# Predicting a single response variable with a multivariate regression model

For my research I am interested in performing inferential and predictive analysis on a single response variable. I have already made one univariate and one multivariate regression model. In the multivariate model, the response variable which I am particularly interested in is modeled together with another response variable. The multivariate model that I use can be described as a vector autoregressive model with random effects.

For the inferential part of my research, I am already aware that my multivariate model has some advantages over the univariate model. However, I would like to ask the following:

'Does it make any sense to predict the single response variable I am interested in with my multivariate regression model rather than with my univariate regression model?'

I have some read some research where authors show that the prediction of for example multiple macroeconomic series can benefit from predicting them with a multivariate model rather than multiple univariate models. In my case I have a multivariate regression model in which I am actually only interested in one of the two response variables. The only reason why I made a multivariate model in the first case was because of the inferential part of my research.

• Since you have only 1 response variable, you probably mean multiple regression NOT multivariate regression. – mkt May 28 '18 at 14:01
• I have actually two response variables in my multivariate model. However, I am only interested in predicting one of the two. I initially made the multivariate model in order to asses the relation between the two response variables by means of a mixed effects vector autoregressive model. My question now is: 'is it useful to use this multivariate model to predict the outcome of the one response variable?'. I am aware that if I would forecast using this multivariate model, I get two predictions, one for each response variable. But, as mentioned, I am only interested in one of them. – dubvice May 29 '18 at 9:55
• The question is a bit confusing: how many and what input variables do you have? If you want to model (abstract) functional behaviour, it's a regression task. If you have more than one input variable is multivariate, otherwise is univariate. The choice of model should be drawn from the type of regression, the data and possibly other constraints. Could you please provide more information and make the question more precise? – cherub May 29 '18 at 14:05
• My answer is yes! and I am eager to hear what makes you doubt. With one predictor, you basically have one variable on the right side of the equation, so what is the problem? – Million May 29 '18 at 14:09
• If you have a vector autoregression (VAR), then only one equation matters for forecasting one step ahead but all equations matter for forecasting further into the future. Furthermore, if the regressors are the same in all equations, equation-by-equation OLS estimation is optimal, and hence the estimates of the equation of interest are the same as if the equation was estimated separately by OLS. Otherwise, feasible GLS is (asymptotically) better than eq.-by-eq. OLS, and so the estimates of the single equation are different when obtained from the system than obtained independently. Clearer now? – Richard Hardy May 29 '18 at 16:43

## 1 Answer

If you have a vector autoregression (VAR), then only one equation matters for forecasting one step ahead but all equations matter for forecasting further into the future. Furthermore, if the regressors are the same in all equations, equation-by-equation OLS estimation is optimal, and hence the estimates of the equation of interest are the same as if the equation was estimated separately by OLS. Otherwise, feasible GLS is (asymptotically) better than eq.-by-eq. OLS, and so the estimates of the single equation are different when obtained from the system than obtained independently. In small samples feasible GLS may or may not be better than eq.-by-eq. OLS. (This should also answer the question in the comments: When I compare the univariate parameter estimates with the multivariate estimates, could I expect the multivariate estimates to be more accurate (and thus more useful for forecasting)?)