Markov Process

A latent state at time $t$ is unobservable to an econometrician and is denoted by $s_t$. It takes the value $k\in \left\{1,2,,K\right\}$, where $K$ is a positive integer representing the total number of states. The latent variable $s_t$ indicates in which state the current system is at time $t$. Hence, it can be called a state indicator or regime indicator. The terms state and regime are often used interchangeably.

The dynamics of the state indicator are governed by a Markov process. The probability distribution of $s_t$ given the whole path $\left\{s_{t-1},s_{t-2},\dots ,s_1\right\}$ only depends on the most recent state $s_{t-1}$. Define a transition probability as

where $i,j=1$,…,$K$. Given that in period $t-1$ the process was in state $i$, the probability that the state will switch to state $j$ in period $t$ is equal to $p_{ij}$. A transtion matrix organizes these transition probabilities in a $K\times K$ matrix and is definded as

where $p_{ij}$ is the element on the ith row and jth column such that the elements in each of the rows of matrix $P$ sum to one. The diagonal elements of the transition matrix determine the expected state duration that is equal to ${\left(1-p_{ii}\right)}^{-1}$ for state $i$.

A vector of unconditional state probabilities, denoted by $\pi \equiv Pr\left[s_t\right]$, is defined by

This equation indicates time invariance of distribution —that is, an iteration over one period performed by the premultiplication by the transition probability $P^{\prime }$ does not change vector $\pi$. The solution to equation (3) for $\pi$ given ${l^{\prime }}_K\pi =1$, where ${\iota }_K$ is a $K$-vector of ones, is given in Hamilton (1994, Chapter 22) and expresses $\pi$ as a function of $P$. Define $a\left(K+1\right)\times K$ matrix $\mathbf{\text{P}}=\left[\begin{array}{c}{I}_K-P^{\prime }\\ {{\iota }^{\prime }}_K\end{array}\right]$, where $I_K$ denotes an indentity matrix of order $K$. Hamilton’s solution for $\pi$ is given by the $\left(K+1\right)\text{th}$ column of ${\left({\mathbf{\text{P}}}^{\prime }\mathbf{\text{P}}\right)}^{-1}{\mathbf{\text{P}}}^{\prime }$.

The distribution of the initial state at $t=0$, denoted by $s_0$, is represented by the following $K$vector:

For an ergodic Markov process, the initial distribution can be simply set to the stationary distribution $\pi$. If the Markov process is nonstationary, there typically exists a theory-guided choice for ${\pi }_0$. For example, a change-point model requires that at the initial period the process is in the first regime, $s_0=1$.

Measurement Equation

A measurement equation lays out the probability law of the observations and is given by

where $F$ represents the distribution of $y_t$ conditional on the observation $x_t$ and the latent state $s_t$. $F$ should be specified in applications and depends on the structure of the data $y_t$. Therefore, it can be given by a discrete, continuous, or mixture distribution. For example, consider a time series of financial asset returns and assume that each state admits an autoregressive process of order one with Gaussian innovations. Then $F$ can be expressed as

where ${\mu }_{s_t}+{\beta }_{s_t}{y}_{t-1}$ is the mean and ${\sigma }_{s_t}^2$ the variance of this conditional distribution given $s_t$ and $y_{t-1}$. The same assumption also implies a regression form of equation (5) given by

Equations (1) and (5) comprise the foundation of the MS framework. Combining them with the initial condition (4) yields the likelihood function $p\left(Y|\Theta \right)$ that is available through the filter technique from Hamilton (1989) called the Hamilton filter, where $Y$ denotes a collection of all $y_t$ for $t=1$,…,$T$; and $\Theta$ is the collection of all time-invariant parameters. For instance, in the example from equation (6), $\Theta \equiv {\left\{{\mu }_k,{\beta }_k{\sigma }_k^2\right\}}_{k=1}^K$.

