Presence Of Nuisance Parameters Research Articles

Testing in GARCH‐X models: boundary, correlations and bootstrap theory

In this article, we consider the so‐called Fixed Shrinkage (FS) bootstrap for the class of GARCH models with explanatory variables (GARCH‐X). Under the assumption of stationary covariates, the proposed FS bootstrap does not require modeling the covariates, as these are kept fixed in the bootstrap generating process. Our main focus is on testing whether one or more of the covariates can be excluded in the GARCH‐X model. As is well‐known the limiting distribution of the likelihood‐ratio (LR) statistic in this setting is non‐standard and depends in particular on whether nuisance parameters are on the boundary or in the interior of the parameter space. In particular, and as detailed here, the non‐standard limiting distribution depends on correlations, or dependence, between the explanatory variables. The FS bootstrap takes the presence of nuisance parameters into account by implementing shrinking as proposed in Cavaliere et al. (2022) for pure ARCH models. We establish asymptotic validity of the FS bootstrap for GARCH‐X models, and demonstrate by simulations that the bootstrap‐based test performs extremely well even when nuisance parameters lie on the boundary of the parameter space. The empirical illustration amplifies that the presence of nuisance parameters (especially whether or not on the boundary) are vital for interpreting the dynamics of conditional volatility in financial stock market indices.

Multiscale scanning with nuisance parameters

Abstract We develop a multiscale scanning method to find anomalies in a d-dimensional random field in the presence of nuisance parameters. This covers the common situation that either the baseline-level or additional parameters such as the variance are unknown and have to be estimated from the data. We argue that state of the art approaches to determine asymptotically correct critical values for multiscale scanning statistics will in general fail when such parameters are naively replaced by plug-in estimators. Instead, we suggest to estimate the nuisance parameters on the largest scale and to use (only) smaller scales for multiscale scanning. We prove a uniform invariance principle for the resulting adjusted multiscale statistic, which is widely applicable and provides a computationally feasible way to simulate asymptotically correct critical values. We illustrate the implications of our theoretical results in a simulation study and in a real data example from super-resolution STED microscopy. This allows us to identify interesting regions inside a specimen in a pre-scan with controlled family-wise error rate.

A new Bayesian discrepancy measure

The aim of this article is to make a contribution to the Bayesian procedure of testing precise hypotheses for parametric models. For this purpose, we define the Bayesian Discrepancy Measure that allows one to evaluate the suitability of a given hypothesis with respect to the available information (prior law and data). To summarise this information, the posterior median is employed, allowing a simple assessment of the discrepancy with a fixed hypothesis. The Bayesian Discrepancy Measure assesses the compatibility of a single hypothesis with the observed data, as opposed to the more common comparative approach where a hypothesis is rejected in favour of a competing hypothesis. The proposed measure of evidence has properties of consistency and invariance. After presenting the definition of the measure for a parameter of interest, both in the absence and in the presence of nuisance parameters, we illustrate some examples showing its conceptual and interpretative simplicity. Finally, we compare a test procedure based on the Bayesian Discrepancy Measure, with the Full Bayesian Significance Test, a well-known Bayesian testing procedure for sharp hypotheses.

On the Nuisance Parameter Elimination Principle in Hypothesis Testing.

The Non-Informative Nuisance Parameter Principle concerns the problem of how inferences about a parameter of interest should be made in the presence of nuisance parameters. The principle is examined in the context of the hypothesis testing problem. We prove that the mixed test obeys the principle for discrete sample spaces. We also show how adherence of the mixed test to the principle can make performance of the test much easier. These findings are illustrated with new solutions to well-known problems of testing hypotheses for count data.

A smoothed p-value test when there is a nuisance parameter under the alternative

We present a new test when there is a nuisance parameter λ under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the subset of λ on which a p-value test based on a test statistic Tn(λ) rejects the null hypothesis. Key contributions are: (i) An asymptotic critical value upper bound for our test is the significance level α, making inference easy. (ii) We only require Tn(λ) to have a known or bootstrappable limit distribution, hence we do not require n-Gaussian asymptotics, allowing for weak or non-identification, boundary values, heavy tails, infill asymptotics, and so on. (iii) A test based on the transformed p-value supλ∈Λpn(λ) may be conservative and in some cases have trivial power, while the PVOT naturally controls for this by smoothing over the nuisance parameter space. Finally, (iv) the PVOT uniquely allows for bootstrap inference in the presence of nuisance parameters when some estimated parameters may not be identified.

Short and Simple Confidence Intervals When the Directions of Some Effects Are Known

Abstract 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.

