log differencing and VAR model

I am working on a VAR model to forecast inflation using variables like CPI prices, oil prices, unemployment rates, PMI, inflation expectations, policy rates, and GDP.

To use the VAR model for my forecast, I had to make all the variables stationary. To do so, I applied log differencing to the data for all the variables.

I'm a bit confused about how this affects the interpretation of the model, the forecasts, and the results.

When I take the log difference of a certain variable, then I'm getting the continuous compounded growth rate of it. If I take the log difference of some variables but not all, what does my forecast mean? Is it giving the forecast for the variables in terms of the conversions done? So if I took log difference of CPI and Oil prices, just differenced unemployment rate, and left the policy rate as is, how would this affect the forecast interpretation? Would CPI and Oil forecasts be in terms of log differences, while the unemployment rate forecast is in terms of difference, and the policy rate forecast is just the policy rate value?

If I take the log difference of all variables, then the forecast will give CPI in terms of log differences (right?). And would CPI in terms of log differences give the values for inflation rates?

If I wanted to convert the forecast values for CPI and the variables back to their initial measures and units before any sort of conversion was done for stationarity, how would I do so?

• Would CPI and Oil forecasts be in terms of log differences, while the unemployment rate forecast is in terms of difference, and the policy rate forecast is just the policy rate value? Yes. Commented Oct 24, 2022 at 8:38
• @RichardHardy would the results for CPI in terms of log differences give inflation rate? Commented Oct 24, 2022 at 13:21
• No, I do not think so, because inflation is not defined as log-difference of CPI. Commented Oct 24, 2022 at 13:56
• @RichardHardy: in most cases, the 12-month log difference of the CPI levels is a good enough approximation of the 12-month growth rate of prices, so can be interpreted as inflation, possibly multiplied by 100 to give percentage points.
– BrsG
Commented Oct 24, 2022 at 15:32
• @BrsG, you are right. On the other hand, why use an approximation when you can get the exact figure so easily? Commented Oct 24, 2022 at 15:58