Filtering and Smoothing the States

The Hamilton filter gives the conditional distribution of the state $s_t$ given the data up to time $t$, denoted by $p\left(s_t|Y_{1:t}\right)$. It is used to compute the one-period-ahead probability density of $y_t$ from the following formula:

where $p\left(y_{t+1}|s_{t+1},Y_{1:t}\right)$ is obtained from the measurement equation (5) and $p\left(s_{t+1}|s_t\right)$ is the transition probability in equation (1). Equation (7b) includes the forecasted state probability that is conveniently decomposed into the transition and filtered probabilities according to

The filtered probability for $s_{t+1}$ is given by

The filtered probability of $s_{t+1}$ is easy to compute as long as the filtered probability of $s_t$ is known. The conditional density $p\left(y_{t+1}|Y_{1:t}\right)$ is derived in equation (7) and can be easily computed in an iterative procedure. The filtering algorithm proceeds iteratively for $t=1$,…,$T$ by the application of equation (9c) that describes the progression of the filtered probabilities from $p\left(s_t|Y_{1:t}\right)$ to $p\left(s_{t+1}|Y_{1:t+1}\right)$. Note that the filtering step includes a prediction step consisting of computing the forecasted probability $p\left(s_{t+1}|Y_{1:t}\right)$.

The computation of the smoothed probabilities denoted by $p\left(s_t|Y_{1:T}\right)$ for all $t$ is performed by an iterative backward smoothing procedure proposed by Chib (1996). It is initiated at $t=T$, where the last filtered density is equal to the smoothed probability at time $T,p\left(s_T|Y_{1:T}\right)$, and then proceeds for $t=T-1$,…, 1 by applying:

A comprehensive exposition on the forward filtering and the backward smoothing procedures can be found in Frühwirth-Schnatter (2006, Chapter 13).

Finally, the likelihood function is constructed as the product of conditional probability densities:

where $Y_{1:0}$ is treated as known and the state indicators, $s_t$ for $t=1$,…,$T$, are integrated out. An analytical expression for the likelihood function is based on equations (11) and (7b) and is written out in Frühwirth-Schnatter (2006).

The presented derivations are valid for homogenous Markov switching models—that is, the models for which $p\left(s_{t+1}|s_t,Y_{1:t}\right)=P$. The filtering and smoothing algorithms complicate for dynamic models with path dependence in which the distribution of the current state depends on the path of all of the past latent state indicators. Some early solutions were proposed by Billio and Monfort (1998) and Billio et al. (1999) for autoregressive moving average models and by Haas et al. (2004) for generalized autoregressive conditional heteroskedasticity (GARCH) models. The application of sequential Monte Carlo methods was proposed by Bauwens et al. (2014).

The interpretation of the MS models is based on the combined analysis of the parameter estimates and the estimate of the state probabilities. The latter should be chosen among the forecasted, filtered, or smoothed probabilities, denoted by $p\left(s_t|Y_{1:t-1}\right)$, $p\left(s_t|Y_{1:t}\right)$, and $p\left(s_t|Y_{1:T}\right)$, respectively, depending on a particular application and the objective of investigation.

Parameter Estimation

The maximum likelihood estimation is based on an iterative expectation-maximization algorithm (see Hamilton, 1990). In each of its iterations, a filtering-smoothing algorithm is used to propose the current estimate of $S$, a collection of all $s_t$ for $t=1$,…,$T$, and a maximization step is applied to compute an estimate of $\Theta$ and $P$. The maximum likelihood estimation is straightforward if regularity conditions are satisfied. However, for larger models, it can become cumbersome due to unbounded likelihood function.

The first results for the asymptotic normality of the maximum likelihood estimator of the MS model parameters were proposed by Lindgren (1978) and Lehmann (1983), whereas the finite sample properties of this estimator were analyzed in Psaradakis and Sola (1998). The latter study using Monte Carlo simulations for a simple univariate MS model documents strong irregularities in the shape of the small sample distribution of the parameters, including larger estimation standard errors and skewness, in samples of up to 400 observations, especially for variance parameters and transition probabilities. Such irregularities were much less pronounced for the conditional mean parameters and usually disappeared for samples exceeding 800 observations.