A Discussion of “A Note on Universal Inference” by Tse and Davison

We congratulate the authors for their insightful contribution to universal inference, as proposed in Wasserman et al. (2020). The split likelihood ratio test (SLRT) provided by the universal inference framework is valid in finite samples and requires essentially no regularity conditions. However, this amazing universality also comes at a price as the SLRT may be highly conservative. In their article, Tse and Davison (2022) explore how nuisance parameters impact the conservativeness of the SLRT. They study the loss of power in the basic normal location model, taking up the analysis in Dunn et al. (2022). Tse and Davison also propose a modification to reduce the conservatism of the SLRT. Universal inference is justified by an application of Markov's inequality, and the modification is based on the clever observation that, in the normal location model, the expectation to be considered may be more tightly controlled by exploiting the independence of two terms that emerge in the presence of nuisance parameters. The impact of using this trick for irregular testing problems is illustrated in an example based on a Gaussian mixture model. An independent recent work of our group studies universal inference from a related but also different angle (Strieder & Drton, 2022). We would like to take the opportunity of this discussion to point out similarities and also differences between this work and the paper of Tse and Davison. The main results in Strieder and Drton (2022) provide the large-sample theory for the SLRT under both the null hypothesis and local alternatives assuming the classical regularity conditions that lead to Wilks' theorem for the standard likelihood ratio test (see Theorem 3.1 and related Corollaries/Remarks in Strieder & Drton, 2022). The setup allows for general composite null hypotheses. The limiting distributions of the SLRT are then used to illustrate how (loss of) power of the test is impacted by dimensionality of the hypotheses. These limiting distributions coincide with the distributions that are considered in the paper of Tse and Davison (e.g. Equations 8, 13–14). The two articles differ in the way the (limiting) distribution of the SLRT is used in order to improve power. While Tse and Davison focus on using the independence of the two parts stemming from the parameter of interest and the nuisance parameter, Strieder and Drton concentrate on highlighting the impact of the data splitting. Tse and Davison utilize insights from the regular setting under the perspective of nuisance parameters to motivate the use of a smaller critical value. Such a smaller critical value obviously increases power and continues to control the size in the regular case. However, with the modified critical value, one no longer has a universal guarantee to control the size in general irregular problems. This raises the question: How far should we go in terms of lowering the critical value once we start setting it based on asymptotic distributions derived for regular testing problems? While this would be easy to do, we certainly do not want to go all the way to the relevant quantile of the regular limiting distribution. But then which principles should we invoke to justify a particular lowered critical value when applying it to irregular problems? As mentioned, the work of Strieder and Drton pursues a different goal and proposes a different modification. Instead of changing the critical value, this modification employs the regular limiting distribution merely to tune the data splitting ratio (at a target power level). Tuning the splitting ratio has a more limited impact on power, but importantly, an SLRT with tuned splitting remains universally valid in finite samples—no matter whether the problem is regular or not. Tse and Davison also investigate the power of the SLRT across various splitting ratios with the conclusion that a ratio of γ = 2 / 3 $$ \gamma =2/3 $$ performs best. We emphasize that this is a feature of the considered low-dimensional setting. Contrasting Figure 2 from the article of Tse and Davison, Figure 1 here displays the power and size of the SLRT when testing a 45-dimensional hypothesis in a 50-dimensional space and shows that in a higher-dimensional setting, a splitting ratio below 0.5 performs best. For a detailed analysis of the optimal choice of the splitting ratio based on the dimensionality of tested null and alternative hypotheses, we refer to Strieder and Drton (2022). Open Access funding enabled and organized by Projekt DEAL.

A note on universal inference

Universal inference enables the construction of confidence intervals and tests without regularity conditions by splitting the data into two parts and appealing to Markov's inequality. Previous investigations have shown that the cost of this generality is a loss of power in regular settings for testing simple hypotheses. The present paper makes three contributions. We first clarify the reasons for the loss of power and use a simple illustrative example to investigate how the split proportion optimizing the power depends on the nominal size of the test. We then show that the presence of nuisance parameters can severely impact the power and suggest a simple asymptotic improvement. Finally, we show that combining many data splits can also sharply diminish power.

Smooth Transition Simultaneous Equation Models

This paper proposes a generalization of the nonlinear simultaneous equation model of Pesaran and Pick (2007) by modelling the comovement between the two endogenous variables as a smooth function of the magnitude of the endogenous variable rather than a step function. The threshold and the speed at which a shock is transmitted are estimated with the other parameters of the model. We investigate the properties of an accurate estimation method which takes into account endogeneity, and a testing procedure for simultaneity in the presence of nuisance parameters under the null hypothesis. We study the conditions on the parameters that ensure the uniqueness of the implicit reduced form of the model. We apply this methodology to the comovement between the sovereign and banking sectors of nine developed countries.

