Can you recursively forecast one variable using two variables?

Question

My question is probably very elementary but I haven't been able to find an explanation of recursive forecasting that I fully understand.

I've read a journal article that seemed to recursively forecast unemployment with an AR(2) process augmented by a variable for depth of recession lagged by one period. It seems to me like that wouldn't be possible if the model doesn't produce forecasts for the other variable to include in the forecast for the subsequent period as well.

So does recursive forecasting use actual values for each subsequent forecast or forecasted ones? And does it just do that for the auto-regressed variable or both? Is it a standard practice to use forecasted values for one variable and actual ones for another?

I have quarterly data on unemployment (U) and change in total lending (DL) in an economy. I want to estimate an AR(2) for U plus DL at t-1. How can I make out of sample forecasts with this model? Is my only option to use predicted values for DL that are forecasted by a separate model?

Answer 1

If it is possible to have actual values of the other variables for example let’s say variable $y$ is always provided by statistical bureau with 5 year lag but for second variable $x$ the values are always up to date you can use them.

However, if you have data for both $y$ and $x$ up to date and you want to make forecasts for next year and further in the future you will have to do auxiliary forecasts for $x$ as well.

When you are doing some out of sample evaluation and you want to make a thought experiment of how the model forecasts $y$ conditional on knowing what was the $x$ then you can do that, but such evaluation won’t be indicative of actual performance when in real life both future $y$ and $x$ is unknown. This is the biggest downside of multivariate forecasting.

However, you can always make an auxiliary forecast for the $x$. Let’s say with an auxiliary AR model for the $x$ variable.