Bayesian inference for random coefficient dynamic panel data models

We develop a hierarchical Bayesian approach for inference in random coefficient dynamic panel data models. Our approach allows for the initial values of each unit's process to be correlated with the unit-specific coefficients. We impose a stationarity assumption for each unit's process by assuming that the unit-specific autoregressive coefficient is drawn from a logitnormal distribution. Our method is shown to have favorable properties compared to the mean group estimator in a Monte Carlo study. We apply our approach to analyze energy and protein intakes among individuals from the Philippines.     

Kernel estimates of the mean and the volatility functions in a nonlinear autoregressive model with ARCH errors

In this paper, we deal with the nonparametric kernel estimation of the regression and volatility functions pertaining to nonlinear autoregressive model with ARCH errors. Under stationarity and ergodicity, we establish the strong uniform consistency and asymptotic normality of the estimators. Our results hold without any mixing condition and do not require the existence of marginal densities. Furthermore, rates of convergence are obtained.     

On the Performance of Popular Unit-Root Tests Against Various Nonlinear Dynamic Models: A Simulation Study

ABSTRACT

There is a widespread perception that standard unit-root tests have poor discriminatory power when they are applied to time series with nonlinear dynamics. Via Monte Carlo simulations this study re-examines the finite sample properties of selected univariate tests for unit-root and stationarity under a broad class of nonlinear dynamic models. Our simulation experiments produce a couple of interesting findings. First, performance of tests is driven by the degree of underlying persistence rather than the nonlinear dynamics per se. Tests under study exhibit reasonable performance for nonlinear models with mild persistence, while the accuracy of inference deteriorates substantially when the models are highly persistent regardless of the linearity. Second, when it comes to deciding which one to identify first between linearity and stationarity, our results suggest to conduct linearity test first to enhance the reliability of test inference.     

Nonparametric Inference for Time-Varying Coefficient Quantile Regression

The article considers nonparametric inference for quantile regression models with time-varying coefficients. The errors and covariates of the regression are assumed to belong to a general class of locally stationary processes and are allowed to be cross-dependent. Simultaneous confidence tubes (SCTs) and integrated squared difference tests (ISDTs) are proposed for simultaneous nonparametric inference of the latter models with asymptotically correct coverage probabilities and Type I error rates. Our methodologies are shown to possess certain asymptotically optimal properties. Furthermore, we propose an information criterion that performs consistent model selection for nonparametric quantile regression models of nonstationary time series. For implementation, a wild bootstrap procedure is proposed, which is shown to be robust to the dependent and nonstationary data structure. Our method is applied to studying the asymmetric and time-varying dynamic structures of the U.S. unemployment rate since the 1940s. Supplementary materials for this article are available online.     

Bayesian estimation for time-series regressions improved with exact likelihoods

We propose an estimation procedure for time-series regression models under the Bayesian inference framework. With the exact method of Wise [Wise, J. (1955). The autocorrelation function and spectral density function. Biometrika, 42, 151–159], an exact likelihood function can be obtained instead of the likelihood conditional on initial observations. The constraints on the parameter space arising from the stationarity conditions are handled by a reparametrization, which was not taken into consideration by Chib [Chib, S. (1993). Bayes regression with autoregressive errors: A Gibbs sampling approach. J. Econometrics, 58, 275–294] or Chib and Greenberg [Chib, S. and Greenberg, E. (1994). Bayes inference in regression model with ARMA(p, q) errors. J. Econometrics, 64, 183–206]. Simulation studies show that our method leads to better inferential results than their results.     

Interaction Between Autocorrelation and Conditional Heteroscedasticity: A Random-Coefficient Approach

In applied econometrics, we tend to tackle specification problems one at a time rather than considering them jointly. This has serious consequences for statistical inference. One example of this is considering autocorrelation and autoregressive conditional heteroscedasticity (ARCH) separately. In this article we consider a linear regression model with random coefficient autoregressive disturbances that provides a convenient framework to analyze autocorrelation and ARCH simultaneously. Our stationarity conditions and testing results reveal the strong interaction between ARCH and autocorrelation. An empirical example of testing the unbiasedness of experts' expectations of inflation demonstrates that neglecting conditional heteroscedasticity or misspecifying the autocorrelation structure might result in unreliable inference.     