Parameter uncertainties in weighted unbinned maximum likelihood fits

Parameter estimation via unbinned maximum likelihood fits is central for many analyses performed in high energy physics. Unbinned maximum likelihood fits using event weights, for example to statistically subtract background contributions via the sPlot formalism, or to correct for acceptance effects, have recently seen increasing use in the community. However, it is well known that the naive approach to the estimation of parameter uncertainties via the second derivative of the logarithmic likelihood does not yield confidence intervals with the correct coverage in the presence of event weights. This paper derives the asymptotically correct expressions and compares them with several commonly used approaches for the determination of parameter uncertainties, some of which are shown to not generally be asymptotically correct. In addition, the effect of uncertainties on event weights is discussed, including uncertainties that can arise from the presence of nuisance parameters in the determination of sWeights.

Most powerful test against a sequence of high dimensional local alternatives

We develop a powerful quadratic test for the overall significance of many covariates in a dense regression model in the presence of nuisance parameters. By equally weighting the sample moments, the test is asymptotically correct in high dimensions even when the number of coefficients is larger than the sample size. Our theory allows a non-parametric error distribution and weakly exogenous nuisance variables, in particular autoregressors in many applications. Using random matrix theory, we show that the test has the optimal asymptotic testing power among a large class of competitors against local alternatives whose coordinates are dense in the eigenbasis of the high dimensional sample covariance matrix among regressors. The asymptotic results are adaptive to the covariates’ cross-sectional and temporal dependence structure and do not require a limiting spectral law of their sample covariance matrix. In the most general case, the nuisance estimation may play a role in the asymptotic limit and we give a robust modification for these irregular scenarios. Monte Carlo studies suggest a good power performance of our proposed test against high dimensional dense alternative for various data generating processes. We apply the test to detect the significance of over one hundred exogenous variables in the FRED-MD database for predicting the monthly growth in the US industrial production index.

Saddlepoint Approximations for Spatial Panel Data Models

We develop new higher-order asymptotic techniques for the Gaussian maximum likelihood estimator in a spatial panel data model, with fixed effects, time-varying covariates, and spatially correlated errors. Our saddlepoint density and tail area approximation feature relative error of order with n being the cross-sectional dimension and T the time-series dimension. The main theoretical tool is the tilted-Edgeworth technique in a nonidentically distributed setting. The density approximation is always nonnegative, does not need resampling, and is accurate in the tails. Monte Carlo experiments on density approximation and testing in the presence of nuisance parameters illustrate the good performance of our approximation over first-order asymptotics and Edgeworth expansion. An empirical application to the investment–saving relationship in OECD (Organisation for Economic Co-operation and Development) countries shows disagreement between testing results based on the first-order asymptotics and saddlepoint techniques. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

PARAMETRIC CONDITIONAL MEAN INFERENCE WITH FUNCTIONAL DATA APPLIED TO LIFETIME INCOME CURVES

AbstractWe propose a framework for estimation of the conditional mean function in a parametric model with function space covariates. The approach employs a functional mean squared error objective criterion. Under regularity conditions, consistency and asymptotic normality are established. The analysis extends to situations where the asymptotic properties are influenced by estimation errors arising from the presence of nuisance parameters. Wald, Lagrange multiplier, and quasi‐likelihood ratio statistics are studied. An empirical application conducts lifetime income path comparisons across different demographic groups according to years of work experience.

On the testing hypothesis in uniform family of distributions with nuisance parameter

Abstract The problem of finding optimal tests in the family of uniform distributions is investigated. The general forms of the uniformly most powerful and generalized likelihood ratio tests are derived. Moreover, the problem of finding the uniformly most powerful unbiased test for testing two-sided hypothesis in the presence of nuisance parameter is investigated, and it is shown that such a test is equivalent to the generalized likelihood ratio test for the same problem. The simulation study is performed to evaluate the performance of power function of the tests.

Unified Bayesian theory of sparse linear regression with nuisance parameters

We study frequentist asymptotic properties of Bayesian procedures for high-dimensional Gaussian sparse regression when unknown nuisance parameters are involved. Nuisance parameters can be finite-, high-, or infinite-dimensional. A mixture of point masses at zero and continuous distributions is used for the prior distribution on sparse regression coefficients, and appropriate prior distributions are used for nuisance parameters. The optimal posterior contraction of sparse regression coefficients, hampered by the presence of nuisance parameters, is also examined and discussed. It is shown that the procedure yields strong model selection consistency. A Bernstein-von Mises-type theorem for sparse regression coefficients is also obtained for uncertainty quantification through credible sets with guaranteed frequentist coverage. Asymptotic properties of numerous examples are investigated using the theory developed in this study.

Approximations of the power functions for Wald, likelihood ratio, and score tests and their applications to linear and logistic regressions.

