I have monthly data for about the last 2.5 years for an economic index as well as external regressors for each month. Is it feasible to treat the months as independent and just fit a multiple regression model regressing the external data (3-5 variables) on the index? I have tried fitting an ARIMA (1,0,0) and with external regressors and the auto regressive term is not significant. Is there a test that will determine whether a time series is independent given external data? Also, given my small amount of data what is the best way to validate any models.
- No it is usually not correct to just treat the samples as a cross-sectional one. ESPECIALLY in small samples.
- Fitting the AR(1) model with external regressors is usually called the ADL-model (autoregressive, distributed lags). You should use your statistical program to look at the ACF and PACF plots of your regressors and dependant variable and see which lags could be significant. However, fitting an ADL model instead of pure ARMA models can be very tricky.
First, just like in any ordinary linear regression, you need to include all determinant factors (including lags), otherwise you don't even have asymptotic/large sample consistency of the model. This can be already impossible for the data, but you should try to do it anyway. Second, you might need to get include ARMA terms to get rid of autocorrelation (of the error terms, use the Durbin-Watson Test). If you do this and everything goes perfectly, you now have a consistent estimate. However usually you still face issues such as nonstationarity, trends, cointegration etc, which is why the ADL model rarely works. There is no quick solution other than doing a time-series course.
In general, seeing as you are trying to do it by fitting a linear model, you would try to regain some resemblance of normal distribution in your residuals. If you have that, you can in general feel pretty good about your model. At the same time try to include significant lags and maybe appropriate structural breaks.
You should know that this is not an easy topic and you will most likely fail in the above. There is a pretty definite progression on time series and econometric literature which goes follows:
1. Intro book - I recommend Stock & Watson, which talks about almost every topic 2. Intermediate book - Either Green or Hayashi 3. Hamilton - Time Series Analysis by Hamilton. Everything you ever wanted to know.
If you are lucky and your foundational model is good, you might succeed with an ADL model. However seeing as you are looking at an index value, I'd say further econometric research is in order. There is a reason why ARIMA and VAR models are usually separated from models with an explanatory model such as ordinary linear regression. Their objective is different. I am not going to write a complete time series course down here but this might give you some points to start on.