# Does a Vector Autoregression Model truly avoid circular function issues?

Let's say you have a VAR model that estimates GDP and Unemployment among many other variables using a certain number of lags. This VAR model can estimate or regress GDP using Unemployment. And, it can do the reverse too. It can also do Granger Causality analysis between these two variables to analyze the direction of the causality and Impulse Response Function to measure the intensity of this directional causality. But, can it truly forecast several periods out both GDP and Unemployment (Hold Out test situation)?

I would think once you run out of lags, this VAR model could not forecast for periods beyond the lags because then it has to rely on estimates of both variables to forecast out the specific variable you want to forecast. And, at this stage you run into a circular function situation.

With VAR models you can predict as many steps ahead as you like. When the future horizon increases, you will eventually run out of observed values to base your forecasts on. Then you just substitute the required observed values with their forecasts and iterate forward. Here is an example of VAR(1) (I skip the intercept for brevity): \begin{aligned} x_t &= \Phi x_{t-1}+\varepsilon_t \\ \hat x_{t+1|t} &= \Phi x_{t} \quad \text{(have not run out of observed values yet)}\\ \hat x_{t+2|t} &= \Phi \hat x_{t+1} = \Phi^2 x_{t} \quad \text{(have run out of observed values, thus substitute)}\\ \hat x_{t+3|t} &= \Phi \hat x_{t+2} = \cdots = \Phi^3 x_{t} \\ &\cdots \\ \hat x_{t+h|t} &= \Phi \hat x_{t+h-1} = \cdots = \Phi^h x_{t} \\ \end{aligned} where $$x_t$$ is a multivariate time series of interest, $$\Phi$$ is a square matrix of coefficients and $$\varepsilon_t$$ is a multivariate time series of errors (innovations, shocks).

Update to include some information from the comments:
We are doing the best we can given that we only have current information but not future information. The fact that forecasts are mean reverting reflects an assumption on the data generating process (DGP). As Chris Haug points out, this is a consequence of the linearity of the model, but it is not an approximation. The behavior of the forecasts is actually optimal given the DGP. And as long as VAR is a good approximation of the DGP, I would not expect alternative forecasts to beat the ones by VAR, unless they employ some future information. But in the latter case, a fair comparison would add this information to the VAR model in terms of exogenous variables or the like.

• when "you just substitute the required observed values with their forecasts and iterate forward"... Isn't this a sort of circular function issue. You have forecasts that end up just using forecasts as their independent variable inputs. And, as you go forward you are using forecasts of forecasts of forecasts to estimate further forecasts. Mathematically, does not this result in some sort of flattening out every forecasted value to some sort of average historical value? And, by doing so you loose all real sensitivity to the economic environment. Nov 4, 2019 at 22:01
• @Sympa, as you go forward you are using forecasts of forecasts of forecasts to estimate further forecasts - right. Also, the answer to Mathematically, does not this result in some sort of flattening out every forecasted value to some sort of average historical value? is a Yes. However, in which sense do you think that you loose all real sensitivity to the economic environment? After all, we only have information as of time $t$, and we incorporate it (so we do not lose sensitivity there). However, we cannot incorporate future information about the economic environment that we do not have. Nov 5, 2019 at 7:49
• @Sympa, so briefly, we are doing the best we can given that we only have current information but not future information. The flattening out every forecasted value to some sort of average historical value is essentially caused by the forecasting problem being tough, not the forecast method being poor. Nov 5, 2019 at 7:51
• thanks for all the insights. The forecasting environment I am in is where we are provided forecast-scenarios for a bunch of macroeconomic variables. And, we are modeling-forecasting a dependent variable using those scenarios. And, what we are considering is forecast two dependent variables, let's say Y1 and Y2. Given this framework, a VAR model forecast would work as long as you do not run out of lags. Once you do, you would run into the mentioned issues. And, you would be better off regressing Y1 and Y2 separately using a multiple regression framework. Nov 5, 2019 at 19:15
• @Sympa, I see. It is hard to comment on that without some more details, though. Is there anything else I should address in my answer? Nov 5, 2019 at 23:26

After giving it some thought, I think the straight forward answer is that VAR does not resolve the circular function situation as framed in my question. And, frankly I don't think any other methodology can. You can't use A to forecast B at the same time as you use B to forecast A once you go beyond any related lagged variables.

The key is "at the same time" and "once you go beyond any related lagged variables."

Practitioners often think they can by relying on forecasts once you go beyond the lags. But, that is not the same thing. You very soon are relying on the forecast of a forecast to forecast the next period. You end up with some mean-reverting process whereby you probably would be better off to simply use a simple average as your long term forecast.

• The conditional expectation $h$ steps ahead for a VAR($p$) with $h>p$ is exact and does not "rely" on forecasts. The fact that the conditional expectation is the same as applying the matrix $\Phi$ to the forecasts is simply a consequence of the linearity of the model, but it is not an approximation. There is nothing circular about this situation. Nov 22, 2019 at 0:26
• @ChrisHaug, excellent point (that I missed in my answer). Nov 22, 2019 at 6:00
• I am unclear how you could not "rely" on forecast going forward as specified. Similarly, the conditional expectation I would think would become increasingly mean-reverting the further you go out on your forecast. Nov 22, 2019 at 18:03
• The two-step forecast for a VAR(1) is just $\Phi^2 x_t$, so what you are relying on is your estimate for $\Phi$. Phrasing it as a "forecast of a forecast of a forecast" makes it sound like some disastrous sequence of approximations on top of each other, when in fact the only approximation here is the standard one that you make whenever you fit any model. It's also not clear to me why you think that the forecast being mean-reverting is somehow bad; if the data is well-represented by a VAR(p), then that is exactly what should happen. Nov 22, 2019 at 18:46
• "The standard one you make whenever you fit any model" is not true for any non autoregressive models. They do not rely on any estimation of either Y or X variables to forecast Y. Also, the "data is well-represented ... would entail that it is mean-reverting" is only true for the history or learning sample. It is not true for any forecast. Nov 25, 2019 at 4:30