Testing Many Restrictions Under Heteroskedasticity

Abstract

We propose a hypothesis test that allows for many tested restrictions in a heteroskedastic linear regression model. The test compares the conventional F-statistic to a critical value that corrects for many restrictions and conditional heteroskedasticity. The correction utilizes leave-one-out estimation to recenter the conventional critical value and leave-three-out estimation to rescale it. Large sample properties of the test are established in an asymptotic framework where the number of tested restrictions may grow in proportion to the number of observations. We show that the test is asymptotically valid and has non-trivial asymptotic power against the same local alternatives as the exact F test when the latter is valid. Simulations corroborate the relevance of these theoretical findings and suggest excellent size control in moderately small samples also under strong heteroskedasticity.

It ain’t where you’re from, it’s where you’re at: Hiring origins, firm heterogeneity, and wages

“…These standard errors were constructed according to equation 7 ofKline et al (2020). We opt not to implement the small sample correction to these standard errors developed byAnatolyev and Sølvsten (2020) as the effective degrees of freedom in our dataset is very large.13 Because estimating the sampling covariance between the projections corresponding to the first stage and reduced form of this system is computationally burdensome, we refrain from reporting standard errors on these coefficients.…”

Systemic Discrimination Among Large U.S. Employers

“…Applying the bias correction of Kline, Saggio and Sølvsten (2020), the estimated standard deviation of firm effects across jobs is 0.015, while the estimated variance of SOC-3 job title effects is negative. Using the procedure of Anatolyev and Sølvsten (2020) to test that the job title effects are jointly zero yields a p-value of 0.33, suggesting that job title effects are not a major source of variation in firm contact gaps in our experiment. 11 The firm effects, by contrast, are strongly significant (p < 0.001).…”

“…The estimated variance of state gender gap fixed effects is actually negative, suggesting that this component is very small or zero. To formally test whether the state fixed effects can be distinguished from noise we employ the high dimensional heteroscedasticity-robust testing procedure of Anatolyev and Sølvsten (2020), which yields joint p-values of 0.19 and 0.48 for the state race and gender gap fixed effects, respectively. By contrast, the null hypothesis that the firm fixed effects jointly equal zero is decisively rejected for both race and gender (p < 0.001).…”

“…"N job titles / states > 1 firm" is the number of states or job titles in the connected set observed at 2 or more firms. The final two rows report p-values from tests of the joint hypothesis that all firm or job title / state fixed effects equal zero, computed using the heteroscedasticity robust procedure ofAnatolyev and Sølvsten (2020).…”