Journal Article Analysis of Covariance with Qualitative Data Get access Gary Chamberlain Gary Chamberlain University of Wisconsin—Madison Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 47, Issue 1, 1980, Pages 225–238, https://doi.org/10.2307/2297110 Published: 01 January 1980

This paper considers a panel data model for predicting a binary outcome. The conditional probability of a positive response is obtained by evaluating a given distribution function (F) at a linear combination of the predictor variables. One of the predictor variables is unobserved. It is a random effect that varies across individuals but is constant over time. The semiparametric aspect is that the conditional distribution of the random effect, given the predictor variables, is unrestricted.

This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced-form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (ρ, φ, ω), where φis the parameter of interest and is normalized to be a point on the unit circle. The reduced-form coefficients on the instrumental variables are split into a scalar parameter ρand a parameter vector ω, which is normalized to be a point on the (k−1)-dimensional unit sphere; ρmeasures the strength of the association between the endogenous variables and the instrumental variables, and ωis a measure of direction. A prior distribution is introduced for the IV model. The parameters φ, ρ, and ωare treated as independent random variables. The distribution for φis uniform on the unit circle; the distribution for ωis uniform on the unit sphere with dimension k-1. These choices arise from the solution of a minimax problem. The prior for ρis left general. It turns out that given any positive value for ρ, the Bayes estimator of φdoes not depend on ρ; it equals the maximum-likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. The same general concepts are applied to obtain confidence intervals. The prior distribution is used in two ways. The first way is to integrate out the nuisance parameter ωin the IV model. This gives an integrated likelihood function with two scalar parameters, φand ρ. Inverting a likelihood ratio test, based on the integrated likelihood function, provides a confidence interval for φ. This lacks finite sample optimality, but invariance arguments show that the risk function depends only on ρand not on φor ω. The second approach to confidence sets aims for finite sample optimality by setting up a loss function that trades off coverage against the length of the interval. The automatic uniform priors are used for φand ω, but a prior is also needed for the scalar ρ, and no guidance is offered on this choice. The Bayes rule is a highest posterior density set. Invariance arguments show that the risk function depends only on ρand not on φor ω. The optimality result combines average risk and maximum risk. The confidence set minimizes the average—with respect to the prior distribution for ρ—of the maximum risk, where the maximization is with respect to φand ω.

Efficiency bounds for conditional moment restrictions with a nonparametric component are derived. There is a given function of the data (a random sample from a distribution F) and a parameter. The restriction is that a conditional expectation of this function is zero at some point in the parameter space. The parameter has two parts: a finite-dimensional component and a general function evaluated at a subset of the conditioning variables. An example is a regression function that is additive in parametric and nonparametric component, as arises in sample selection models.

[We present a definition of factor structure that is less restrictive than the one typically used in arbitrage pricing models. Our factor structure restrictions build on the following intuitive distinctions between factor variance and idiosyncratic variance: (i) A well-diversified portfolio contains only factor variance. (ii) If a portfolio is uncorrelated with the well-diversified portfolios, then it contains only idiosyncratic variance; so if a sequence of such portfolios becomes well-diversified, the limiting variance should be zero. Our factor structure restrictions imply Ross' [5] arbitrage pricing formula. We obtain upper and lower bounds on the approximation error in that formula; these bounds may be useful in empirical work. They imply that arbitrage pricing is exact if and only if there is a risky, well-diversified portfolio on the mean-variance frontier. If all mean-variance efficient portfolios are well-diversified, then the well-diversified portfolios provide mutual fund separation. Our factor structure restrictions are satisfied (with K factors) if and only if the covariance matrix of asset returns has only K unbounded eigenvalues as the number of assets increases.]

Chetty et al. (2014) document variation across commuting zones in intergenerational mobility. With over 700 commuting zones, the task of estimating place effects involves a high-dimension parameter space. I develop a fixed-effects model along with an oracle bound on the risk of invariant estimators. The oracle estimator uses an invariant prior, which I have incorporated into a random-effects model to obtain a feasible estimator. This estimator almost achieves the oracle bound over the relevant part of the (fixed-effects) parameter space in the empirical application. There is substantial reduction in risk compared with the least-squares estimator.

Consider an individual making a portfolio choice at date T involving two assets. The (gross) returns at t per unit invested at t 1 are Yit and Y2t* The individual has observed these returns from t = 0 to t = T. He has also observed the values of the variables Y3t',... YKt, which are thought to be relevant in forecasting future returns. Thus, the information available to him when he makes his portfolio choice is z = {(Yt ... YKt)} t=0. He invests one unit, divided between an amount a in asset 1 and an amount 1 a in asset 2, and he then holds on to the portfolio until date T + H. Let w = {(Yit, Y2t)}T:H+ 1 and let h(w, a) denote the value of the portfolio at t T + H:

This paper applies some general concepts in decision theory to a linear panel data model. A simple version of the model is an autoregression with a separate intercept for each unit in the cross section, with errors that are independent and identically distributed with a normal distribution. There is a parameter of interest Gamma and a nuisance parameter τ, a N×K matrix, where N is the cross-section sample size. The focus is on dealing with the incidental parameters problem created by a potentially high-dimension nuisance parameter. We adopt a "fixed-effects" approach that seeks to protect against any sequence of incidental parameters. We transform tau to (delta, rho, omega), where delta is a J x K matrix of coefficients from the least-squares projection of tau on a N x J matrix x of strictly exogenous variables, rho is a K x K symmetric, positive semidefinite matrix obtained from the residual sums of squares and cross-products in the projection of tau on x, and omega is a (N - J) x K matrix whose columns are orthogonal and have unit length. The model is invariant under the actions of a group on the sample space and the parameter space, and we find a maximal invariant statistic. The distribution of the maximal invariant statistic does not depend upon omega. There is a unique invariant distribution for omega. We use this invariant distribution as a prior distribution to obtain an integrated likelihood function. It depends upon the observation only through the maximal invariant statistic. We use the maximal invariant statistic to construct a marginal likelihood function, so we can eliminate omega by integration with respect to the invariant prior distribution or by working with the marginal likelihood function. The two approaches coincide.

In this paper we propose a new estimator for a model with one endogenous regressor and many instrumental variables. Our motivation comes from the recent literature on the poor properties of standard instrumental variables estimators when the instrumental variables are weakly correlated with the endogenous regressor. Our proposed estimator puts a random coefficients structure on the relation between the endogenous regressor and the instruments. The variance of the random coefficients is modelled as an unknown parameter. In addition to proposing a new estimator, our analysis yields new insights into the properties of the standard two-stage least squares (TSLS) and limited-information maximum likelihood (LIML) estimators in the case with many weak instruments. We show that in some interesting cases, TSLS and LIML can be approximated by maximizing the random effects likelihood subject to particular constraints. We show that statistics based on comparisons of the unconstrained estimates of these parameters to the implicit TSLS and LIML restrictions can be used to identify settings when standard large sample approximations to the distributions of TSLS and LIML are likely to perform poorly. We also show that with many weak instruments, LIML confidence intervals are likely to have under-coverage, even though its finite sample distribution is approximately centered at the true value of the parameter. In an application with real data and simulations around this data set, the proposed estimator performs markedly better than TSLS and LIML, both in terms of coverage rate and in terms of risk. Copyright Econometric Society 2004.