Inference in Approximately Sparse Correlated Random Effects Probit Models With Panel Data

Abstract

We propose a simple procedure based on an existing “debiased” l₁-regularized method for inference of the average partial effects (APEs) in approximately sparse probit and fractional probit models with panel data, where the number of time periods is fixed and small relative to the number of cross-sectional observations. Our method is computationally simple and does not suffer from the incidental parameters problems that come from attempting to estimate as a parameter the unobserved heterogeneity for each cross-sectional unit. Furthermore, it is robust to arbitrary serial dependence in underlying idiosyncratic errors. Our theoretical results illustrate that inference concerning APEs is more challenging than inference about fixed and low-dimensional parameters, as the former concerns deriving the asymptotic normality for sample averages of linear functions of a potentially large set of components in our estimator when a series approximation for the conditional mean of the unobserved heterogeneity is considered. Insights on the applicability and implications of other existing Lasso-based inference procedures for our problem are provided. We apply the debiasing method to estimate the effects of spending on test pass rates. Our results show that spending has a positive and statistically significant average partial effect; moreover, the effect is comparable to found using standard parametric methods.     

Non-stationary partition modeling of geostatistical data for malaria risk mapping

The most common assumption in geostatistical modeling of malaria is stationarity, that is spatial correlation is a function of the separation vector between locations. However, local factors (environmental or human-related activities) may influence geographical dependence in malaria transmission differently at different locations, introducing non-stationarity. Ignoring this characteristic in malaria spatial modeling may lead to inaccurate estimates of the standard errors for both the covariate effects and the predictions. In this paper, a model based on random Voronoi tessellation that takes into account non-stationarity was developed. In particular, the spatial domain was partitioned into sub-regions (tiles), a stationary spatial process was assumed within each tile and between-tile correlation was taken into account. The number and configuration of the sub-regions are treated as random parameters in the model and inference is made using reversible jump Markov chain Monte Carlo simulation. This methodology was applied to analyze malaria survey data from Mali and to produce a country-level smooth map of malaria risk.     

BAYESIAN ANALYSIS OF A FRACTIONAL COINTEGRATION MODEL

The concept of fractional cointegration, whereby deviations from an equilibrium relationship follow a fractionally integrated process, has attracted some attention of late. The extended concept allows cointegration to be associated with mean reversion in the error, rather than requiring the more stringent condition of stationarity. This paper presents a Bayesian method for conducting inference about fractional cointegration. The method is based on an approximation of the exact likelihood, with a Jeffreys prior being used to offset identification problems. Numerical results are produced via a combination of Markov chain Monte Carlo algorithms. The procedure is applied to several purchasing power parity relations, with substantial evidence found in favor of parity reversion.     

Generalized autoregressive and moving average models: multicollinearity,interpretation and a new modified model

In this paper, we call attention of two observed features in practical applications of the Generalized Autoregressive Moving Average (GARMA) model due to the structure of its linear predictor. One is the multicollinearity which may lead to a non-convergence of the maximum likelihood, using iteratively reweighted least squares, and the inflation of the estimator's variance. The second is that the inclusion of the same lagged observations into the autoregressive and moving average components confounds the interpretation of the parameters. A modified model, GAR-M, is presented to reduce the multicollinearity and to improve the interpretation of the parameters. The expectation and variance under stationarity conditions are presented for the identity and logarithm link function. In a general sense, simulation studies show that the maximum likelihood estimators based on the GARMA and GAR-M models are equivalent but the GAR-M estimators presented a little lower standard errors and some restrictions in the parametric space are imposed to guarantee the stationarity of the process. Also, a real data analysis illustrates the GAR-M fit for daily hospitalization rates of elderly people due to respiratory diseases from October 2012 to April 2015 in São Paulo city, Brazil.     

A note on fourth moment-based direction dependence measures when regression errors are non normal

Measures of direction dependence enable researchers to determine the directionality of linear effects in bivariate data. Existing fourth moment-based approaches assume that regression errors are at least mesokurtic. Direction dependence measures based on the co-kurtosis of variables are proposed that relax this assumption. Simulations suggest that co-kurtosis-based measures perform equally well as existing kurtosis-based methods when distributional assumptions of the latter are fulfilled. However, kurtosis-based approaches are sensitive to platy- or leptokurtic errors, while co-kurtosis-based measures protect Type I error and power rates. Data requirements necessary for causal inference and recommendations for selecting proper direction dependence measures are discussed.     

