We offer a brief background for switching VAR models. Recall the standard -dimensional, VAR(p) process

The vector of innovations are contemporaneously correlated with full rank matrix , but are uncorrelated with their leads and lags of the innovations and assumed to be uncorrelated with all of the right-hand side variables.

Following Krolzig, we modify Equation (50.1) to allow for regime change so that follows a VAR process that depends on the value of an unobserved discrete state variable . We assume there are possible regimes, and we are said to be in regime in period when .

Further, we will assume that the errors are distributed as for , with density function

Common practice divides the parameters in the VAR specification into three groups: the intercept parameters , the endogenous variables parameters , the error variance parameters , Typically, only a subset of the groups is allowed to vary across regimes. For example, a common restriction is that only the intercepts, or only the intercepts and the error variances are regime specific.

so the are parameterized in terms of exogenous variable parameters . The remainder of our discussion will be in terms of , but the analysis can extend to underlying parameters.

This difference creates some notational challenges. To facilitate discussion, the remainder of our discussion will organized around a new variable that is defined in terms of current and lagged and has possible states.

We define for both specifications in the discussion below.

Note that the expression for depends only on the current state. Accordingly, we have and . It follows that is equivalent to the statement .

where is a dimensional state vector representing the current and previous regimes, with .

To simplify notation, in switching mean specifications should be interpreted as shorthand for being equal to the -th possible realization of the dimensional vector, as in

where is the value of the -th lagged state in the -th possible realization, for.

where is obtained from the regime specific specifications Equation (50.6) and Equation (50.7).

, , , are the VAR parameters, are parameters that determine the regime probabilities.

Defining , we have the full normal mixture log-likelihood

which may be maximized with respect to .

More generally, we may allow for varying probabilities by assuming that is a function of vectors of exogenous observables and coefficients parameterized using a multinomial logit specification:

for with the identifying normalization . The special case of constant probabilities is handled by choosing to be identically equal to 1.

Typically, these transition probabilities are assumed to be time-invariant so that for all , but this restriction is not required.

where the -th element represents the probability of transitioning from regime in period to regime in period . (Note that some authors use the transpose of so that all of their indices are reversed from those used here.)

As in the simple switching model, we may parameterize the probabilities in terms of a multinomial logit. Note that since each row of the transition matrix specifies a full set of conditional probabilities, we define a separate multinomial specification for each row of the matrix

for and with the normalizations .

The likelihood function in Equation (50.10) depends on the one-step ahead predicted probabilities of being in a regime: . Obtaining these predicted probabilities is central to the evaluation of the likelihood.

Of related interest are the contemporaneous estimates of the regime probabilities: . The observed value of the dependent variable provides information about which regime is in effect in a given period, and we may use this contemporaneous information to updated our estimates of the regime probabilities. The process by which the predicted probability estimates are updated to form is commonly termed filtering.

Note that in the switching model setting, the state variable is -dimensional, so the above relationship does not apply. We must instead treat this model as a restricted form of the Markov switching model (as described below) where there no state dependence in the probability function.

Briefly, each recursion step begins with filtered estimates of the regime probabilities for the previous period. Given filtered probabilities, , the recursion may broken down into three steps:

These steps are repeated successively for each period, . All that we require for implementation are the initial filtered probabilities, , or alternately, the initial one-step ahead regime probabilities . See “Initial Probabilities” for discussion.

The likelihood may be maximized with respect to the parameters using iterative methods. Coefficient covariances may be estimated using standard approaches.

In the switching intercept form, the Markov switching filter requires initialization of the filtered regime probabilities in period 0, .

In the switching means setting, the Markov switching filter requires initialization of the vector of probabilities associated with the dimensional state vector. We may proceed as in the uncorrelated model by setting initial probabilities in period as described above, and recursively applying Markov transition updates to obtain the joint initial probabilities for the dimensional initial probability vector in period .

Again note that the initialization to steady state values using the period information is somewhat arbitrary in the case of time-varying transition probabilities.

For the Markov switching specification, estimates of the regime probabilities may be improved by using all of the information in the sample. The smoothed estimates for the regime probabilities in period use the information set in the final period, , in contrast to the filtered estimates which employ contemporaneous information, .

Intuitively, using information about future realizations of the dependent variable () improves our estimates of being in regime in period because the Markov transition probabilities link together the likelihood of the observed data in different periods.

The key in moving from the first to the second line of Equation (50.23) is the fact that under appropriate assumptions, if were known, there is no additional information about in the future data .

The smoothed probability in period is then obtained by marginalizing the joint probability with respect to :

Note that apart from the smoothed probability terms, , all of the terms on the right-hand side of Equation (50.23) are obtained as part of the filtering computations. Given the set of filtered probabilities, we initialize the smoother using , and iterate computation of Equation (50.23) and Equation (50.24) for to obtain the smoothed values.