Publications - Michael Jansson

Nearly Efficient Likelihood Ratio Tests of a Unit Root in an Autoregressive Model of Arbitrary Order

Brien, S., Jansson, M., Nielsen, M. Ø. — Mon, 01 Aug 2022 09:53:28 +0200

Simple Local Polynomial Density Estimators

Cattaneo, M. D., Jansson, M., Ma, X. — Wed, 01 Jul 2020 09:53:28 +0200

This article introduces an intuitive and easy-to-implement nonparametric density estimator based on local polynomial techniques. The estimator is fully boundary adaptive and automatic, but does not require prebinning or any other transformation of the data. We study the main asymptotic properties of the estimator, and use these results to provide principled estimation, inference, and bandwidth selection methods. As a substantive application of our results, we develop a novel discontinuity in density testing procedure, an important problem in regression discontinuity designs and other program evaluation settings. An illustrative empirical application is given. Two companion Stata and R software packages are provided.

Bootstrap-Based Inference for Cube Root Asymptotics

Cattaneo, M. D., Jansson, M., Nagasawa, K. — Tue, 01 Sep 2020 09:53:28 +0200

This paper proposes a valid bootstrap-based distributional approximation for M-estimators exhibiting a Chernoff (1964)-type limiting distribution. For estimators of this kind, the standard nonparametric bootstrap is inconsistent. The method proposed herein is based on the nonparametric bootstrap, but restores consistency by altering the shape of the criterion function defining the estimator whose distribution we seek to approximate. This modification leads to a generic and easy-to-implement resampling method for inference that is conceptually distinct from other available distributional approximations. We illustrate the applicability of our results with four examples in econometrics and machine learning.

Two-step estimation and inference with possibly many included covariates

Cattaneo, M. D., Jansson, M., Xinwei, M. A. — Tue, 01 Jan 2019 09:53:28 +0100

We study the implications of including many covariates in a first-step estimate entering a two-step estimation procedure. We find that a first-order bias emerges when the number of included covariates is “large” relative to the square-root of sample size, rendering standard inference procedures invalid. We show that the jackknife is able to estimate this “many covariates” bias consistently, thereby delivering a new automatic bias-corrected two-step point estimator. The jackknife also consistently estimates the standard error of the original two-step point estimator. For inference, we develop a valid post-bias-correction bootstrap approximation that accounts for the additional variability introduced by the jackknife bias-correction. We find that the jackknife bias-corrected point estimator and the bootstrap post-bias-correction inference perform excellent in simulations, offering important improvements over conventional two-step point estimators and inference procedures, which are not robust to including many covariates. We apply our results to an array of distinct treatment effect, policy evaluation, and other applied microeconomics settings. In particular, we discuss production function and marginal treatment effect estimation in detail.

Improved likelihood ratio tests for cointegration rank in the VAR model

Boswijk, H. P., Jansson, M., Nielsen, M. Ø. — Thu, 01 Jan 2015 09:53:28 +0100

We suggest improved tests for cointegration rank in the vector autoregressive (VAR) model and develop asymptotic distribution theory and local power results. The tests are (quasi-)likelihood ratio tests based on a Gaussian likelihood, but as usual the asymptotic results do not require normally distributed innovations. Our tests differ from existing tests in two respects. First, instead of basing our tests on the conditional (with respect to the initial observations) likelihood, we follow the recent unit root literature and base our tests on the full likelihood as in, e.g., Elliott et al. (1996). Second, our tests incorporate a “sign” restriction which generalizes the one-sided unit root test. We show that the asymptotic local power of the proposed tests dominates that of existing cointegration rank tests.

Manipulation Testing Based on Density Discontinuity

Cattaneo, M. D., Jansson, M., Ma, X. — Mon, 01 Jan 2018 09:53:28 +0100

In this article, we introduce two community-contributed commands, rddensity and rdbwdensity, that implement automatic manipulation tests based on density discontinuity and are constructed using the results for local-polynomial density estimators in Cattaneo, Jansson, and Ma (2017b, Simple local polynomial density estimators, Working paper, University of Michigan). These new tests exhibit better size properties (and more power under additional assumptions) than other conventional approaches currently available in the literature. The first command, rddensity, implements manipulation tests based on a novel local-polynomial density estimation technique that avoids prebinning of the data (improving size properties) and allows for restrictions on other features of the model (improving power properties). The second command, rdbwdensity, implements several bandwidth selectors specifically tailored for the manipulation tests discussed herein. We also provide a companion R package with the same syntax and capabilities as rddensity and rdbwdensity.

