Hrishikesh D. Vinod

Verifying the Solution from a Nonlinear Solver: A Case Study

We are pleased to confirm that any doubt our article (McCullough and Vinod, 2003; hereafter “MV03”) may have cast on Ron Shachar and Barry Nalebuff (1999; hereafter “SN99”) must be removed. We are especially pleased because we thought it quite unfair that other researchers were able to exempt themselves from such detailed scrutiny. It appears that such researchers no longer will have the luxury of reneging on their agreement to honor the replication policy, as this journal now requires authors of accepted empirical papers to provide all programs and data files for posting on the AER Web site as a precondition of publication. The primary aim of our article (MV03) was to provide a four-part methodology for verifying the solution from a nonlinear solver: check the gradient, examine the trace, analyze the Hessian, and profile the likelihood. We adduced copious evidence (MV03, p. 873) that solvers used by economists can produce inaccurate answers, gave examples of different packages giving different answers to the same nonlinear problems (MV03, p. 874), and showed (MV03, pp. 873–74), at least in this journal, that researchers make no effort to verify the solutions from the solvers that they use. We believe this uncritical acceptance of solutions from nonlinear solvers to be a systemic problem in economic research; that is why we wrote the article—certainly, econometrics texts do not show how to verify the solution from a nonlinear solver. In passing, we also showed how a problem can be too large for conventional PC methods, and indicated the failure of replication policies in this journal and other journals. We used the data and likelihood function from SN99 to illustrate the methodology. In the course of this illustration, we noted that the Hessian was ill-conditioned, suggested that there might exist multiple optima, and that inference based on the Wald statistic was not appropriate. We concluded that the solution we found was, at best, a tentative solution. However, as shown in Shachar and Nalebuff (2004; hereafter SN04), when the problem is rescaled the Hessian is not ill-conditioned; they have correctly identified the difference between the condition number of the badly scaled version of the problem that we analyzed in MV03 and the well-scaled problem that they have analyzed. When the problem is correctly scaled, the Hessian is well-conditioned, the model is locally identifiable, the problem can be solved on a PC, a solution to the problem exists, and SN present it in their Table 1. Though we were aware that rescaling could ameliorate ill-conditioning (MV03, p. 882), we were unaware of the distinction between artificial ill-conditioning and inherent ill-conditioning, so the method we suggested for analyzing the Hessian contained an error of omission. This error caused us to reach an incorrect conclusion concerning the existence of a solution to the problem. We apologize to Professors Shachar and Nalebuff, and we thank them for their gracious understanding in this regard. Accordingly, we have amended our prescription for analyzing the Hessian—see our nearby exchange with David M. Drukker and Vince Wiggins (2004; hereafter “DW”) for complete details. Our

The role of data/code archives in the future of economic research

This essay examines the role of data and program‐code archives in making economic research ‘replicable.’ Replication of published results is recognized as an essential part of the scientific method. Yet, historically, both the ‘demand for’ and ‘supply of’ replicable results in economics has been minimal. ‘Respect for the scientific method’ is not sufficient to motivate either economists or editors of professional journals to ensure the replicability of published results. We enumerate the costs and benefits of mandatory data and code archives, and argue that the benefits far exceed the costs. Progress has been made since the gloomy assessment of Dewald, Thursby and Anderson some 20 years ago in the American Economic Review, but much remains to be done before empirical economics ceases to be a ‘dismal science’ when judged by the replicability of its published results.

Implementing the Double Bootstrap

The single bootstrap already is popular in economics, though the double bootstrap has better convergence properties. We discuss the theory and implementation of the double bootstrap, both with and without the pivotal transformation, and give detailed examples of each. One example is a nonlinear double bootstrap of a Cobb-Douglas production function, and explains the use of Gauss-Newton Regressions as a device to decrease computational time. Another example is double bootstrapping elasticities from a translog production function.