Bayesian estimation relies on a data augmentation technique that requires the specification of a complete-data likelihood function $p\left(Y,S|\Theta \right)=p\left(Y|S,\Theta \right)p\left(S|\Theta \right)$, where the objects on the righthand side of the expression are easily obtainable without integration. The complete data likelihood function is subsequently used to specify the full conditional posterior distributions $p\left(\Theta |Y,S\right)$ and $p\left(S|Y,\Theta \right)$. Sampling from the joint posterior distribution of the parameters and states is performed via Markov Chain Monte Carlo (MCMC) methods. Since the former conditional distribution takes $S$ as given, it can be sampled from using standard techniques. The sampling from the latter relies on an iterative procedure, the forward filtering and backward sampling (FFBS) method proposed by Chib (1996), as the latent state indicators are not conditionally a posteriori independent. In order to draw samples from $p\left(S|Y,\Theta \right)$, apply the forward filtering algorithm and save the filtered probabilities $p\left(s{}_t|Y_{1:t}\right)$ for all $t$. Then, at the $m\text{th}$ iteration of the MCMC algorithm sample $S_T^{\left(m\right)}$ from $p\left(s{}_T|Y_{1:T}\right)$, and sample $S_t^{\left(m\right)}$ backwards for each $t=T-1$,…,1 from

Frühwirth-Schnatter (2006) surveys various versions of this algorithm.

Inference on the Number of States

The frequentist solution to a problem of selection of the number of states of the Markov process $K$ relies on information criteria such as the Akaike Information Criterion (see Psaradakis & Spagnolo, 2003). Testing of a hypothesis that $K=1$ against an alternative hypothesis that $K=2$ is highly cumbersome because the MS model is not identified under the null hypothesis and the solutions require sophisticated inferential methods, some of which are provided by Carrasco et al. (2014) and Meitz and Saikkonen (in press).

Bayesian model selection based on marginal data densities provides the solution to the number of states determination problem, including a model with $K=1$ (see Frühwirth-Schnatter, 2006). However, the main challenge for a correct Bayesian inference in MS models is the problem of label switching, which is defined as the invariance of the likelihood function to various labelings of the states. Consider an example in equation (6) with two states characterized by state-specific parameter vectors of state $A$ and state $B$, denoted respectively by ${\Theta }_A$ and ${\Theta }_B$. The label switching problem states that irrespective of whether the vector of parameters $\left({\Theta }_1,{\Theta }_2\right)$ is set to $\left({\Theta }_A,{\Theta }_B\right)$ or $\left({\Theta }_B,{\Theta }_A\right)$ the value of the likelihood function evaluated at these parameters stays invariant. Frühwirth-Schnatter (2001) analyzed a multimodal global shape of the likelihood function and the posterior distribution (see Frühwirth-Schnatter, 2004, for the application of this approach to the computation of marginal data densities). Implementing ordering restrictions on the state-dependent parameters of the model that would provide a unique classification of the states is a solution that is only applicable if such restrictions do not numerically bind the posterior distribution. Alternatively, Geweke (2007) proposed basing the inference on label-switching invariant characteristics such as predictive densities.

Forecasting

MS models offer an improved forecasting accuracy due to assigning a significant weight to the most recent regime, if the states are persistent, and by incorporating the possibility of switching to other states over the forecast period. Consider forecasted state probabilities for an h-period-ahead forecast that are given by

where the last equality holds for the homogenous MS models. These forecasted probabilities play the role of the weight assigned to the regime-dependent predictive densities. Therefore, the joint predictive density for all the forecasted variables up to forecast horizon $h$ conditional on the data, $Y_{1:t}$, and parameters, $\Theta$, is as follows:

Given specific assumptions of the considered models, it is often possible to derive analytical formulae for the conditional expected value point forecast and the associated forecast error variance. For instance, the moments of the one-period-ahead forecast of the model in equation (6) are given by:

where ${\pi }_{t+h|t.s_{t+h}}$ denotes the $s_{t+h}\text{th}$ element of vector ${\pi }_{t+h|t}$. Note that the predictive density is not normal but a mixture of normal distributions that might exhibit such features as leptokurtosis, multimodality, and asymmetry. For more complicated models, the employment of numerical integration methods might be necessary. This and several other aspects of forecasting with MS models are presented in Krolzig (1997) and Teräsvirta (2006).

