This paper analyzes the properties of subsampling, hybrid subsampling, and size-correction methods in two non-regular models. The latter two procedures are introduced in Andrews and Guggenberger (2005b). The models are non-regular in the sense that the test statistics of interest exhibit a discontinuity in their limit distribution as a function of a parameter in the model. The first model is a linear instrumental variables (IV) model with possibly weak IVs estimated using two-stage least squares (2SLS). In this case, the discontinuity occurs when the concentration parameter is zero. The second model is a linear regression model in which the parameter of interest may be near a boundary. In this case, the discontinuity occurs when the parameter is on the boundary. The paper shows that in the IV model one-sided and equal-tailed two-sided subsampling tests and confidence intervals (CIs) based on the 2SLS t statistic do not have correct asymptotic size. This holds for both fully- and partially-studentized t statistics. But, subsampling procedures based on the partially-studentized t statistic can be size-corrected. On the other hand, symmetric two-sided subsampling tests and CIs are shown to have (essentially) correct asymptotic size when based on a partially-studentized t statistic. Furthermore, all types of hybrid subsampling tests and CIs are shown to have correct asymptotic size in this model. The above results are consistent with "impossibility" results of Dufour (1997) because subsampling and hybrid subsampling CIs are shown to have infinite length with positive probability. Subsampling CIs for a parameter that may be near a lower boundary are shown to have incorrect asymptotic size for upper one-sided and equal-tailed and symmetric two-sided CIs. Again, size-correction is possible. In this model as well, all types of hybrid subsampling CIs are found to have correct asymptotic size.

Applications of Subsampling, Hybrid, and Size-Correction Methods (joint with D.W.K. Andrews), 2005, this version May 2007

This article studies the introduction of an EU-wide emission standard on the automobile market. Using panel data from 1998 to 2011, I find that firms decreased emission ratings by 14%. Firms use technology adoption and gaming of emission tests to decrease emissions, rather than shifting the sales mix or downsizing. I find that the standard missed its emission target, and from estimating a structural model, I find that the standard was not welfare improving. The political environment in the EU shaped the design and weak enforcement and resulted in firms’ choices for abatement by technology adoption and gaming.

/pdf/abatement-strategies-and-the-cost-of-environmental-3u2v2mmz21.pdf

Abatement Strategies and the Cost of Environmental Regulation: Emission Standards on the European Car Market

We present novel theory for testing for reduction of GARCH-X type models with an exogenous (X) covariate to standard GARCH type models. To deal with the problems of potential nuisance parameters on the boundary of the parameter space as well as lack of identification under the null, we exploit a noticeable property of specific zero-entries in the inverse information of the GARCH-X type models. Specifically, we consider sequential testing based on two likelihood ratio tests and as demonstrated the structure of the inverse information implies that the proposed test neither depends on whether the nuisance parameters lie on the boundary of the parameter space, nor on lack of identification. Asymptotic theory is derived essentially under stationarity and ergodicity, coupled with a regularity assumption on the exogenous covariate X. Our general results on GARCH-X type models are applied to Gaussian based GARCH-X models, GARCH-X models with Student’s t-distributed innovations as well as integer-valued GARCH-X (PAR-X) models.

/pdf/testing-garch-x-type-models-3xgfaec9m6.pdf

Testing garch-x type models

Bootstrap inference on the boundary of the parameter space, with application to conditional volatility models

Indirect Inference (I-I) estimation of structural parameters $\theta$ {requires matching observed and simulated statistics, which are most often generated using an auxiliary model that depends on instrumental parameters $\beta$.} {The estimators of the instrumental parameters will encapsulate} the statistical information used for inference about the structural parameters. As such, artificially constraining these parameters may restrict the ability of the auxiliary model to accurately replicate features in the structural data, which may lead to a range of issues, such as, a loss of identification. However, in certain situations the parameters $\beta$ naturally come with a set of $q$ restrictions. Examples include settings where $\beta$ must be estimated subject to $q$ possibly strict inequality constraints $g(\beta) > 0$, such as, when I-I is based on GARCH auxiliary models. In these settings we propose a novel I-I approach that uses appropriately modified unconstrained auxiliary statistics, which are simple to compute and always exists. We state the relevant asymptotic theory for this I-I approach without constraints and show that it can be reinterpreted as a standard implementation of I-I through a properly modified binding function. Several examples that have featured in the literature illustrate our approach.

Indirect Inference With(Out) Constraints

Subvector inference when the true parameter vector may be near or at the boundary

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.

On asymptotic size distortions in the random coefficients logit model

Testing overidentifying restrictions with a restricted parameter space

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).

Philipp Ketz

Papers

Subvector inference when the true parameter vector may be near or at the boundary

Subvector inference when the true parameter vector may be near or at the boundary

On asymptotic size distortions in the random coefficients logit model

Testing overidentifying restrictions with a restricted parameter space

Allowing for weak identification when testing GARCH-X type models