Lesson M08.L01: The Nature of Time Series Data

Field	Detail
Module	08 — Time Series Basics
Level	Graduate (ECMT5001 S1 2027)
Duration	30–35 min
Learning Objective	Understand how time series data differ from cross-sections; define strict vs contemporaneous exogeneity; introduce the finite distributed lag model and lag operator notation.
Provenance	Wooldridge, Introductory Econometrics, Ch 10.1

Explanation

1. What Is Time Series Data?

A time series is a sequence of observations on one or more variables taken at equally spaced points in time:

\[\{y_t\}_{t=1}^{T} = \{y_1, y_2, \ldots, y_T\}\]

Examples from economics and finance:

Series	Frequency	Key feature
Australian GDP growth (%)	Quarterly	Mean-reverting, business-cycle fluctuations
RBA cash rate (%)	Monthly	Persistent, slow-moving
CPI inflation (%)	Monthly	Some persistence; mean-shifts over policy regimes
Unemployment rate (%)	Monthly	Persistent; rises fast, falls slowly
ASX 200 daily returns	Daily	Near-serially-uncorrelated, volatility clustering

2. Key Difference from Cross-Sectional Data

In a cross-section, observations are drawn independently from a population: knowing respondent $i$'s income tells us nothing about respondent $j$'s income.

In a time series, observations are temporally ordered and dependent: today's GDP growth, interest rate, and inflation all depend on yesterday's values. This serial dependence (autocorrelation) is the defining feature of time series, and it fundamentally changes what assumptions are required for valid OLS inference.

Formally, for a stationary time series we expect:

\[\text{Cov}(y_t, y_{t-h}) \neq 0 \quad \text{for small } h\]

This means the i.i.d. assumption that underlies cross-sectional OLS asymptotics does not hold for time series without additional structure.

3. Strict Exogeneity

In cross-sectional OLS, the key assumption is contemporaneous exogeneity: $E(u_i \mid x_i) = 0$.

For time series, the stronger assumption of strict exogeneity requires that the error at time $t$ is uncorrelated with every regressor at every time period — past, present, and future:

\[E(u_t \mid x_1, x_2, \ldots, x_T) = 0 \quad \text{for all } t\]

This means: - $u_t$ is uncorrelated with current $x_t$ (contemporaneous). - $u_t$ is uncorrelated with past regressors $x_{t-1}, x_{t-2}, \ldots$ (no feedback from past errors to current regressors). - $u_t$ is uncorrelated with future regressors $x_{t+1}, x_{t+2}, \ldots$ — the most demanding part.

Why is strict exogeneity so strong? Future regressors must be unaffected by current errors. This rules out: - Feedback systems: If $y$ affects $x$ in subsequent periods (e.g. income today affects savings next period, which affects income next period), strict exogeneity fails. - Lagged dependent variables as regressors: $y_{t-1}$ as a regressor automatically violates strict exogeneity because $u_{t-1}$ affects $y_{t-1}$, which is a regressor at time $t$.

Under strict exogeneity, OLS is unbiased for finite $T$. Under weaker (contemporaneous) exogeneity, OLS is consistent as $T \to \infty$ but not unbiased in finite samples.

4. Economic Time Series: Autocorrelation Is the Norm

Consider the RBA cash rate ($r_t$). After the RBA sets the rate in period $t$, it tends to keep it at a similar level next period — changing rates by 25 basis points at a time, infrequently. Correlation between $r_t$ and $r_{t-1}$ is typically above 0.95. This is not noise — it reflects the persistence of monetary policy.

Similarly, Australian GDP growth ($g_t$) is positively autocorrelated at short lags (expansions last multiple quarters) but can show negative autocorrelation at business-cycle frequency.

The practical implication: regressing one time series on another without accounting for this persistence can produce misleading results (see M08.L03 on spurious regression).

5. Finite Distributed Lag (FDL) Models — Preview

A finite distributed lag model allows current and past values of a regressor $z$ to affect the current outcome $y$:

\[y_t = \alpha + \delta_0 z_t + \delta_1 z_{t-1} + \delta_2 z_{t-2} + \cdots + \delta_q z_{t-q} + u_t\]

This is natural in economics: a change in the interest rate today affects consumption over the next several quarters, not just this quarter. The subscripts on $\delta_j$ represent the lag — how many periods back the effect applies.

Full treatment of FDL models is in M08.L02.

