How to handle a structural break in VAR model? I want to construct a VAR model of three time series: Inflation, GDP growth and Unemployment from 1963 to 2018. I have found a structural break around the year 2007 (2007-2008 financial crisis). I do not want to estimate two different VAR models, i.e. one for the crisis and one after the crisis. I am considering using a dummy variable for the crisis as an exogeneous variable to solve the problem, but I do not like this particular solution. Does anyone know how I could build my model such that the structural break is incorporated?
 A: Answer
One approach is to estimate a Markov-switching VAR (MS-VAR). Here are some key references:


*

*Hamilton (Econometrica 1989)

*Sims and Zha (American Economic Review 2006)

*Sims, Waggoner, and Zha (Journal of Econometrics 2008)

*Bognanni and Herbst (Journal of Applied Econometrics 2017)
Some Details
The basic VAR($p$) model is 
$$\mathbf{y}_t'=\mathbf{x}_t'\mathbf{B}+\mathbf{u}_t'\quad \mathbf{u}_t'\sim N(\mathbf{0},\,\boldsymbol{\Sigma}),$$
where $\mathbf{y}_t$ is $n\times 1$ and $\mathbf{x}_t=\begin{bmatrix}\mathbf{y}_{t-1}'&\mathbf{y}_{t-2}'&\cdots&\mathbf{y}_{t-p}'&1\end{bmatrix}'$. 
The MS-VAR introduces a time-varying regime indicator $s_t\in\{1,\,2,\,...,\,m\}$ and allows the VAR coefficients $\mathbf{B}$ and error covariances $\mathbf{\Sigma}$ to change depending on the value of $s_t$:
$$\mathbf{y}_t'=\mathbf{x}_t'\mathbf{B}_{s_t}+\mathbf{u}_t'\quad \mathbf{u}_t'\sim N(\mathbf{0},\,\boldsymbol{\Sigma}_{s_t}).$$
So if in period $t$ the economy is in regime 1, then $s_t=1$, and the VAR parameters take on values $\mathbf{B}_1$ and $\boldsymbol{\Sigma}_1$ in period $t$. If we're in regime 2, then $s_t=2$, and the VAR parameters take on values $\mathbf{B}_2$ and $\boldsymbol{\Sigma}_2$ in that period. And so on for each of the $m$ regimes.
This is a state space model where $s_t$ is treated as an unobserved latent state, and to complete the model we specify a Markov transition distribution for $s_t$. That is, we let $s_t$ evolve according to a discrete-time, discrete-state Markov chain.
In full generality, you jointly estimate the VAR parameters in every regime $\{\mathbf{B}_j,\,\boldsymbol{\Sigma}_j\}_{j=1}^m$ and the prevailing regime in each period $s_1$, $s_2$, ..., $s_T$. This is done in a Bayesian mode using Monte Carlo methods like the Gibbs sampler. 
