Transfer Function Models :: Arimax in TSA

ARIMAX models can be a bit difficult to implement/interpret in R. In this case there are a few things that tripped you up. Here they are in no particular order:

"mean-delete" is another way of saying "remove the mean". In this case, it refers to the covariate lead_140. So, start with

lead_140_Z <- lead_140 - mean(lead_140).

The order of the ARIMAX model you are trying to fit is (0,1,1), which is the same as ARMAX(0,1) on the first-differenced data. So, rather than work with the differencing inside of the model, just do so beforehand:

sales_140_D <- diff(sales_140)lead_140_D <- diff(lead_140_Z)

In this case, the order of the transfer function is actually (1,3), but the first, second and third MA parameters (MA0, MA1 and MA2) are fixed at 0 (ie, only B^3 appears in the numerator). To address this, you need to use the fixed argument in ARIMAX() and specify NA for those params to estimate and 0 for those to omit.

You don't need anything for xreg as the covariate occurs in the transfer.

mod <- arimax(sales_140_D,   order=c(0,0,1),   include.mean=TRUE,   fixed=c(NA,NA,NA,0,0,0,NA),   xtransf=lead_140_D,   transfer=list(c(1,3)),   method="ML")mod# Coefficients:          ma1  intercept  T1-AR1  T1-MA0  T1-MA1  T1-MA2  T1-MA3      -0.5791     0.0286  0.7255       0       0       0  4.7092s.e.   0.0756     0.0090  0.0040       0       0       0  0.0551

The results aren't exact, but they are quite close.