MATH7970 Chap.7 ARIMA and Seasonal Models
ARIMA and Seasonal Models
Difference before identifying dependence
ARIMA separates autoregressive order, differencing order and moving-average order. Differencing changes the scale being modelled and must be settled before p and q are interpreted.
A second seasonal clock
Seasonal orders describe dependence at a known period.
Ordinary and seasonal differences answer different questions, and applying both means modelling changes in seasonal changes.
Read each order as a separate statement
The p order describes autoregressive dependence after transformation, d counts ordinary differences and q describes moving-average dependence. Seasonal orders repeat these roles at period s.
Translate notation into a sentence before interpreting coefficients. This prevents an effect on first differences from being reported as a direct level effect and keeps ordinary and seasonal clocks distinct.
Recognise integrated level behaviour
A random walk carries innovations permanently into its level, producing slow ACF decay and forecast uncertainty that grows with horizon.
One ordinary difference exposes the innovations. A drift term becomes the mean of the differences and creates a linear expected level path. This transformation changes the object being modelled, so forecast interpretation must return deliberately from change to level.
Difference conservatively
Begin with the smallest operation supported by plots and lag evidence.
After every ordinary or seasonal difference, inspect the new series. Strong negative lag-one correlation, extra noise and worse validation performance can indicate over-differencing.
Reversing an unnecessary difference is better than adding MA terms solely to repair dependence the transformation itself created.
Combine seasonal and ordinary evidence
Seasonal spikes at s and its multiples guide seasonal orders; early lags guide non-seasonal terms. Their shapes interact after differencing, so propose a small candidate set rather than selecting one order mechanically.
Compare candidates on the identical transformed sample, then require residual correlation to disappear.
Accumulate forecasts, reattach seasonal history and handle logged scales carefully so final levels trace back to the modelled series.
Guard against automatic order search
Software can examine many combinations and return one with the smallest criterion, but that search does not decide whether the transformation is justified or whether the final dynamics are interpretable.
Constrain orders using plots, correlation patterns and the published modelling sequence. Inspect near-ties and prefer the simpler admissible candidate when forecast performance is indistinguishable. Record the candidate set and holdout before fitting so the selected model is not the product of repeated adaptation to the same evaluation observations.
What this chapter covers
- 01
Integrated process
- 02
Random walk
- 03
Ordinary difference
- 04
Seasonal difference
- 05
Seasonal ARIMA
- 06
Box-Jenkins identification
Worked example · free
Accumulate forecast differences
- 2Forecast each change.
- 3Accumulate to levels.
- 1Interpret uncertainty.
Key terms
- Integrated Process
- A process requiring differencing to obtain a stationary representation.
- Random Walk
- A level equal to its previous value plus a new innovation.
- Differencing Order
- The number of ordinary differences represented by d in ARIMA notation.
- Seasonal Period
- The number of observations in one repeating calendar cycle.
- Seasonal ARIMA
- An ARIMA model with additional seasonal orders at a specified period.
ARIMA and Seasonal Models FAQ
What does ARIMA(1,1,0) model?
It models first differences with first-order autoregressive dependence, then accumulates forecasts back to the level scale. Interpretation must therefore state that coefficients describe changes rather than raw levels.
Why can over-differencing hurt?
It can inject avoidable negative dependence and widen forecasts by transforming an already stable series. Always inspect the transformed plot and ACF before increasing the differencing order.
What seasonal period fits monthly annual data?
A period of twelve, because corresponding months are separated by twelve observations. Seasonal lags then occur at twelve, twenty-four and further annual multiples in the correlation structure.
Exam move
Translate every ARIMA order into words about transformation and dependence. Practise moving from levels to differences and back without skipping the accumulation step. Translate several ARIMA and seasonal notations into sentences naming the modelled scale and every order. For short series, calculate ordinary and seasonal differences by hand and rebuild the original level.
Sketch ACFs before and after a justified difference, watching for over-differencing. Design two parsimonious candidates from one transformed plot and compare residuals and validation errors. End by explaining how drift on the difference scale becomes trend after accumulation and where uncertainty enters the level forecast. Keep the final level reconstruction beside the differenced forecast.
Verify every accumulated or seasonal component uses the correct date, because a correct transformed forecast can return to the wrong level. State the final forecast scale in words beside every back-transformed numerical answer.
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