Time Series

Lecture 8

What is time series data?

A single unit observed repeatedly over time.

Indexed by t = 1, 2, ..., T. Order matters, this is not a random sample.
Examples: quarterly GDP, monthly unemployment, daily stock prices, annual crime rates.

The key challenge: observations are not independent.

Today's value is typically correlated with yesterday's value (autocorrelation).
OLS remains unbiased under autocorrelation, but standard errors are wrong, just like heteroskedasticity.

Two additional complications beyond cross-section.

Trends: many economic series drift upward over time, creating spurious correlations.
Seasonality: systematic within-year patterns (retail sales, weather-sensitive output).

Stationarity is the time-series analogue of having a stable population to draw from.

Stationary

Mean, variance, and autocovariances do not depend on t. The series fluctuates around a fixed level. Classical inference applies.

Non-stationary

Mean or variance drifts over time. Series may wander without bound. Regressions can be spurious. Standard t-statistics are unreliable.

Weak (covariance) stationarity requires: (1) E[Y_t] = μ for all t, (2) Var(Y_t) = σ² for all t, (3) Cov(Y_t, Y_t−k) depends only on k, not on t.

Most asymptotic theory for time series requires stationarity (or a transformation to achieve it). Always plot your series and check before running regressions.

Autocorrelation: the correlation between a series and its own past values.

ρ_k = Corr(Y_t, Y_t−k) = Cov(Y_t, Y_t−k) / Var(Y_t)

k is the lag. The autocorrelation function (ACF) plots ρ_k against k. A slowly decaying ACF signals a non-stationary or long-memory series.

Consequences for OLS: With autocorrelated errors, the OLS estimator is still unbiased and consistent, but the usual standard errors are biased downward, t-statistics are too large, leading to over-rejection of the null.

Breusch-Godfrey test: regress ê_t on X_t and p lags of the residual. The nR² statistic is χ²(p). Preferred to the Durbin-Watson test because it works with lagged dependent variables.

The AR(1) is the simplest time series model.

Y_t = ρY_t−1 + e_t, e_t ~ i.i.d.(0, σ²)

ρ is the autocorrelation coefficient. It controls how much today's value depends on yesterday's.

If |ρ| < 1: the series is stationary. Shocks dissipate over time.
If ρ = 1: unit root. The series is a random walk, shocks are permanent.
If |ρ| > 1: explosive. The series diverges. Rare in economic data.

An AR(p) model includes p lags: Y_t = ρ₁Y_t−1 + … + ρ_pY_t−p + e_t. Use AIC/BIC to select p.

A random walk has no tendency to revert to any long-run mean.

Y_t = Y_t−1 + e_t ⇒ Y_t = Y₀ + ∑_s=1^t e_s

Variance grows without bound: Var(Y_t) = tσ². This violates stationarity. Any shock is permanently embedded in all future values.

Economic examples of likely unit roots: GDP, price levels, nominal exchange rates, asset prices. Their log-differences (growth rates, returns) are typically stationary.

A random walk with drift adds a constant: Y_t = μ + Y_t−1 + e_t. The series trends upward (or downward) on average while still wandering.

If two series both have unit roots, regressing one on the other can produce a high R² and a significant t-statistic even when they are completely unrelated.

Spurious regression: non-stationary series that share a common trend look correlated, but the relationship is meaningless.

Granger and Newbold (1974) showed that regressing two independent random walks typically produces R² > 0.9 and |t| > 2 even when the true coefficient is zero. As T ︎→︎ ∞, the t-statistic diverges rather than converging to its null distribution.

Warning

A high R² with non-stationary variables is a red flag, not evidence of a real relationship. Always test for unit roots before regressing time series on each other.

The classic example: regressing the number of Nicolas Cage films on US drowning deaths yields a near-perfect fit. Both series happen to trend together by coincidence.

Remedy: either difference the series to induce stationarity, or test for cointegration before treating the levels regression as meaningful.

The Augmented Dickey-Fuller (ADF) test tests the null hypothesis of a unit root.

Rewrite the AR(1) by subtracting Y_t−1 from both sides:

ΔY_t = δY_t−1 + e_t, δ = ρ − 1

H₀: δ = 0 (unit root) vs. H₁: δ < 0 (stationary).

The augmented version adds lags of ΔY_t to absorb serial correlation in the residuals. The test statistic does not follow a standard t-distribution under H₀, use Dickey-Fuller critical values.

Variants: include a constant (drift), or a constant and a trend. Match the specification to the behavior of the series.

KPSS test reverses the null: H₀ is stationarity. Using both ADF and KPSS together gives more confidence in the conclusion.

Differencing transforms a non-stationary series into a stationary one.

First difference: ΔY_t = Y_t − Y_t−1.

If Y_t is a random walk, then ΔY_t = e_t, white noise, stationary.
We say Y_t is integrated of order 1, or I(1).
GDP in levels is I(1), GDP growth is I(0) (stationary).

