Answer-first
Autocorrelation breaks OLS efficiency because the error terms are no longer independent. When errors are correlated across time, OLS still produces unbiased slope estimates, but they are no longer the most efficient, and the usual standard errors become invalid. This makes hypothesis tests unreliable and forecasts misleading.
Warm intro (and where to find the “why” pages)
If you’re staring at an autocorrelation question at 11:47pm, feeling stuck, behind, or low‑key panicking because the “errors depend on past errors” explanation feels circular, you’re not alone. These questions look simple—“Durbin‑Watson < 2”—but under exam pressure, students freeze when asked to explain why autocorrelation breaks OLS efficiency and what actually goes wrong inside the variance formula.
If you’re rebuilding your foundation across topics, start at our Why Hub. If you need the full econometrics & time series roadmap for last-minute studying or troubleshooting, see Econometrics & Time Series.
Problem setup
Consider the regression model:
\[ y_t = \beta_0 + \beta_1 x_t + u_t. \]
OLS assumes:
\[ \text{Cov}(u_t, u_{t-1}) = 0. \]
Autocorrelation occurs when:
\[ u_t = \rho u_{t-1} + \varepsilon_t, \]
with \(|\rho| > 0\).
Step-by-step solution (WordPress-safe MathJax)
Step 1: OLS remains unbiased
Even with autocorrelation:
\[ \mathbb{E}[\hat{\beta}_{OLS}] = \beta. \]
So the slope is still centered on the truth.
Step 2: But OLS is no longer efficient
OLS is BLUE only if errors are independent. Autocorrelation violates this, so OLS no longer minimizes variance.
Step 3: Standard errors become wrong
The usual OLS variance formula assumes independence:
\[ \text{Var}(\hat{\beta}) = \sigma^2 (X’X)^{-1}. \]
With autocorrelation, the true variance is:
\[ \text{Var}(\hat{\beta}) = (X’X)^{-1} X’ \Omega X (X’X)^{-1}, \]
where \(\Omega\) contains autocorrelated errors.
Step 4: t‑tests and F‑tests break
Standard errors are underestimated or overestimated, making inference invalid.
Step 5: State the identifying assumption clearly
OLS requires:
\[ \text{Cov}(u_t, u_{t-k}) = 0 \quad \text{for all } k>0. \]
Autocorrelation violates this assumption.
Intuition
Autocorrelation means today’s error contains yesterday’s error. This creates “momentum” in the noise. OLS thinks each error is new information, but in reality, the errors are repeating themselves. This makes OLS overconfident and underestimates uncertainty.
Common exam mistakes
- Claiming autocorrelation biases OLS. It does not.
- Using non‑robust standard errors.
- Confusing autocorrelation with multicollinearity.
- Ignoring Durbin‑Watson or Breusch‑Godfrey tests.
- Misinterpreting AR(1) as a model for the dependent variable.
Why this matters
Autocorrelation is everywhere in time series—GDP, inflation, stock returns, energy demand. If you ignore it, your standard errors collapse, your inference becomes meaningless, and your forecasts drift.
Final summary
- Autocorrelation leaves OLS unbiased but inefficient.
- Standard errors become wrong under autocorrelation.
- Inference breaks unless you use HAC (Newey‑West) SEs.
- AR(1) and GLS methods fix the problem.
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