Maximum-likelihood estimation of nonlinear models with fixed effects is subject to the incidental-parameter problem. This typically implies that point estimates suffer from large bias and confidence intervals have poor coverage. This paper presents a jackknife method to reduce this bias and to obtain confidence intervals that are correctly centered under rectangular-array asymptotics. The method is explicitly designed to handle dynamics in the data, and yields estimators that are straightforward to implement and can be readily applied to a range of models and estimands. We provide distribution theory for estimators of model parameters and average effects, present validity tests for the jackknife, and consider extensions to higher-order bias correction and to two-step estimation problems. An empirical illustration relating to female labor-force participation is also provided. solutions generally do not give guidance for estimating average marginal effects, which are quantities of substantial interest. Furthermore, they typically restrict the fixed effects to be univariate, often entering the model as location parameters. Arellano and Honoré (2001) provide an overview of these methods. Browning and Carro (2007), Browning, Ejrnaes, and Alvarez (2010), and Arellano and Bonhomme (2012) discuss several examples where unit-specific location parameters cannot fully capture the unobserved heterogeneity in the data. Hospido (2012) and Carro and Traferri (2012) present empirical applications using models with multivariate fixed effects. The incidental-parameter problem is most severe in short panels. Fortunately, in recent decades longer data sets are becoming available. For example, the Panel Study of Income Dynamics has been collecting waves since 1968 and the British Household Panel Survey since 1991. They now feature a time-series dimension that can be considered statistically informative about unit-specific parameters. The availability of more observations per unit does not necessarily solve the inference problem, however, because confidence intervals centered at the maximum-likelihood estimate are incorrect under rectangular-array asymptotics, i.e., as N, T → ∞ at the same rate (see, e.g., Li, Lindsay, and Waterman 2003). It has, however, motivated a recent body of literature in search of bias corrections to maximum likelihood that have desirable properties under rectangular-array asymptotics for a general class of fixed-effect models. Hahn and Newey (2004) and Hahn and Kuersteiner (2011) provide such corrections for static and dynamic models, respectively. Lancaster (2002), Woutersen (2002), Arellano and Hahn (2006), and Arellano and Bonhomme (2009) propose estimators that maximize modified objective functions and enjoy the same type of asymptotic properties. The primary aim of these methods is to remove the leading bias from the maximum-likelihood estimator and, thereby, to recenter its asymptotic distribution. The main difference between the various methods lies in how the bias is estimated.With the exception of th...
This paper studies inference on fixed effects in a linear regression model estimated from network data. An important special case of our setup is the two-way regression model, which is a workhorse method in the analysis of matched data sets. Networks are typically quite sparse and it is difficult to see how the data carry information about certain parameters. We derive bounds on the variance of the fixed-effect estimator that uncover the importance of the structure of the network. These bounds depend on the smallest non-zero eigenvalue of the (normalized) Laplacian of the network and on the degree structure of the network. The Laplacian is a matrix that describes the network and its smallest non-zero eigenvalue is a measure of connectivity, with smaller values indicating less-connected networks. These bounds yield conditions for consistent estimation and convergence rates, and allow to evaluate the accuracy of first-order approximations to the variance of the fixed-effect estimator. The bounds are also used to assess the bias and variance of estimators of moments of the fixed effects.
We consider a statistical model for network formation that features both node-specific heterogeneity parameters and common parameters that reflect homophily among nodes. The goal is to perform statistical inference on the homophily parameters while allowing the distribution of the node heterogeneity to be unrestricted, that is, by treating the node-specific parameters as fixed effects. Jointly estimating all the parameters leads to asymptotic bias that renders conventional confidence intervals incorrectly centered. As an alternative, we develop an approach based on a sufficient statistic that separates inference on the homophily parameters from estimation of the fixed effects. This estimator is easy to compute and is shown to have desirable asymptotic properties. In numerical experiments we find that the asymptotic results provide a good approximation to the small-sample behavior of the estimator. As an empirical illustration, the technique is applied to explain the import and export patterns in a cross-section of countries.
We calculate the bias of the profile score for the regression coefficients in a multistratum autoregressive model with stratum-specific intercepts. The bias is free of incidental parameters. Centering the profile score delivers an unbiased estimating equation and, upon integration, an adjusted profile likelihood. A variety of other approaches to constructing modified profile likelihoods are shown to yield equivalent results. However, the global maximizer of the adjusted likelihood lies at infinity for any sample size, and the adjusted profile score has multiple zeros. Consistent parameter estimates are obtained as local maximizers inside or on an ellipsoid centered at the maximum likelihood estimator.
Empirical models for dyadic interactions between n agents often feature agent-specific parameters. Fixed-effect estimators of such models generally have bias of order n −1 , which is nonnegligible relative to their standard error. Therefore, confidence sets based on the asymptotic distribution have incorrect coverage. This paper looks at models with multiplicative unobservables and fixed effects. We derive moment conditions that are free of fixed effects and use them to set up estimators that are n-consistent, asymptotically normally distributed, and asymptotically unbiased. We provide Monte Carlo evidence for a range of models. We estimate a gravity equation as an empirical illustration.
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