Log-differencing.

Δ log Y_t ≈ (Y_t − Y_t−1) / Y_t−1, the growth rate.
Stabilizes variance and linearizes multiplicative trends. Preferred for price and output series.

Cost of differencing: you lose the long-run relationship.

Differenced regressions identify short-run dynamics only.
If two series are cointegrated, differencing is inefficient. Use an error-correction model instead.

Cointegration: two I(1) series that share a common stochastic trend.

Even though X_t and Y_t individually wander, the linear combination Y_t − βX_t is stationary. They are “tied together” in the long run.

Examples: consumption and income, prices in different markets (law of one price), money supply and price level.

Testing: Engle-Granger two-step, regress Y_t on X_t in levels, then test the residuals for a unit root. If the residuals are stationary, the series are cointegrated.

Error-correction model (ECM): if cointegrated, short-run dynamics plus error-correction term:

ΔY_t = α(Y_t−1 − βX_t−1) + γΔX_t + e_t

α < 0 is the speed of adjustment back to the long-run equilibrium.

When errors are autocorrelated, use HAC (heteroskedasticity and autocorrelation consistent) standard errors.

Newey and West (1987) proposed a covariance estimator that is robust to both heteroskedasticity and serial correlation of unknown form:

V̂_HAC = (X^TX)⁻¹ Ŝ (X^TX)⁻¹

where Ŝ is the Newey-West long-run variance estimate that uses a kernel (Bartlett weights) to down-weight distant lags. The number of lags included is the bandwidth.

Bandwidth choice: a common rule of thumb is m = 0.75 T^1/3. Too few lags: SEs remain too small. Too many: SEs are over-corrected.

HAC SEs are the time-series counterpart of HC robust SEs in cross-section. Use them by default whenever you are unsure whether errors are serially correlated.

Distributed lag models capture delayed effects of X on Y.

Finite distributed lag (FDL) model.

Y_t = α + β₀X_t + β₁X_t−1 + … + β_qX_t−q + u_t
β₀ = impact multiplier: immediate effect of a unit increase in X.
β₀ + … + β_q = long-run multiplier: total cumulative effect after q periods.

Autoregressive distributed lag (ARDL) model.

Adds lags of Y on the right-hand side. Parsimonious way to capture rich dynamics.
Nests the FDL model and the error-correction model as special cases.

Multicollinearity in lag models.

Lags of the same variable are highly correlated. Individual β estimates can be imprecise even when the long-run sum is well-estimated.

Controlling for trends and seasonality.

Linear time trend.

Include t as a regressor. Removes a deterministic linear trend from both Y and X.
Equivalent to Frisch-Waugh: regressing detrended Y on detrended X.

Quadratic or log trend.

Include t² or log(t) for series with non-linear trends.
But trends can also be stochastic (unit root) rather than deterministic. Distinguish by unit root testing.

Seasonal dummies.

Include Q − 1 dummy variables for quarterly data (or 11 dummies for monthly).
Alternatively, use seasonally adjusted data from official sources (BLS, BEA).

Failing to detrend leads to spurious correlations.

Two trending series look correlated even when the detrended residuals are not.

Forecasting with time series models.

One-step-ahead forecast from AR(1).

Ŷ_T+1|T = ρ̂Y_T. Forecast uncertainty grows as the horizon extends.

Model selection for forecasting.

Use AIC (Akaike) or BIC (Schwarz/Bayesian) to choose lag length.
BIC penalizes extra parameters more heavily and tends to select more parsimonious models.

Evaluate forecast performance out-of-sample.

In-sample fit is not forecast accuracy. A complex model can overfit.
Use RMSE or MAE on a held-out test set. A common benchmark is the random-walk forecast: Ŷ_T+1 = Y_T.

Diebold-Mariano test.

Tests whether two forecasting models have equal predictive accuracy based on their forecast error sequences.

Common mistakes in time series analysis.

Regressing non-stationary series in levels without testing.

High R² and significant t-stats can be entirely spurious. Run ADF first.

Using OLS standard errors with autocorrelated errors.

Even a small autocorrelation in errors can severely distort t-statistics. Use HAC SEs.

Differencing away cointegrated series.

If two series are cointegrated, differencing discards the long-run relationship. Use ECM.

Evaluating forecast models in-sample.

Always assess forecast accuracy on data not used in estimation. Time-ordered splitting matters, never shuffle time series data.

Practice Questions

Question 1 of 4

Time series introduces new hazards: autocorrelation, non-stationarity, and spurious regression.

Autocorrelated errors do not bias OLS, but they invalidate standard errors. Use HAC (Newey-West) SEs.
Non-stationary (I(1)) series must be differenced or tested for cointegration before regression.
Always run an ADF test before regressing trending series on each other.