Estimating aggregate autoregressive processes when only macro data are available

The aggregation of individual random AR(1) models generally leads to an AR(∞$\infty$) process. We provide two consistent estimators of aggregate dynamics based on either a parametric regression or a minimum distance approach for use when only macro data are available. Notably, both estimators allow us to recover some moments of the cross-sectional distribution of the autoregressive parameter. Both estimators perform very well in our Monte-Carlo experiment, even with finite samples.     

Estimation of Censored Regression Model: A Simulation Study

We investigate the finite sample performance of several estimators proposed for the panel data Tobit regression model with individual effects, including Honoré estimator, Hansen’s best two-step GMM estimator, the continuously updating GMM estimator, and the empirical likelihood estimator (ELE). The latter three estimators are based on more conditional moment restrictions than the Honoré estimator, and consequently are more efficient in large samples. Although the latter three estimators are asymptotically equivalent, the last two have better finite sample performance. However, our simulation reveals that the continuously updating GMM estimator performs no better, and in most cases is worse than Honoré estimator in small samples. The reason for this finding is that the latter three estimators are based on more moment restrictions that require discarding observations. In our designs, about seventy percent of observations are discarded. The insufficiently few number of observations leads to an imprecise weighted matrix estimate, which in turn leads to unreliable estimates. This study calls for an alternative estimation method that does not rely on trimming for finite sample panel data censored regression model.     

Hierarchically spatial autoregressive and moving average error model

This paper considers a hierarchically spatial autoregressive and moving average error (HSEARMA) model. This model captures the spatially autoregressive and moving average error correlation, the county-level random effects, and the district-level random effects nested within each county. We propose optimal generalized method of moments (GMM) estimators for the spatial error correlation coefficient and the error components' variances terms, as well as a feasible generalized least squares (FGLS) estimator for the regression parameter vector. Further, we prove consistency of the GMM estimator and establish the asymptotic distribution of the FGLS estimator. A finite-scale Monte Carlo simulation is conducted to demonstrate the good finite sample performances of our GMM-FGLS estimators.     

Rationality tests in the presence of instabilities in finite samples

The importance of expectations in modern macroeconomic models and in particular of policy makers expectations for forward looking policy rules has generated a lot of interest in time series of professional forecasts (including central bank staff forecasts). This has spawned a large literature on the evaluation of forecasts that are not model based or where the model is unknown. Although, the available time series of historical forecasts are typically short, this literature has so far mostly disregarded the small sample properties of the proposed tests and estimators. In this paper we fill this gap in the literature, focusing on a set of recently proposed rationality tests for unstable environments. Using a Monte Carlo study we demonstrate that the asymptotic tests are substantially oversized in finite samples including any sample size that is practically available. We provide finite sample adjusted critical values, that allow those tests to be properly applied to sample sizes of typically available forecasts such as the Survey of Professional Forecasters, the Federal Open Market Committee. The critical values we provide will help to avoid false rejections using those data.     

Fractional integration with drift: estimation in small samples

We examine the finite-sample behavior of estimators of the order of integration in a fractionally integrated time-series model. In particular, we compare exact time-domain likelihood estimation to frequency-domain approximate likelihood estimation. We show that over-differencing is of critical importance for time-domain maximum-likelihood estimation in finite samples. Overdifferencing moves the differencing parameter (in the over-differenced model) away from the boundary of the parameter space, while at the same time obviating the need to estimate the drift parameter. The two estimators that we compare are asymptotically equivalent. In small samples, however, the time-domain estimator has smaller mean squared error than the frequency-domain estimator. Although the frequency-domain estimator has larger bias than the time-domain estimator for some regions of the parameter bias, it can also have smaller bias. We use a simulation procedure which exploits the approximate linearity of the bias function to reduce the bias in the time-domain estimator.     

SPECTRAL DENSITY ESTIMATION AND ROBUST HYPOTHESIS TESTING USING STEEP ORIGIN KERNELS WITHOUT TRUNCATION*

A new class of kernels for long‐run variance and spectral density estimation is developed by exponentiating traditional quadratic kernels. Depending on whether the exponent parameter is allowed to grow with the sample size, we establish different asymptotic approximations to the sampling distribution of the proposed estimators. When the exponent is passed to infinity with the sample size, the new estimator is consistent and shown to be asymptotically normal. When the exponent is fixed, the new estimator is inconsistent and has a nonstandard limiting distribution. It is shown via Monte Carlo experiments that, when the chosen exponent is small in practical applications, the nonstandard limit theory provides better approximations to the finite sample distributions of the spectral density estimator and the associated test statistic in regression settings.     