Bandwidth selection through cross-validation for semi-parametric varying-coefficient partially linear models

We study bandwidth selection for a class of semi-parametric models. The proper choice of optimal bandwidth minimizes the prediction errors of the model. We provide detailed derivation of our procedure and the corresponding computation algorithms. Our proposed method simplifies the computation of the cross-validation criteria and facilitates more complicated inference and analysis in practice. A data set from Wisconsin Diabetes Registry has been analysed as an illustration.     

Testing for stationarity in large panels with cross-dependence,and US evidence on unit labor cost

A new stationarity test for heterogeneous panel data with large cross-sectional dimension is developed and used to examine a panel with growth rates of unit labor cost in the USA. The test allows for strong cross-unit dependence in the form of unbounded long-run correlation matrices, for which a simple parameterization is proposed. A KPSS-type distribution results asymptotically if letting T→∞ be followed by N→∞. Some evidence against stationarity (short memory) is found for the examined series.     

Bootstrapping Noncausal Autoregressions: With Applications to Explosive Bubble Modeling

In this article, we develop new bootstrap-based inference for noncausal autoregressions with heavy-tailed innovations. This class of models is widely used for modeling bubbles and explosive dynamics in economic and financial time series. In the noncausal, heavy-tail framework, a major drawback of asymptotic inference is that it is not feasible in practice as the relevant limiting distributions depend crucially on the (unknown) decay rate of the tails of the distribution of the innovations. In addition, even in the unrealistic case where the tail behavior is known, asymptotic inference may suffer from small-sample issues. To overcome these difficulties, we propose bootstrap inference procedures using parameter estimates obtained with the null hypothesis imposed (the so-called restricted bootstrap). We discuss three different choices of bootstrap innovations: wild bootstrap, based on Rademacher errors; permutation bootstrap; a combination of the two (“permutation wild bootstrap”). Crucially, implementation of these bootstraps do not require any a priori knowledge about the distribution of the innovations, such as the tail index or the convergence rates of the estimators. We establish sufficient conditions ensuring that, under the null hypothesis, the bootstrap statistics estimate consistently particular conditionaldistributions of the original statistics. In particular, we show that validity of the permutation bootstrap holds without any restrictions on the distribution of the innovations, while the permutation wild and the standard wild bootstraps require further assumptions such as symmetry of the innovation distribution. Extensive Monte Carlo simulations show that the finite sample performance of the proposed bootstrap tests is exceptionally good, both in terms of size and of empirical rejection probabilities under the alternative hypothesis. We conclude by applying the proposed bootstrap inference to Bitcoin/USD exchange rates and to crude oil price data. We find that indeed noncausal models with heavy-tailed innovations are able to fit the data, also in periods of bubble dynamics. Supplementary materials for this article are available online.     

Ecological inference techniques: an empirical evaluation using data describing gender and voter turnout at New Zealand elections, 1893–1919

Summary.  The difference, if any, between men's and women's voting patterns is of particular interest to historians of gender and politics. For elections that were held before the introduction of opinion surveying in the 1940s, little data are available with which to estimate such differences. We apply six methods for ecological inference to estimate men's and women's voting rates in New Zealand (NZ), 1893–1919. NZ is an interesting case-study, since it was the first self-governing country where women could vote. Furthermore, NZ officials recorded the voting rates of men and women at elections, making it possible to compare estimates produced by methods for ecological inference with known true values, thus testing the efficacy of different methods for ecological inference for this data set. We find that the most popular methods for ecological inference, namely Goodman's ecological regression and King's parametric method, give poor estimates, as does the much debated neighbourhood method. However, King's non-parametric method, Chambers and Steel's semiparametric method and the Steel, Beh and Chambers homogeneous approach all gave good estimates that were close to the known values, with the homogeneous approach performing best overall. The success of these methods in this example suggests that ecological inference may be a viable option when investigating gender and voting. Moreover, researchers using ecological inference in other fields may do well to consider a range of statistical methods. This work is a significant NZ contribution to historical politics and the first quantitative contribution, in the area of NZ gender and politics.     

