ECON4916 Chap.2 Stationarity, Dependence and Forecasts
Stationarity, Dependence and Forecasts
A time series supplies one realised path through an ordered collection of random variables. Inference becomes possible only by placing defensible structure across dates. This chapter contrasts strict and weak stationarity, explains why ergodicity is needed for sample averages to learn population moments, and derives autocovariance and autocorrelation as lag-based dependence measures.
Moving-average examples show how shared innovations create covariance, while conditional-expectation arguments establish the optimal forecast under squared loss. The chapter repeatedly separates population properties from noisy sample estimates and attaches every forecast to an information set.
That discipline prevents three common errors: treating a stable-looking chart as proof of stationarity, erasing a currently known innovation, and reading a sample autocorrelation spike as a complete model diagnosis.
What this chapter covers
- 01
Time series and observed sample paths
- 02
Strict and weak stationarity
- 03
Ergodicity and time averages
- 04
Autocovariance and autocorrelation
- 05
Conditional expectations
- 06
Mean-squared-error forecasts
Worked example · free
Stationarity of a moving-average process
- +1Use the zero innovation mean to obtain E[X_t] = 2 for every date.
- +1Add independent shock variances: gamma_0 = 9 + 0.4 squared times 9 = 10.44.
- +1Identify the shared u_(t-1) term to obtain gamma_1 = 0.4 times 9 = 3.6.
- +1At lags of two or more the expressions share no innovation, so gamma_k = 0.
- +1Conclude that the finite mean is constant and covariance depends only on lag, establishing weak stationarity.
Key terms
- Sample Path
- One realised sequence drawn from the random variables that constitute a time series.
- Strict Stationarity
- Invariance of every finite joint distribution to a common shift in calendar time.
- Weak Stationarity
- Finite second moments, a constant mean and covariance determined only by lag.
- Ergodicity
- A property limiting temporal dependence enough for time averages to converge toward population moments.
- Autocovariance
- The covariance between two values of a stationary process separated by a specified lag.
- Information Set
- The collection of observations and variables available to construct a forecast at a given origin.
Stationarity, Dependence and Forecasts FAQ
Does a flat-looking plot prove stationarity?
No. A chart may suggest a stable mean while hiding changes in variance, dependence or higher moments. Weak stationarity is a population claim about finite second moments, constant mean and lag-based covariance. Use plots for diagnosis, then examine moments, breaks and model assumptions rather than treating appearance as proof.
Why is ergodicity different from stationarity?
Stationarity provides time invariance, while ergodicity controls dependence strongly enough for observations along one path to reveal population features. A stationary process can retain dependence that prevents a time average from converging usefully. In short, stationarity makes moments comparable across dates; ergodicity helps sample moments learn them.
Why is conditional expectation the optimal forecast?
Under squared-error loss, the conditional forecast error separates into irreducible conditional variance plus the squared distance between the chosen forecast and the conditional mean. The first part cannot be reduced with current information, and the second is smallest when the forecast equals the conditional expectation.
Can a sample autocorrelation equal the population value?
It can by chance, but exact equality is not expected in a finite sample. A sample autocorrelation is an estimator and varies across realised paths. Long-lag estimates often use fewer overlapping pairs and can be noisy. Compare the pattern with model implications and inspect residual dependence after estimation.
Exam move
Use a three-column time-series notebook. In the first column, write the population process and derive its mean, variance and autocovariance. In the second, record what is actually observed in one sample path and which sample analogue estimates each population quantity. In the third, list the conditions needed for that estimator or forecast to work.
This layout prevents strict stationarity, weak stationarity and ergodicity from collapsing into one vague stability claim. Practise moving-average covariance by expanding adjacent dates and identifying shared innovations. Practise conditional forecasts by listing the information set and crossing out only future zero-mean shocks, never the currently recovered shock.
When reading a sample ACF, state the population pattern expected under the candidate model and the sampling uncertainty that can disturb it. Use MATLAB to plot paths and correlograms, but verify one or two lags by hand. Close each answer with a qualified statement such as weakly stationary under the stated white-noise assumptions or optimal under squared loss given the specified information set.
These qualifications are part of the result, not optional decoration.
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