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I'm conducting an economic forecast based on an ARIMA time series model with multiple independent variables. I'm using a daily time series data that contains about 2 years of daily data inputs for a sum of 8 different regressors.

My ARIMA fit is so good so far but I'm wondering why the external regressors are not of a great effect to the predicted values. In my model the predicted values and $R^2$ are only slightly affected regardless of whether I include the external regressors or not.

In both cases the number of observations is equal and the adjusted R2=0.97 (Predicted value is not of a great change).

Could anybody help me with the significance of this phenomenon?

Update:
The case is now solved as shown in the following graphs:
1: The out of sample prediction without regressors
enter image description here

2: The out of sample prediction with signficative regressors
enter image description here

Thanks Mr. Richard Hardy for your help. It is highly appreciated

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – whuber ♦
    Mar 13, 2016 at 3:58
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    $\begingroup$ Your first graph doesn't make any sense. Judging by the historical plot, the forecast is inconsistent with training data set. There is not a significant drift in historicals, so there shouldn't be in forecast either $\endgroup$
    – Aksakal
    Mar 15, 2016 at 17:49

1 Answer 1

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Here is a summary of how the issue was presented and how it was addressed.

  1. Comparing two models with and without certain exogenous variables showed the following:
    (A) inclusion of exogenous variables lowers information criteria (AIC, BIC) considerably...
    (B) ...but barely affects the $R^2_{adj.}$ (that is very high anyway at about 0.97).
  2. Points of concern were:
    (A) dependent variable was integrated; hence, $R^2_{adj.}$ was little relevant;
    (B) exogenous variables were integrated, potentially leading to unbalanced regression;
    (C) out-of-sample forecasts were not assessed.
  3. Points of concern were addressed as follows:
    (A) $R^2_{adj.}$ not given consideration;
    (B) exogenous variables were differenced;
    (C) out of sample forecasts were assessed and found to differ significantly between the models with and without the exogenous variables.
  4. The finding in 3. (C) is now in line with the observation in 2. (A). Puzzle solved.

As of now, the forecast without regressors is going down in a weird way. Perhaps both pictures depict the forecasts from the same model that includes the exogenous variables -- but in the first picture the future values of the exogenous variables are set to zero? The idea was to have two different models, not the same model with two different sets of future values of the exogenous variables.

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    $\begingroup$ Either OP's forecast or the model specification is messed up. Otherwise, there's no way to get a forecast like on his first graph from a reasonable ARIMA model. The prediction can't just drop like a stone when the historicals seem to be rather stationary in mean. $\endgroup$
    – Aksakal
    Mar 15, 2016 at 17:52
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    $\begingroup$ @Aksakal, right... See my updated answer (which now does not quite qualify as a full answer now). $\endgroup$
    – Richard Hardy
    Mar 15, 2016 at 18:10
  • $\begingroup$ It's a good guess that maybe exogenous vars are set to zeros. $\endgroup$
    – Aksakal
    Mar 15, 2016 at 18:35
  • $\begingroup$ In the case of the first graph, the exogenous variables are not included but a constant is included. Taking out the constant the prediction goes to be a straight line and not drops in that way $\endgroup$
    – Mohammed Elshendy
    Mar 16, 2016 at 17:14
  • $\begingroup$ @MohammedElshendy, So have you done things this way: (1) estimated a model without regressors and predicted from it; (2) estimated a model with regressors and predicted from it, which requires supplying predictions of the regressors (and "sensible" predictions were supplied)? $\endgroup$
    – Richard Hardy
    Mar 16, 2016 at 19:20

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