**Working Papers**

**The Size–Power Tradeoff in HAR Inference**, with James H. Stock and Daniel J. Lewis, August 2020

Abstract **+** | Online Appendix | Replication FilesAbstract **×** |

Heteroskedasticity and autocorrelation-robust (HAR) inference in time series regression typically involves kernel estimation of the long-run variance. Conventional wisdom holds that, for a given kernel, the choice of truncation parameter trades off a test’s null rejection rate and power, and that this tradeoff differs across kernels. We formalize this intuition: using higher-order expansions, we provide a unified size-power frontier for both kernel and weighted orthonormal series tests using nonstandard “fixed-b” critical values. We also provide a frontier for the subset of these tests for which the fixed-b distribution is t or F. These frontiers are respectively achieved by the QS kernel and equal-weighted periodogram. The frontiers have simple closed-form expressions, which upon evaluation show that the price paid for restricting attention to tests with t and F critical values is small. The frontiers are derived for the Gaussian multivariate location model, but simulations suggest the qualitative findings extend to stochastic regressors.

Conditionally accepted, * Econometrica *

**Duration-Driven Returns**, with Niels J. Gormsen, May 2020

Abstract **+**Abstract **×**

We propose a duration-based explanation for the major equity risk factors, including value, profitability, investment, low-risk, and payout factors. Both in the U.S. and globally, these factors invest in firms that earn most of their cash flows in the near future. The factors could therefore be driven by a premium on near-future cash flows. We test this hypothesis using a new dataset of single-stock dividend futures, which are claims on annual dividends for individual firms. Consistent with our hypothesis, risk-adjusted returns are higher on near- than on distant-future cash flows. In addition, the returns on individual cash flows do not vary across firms once controlling for maturity.

Revise and resubmit, * Journal of Finance *

**Restrictions on Asset-Price Movements Under Rational Expectations: Theory and Evidence**, with Ned Augenblick, March 2019

Abstract **+**Abstract **×**

How restrictive is the assumption of rational expectations in asset markets? We provide two contributions to address this question. First, we derive restrictions on the admissible variation in asset prices in a general class of rational-expectations equilibria. The challenge in this task is that asset prices reflect both beliefs and preferences. We gain traction by considering market-implied, or risk-neutral, probabilities of future outcomes, and we provide a mapping between the variation in these probabilities and the minimum curvature of utility — or, more generally, the slope of the stochastic discount factor — required to rationalize the marginal investor’s beliefs. Second, we implement these bounds empirically using S&P 500 index options. We find that very high utility curvature is required to rationalize the behavior of risk-neutral beliefs, and in some cases, no stochastic discount factor in the class we consider is capable of rationalizing these beliefs. This provides evidence of overreaction to new information relative to the rational benchmark. We show further that this overreaction is strongest for beliefs over prices at distant horizons, and that our findings cannot be explained by factors specific to the option market.

**Horizon-Dependent Risk Pricing: Evidence from Short-Dated Options**, March 2019

Abstract **+**Abstract **×**

I present evidence from index options that the price of risk over the value of the S&P 500 increases as the investment horizon becomes shorter. I show first how these risk prices may be estimated from the data, by translating the risk-neutral probabilities implied by options prices into physical probabilities that must provide unbiased forecasts of the terminal outcome. The risk price can be interpreted as the marginal investor’s effective risk aversion, and estimating this value over different option-expiration horizons for the S&P, I find that risk aversion is reliably higher for near-term outcomes than for longer-term outcomes: the market’s relative risk aversion over terminal index values decreases from around 15 at a one-week horizon to around 3 at a 12-week horizon. It is difficult to reconcile these findings with leading asset-pricing models, and I discuss necessary conditions for any such rational model to produce such a pattern. Models with dynamically inconsistent risk preferences, however, are capable of straightforwardly producing the findings presented here, and I discuss possible specifications of such models and their applicability to related results from previous literature.

**Publications**

**HAR Inference: Recommendations for Practice**, with James H. Stock, Daniel J. Lewis, and Mark W. Watson, October 2018

Abstract **+** | Replication FilesAbstract **×** |

The classic papers by Newey and West (1987) and Andrews (1991) spurred a large body of work on how to improve heteroscedasticity- and autocorrelation-robust (HAR) inference in time-series regression. This literature finds that using a larger-than-usual truncation parameter to estimate the long-run variance, combined with Kiefer–Vogelsang (2002, 2005) fixed-*b* critical values, can substantially reduce size distortions, at only a modest cost in (size-adjusted) power. Empirical practice, however, has not kept up. This article therefore draws on the post-Newey–West/Andrews literature to make concrete recommendations for HAR inference. We derive truncation parameter rules that choose a point on the size-power tradeoff to minimize a loss function. If Newey–West tests are used, we recommend the truncation parameter rule *S* = 1.3*T*^{1/2} and (nonstandard) fixed-*b* critical values. For tests of a single restriction, we find advantages to using the equal-weighted cosine (EWC) test, where the long run variance is estimated by projections onto Type-II cosines, using ν = 0.4*T*^{2/3} cosine terms; for this test, fixed-*b* critical values are, conveniently, *t*_{ν} or *F*. We assess these rules using first an ARMA/GARCH Monte Carlo design, then a dynamic factor model design estimated using 207 quarterly U.S. macroeconomic time series.

* Journal of Business & Economic Statistics* (2018), Vol. 36, No. 4, 541–559