Time Seriesadvanced

The Time Series Assumptions (TS.1-TS.6)

OLS on time series is justified by the Gauss-Markov assumptions TS.1 through TS.5 for unbiasedness and efficiency, plus TS.6 (normality) for exact inference. The most demanding is strict exogeneity, $E(u_t \mid X) = 0$ , which says the error in every period is unrelated to the regressors in all periods, past, present, and future. Under TS.1-TS.3 OLS is unbiased; adding TS.4 (no serial correlation) and TS.5 (homoskedasticity) makes it BLUE. For large-sample inference these strong conditions relax to stationarity, weak dependence, and contemporaneous exogeneity, which together keep OLS consistent and asymptotically normal in realistic time-series data.

Why it matters

Strict exogeneity is a tall order because it rules out feedback: a shock to $y$ today must not influence future regressors. That fails whenever the model includes a lagged dependent variable, which is why the large-sample story matters. There we ask only that the series be stationary (its distribution does not drift over time) and weakly dependent (observations far apart are nearly unrelated), and those two together let the law of large numbers and central limit theorem do their job.

Formulas

Strict exogeneity (TS.3)

E(u_t \mid X) = 0, \quad t = 1, \dots, n

The error each period is mean-independent of regressors in every period, not just the current one.

No serial correlation (TS.4)

\mathrm{Corr}(u_t, u_s \mid X) = 0, \quad t \neq s

Errors are pairwise uncorrelated; combined with TS.5 (homoskedasticity) this delivers BLUE.

Weak dependence

\mathrm{Corr}(y_t, y_{t+h}) \to 0 \ \text{as} \ h \to \infty

Weak dependence lets the LLN and CLT apply to dependent data and relaxes strict exogeneity to contemporaneous exogeneity, but an exogeneity condition is still required.

Worked examples

Scenario

You estimate a static Phillips-curve regression and want to know whether the classical standard errors are trustworthy.

Solution

Run `regress inflation unemp` and then check the assumptions in turn. Strict exogeneity rules out feedback from inflation shocks to future unemployment, TS.4 rules out serially correlated errors, and TS.5 rules out heteroskedasticity. If TS.4 or TS.5 is doubtful, the usual standard errors are invalid even though the coefficients may still be unbiased.

NoteFailing TS.4 or TS.5 breaks inference, not unbiasedness, provided strict exogeneity holds.

Common mistakes

✗Strict exogeneity is the same as contemporaneous exogeneity. Strict exogeneity is much stronger: it requires no correlation with regressors in any period, including future ones.
✗A model with a lagged dependent variable can satisfy strict exogeneity. It generally cannot, because $u_t$ feeds into $y_t$ , which becomes a future regressor; we then rely on the weak-dependence large-sample results.
✗Violating TS.4 or TS.5 makes OLS biased. With strict exogeneity, OLS stays unbiased; what fails is the validity of the usual standard errors and tests.
✗Stationarity and weak dependence are the same property. Stationarity is about a stable distribution over time; weak dependence is about correlations dying out as observations move apart.

Revision bullets

•TS.1-TS.5 are the Gauss-Markov conditions for time series; TS.6 adds normality
•Strict exogeneity $E(u_t \mid X) = 0$ holds across all periods
•TS.1-TS.3 give unbiasedness; adding TS.4-TS.5 gives BLUE
•Large samples relax these to stationarity + weak dependence
•A lagged dependent variable breaks strict exogeneity

Quick check

Strict exogeneity in the time series model requires that the error $u_t$ be uncorrelated with:

In large samples, the strong TS assumptions are commonly replaced by:

Connected topics

Gauss-Markov Asymptotics Heteroskedastic TS Data Serial Corr.Unit Roots

Sources

Wooldridge (2019), §10.3 and Ch. 11
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
States TS.1-TS.6 and the finite-sample properties of OLS, then develops stationarity and weak dependence for the large-sample case.
Hamilton (1994), Ch. 7
Hamilton, James D. Time Series Analysis. Princeton University Press, 1994.
Graduate-level reference for stationarity, ergodicity, and the asymptotics underlying time-series OLS.

How to cite this page

Dr. Phil's Quant Lab. (2026). The Time Series Assumptions (TS.1-TS.6). Derivatives Atlas. https://phucnguyenvan.com/concept/efm-ts-assumptions

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