Category A COMPANION TO Theoretical Econometrics

Uniform linear hypothesis in multivariate regression models

Multivariate linear regression (MLR) models involve a set of p regression equa­tions with cross-correlated errors. When regressors may differ across equations, the model is known as the seemingly unrelated regression model (SUR or SURE; Zellner, 1962). The MLR model can be expressed as follows: Y = XB + U,

where Y = (Y1,… ,Yp) is an n x p matrix of observation on p dependent variables, X is an n x k full-column rank matrix of fixed regressors, B = [pv…, Pp] is a k x p matrix of unknown coefficients and U = [Uv…, Up] = [Q1,…, Un]’ is an n x p matrix of random disturbances with covariance matrix X where det (X) Ф 0. To derive the distribution of the relevant test statistics, we also assume the following:

A = W, i = 1,…, n, (23.19)

where the vector w = vec(W1,…, Wn) has a known distribution...

Differencing the data

A question that arises in practice is whether to difference the data prior to con­struction of a forecasting model. This arises in all the models discussed above, but for simplicity it is discussed here in the context of a pure AR model. If one knows a priori that there is in fact a unit autoregressive root, then it is efficient to impose this information and to estimate the model in first differences. Of course, in practice this is not known. If there is a unit autoregressive root, then estimates of this root (or the coefficients associated with this root) are generally biased towards zero, and conditionally biased forecasts can obtain...

Higher order cointegrated systems

The statistical theory of I(d) systems with d = 2, 3,…, is much less developed than the theory for the I(1) model, partly because it is uncommon to find time series, at least in economics, whose degree of integration higher than two, partly because the theory is quite involved as it must deal with possibly multicointegrated cases where, for instance, linear combinations of levels and first differences can achieve stationarity. We refer the reader to Haldrup (1999) for a survey of the statistical treatment of I(2) models, restricting the discussion in this chapter to the basics of the CI(2, 2) case.

Assuming, thus, that yt ~ CI(2, 2), with Wold representation given by

(1 – L)2 y = C(L)e f, (30.23)

then, by means of a Taylor expansion, we can write C(L) as C(L) = C(1) – C *(1)(1 – L) + C(L)(...

Parametric and. Nonparametric Tests. of Limited Domain and. Ordered Hypotheses. in Economics

Esfandiar Maasoumi*

1 Introduction

In this survey, technical and conceptual advances in testing multivariate linear and nonlinear inequality hypotheses in econometrics are summarized. This is discussed for economic applications in which either the null, or the alternative, or both hypotheses define more limited domains than the two-sided alternatives typically tested. The desired goal is increased power which is laudable given the endemic power problems of most of the classical asymptotic tests. The impedi­ments are a lack of familiarity with implementation procedures, and characteri­zation problems of distributions under some composite hypotheses.

Several empirically important cases are identified in which practical "one-sided" tests can be conducted by either the %2-distribution, or th...

Basic duration distributions

In this section we introduce some parametric families of duration distributions. Exponential family

The exponentially distributed durations feature no duration dependence. In consequence of the time-independent durations, the hazard function is constant, X(y) = X. The cdf is given by the expression F(y) = 1 – exp(-Xy), and the survivor function is S( y) = exp(-Xy).

The density is given by:

f(У) = X exp^Xy^ У > °. (21.5)

This family is parametrized by the parameter X taking strictly positive values.

An important characteristic of the exponential distributions is that the mean and standard deviation are equal, as implied by EY = |, VY = ^ ... Уі = /X; P) + Vi _ Zi.