Nonlinear Regime-Switching State-Space (RSSS) Models

Abstract

Nonlinear dynamic factor analysis models extend standard linear dynamic factor analysis models by allowing time series processes to be nonlinear at the latent level (e.g., involving interaction between two latent processes). In practice, it is often of interest to identify the phases—namely, latent “regimes” or classes—during which a system is characterized by distinctly different dynamics. We propose a new class of models, termed nonlinear regime-switching state-space (RSSS) models, which subsumes regime-switching nonlinear dynamic factor analysis models as a special case. In nonlinear RSSS models, the change processes within regimes, represented using a state-space model, are allowed to be nonlinear. An estimation procedure obtained by combining the extended Kalman filter and the Kim filter is proposed as a way to estimate nonlinear RSSS models. We illustrate the utility of nonlinear RSSS models by fitting a nonlinear dynamic factor analysis model with regime-specific cross-regression parameters to a set of experience sampling affect data. The parallels between nonlinear RSSS models and other well-known discrete change models in the literature are discussed briefly.

Appendix

1.1 A.1 The Extended Kim Filter and Extended Kim Smoother

We outline the key procedures for implementing the extended Kim filter and extended Kim smoother here. Besides the modifications added to accommodate linearization constraints, the estimation process is identical to that associated with the linear Kim filter and Kim smoother. Interested readers are referred to Kim and Nelson (1999) for further details.

The extended Kim filter algorithm can be decomposed into three parts: the extended Kalman filter (for latent variable estimation), the Hamilton filter (to estimate the latent regime indicator, S _it ) and a collapsing procedure (to consolidate regime-specific estimates to reduce computational burden). For didactic reasons, we will describe the extended Kalman filter followed by the collapsing process and finally, the Hamilton filter. In actual implementation, however, the Hamilton filter step has to be executed before the collapsing process takes place. MATLAB scripts for implementing these procedures with annotated comments can be downloaded from the first author’s website at http://www.personal.psu.edu/quc16/.

1.2 A.2 The Extended Kalman Filter (EKF)

The extended Kalman filter (EKF) essentially provides a way to derive longitudinal factor or latent variable scores in real time as a new observation, y _it , is brought in. Let $\boldsymbol {\eta }_{i,t|t-1}^{j,k}=E(\boldsymbol {\eta}_{\mathit{it}}|S_{\mathit{it}}=k,S_{i,t-1}=j, \mathbf {Y}_{i,t-1})$, $\mathbf {P}_{i,t|t-1}^{j,k} = \mathop {\mathrm {Cov}}(\boldsymbol {\eta}_{\mathit{it}}|S_{\mathit{it}}=k,S_{i,t-1}=j, \mathbf {Y}_{i,t-1})$, $\boldsymbol {v}_{\mathit{it}}^{j,k}$ is the one-step-ahead prediction errors and $\mathbf {F}_{\mathit{it}}^{j,k}$ is the associated covariance matrix; j and k are indices for the previous regime and current regime, respectively. The extended Kalman filter can be expressed as

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

where $\mathbf {K}_{k,it}= \mathbf {P}_{i,t|t-1}^{j,k} \boldsymbol {\Lambda }_{k}' [ \mathbf {F}_{\mathit{it}}^{j,k}]^{-1}$ is called the Kalman gain function; B _k,it is the Jacobian matrix that consists of differentiations of the dynamic functions around the latent variable estimates, $\boldsymbol {\eta }_{i,t-1|t-1}^{j}$, namely, $\mathbf {B}_{k,it} = \frac {\partial \mathbf {b}_{k}( \boldsymbol {\eta }_{i,t-1|t-1}^{j}, \boldsymbol {x}_{\mathit{it}})}{\partial \boldsymbol {\eta }_{i,t-1|t-1}^{j}}$, with the time-varying covariates, x _it , fixed at their observed values. The gth row and lth column of B _k,it carries the partial derivative of the gth dynamic function characterizing regime k with respect to the lth latent variable, evaluated at subject i’s conditional latent variable estimates from time t−1, $\boldsymbol {\eta }_{i,t-1|t-1}^{j}$, from the jth regime. The subject index in B _k,it is used to indicate that the Jacobian matrix has different numerical values because it is evaluated at each person’s respective latent variable estimates, not that the dynamic functions are subject-dependent. Because our hypothesized measurement functions are linear, no linearization of the measurement functions is needed.

The EKF summarized in Equations (A.1–A.6) works recursively (i.e., one time point at a time) from time 1 to T and i=1,…,n until $\boldsymbol {\eta }_{i,t|t}^{j,k}$ and $\mathbf {P}_{i,t|t}^{j,k}$, have been computed for all time points and people. To start the filter, the initial latent variable scores at time t = 0, η ₀, are assumed to be distributed as η ₀∼ MVN(η _0|0,P _0|0). Typically, η ₀ is assumed to have a diffuse density, that is, η _0|0 is fixed to be a vector of constant values (e.g., a vector of zeros) and the diagonal elements of the covariance matrix P _0|0 are set to some arbitrarily large constants.

