Non-asymptotic tests of model performance

This paper describes a non-asymptotic approach to the problem of selection bias in economic forecasting. By using non-asymptotic measure concentration results, it is possible to deal with settings in which the class of potential models is large with respect to the number of data points. The bounds on p values obtained by these methods are necessarily conservative, but they provide a useful benchmark for model selection in settings where asymptotics may not apply.     

Robust covariance matrix estimation and identification of unusual data points: New tools

Most consistent estimators are prone to total breakdown in the presence of a handful of unusual data points (UDPs). This compromises inference. Robust estimation is a (seldom-used) solution; but methods commonly-used in applied research have severe drawbacks. In this paper, building upon methods that are relatively unknown outside of the robust statistics literature, we provide an enhanced tool for robust estimates of mean and covariance, useful both for robust estimation and for detection of unusual data points. It is relatively fast and useful for large data sets. We also provide a new robust cluster method, an input to our broader method, but also useful for standalone UDP detection or cluster analysis. We provide a comparative study of numerous methods that is not available in the current literature. Testing indicates that our method performs at par with, and often better than, two of the currently best available methods. We also demonstrate that the issues we discuss are not merely hypothetical, by applying our tools to real world data, and to re-examine two prominent economic studies. Our methods reveal that their central results are driven by a set of unusual points.     

A Note on Covariance Matrix Estimation in Quantile Regressions

This note discusses some issues related to bandwidth selection based on moment expansions of the mean squared error （MSE） of the regression quantile estimator. We use higher order expansions to provide a way to distinguish among asymptotically equivalent nonparametric estimators. We derive approximations to the （standardized） MSE of the covariance matrix estimation. This facilitates a comparison of different estimators at the second order level, where differences do occur and depend on the bandwidth choice. A method of bandwidth selection is defined by minimizing the second order effect in the mean squared error.     

Semiparametric Estimation of Simultaneous-Equation Microeconometric Models with Index Restrictions

This article introduces semiparametric methods for the estimation of simultaneous-equation microeconometric models with index restrictions. The methods are motivated by a semiparametric minimum-distance procedure, which unifies the estimation of both regression-type and linear or nonlinear simultaneous-equation models without emphasis on the construction of instrumental variables. Single-equation and systematic estimation methods and optimal weighting procedures are considered. The estimators are √ n -consistent and asymptotically normal. For the estimation of nonparametric regression and some sample selection models where the variances of disturbances are functions of the same indices, the optimal weighted estimator attains Chamberlain's efficient bound for models with conditional moment restrictions. The weighted estimator is shown to be optimal within a class of semiparametric instrumental variables estimators.
JEL classification numbers: C14, C24, C34.     

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.     

When outcome heterogeneously matters for selection: a generalized selection correction estimator

The classical Heckman (1976, 1979) selection correction estimator (heckit) is misspecified and inconsistent, if an interaction of the outcome variable with an explanatory variable matters for selection. To address this specification problem, a full information maximum likelihood (FIML) estimator and a simple two-step estimator are developed. Monte Carlo (MC) simulations illustrate that the bias of the ordinary heckit estimator is removed by these generalized estimation procedures. Along with OLS and ordinary heckit, we apply these estimators to data from a randomized trial that evaluates the effectiveness of financial incentives for reducing obesity. Estimation results indicate that the choice of the estimation procedure clearly matters.     

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.     

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.     

Misspecification in Dynamic Panel Data Models and Model‐Free Inferences

This paper discusses the issue of model misspecification and model‐free methods in dynamic panel data analysis. We primarily review existing results, but also provide several new results. When the dynamics are homogeneous, we show that several widely used estimators for panel first‐order autoregressive AR(1) models converge to first‐order autocorrelation, even under misspecification. Under heterogeneity, these estimators converge to the ratio of the means of the first‐order autocovariances and variances. We also discuss the estimation of autocovariances, the estimation of panel AR(∞) models, and the estimation of the distribution of the heterogeneous mean and autocovariances.     

Steady state adjusting trends using a data-driven local polynomial regression

Economic variables usually follow a dynamic trend pattern. However, it is difficult to estimate this trend precisely as numerous economically- and statistically-based estimation methods exist. This contribution proposes a data-driven nonparametric trend that is local polynomial, to improve arbitrary trend estimations of commonly used methods concerning the selection of the smoothing parameter and the dependence structure. An iterative plug-in (IPI) algorithm determines the bandwidth endogenously and allows a theory-based interpretation of the length of growth processes. This length of the bandwidth reflects the lengths of the steady state periods. Consequently, the bandwidth identifies the time period of stable economic conditions and can detect economic changes. To demonstrate the power of this estimation approach, an extensive simulation study is performed. Furthermore, examples using US and UK GDP data along with a guide for the optimal choice of algorithms for empirical applications are provided. This proposed method yields new insights for growth dynamics, cyclical movements and their dependence.     

