ETC3450 Chap.4 Autoregression, Moments and Forecasts

Autoregression, Moments and Forecasts

The AR(1) model combines an intercept, a lagged observation and a new innovation. Conditional on current information, the next mean is the intercept plus the autoregressive coefficient times the latest value. The innovation variance is conditional forecast uncertainty for one step, not the unconditional variance of the process.

When the autoregressive root is stable, repeated substitution makes the effect of initial conditions decay. The unconditional mean and variance then exist. As persistence approaches the boundary, shocks decay slowly and long-run variance becomes large. With a positive coefficient, a value above the long-run mean predicts another above-mean value, but closer to the centre when the coefficient is below one.

A negative coefficient implies alternating adjustment. Derive the long-run mean by equating expected values only after asserting stationarity. For variance, keep innovation independence explicit and reject a denominator that is nonpositive. Forecast reasoning distinguishes an information set from future innovations.

Write the projection for AR(1), set the conditional mean of the new disturbance to zero only when the assumptions justify it, and propagate persistence recursively. Uncertainty usually accumulates differently from the point forecast. Check this against the innovation assumptions. Root mean squared error summarises the magnitude of point forecast errors and penalises large misses.

Coverage asks how often realised values fall inside prediction intervals. A wide interval can attain nominal coverage while providing little precision, so calibration alone does not rank all forecasts. For an autoregression, a one-step forecast uses the latest value and one new innovation.

Longer horizons replace unknown future values recursively and accumulate innovation uncertainty while the conditional mean moves toward its long-run centre. A conditional forecast and an unconditional-mean forecast can both achieve approximate interval coverage when their intervals use the matching variance. The conditional forecast may still deliver smaller point errors because it uses recent information.

Report horizon, RMSE, coverage and average interval width together. Use an out-of-sample sequence and ensure estimates are based only on data available at each forecast origin. Model comparison should not reduce to selecting the largest fit statistic. Ask whether RMSE leaves serial structure in the residuals, whether coverage is parsimonious enough for the sample, and whether parameters remain stable over the forecast origin.

Interpret forecast horizon against a substantive dynamic mechanism. If rival models perform similarly, report that uncertainty and prefer a specification whose assumptions can be checked transparently. Separate three uncertainty layers around RMSE: future innovations, estimated parameters and uncertainty about the model form itself.

A standard conditional forecast may represent the first while treating coverage and the specification as fixed.

What this chapter covers

• 01

  AR(1) turns the latest level into a conditional mean

• 02

  Forecast quality has accuracy and calibration dimensions

When the forecast depends on AR(1) turns the latest level into a conditional mean

Key terms

AR(1) conditional mean

AR(1) turns the latest level into a conditional mean — The AR(1) model combines an intercept, a lagged observation and a new innovation. Conditional on current information, the next mean is the intercept plus the autoregressive coefficient times the latest value. The innovation variance is conditional forecast uncertainty for one step, not the unconditional variance of the process. Derive the long-run mean by equating expected values only after asserting stationarity. For variance, keep innovation independence explicit and reject a denominator that is nonpositive.

Forecast accuracy and calibration

Forecast quality has accuracy and calibration dimensions — Root mean squared error summarises the magnitude of point forecast errors and penalises large misses. Coverage asks how often realised values fall inside prediction intervals. A wide interval can attain nominal coverage while providing little precision, so calibration alone does not rank all forecasts. Report horizon, RMSE, coverage and average interval width together. Use an out-of-sample sequence and ensure estimates are based only on data available at each forecast origin.

Autoregression, Moments and Forecasts FAQ

Exam move

Keep a derivation and diagnostic sheet for Autoregression, Moments and Forecasts. Write every process assumption before calculating a moment, response or forecast. Begin with AR(1) and reconstruct the reasoning without looking at the worked response. Then change one condition in the example and decide whether persistence still explains the outcome.

Use the chapter questions to compare direct observation with inference, and write the strongest rival account in full. Before closing the chapter, return to forecast horizon and state the precise boundary it places on transfer. Check that every conclusion names an observable consequence and that uncertainty is attached to the step it affects.

A final retrieval pass should be fast enough to reproduce the method from headings and diagrams while leaving the detailed prose for checking nuance.

ETC2410 Introductory Econometrics · ETC2420 Statistical Thinking · ETC3460 Financial Econometrics