ECON4916 Chap.4 Autoregressive and Moving-Average Dynamics
Autoregressive and Moving-Average Dynamics
AR and MA models encode different forms of financial time dependence. An autoregression places lagged outcomes in the conditional mean, so one innovation travels forward through recursive powers of the AR coefficient. A finite moving average places current and lagged innovations directly in the observed value, producing a shock response that ends after the last included lag.
This chapter derives stationary AR moments, shows how the sign and magnitude of the coefficient determine reversion and oscillation, and calculates MA covariance by identifying shocks shared across dates. Forecast comparisons connect those population dynamics to the information set available at the forecast origin.
The final diagnostic section treats theoretical ACF patterns as candidate-generating clues and gives residual dependence, rather than a visual slogan, the final audit role.
What this chapter covers
- 01
AR recursion and innovation propagation
- 02
Stationary AR moments
- 03
AR conditional forecasts
- 04
MA covariance cutoffs
- 05
Stationarity and invertibility
- 06
Residual dependence diagnostics
Worked example · free
Stationary AR moments and forecast
- +1Verify that the absolute AR coefficient is 0.6, below one, so the stationary solution exists.
- +1Solve mu = 1.2 + 0.6mu to obtain the stationary mean mu = 3.
- +1Use variance 4 divided by 1 minus 0.6 squared to obtain 6.25.
- +1Use rho_2 = phi squared to obtain lag-two autocorrelation 0.36.
- +1Calculate 1.2 + 0.6 times 4 = 3.6 and verify that the forecast moves toward the mean from above.
Key terms
- Autoregression
- A dynamic specification whose conditional mean depends on lagged values of the modelled variable.
- Moving Average
- A dynamic specification built from current and finitely many lagged innovations.
- Innovation
- The new unpredictable component arriving at a date relative to the model's prior information.
- Shock Response
- The sequence of future effects generated by one innovation while later innovations are held at zero.
- Invertibility
- A restriction supporting a unique, stable recovery of innovations from the observed moving-average process.
- Forecast Error
- The realised future value minus the forecast constructed at an earlier information set.
Autoregressive and Moving-Average Dynamics FAQ
Why does an AR shock persist indefinitely?
Recursive substitution carries the innovation into every future outcome through powers of the AR coefficient. Under stationarity those powers shrink toward zero, so the response becomes small but does not reach a structural cutoff. A negative coefficient also alternates the response sign across successive horizons.
Why does an MA shock stop after a finite lag?
A finite moving-average equation includes only current and a fixed number of lagged innovations. Once the shock falls outside that listed window, its coefficient is exactly zero. This is a model cutoff, not merely a response that became numerically small.
Is invertibility the same as stationarity?
No. Stationarity concerns stable moments and shock propagation in the observed process. Invertibility concerns recovering a unique innovation sequence from observations. A finite MA process with finite-variance white noise is stationary even when a common invertibility magnitude restriction is not imposed.
Can the ACF identify the correct model by itself?
The population ACF provides useful patterns, such as geometric decay for a stationary AR model and cutoff for a finite MA model. A sample ACF contains estimation noise and mixed models can blur those shapes. Use it to propose candidates, then estimate, inspect residuals and validate forecasts.
Exam move
Study ARMA dynamics by tracing one innovation, not by memorising a catalogue of ACF pictures. For an AR equation, repeatedly substitute lagged outcomes until the pattern of coefficient powers is visible. Record the stationary condition, long-run mean, variance and autocorrelation beside that recursion.
For an MA equation, write adjacent dates next to each other and mark the innovations they share; the covariance becomes the product of coefficients on those shared terms times innovation variance. Keep stationarity and invertibility on separate lines of the notes sheet. Forecast practice should always identify the information set, recover any known current shock and set only future innovations to their conditional means.
Use signs as a fast audit: a negative AR coefficient alternates responses, and a negative MA coefficient can make the first autocorrelation negative. After estimation, inspect residual autocorrelation and remember that a clean residual ACF does not establish normality, constant conditional variance or structural stability.
Use MATLAB simulations to compare theory with sample paths, but derive at least one moment and one multi-step forecast by hand for every model form.
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