# Seasonal ARIMA estimate with non-significant variance

I'm trying to fit a seasonal ARIMA model for a real data time series, 300 samples long, sawtooth looking. I'll write down a resume of my work trying to be clear without many plots.

The ACF of the original data slowly tapering towards 0 and the value PACF(1)=0.8 both suggest a non-seasonal differencing of order 1. As a consequence, both ACF and PACF correlations of the differencied data fall in the 95% confidence interval except for ACF(43)=+0.5 and PACF(~40)=-0.3. This value roughly corresponds to the period of my spikes in the signal.

I therefore applied a seasonal differencing for seasonality=43 multiplied by the non-seas. differencing. The correlograms follow:

What are the negative correlations at lag=1 telling me, now? Maybe differencing is too much and an AR(1) should be employed?

However, I tried some models with combinations of AR(1), MA(1), SAR(1) and SMA(1) and I got from minimization of AIC and BIC the following best models:

$$(1,0,0)\times(0,1,1)_{43}$$ $$(1,0,1)\times(0,1,0)_{43}$$ $$(1,0,0)\times(0,1,0)_{43}$$

For example, the residuals correlograms for the latter:

which look not bad to me even though some significant spikes around the lag of seasonality remain. Can be this due to the imperfect seasonality (data are from extremely fine sampled measurements!)? Anyway, I have other two concerns:

• the estimated variance of the model is $\sigma=1.7\times 10^{-10}$ with st. error$=4.6\times 10^{-8}$ which gives a t-statistic$=0.002$ really low!

• the qq-plot of residuals shows a non-agreement with the Gaussian distribution:

In conclusion, how much can I trust my estimated model given these two problems?

Probably not at all ! Your data might disclose changes in parameters over time or deterministic changes in error variance over time suggesting Weighted Least Squares or the need to identify and incorporate latent deterministic structure like pulses, step/level shifts or local time trends OR the need to incorporate seasonal dummies to deal with seasonal structure (often a good choice ! ).

EDITED AFTER RECEIPT OF YOUR DATA:

To review , auto.arima in a brute force list-based procedure that tries a fixed set of models and selects the calculated AIC based upon estimated parameters. The AIC should be calculated from residuals using models that control for intervention administration, otherwise the intervention effects are taken to be Gaussian noise, underestimating the actual model's autoregressive effect and thus miscalculates the model parameters which leads directly to an incorrect error sum of squares and ultimately an incorrect AIC.

The acf of the original is useful in assessing ARIMA structure ( that's the big print ! ) when there are no deterministic factors/features in play like period # ( that's the small print ! ) . Here is the acf of the original data .

Your 301 values are indeed non-stationary BUT there is more than 1 remedy for non-stationarity. Your series has deterministic effects for certain periods in the 43 period cycle . See I have correlogram ACF and PACF below for a temperature time series. Can I say it is MA(2) from ACF? What about AR? for a similar but different but related discussion.

I took your 301 values into AUTOBOX a piece of software that I had helped to develop and I asked it to tell me what it could find out by studying the 301 historical values. It hesitated for a very brief amount of time and told me the following.

Here is the Actual /Fit and Forecast graph

3: https://i.sstatic.net/lbD0m.png and a less busy Actual/Forecast . The USEFUL equation that was found ( Note all models are "rong" but some are useful and some are very wrong ) is here in two parts. and providing clarity that the arima portion is simply (1,0,0) .

The statistics for the model are presented here

The residuals look pristine with a very clean acf .

A closeup of the forecasts is here ....

You said " I therefore applied a seasonal differencing for seasonality=43 " ... and I say that the data says no .

Periods 1-8 AND 36-43 ... are definitely non-players as they have little or no impact on what you are measuring.