Category A COMPANION TO Theoretical Econometrics

Artificial Regressions

Russell Davidson and James G. MacKinnon

1 Introduction

All popular nonlinear estimation methods, including nonlinear least squares (NLS), maximum likelihood (ML), and the generalized method of moments (GMM), yield estimators which are asymptotically linear. Provided the sample size is large enough, the behavior of these nonlinear estimators in the neighborhood of the true parameter values closely resembles the behavior of the ordinary least squares (OLS) estimator. A particularly illuminating way to see the relationship between any nonlinear estimation method and OLS is to formulate the artificial regression that corresponds to the nonlinear estimator.

An artificial regression is a linear regression in which the regressand and re­gressors are constructed as functions of the data and parame...

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Neyman-Pearson generalized lemma and its applications

The lemma can be stated as follows:

Let g1, g,…, gm, gm+1 be integrable functions and ф be a test function over S such that 0 < ф < 1, and

Подпись: (2.6)фgidy = ci i = 1, 2,…, m,

where c1, c2,…, cm are given constants. Further, let there exist a ф* and constants k1, k2,…, km such that ф* satisfies (2.6), and

Ф* = 1 if gm+1 > X



Подпись: (2.7)= 0 if gm+1 < X kigi



Ф*gm+1dУ ^

the function ф*(у) defined as

ф*(у) = 1 when = 0 when

Ф*(у)Ц91)йу ^

that is, ф*(у) will provide the MP test. Therefore, in terms of critical region,


where k is such that Pr{o | H0} = a, is the MP critical region.

The N-P lemma also provides the logical basis for the LR test...

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The Concept of an Artificial Regression

Consider a fully parametric, nonlinear model that is characterized by a para­meter vector 0 which belongs to a parameter space 0 C R k and which can be estimated by minimizing a criterion function Q(0) using n observations. In the case of a nonlinear regression model estimated by nonlinear least squares, Q(0) would be one half the sum of squared residuals, and in the case of a model estim­ated by maximum likelihood, Q(0) would be minus the loglikelihood function.

If an artificial regression exists for such a model, it always involves two things: a regressand, r(0), and a matrix of regressors, R(0). The number of regressors for the artificial regression is equal to k, the number of parameters...

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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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