Heteroskedasticity and Autocorrelation Consistent (HAC) Standard Errors in Regression Analysis

Regression Analysis

Heteroskedasticity and Autocorrelation Consistent (HAC) Standard Errors

In econometric analysis, especially with time series data, it is crucial to account for both heteroskedasticity (non-constant variance of errors) and autocorrelation (correlation of error terms across time). HAC standard errors provide a robust method for estimating the variance of regression coefficients when these issues are present.

• Heteroskedasticity: Occurs when the variance of the error term ut is not constant across observations.

• Autocorrelation: Occurs when the error terms are correlated across time, i.e., Cov(ut, us) ≠ 0 for t ≠ s.

• HAC Standard Errors: These are standard errors that are robust to both heteroskedasticity and autocorrelation, ensuring valid inference in time series regressions.

OLS Estimator and Variance in the Presence of HAC

Consider the simple linear regression model with a single regressor:

• Model: $Y_t = \beta_0 + \beta_1 X_t + u_t$

• OLS estimator for $\beta_1$ (from SW Appendix 4.3):

$\hat{\beta}_1 - \beta_1 = \frac{1}{T} \sum_{t=1}^T \frac{(X_t - \bar{X})u_t}{\frac{1}{T} \sum_{t=1}^T (X_t - \bar{X})^2} \approx \frac{1}{T} \sum_{t=1}^T \frac{v_t}{\sigma_X^2}$

• Where $v_t = (X_t - \bar{X})u_t$

Variance of the OLS Estimator

The variance of the OLS estimator depends on the properties of the error terms:

• General formula for variance:

$\mathrm{var}(\hat{\beta}_1) = \mathrm{var}\left(\frac{1}{T} \sum_{t=1}^T v_t\right) / (\sigma_X^2)^2$

• Expanded for general T:

$\mathrm{var}(\hat{\beta}_1) = \frac{1}{T^2} \sum_{t=1}^T \sum_{s=1}^T \mathrm{cov}(v_t, v_s) / (\sigma_X^2)^2$

• For i.i.d. (independent and identically distributed) cross-sectional data, $\mathrm{cov}(v_t, v_s) = 0$ for $t \neq s$:

$\mathrm{var}(\hat{\beta}_1) = \frac{\sigma_v^2}{T (\sigma_X^2)^2}$

• This is the standard cross-sectional result.

Time Series Data: The Need for HAC

In time series data, error terms may be autocorrelated, so $\mathrm{cov}(v_t, v_s) \neq 0$ in general. For example, with T = 2:

$\mathrm{var}\left(\frac{1}{2}(v_1 + v_2)\right) = \frac{1}{4}[\mathrm{var}(v_1) + \mathrm{var}(v_2) + 2\mathrm{cov}(v_1, v_2)]$

$= \frac{1}{2}\sigma_v^2 + \frac{1}{2}\rho_1\sigma_v^2$

• Where $\rho_1 = \mathrm{corr}(v_1, v_2)$

• Let $f_2 = (1 + \rho_1)$, so variance is $\frac{1}{2}f_2\sigma_v^2$

• If $\rho_1 = 0$ (i.i.d. case), $f_2 = 1$, recovering the usual formula.

• If $\rho_1 \neq 0$, the variance is not given by the usual formula.

General Formula for HAC Variance (Any T)

For general T, the variance formula incorporates autocorrelation:

$\mathrm{var}\left(\frac{1}{T} \sum_{t=1}^T v_t\right) = \frac{\sigma_v^2}{T} f_T$

So,

$\mathrm{var}(\hat{\beta}_1) = \frac{1}{T} \frac{\sigma_v^2}{(\sigma_X^2)^2} f_T$

Where

$f_T = 1 + 2 \sum_{j=1}^{T-1} \frac{T-j}{T} \rho_j$

• $\rho_j$ is the autocorrelation at lag j.

Implications and Applications

• Conventional heteroskedasticity-robust OLS standard errors are incorrect when ut is serially correlated.

• The OLS standard errors are off by the factor fT.

• It is necessary to use the augmented HAC standard error formula for valid inference in time series regressions.

Summary Table: Comparison of Standard Error Formulas

Data Type	Variance Formula	Assumptions
Cross-sectional (i.i.d.)	$\frac{\sigma_v^2}{T (\sigma_X^2)^2}$	No autocorrelation, possibly heteroskedastic
Time Series (HAC)	$\frac{1}{T} \frac{\sigma_v^2}{(\sigma_X^2)^2} f_T$	Allows for autocorrelation and heteroskedasticity

Example Application

• Suppose you are estimating the effect of an economic variable Xt on Yt using quarterly data. If the error terms are autocorrelated (e.g., due to business cycles), using conventional OLS standard errors will underestimate the true variability of your estimator. HAC standard errors correct for this, providing more reliable confidence intervals and hypothesis tests.