Considera sample of 50 observations from a stationary process with

$$\begin{array}{c|c|c|} \text{lag} & \text{acf} & \text{pacf} \\ \hline 1 & 0.9 & 0.9 \\ \hline 2& 0.85 & 0.4\\ \hline 3&0.7&0.1\\ \hline \end{array}$$

How do I suggest an appropriate model using the above information.

From my understanding, significance of ACF and PACF values were checked using:$ H_{0}: \rho=0$ rejected at 5% level of significance if $|\rho| >\frac{1.96}{\sqrt(50)}$. I found that only the last value of pacf is non significant.

(1)Can I say that the model is ARMA(2,3)? (2) how to decide if it is AR, MA or ARMA?

NB: this was a question asked in an examination. To be worked out manually.


1 Answer 1



Since the ACF is dominant i.e.has the most significant values the process is autoregressive (AR) . The order of the AR model is determined by the # of significant values in the subordinate structure ... in this case the PACF .. thus 2.

The answer is (2,0,0)


When does this simple strategy work ...

1) When there are no step/level shifts in the data

2) When there are no deterministic trends in the data

3) When there are no pulses in the data

4) When there are no seasonal pulses in the data

5) when the suggested model has constant parameters and constant error variance over time

  • $\begingroup$ Thank you. So if pacf had more number of significant values, would it be an MA process? When can one conclude that a process is ARMA? $\endgroup$
    – Harry
    Jul 17, 2019 at 9:45
  • $\begingroup$ Yes ... if PACF has more significant values than the ACF it would be an MA process of some order Q (GIVEN that you are not including possible seasonal structure) . An ARMA structure could be initially identified if there was no clear cut dominant structure i.e. a tie . More correctLy after estimating an AR MODEL the model residuals might suggest the need to add MA structure. Model identification is an iterative process NOT just a 1 step process. $\endgroup$
    – IrishStat
    Jul 17, 2019 at 9:48

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