AR(2) model with and without mean

Question

I am fitting a time series data by AR(2). After I subtracted the mean of the whole dataset, I expect the command include.mean = T and include.mean = F to give the same zero intercept. However, it seems its not the case. Could anyone explain why?

The following is the code:

sspar2 <- arima(ssp-mean(ssp$X101), order=c(2,0,0), include.mean = F, method = "ML")

with output

Call:
arima(x = ssp - mean(ssp$X101), order = c(2, 0, 0), include.mean = F, method = "ML")

Coefficients:
         ar1      ar2
      1.3998  -0.7094
s.e.  0.0707   0.0701

sigma^2 estimated as 227.1:  log likelihood = -410.29,  aic = 826.59

and

sspar2 <- arima(ssp-mean(ssp$X101), order=c(2,0,0), include.mean = T, method = "ML")
sspar2

with output

Call:
arima(x = ssp - mean(ssp$X101), order = c(2, 0, 0), include.mean = T, method = "ML")

Coefficients:
         ar1      ar2  intercept
      1.3999  -0.7092     1.1877
s.e.  0.0706   0.0701     4.9039

sigma^2 estimated as 227:  log likelihood = -410.26,  aic = 828.53

Answer 1

From R documentation, include.mean definition is "Should the ARMA model include a mean/intercept term? The default is TRUE for undifferenced series, and it is ignored for ARIMA models with differencing." You have an undifferenced series as your d=0 in Arima(p,d,q). Therefore you have an intercept term unless you specify otherwise.