Philipp Ketz

This paper demonstrates that sex-selective abortion induces a correlation between birth interval length and the sex of the next-born child. Using a statistical model, we show that shorter birth intervals for next-born girls indicate repeated sex-selective abortions between consecutive births. Analyzing data from India, we find evidence of repeated sex-selective abortions at birth order 2 when the first child is a girl, and strong evidence at birth order 3 when the first two children are girls. To quantify the extent of repeated abortions, we propose a maximum likelihood estimator that provides the number of women who abort and their likelihood of performing repeated abortions. Our estimation results reveal significant heterogeneity across birth orders, sibling compositions, and socio-demographic and geographic groups. Notably, literate and urban women who first had a girl rarely abort a second time, whereas women in northern India who first had two girls show a 13% likelihood of repeated sex-selective abortion. In this group, the estimated number of aborted female fetuses-the standard measure of sex-selective abortion-is 50% higher than the number of women who abort.

We provide adaptive confidence intervals on a parameter of interest in the presence of nuisance parameters when some of the nuisance parameters have known signs. The confidence intervals are adaptive in the sense that they tend to be short at and near the points where the nuisance parameters are equal to zero. We focus our results primarily on the practical problem of inference on a coefficient of interest in the linear regression model when it is unclear whether or not it is necessary to include a subset of control variables whose partial effects on the dependent variable have known directions (signs). Our confidence intervals are trivial to compute and can provide significant length reductions relative to standard confidence intervals in cases for which the control variables do not have large effects. At the same time, they entail minimal length increases at any parameter values. We prove that our confidence intervals are asymptotically valid uniformly over the parameter space 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.

Current measurement of sex-selective abortion is based on observing an imbalance be tween the sex ratio at birth and the natural sex ratio, providing us with the number of missing female foetuses. However, this measure does not tell us how widespread this phenomenon is, i.e., how many women abort, which will not be equal to the number of sex-selective abortions if there is repeated sex-selective abortion. In this paper, we show that the number of women that abort between two consecutive births and whether they do so repeatedly can be inferred using sex ratios and information on birth spacing. We apply our model to Indian DHS data to estimate how many women abort and to assess whether they do so repeatedly between two births. The results depend on the birth order and siblings composition: For example, we find that women whose first born is a girl abort at most once before the birth of the second child, i.e., (almost) none of them abort a second time if again pregnant with a girl after a first abortion. In contrast, we find evidence of repeated sex-selective abortion before the birth of the third child among women whose first two children are girls. We also introduce a novel constrained maximum likelihood estimator that imposes a (set of) random constraint(s) and that may be of independent interest.

We show that the standard test for testing overidentifying restrictions, which compares the J-statistic (Hansen, 1982) to 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 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.