, we establish consistency and obtain limit distribution. The limit distribution is generally non-Gaussian and represented as a functional of Brownian motions. However, it becomes Gaussian if the covariate has innovation uncorrelated with the squared innovation of the model or the volatility function is linear in parameter. We provide a simulation study to demonstrate the relevance and usefulness of our asymptotic theory.     

Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE

In generalized autoregressive conditional heteroskedastic (GARCH) models, the standard identifiability assumption that the variance of the iid process is equal to 1 can be replaced by an alternative moment assumption. We show that, for estimating the original specification based on the standard identifiability assumption, efficiency gains can be expected from using a quasi-maximum likelihood (QML) estimator based on a non Gaussian density and a reparameterization based on an alternative identifiability assumption. A test allowing to determine whether a reparameterization is needed, that is, whether the more efficient QMLE is obtained with a non Gaussian density, is proposed.     

Bias in the estimation of the mean reversion parameter in continuous time models

It is well known that for continuous time models with a linear drift standard estimation methods yield biased estimators for the mean reversion parameter both in finite discrete samples and in large in-fill samples. In this paper, we obtain two expressions to approximate the bias of the least squares/maximum likelihood estimator of the mean reversion parameter in the Ornstein–Uhlenbeck process with a known long run mean when discretely sampled data are available. The first expression mimics the bias formula of Marriott and Pope (1954) for the discrete time model. Simulations show that this expression does not work satisfactorily when the speed of mean reversion is slow. Slow mean reversion corresponds to the near unit root situation and is empirically realistic for financial time series. An improvement is made in the second expression where a nonlinear correction term is included into the bias formula. It is shown that the nonlinear term is important in the near unit root situation. Simulations indicate that the second expression captures the magnitude, the curvature and the non-monotonicity of the actual bias better than the first expression.     

Matrix exponential GARCH

I propose a new multivariate GARCH specification that maintains positive definiteness of the conditional covariance matrix. The idea is to specify the dynamics in the matrix logarithm of the conditional covariance. Because the matrix exponential transformation ensures positive definiteness, the dynamics can be specified without the positive definiteness constraint. This affords a variety of specifications and, in particular, we can specify element-by-element the dynamics of the matrix logarithm. I discuss specifications with leverage effects, estimation with multivariate Gaussian and t-distributions, and diagnostics that evaluate the appropriateness of the matrix exponential specification.     

Modelling Regime‐Specific Stock Price Volatility*

Single‐state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state‐dependent behaviour of volatility nor explain why the equity index skew persists into long‐dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state‐dependent leverage effect to NM‐GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time‐varying volatility risk premia. An empirical study on European equity indices identifies two‐state asymmetric NM‐GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean‐reversion and leverage effects during crash markets are quite different from those in the stable regime.     

Estimation and tests for power-transformed and threshold GARCH models

Consider a class of power-transformed and threshold GARCH(p,q)$(p,q)$ (PTTGRACH(p,q)$(p,q)$) model, which is a natural generalization of power-transformed and threshold GARCH(1,1) model in Hwang and Basawa [2004. Stationarity and moment structure for Box–Cox transformed threshold GARCH(1,1) processes. Statistics & Probability Letters 68, 209–220.] and includes the standard GARCH model and many other models as special cases. We first establish the asymptotic normality for quasi-maximum likelihood estimators (QMLE) of the parameters under the condition that the error distribution has finite fourth moment. For the case of heavy-tailed errors, we propose a least absolute deviations estimation (LADE) for PTTGARCH(p,q)$(p,q)$ model, and prove that the LADE is asymptotically normally distributed under very weak moment conditions. This paves the way for a statistical inference based on asymptotic normality for heavy-tailed PTTGARCH(p,q)$(p,q)$ models. As a consequence, we can construct the Wald test for GARCH structure and discuss the order selection problem in heavy-tailed cases. Numerical results show that LADE is more accurate than QMLE for heavy-tailed errors. Furthermore, the theory is applied to the daily returns of the Hong Kong Hang Seng Index, which suggests that asymmetry and nonlinearity could be present in the financial time series and the PTTGARCH model is capable of capturing these characteristics. As for the probabilistic structure of PTTGARCH(p,q)$(p,q)$ model, we give in the appendix a necessary and sufficient condition for the existence of a strictly stationary solution of the model, the existence of the moments and the tail behavior of the strictly stationary solution.     

Model selection criteria in multivariate models with multiple structural changes

This paper considers the issue of selecting the number of regressors and the number of structural breaks in multivariate regression models in the possible presence of multiple structural changes. We develop a modified Akaike information criterion (AIC), a modified Mallows’ C_p criterion and a modified Bayesian information criterion (BIC). The penalty terms in these criteria are shown to be different from the usual terms. We prove that the modified BIC consistently selects the regressors and the number of breaks whereas the modified AIC and the modified C_p criterion tend to overfit with positive probability. The finite sample performance of these criteria is investigated through Monte Carlo simulations and it turns out that our modification is successful in comparison to the classical model selection criteria and the sequential testing procedure robust to heteroskedasticity and autocorrelation.     

Risk-parameter estimation in volatility models

This paper introduces the concept of risk parameter in conditional volatility models of the form _?t=_σt(_θ0)_ηt${?}_t={\sigma }_t\left({\theta }_0\right){\eta }_t$ and develops statistical procedures to estimate this parameter. For a given risk measure r$r$, the risk parameter is expressed as a function of the volatility coefficients _θ0${\theta }_0$ and the risk, r(_ηt)$r\left({\eta }_t\right)$, of the innovation process. A two-step method is proposed to successively estimate these quantities. An alternative one-step approach, relying on a reparameterization of the model and the use of a non Gaussian QML, is proposed. Asymptotic results are established for smooth risk measures, as well as for the Value-at-Risk (VaR). Asymptotic comparisons of the two approaches for VaR estimation suggest a superiority of the one-step method when the innovations are heavy-tailed. For standard GARCH models, the comparison only depends on characteristics of the innovations distribution, not on the volatility parameters. Monte-Carlo experiments and an empirical study illustrate the superiority of the one-step approach for financial series.     

Markov-switching models with endogenous explanatory variables II: A two-step MLE procedure

This paper proposes a two-step maximum likelihood estimation (MLE) procedure to deal with the problem of endogeneity in Markov-switching regression models. A joint estimation procedure provides us with an asymptotically most efficient estimator, but it is not always feasible, due to the ‘curse of dimensionality’ in the matrix of transition probabilities. A two-step estimation procedure, which ignores potential correlation between the latent state variables, suffers less from the ‘curse of dimensionality’, and it provides a reasonable alternative to the joint estimation procedure. In addition, our Monte Carlo experiments show that the two-step estimation procedure can be more efficient than the joint estimation procedure in finite samples, when there is zero or low correlation between the latent state variables.