0.1 Regime-Switching Models
Linear models (ARMA, GARCH) assume one fixed dynamic regime throughout the sample. Many economic and financial series instead alternate between qualitatively different regimes (expansion/recession, calm/crisis volatility), motivating nonlinear time series models that allow parameters to switch.
Let \(S_t \in \{1,\dots,K\}\) be an unobserved discrete regime following a first-order Markov chain with transition matrix \(P_{ij} = \mathbb{P}(S_t=j\mid S_{t-1}=i)\). Conditional on \(S_t\), the series follows a regime-specific model, e.g. a Markov-switching AR(p): \[ X_t = \mu_{S_t} + \phi_{1,S_t}X_{t-1}+\dots+\phi_{p,S_t}X_{t-p} + \sigma_{S_t}Z_t. \] Since \(S_t\) is unobserved, this is a hidden Markov model with continuous-valued emissions; estimation uses the Hamilton filter (a discrete-state analogue of the Kalman filter) to recursively compute filtered regime probabilities \(\mathbb{P}(S_t\mid X_1,\dots,X_t)\) and evaluate the likelihood by MLE, with smoothed probabilities from a backward pass.
Threshold and smooth-transition models. An alternative to a latent Markov regime is to let the regime be an observable (deterministic) function of the data itself: - TAR (Threshold Autoregressive): regime determined by whether a lagged value \(X_{t-d}\) crosses a threshold \(\gamma\), giving piecewise-linear AR dynamics — e.g. SETAR (self-exciting TAR) when the threshold variable is the series itself. - STAR (Smooth Transition AR): replaces the hard threshold indicator with a smooth logistic/exponential transition function, avoiding a discontinuous switch.
Applications. Business-cycle dating (NBER-style expansion/recession classification), volatility regime identification (calm vs. crisis, complementing GARCH/stochastic volatility), and interest-rate/exchange-rate models with regime-dependent mean reversion.