Solved – Coefficient Significance in Regression with Arima Errors

arimaforecastingrregressionregression coefficients

In the R package forecast, when you run dynamic regression (regression with arima errors), the coefficients and their standard error are output, but there is no significance test available for the coefficients. I am wondering why this is– does variable significance in the typical sense (t-test) not hold when adding in Arima terms to the model?

An example of this would be

library(fpp2)
library(forecast)

auto.arima(uschange[, "Consumption"],
       xreg = uschange[, "Income"])


Series: uschange[, "Consumption"] 
Regression with ARIMA(1,0,2) errors 

Coefficients:
        ar1      ma1     ma2  intercept    xreg
      0.6922  -0.5758  0.1984     0.5990  0.2028
s.e.  0.1159   0.1301  0.0756     0.0884  0.0461

sigma^2 estimated as 0.3219:  log likelihood=-156.95
AIC=325.91   AICc=326.37   BIC=345.29

Best Answer

The forecast package does forecasting. For that purpose, the significance of variables is irrelevant. What matters is whether a variable is useful for forecasting. The AIC is a good guide for selecting variables for forecasting, so the package minimizes the AIC. If you really want to do a significance test on a variable, just compute the t-statistics from the output.

In the example provided, the t-statistic for income is 0.2028/0.0461 = 4.4. The p-value is 2*(1-pt(0.2028/0.0461, NROW(fpp2::uschange)-5)) = 1.8e-5

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