# Panel VAR impulse response interpretation?

Quick question on panel VARs. The equation is:

$$Y_{it}=a_i+\Pi Y_{it-1}+\epsilon_{it}$$

In estimating these models the fixed effect $$a_i$$ is often removed by differencing or forward demeaning, and then the coefficients are estimated and some restriction imposed so that the error terms are not correlated.

Now the question, is it not a problem in the interpretation of the impulse responses that the fixed effects are not accounted in any way? The shocks have different interpretation if some fixed effect is included correct?

For regular (non-panel) VAR as far as I understand the constant is included when calculating the impulse response in comparison.

## 1 Answer

I think you are incorrect about the constant usually being included when calculating the impulse response function. Ignoring the panel aspect for now, the impulse response function for $$k$$ periods ahead is defined as $$\frac{\partial Y_{t+k}}{ \partial \epsilon_{t}}$$. Obviously $$\epsilon$$ has no effect on $$Y$$ via the constant.

An equivalent definition is given in these lecture notes from Eric Sims as $$IRF(k) = E_{t}[Y_{t+k}] - E_{t-1}[Y_{t+k}] | \epsilon_{t}=e$$. You have in mind the impulse response function as just $$E_{t}[Y_{t+k}]$$ but the constant term is cancelled out by $$- E_{t-1}[Y_{t+k}]$$. The documentation for MATLAB's armaIRF makes clear that it uses this definition.

You will of course need to use the constant in some applications such as forecasting.