A concept of Granger causality relates the causal link between variables to their predictive power and was proposed by Granger (1969) and Sims (1980). An analysis for a specific state of an MS vector autoregressive model was proposed by Psaradakis et al. (2005). The framework that investigates Granger causality in the mean unconditionally on the states and treats the MS vector autoregressive model as a stochastic process was proposed by Droumaguet et al. (2017), who also introduced parametric conditions under which a variable does not affect the forecast of the hidden Markov process. Warne (2000) proposed a related analysis of Granger causality in the variance.

Selected Applications in Economics and Finance

The first application of the MS models in economics was proposed by Hamilton (1989) and consisted of the analysis of business cycles. Consider data on the gross domestic product growth rates to which the following autoregressive model is fitted:

where the error term is conditionally normally distributed. The business cycle interpretation of the model relied on the combined analysis of the signs of the regime-specific intercept terms and the historical narrative about the periods with high values of the smoothed state probabilities for each of the regimes. Accordingly, a negative value of the intercept term coincided with the periods of economic recessions, whereas its positive value was associated with economic expansions.

Similar reasoning was applied to financial markets characterized by bull and bear markets occurring one after another by Hamilton and Lin (1996). An important feature of the model in equation (16) is that it implies autocorrelations that are functions not only of the autoregressive parameter, as is the case in linear autoregressive models with $K=1$, but also of the elements of the transition matrix (see Krolzig, 1997).

Another example includes a multivariate model of the effects of monetary policy on the real economy in the United States with conditional heteroskedasticity modeled with the MS process proposed by Sims and Zha (2006). In this model, the volatility states occurred to have high values of smoothed probabilities in the periods corresponding to the terms of subsequent chairs of the Federal Reserve. For instance, the state with the highest value of the volatility had high probabilities of occurrence in the period of Paul Volcker’s chairmanship. In contrast, the lowest volatility state spanned Alan Greenspan’s term. Lanne et al. (2010) and Lütkepohl and Woźniak (2020) obtained a similar interpretation of MS heteroskedastic states and used it to identify a monetary policy shock in a structural dynamic model. In another example of multivariate analysis, an explicit form of dependence between two or more Markov processes describing country or regional business cycles was proposed by Owyang et al. (2005), Hamilton and Owyang (2012), and Leiva-Leon (2017).

An example of introducing MS in a structural model of an economy was introduced in a new class of structural MS rational expectations models. These models use the MS rule to determine the time variation of the structural shock variances of the dynamic stochastic general equilibrium model and to model the time-varying inflation target of a central bank. In this framework, the considered agents are rational, and therefore they know the MS rule and, consequently, take it into account in their decision-making problem. Farmer et al. (2009) provided the theory behind such a formulation of the model while Liu et al. (2011) provided a model for macroeconomic fluctuations. Farmer et al. (2011) proposed a method of solving these models. Waggoner and Zha proposed a novel approach to the temporal model selection problem in this context. They set as two states of an MS model two macroeconomic models, a DSGE model and the corresponding VAR specification, allowing the state allocation to decide which of these two models fits the data better in each of the sample periods.

The time variation of the parameters according to the MS rule was applied with success to dynamic models of conditional variances of financial asset returns. This extension is essential from the point of view of making statements about the persistence of the volatility process that changes over time. Hamilton and Susmel (1994) and Kaufmann and Frühwirth-Schnatter (2002) introduced MS in the parameters of the autoregressive conditional heteroskedasticity (ARCH) model, whereas Bauwens et al. (2014) and Augustyniak et al. (2019) applied such a process to the parameters of the GARCH models and proposed solutions to the path dependence problem. Finally, So et al. (1998) and Carvalho and Lopes (2007) implemented the MS mechanism to the parameters of the state equation for the logarithm of conditional variance of financial returns in a stochastic volatility model.

