How to analyse an impulse response function with more than 2 variables?

Question

I am running an impulse response function in R, using the package vars.

My data has 3 variables, the inflation (Brazilian CPI, or IPCA), the exchange rate and the output gap.

My goal is to calculate the exchange rate pass-through (both the maximum impact and the lag), and I am following and academic recommendation to add the output gap (as the monthly industrial production with HP filter).

The pass-through I am interested in is exchange rate -> CPI. The output gap is of my interest only in the way it impacts this pass-through relation. So I wrote the code as:

model_irf  <- vars::irf(model_var,  
                      impulse = "Exchange Rate", 
                      response = "CPI", 
                      n.ahead = 12, 
                      cumulative = TRUE) 

This gives me the expected response of variable “CPI” t+12 to a unit change in variable “Exchange Rate”.

I imagine (from macroeconomic theory) the output gap impacts the magnitude of the pass through, so in periods of larger output gap companies have less space to increase prices; relation that is not visible in this model I wrote.

My question is: How is the output gap related to the IRF I calculated? (Or if the model is wrong and I should write it differently to test this assumption)

Thank you very much for your time!

Answer 1

A VAR model consists of several Equations which are estimated at the same time to get an idea about the interatction effects in your model. Here you use 3 variables. therefore, your model contains one equation for CPI, one for output gap and one for exchange rate. Each variable will be explained by all other variables and their lags. You can get an idea about the coefficents with summary(model_var). The vars::irf just calculates the impulse responses with cholesky decomposition. Maybe you should take a look at all impulse responses together: vars::irf(model_var,n.ahead = 12,cumulative = TRUE). You will see the 3 columns where the column with the row heading "exchange rate" is most important for you.

Notes:

Mostly IRFs are calculated as non cumulative (cumulattive=F).

To investigate if the magnitude of the outputgap has an influence on the pass through you might need special models