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Lesson M09.L01: OLS and Serial Correlation

Field Detail
Module M09 — Serial Correlation and Time Series Inference
Level Graduate (ECMT5001 S1 2027)
Duration 30–35 min
Learning Objective Understand why OLS remains consistent under contemporaneous exogeneity but produces invalid standard errors when errors are serially correlated; quantify the direction of the bias in OLS SEs under AR(1) errors
Provenance Wooldridge, Introductory Econometrics, Ch 12.1

Explanation

Time Series Assumptions: TS.1′–TS.5′

In cross-section, OLS requires strict exogeneity (MLR.4): \(E(u_i \mid \mathbf{x}_1, \ldots, \mathbf{x}_n) = 0\). In time series, strict exogeneity is often implausible — it rules out lagged dependent variables and feedback from \(y\) to future \(x\). The relaxed assumptions TS.1′–TS.5′ replace strict with contemporaneous exogeneity:

Assumption Statement
TS.1′ Linearity in parameters (same as before)
TS.2′ No perfect collinearity among regressors
TS.3′ Contemporaneous exogeneity: \(E(u_t \mid \mathbf{x}_t) = 0\) — the error is mean-zero given current regressors only
TS.4′ Homoskedasticity: \(\text{Var}(u_t \mid \mathbf{x}_t) = \sigma^2\)
TS.5′ No serial correlation: \(\text{Cov}(u_t, u_s \mid \mathbf{x}_t, \mathbf{x}_s) = 0\) for \(t \neq s\)

Under TS.1′–TS.3′ alone, OLS is consistent for \(\boldsymbol{\beta}\). Under TS.1′–TS.5′, OLS is BLUE and standard \(t\)- and \(F\)-statistics are valid.

Key distinction:
- Strict exogeneity → OLS unbiased (stronger, harder to satisfy)
- Contemporaneous exogeneity → OLS consistent (weaker, sufficient for large-sample validity if TS.5′ also holds)

What Happens When TS.5′ Fails?

When \(\text{Cov}(u_t, u_s) \neq 0\), serial correlation is present. Under TS.1′–TS.3′ (contemporaneous exogeneity), OLS is still consistent — the coefficient estimates converge to the true \(\boldsymbol{\beta}\) as \(n \to \infty\). But the OLS standard errors are wrong because their derivation assumes TS.5′.

Serial correlation does NOT bias OLS estimates (under contemporaneous exogeneity). It only invalidates inference.

AR(1) Error Process

The simplest and most common serial correlation structure is the first-order autoregressive process:

\[u_t = \rho\, u_{t-1} + e_t, \qquad e_t \sim \text{i.i.d.}(0, \sigma_e^2), \quad |\rho| < 1\]
  • \(\rho\) is the autocorrelation coefficient of the errors
  • \(e_t\) is a white-noise innovation — uncorrelated with everything up to period \(t-1\)
  • Stationarity requires \(|\rho| < 1\)

The variance and autocovariances of \(u_t\) under AR(1):

\[\text{Var}(u_t) = \frac{\sigma_e^2}{1 - \rho^2}, \qquad \text{Cov}(u_t, u_{t-s}) = \rho^s \cdot \frac{\sigma_e^2}{1 - \rho^2}\]

The autocorrelations decay geometrically:

\[\text{Corr}(u_t, u_{t-s}) = \rho^s\]

Effect of Serial Correlation on OLS Variance

Consider a simple time-series regression \(y_t = \beta_0 + \beta_1 x_t + u_t\) with AR(1) errors. The true asymptotic variance of \(\hat{\beta}_1^{\text{OLS}}\) is larger than the standard OLS formula assumes.

Under positive serial correlation (\(\rho > 0\)):

\[\text{True Var}(\hat{\beta}_1^{\text{OLS}}) > \text{Computed Var}_{\text{OLS}}(\hat{\beta}_1)\]

Intuition: Positively correlated errors mean consecutive observations carry less independent information than \(n\) i.i.d. draws. The "effective sample size" is smaller than \(n\), but the standard OLS SE formula pretends it equals \(n\) — so it understates the true variance.

Consequences of understated SEs when \(\rho > 0\):

Statistic Effect
Standard errors Too small
\(t\)-statistics Too large (inflated)
Confidence intervals Too narrow
Hypothesis tests Over-reject \(H_0\) — spurious significance

Why Positive Serial Correlation is Common in Economics

Economic time series exhibit persistence for structural reasons:

  • Technology shocks take time to diffuse through the economy
  • Policy changes (monetary, fiscal) have lagged multiplier effects
  • Demand shocks (oil prices, consumer confidence) propagate through supply chains
  • Habit formation in consumption: \(c_t\) depends on \(c_{t-1}\)

Positive \(\rho\) is the rule rather than the exception in macro time series.