The MS model was also used to capture time-varying contagion and systematic risk on financial markets as in Gallo and Otranto (2008) and Casarin et al. (2018).

A Family of Markov Switching Models

A general stationary and aperiodic MS process in which each of the states can be revisited at any time $t$ presumes that all of the $K^2$ elements of the transition matrix are estimated. In such a model, there is no absorbing state, all of the elements of the ergodic probabilities vector $\pi$ are greater than zero, and the probabilities of the initial state are most often set to $\pi$. This is the most frequently applied MS model in economics and finance.

A class of finite mixture models is nested within the MS models by setting each of the rows of the transition matrix to ${\pi }^{\prime }$, where all the elements of $\pi$ are strictly positive. In this model, the forecasted state probabilities are time invariant and equal to $p\left(s_{t+1}|Y_{1:t}\right)=\pi$. However, the classification of observations is facilitated through the smoothed probabilities that change over time. Finite mixture models provide a convenient way of modeling nonstandard distributions that are often required for the error terms in economic and finance applications. It can be shown that any distribution of a random variable defined on a real scale can be approximated by a mixture of normal distributions, while distribution of a random variable defined on a positive real scale can be approximated by a mixture of gamma distributions (see Norets, 2010).

Change-point models can be used to introduce monotonic regime changes as in a model proposed by Chib (1998). The process is initiated in the first state $s_0=1$. With probability $p_{11}$ it remains unchanged, and with probability $p_{21}$ it switches to the other regime. The first state is never to be revisited and, thus, $p_{12}=0$. In general, given that at some period $t$ the Markov process is in state $s_t=k<K$, it remains in this state with probability $p_{kk}$ and is only allowed to switch to the next regime with probability $p_{k+1.k}$. Finally, when the process reaches the Kth state, it stays there forever. An example of a transition matrix for such a process with the number of states $K=3$ is given by

Therefore, these models are capable of estimating the time at which the regime changes occur.

An interesting extension of this model was proposed by Pesaran et al. (2006), which allows new regimes to occur in the forecasted sample. Finally, the change-point models introduce nonstationarity in the Markov process and, thus, their ergodic probabilities are all equal to zero except for the last element of $\pi$ that is equal to one. Therefore, the last state is the absorbing state that gains 100% of the probability mass asymptotically with $T\to \infty$. Finally, Frühwirth-Schnatter (2006) provided a detailed discussion of the nuances of the estimation of stationary and nonstationary Markov processes.

In simple and popular deterministic change-point models, $s_t$ is assumed to be known and provided by the econometrician. In many applications, this model is used to estimate state-dependent parameters in samples predetermined by the investigator or using a test for multiple breaking points for stationary data (see, e.g., Bai & Perron, 1998). Moreover, it is straightforward to set $s_t$ to obtain monotonic regime changes. Nevertheless, in the deterministic change-point models the transition matrix is redundant given that $s_t$ is known. The regime-change dates are not estimated and, unless the econometrician knows the data generating process and sets $s_t$ accordingly, the model fit deteriorates heavily compared to other models considered in this section.

A more elaborate form of the transition matrix may lead to the desired application-specific properties of the Markov process. Consider a model used by Sims (2001), who introduced symmetric jumping among adjacent regimes. In this example, the desired transition matrix for the case of $K=4$ states is

Sims et al. (2008) considered a generalization of this model inspired by the developments in Cogley and Sargent (2005). This model uses a scarce parameterization of the transition matrix that is capable of capturing occasional discontinuous shifts in the values of the regime-dependent parameters when $K$ is small, as well as frequent, incremental changes in these parameters for larger $K$. A general way of imposing restrictions on the transition matrix was proposed by Sims et al. (2008) and Woźniak and Droumaguet (2015).