Traditionally, asymptotic tests are studied and applied under local alternative (Aivazian, et al., 1985). There exists a widespread opinion that the Wald, likelihood ratio, and score tests are asymptotically equivalent. We dispel this myth by showing that These tests have different statistical power in the presence of nuisance parameters. The local properties of the tests are described in terms of the first and second derivative evaluated at the null hypothesis. The comparison of the tests are illustrated with two popular regression models: linear regression with random predictor and logistic regression with binary covariate. We study the aberrant behavior of the tests when the distance between the null and alternative does not vanish with the sample size. We demonstrate that these tests have different asymptotic power. In particular, the score test is generally asymptotically biased but slightly superior for linear regression in a close neighborhood of the null. The power approximations are confirmed through simulations.

Low-complexity detection of small frequency deviation by the generalized LMPU test

In this paper, we consider the detection of a small deviation of the frequency of sinusoidal signals, which arises in various signal processing applications. The generalized likelihood ratio test (GLRT) for this problem uses the maximum likelihood (ML) estimator of the frequency, and therefore suffers from high computational complexity. In addition, the GLRT is not necessarily optimal and its performance may degrade for non-asymptotic scenarios that are characterized by close hypotheses and small sample sizes. In this paper we propose a new detection method, named the generalized locally most powerful unbiased (GLMPU) test, which is a general method for local detection in the presence of nuisance parameters. A closed-form expression of the GLMPU test is developed for the detection of frequency deviation in the case where the complex amplitudes of the measured signals and the noise variance are unknown. Numerical simulations show improved performance over the GLRT in terms of probability of detection performance and computational complexity.

Estimation and Testing of a Smooth Transition Simultaneous Equation Model

This paper proposes a generalization of the nonlinear simultaneous equation model of Pesaran and Pick (2007) by modelling the comovement between the two endogenous variables as a smooth function of the magnitude of the endogenous variable rather than a step function. The threshold and the speed at which a shock is transmitted are estimated with the other parameters of the model. We investigate the properties of an accurate estimation method which takes into account endogeneity, and a testing procedure for simultaneity in the presence of nuisance parameters under the null hypothesis. We study the conditions on the parameters that ensure the uniqueness of the implicit reduced form of the model. We apply this methodology to the comovement between the sovereign and banking sectors of nine developed countries.

Inference of local regression in the presence of nuisance parameters

We consider inference based on local estimating equations in the presence of nuisance parameters. The framework is useful for a number of applications including those in economic policy evaluation based on discontinuities or kinks and in real-time financial risk management. We focus on the criterion-function-based (in particular, empirical likelihood-based) inference, and establish conditions under which the test statistic has a pivotal asymptotic distribution. In the key step of eliminating nuisance parameters in the (possibly non-smooth) criterion function, we consider two different approaches based on either concentration or Laplace-type plug-in estimation. The former is natural, and the latter does not require optimization and can be computationally attractive in applications. Our framework can easily incorporate bias correction induced by localization, and the inference is robust to the identification strength of the parameter of interest. The high-level assumptions are illustrated with several examples. We also conduct Monte Carlo simulations and provide an empirical application which assesses heterogeneous effects of academic probation in college and gender differences under the quantile regression discontinuity design.

POWER PROPERTIES OF EMPIRICAL LIKELIHOOD FOR STATIONARY PROCESSES

In this paper we discuss the second order power properties of empirical likelihood for stationary processes. The asymptotic distribution of empirical likelihood ratio statistics under a sequence of local alternatives is given. 1. Introduction. Since its introduction by Owen (1988, 1990), empirical likelihood has become a useful tool for nonparametric inference. It is well known that the empirical likelihood ratio statistic inherits a number of properties of the parametric likelihood ratio statistic. Owen has shown that the empirical likelihood ratio statistic has limiting chi- squared distribution. Qin and Lawless (1994) connected the theories of empirical likelihood and general estimating equations. Hence using empirical likelihood ratio statistics it is possible to obtain tests and confidence regions for a wide range of problems, including linear models (Owen, 1991). Another property of empirical likelihood which also resembles that of a parametric likelihood is Bartlett correction; see for example Hall and La Scala (1990) for the case of the mean parameters, DiCiccio, et al (1991) for the case of smooth functions of means, Chen (1993, 1994) for linear regression model, Chen and Cui (2006) in the presence of nuisance parameters. For dependent data, Monti (1997) applied the empirical likelihood approach to the derivative of the Whittle likelihood. Kitamura (1997) considered blockwise empirical likeli- hood ratios based on data blocks rather than individual observations. Recently, Nordman and Lahiri (2006) introduced a version of empirical likelihood based on the periodogram and spectral estimating equations. They elucidated the asymptotic properties of frequency domain empirical likelihood for linear processes exhibiting both short- and long-range de- pendence. In this paper we consider the second order power properties of empirical likelihood for stationary processes. Section 2 provides a survey of frequency domain empirical likelihood which is due to Nordman and Lahiri (2006). Section 3 gives the asymptotic distribution of empirical likelihood ratio statistics under a sequence of local alternatives. The proofs of results are relegated to Section 4.