ECMT6002 Chap.7 Time Series Models and Forecasting
Time Series Models and Forecasting
Time series methods use the ordering of observations rather than treating rows as exchangeable. This chapter develops autoregressive and distributed lag models, stability and stationarity, one step and multi step forecasts, and forecast errors. Dynamic multipliers distinguish an immediate effect from the accumulated long run response.
Model evaluation must use information genuinely available at the forecast origin and preferably an out of sample comparison. Weak stationarity stabilises the mean, variance and lag-dependent covariance structure. A moving-average process transmits each innovation for a finite number of periods, producing an autocorrelation cutoff, while a stationary autoregression recycles past outcomes and produces gradual geometric decay.
A unit root breaks that mean-reverting pattern because shocks do not fade and variance grows through time. Differencing can shift the target from a level to a change; log differencing often yields an approximate growth rate. Dynamic regression separates the impact response from delayed effects and sums a stable propagation path into a long-run multiplier. Forecasting applies the same dynamics conditionally.
The one-step forecast uses observed lags, but longer horizons must substitute earlier forecasts for future values that are not yet observed. Forecast errors need a declared sign convention and a genuine information set. In-sample fit is insufficient for selecting a prediction rule because it rewards flexibility used during estimation.
Rolling or held-out forecasts compare candidate models on common dates with a loss function chosen for the costs of the application.
What this chapter covers
- 01
Lag operators and dynamic equations
- 02
Autoregressive models
- 03
Stability and stationarity
- 04
Dynamic multipliers
- 05
Recursive forecasting
- 06
Forecast errors and loss
Worked example · free
Two step forecasting in an AR model
- +1Use the observed current value for the one step forecast: 2 + 0.5 times 6 = 5.
- +1For the next horizon, the future lag is not observed at the forecast origin.
- +1Insert the one step forecast recursively: 2 + 0.5 times 5.
- +1The two step forecast equals 4.5 and moves toward the long run mean of 4.
Key terms
- Autoregression
- A time series model using lagged values of the dependent variable as predictors.
- Stationarity
- Stability of specified distributional features of a process over time.
- Dynamic multiplier
- The effect of a regressor change at a specified horizon in a dynamic model.
- Forecast error
- The realised outcome minus its forecast under a stated sign convention.
Time Series Models and Forecasting FAQ
Why do forecasts converge to the long run mean?
In a stable autoregression, repeated powers of the autoregressive coefficient shrink toward zero. The influence of the current deviation therefore fades while the intercept accumulation approaches the unconditional mean.
What is the difference between a residual and forecast error?
A fitted residual is calculated using a model and sample fit, often in sample. A genuine forecast error compares a future realised outcome with a prediction made from the information available at the forecast origin.
Why prefer out of sample evaluation?
In sample fit rewards flexibility used to estimate the model and can overstate predictive performance. A held out or rolling forecast comparison better measures how the model behaves on observations not used for fitting.
How do I avoid information leakage in forecasting?
Mark the forecast origin and list what is known on that date. At longer horizons, replace unavailable future lags with earlier forecasts rather than realised values. Estimate every competing model without the evaluation outcomes and score them on identical dates, horizons and loss functions. Otherwise the reported accuracy benefits from information unavailable to a real forecaster.
Exam move
Draw a time line for every forecast problem and mark what is known at the forecast origin. Memorise the stability condition through its implication for repeated powers, not as an isolated inequality. Practise recursive forecasts and long run multipliers side by side, then evaluate competing models with the same forecast window and loss function.
Build an ACF recognition sheet that contrasts white noise, finite moving-average cutoffs, autoregressive decay and unit-root persistence. Derive the stationary AR mean by taking expectations and its forecast path by repeated substitution, so both formulas share the same logic. When differencing, write the units of the transformed series and explain how level forecasts would be reconstructed.
Generate a multi-step forecast in a table with columns for horizon, available lag, inserted forecast and resulting prediction; this prevents future-value leakage. Score two models with squared and absolute error to see how loss functions can select different winners. Keep estimation and evaluation samples visibly separate, and use the same dates for every competitor.
For distributed lags, list impact, interim and cumulative effects rather than reporting only one coefficient. End each exercise by saying whether the result describes a level, change, growth rate, conditional mean or realised forecast error.
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