Finally, the Markov property of the latent process might be extended by introducing the dependence of the current state, $s_t^{*}$, on its several recent realizations. For instance, the original model by Hamilton (1989) assumed the dependence of the model parameters on the current and previous regime, $s_t^{*}$ and $s_{t-1}^{*}$, respectively. This dependence can be modeled by a new four-state Markov process, $s_t$, through the following state representation:

and an appropriate form of the transition matrix:

Independent Markov Processes

In this class of MS models, various groups of parameters of the model depend on separate and independent Markov processes (see Phillips, 1991; Ravn & Sola, 1995, for some early applications). Examples include models in which the parameters of the conditional mean process depend on a different Markov process than the conditional variances (see Sims et al., 2008), structural models in which the money demand equation depends on a different Markov process than other parameters of the models (see Sims & Zha, 2006), and panel MS models that include a separate Markov process for parameters of equations for each individual (see Billio et al., 2016; Kaufman, 2010)). Consider $L$ such independent processes $s_{lt}$ each parameterized by a $K^l\times K^l$ transition matrix $P^l$, for $l=1$,…,$L$. This model can be represented by a composite Markov process $s_t=\left(s_{1t},,s_{Lt}\right)$ with $\prod _{l=1}^L{K}^l$ states and the corresponding transition matrix given by

where $\otimes$ denotes the Kronecker product. The gain from the tensor product representation of the transition matrix in equation (17), introducing nonlinear restrictions, is an economic parameterization facilitating the estimation.

Nonhomogenous Markov Switching

Finally, this survey of parameterizations of transition matrices is concluded by the presentation of a nonhomogenous MS model in which the transition probabilities change over time (see Diebold et al., 1994; Filardo, 1994). The introduction of this time variation is often combined with the dependence on some variables $v_t$ that might contain $x_t$ (e.g., Filardo, 1994), the state indicators $s_t$ (see Otranto, 2005), or both (see Billio et al., 2016), as well as a measure of a duration of the state (see Durland & McCurdy, 1994; Sichel, 1991). The restriction imposed on the rows of the transition matrix leads to the parameterization of the transition probabilities through the multinomial logistic regression as proposed by Meligkotsidou, Dellaportas (2011) in the following form:

where ${\gamma }_{ij}$ are parameter vectors to be estimated, the dimensions of which correspond to vector $v_t$. Note that for the identification of transition probabilities for each $i$ there is a $j$ so that ${\gamma }_{ij}$ is a vector of zeros (see Kaufmann, 2015; Koki et al., 2020, for the detailed model specification and estimation procedures). The selection of the variables in $v_t$ determines the time dependence in transition probabilities and is subject to empirical verification. Finally, Billio and Casarin (2010) extended this specification by considering a two-state MS model $K=2$ and setting the transition probabilities $p_{ii}$ to follow beta distribution with its parameters depending on $v_t$.

Estimation

Due to the parameter saturation problem, the IHMM cannot be estimated by classical methods without regularization. On the contrary, the Bayesian approach is coherent and more appropriate for inference, and, thus, most research on the IHMM uses this framework.

Three alternative approaches can be used to draw inference from the IHMM. The first is to integrate out the transition probability $P$ based on the Chinese restaurant representation of the Dirichlet process as in Fox et al. (2011). This method works directly on the states, but its derivation is complicated. The second is to apply the beam sampler to truncate stochastically the number of states to a finite one during the MCMC as in Van Gael et al. (2008). This method provides an exact inference similar to the first method. It also utilizes the conditional independence by keeping the transition matrix to a finite dimension, which allows partial parallel computations and is usually much faster than the first approach.

The last method applies the degree-K weak limit approximation from Ishwaran and Zarepour (2002) as in Bauwens et al. (2017). It uses a truncated Dirichlet process so that the IHMM resembles an appropriate finite-state MS model. In practice, Song (2014) found that standard MS models with a large number of states performed similarly to the IHMM, where the number of inactive states should be nonzero and where an inactive state is a state with no data assigned to it. This approach renders the IHMM easier to execute, although two caveats exist. First, the prior distribution on the transition matrix must be chosen so that the concentration parameter from the truncation approximation is consistent with the concentration parameter from the IHMM (see Ishwaran & Zarepour, 2002). Otherwise, the approximation is not valid. Second, the number of states in the MCMC must be monitored to avoid poor approximation. A simple rule would be that the number of active states—that is, those with a nonzero number of observations classified into it—must always be less than the total number states $K$ in the approximation (see the implementation by Bauwens et al., 2017).