Feasible bias-corrected OLS, within-groups, and first-differences estimators for typical micro and macro AR(1) panel data models

In this paper we suggest several alternative ways of constructing feasible bias-corrected (FBC) pooled least squares, within-groups, and first-differences estimators for AR(1) panel data models. In a Monte Carlo simulation study involving data with the qualities normally encountered by both microeconomists and macroeconomists we found that the estimators proposed seem to possess better finite sample properties than the GMM estimators usually employed in this setting: most FBC estimators are unbiased, even when the time series is highly persistent, display less variability, and are not affected by the relative magnitude of the variances for the individual effect and the idiosyncratic error.     

Comparing features of convenient estimators for binary choice models with endogenous regressors

Abstract We discuss the relative advantages and disadvantages of four types of convenient estimators of binary choice models when regressors may be endogenous or mismeasured or when errors are likely to be heteroscedastic. For example, such models arise when treatment is not randomly assigned and outcomes are binary. The estimators we compare are the two‐stage least squares linear probability model, maximum likelihood estimation, control function estimators, and special regressor methods. We specifically focus on models and associated estimators that are easy to implement. Also, for calculating choice probabilities and regressor marginal effects, we propose the average index function (AIF), which, unlike the average structural function (ASF), is always easy to estimate.     

The combined estimator approach to model transferability and updating

The idea of transferability is to employ in model estimation, fitted model parameters computed from a different data set. Thecombined estimator approach to the transferability problem is expressed as a linear combination of the unbiased direct estimators on the two data sets. The major gain is in variance reduction. The combined estimator is shown to have superior accuracy, in a Mean Square Error sense, to a unbiased direct estimator whenever the transfer bias is relatively small. A test that indicates if the combined estimator is superior to the direct estimator is provided. Variances of the direct estimators are assumed to be known. Monte Carlo experiments are performed to assess the quality of the approximations. The results show that the approximations used are highly conservative. An empirical example of the combined estimator applied to a discrete choice problem is presented.     

Estimating average economic growth in time series data with persistency

This paper studies estimation of average economic growth in time series models with persistency. In particular, a joint estimation of the trend coefficient and the autoregressive parameter is considered. An analysis on the proposed estimator is provided. Our analysis is also extended to the case with general disturbance distributions. A nonlinear M estimator and a class of partially adaptive M estimators which adapt themselves with respect to a measure of the tailthickness are considered. The joint estimator and its partially adapted version are compared with several conventional estimators. Monte Carlo experiments indicate that the proposed estimators have good finite sample performance. We use the proposed estimation procedure to estimate the growth rates for real GNP and consumer price index in 40 countries.     

Stochastic imitation in finite games

In this paper we model an evolutionary process with perpetual random shocks, where individuals sample population-specific strategy-payoff pairs and imitate the most successful behavior. For finite n-player games we prove that in the limit, as the perturbations tend to zero, only strategy-tuples in minimal sets closed under single better replies will be played with positive probability. If the strategy-tuples in one such minimal set have strictly higher payoffs than all outside strategy-tuples, then the strategy-tuples in this set will be played with probability one in the limit, provided the minimal set is a product set and the sample is sufficiently large.     

RIDGE ESTIMATORS FOR PROBIT REGRESSION: WITH AN APPLICATION TO LABOUR MARKET DATA

In this paper we propose ridge regression estimators for probit models since the commonly applied maximum likelihood (ML) method is sensitive to multicollinearity. An extensive Monte Carlo study is conducted where the performance of the ML method and the probit ridge regression (PRR) is investigated when the data are collinear. In the simulation study we evaluate a number of methods of estimating the ridge parameter k that have recently been developed for use in linear regression analysis. The results from the simulation study show that there is at least one group of the estimators of k that regularly has a lower mean squared error than the ML method for all different situations that have been evaluated. Finally, we show the benefit of the new method using the classical Dehejia and Wahba dataset which is based on a labour market experiment.     

Simulated Non-Parametric Estimation of Dynamic Models

This paper introduces a new class of parameter estimators for dynamic models, called simulated non-parametric estimators (SNEs). The SNE minimizes appropriate distances between non-parametric conditional (or joint) densities estimated from sample data and non-parametric conditional (or joint) densities estimated from data simulated out of the model of interest. Sample data and model-simulated data are smoothed with the same kernel, which considerably simplifies bandwidth selection for the purpose of implementing the estimator. Furthermore, the SNE displays the same asymptotic efficiency properties as the maximum-likelihood estimator as soon as the model is Markov in the observable variables. The methods introduced in this paper are fairly simple to implement, and possess finite sample properties that are well approximated by the asymptotic theory. We illustrate these features within typical estimation problems that arise in financial economics.     

