2.15.1. Filtering and decompositions

Linear filtration

Filtering with constant parameters can be done efficiently with algorithms presented in e.g. [De Jong and Chu‐Chun‐Lin, 2003], [Koopman and Durbin, 2000], [Koopman and Durbin, 2003], [Durbin and Koopman, 2012]. Unfortunately those procedures cannot be used in a regime-switching context.

The main references in the economics literature for the filtration of Markov-switching state-space models are [Kim and Nelson, 1999] and [Kim and Nelson, 2001], which combine the Kalman and Hamilton filters. Similar algorithms can be found in the engineering literature. See e.g. [Bar-Shalom et al., 2001], [bar-shalom et al., 2004].

RISE uses an extension of the algorithms by [Durbin and Koopman, 2012] except for the initialization process, which in the case of nonstationarity, hasn’t been extended to regime-switching models. In the case of nonstationarity, RISE simply applies an arbitrary variance for the initialization process.

Also, for efficiency reasons, RISE deviates from [Kim and Nelson, 2001] when it comes to the collapsing of the regimes. While [Kim and Nelson, 2001] collapse the updates, RISE collapse the forecasts. The two procedures yield numerically similar results but the procedure in RISE is faster.

Historical Decomposition

Warning

Exact only for constant-parameter linear models.

Observables decomposition

Important

Available only for constant-parameter models

Conditional forecasting via filtering

Conditional forecasting can be implemented using filtering algorithms.

Suppose we want to condition on a variable X0.

Hard conditions

We simply smooth over an extended sample.

Soft conditions

We relate X0 to some variable X in the model as \(X0_t=X_t+\sigma \varepsilon_t\).

And we let \(\sigma\) control the softness of the condition :

If \(\sigma\) tends to \(\infty\) the condition is irrelevant

If \(\sigma\) tends to 0 we have hard conditions

If \(\sigma\) is greater than 0 but finite (small enough) we have soft conditions.