General Nonparametric Regression Estimation and Testing in Econometrics

Publisher Summary This chapter discusses the general nonparametric regression estimation and testing in econometrics. The basic computer algorithm in nonparametric density estimation replaces an estimate of a density f(x) by a weighted sum, where weights can be complicated functions w(x, h). These weighted sums are used to unify the vast literature dealing with nonparametric estimation and inference. Description of economic data requires estimation of higher order moments and quantiles, which can be made richer with the help of nonparametric methods. It is shown that the nonparametric weighted sums can offer truly flexible functional forms as they are feasible in the estimation of the regression function m(x) and in the computation of the partial derivatives of m(x) with respect to the regressors. Econometrics is concerned with computation of marginal productivities and elasticities, which are related to partial derivatives of m(x). The initial appeal of these nonparametric estimators is that one need not specify the functional forms of m(x) to estimate the partials of E(y│x) with respect to x. And, empirical econometrics cannot be free from parametric forms, especially for small sample sizes with dependent observations and several regressors.

Exact Moments for Autor1egressive and Random walk Models for a Zero or Stationary Initial Value

For a first-order autoregressive AR(1) model with zero initial value, x i = ax i−1 ,_, + e i , we provide the bias, mean squared error, skewness, and kurtosis of the maximum likelihood estimator â. Brownian motion approximations by Phillips (1977, Econometrica 45, 463–485; 1978, Biometrika 65, 91–98; 1987, Econometrica 55, 277–301), Phillips and Perron (1988, Biometrika 75, 335–346), and Perron (1991, Econometric Theory 7, 236–252; 1991, Econometrica 59, 211–236), among others, yield an elegant unified theory but do not yield convenient formulas for calibration of skewness and kurtosis. In addition to the usual stationary case |α|

Double bootstrap for shrinkage estimators

Computer-intensive bootstrap methods are potentially problematic when applied to biased estimators similar to ridge regression, because biased estimators may lack a pivot. We propose a modification for Efrons bias corrected percentiles and suggest an approximate pivoting transformation for ridge regression. We indicate practical aspects of Berans double bootstrap designed specifically for situations where there is nonnormality and a lack of a pivot. Econometric applications to estimating consumption functions, a simulation with multicollinear data and new graphical tools are included.

Statistical Analysis of Corruption Data and Using the Internet to Reduce Corruption

Abstract Corruption is a serious problem in Asia and elsewhere. The Harrison and Vinod (1992) confidence interval for the marginal excess burden (MEB) of taxation is used to estimates the economic harm arising from corruption. One dollar of corruption is estimated to impose a burden of

BOUNDS ON THE VARIANCE OF REGRESSION COEFFICIENTS DUE TO HETEROSCEDASTIC OR AUTOREGRESSIVE ERRORS

1.67, which becomes very large when compounded over time. After a brief review of economic theory, this paper uses data on sixteen socio-economic and political variables. A cross-sectional study reveals the relevance of “red tape” and “efficiency of judiciary.” A subset regression using Mallows’ Cp and Akaike information criteria reveals relevance of schooling and income inequality. International aid and cooperation in exposing and fighting corruption and innovative uses of the Internet for information exchange are claimed to be hopeful new tools to fight corruption in the new century.

CEO Age and Outside Directors: A Hazard Analysis

In applications of linear regression analysis, the unknown error covariance matrix has to be somehow estimated. This can lead to biased estimates of the covariance matrix of the regression coefficients. Since such bias is difficult to eliminate completely, its sensitivity to alternative estimates of error covariances is studied by Watson, Theil, Malinvaud, and others with the help of bounds on the bias derived under certain assumptions. This paper gives similar bounds under less restrictive assumptions, and illustrates them in the context of heteroscedasticity and autocorrelation problems. In particular, for the first order error autocorrelation coefficient of p the upper bound on proportionate bias is shown to be reasonably approximated by (1 + p)/(l - p) - 1.

Measurement of Economic Distance Between Blacks and Whites

This paper examines the relationship between CEO tenure, CEO age, the firms industry group, the proportion of directors from outside the firm, and the cost of firing the CEO. A Cox proportional hazard model of CEO survival is used to study the length of the CEOs stay at the firm. We find that, contrary to previous studies, a greater proportion of outsiders has a positive effect on CEO tenure. The significance of this result is however sensitive to the inclusion of age and performance variables. We test for the effects of heterogeneity of industry, and find that firms in homogeneous industries exhibit lower durations. As the cost of firing the CEO rises, tenure also rises.