6. The Lag Operator

The lag operator $L$ (also written $B$ in some textbooks) is a convenient notation for time-shifted series:

\[L y_t = y_{t-1}$$ $$L^2 y_t = y_{t-2}$$ $$L^j y_t = y_{t-j}\]

The first difference operator $\Delta$:

\[\Delta y_t = y_t - y_{t-1} = (1 - L) y_t\]

Using lag operator notation, the FDL model becomes:

\[y_t = \alpha + (\delta_0 + \delta_1 L + \delta_2 L^2 + \cdots + \delta_q L^q) z_t + u_t = \alpha + \delta(L) z_t + u_t\]

where $\delta(L) = \sum_{j=0}^q \delta_j L^j$ is a lag polynomial.

Worked Example

Setting up an Australian monetary policy time series model.

Suppose we want to model Australian headline CPI inflation ($\pi_t$, %) as a function of the RBA cash rate ($r_t$, %). Using monthly RBA data (January 2000 – December 2023):

Static model:

\[\pi_t = \alpha + \beta_0 r_t + u_t\]

Problem with the static model: Rate changes take time to filter through the economy (via bank lending rates, mortgage repayments, business investment, etc.). The contemporaneous relationship alone is insufficient.

FDL model (12 lags):

\[\pi_t = \alpha + \sum_{j=0}^{12} \delta_j r_{t-j} + u_t\]

Here: - $\delta_0$: immediate effect of a 1 pp rate rise on inflation this month. - $\delta_3$: effect 3 months later. - $\sum_{j=0}^{12} \delta_j$: total (long-run) effect after one year.

Strict exogeneity concern: Does the RBA set $r_t$ partly in response to past inflation errors $u_{t-1}$? Almost certainly yes — the RBA targets inflation. This means strict exogeneity is likely violated; OLS may still be consistent (under sequential exogeneity), but finite-sample properties are less clean.

Common Misconception

"OLS works the same for time series as for cross-sectional data."

This is false in several important ways:

Temporal dependence invalidates i.i.d.-based inference. Standard errors computed assuming i.i.d. errors are incorrect in the presence of serial correlation. You need HAC (heteroskedasticity and autocorrelation consistent) standard errors or GLS.
Strict exogeneity is a much stronger assumption than contemporaneous exogeneity and often fails in feedback models common in macroeconomics.
Non-stationarity creates entirely new problems (spurious regressions, invalid t-statistics, diverging variance) that do not arise in cross-sections. See M08.L03 and M08.L04.
Finite-sample distributions of OLS estimators may differ from their cross-sectional counterparts even when asymptotic properties hold.

Time series econometrics requires its own toolkit: HAC SEs, unit root tests, cointegration, ARIMA/VAR models.

Practice Prompts

Strict vs contemporaneous exogeneity. An analyst estimates a model where quarterly retail sales ($y_t$) are regressed on the consumer sentiment index ($x_t$). (a) What does strict exogeneity require of the error $u_t$ relative to all values of $x$? (b) Explain why strict exogeneity might be violated if the RBA uses retail sales data to set interest rates, which in turn affect consumer sentiment. (c) Is OLS necessarily inconsistent if strict exogeneity fails? What weaker assumption would still give consistency?
Lag operator practice. Let $y_t = 2 + 0.6 y_{t-1} + e_t$ (an AR(1) process). (a) Express this model using the lag operator $L$. (b) Write out $\Delta y_t$ using $L$. (c) If $y_{100} = 5.0$ and $e_{101} = 0.2$, compute $y_{101}$.
FDL interpretation. A researcher estimates a quarterly FDL model of the unemployment rate ($u_t$) on GDP growth ($g_t$) with 4 lags:

$$u_t = 6.1 - 0.3 g_t - 0.5 g_{t-1} - 0.4 g_{t-2} - 0.2 g_{t-3} + \text{error}$$

(a) Interpret $\delta_0 = -0.3$. (b) Interpret $\delta_2 = -0.4$. (c) Compute the long-run propensity (LRP). What does it mean?

Further Resources

Wooldridge, Introductory Econometrics (7e), Ch 10 — Basic Regression Analysis with Time Series Data.
RBA Statistical Tables (available at rba.gov.au) — for Australian cash rate, CPI, and GDP series.
Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press. — Graduate-level reference.
Stock, J.H. & Watson, M.W. (2020). Introduction to Econometrics (4e), Ch 14–16 — complementary time series treatment.