Jeffrey M. Wooldridge

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR a...

A Capital Asset Pricing Model with Time Varying Covariances

The capital asset pricing model provides a theoretical structure for the pricing of assets with uncertain returns. The premium to induce risk-averse investors to bear risk is proportional to the nondiversifiable risk, which is measured by the covariance of the asset return with the market portfolio return. In this paper a multivariate generalized autoregressive conditional heteroscedastic process is estimated for returns to bills, bonds, and stock where the expected return is proportional to the conditional convariance of each return with that of a fully diversified or market portfolio. It is found that the conditional covariances are quite variable over time and are a significant determinant of time-varying risk premia. The implied betas are also time-varying and forecastable. However, there is evidence that other variables including innovations in consumption should also be considered in the investors information set when estimating the conditional distribution of returns.

ECONOMETRIC METHODS FOR FRACTIONAL RESPONSE VARIABLES WITH AN APPLICATION TO 401 (K) PLAN PARTICIPATION RATES

We offer simple quasi-likelihood methods for estimating regression models with a fractional dependent variable and for performing asymptotically valid inference. Compared with log-odds type procedures, there is no difficulty in recovering the regression function for the fractional variable, and there is no need to use ad hoc transformations to handle data at the extreme values of zero and one. We also offer some new, simple specification tests by nesting the logit or probit function in a more general functional form. We apply these methods to a data set of employee participation rates in 401(k) pension plans.

Cluster-Sample Methods in Applied Econometrics

Inference methods that recognize the clustering of individual observations have been available for more than 25 years. Brent Moulton (1990) caught the attention of economists when he demonstrated the serious biases that can result in estimating the effects of aggregate explanatory variables on individual-specific response variables. The source of the downward bias in the usual ordinary least-squares (OLS) standard errors is the presence of an unobserved, state-level effect in the error term. More recently, John Pepper (2002) showed how accounting for multi-level clustering can have dramatic effects on t statistics. While adjusting for clustering is much more common than it was 10 years ago, inference methods robust to cluster correlation are not used routinely across all relevant settings. In this paper, I provide an overview of applications of cluster-sample methods, both to cluster samples and to panel data sets.

Selection corrections for panel data models under conditional mean independence assumptions

Some new methods for testing and correcting for sample selection bias in panel data models are proposed. The assumptions allow the unobserved effects in both the regression and selection equations to be correlated with the observed variables; the error distribution in the regression equation is unspecified; arbitrary serial dependence in the idiosyncratic errors of both equations is allowed; and all idiosyncratic errors can be heterogeneously distributed. Compared with maximum likelihood and other estimators derived under fully parametric assumptions, the new estimators are much more robust and have significant computational advantages.

Distribution-free estimation of some nonlinear panel data models

This paper studies distribution-free estimation of some multiplicative unobserved components panel data models. One class of estimators requires only specification of the conditional mean; in particular, the multinomial quasi-conditional maximum likelihood estimator is shown to be consistent when only the conditional mean in the unobserved effects model is correctly specified. Additional orthogonality conditions can be used in a method of moments framework. A second class of problems specifies the conditional mean, conditional variances, and conditional covariances. Using the notion of a conditional linear predictor, it is shown that specification of conditional second moments implies further orthogonality conditions in the observable data that can be exploited for efficiency gains. This has applications to both count and gamma-type panel data regression models.

What are We Weighting for

When estimating population descriptive statistics, weighting is called for if needed to make the analysis sample representative of the target population. With regard to research directed instead at estimating causal effects, we discuss three distinct weighting motives: (1) to achieve precise estimates by correcting for heteroskedasticity; (2) to achieve consistent estimates by correcting for endogenous sampling; and (3) to identify average partial effects in the presence of unmodeled heterogeneity of effects. In each case, we find that the motive sometimes does not apply in situations where practitioners often assume it does.

A Unified Approach to Robust, Regression-Based Specification Tests

This paper develops a general approach to robust, regression-based specification tests for (possibly) dynamic econometric models. A useful feature of the proposed tests is that, in addition to estimation under the null hypothesis, computation requires only a matrix linear least-squares regression and then an ordinary least-squares regression similar to those employed in popular nonrobust tests. For the leading cases of conditional mean and/or conditional variance tests, the proposed statistics are robust to departures from distributional assumptions that are not being tested, while maintaining asymptotic efficiency under ideal conditions. Moreover, the statistics can be computed using any √ T -consistent estimator, resulting in significant simplifications in some otherwise difficult contexts. Among the examples covered are conditional mean tests for models estimated by weighted nonlinear least squares under misspecification of the conditional variance, tests of jointly parameterized conditional means and variances estimated by quasi-maximum likelihood under nonnormality, and some robust specification tests for a dynamic linear model estimated by two-stage least squares.

On the application of robust, regression- based diagnostics to models of conditional means and conditional variances

Abstract A strategy is proposed for applying a class of robust, regression-based diagnostics to nonlinear models of conditional means and conditional variances for cross-section or time-series data. The distinguishing feature of the current approach, which builds on already popular residual-based procedures, is that no auxiliary assumptions are imposed at any testing stage. Consequently, the statistics are guaranteed to have the correct asymptotic size under the relevant null hypothesis. Several new, regression-based conditional mean and conditional variance diagnostics are proposed. The case of incompletely specified dynamics in time-series models is explicitly covered.

Chapter 45 Estimation and inference for dependent processes

This chapter provides an overview of asymptotic results available for parametric estimators in dynamic models. Three cases are treated: stationary (or essentially stationary) weakly dependent data, weakly dependent data containing deterministic trends, and nonergodic data (or data with stochastic trends). Estimation of asymptotic covariance matrices and computation of the major test statistics are covered. Examples include multivariate least squares estimation of a dynamic conditional mean, quasi-maximum likelihood estimation of a jointly parameterized conditional mean and conditional variance, and generalized method of moments estimation of orthogonality conditions. Some results for linear models with integrated variables are provided, as are some abstract limiting distribution results for nonlinear models with trending data.