MATH7970 Chap.3 Trend, Seasonality and Stationarity
Trend, Seasonality and Stationarity
Diagnose before transforming
Trend changes long-run level, seasonality repeats on a known calendar and stochastic trend accumulates shocks. The plot should determine which transformation is plausible.
Stable probabilistic structure
Weak stationarity requires stable mean and variance and covariance governed by lag.
It is not the same as a flat graph, and one unnecessary difference can create dependence rather than remove it.
Let the plot name the problem
Mark the opening and closing level, repeated peaks, changing amplitude and known intervention dates. Trend, seasonality, variance and breaks are separate observations.
A policy change can shift a deterministic mean without creating a stochastic trend, while a measurement redesign can change variance without altering the underlying process.
One global model is doubtful when the definition or mechanism changes across a visible break, and ordinary transformation cannot repair that mismatch.
Distinguish permanent from transitory movement
A deterministic trend leaves deviations around a fixed function of time, so removing that function can stabilise the mean.
A stochastic trend accumulates innovations, giving shocks permanent effects and motivating differencing. Short samples can make both paths look similar. Use ACF decay, behaviour after shocks and forecast uncertainty to support the choice.
A random-walk level has increasingly uncertain long-horizon forecasts even when its expected change is simple.
Seasonal operations have different jobs
Seasonal differencing removes a repeating level relation by comparing corresponding points one cycle apart. Seasonal AR and MA terms describe dependence that remains after the seasonal level has been handled.
Apply the difference only when the plot and correlations justify it, then inspect the transformed series again.
Repeated spikes at the seasonal lag can reflect remaining dependence rather than a need for another difference, so transformation and modelling decisions should not be collapsed.
Variance and mean require separate checks
When fluctuations grow with the level, a log or related transformation may stabilise relative variation. That does not automatically remove trend or seasonality.
Compare the beginning and end of the transformed series, then examine its correlation pattern. Report what the operation removed and what remained. The target is stable moments with interpretable dependence, not a visually flat line.
If the series definition changed, model segments or intervention effects rather than treating the break as noise.
Use domain timing to check seasonality
A numerical peak every fixed number of observations is only statistically seasonal when that lag corresponds to a plausible calendar mechanism. Trading days, school terms and holidays may create unequal or shifting intervals that a simple period does not capture.
Write the sampling frequency and seasonal period together, mark missing dates and check whether calendar alignment is preserved. When seasonal timing changes after a break, a single difference at the old period can leave structured residuals and mislead later identification even though the arithmetic transformation was performed correctly.
What this chapter covers
- 01
deterministic trend
- 02
stochastic trend
- 03
seasonal period
- 04
weak stationarity
- 05
ordinary differencing
- 06
seasonal differencing
Worked example · free
Remove quarterly seasonality
- 4Use lag four pairs.
- 2Describe the remaining movement.
Key terms
- Deterministic Trend
- A systematic mean path represented by a function of time.
- Stochastic Trend
- A level whose shocks accumulate permanently through time.
- Seasonality
- A recurring pattern tied to a fixed calendar period.
- Weak Stationarity
- Constancy of mean and variance with covariance depending only on lag.
- Over-differencing
- Applying more differencing than the structure requires and inducing avoidable dependence.
Trend, Seasonality and Stationarity FAQ
Does a stationary series have to look flat?
No. It can fluctuate sharply while retaining stable moments and lag dependence. Stationarity is probabilistic structure, not visual calm. The time plot and lag dependence must both support the claimed stable representation.
When is seasonal differencing useful?
It is useful when corresponding positions in consecutive seasons share a repeating level pattern that remains after other transformation. Inspect the remainder to verify that the calendar pattern was actually reduced.
How can over-differencing be detected?
A strong negative lag-one correlation and unnecessarily noisy forecasts are common warning signs after too many differences. Forecast variance may also grow without a corresponding gain in residual cleanliness.
Exam move
For each plot, write trend, seasonality and variance observations separately. Predict what ordinary, seasonal or log transformation will remove, then inspect the result again. Annotate several real series with level, trend, seasonality, variance and breaks. For each, write the feature a proposed transformation should remove before applying it.
Compare transformed early and late segments, then inspect the ACF for slow decay or seasonal spikes. Practise explaining why deterministic trend calls for detrending while accumulated shocks call for differencing. Include one counterexample where a structural break makes a single stationary model misleading. This turns transformation into a testable claim rather than a routine menu choice.
Keep original and transformed plots side by side. Annotate which feature changed and which remained, so later model selection cannot forget the scale and calendar relation created during preprocessing. Note any break that makes comparisons across the full sample scientifically doubtful.
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