8.3.2 Empirical Example: Two-State Regime Switching Models

8.3.2 Empirical Example: Two-State Regime Switching Models

As an empirical illustration, we fitted two-state regime switching models to the four variables studied previously, i.e., inflation, stock returns, unemployment and interest rates, using a simple model with state-dependent intercepts, first-order autoregressive terms and volatilities:

$$y_{t+1}={\mu }_{s_{t+1}}+{\varphi }_{s_{t+1}}{y}_t+{\sigma }_{s_{t+1}}{\epsilon }_{t+1},{\epsilon }_{t+1}\sim \mathrm{N}(0,1).\text{\hspace{1em}(8.25)}$$

Figure 8.3 plots the smoothed-state probabilities, i.e., the states inferred from the full data sample. Two highly persistent states are identified for both the inflation and the interest rate series—for the latter variable, one state picks up the changes to the Fed’s monetary policy during 1979–82. For stock returns, shorter-lived regimes with high and low volatility are identified by the two-state models.

Table 8.1 reports full-sample coefficient estimates for the Markov switching model (8.25) fitted to the four series. For the inflation rate, outcomes in the first state are largely serially uncorrelated, whereas the second state sees strongly autocorrelated outcomes. Both states have high persistence, with “stayer” probabilities of 0.75 and 0.96, respectively. This means that the expected duration of the second state exceeds 20 quarters. The first state has a large intercept, while the second state has a small intercept, suggesting that the predicted inflation rate will vary substantially depending on the initial state. Inflation rate uncertainty also varies considerably across the two states, with the volatility parameter in the first state $\left({\sigma }_1\right)$ being two and a half times greater than that in the second state $\left({\sigma }_2\right)$

TABLE 8.1:
Maximum likelihood estimates of the parameters of two-state regime switching models: ${\sigma }_1$ and ${\sigma }_2$ measure the volatility parameters in states 1 and $2;{\mu }_1$ and ${\mu }_2$ measure the intercept in states 1 and 2; ${\rho }_1$ and ${\rho }_2$ measure the autoregressive coefficient in the two states. Finally, $p_{11}$ and $bP22$ are the probabilities of remaining in states 1 and 2, respectively.

For stock returns, the model identifies a low volatility state with a high mean (state 1) and a high volatility state with a low mean (state 2), with no evidence of serial correlation in either state. These states are only moderately persistent, and so we observe many regime switches for this series.

The regimes identified for the unemployment rate series suggest the presence of two moderately persistent states, one of which (state 1) has a small intercept and low volatility, while the other (state 2) has a higher intercept and high volatility. Both states have stayer probabilities around 0.90.

The interest rate process is mildly explosive in the first state and highly persistent but stationary in the second state. Again, both states have stayer probabilities close to 0.90. As shown in figure 8.3, the mildly explosive behavior in the interest rate is associated with the sharp reduction in interest rates following the Fed’s policy actions during 1979–82.

Recursively generated one-step-ahead forecasts from the two-state Markov switching models are shown in figure 8.4. Despite the very different functional form of the regime switching model in (8.25) versus the simple autoregressive model, the point forecasts for the three persistent variables (inflation, unemployment and interest rates) are quite similar to the ones generated by the linear AR models.