Is the the dependence of the residual of a ARMA time series model only based on AR term?

Question

Lets suppose we fit two time series models AR(1) and ARMA(1,1) to a data series. Should be the results of the ljung-Box test for the residuals be the same for these models? I mean does MA term affect the residual dependence?

Answer 1

Yes, why not? Via the additional MA term you can mop up more of the dependence in the series, such that the residuals will look "more" uncorrelated. Indeed, an $ARMA(1,1)$ can be written as an $AR(\infty)$ (subject to invertibility conditoins), so that it implicitly allows for many AR terms.

Try

ts <- arima.sim(n = 100, list(ar = c(0.5, 0.4), ma = c(0.4, 0.2)))
fit1 <- arima(ts, order = c(1,0,0))
fit2 <- arima(ts, order = c(1,0,1))

Box.test(fit1$residuals, lag = 10, type = "Ljung-Box", fitdf = 1)
Box.test(fit2$residuals, lag = 10, type = "Ljung-Box", fitdf = 2)

Answer 2

As Christopher said this portmandeau test will test against several types of linear structure in the residual or time series. It is based on significance of autocorrelations and mixed process can easily cause autocorrelation structure which is similar for several types of procesesses.