# 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'.

## 1 Answer

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