Alternative asymptotics and the partially linear model with many regressors

Cattaneo, M. D., Jansson, M., Newey, W. K. — Mon, 01 Jan 2018 09:53:28 +0100

Many empirical studies estimate the structural effect of some variable on an outcome of interest while allowing for many covariates. We present inference methods that account for many covariates. The methods are based on asymptotics where the number of covariates grows as fast as the sample size. We find a limiting normal distribution with variance that is larger than the standard one. We also find that with homoskedasticity this larger variance can be accounted for by using degrees-of-freedom-adjusted standard errors. We link this asymptotic theory to previous results for many instruments and for small bandwidth(s) distributional approximations.

Kernel-Based Semiparametric Estimators: Small Bandwidth Asymptotics and Bootstrap Consistency

Cattaneo, M. D., Jansson, M. — Mon, 01 Jan 2018 09:53:28 +0100

This paper develops asymptotic approximations for kernel-based semiparametric estimators under assumptions accommodating slower-than-usual rates of convergence of their nonparametric ingredients. Our first main result is a distributional approximation for semiparametric estimators that differs from existing approximations by accounting for a bias. This bias is nonnegligible in general, and therefore poses a challenge for inference. Our second main result shows that some (but not all) nonparametric bootstrap distributional approximations provide an automatic method of correcting for the bias. Our general theory is illustrated by means of examples and its main finite sample implications are corroborated in a simulation study.

Inference in Linear Regression Models with Many Covariates and Heteroscedasticity

Cattaneo, M. D., Jansson, M., Newey, W. K. — Tue, 03 Jul 2018 09:53:28 +0200

The linear regression model is widely used in empirical work in economics, statistics, and many other disciplines. Researchers often include many covariates in their linear model specification in an attempt to control for confounders. We give inference methods that allow for many covariates and heteroscedasticity. Our results are obtained using high-dimensional approximations, where the number of included covariates is allowed to grow as fast as the sample size. We find that all of the usual versions of Eicker–White heteroscedasticity consistent standard error estimators for linear models are inconsistent under this asymptotics. We then propose a new heteroscedasticity consistent standard error formula that is fully automatic and robust to both (conditional) heteroscedasticity of unknown form and the inclusion of possibly many covariates. We apply our findings to three settings: parametric linear models with many covariates, linear panel models with many fixed effects, and semiparametric semi-linear models with many technical regressors. Simulation evidence consistent with our theoretical results is provided, and the proposed methods are also illustrated with an empirical application. Supplementary materials for this article are available online.

Bootstrap-Based Inference for Cube Root Consistent Estimators

Cattaneo, M. D., Jansson, M., Nagasawa, K. — Fri, 05 May 2017 09:53:28 +0200

This note proposes a consistent bootstrap-based distributional approximation for cube root consistent estimators such as the maximum score estimator of Manski (1975) and the isotonic density estimator of Grenander (1956). In both cases, the standard nonparametric bootstrap is known to be inconsistent. Our method restores consistency of the nonparametric bootstrap by altering the shape of the criterion function defining the estimator whose distribution we seek to approximate. This modification leads to a generic and easy-to-implement resampling method for inference that is conceptually distinct from other available distributional approximations based on some form of modified bootstrap. We offer simulation evidence showcasing the performance of our inference method in finite samples. An extension of our methodology to general M-estimation problems is also discussed.

Treatment Effects with Many Covariates and Heteroskedasticity

Cattaneo, M. D., Jansson, M., Newey, W. K. — Mon, 03 Aug 2015 09:53:28 +0200

The linear regression model is widely used in empirical work in Economics. Researchers often include many covariates in their linear model specification in an attempt to control for confounders. We give inference methods that allow for many covariates and heteroskedasticity. Our results are obtained using high-dimensional approximations, where the number of covariates are allowed to grow as fast as the sample size. We find that all of the usual versions of Eicker-White heteroskedasticity consistent standard error estimators for linear models are inconsistent under this asymptotics. We then propose a new heteroskedasticity consistent standard error formula that is fully automatic and robust to both (conditional) heteroskedasticity of unknown form and the inclusion of possibly many covariates. We apply our findings to three settings: (i) parametric linear models with many covariates, (ii) semiparametric semi-linear models with many technical regressors, and (iii) linear panel models with many fixed effects

Small bandwidth asymptotics for density-weighted average derivatives

Cattaneo, M. D., Crump, R. K., Jansson, M. — Wed, 01 Jan 2014 09:53:28 +0100

This paper proposes (apparently) novel standard error formulas for the density-weighted average derivative estimator of Powell, Stock, and Stoker (Econometrica 57, 1989). Asymptotic validity of the standard errors developed in this paper does not require the use of higher-order kernels, and the standard errors are robust in the sense that they accommodate (but do not require) bandwidths that are smaller than those for which conventional standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that the finite sample coverage rates of confidence intervals constructed using the standard errors developed in this papercoincide (approximately) with the nominal coverage rates across a nontrivial range of bandwidths.

