I am reviewing the advanced section of the otext book on forecasting (https://www.otexts.org/fpp/9/1) and am hoping to obtain a clarification. Example 9.1 aims to model consumption based on income. Both series shown are quarterly % changes in US consumption, e.g. (Q3-Q2)/Q2 %, both are stationary. The parameter estimate for the fit2 model (copied from the otext to below) shows that income 2 quarters ago is related to consumption in the current period. The point estimate for the parameter is 0.242. So, what is the correct interpretation of this parameter in this specific case?
Model output (from link above):
> (fit2 <- Arima(usconsumption[,1], xreg=usconsumption[,2], order=c(1,0,2))) Series: usconsumption[, 1] ARIMA(1,0,2) with non-zero mean Coefficients: ar1 ma1 ma2 intercept usconsumption[, 2] 0.6516 -0.5440 0.2187 0.5750 0.2420 s.e. 0.1468 0.1576 0.0790 0.0951 0.0513 sigma^2 estimated as 0.3396: log likelihood=-144.27 AIC=300.54 AICc=301.08 BIC=319.14
My own interpretation is that a 1% increase in the percentage change in income (in absolute terms, not multiplicatively) is associated with a 0.24% increase in the percentage change consumption (again in absolute terms) two quarters later. So, for example when the % change in income goes from 1 to 2 % (say between Q4 and Q1) then we expect the % change in consumption to increase by 0.24% two quarters later (in Q3).
Is my interpretation correct? Note - While this topic has been raised before (e.g. Interpretation of regression with ARIMA(0,1,1) error) I am yet to find a succinct and satisfying answer.