Estimation of Dynamic Panel Data Sample Selection Models

This paper considers the problem of identification and estimation in panel data sample selection models with a binary selection rule, when the latent equations contain strictly exogenous variables, lags of the dependent variables, and unobserved individual effects. We derive a set of conditional moment restrictions which are then exploited to construct two-step GMM-type estimators for the parameters of the main equation. In the first step, the unknown parameters of the selection equation are consistently estimated. In the second step, these estimates are used to construct kernel weights in a manner such that the weight that any two-period individual observation receives in the estimation varies inversely with the relative magnitude of the sample selection effect in the two periods. Under appropriate assumptions, these "kernel-weighted" GMM estimators are consistent and asymptotically normal. The finite sample properties of the proposed estimators are investigated in a small Monte-Carlo study.     

Enhancing the local power of IVX-based tests in predictive regressions

IVX estimation is used increasingly often in predictive regressions with regressors of unknown persistence. While not exhibiting the second-order bias the OLS estimator has in this setup, IVX estimators have reduced rates of convergence when the regressors are highly persistent. The reduced convergence rates may sometimes lead to power losses in finite samples when testing for no predictability, for instance. The note discusses a simple way of improving the local power of IVX-based tests, consisting of augmenting the predictive regression with the lagged dependent variable. This implies a feed-back loop which strengthens the signal of the IVX instrument without changing its dynamic properties. The proposed augmentation works best when the power loss of IVX would have been maximal compared to the infeasible OLS-based test.     

Small-sample inference with spatial HAC estimators

This paper examines the small-sample performance of spatial HAC (SHAC) estimators of the standard errors on parameters. We find that, in small to moderately-sized datasets, the use of HAC estimators may be recommended only with a relatively large degree of cross-sectional interdependence.     

PANEL DATA UNIT ROOT TEST WITH FIXED TIME DIMENSION

The consideration of the limit theory in which T is fixed and N is allowed to go to infinity improves the finite‐sample properties of the tests and avoids the imposition of the relative rates at which T and N go to infinity.     

The small sample properties of several limited information estimators in interdependent structures with scalar and vector autoregressive errors

This study is concerned with an examination of the finite sample behaviour of several limited information estimators in interdependent structures with error terms related over time and in certain specifications across equations. The Monte Carlo or simulation approach is adopted and applied to computationally manageable structures containing lagged dependent variables. The analysis of the Monte Carlo experiments is formulated in terms of estimating response functions, the dependent variables of which are the first two moments of target model estimators. In addition to the impact of simultaneity, autocorrelation and lagged dependent variables on the estimators, evidence is also accumulated on the small sample effects of misspecification in terms of the faulty inclusion and deletion of regressors. The results of the experiments revealed the substantial impact which autocorrelation can have on ordinary least squares (OLS) and two-stage least squares (2SLS) in terms of efficiency loss. Averaging over all the coefficients in the models, estimators which take account of both autocorrelation and simultaneity had a relative efficiency factor of about 1.5 to 1.9. Many of the parameters in the Monte Carlo model (including misspecification errors, multicollinearity) had qualitatively the same effect on bias and dispersion properties of the estimators.     

Maximum likelihood estimators of the dagum model parameters

This paper will show the sample size needed to provide good maximum likelihood estimators of the Dagum model parameters. The principal goal of this study is to verify the asymptotic properties of these estimators for finite sample sizes comparable to the ones employed in the real surveys (for example, the Labor Force Survey).     

TESTING FOR STATIONARITY WITH A BREAK IN PANELS WHERE THE TIME DIMENSION IS FINITE

In this paper, we consider the case of finite time dimension in the panel stationarity tests with structural breaks. By fixing T, the finite sample properties of the tests for both micro (T small and N large) and macro (both T and N large) panel data are generally greatly improved. More importantly, the derivation of the tests for finite T and , as opposed to joint asymptotic where N and simultaneously, avoids the imposition of the rate condition making the test valid for any (T, N) blend. Four models corresponding to the usual combination of breaks are considered. The asymptotic distributions of the test are derived under the null hypothesis and are shown to be normally distributed. Their moments for T fixed are derived analytically employing Ghazal’s corollary 1. The case with unknown breaks is also considered. The proposed tests have generally empirical sizes that are very close to the nominal size. The Monte Carlo simulations show that the power of the test statistics increases substantially with N and T.