Bootstrapping density-weighted average derivatives

Cattaneo, M. D., Crump, R. K., Jansson, M. — Wed, 01 Jan 2014 09:53:28 +0100

We investigate the properties of several bootstrap-based inference procedures for semiparametric density-weighted average derivatives. The key innovation in this paper is to employ an alternative asymptotic framework to assess the properties of these inference procedures. This theoretical approach is conceptually distinct from the traditional approach (based on asymptotic linearity of the estimator and Edgeworth expansions), and leads to different theoretical prescriptions for bootstrap-based semiparametric inference. First, we show that the conventional bootstrap-based approximations to the distribution of the estimator and its classical studentized version are both invalid in general. This result shows a fundamental lack of robustness of the associated, classical bootstrap-based inference procedures with respect to the bandwidth choice. Second, we present a new bootstrap-based inference procedure for density-weighted average derivatives that is more robust to perturbations of the bandwidth choice, and hence exhibits demonstrable superior theoretical statistical properties over the traditional bootstrap-based inference procedures. Finally, we also examine the validity and invalidity of related bootstrap-based inference procedures and discuss additional results that may be of independent interest. Some simulation evidence is also presented.

Bootstrapping Kernel-Based Semiparametric Estimators

Cattaneo, M. D. ., Jansson, M. — Mon, 25 Aug 2014 09:53:28 +0200

This paper develops alternative asymptotic results for a large class of two-step semiparametric estimators. The first main result is an asymptotic distribution result for such estimators and differs from those obtained in earlier work on classes of semiparametric two-step estimators by accommodating a non-negligible bias. A noteworthy feature of the assumptions under which the result is obtained is that reliance on a commonly employed stochastic equicontinuity condition is avoided. The second main result shows that the bootstrap provides an automatic method of correcting for the bias even when it is non-negligible.

Generalized Jackknife Estimators of Weighted Average Derivatives

Cattaneo, M. D. ., Crump, R. K., Jansson, M. — Tue, 01 Jan 2013 09:53:28 +0100

Nearly efficient likelihood ratio tests of the unit root hypothesis

Jansson, M., Nielsen, M. Ø. — Sat, 01 Sep 2012 09:53:28 +0200

Optimal Inference for Instrumental Variables Regression with non-Gaussian Errors

Cattaneo, M. D., Crump, R. K., Jansson, M. — Sun, 01 Jan 2012 09:53:28 +0100

Improved Likelihood Ratio Tests for Cointegration Rank in the VAR Model

Boswijk, H. P., Jansson, M., Nielsen, M. Ø. — Fri, 21 Sep 2012 09:53:28 +0200

We suggest improved tests for cointegration rank in the vector autoregressive (VAR) model and develop asymptotic distribution theory and local power results. The tests are (quasi-)likelihood ratio tests based on a Gaussian likelihood, but of course the asymptotic results apply more generally. The power gains relative to existing tests are due to two factors. First, instead of basing our tests on the conditional (with respect to the initial observations) likelihood, we follow the recent unit root literature and base our tests on the full likelihood as in, e.g., Elliott, Rothenberg, and Stock (1996). Secondly, our tests incorporate a “sign”restriction which generalizes the one-sided unit root test. We show that the asymptotic local power of the proposed tests dominates that of existing cointegration rank tests.

Generalized Jackknife Estimators of Weighted Average Derivatives

Cattaneo, M. D. ., Crump, R. K., Jansson, M. — Sat, 01 Jan 2011 09:53:28 +0100

With the aim of improving the quality of asymptotic distributional approximations for nonlinear functionals of nonparametric estimators, this paper revisits the large-sample properties of an important member of that class, namely a kernel-based weighted average derivative estimator. Asymptotic linearity of the estimator is established under weak conditions. Indeed, we show that the bandwidth conditions employed are necessary in some cases. A bias-corrected version of the estimator is proposed and shown to be asymptotically linear under yet weaker bandwidth conditions. Consistency of an analog estimator of the asymptotic variance is also established. To establish the results, a novel result on uniform convergence rates for kernel estimators is obtained.

Bootstrapping Density-Weighted Average Derivatives

Cattaneo, M. D., Crump, R. K., Jansson, M. — Fri, 01 Jan 2010 09:53:28 +0100

Employing the "small bandwidth" asymptotic framework of Cattaneo, Crump, and Jansson (2009), this paper studies the properties of a variety of bootstrap-based inference procedures associated with the kernel-based density-weighted averaged derivative estimator proposed by Powell, Stock, and Stoker (1989). In many cases validity of bootstrap-based inference procedures is found to depend crucially on whether the bandwidth sequence satisfies a particular (asymptotic linearity) condition. An exception to this rule occurs for inference procedures involving a studentized estimator employing a "robust" variance estimator derived from the "small bandwidth" asymptotic framework. The results of a small-scale Monte Carlo experiment are found to be consistent with the theory and indicate in particular that sensitivity with respect to the bandwidth choice can be ameliorated by using the "robust"variance estimator

