MATH7970 Chap.6 Moving Average and ARMA Models
Moving Average and ARMA Models
Finite innovation memory
An MA process combines current and recent innovations, so one shock has a direct effect for a fixed number of observations. Invertibility selects a recoverable representation.
Mixed dependence
ARMA models combine recursive feedback and finite shock terms.
Both ACF and PACF can tail off, making estimation, residual checks and parsimony more important than mechanical pattern matching.
Map innovations by date
An MA equation becomes clearer in a grid with observation dates across columns and innovations down rows. Mark where one shock enters and where it disappears.
Adjacent observations correlate while their innovation windows overlap; beyond q lags they share no shocks under the white-noise assumptions. This construction explains the ACF cutoff and prevents vague claims that moving average means averaging observed values.
Preserve the equation convention
Some notes write lagged innovations with plus signs and others with minus signs.
Equivalent models then carry parameters with opposite signs. Copy the working equation before deriving covariance, forecasting or checking invertibility. An answer that imports a memorised sign rule from another convention can reverse the autocorrelation while every later calculation looks internally tidy.
Use invertibility as identification
Different MA parameterisations can generate the same autocovariances.
Invertibility selects the representation whose innovations can be recovered stably from observed history, which makes estimation and forecasting operational. A non-invertible equation may still generate a process, but its shock sequence is not the unique recoverable innovation representation.
Check the relevant roots before interpreting a coefficient as a particular shock response.
Keep mixed models parsimonious
ARMA combines recursive feedback with finite shock memory when both mechanisms remain. ACF and PACF may tail off together, so orders are not read mechanically. Estimate a small candidate set, inspect root conditions and residuals, and compare held-out forecasts.
Near cancellation between AR and MA terms produces unstable estimates and a model that behaves like a simpler process.
Prefer the smaller representation when its diagnostics and forecasting are equally credible.
Check recoverability with forecasts
After estimating an invertible MA or ARMA model, reconstruct recent innovations from observed history and verify that their signs and magnitudes produce the fitted values. Then advance the forecast origin and watch known innovation terms leave the equation.
This exercise connects an abstract root condition with the practical information used at each horizon. If large reconstructed shocks appear only because AR and MA terms nearly cancel, compare a simpler specification and inspect whether its innovations are more stable while residual and validation performance remain comparable.
What this chapter covers
- 01
MA innovation filter
- 02
Finite memory
- 03
Invertibility
- 04
ACF cutoff
- 05
ARMA mixture
- 06
Parameter cancellation
Worked example · free
Compute an MA path
- 3Calculate x_t.
- 3Calculate x_{t+1} and interpret memory.
Key terms
- Moving Average Process
- A process formed from current and finitely many lagged innovations.
- Invertibility
- A condition allowing innovations to be recovered from observed history.
- Finite Memory
- A direct shock effect ending after the model's largest MA lag.
- ARMA Model
- A stationary model combining autoregressive and moving-average terms.
- Cancellation
- Near-offsetting AR and MA effects that can hide a simpler representation.
Moving Average and ARMA Models FAQ
How long does an MA shock last?
An MA(q) innovation affects its arrival observation and the next q observations, then leaves the direct representation. Adjacent observations can still correlate while their innovation sets overlap.
Why is invertibility needed?
It selects an identifiable representation whose innovations can be reconstructed from the observable series. Without it, different parameterisations can imply the same observed covariance pattern.
What suggests an ARMA model?
Gradual patterns in both ACF and PACF can motivate a mixed candidate, but residual evidence must confirm the choice. Near cancellation between terms is a warning that a simpler model may suffice.
Exam move
Draw which innovations enter each adjacent observation. Then mark when one chosen shock disappears and connect that overlap to the ACF cutoff. Make an innovation-date grid for MA(1), MA(2) and one ARMA process. Shade every observation touched by a chosen shock and use overlaps to derive where autocovariance becomes zero. Check the sign convention before comparing formulas.
For the mixed model, inspect whether AR and MA terms nearly cancel and whether a simpler candidate leaves equally clean residuals. Forecast enough horizons to see finite MA information disappear while AR feedback decays, then explain the contrast without notation alone. Add a final row stating when the chosen innovation disappears.
This distinguishes finite moving-average memory from an autoregressive tail more clearly than another page of notation. Compare reconstructed innovations across candidates before interpreting any individual shock coefficient.
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