Philipp Ketz

We introduce adaptive confidence intervals on a parameter of interest in the presence of nuisance parameters, such as coefficients on control variables, with known signs. Our confidence intervals are trivial to compute and can provide significant length reductions relative to standard ones when the nuisance parameters are small. At the same time, they entail minimal length increases at any parameter values. We apply our confidence intervals to the linear regression model, prove their uniform validity and illustrate their length properties in an empirical application to a factorial design field experiment and a Monte Carlo study calibrated to the empirical application.

We show that the standard test for testing overidentifying restrictions, which compares the J-statistic (Hansen, 1982) to χ2 critical values, does not control asymptotic size when the true parameter vector is allowed to lie on the boundary of the (optimization) parameter space. We also propose a modified J-statistic that, under the null hypothesis, is asymptotically χ2 distributed, such that the resulting test does control asymptotic size.

We show that, in the random coefficients logit model, standard inference procedures can suffer from asymptotic size distortions. The problem arises due to boundary issues and is aggravated by the standard parameterization of the model, in terms of standard deviations. For example, in case of a single random coefficient, the asymptotic size of the nominal 95% confidence interval obtained by inverting the two-sided t-test for the standard deviation equals 83.65%. In seeming contradiction, we also show that standard error estimates for the estimator of the standard deviation can be unreasonably large. This problem is alleviated if the model is reparameterized in terms of variances. Furthermore, a numerical evaluation of a conjectured lower bound suggests that the asymptotic size of the nominal 95% confidence interval obtained by inverting the two-sided t-test for variances (means) is within 0.5 percentage points of the nominal level as long as there are no more than five (four) random coefficients and as long as an optimal weighting matrix is employed.

Extremum estimators are not asymptotically normally distributed when the estimator satisfies the restrictions on the parameter space—such as the non-negativity of a variance parameter—and the true parameter vector is near or at the boundary. This possible lack of asymptotic normality makes it difficult to construct tests for testing subvector hypotheses that control asymptotic size in a uniform sense and have good local asymptotic power irrespective of whether the true parameter vector is at, near, or far from the boundary. We propose a novel estimator that is asymptotically normally distributed even when the true parameter vector is near or at the boundary and the objective function is not defined outside the parameter space. The proposed estimator allows the implementation of a new test based on the Conditional Likelihood Ratio statistic that is easy-to-implement, controls asymptotic size, and has good local asymptotic power properties. Furthermore, we show that the test enjoys certain asymptotic optimality properties when the parameter of interest is scalar. In an application of the random coefficients logit model (Berry, Levinsohn, and Pakes, 1995) to the European car market, we find that, for most parameters, the new test leads to tighter confidence intervals than the two-sided t-test commonly used in practice.

In this paper, we use the results in Andrews and Cheng (2012), extended to allow for parameters to be near or at the boundary of the parameter space, to derive the asymptotic distributions of the two test statistics that are used in the two-step (testing) procedure proposed by Pedersen and Rahbek (2019). The latter aims at testing the null hypothesis that a GARCH-X type model, with exogenous covariates (X), reduces to a standard GARCH type model, while allowing the "GARCH parameter" to be unidentified. We then provide a characterization result for the asymptotic size of any test for testing this null hypothesis before numerically establishing a lower bound on the asymptotic size of the two-step procedure at the 5% nominal level. This lower bound exceeds the nominal level, revealing that the two-step procedure does not control asymptotic size. In a simulation study, we show that this finding is relevant for finite samples, in that the two-step procedure can suffer from overrejection in finite samples. We also propose a new test that, by construction, controls asymptotic size and is found to be more powerful than the two-step procedure when the "ARCH parameter" is "very small" (in which case the two-step procedure underrejects).

Current measurement of sex-selective abortion is based on a comparison of the sex ratio at birth with the natural sex ratio and provides us with the number of aborted female foetuses. This, however, does not tell us how prevalent the phenomenon is, i.e., how many women abort, which will differ from the number of aborted female foetuses when there is repeated sex-selective abortion. In this paper, we show that the number of women that abort between two consecutive births can be inferred using the sex ratio at birth and information on birth spacing. We use this result to study sex-selective abortion patterns in India over time and across birth orders, sibling compositions, and different socio-demographic/geographic groups. We find evidence of strong heterogeneity across samples, with the number of aborted female foetuses being up to 50% larger than the number of women that abort. Our findings suggest possible improvements in targeting of policies that aim at reducing sex-selective abortion.