Nearly Efficient Likelihood Ratio Tests for Seasonal Unit Roots

Jansson, M., Nielsen, M. Ø. — Thu, 01 Jan 2009 09:53:28 +0100

In an important generalization of zero frequency autore-
gressive unit root tests, Hylleberg, Engle, Granger, and Yoo (1990) developed
regression-based tests for unit roots at the seasonal frequencies in quarterly time
series. We develop likelihood ratio tests for seasonal unit roots and show that
these tests are "nearly efficient" in the sense of Elliott, Rothenberg, and Stock
(1996), i.e. that their local asymptotic power functions are indistinguishable
from the Gaussian power envelope. Currently available nearly efficient testing
procedures for seasonal unit roots are regression-based and require the choice
of a GLS detrending parameter, which our likelihood ratio tests do not.

Robust Data-Driven Inference for Density-Weighted Average Derivatives

Cattaneo, M. D., Crump, R. K., Jansson, M. — Thu, 01 Jan 2009 09:53:28 +0100

This paper presents a new data-driven bandwidth selector compatible with the
small bandwidth asymptotics developed in Cattaneo, Crump, and Jansson (2009) for density-
weighted average derivatives. The new bandwidth selector is of the plug-in variety, and is
obtained based on a mean squared error expansion of the estimator of interest. An extensive
Monte Carlo experiment shows a remarkable improvement in performance when the bandwidth-
dependent robust inference procedure proposed by Cattaneo, Crump, and Jansson (2009) is
coupled with this new data-driven bandwidth selector. The resulting robust data-driven confi-
dence intervals compare favorably to the alternative procedures available in the literature.

Nearly Efficient Likelihood Ratio Tests of the Unit Root Hypothesis

Jansson, M., Nielsen, M. Ø. — Thu, 01 Jan 2009 09:53:28 +0100

Seemingly absent from the arsenal of currently available
"nearly efficient" testing procedures for the unit root hypothesis, i.e. tests
whose local asymptotic power functions are indistinguishable from the Gaussian
power envelope, is a test admitting a (quasi-)likelihood ratio interpretation. We
show that the likelihood ratio unit root test derived in a Gaussian AR(1) model
with standard normal innovations is nearly efficient in that model. Moreover,
these desirable properties carry over to more complicated models allowing for
serially correlated and/or non-Gaussian innovations.

Small Bandwidth Asymptotics for Density-Weighted Average Derivatives

Cattaneo, M. D., Crump, R. K., Jansson, M. — Tue, 01 Jan 2008 09:53:28 +0100

This paper proposes (apparently) novel standard error formulas for the density-weighted
average derivative estimator of Powell, Stock, and Stoker (1989). Asymptotic validity of
the standard errors developed in this paper does not require the use of higher-order
kernels and the standard errors are "robust" in the sense that they accommodate
(but do not require) bandwidths that are smaller than those for which conventional
standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that
the finite sample coverage rates of con…dence intervals constructed using the standard
errors developed in this paper coincide (approximately) with the nominal coverage rates
across a nontrivial range of bandwidths.

Semiparametric Power Envelopes for Tests of the Unit Root Hypothesis

Jansson, M. — Mon, 01 Jan 2007 09:53:28 +0100

This paper derives asymptotic power envelopes for tests of
the unit root hypothesis in a zero-mean AR(1) model. The power envelopes are
derived using the limits of experiments approach and are semiparametric in the
sense that the underlying error distribution is treated as an unknown infinitedimensional
nuisance parameter. Adaptation is shown to be possible when the
error distribution is known to be symmetric and to be impossible when the
error distribution is unrestricted. In the latter case, two conceptually distinct
approaches to nuisance parameter elimination are employed in the derivation
of the semiparametric power bounds. One of these bounds, derived under an
invariance restriction, is shown by example to be sharp, while the other, derived
under a similarity restriction, is conjectured not to be globally attainable.

Optimal Inference for Instrumental Variables Regression with non-Gaussian Errors

Cattaneo, M. D., Crump, R. K., Jansson, M. — Mon, 01 Jan 2007 09:53:28 +0100

This paper is concerned with inference on the coefficient on
the endogenous regressor in a linear instrumental variables model with a single
endogenous regressor, nonrandom exogenous regressors and instruments,
and i.i.d. errors whose distribution is unknown. It is shown that under mild
smoothness conditions on the error distribution it is possible to develop tests
which are "nearly" efficient when identification is weak and consistent and asymptotically
optimal when identification is strong. In addition, an estimator
is presented which can be used in the usual way to construct valid (indeed,
optimal) confidence intervals when identification is strong. The estimator is of
the two stage least squares variety and is asymptotically efficient under strong
identification whether or not the errors are normal.