Newey-West HAC Adjustment for Information Coefficient
A daily alpha signal $a_t$ is evaluated against returns $r_t$. Define the (Pearson) information coefficient as $\text{IC} = \text{Corr}(a_t, r_t)$.
Assume $a_t$ and $r_t$ are covariance-stationary and that the product series $z_t = \tilde{a}_t \tilde{r}_t$ (where tildes denote demeaned values) has AR(1) residual autocorrelation $\rho$.
- Derive the Newey-West $L = 1$ HAC estimator of $\text{Var}(\widehat{\text{IC}})$.
- Construct the corresponding $t$-statistic for testing $H_0: \text{IC} = 0$.
- State a decision rule at the 5% significance level for an in-sample window of length $T$.
Hints
- The naive IC $t$-statistic assumes independence of the product series $z_t = \tilde{a}_t \tilde{r}_t$. What goes wrong when $z_t$ is autocorrelated?
- The Bartlett kernel for lag $j$ with bandwidth $L$ assigns weight $w_j = 1 - j/(L+1)$. Apply this with $L = 1$ to the sample autocovariances $\hat{\gamma}_0$ and $\hat{\gamma}_1$.
- Write the HAC variance as $\frac{1}{T}(\hat{\gamma}_0 + 2 w_1 \hat{\gamma}_1)$ and note that the IC is just $\bar{z}$ rescaled by $\hat{\sigma}_a \hat{\sigma}_r$. Form $t = \widehat{\text{IC}} / \sqrt{\widehat{\text{Var}}_{\text{HAC}}}$.
Worked Solution
How to Think About It: In quant research, you compute an IC (correlation between your signal and forward returns) and want to know if it is statistically significant. The naive $t$-stat is $\text{IC} \cdot \sqrt{T}$, which assumes the observations are independent. But in practice, if your signal or returns have serial correlation, the products $z_t = \tilde{a}_t \tilde{r}_t$ are autocorrelated, and the naive standard error understates the true uncertainty. Newey-West fixes this by inflating the variance estimate to account for lagged covariances. With lag $L = 1$, you only need to estimate the first-order autocovariance of $z_t$.
Quick Estimate: Suppose $T = 252$ (one year of daily data), $\widehat{\text{IC}} = 0.05$, and $\rho \approx 0.1$. The naive variance is
Approach: We derive the HAC variance using the Bartlett kernel with one lag, then form the $t$-statistic.
Formal Solution:
Define $z_t = \tilde{a}_t \tilde{r}_t$ where $\tilde{a}_t = a_t - \bar{a}$ and $\tilde{r}_t = r_t - \bar{r}$. Under $H_0: \text{IC} = 0$, the sample IC is approximately
$\widehat{\text{IC}} = \frac{\bar{z}}{\hat{\sigma}_a \hat{\sigma}_r}$
where $\bar{z} = \frac{1}{T}\sum_{t=1}^{T} z_t$.
The key quantity is $\text{Var}(\bar{z})$. Define the autocovariance function:
$\hat{\gamma}_j = \frac{1}{T} \sum_{t=j+1}^{T} (z_t - \bar{z})(z_{t-j} - \bar{z})$
The Newey-West HAC estimator with $L = 1$ uses the Bartlett kernel:
$\hat{\sigma}_{\text{HAC}}^2 = \frac{1}{T}\left[\hat{\gamma}_0 + 2 \cdot \frac{1}{2} \cdot \hat{\gamma}_1\right] = \frac{1}{T}\left[\hat{\gamma}_0 + \hat{\gamma}_1\right]$
The Bartlett weight for lag $j$ with $L = 1$ is $w_j = 1 - j/(L+1)$, so $w_1 = 1 - 1/2 = 1/2$. Accounting for both sides of the autocovariance (positive and negative lags):
$\widehat{\text{Var}}_{\text{HAC}}(\bar{z}) = \frac{1}{T}\left[\hat{\gamma}_0 + 2 \cdot \frac{1}{2} \cdot \hat{\gamma}_1\right] = \frac{1}{T}\left[\hat{\gamma}_0 + \hat{\gamma}_1\right]$
Since $\widehat{\text{IC}} \approx \bar{z}/(\hat{\sigma}_a \hat{\sigma}_r)$, the HAC variance of the IC is:
$\widehat{\text{Var}}_{\text{HAC}}(\widehat{\text{IC}}) = \frac{\hat{\gamma}_0 + \hat{\gamma}_1}{T \cdot \hat{\sigma}_a^2 \hat{\sigma}_r^2}$
The $t$-statistic is:
$t = \frac{\widehat{\text{IC}}}{\sqrt{\widehat{\text{Var}}_{\text{HAC}}(\widehat{\text{IC}})}} = \frac{\widehat{\text{IC}} \cdot \hat{\sigma}_a \hat{\sigma}_r \cdot \sqrt{T}}{\sqrt{\hat{\gamma}_0 + \hat{\gamma}_1}}$
Note that if $\hat{\gamma}_1 = 0$ (no autocorrelation), this collapses to the naive $t = \widehat{\text{IC}} \cdot \sqrt{T}$ (since $\hat{\gamma}_0 = \hat{\sigma}_a^2 \hat{\sigma}_r^2$ under $H_0$).
If the residual autocorrelation is $\rho$, then $\hat{\gamma}_1 \approx \rho \cdot \hat{\gamma}_0$, and the inflation factor is:
$\text{Inflation} = \frac{1}{\sqrt{1 + \rho}}$
so $t_{\text{HAC}} \approx t_{\text{naive}} / \sqrt{1 + \rho}$.
Decision Rule: At 5% significance (two-sided), reject $H_0: \text{IC} = 0$ if $|t| > 1.96$. Equivalently, the IC is statistically significant if:
$|\widehat{\text{IC}}| > \frac{1.96 \cdot \sqrt{\hat{\gamma}_0 + \hat{\gamma}_1}}{\hat{\sigma}_a \hat{\sigma}_r \cdot \sqrt{T}}$
Answer: The Newey-West $L=1$ HAC variance of the IC is $\widehat{\text{Var}}_{\text{HAC}}(\widehat{\text{IC}}) = (\hat{\gamma}_0 + \hat{\gamma}_1) / (T \hat{\sigma}_a^2 \hat{\sigma}_r^2)$, and the $t$-statistic is $t = \widehat{\text{IC}} / \sqrt{\widehat{\text{Var}}_{\text{HAC}}}$. Reject $H_0$ at 5% if $|t| > 1.96$.
Intuition
The core issue is that serial correlation in your signal-return products inflates the true sampling variance of the IC beyond what the naive
In practice, this matters enormously for alpha research. A strategy that looks like it has a $t$-stat of 3 under the naive formula might drop to 2.2 after HAC correction, which changes how much capital you would allocate. Experienced quant researchers always report HAC-adjusted $t$-statistics, and many firms have been burned by ignoring serial correlation in backtested IC series.