Worked Example

Setup: Consider a monthly model of consumer spending on income:

\[\text{cons}_t = \beta_0 + \beta_1 \text{income}_t + u_t\]

Suppose the true error process is AR(1) with \(\rho = 0.6\), \(\sigma_e^2 = 1\), \(n = 120\), and \(\text{Var}(x) = 4\).

Step 1: True error variance

\[\text{Var}(u_t) = \frac{\sigma_e^2}{1-\rho^2} = \frac{1}{1-0.36} = \frac{1}{0.64} = 1.5625\]

Step 2: OLS-computed variance of \(\hat{\beta}_1\) (incorrectly assuming i.i.d. errors)

\[\widehat{\text{Var}}_{\text{OLS}}(\hat{\beta}_1) = \frac{\hat{\sigma}^2}{\sum(x_t - \bar{x})^2} \approx \frac{1.5625}{n \cdot \text{Var}(x)} = \frac{1.5625}{120 \times 4} = 0.003255\]
\[\widehat{\text{SE}}_{\text{OLS}}(\hat{\beta}_1) \approx 0.057\]

Step 3: True asymptotic variance (accounting for AR(1) correlation)

The true variance involves a correction factor. For a simple illustration, the ratio of true to OLS variance under AR(1) is approximately:

\[\frac{\text{True Var}}{\text{OLS Var}} \approx 1 + 2\rho \cdot r_x\]

where \(r_x = \text{Corr}(x_t, x_{t-1})\). If \(r_x = 0.5\) (income is also persistent):

\[\text{Inflation factor} \approx 1 + 2(0.6)(0.5) = 1.6\]
\[\text{True SE} \approx 0.057 \times \sqrt{1.6} = 0.057 \times 1.265 \approx 0.072\]

Step 4: Interpret

The OLS standard error (0.057) is 21% smaller than the true standard error (0.072). If the true \(t\)-statistic were \(0.072/\text{s.e.} = 1.8\), the OLS procedure would report \(t = 0.057^{-1} \times \hat{\beta}_1\) and classify it as significant at 5% when it should not be. This is the mechanism of spurious significance due to serial correlation.

Common Misconception

"Serial correlation biases OLS coefficient estimates."

This is false under contemporaneous exogeneity. If \(E(u_t \mid \mathbf{x}_t) = 0\) (TS.3′), OLS is consistent regardless of serial correlation in \(u_t\). The coefficients converge to the true \(\beta\) in large samples. The only problem is that the OLS standard errors are computed under the false assumption that errors are uncorrelated — so inference (t-tests, confidence intervals, F-tests) is invalid.

The correct statement: Serial correlation leaves OLS point estimates consistent but renders inference unreliable. The fix is corrected standard errors (HAC) or GLS — not a new estimation strategy to unbias \(\hat{\beta}\).

Practice Prompts

  1. A researcher estimates a quarterly GDP growth regression and finds \(\hat{\rho} = 0.7\) from the residuals. The OLS \(t\)-statistic on the lagged policy variable is 2.4. Should the researcher conclude the policy variable is significant at the 5% level? Explain your reasoning in terms of what \(\hat{\rho} = 0.7\) implies for the OLS SE.

  2. Under AR(1) errors with \(\rho = -0.4\) (negative serial correlation), in which direction are OLS standard errors biased — understated or overstated? What does this imply for Type I and Type II errors in hypothesis testing?

  3. Consider the regression \(\log(\text{wage}_t) = \beta_0 + \beta_1 \text{edu}_t + u_t\) estimated on annual panel data. Explain why contemporaneous exogeneity (TS.3′) is a more defensible assumption than strict exogeneity for this model. Does contemporaneous exogeneity alone guarantee valid OLS \(t\)-statistics?

Further Resources

  • Wooldridge, Introductory Econometrics (7th ed.), Ch 12.1 — Time Series Assumptions and Serial Correlation
  • Greene, Econometric Analysis (8th ed.), Ch 20 — Autocorrelation
  • Stock & Watson, Introduction to Econometrics, Ch 15 — Regression with Time Series Data
  • statsmodels documentation — AR model fitting in Python