Linear VAR impulse responses - sensitivity of confidence interval bands to shock size

Question

My main question is: is the statistical significance of an impulse response in a linear VAR dependent on the size of the shock? Or put alternatively, how do the upper/lower confidence interval bands for an impulse response function in a linear VAR scale with the size of the shock?

For example, are the confidence interval bands for a two standard deviation shock simply twice the confidence interval bands for a one standard deviation shock? If yes, this would imply that the statistical significance of the impulse response would not be dependent on the size of the shock. If instead the scaling is more complicated, the statistical significance would be dependent on the shock size.

Lutkepohl (2000) notes that "...the direction or the size of a shock do not have an impact on the shape of the response" (link). However, this refers to the impact response itself, rather than the confidence intervals.

I am asking this question as in standard econometrics packages (e.g. STATA, JMulTi), the default setting for impulse responses is for a one standard deviation shock and it appears very difficult (or impossible) to customize this.

Answer 1

Impulse response for VAR process at a horizon $s$ defined as an effect of increase of element of $\varepsilon_t$ by 1 on response $y_{t+s}$. Impulse responses are simply elements of matrices $\Psi_j$ in $MA(\infty)$ representation of VAR process:

$$y_t=\mu+\varepsilon_t+\Psi_1\varepsilon_{t-1}+\Psi_2\varepsilon_{t-2}+...,$$

since

$$\frac{\partial y_{t+s}}{\partial \varepsilon_t'}=\Psi_s$$

Given this definition it is clear that statistical significance does not depend on the shock size, since if we increase the shock size, the impulse response increases proportionately.

This answers your initial question. However your last sentence implies that you misunderstand what impulse response for one standard deviation shock means.

Usually it is assumed that $\varepsilon_t$ is multivariate normal with covariance matrix an identity or any positive definite matrix $\Omega$. Then for the first case increase of 1 for the element of $\varepsilon_t$ means increase of one standard deviation, since standard deviation is 1. For the second case we write $\varepsilon_t=Fu_t$, where $F'F=\Omega^{-1}$, hence $cov(u_t)=I$, where $I$ is the identity matrix. Then $MA(\infty)$ decomposition looks like this:

$$y_t=\mu+Fu_t+\Psi_1Fu_{t-1}+\Psi_2u_{t-2}+...,$$

and we have new impulse responses $\Psi_sF$. Now these impulse response correspond to orthogonalized one deviation shock. Again the statistical significance for these new impulse responses does not depend on the shock size. That is why the software packages have the default setting of one deviation shock and this setting is not customizable.