Confidence Sets for Cointegrating Coefficients Based on Stationarity Tests

Standard methods for inference in cointegrating systems require all the variables to have exact unit roots and are not at all robust even to slight violations of this condition. In this article, I consider an alternative approach to inference in a cointegrating system. This involves testing the hypothesis that a cointegrating vector takes on a specified value by testing for the stationarity of the associated residual. Confidence sets for the cointegrating vector can be constructed by exploiting the equivalence between tests and confidence sets. This method has the advantage that it remains valid even if the regressors have roots that are not exactly equal to unity.     

Approximate inference in heteroskedastic regressions: A numerical evaluation

The commonly made assumption that all stochastic error terms in the linear regression model share the same variance (homoskedasticity) is oftentimes violated in practical applications, especially when they are based on cross-sectional data. As a precaution, a number of practitioners choose to base inference on the parameters that index the model on tests whose statistics employ asymptotically correct standard errors, i.e. standard errors that are asymptotically valid whether or not the errors are homoskedastic. In this paper, we use numerical integration methods to evaluate the finite-sample performance of tests based on different (alternative) heteroskedasticity-consistent standard errors. Emphasis is placed on a few recently proposed heteroskedasticity-consistent covariance matrix estimators. Overall, the results favor the HC4 and HC5 heteroskedasticity-robust standard errors. We also consider the use of restricted residuals when constructing asymptotically valid standard errors. Our results show that the only test that clearly benefits from such a strategy is the HC0 test.     

Maximum-Entropy Prior Uncertainty and Correlation of Statistical Economic Data

Empirical estimates of source statistical economic data such as trade flows, greenhouse gas emissions, or employment figures are always subject to uncertainty (stemming from measurement errors or confidentiality) but information concerning that uncertainty is often missing. This article uses concepts from Bayesian inference and the maximum entropy principle to estimate the prior probability distribution, uncertainty, and correlations of source data when such information is not explicitly provided. In the absence of additional information, an isolated datum is described by a truncated Gaussian distribution, and if an uncertainty estimate is missing, its prior equals the best guess. When the sum of a set of disaggregate data is constrained to match an aggregate datum, it is possible to determine the prior correlations among disaggregate data. If aggregate uncertainty is missing, all prior correlations are positive. If aggregate uncertainty is available, prior correlations can be either all positive, all negative, or a mix of both. An empirical example is presented, which reports relative uncertainties and correlation priors for the County Business Patterns database. In this example, relative uncertainties range from 1% to 80% and 20% of data pairs exhibit correlations below ?0.9 or above 0.9. Supplementary materials for this article are available online.     

Stochastic Model Specification Search for Time-Varying Parameter VARs

This article develops a new econometric methodology for performing stochastic model specification search (SMSS) in the vast model space of time-varying parameter vector autoregressions (VARs) with stochastic volatility and correlated state transitions. This is motivated by the concern of overfitting and the typically imprecise inference in these highly parameterized models. For each VAR coefficient, this new method automatically decides whether it is constant or time-varying. Moreover, it can be used to shrink an otherwise unrestricted time-varying parameter VAR to a stationary VAR, thus providing an easy way to (probabilistically) impose stationarity in time-varying parameter models. We demonstrate the effectiveness of the approach with a topical application, where we investigate the dynamic effects of structural shocks in government spending on U.S. taxes and gross domestic product (GDP) during a period of very low interest rates.     

Confidence intervals for the mean and a percentile based on zero-inflated lognormal data

The problems of estimating the mean and an upper percentile of a lognormal population with nonnegative values are considered. For estimating the mean of a such population based on data that include zeros, a simple confidence interval (CI) that is obtained by modifying Tian's [Inferences on the mean of zero-inflated lognormal data: the generalized variable approach. Stat Med. 2005;24:3223—3232] generalized CI, is proposed. A fiducial upper confidence limit (UCL) and a closed-form approximate UCL for an upper percentile are developed. Our simulation studies indicate that the proposed methods are very satisfactory in terms of coverage probability and precision, and better than existing methods for maintaining balanced tail error rates. The proposed CI and the UCL are simple and easy to calculate. All the methods considered are illustrated using samples of data involving airborne chlorine concentrations and data on diagnostic test costs.