# Multivariate time series forecasting: Endogenous vs Exogenous

I recently came across a problem where I want to predict a timeseries let's refer to it as T1 but it seems that two other time series, let's call them T2 and T3 have some predictive power on the T1 time series.

Now the question came up, I can use either multivariate analysis and include the series T2 and T3 as endogenous variables or I can include them as exogenous. I have been reading around here and there but I did not come up with a clear answer on when to use which and given my data which approach is more suitable.

I would also be interested in the details on how SARIMA (for example) treats endogenous vs exogenous variables and how these choices influence the forecasting of my main time series T1.

I hope everything was clear. Thanks a lot and please let me know if there is already some answer to that!

The distinction between multivariate and exogenous seems to be a false dichotomy. If your model contains more than one variable or more than one time series, it is a multivariate model. The relevant distinction is between treating all variables as endogenous vs. treating only one of them as endogenous and the rest as exogenous.

Now, which approach would deliver more accurate forecasts of the series T1? If we are talking about one-step forecasts, there need not be a difference. The result of interest will have T1 as the dependent variable and T2 and T3 (and probably some lags) as predictors. Whether you take a model with this single equation or a model with multiple equations (one for each endogenous variable), there will still be just one equation you focus on. So there need not be a difference.

Regarding multiple-step forecasts, you may need to predict the predictors themselves, and for that you will need equations for them. Thus a multi-equation model with more than one endogenous variable may be beneficial. But there is always the bias-variance trade-off, so we cannot be sure ahead of time regarding which model will perform better.

Regarding SARIMAX or regression with SARIMA errors (these are different things), it has a single dependent (endogenous) variable and some predictors (exogenous variables). You will find a brief overview with concrete equations and their interpretations by following the link above.

• Nice i really thank you for such a great explanation and for guiding my through my missconception of multivariate and exogenous. Then from here follows another question, for what i have understood exogenous variables by definition can't be predicted. Therefore if i have three timeseries to train my model with data from t-n until t and i want to predict something with the exogenous data at 't+3' i need the exogenous data at t+2 right? And the only way to do that is actually having a prediction of the exogenous data at t+2 therefore must be endogenous instead of exogenous. Right?
– erni
Apr 16 at 11:38
• @erni, that depends on your model. E.g. $y_t=\alpha_0+\beta_3 x_{t-3}+u_t$ would allow you to predict $y_{t+3}$ using $x_t$, but $y_t=\alpha_0+\beta_1 x_{t-1}+v_t$ would require a model for $x_t$. Relevant keywords are direct vs. iterative/iterated multi-step forecasting. Apr 16 at 14:27
• I see, makes total sense. So simply the conclusion of all this is: exogenous variables are variables that have predictive power to whatever our prediction is, but our model can't predict them. On the other side endogenous variables are all the variables that our model is able to predict, regardless if they have or not predictive power. Right?
– erni
Apr 16 at 21:39
• @erni, you could say that, though I would replace "can [predict]" or "is able to [predict]" with "tries to [predict]". For endogeneity/exogeneity what matters is whether we try to model some variables, not how successful we are at it. Apr 17 at 5:19
• " If your model contains more than one variable or more than one time series, it is a multivariate model." Are you conflating "multivariate" (i.e. multiple variables on the left hand side of the equation) with "multiple" (i.e. more than one variable on the right hand side of the equation)? Sep 15 at 4:04