# Fitting an ARIMA Model to Seasonal Data [duplicate]

I am trying to attain an ARIMA model for the following Time Series Data:

There is quite obviously a seasonal component - as the plot seems to oscillate between smaller and larger peaks, its seems to suggest that the pattern repeats every 48 timesteps.

I then took the seasonal difference and plotted the ACF & PACF:

From the ACF & PACF plots, I concluded that the model will most likely fit a $$\text{ARIMA}(0,0,1)(0,1,1)_{48}$$ model due the the shape of the PACF and the peaks at the ACF plots.

However, when trying to fit the data onto such model, the estimated coefficient for the regular fails the hypothesis test (roughly if $$\mid\frac{\hat{\theta}}{\hat{\sigma}_\theta}\mid > 1.96$$, then accept for some coefficient $$\theta$$).

I have tried other variations, but they consistently fail. I have come to the conclusion that I've made a grave error or I am missing a step. Note that when using Regular Differencing, the model does seem to fit better - but I don't know how to justify using Regular Differencing.

I have come to one of two conclusions: Either taking the seasonal difference of 48 is too high, or, the plot for PACF shows that the data may be non-stationary - therefore needing some Regular differencing.

Thank you

## marked as duplicate by Stephan Kolassa, Michael Chernick, whuber♦May 3 at 20:34

• Your time series plot is pretty much an exact copy of the one in the proposed duplicate. Are the two of you in the same course? – Stephan Kolassa May 3 at 19:24
• stats.stackexchange.com/users/1352/stephan-kolassa this has magic around 48,24 and 12 the one u are referring to just had 24 and 12 ...I believe – IrishStat May 3 at 19:31
• It's seems like very similar data - but in my course we were all given different data to play with, so at most it might be very similar. @irishstat I'm really sorry to be a nuisance, but I don't know how to upload the data here (or an online platform), could you just direct me on how to do so, thank you – Naji May 3 at 19:35

If you are using R statistical software you can take advantage of some excellent time-series libraries such as 'forecast' package (by Prof. R. Hyndman et al.). In that package there is auto.arima function which determines the best ARIMA model.

This may be worth checking out:

https://robjhyndman.com/hyndsight/arimaconstants/

• I would be interested in comparing your results to what AUTOBOX automatically delivered. – IrishStat May 3 at 20:04
• There are some results/plots based on auto.arima on a very similar time series, as pointed in the comments above, and here iis the link again: stats.stackexchange.com/questions/405204/…. Cheers. – dnqxt May 3 at 20:18
• Thank you for your help, I am having some problems with the auto.arima() function as although it takes into account seasonality, it thinks that the model is ARIMA(3,0,2)(0,0,1)_48 - which doesnt make sense, I would thought there would have to be a "1" for the seasonal differencing part – Naji May 3 at 21:30
• As you may have seen already, auto.arima goes through a fairly elaborate procedure checking many models and chooses a best one according to the smallest AIC. Auto.arima automates the task of model selection and identifies the best model among all that have been tried. Although the "best", the model may not be good enough still. There is a fair amount of uncertainty present in model selection as there is in the estimated model parameters. Once in stats world, one has to reconcile with the certainty of the presence of uncertainty. New R.Hyndman's book on forecasting principles may help. – dnqxt May 3 at 23:23
• there is no need whatsoever for any ar / ma structure . Your results simply reflect 6 redundant (unnecesary) structure/coefficients . The memory model is simply (0,0,0)(0,1,0)48 with some very minor adjustments for 4 periods. Your (3,0,2)(0,0,1) is way over-parameterized and should be simplified to (0,0,0)(0,1,0)48 . Fitting is not the same as modelling which examines sufficiency and necessity . – IrishStat May 4 at 9:51

I took your 240 values and introduced them to AUTOBOX and obtained the following Actual and Forecast graph . The Actual/Fit and Forecast is busier and here . with white-noise confirming residual ACF here .

The Actual/Cleansed plot is always informative about the anomalies

Any questions ... I would be glad to answer ...Note that even though the data is monthly there are strong model components at 48 as was suggested by the OP.

In terms of statistical characteristics ....

The residual plot is here

The equation is here

• Thank you - the coefficients do seem unusually large however – Naji May 3 at 21:31
• these are estimates of the pulses i.e. the adjustments necessary they are not ar or ma coefficients. – IrishStat May 4 at 9:45