# Is the R-square valid in a Regression model with Lag values of the Dependent Variables

I was working on some Time dependent data. Due to Client requirements I am forced to use LInear REgression for the modelling instead of Time series regression techniques like ARIMA. In order to not offend the Gods of data analysis, and also the Client, I was trying to cast a ARIMA model into a LInear Regression kind of framework. For that, I ended up including lagged values of the dependent variables (To imitate AR terms). End goal is to add terms so that the error terms are white noise.

Since what we have is not a text-book Linear Regression model, I had some concerns if Rsquare would be valid in such a situation. Could somebody please shed some light on the appropriateness of Rsquare in this situation of a Linear REgression with lagged dependent variable?

• Is the data actually integrated or have a moving average component? If it only has autoregressive components, it already is linear regression – shadowtalker Dec 24 '14 at 14:40
• Hi ssdecontrol... I built Arima models first and tried to build analogous Regression models. There wasn't a significant MA component. My question however is if the Rsquare is valid in case we include a Lag term (corresponding to, say, an AR(1) term)? – xxxmsi Dec 24 '14 at 15:19
• @RichardHardy post that as an answer – shadowtalker Jan 13 '15 at 15:54

Yes, the $R^2$ is valid.

Autoregressive models (with or without exogenous regressors) can be estimated using OLS as they satisfy the standard regression assumptions (where the requirement of independent regressors is replaced by a requirement for predetermined regressors). As far as I know, AR(X) and VAR(X) models are often estimated by OLS, and $R^2$ is used without problems.

I agree with @ssdecontrol that If it only has autoregressive components, it already is linear regression.

I think you have two options to consider if you do not want to go with Time Series like ARIMA. The first one is Generalized Linear Models, which is basically a linear regression that allows you to characterize the residuals with other distributions other than Normal. This includes lots of methods such as Ridge Regression, LASSO, and Automatic Relevance Determination Regression.

The second is Exponential smoothing where you average out the smooth out the previous set of data. This includes Holt winters (for trends), Brown model, and there is also a dampened model for things like growth and decay.

Not sure if this helps but you can also put prediction bands on data or use something like GARCH to have a range for your predictions.

• Thanks ccsv - I believe much of the variation and serial dependence can be captured using lags of the dependent variable. At least the errors clear the white noise tests. Your suggestions of GLS and Smoothing would not work, because... client :D. Hence we are trying to 'hack' (if you will) the usual regression to fit the data as correctly as possible. My question is, is the Rsquare valid if we use lags of dependent variables in the right hand side of the regression model. – xxxmsi Dec 31 '14 at 11:17