# Can we forecast a random-walk time series with another one?

I have two time series that follow a random walk behaviour. I would like to use the one I know to forecast the other one.

Suppose there is a strong correlation between them with $\text{lag}=k$, for example. This lag would permit me to forecast one because I have the information $k$ days before from the other one, even if it is a random walk.

Does it make sense? The problem I intuitively observe is that I'm trying to forecast a random evolution.

• I think you want a var model. – gung - Reinstate Monica Dec 12 '15 at 1:17
• With random walks, it's important to take first differences and avoid some possibly common pitfalls... There's a well known result that if you have two randomly generated random walks, you can regress one on the other and consistently get "statistically significant" results using classic regression techniques even though there's 0 actual relationship. – Matthew Gunn Dec 12 '15 at 7:14
• Actually, if you have two non-stationary which cointegrate then you should a) look at a VECM or b) First you'll have to check whether or not the series cointegrate. If they do not cointegrate you should take first differences of the variables and set-up a VAR for first differences. If they do cointegrate you can continue with the VAR in levels. – Plissken Dec 12 '15 at 10:32
• @MatthewGunn. That does not hold if the series cointegrate. – Plissken Dec 12 '15 at 10:33
• @Plissken Yeah, good points. How one should proceed depends on whether the series are cointegrated. There are formal tests, but often, some knowledge/intuition about the structure of the problem can help guide (eg. aggregate consumption and aggregate dividends are often assumed to be cointegrated). If two series are cointegrated, a VAR in first differences will miss that the cointegrating relationship will tend to push the two series back together if they drift too far apart in levels. – Matthew Gunn Dec 12 '15 at 11:16