# Different estimated parameters in similar models in R

A particular series (std), seems to exhibit a trend-like behavior. According to the ADF test for this series:

Dickey-Fuller = -2.8618, Lag order = 6, p-value = 0.2131


Therefore, I am taking the first difference of std with this code

stddif1<- diff(std)


Here is the tricky part, the acf and pacf suggest that this would be an ARMA process (2,1), with a d=1. But the code shows both different estimates and different AIC values, when (I think) this shouldn't be the case:

For std with no difference:

> arima(std, order=c(2,1,1))

Call:
arima(x = std, order = c(2, 1, 1))

Coefficients:
ar1     ar2      ma1
0.5206  0.2697  -0.7638
s.e.  0.1218  0.0552   0.1153

sigma^2 estimated as 0.06355:  log likelihood = -13.3,  aic = 34.6


And, for the differenced std (stddif):

> arima(stddif, order=c(2,0,1))

Call:
arima(x = stddif, order = c(2, 0, 1))

Coefficients:
ar1     ar2      ma1  intercept
0.5188  0.2695  -0.7620    -0.0003
s.e.  0.1223  0.0554   0.1159     0.0157

sigma^2 estimated as 0.06355:  log likelihood = -13.3,  aic = 36.6


The values for the AR1, AR2, MA1 as well as the AIC are different. Why is this?

This was all done in R, the relevant package is 'tseries'.

This is because you are fitting different models! The 2nd model has been fitted with an intercept. Try:

arima(stddif, order=c(2,0,1),include.mean = FALSE)


and you should get the same resutls. Here is another example:

> require(tseries)
> arima(USAccDeaths, order = c(2,1,1))

Call:
arima(x = USAccDeaths, order = c(2, 1, 1))

Coefficients:
ar1      ar2      ma1
0.8526  -0.1838  -1.0000
s.e.  0.1173   0.1171   0.0634

sigma^2 estimated as 438908:  log likelihood = -563.42,  aic = 1134.83
> arima(diff(USAccDeaths), order = c(2,0,1),include.mean = FALSE)

Call:
arima(x = diff(USAccDeaths), order = c(2, 0, 1), include.mean = FALSE)

Coefficients:
ar1      ar2      ma1
0.8526  -0.1838  -1.0000
s.e.  0.1173   0.1171   0.0634

sigma^2 estimated as 438908:  log likelihood = -563.42,  aic = 1134.83