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 

         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.


Yes, that's correct. The only change I'd make is to include "on average". Here is how we describe the same problem in the upcoming 2nd edition of our book:

The slope coefficient shows that a one unit increase in $x$ (a 1% increase in personal disposable income) results on average in 0.24 units increase in $ y$ (an average 0.24% increase in personal consumption expenditure).

As you have recognized, this example is a little tricky to describe because the two variables are percentage changes.


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