Category A COMPANION TO Theoretical Econometrics

Methods for introducing inexact nonsample information

Economists usually bring general information about parameters to the estimation problem, but it is not like the exact restrictions discussed in the previous section. For example, we may know the signs of marginal effects, which translate into inequality restrictions on parameters. Or we may think that a parameter falls in the unit interval, and that there is a good chance it falls between 0.25 and 0.75. That is, we are able to suggest signs of parameters, and even perhaps ranges of reasonable values. While such information has long been available, it has been difficult to use in applications...

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Essentials of Count. Data Regression

A. Colin Cameron and Pravin K. Trivedi

1 Introduction

In many economic contexts the dependent or response variable of interest (y) is a nonnegative integer or count which we wish to explain or analyze in terms of a set of covariates (x). Unlike the classical regression model, the response variable is discrete with a distribution that places probability mass at nonnega­tive integer values only. Regression models for counts, like other limited or discrete dependent variable models such as the logit and probit, are nonlinear with many properties and special features intimately connected to discreteness and nonlinearity.

Let us consider some examples from microeconometrics, beginning with samples of independent cross section observations...

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Estimation of simultaneous equation sample selection model

A two-stage estimation method can be easily generalized for the estimation of a simultaneous equation model. Consider the linear simultaneous equation y* = y*B + xC + u, which can be observed only if zY > e. For the estimation of structural parameters, consider the first structural equation y* = y(1) ри + x151 + ии where y*(1) consists of included endogenous variables on the right-hand side of the structural equation. The bias-corrected structural equation is уи = y(1p1 + x181 + o1e (—-фт|)-) + . The system implies the reduced form equations y* = хП + v.

Lee, Maddala, and Trost (1980) suggest the estimation of the reduced form parameters П by the Heckman two-stage method, and used the predicted y(1) to estimate the bias-corrected structural equation similar to Theil’s two-stage...

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The Neyman-Pearson lemma and the Durbin-Watson test

The first formal specification test in econometrics, the Durbin-Watson (DW) (1950) test for autocorrelation in the regression model has its foundation in the UMP test principle via a theorem of Anderson (1948). Most econometrics textbooks provide a detail discussion of the DW test but do not mention its origin. Let us consider the standard linear regression model with autocorrelated errors:

Vt = 4е + et (2.54)

£t = pet-1 + Ut, (2.55)

where Vt is the tth observation on the dependent variable, xt is the tth observation on k strictly exogenous variables, | p | < 1 and ut ~ iidN(0, о2), t = 1, 2,…, n. The problem is testing H0 : p = 0. Using the N-P lemma, Anderson (1948) showed that UMP tests for serial correlation can be obtained against one-sided alternatives...

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The Gauss-Newton Regression

Associated with every nonlinear regression model is a somewhat nonstandard artificial regression which is probably more widely used than any other. Con­sider the univariate, nonlinear regression model

yt = vt(P) + ut, ut ~ iid(0, о2), t = 1,…, n, (1.2)

where yt is the tth observation on the dependent variable, and в is a ^-vector of parameters to be estimated. The scalar function vt(P) is a nonlinear regression function. It determines the mean value of yt as a function of unknown parameters в and, usually, of explanatory variables, which may include lagged dependent variables. The explanatory variables are not shown explicitly in (1.2), but the t subscript on х((в) reminds us that they are present. The model (1.2) may also be written as

y = x(e) + u, u ~ iid(0, о2I), (1.3)

where y i...

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Hypothesis Testing with Artificial Regressions

Artificial regressions like the GNR are probably employed most frequently for hypothesis testing. Suppose we wish to test a set of r equality restrictions on 9. Without loss of generality, we can assume that these are zero restrictions. This allows us to partition 9 into two subvectors, 91 of length k – r, and 92 of length r, the restrictions being that 92 = 0. If the estimator 0 is not only root-n consistent but also asymptotically normal, an appropriate statistic for testing these restric­tions is

0ї(^г (02))-102, (1.19)

which will be asymptotically distributed as x2(r) under the null if ^«(02) is a suit­able estimate of the covariance matrix of 02.

Suppose that r(9) and R(9) define an artificial regression for the estimator 0...

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