Estimating logit models with small samples

In small samples, maximum likelihood (ML) estimates of logit model coefficients have substantial bias away from zero. As a solution, we remind political scientists of Firth's (1993, Biometrika, 80, 27–38) penalized maximum likelihood (PML) estimator. Prior research has described and used PML, especially in the context of separation, but its small sample properties remain under-appreciated. The PML estimator eliminates most of the bias and, perhaps more importantly, greatly reduces the variance of the usual ML estimator. Thus, researchers do not face a bias-variance tradeoff when choosing between the ML and PML estimators—the PML estimator has a smaller bias and a smaller variance. We use Monte Carlo simulations and a re-analysis of George and Epstein (1992, American Political Science Review, 86, 323–337) to show that the PML estimator offers a substantial improvement in small samples (e.g., 50 observations) and noticeable improvement even in larger samples (e.g., 1000 observations).

Fractional unit-root tests allowing for a fractional frequency flexible Fourier form trend: predictability of Covid-19

In this study we propose a fractional frequency flexible Fourier form fractionally integrated ADF unit-root test, which combines the fractional integration and nonlinear trend as a form of the Fourier function. We provide the asymptotics of the newly proposed test and investigate its small-sample properties. Moreover, we show the best estimators for both fractional frequency and fractional difference operator for our newly proposed test. Finally, an empirical study demonstrates that not considering the structural break and fractional integration simultaneously in the testing process may lead to misleading results about the stochastic behavior of the Covid-19 pandemic.

QANOVA: quantile-based permutation methods for general factorial designs

Population means and standard deviations are the most common estimands to quantify effects in factorial layouts. In fact, most statistical procedures in such designs are built toward inferring means or contrasts thereof. For more robust analyses, we consider the population median, the interquartile range (IQR) and more general quantile combinations as estimands in which we formulate null hypotheses and calculate compatible confidence regions. Based upon simultaneous multivariate central limit theorems and corresponding resampling results, we derive asymptotically correct procedures in general, potentially heteroscedastic, factorial designs with univariate endpoints. Special cases cover robust tests for the population median or the IQR in arbitrary crossed one-, two- and higher-way layouts with potentially heteroscedastic error distributions. In extensive simulations, we analyze their small sample properties and also conduct an illustrating data analysis comparing children’s height and weight from different countries.

Testing financial time series for autocorrelation: Robust Tests

Two modified Portmanteau statistics are studied under dependence assumptions common in financial applications which can be used for testing that heteroskedastic time series are serially uncorrelated without assuming independence or Normality. Their asymptotic distribution is found to be null and their small sample properties are examined via Monte Carlo. The power of the tests is studied under the MA and GARCH-in-mean alternatives. The tests exhibit an appropriate empirical size and are seen to be more powerful than a robust Box-Pierce to the selected alternatives. Real data on daily stock returns and exchange rates is used to illustrate the tests.

Testing identification via heteroskedasticity in structural vector autoregressive models

Summary Tests for identification through heteroskedasticity in structural vector autoregressive analysis are developed for models with two volatility states where the time point of volatility change is known. The tests are Wald-type tests for which only the unrestricted model, including the covariance matrices of the two volatility states, has to be estimated. The residuals of the model are assumed to be from the class of elliptical distributions, which includes Gaussian models. The asymptotic null distributions of the test statistics are derived, and simulations are used to explore their small-sample properties. Two empirical examples illustrate the usefulness of the tests in applied work.

RANDOMIZATION TESTS OF COPULA SYMMETRY

New nonparametric tests of copula exchangeability and radial symmetry are proposed. The novel aspect of the tests is a resampling procedure that exploits group invariance conditions associated with the relevant symmetry hypothesis. They may be viewed as feasible versions of randomization tests of symmetry, the latter being inapplicable due to the unobservability of marginals. Our tests are simple to compute, control size asymptotically, consistently detect arbitrary forms of asymmetry, and do not require the specification of a tuning parameter. Simulations indicate excellent small sample properties compared with existing procedures involving the multiplier bootstrap.

A regularization approach to the minimum distance estimation: application to structural macroeconomic estimation using IRFs

This article considers the invertibility problem of the optimal weighting matrix encountered during Impulse Response Function Matching Estimation (IRFME) of Dynamic Stochastic General Equilibrium (DSGE) Models. We propose to use a regularized inverse and derive the asymptotic properties of the estimator. We show that the asymptotic distribution of our estimator converges to that of the optimal estimator which has important implications for testing the fit of the model. We demonstrate the small sample properties of the estimator by Monte Carlo simulation exercises. Finally, we use our estimator to estimate the model in Altig et al.

Monte Carlo comparison of LCCA- and ML-based cointegration tests for panel var process with cross-sectional cointegrating vectors

Small-sample properties of bootstrap cointegration rank tests for unrestricted panel VAR process are considered when long-run cross-sectional dependencies occur. It is shown that the bootstrap cointegration rank tests for the panel VAR model based on levels canonical correlation analysis are oversized, whereas the bootstrap cointegration rank tests based on maximum likelihood framework are undersized. Moreover, the former tests are in general outperformed by the latter in terms of performance. The results of the investigation indicate that the ML-based bootstrap cointegration rank tests perform well in small samples for small-sized panel VAR models with a few cross-sections.

Misspecified Discrete Choice Models and Huber-White Standard Errors

I analyze properties of misspecified discrete choice models and the efficacy of Huber-White (sometimes called ‘robust’) standard errors. The Huber-White correction provides asymptotically correct standard errors for a consistent estimator from a misspecified model. There is little justification for using Huber-White standard errors in discrete choice models since misspecification usually leads to inconsistent estimators. I derive necessary and sufficient conditions for consistency of the maximum likelihood estimator of any potentially misspecified random utility model (e.g. conditional logit). I also derive (easily satisfied) sufficient conditions for consistent estimation of the sign of the data generating parameter. It follows the researcher can consistently test the sign (or nullity) of the parameter from the data generating process using the (possibly) misspecified conditional logit. I investigate small sample properties of the Huber-White estimator via a simulation study and find the correction provides little to no improvement for inferences.