Flexibility of the IHMM

The IHMM originated from the machine learning literature where it was used to reveal dynamic clustering in applications, including dialogue summary and motion capture. Subsequently, it has been applied to various fields in economics and finance. The seminal developments include Song (2014), Jochmann (2015), and Dufays (2015). The motivation for using the IHMM is that it demonstrates well the trade-off between heuristic economic interpretations and competitive forecast accuracy. This feature constitutes an advantage over many of the machine learning methods that hardly allow for structural interpretations.

Similar to the conventional MS models, the IHMM maintains its first-order Markov chain property. However, due to the differences in the prior distribution setup, and the problem of doubling the states, a phenomenon consisting of the possibility of producing an additional state that mimics an already existing one, the IHMM should not be considered a device for detecting the number of states of the mixture or MS models, as was suggested in some early approaches (e.g., Otranto & Gallo, 2002). Miller and Harrison (2013) stated the argument formally.

An attractive feature of the IHMM is that it jointly captures regime-switching and structural break. Consequently, it grants more flexibility and accommodates data dynamics upon the arrival of new observations. The regime-switching module pools data with similar behavior to borrow statistical strength from each other. In addition, the structural break dynamics can generate a new state when the arriving new observations exhibit a new law of motion. A well-known example is the unprecedented global financial crisis in 2007 and 2008. It began with the subprime mortgage crisis in the United States and became the most severe financial crisis since 1930. Any standard MS model for bull and bear markets such as Maheu and McCurdy (2000), Lunde and Timmermann (2004), and Maheu et al. (2012) is incapable of capturing the new phenomenon because they are limited by the data history and do not allow structural changes. The IHMM is an appropriate vehicle to achieve such a goal because the capability of generating a new unprecedented state is a feature of the latent process.

Another advantage of the IHMM lies in its superior forecasting performance, which is an empirical observation with plausible intuition. Benchmark models such as autoregressions or linear regression have a rigid assumption of the functional forms. Instead, the IHMM is more flexible and hence robust to model misspecification. Its flexibility comes from its ability to sort data into clusters endogenously, so that behaviorally different data will not affect the state inference. Two vital components to achieve improved forecasting performance are the hierarchical prior structure (Song, 2014) and certain regime persistence (Fox et al., 2011) in economic and finance applications. The hierarchical structure exploits more data information by letting the regimes inform one another. At the same time, regime persistence reflects salient stylized facts that economic time series are prone to local dynamic stability.

The IHMM provides a convenient tool for a control approach in grand modeling frameworks. Consider a modeling framework that utilizes independent variables $x_t$ and error term ${\epsilon }_t$. Any incorrect distributional assumption about ${\epsilon }_t$ could potentially adversely affect the inference. If this distribution is not the focus of the application, simple estimators are applicable. However, if the distributional assumptions become essential as in the risk analysis, the semiparametric approach has the advantage of imposing a nonparametric distribution on ${\epsilon }_t$. This assumption releases ${\epsilon }_t$ from any potential misspecification. The examples of such applications are provided by Jensen and Maheu (2010), who modeled ${\epsilon }_t$ using the DPM, as well as Hou (2017) and Dufays et al. (2019), who applied the IHMM.

Extensions of the IHMM

The IHMM provides a basis for the burgeoning academic literature on its extensions. An IHMM with DPM emission was proposed in Fox et al. (2011) and further generalized to a block IHMM with more extensive in-state dynamics by Stepleton et al. (2009). In this approach, each state is a distinct MS model, which makes it suitable to the bull and bear market modeling as in Maheu et al. (2012). To capture long memory, Van Gael et al. (2009) proposed the factorial IHMM.

