ETC3450 Chap.2 Moments, Dependence and Stationarity

Moments, Dependence and Stationarity

The mean locates the process, variance measures dispersion at a time, and autocovariance connects observations separated by a lag. Time indexes cannot be dropped casually: a nonstationary process can have a mean or variance that changes with the date, making one global summary misleading. Dividing autocovariance by the appropriate standard deviations creates a unit-free coefficient.

The sign describes direction and magnitude describes linear association, not causation. Sample estimates remain noisy, particularly at long lags with fewer paired observations. Two series share the same marginal mean and variance, yet one alternates signs and the other forms long runs. Their autocovariances expose dynamics that unordered summaries cannot see. Write the pair of times behind every covariance.

Before interpreting a lag statistic, check that the relevant moments exist and that stationarity assumptions justify pooling across dates. Econometric interpretation begins by fixing the stochastic object behind expected value: its conditioning information, time index and maintained assumptions. Derive the implication for variance before inserting estimated values, because a numerical answer can conceal a wrong model.

Diagnostics then ask whether residual behaviour is compatible with the specification. Weak stationarity requires the unconditional mean and variance not to depend on calendar time, while covariance depends only on the separation between observations. It does not require every realised path to look flat or every distributional feature to be constant. A stationary series can contain long runs or extreme values by chance.

Conversely, a visually calm segment of a random walk remains generated by a nonstationary process. Theory and diagnostics must be read together. An AR process with a coefficient close to one may appear to wander for a finite sample although it is stationary. A random walk may temporarily oscillate near zero although its variance grows with time. Test the stated model conditions before judging the plot.

If moments depend on initial conditions or time, say which stationarity requirement fails rather than using 'nonstationary' as an unexplained label. A time-series calculation should be accompanied by a probability statement. For weak stationarity, separate a realised path from the process that could generate many paths. State which moments are stable and how constant variance changes with the lag.

When the result depends on lag dependence, identify the parameter region and initial condition explicitly. The limiting case often reveals whether the claimed long-run interpretation is mathematically coherent. Turn weak stationarity into a diagnostic sequence. Plot the series and transformations, inspect dependence over several lags, estimate a defensible candidate and then examine the residuals for structure left behind.

A conditional mean uses the current information set and changes with observed history. An unconditional mean averages over possible histories and, for a stationary process, describes its long-run centre. The conditional variance measures uncertainty around a forecast; the unconditional variance describes dispersion of the process itself. A long stationary ergodic realisation can support estimation of unconditional moments.

It does not estimate a conditional mean that varies with the latest lag unless the conditioning relation is modelled. For a stationary AR(1), the next expected value depends on the current observation, while the long-run mean is constant. Forecasting every date with the long-run mean ignores useful lag information even though the mean itself is correct.

Write conditioning bars and information sets until the distinction becomes automatic. Compare a forecast error with conditional variance, not with the wider unconditional variance. Forecast reasoning distinguishes an information set from future innovations.

Write the projection for conditional mean, set the conditional mean of the new disturbance to zero only when the assumptions justify it, and propagate unconditional mean recursively. Uncertainty usually accumulates differently from the point forecast. Use ergodicity to explain what the interval covers and why a narrow interval cannot rescue a misspecified dynamic model.

Compare two specifications for conditional mean using the same estimation sample and forecast origin. Ask whether adding unconditional mean removes a residual pattern, changes uncertainty materially or merely improves in-sample fit.

What this chapter covers

• 01

  Moments describe level, dispersion and co-movement

• 02

  Weak stationarity preserves selected moments

• 03

  Conditional and unconditional moments serve different tasks

What to derive after Moments describe level, dispersion and co-movement

Key terms

Mean, variance and covariance

Moments describe level, dispersion and co-movement — The mean locates the process, variance measures dispersion at a time, and autocovariance connects observations separated by a lag. Time indexes cannot be dropped casually: a nonstationary process can have a mean or variance that changes with the date, making one global summary misleading. Write the pair of times behind every covariance. Before interpreting a lag statistic, check that the relevant moments exist and that stationarity assumptions justify pooling across dates.

Weak stationarity

Weak stationarity preserves selected moments — Weak stationarity requires the unconditional mean and variance not to depend on calendar time, while covariance depends only on the separation between observations. It does not require every realised path to look flat or every distributional feature to be constant. Test the stated model conditions before judging the plot. If moments depend on initial conditions or time, say which stationarity requirement fails rather than using 'nonstationary' as an unexplained label.

Conditional and unconditional moments

Conditional and unconditional moments serve different tasks — A conditional mean uses the current information set and changes with observed history. An unconditional mean averages over possible histories and, for a stationary process, describes its long-run centre. The conditional variance measures uncertainty around a forecast; the unconditional variance describes dispersion of the process itself. Write conditioning bars and information sets until the distinction becomes automatic. Compare a forecast error with conditional variance, not with the wider unconditional variance.

Moments, Dependence and Stationarity FAQ

Exam move

Keep a derivation and diagnostic sheet for Moments, Dependence and Stationarity. Write every process assumption before calculating a moment, response or forecast. Begin with expected value and reconstruct the reasoning without looking at the worked response. Then change one condition in the example and decide whether variance 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 ergodicity 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.

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