1.3 A.3 The Collapsing Process

At each t, the EKF procedures utilize only the marginal estimates, $\boldsymbol {\eta }_{i,t-1|t-1}^{j}$ and $\mathbf {P}_{i,t-1|t-1}^{j}$, from the previous time point. This is because to ease computational burden, a collapsing procedure is performed on $\boldsymbol {\eta }_{i,t|t}^{j,k}$ and $\mathbf {P}_{i,t|t}^{j,k}$ after each EKF step to yield $\boldsymbol {\eta }_{i,t|t}^{k}$ and $\mathbf {P}_{i,t|t}^{k}$. Given a total of M regimes, if no collapsing is used, the M sets of computations involving $\boldsymbol {\eta }_{i,t-1|t-1}^{j}$ and $\mathbf {P}_{i,t-1|t-1}^{j}$ in Equations (A.1–A.2) would have to be performed using $\boldsymbol {\eta}^{j,k}_{i,t-1|t-1}$ and $\mathbf {P}^{j,k}_{i,t-1|t-1}$ for every possible value of j and k. As a result, the number of possible values of filtered estimates increases directly with time, leading to considerable computational and storage burden if T is large. To circumvent this computational issue, Kim and Nelson (1999) proposed collapsing the M×M sets of new $\boldsymbol {\eta }_{i,t|t}^{j,k}$ and $\mathbf {P}_{i,t|t}^{j,k}$ at each t as

(A.7)

where W _it is called the weighting factor, the elements of which are computed using the Hamilton filter.

1.4 A.4 The Hamilton Filter

The Hamilton filter is also a recursive process and it can be expressed as:

(A.8)

where Pr[S _it =k|S _i,t−1=j] are elements of the transition probability matrix shown in Equation (8). f(y _it |S _it =k,S _i,t−1=j,Y _i,t−1) is a multivariate normal likelihood function expressed as

f(y _it |Y _i,t−1) in Equation (A.8) is often referred to as the prediction error decomposition function (Schweppe 1965). Taking the log of this value, log[f(y _it |Y _i,t−1)], and subsequently summing over t=1,…,T and then i=1,…,n yields the overall log-likelihood value, denoted herein as log[f(Y|θ)], that can then be maximized using an optimization procedure of choice (e.g., Newton–Raphson) to obtain estimates of θ. Standard errors associated with θ can be obtained by taking the square root of the diagonal elements of I ⁻¹ at the convergence point, where I is the observed information matrix, obtained by computing the negative numerical Hessian matrix of log[f(Y|θ)]. Information criterion measures such as the AIC (Akaike 1973) and BIC (Schwarz 1978) can be computed using log[f(y _it |Y _i,t−1)] as (see p. 80, Harvey 2001):

where q is the number of parameters in a model.

Since the prediction error decomposition function in Equation (A.8) is essentially a raw data likelihood function, missing values can be readily accommodated by using only the nonmissing observed elements of y _it in computing the prediction errors, $\boldsymbol {v}_{\mathit{it}}^{j,k}$ and their associated covariance matrix. To handle missing data in the EKF, we used the approach suggested by Hamaker and Grasman (2012), that is, to only update the estimates in $\{ \boldsymbol {\eta }_{i,t|t}^{j,k},\ \mathbf {P}_{i,t|t}^{j,k},\ \boldsymbol {\eta }_{i,t|t}^{k},\ \mathbf {P}_{i,t|t}^{k},\ \mathit{Pr}[S_{i,t-1}=j,S_{\mathit{it}}=k| \mathbf {Y}_{\mathit{it}}],\ \mathit{Pr}[S_{\mathit{it}}=k| \mathbf {Y}_{\mathit{it}}]\}$ using nonmissing elements from each measurement occasion.

1.5 A.5 The Extended Kim Smoother (EKS)

Given estimates from the EKF, the extended Kim smoother (EKS) can be used to obtain more accurate latent variable estimates and regime probabilities based on all observed information from each individual’s entire time series. Using $\boldsymbol {\eta }_{i,t|t-1}^{j,k}, \mathbf {P}_{i,t|t-1}^{j,k}, \boldsymbol {\eta }_{i,t|t}^{k}, \mathbf {P}_{i,t|t}^{k}$, Pr[S _it =k|Y _it ] and Pr[S _it =k|Y _i,t−1], the smoothing procedure can be implemented for t=T−1,…,1 and i=1,…,n as follows. First, smoothed estimates from regime j to regime k are obtained as

(A.9)

where $\widetilde{ \mathbf {P}}_{t}^{k,h}= \mathbf {P}_{i,t|t}^{k}\mathbf {B}_{h,it}'[ \mathbf {P}_{i,t+1|t}^{k,h}]^{-1}$. Similar to the collapsing procedure used in the extended Kim filter, a collapsing process is implemented here as

(A.10)

Finally, smoothed latent variable estimates and their associated covariance matrix are obtained by summing over the M regimes in effect to yield

(A.11)

Equations (A.9–A.11) yield three sets of estimates: η _i,t|T , the smoothed latent variable estimates conditional on all observations, the smoothed covariance matrix, P _i,t|T , and Pr[S _it =k|Y _iT ], the smoothed probability for person i to be in regime k at time t.