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I am estimating this model:

enter image description here

But I want to do some analysis of the variables before. In particular, I am interested in fitting some ARIMA models. First, I am doing it for the inflation rate in Mexico.

  1. For the ARIMA model, do I need to take into account the other variables or only the values of variation of inflation in previous periods?
  2. When I look at the ACF graph it looks like this: enter image description here

    What does it mean to have a lag at 0.5 ? Since I cannot introduce an MA(.5), should I care about it or only take into account lags at t=1 and t=2?

  3. When I look at the PACF it looks like this:

enter image description here

Again, what does it mean to have considerable autocorrelation at t=.5 and t=.8? Since I cannot have an AR(.5), should I pay attention to this lags or only to lags at t=1 and t=2? Why?

  1. When I use the auto.arima function in R, it produces a model ARIMA(0,0,2), so that is no AR term, but is this not a contradiction with the PACF graph? What should I do? Why?
  2. In order to evaluate the goodness of fit I am using Box.test(fit_resid,lag=10,type="Ljung-Box"), but that gives me a p-value of very small, then is that a good fit or not?
  3. Finally, should I repeat that for every variable in the model or not? Why?
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  1. ARIMA only considers one variable and its own lags.
  2. The lags in the plots reflect periodicity of your data. You probably have monthly data with 12 months making up one period. Then in the plots, a lag of 1 means a lag of 1 year (12 months). Thus 0.5 is 6 months. In modelling, you would use AR(12) or MA(12) and AR(6) or MA(6) components in place of AR(1) or MA(1) and AR(0.5) or MA(0.5).
  3. See 2.
  4. I suggest posting this as a separate question. The implicit role of ACF and PACF in the algorithm of auto.arima is nontrivial (and there is no explicit role for them).
  5. A small p-value suggests low support for the null hypothesis of no serial correlation in model residuals. Therefore, a low p-value is bad.
  6. Depends on why you are doing "some analysis of the variables".
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  • $\begingroup$ But how to model that AR(6) in R.The code arima(x,order=c(1,0,1)) would be for AR(12) and MA(12), what is the code for AR(6)? $\endgroup$ – Carlos N Oct 18 '18 at 0:31
  • $\begingroup$ @CarlosN, the code arima(x,order=c(1,0,1)) would be for ARIMA(1,0,1), so AR(1) and MA(1). Code for ARIMA(12,0,12) would be arima(x,order=c(12,0,12)) and for ARIMA(6,0,6) would be arima(x,order=c(6,0,6)). Some coefficients (e.g. lags between 1 to 5) can be restricted to zero by the argument fixed. $\endgroup$ – Richard Hardy Oct 18 '18 at 5:18

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