Examples of factorial IHMM applications include Nakano et al. (2011) and Heller et al. (2009). The factorial IHMM does not impose any restrictions for identification and, thus, structural interpretations are hardly possible, which limits the scope of applications in economics and finance. Another modeling strategy that consists of applying the IHMM structure to existing interpretable parametric models can be found in Liu and Maheu (2018) and Jin et al. (2019). Finally, there is a substantial body of work that extends the IHMM in various directions, as in Shi and Song (2014), Bauwens et al. (2017), Maheu and Yang (2016), Yang (2019), Hou (2017), and Luo et al. (in press).

Introducing Endogeneity

Define the dynamics of the real-valued latent factor by an autoregressive process

where $\alpha$ is the persistence coefficient such that $\left|\alpha \right|\le 1$ and $v_t$ is a standard normal error term. The initial value of the process, $w_0$, is recommended to be normally distributed with the zero mean and variance equal to $1/\left(1-{\alpha }^2\right)$ for $\left|\alpha \right|<1$, or equal to zero, $w_0=0$, for $\alpha =1$. The process in equation (19) is discretized by defining a threshold parameter, $\tau$, and the discrete-valued state indicator for a two-state model, $K=2$, as

Therefore, as long as the process $w_t$ is subject to interpretation, its primary role is to define the Markov process $s_t$.

Chang et al. (2017) defined the measurement equation in a general form that made an explicit link between the potential dependence of the conditional mean and standard deviation on the independent variables, $x_t$, and the latent processes

where ${\epsilon }_t$, conditionally on $x_t$ and $w_t$ (or $s_t$) is a serially uncorrelated standard normal error term. Endogeneity of the Markov process is formally introduced in this model by an appropriate specification of the joint distribution of the error terms from the state and measurement equations (19) and (21), respectively, that is given by

where $\rho$ is the correlation coefficient. It can be shown that the exogenous MS model can be obtained by setting $\rho =0$. In other words, the restricted endogenous and the exogenous MS models are observationally equivalent—that is, they both lead to the same value of the likelihood function given the values of $\Theta$. However, for the values of $\rho$ that are different from zero the relationship between ${\epsilon }_t$ and $w_{t+1}$ and, consequently, between ${\epsilon }_t$ and $s_{t+1}$ is established. Expressions for the implied transition probabilities that are changing over time are given in Chang et al. (2017).

A different form of endogeneity was proposed by Kim et al. (2008) who specified the joint distribution of error terms, such as the one in equation (22), for vector $\left({\epsilon }_t,v_t\right)$. This model presumes a contemporaneous effect of ${\epsilon }_t$ on $w_t$ and $s_t$ which Chang and Kwak (2017) point out to be misspecified. Kim et al. (2008) consider an application to modeling a volatility feedback effect in financial time series.

Interpretations Considering Correlation Coefficient

To illustrate the interpretation of nonzero $\rho$, consider a model in equation (6) with $s_t$ specified by the endogenous Markov process and with $\left|{\beta }_k\right|<1$ for $k=1,2$. A possible application in finance includes the modeling of the leverage effect defined as the negative correlation between the current innovation and future conditional variance of the return on a financial asset. Let $\rho <0$ and ${\sigma }_1<{\sigma }_2$. In this case, a negative realization of ${\epsilon }_t$ increases the probability of the second state in period $t+1$ and leads to an increase in the conditional volatility.

A simple application in time series analysis includes the modeling of the mean reversion that works differently for $\rho$ of different signs. Let ${\mu }_1<{\mu }_2$ and $\rho <0$. Then, a positive realization of ${\epsilon }_t$ decreases the probability of the second state in the period $t+1$. Therefore, the mean reversion is also obtained at the level of the future conditional expected value that now decreases. In the opposite case of $\rho <0$, a positive realization of ${\epsilon }_t$ increases the probability of the second state in period $t+1$ and, consequently, increases the conditional expected value of $y_t$. Therefore, the forecasts of $y$ revert to a mean that is higher, which has a destabilizing effect. Examples of more elaborate applications in economics include the endogenous switching of the parameters modeling the effects of monetary and fiscal policies proposed by Chang et al. (2018) and Chang and Kwak (2017), respectively.