# ARMAX or Dynamic Regression | regression of multiple timeseries

I have the following time series dataset (dependent | independent) :

Sales | Income,Inflation, Interest Rates etc


All of this is dynamic data pertaining to each of 24 months (month:0 to month:24). For 25th month onward I have no data for the independent variables (Income,Inflation, Interest Rates etc), yet I want to be able to predict sales for month:25 +.

I have been trying to figure out models which I can used to implement this scenario including Dynamic Regression and ARMAX/ARIMAX models. However, it seems that to be able to predict sales for the 25th month, i need data for dependent variables (Income,Inflation, Interest Rates etc) for the month (25).

Can I create a model using lagged values of the dependent and independent variables, used together in a regression model? I'm not sure if that makes sense.

This is my first time series model and im not sure if i am on the right track. Please advise.

• You can develop forecasts for the input series and then incorporate their future values and the uncertainty in them thus affecting the output series. If you wish to post your data I can be more specific. Commented Jul 25, 2016 at 19:30
• I understand your point, but wont the errors be amplified by the time I start predicting sales, using the predicted Income,Inflation, Interest Rates etc for month:25. Meanwhile, I'm reading this paper and it suggests that I might be able to work out a solution using multiple lagged independent predictors. [link] lexjansen.com/wuss/2013/49_Paper.pdf Commented Jul 25, 2016 at 19:45
• Vector autoregression (VAR) or vector error correction model (VECM) could be relevant when modelling a system of time series with the goal of forecasting on (or more) of them. Commented Jul 25, 2016 at 19:57
• The procedure that I was suggesting was exactly the one you referred to with some possible significant improvements. There may be deterministic shifts in your data ( level shifts.time trends/seasonal pulses/pulses )which might need to be identified. Furthermore there may be error variance changes over time. The approach you suggest can use previous errors which can often be quite parsimonious rather than possibly unwieldy long-lagged auto-regressive structures. VECM are a restricted form of an ARMAX model . Commented Jul 25, 2016 at 21:13

You could predict the independent variables using separate models or expert forecasts (the latter should be available for variables of such broad interest as income, inflation and interest rates). Then you could use them in the fitted ARMAX model to predict sales.

Alternatively, you could model the variables together using a vector autoregression (VAR) (or a vector error correction model -- a version of VAR suited for cointegrated variables). This would allow forecasting all of the variables within one model. One-step-ahead forecasts from a VAR are straightforward, while multiple-step-ahead forecasts can be constructed iteratively.

See this answer for a comparison of ARMAX and VAR models.

Can I create a model using lagged values of the dependent and independent variables, used together in a regression model?

Yes, you could also do that within an ARMAX or a VAR framework to forecast the dependent variable directly using lags of itself and the other variables. For $h$-step-ahead forecasts you would need to use lags of at least $h$ to enable direct forecasting.

I want to add that dynamic regression models such as ARDL or transfer function models should work too. They all allow you make predictions into the future. There is no difference in terms of whether one method gives you the power to predict and the other does not.

In addition, I think you very explicitly want to have a regression-style model with one DV and multiple IVs. VAR or vector-something model is not for that purpose. They are for multiple DVs.

I have a post further explaining the differences between dynamic regression models (often a result of terminology confusion) and ARIMAX models.