Identify parameters of ARIMA model

Question

I am trying to build ARIMA model, I have 144 terms in my standardized time series, which represent residuals form original time series. This residuals, on which I would like to build ARIMA model, are obtained when I subtracted linear trend and periodical component from original time series, so residuals are stochastic component.

Because of that subtraction I modeled residuals like stationary series (d=0), so model is ARIMA(p,d,q)=ARIMA(?,0,?).

ACF and PACF functions of my residuals are not very clear as cases in literature for identification ARIMA models, and when I choose parameters p and q according to criteria that they are last values outside of confidence interval, I got values p=109, q=97. Matlab gave me error for this case:

Error using arima/estimate (line 386)

Input response series has an insufficient number of observations.

On the other side, when I am looking only to N/4 length of time series for identifying p and q parameters, I got p=36, q=34. Matlab gave me error for this case

Warning: Nonlinear inequality constraints are active; standard errors may be inaccurate.

In arima.estimate at 1113

Error using arima/validateModel (line 1306)

The non-seasonal autoregressive polynomial is unstable.

Error in arima/setLagOp (line 391) Mdl = validateModel(Mdl);

Error in arima/estimate (line 1181) Mdl = setLagOp(Mdl, 'AR' , LagOp([1 -coefficients(iAR)' ], 'Lags', [0 LagsAR ]));

How do I need to correct identify p and q parameters and what is wrong here? And wwhat does it mean in this partial autocorrelation diagram, why are last values so big?

Tommaso Belluzzo · Accepted Answer · 2017-12-25T04:27:09.910

This guide contains a lot of useful information about the correct estimation of ARIMA p and q parameters.

As long as I can remember from my studies, since ACF tails off after lag q - p and PACF tails off after lag p - q, the correct identification of p and q orders is not always straightforward and even the best practices provided by the above guide could not be enough to point you to the right direction.

Usually, a failproof approach is to apply an information criteria (like AIC, BIC or FPE) to several models with different orders of p and q. The model that presents the smallest value of the criterion is the best one. Let's say your maximum q and p desired order is 6 an that k is the number of observations, you could proceed as follows:

ll = zeros(6);
pq = zeros(6);

for p = 1:6
    for q = 1:6
        mod = arima(p,0,q);
        [fit,~,fit_ll] = estimate(mod,Y,'print',false);
        ll(p,q) = fit_ll;
        pq(p,q) = p + q;
     end
end

ll = reshape(ll,36,1);
pq = reshape(pq,36,1);

[~,bic] = aicbic(ll,pq+1,k);
bic = reshape(bic,6,6);

Once this is done, use the indices returned by the min function in order to find the optimal q and p orders.

On a side note, for what concerns your errors... well, the first one is pretty straightforward and is self-explanatory. The second one basically means that a correct model estimation is not possible.

Identify parameters of ARIMA model

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