# 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?

• What is it that you do not like about the dummy solution? Understanding that could help suggest you the relevant alternatives. – Richard Hardy Nov 10 '18 at 16:53
• My understanding is that a dummy variable only accounts for a different mean. So it doesn't adress in higher statistical moments. – Cardinal Nov 11 '18 at 17:43
• Depending on what else you want to allow to change after the break, you could include more dummies or reshape your data matrix to allow for different slopes and the like. Or you could estimate two models, which you say you do not like. May I ask why? Because the answer below is quite like estimating two different VAR models. What is different is that you also estimate the state process. Therefore, I do not quite see why you do not like two VAR models but like an MS-VAR model. – Richard Hardy Nov 11 '18 at 18:55
• I don't consider having just two different VAR models to be sophisticated. That's why I prefer something else. Yes, a dummy variable with different slopes also does seem to do the trick. Thanks for your comments. – Cardinal Nov 12 '18 at 15:32
• I definitely agree with @RichardHardy. Thinking in terms of the MS-VAR, estimating two VARs effectively means that you are deciding up front to fix $s_t=1$ for every $t$ before 2008 and to fix $s_t=2$ for every $t$ after 2008. Conditional on that, all that remains is to estimate $\mathbf{B}_1$, $\boldsymbol{\Sigma}_1$ and $\mathbf{B}_2$, $\boldsymbol{\Sigma}_2$. The reason to go full MS-VAR would be if you prefer to have the data tell you what the $s_t$ should be, as opposed to making that call yourself. There's nothing wrong with making that call. It just depends what you're trying to study. – jcz Nov 15 '18 at 17:28

One approach is to estimate a Markov-switching VAR (MS-VAR). Here are some key references:

## 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.

• That's just great! – Cardinal Nov 10 '18 at 15:24
• @Cardinal Yupp. Be warned: working with these models is pretty computationally involved, and to the best of my knowledge there isn't canned software out there to lighten the load. – jcz Nov 10 '18 at 15:34