Predictive Regressions and Persistent Predictors

Matrix derivations for forecasting regressions, overlapping horizons, persistent predictors, and Newey–West inference.

Predictive Regressions and Persistent Predictors

Forecast first, explain second

A one-step predictive regression is

y_{t+1}=\alpha+\beta x_t+u_{t+1}, \qquad E(u_{t+1}\mid\mathcal F_t)=0.

In finance, $y_{t+1}$ may be an excess return and $x_t$ a valuation ratio or yield spread.
In economics, $y_{t+1}$ may be inflation or output growth and $x_t$ a financial or survey indicator.

The timing is the claim: $x_t$ must be available at the forecast origin. A contemporaneous regression of $y_t$ on a revised $x_t$ is not the same experiment.

Matrix projection

Stack the outcomes and dated predictors:

\mathbf y= \begin{bmatrix}y_2\\y_3\\\vdots\\y_T\end{bmatrix}, \qquad X= \begin{bmatrix} 1&x_1\\ 1&x_2\\ \vdots&\vdots\\ 1&x_{T-1} \end{bmatrix}.

Then

\hat\beta=(X^\top X)^{-1}X^\top\mathbf y, \qquad X^\top\hat{\mathbf u}=0.

The normal equations establish sample orthogonality. They do not prove that $x_t$ was genuinely known, that the relation is stable, or that the forecast has economic value.

Why persistence changes inference

Predictors such as dividend–price ratios and interest-rate spreads are often well approximated by

x_t=\rho x_{t-1}+v_t,\qquad \rho\approx1.

If $u_t$ and $v_t$ are correlated, an unexpectedly high return changes both the dependent variable and the future predictor path. In short samples, the OLS slope can then be biased. Stambaugh derives this problem for return predictability; Campbell and Yogo develop inference designed for highly persistent regressors.

Practical response:

report the estimated persistence and uncertainty around it;
use theory-appropriate inference, not only an IID t-test;
test stability across forecast origins;
compare genuine out-of-sample performance with a simple benchmark;
call the result predictive unless a separate identification argument exists.

Overlapping horizons

For an $h$ -period cumulative target,

y_{t,t+h}=\sum_{j=1}^h y_{t+j}, \qquad y_{t,t+h}=\alpha_h+\beta_hx_t+u_{t+h}^{(h)}.

Adjacent outcomes share $h-1$ observations. Even if one-period innovations are independent, $u_{t+h}^{(h)}$ is serially correlated. The coefficient may remain an OLS projection, but the IID covariance formula is wrong.

HAC covariance in matrix form

Let $\hat u_t$ be the residual and $\mathbf g_t=\mathbf x_t\hat u_t$ . Define

\hat\Gamma_\ell=\sum_{t=\ell+1}^{T}\mathbf g_t\mathbf g_{t-\ell}^\top

and the Bartlett-weighted long-run covariance

\hat S= \hat\Gamma_0+ \sum_{\ell=1}^{L} \left(1-\frac{\ell}{L+1}\right) (\hat\Gamma_\ell+\hat\Gamma_\ell^\top).

The sandwich estimator is

\widehat{\operatorname{Var}}_{\mathrm{HAC}}(\hat\beta)= (X^\top X)^{-1}\hat S(X^\top X)^{-1}.

This is the matrix form of the Newey–West estimator. HAC repairs a covariance estimate under stated dependence conditions; it does not repair look-ahead bias, unstable coefficients, a bad transformation, or data mining.

R laboratory: overlapping forecasts

OLS and Newey-West covariance for overlapping outcomes

The lag-one residual correlation should be large because outcomes overlap. The HAC and IID standard errors therefore need not agree. The final two values verify the projection and covariance algebra.

Finance and economics: same regression, different audit

Audit question	Finance	Economics
target	excess return, premium, volatility, or spread change	growth, inflation, unemployment, recession
predictor timing	close, rebalance, and execution time	release calendar and vintage
friction	turnover, costs, shorting, capacity	publication lag and revisions
benchmark	historical mean or market-implied forecast	AR benchmark, survey, or institutional forecast
value	certainty equivalent, utility, tail loss	real-time RMSE, density score, decision relevance

Practice

For a 12-month rolling return sampled monthly, why is $L=0$ indefensible?
If $\hat\beta$ is significant in-sample but loses to the historical mean out-of-sample, what is supported?
Why can adding the final revised value of a leading indicator inflate performance?

Answers

Adjacent targets share eleven monthly returns, creating serial dependence.
At most an in-sample projection relation; useful forecast improvement has not been shown.
It leaks information unavailable at historical forecast origins.

Next: Volatility, Tails, and Financial Risk.

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