# Coefficient Significance in Regression with Arima Errors

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


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

• I thought that might have been the case, but it is great to hear it from the straight horses mouth. Thanks Dr. Hyndman! – RayVelcoro Nov 27 '18 at 22:11

If you aren't interested in realistic(ie wide) confidence limits then this is ok, but if you want good confidence limits then this has a negative impact. The AR/MA parameters are significant, but the ACF/PACF didn't warrant them and in essence mathematically they cancel each other out so no harm done....for this example.

The variance just with the causal is .359252 and by adding 3 more parameters it becomes .3219 which is a non-significant reduction.

If there had been an outlier in the last period then this will have a big impact in the forecast. There are outliers in this example and if the forecast was right after one of these it would have consequences.

SPSS's Temporal Causal Model discusses this capability as well.

Here is the model from Autobox(a software I am affiliated with) with a denominator lag and numerator operators on the variable income and 16 outlier variables.