The Neglect of Monotone Comparative Statics Methods

Monotone methods enable comparative static analysis without the restrictive assumptions of the implicit-function theorem. Ease of use and flexibility in solving comparative static and game-theory problems have made monotone methods popular in the economics literature and in graduate courses, but they are still absent from undergraduate mathematical economics courses and textbooks. In this article, the authors illustrate the generality of monotone comparative statics relative to the implicit function approach. For example, to sign the effect of a discrete policy shift on a choice variable, the marginal returns will increase with the policy parameter. They also apply monotone methods in game theory settings. As mathematical economics courses and majors gain popularity, incorporating monotone methods into curriculum and textbooks would provide a modern treatment of comparative static analysis.     

Wage Formation and Bargaining Power during the Great Depression*

We present an econometric analysis of wage behaviour in Norway during the interwar years. The analysis is based on a panel of manufacturing industry data using GMM estimation methods. Our empirical analysis shows that wage formation in the interwar period can be understood with the help of modern bargaining theory and well‐established wage equations. We estimate a long‐run wage curve that has all the standard features of being homogeneous in prices, proportional to productivity, and with a negative unemployment elasticity. We also present some new Monte Carlo evidence on the properties of the estimators used.     

On the estimation and forecasting of international migration: how relevant is heterogeneity across countries?

This paper performs a comparative analysis of estimation as well as of out-of-sample forecasting results of more than 20 estimators common in the panel data literature using the data on migration to Germany from 18 source countries in the period 1967–2001. Our results suggest that the choice of an estimation procedure has a substantial impact on the parameter estimates of the migration function. Out-of-sample forecasting results indicate the following: (1) the standard fixed effects estimators clearly outperforms the pooled OLS estimator, (2) both the fixed effects estimators and the hierarchical Bayes estimator exhibit the superior forecast performance, (3) the fixed effects estimators outperform GMM and other instrumental variables estimators, (4) forecasting performance of heterogenous estimators is mediocre in our data set.     

Data-dependent selection of the lag truncation parameter in unit root tests of the Phillips-Perron type

The Andrews (Econometrica, 1991, 59, 817–858) plug-in method of heteroscedastic and autocovariance consistent covariance matrix estimation is used to construct estimators of the long-run variance parameter for use in Phillips-Perron unit root tests. This allows the lag truncation parameter to be data dependent. Monte Carlo size and power estimates are obtained suggesting that this apparently natural approach does not provide significant improvements in test performance.     

A Comparison of Alternative Non-parametric Estimators of the Short Rate Diffusion Coefficient

In this paper we discuss the estimation of the diffusion coefficient in one-factor models for the short rate via non-parametric methods. We test the estimators proposed by Ait-Sahalia (1996) , Stanton (1997) and Bandi and Phillips (2003) on Monte Carlo simulations of the Vasicek and CIR model. We show that the Ait-Sahalia estimator is not applicable for values of the mean reversion coefficient typically displayed by interest rate data, while the Stanton and Bandi–Phillips estimators perform better. Each of the three estimators depends crucially on the choice of the bandwidth parameter. Our analysis shows that the estimators give different results for both the data set analysed by Ait-Sahalia (1996) and by Stanton (1997) . Finally we show that the data sets used by Ait-Sahalia and Stanton are inherently different and, in particular, that very short-term data exhibit characteristics which are inconsistent with a diffusion.     

Further empirical evidence on the forecasting of volatility with smooth transition exponential smoothing

Smooth transition exponential smoothing (STES) uses a logistic function of a user-specified transition variable as adaptive time varying smoothing parameter. This paper empirically addresses three aspects of the use of STES for volatility forecasting. Previous empirical results showed the method performing well in comparison with fixed parameter exponential smoothing and a variety of GARCH models. However, those results related only to forecasting weekly volatility. In this paper, we address the use of STES for forecasting daily volatility. A second issue that we evaluate is the robustness of STES in the presence of extreme outlying observations. The third aspect that we consider is the use of trading volume within a transition variable in the STES method. Our simulation results suggest that STES performs well in terms of robustness, when compared with standard methods and several alternative robust methods. Analysis using stock return data shows that STES has the potential to outperform standard and robust forms of fixed parameter exponential smoothing and GARCH models. The results suggest the use of the sign and size of past shocks as STES transition variables, and provide no clear support for the incorporation of trading volume in a transition variable.     

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.     

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.     

Cross sectional and panel estimation of convergence

Cross sectional estimation of convergence regressions is known to be hazardous if there is convergence towards heterogeneous steady state values. In this paper, Monte Carlo methods are used to investigate the implications of this parameter heterogeneity problem. The cross sectional and pooled OLS estimators are compared with a panel estimator which is unaffected by heterogeneity. If there is heterogeneity, the latter outperforms both the unconditional and conditional cross sectional and pooled OLS estimators.     

Alternative Techniques for Estimation of Cross-Section Gravity Models

This paper compares four different estimators with respect to their suitability for cross‐section gravity model estimation. In many circumstances, a Hausman–Taylor approach can be recommended. This framework may provide consistent parameter estimates, when OLS or the traditional random‐effects model are biased. In contrast to the fixed‐effects approach, it allows to estimate parameters of variables such as GDP or GDP per capita, which vary only in a single dimension. The Hausman–Taylor model deserves attention in the estimation of cross‐sectional gravity models.