Heteroskedasticity and Autocorrelation Consistent (HAC) Standard Errors and Strict Exogeneity in Time Series Regression

Heteroskedasticity and Autocorrelation Consistent (HAC) Standard Errors

The fT Factor in Time Series Regression

In time series econometrics, standard errors must often be adjusted to account for both heteroskedasticity and autocorrelation in the error terms. The adjustment factor, denoted as fT, is crucial for obtaining valid inference when these issues are present.

• Definition: The fT factor adjusts the variance of the estimated coefficients to account for serial correlation and heteroskedasticity in the residuals.

• Panel Data vs. Time Series: In panel data, clustered standard errors implicitly estimate fT (requiring large n). In time series, fT must be estimated explicitly.

• HAC Standard Errors: Standard errors that use consistent estimators of fT are called Heteroskedasticity- and Autocorrelation-Consistent (HAC) standard errors, also known as heteroskedasticity- and autocorrelation robust (HAR) standard errors.

Mathematical Formulation of the fT Factor

The variance of the OLS estimator in the presence of autocorrelation is given by:

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

where:

• T = sample size (number of time periods)

• \(\sigma_v^2\) = variance of the error term

• \(\sigma_X^2\) = variance of the regressor

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

• \(\rho_j\) = autocorrelation at lag j

Newey-West Estimator for fT

The most commonly used estimator for fT is the Newey-West estimator, which is designed to provide consistent standard errors in the presence of both heteroskedasticity and autocorrelation.

The Newey-West estimator is given by:

$\hat{f}_T = 1 + 2 \sum_{j=1}^{m-1} \frac{m-j}{m} \hat{\rho}_j$

• \(\hat{\rho}_j\) = estimated autocorrelation at lag j

• m = truncation parameter (the maximum lag considered)

Choosing the Truncation Parameter (m)

• Truncation Parameter (m): Determines how many lags of autocorrelation are included in the estimator.

• Rule of Thumb: Use $m = 0.75 \times (T^{1/3})$ and test the sensitivity of results to different values of m.

• Balance: Choose m to avoid including too many or too few lags, which can affect the estimator's accuracy.

Estimation with Strictly Exogenous Regressors

Definition and Implications of Strict Exogeneity

A regressor X is said to be strictly exogenous if:

$E(u_t | ..., X_{t+1}, X_t, X_{t-1}, ...) = 0$

• This means the error term at time t is uncorrelated with the regressor at all time periods (past, present, and future).

• This is a very strong assumption and rarely holds in practice, especially in economic time series.

Estimation Methods under Strict Exogeneity

• Generalized Least Squares (GLS): An efficient estimation method when strict exogeneity holds, as it accounts for autocorrelation and heteroskedasticity in the errors.

• Autoregressive Distributed Lag (ADL) Models: Used to estimate dynamic causal effects more efficiently under strict exogeneity.

• Practical Limitation: Because strict exogeneity is rarely plausible, GLS and ADL approaches are not often useful in real-world applications.

Example: In weather and orange juice (OJ) price regressions, strict exogeneity does not hold because future weather can influence current and future prices, violating the assumption.