Cholesky Shock - Interpretation of logs in IRF Models

Question

I came across a few articles here and there that conclude:

When the data (say variables X, Y) for an impulse response function are on log level, the y-axis depicts the % response of Y to a 1% shock by X.

Broadly speaking, I think the shocks in an IRF are one SD Choleski shocks, where the y-axis reflects responses in the units of the dependent variable.

However, can someone explain (mathematically or not) why using logs automatically implies the interpretation of % impulses and % shocks? And: Why do some source say that log value responses are "approximately equal to" the percent change?

Thank you!

Answer 1

If you logarithmize the dependent variable, you will get approximately % changes, see Why is it that natural log changes are percentage changes? What is about logs that makes this so?

We know that the vcv matrix $\Omega$ of the error term can be factorized into $\Omega = ADA'$ with $D$ being a diagonal matrix (and $D$ is the vcv matrix of the "structual innovations"). Alternatively, we can factorize $\Omega$ by the Choleski decomposition into $\Omega = PP'$ where $P$ is lower triangular. So, we have $\Omega = ADA' = PP'$ and moreover, $$ ADA' = \underbrace{AD^{0.5}}_{=P}\underbrace{(D^{0.5})'A'}_{=P'} $$ so $P$ has the standard deviations of the structual innovations on its principal diagonal. Without loss of generalization, one can set the standard deviations of the structural innovations to 1 (i.e. $D = I$). Triangular factorization and Choleski Decomposition are then equal. That is, the interpretation of the Choleski Decomposition changes to: "If $y_{jt}$ is increased by "standard deviation of the $j$th structural innovation" units (but we just set the standard deviation to one and the units are given by %, see the link above), $y_{t+s}$ changes by $\Psi_sp_j$ units (but again, since we use logs the units are %)" ($p_j$ denotes the $j$th column of $P$)