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This article sheds light on the asymptotic behavior of diagonal elements of projection matrices associated with instruments or regressors under many instrument/regressor asymptotics. When the diagonal elements do not exhibit variation asymptotically, certain results in the many instrument/regressor literature lead to elegant solutions and conclusions. We establish conditions when this happens, provide relevant examples, and analyze instrument designs, for which this property does or does not hold.

We develop and evaluate sequential testing tools for a class of nonparametric tests for predictability of financial returns that include, in particular, the directional accuracy and excess profitability tests. Our sequential methods consider in a unified framework both retrospection of a historical sample and monitoring newly arriving data. To this end, we focus on linear monitoring boundaries that are continuations of horizontal lines corresponding to retrospective critical values, elaborating on both two-sided and one-sided testing. We run a simulation study and illustrate the methodology by testing for directional and mean predictability of returns in young stock markets in Eastern Europe.

This paper establishes central limit theorems (CLTs) and proposes how to perform valid inference in factor models. We consider a setting where many counties/regions/assets are observed for many time periods, and when estimation of a global parameter includes aggregation of a cross-section of heterogeneous microparameters estimated separately for each entity. The CLT applies for quantities involving both cross-sectional and time series aggregation, as well as for quadratic forms in time-aggregated errors. This paper studies the conditions when one can consistently estimate the asymptotic variance, and proposes a bootstrap scheme for cases when one cannot. A small simulation study illustrates performance of the asymptotic and bootstrap procedures. The results are useful for making inferences in two-step estimation procedures related to factor models, as well as in other related contexts. Our treatment avoids structural modeling of cross-sectional dependence but imposes time-series independence.

Estimation of GARCH models can be simplified by augmenting quasi-maximum likelihood (QML) estimation with variance targeting, which reduces the degree of parameterization and facilitates estimation. We compare the two approaches and investigate, via simulations, how non-normality features of the return distribution affect the quality of estimation of the volatility equation and corresponding value-at-risk predictions. We find that most GARCH coefficients and associated predictions are more precisely estimated when no variance targeting is employed. Bias properties are exacerbated for a heavier-tailed distribution of standardized returns, while the distributional asymmetry has little or moderate impact, these phenomena tending to be more pronounced under variance targeting. Some effects further intensify if one uses ML based on a leptokurtic distribution in place of normal QML. The sample size has also a more favorable effect on estimation precision when no variance targeting is used. Thus, if computational costs are not prohibitive, variance targeting should probably be avoided.

We propose a market timing test for conditional mean independence of financial returns. The new excess predictability (EP) test statistic has an interpretation of a properly normalized return of a certain trading strategy. We discuss similarities of the EP test to the popular directional accuracy (DA) test of Pesaran and Timmermann. Power properties of the EP test are advantageous, and size properties are comparable to those of the DA test. We illustrate application of the test using weekly data on the S&P500 index.

We consider a standard instrumental variables model contaminated by the presence of a large number of exogenous regressors. In an asymptotic framework where this number is proportional to the sample size, we study the impact of their ratio on the validity of existing estimators and tests. When the instruments are few, the inference using the conventional 2SLS estimator and associated t and J statistics, as well as the Anderson—Rubin and Kleibergen tests, is still valid. When the instruments are many, the LIML estimator remains consistent, but the presence of many exogenous regressors changes its asymptotic variance. Moreover, the conventional bias correction of the 2SLS estimator is no longer appropriate. We provide asymptotically correct versions of bias correction for the 2SLS estimator, derive its asymptotically correct variance estimator, extend the Hansen—Hausman—Newey LIML variance estimator to the case of many exogenous regressors, and propose asymptotically valid modifications of the J overidentification tests based on the LIML and bias-corrected 2SLS estimators.

We develop a test for a restricted functional form of a mean regression when a complex distributional model for all variables is estimated. The test statistic is an average squared deviation from the estimated hypothesized function of the form implied by the estimated parametric model, and is asymptotically distributed as a mixture of distributions. The test is easy to implement using numerical derivatives, and it performs well in samples of typical size. We illustrate the test using data on labor market characteristics of US young men.

For stationary time series models with serial correlation, we consider generalized method of moments (GMM) estimators that use heteroskedasticity and autocorrelation consistent (HAC) positive definite weight matrices and generalized empirical likelihood (GEL) estimators based on smoothed moment conditions. Following the analysis of Newey and Smith (2004) for independent observations, we derive second order asymptotic biases of these estimators. The inspection of bias expressions reveals that the use of smoothed GEL, in contrast to GMM, removes the bias component associated with the correlation between the moment function and its derivative, while the bias component associated with third moments depends on the employed kernel function. We also analyze the case of no serial correlation, and find that the seemingly unnecessary smoothing and HAC estimation can reduce the bias for some of the estimators.

Sequential procedures for the testing for structural stability do not provide enough guidance on the shape of boundaries that are used to decide on acceptance or rejection, requiring only that the overall size of the test is asymptotically controlled. We introduce and motivate a reasonable criterion for the shape of boundaries which requires that the test size be uniformly distributed over the testing period. Under this criterion, we numerically construct boundaries for the most popular sequential tests that are characterized by a test statistic behaving asymptotically either as a Wiener process or Brownian bridge. We handle this problem both in the context of retrospecting a historical sample and in the context of monitoring newly arriving data. We tabulate the boundaries by fitting them to certain flexible yet parsimonious functional forms. Interesting patterns emerge in an illustrative application of sequential tests to the Phillips curve model.

This paper studies the asymptotic validity of the Anderson–Rubin (AR) test and the J test for overidentifying restrictions in linear models with many instruments. When the number of instruments increases at the same rate as the sample size, we establish that the conventional AR and J tests are asymptotically incorrect. Some versions of these tests, which are developed for situations with moderately many instruments, are also shown to be asymptotically invalid in this framework. We propose modifications of the AR and J tests that deliver asymptotically correct sizes. Importantly, the corrected tests are robust to the numerosity of the moment conditions in the sense that they are valid for both few and many instruments. The simulation results illustrate the excellent properties of the proposed tests.