Category A COMPANION TO Theoretical Econometrics

Deterministic seasonality

A common practice is to attempt the removal of seasonal patterns via seasonal dummy variables (see, for example, Barsky and Miron, 1989; Beaulieu and Miron, 1991; Osborn, 1990). The interpretation of the seasonal dummy approach is that seasonality is essentially deterministic so that the series is stationary around seasonally varying means. The simplest deterministic seasonal model is

s

yt = 1 bstms + ef (31.14)

s=1

where 5s t is the seasonal dummy variable which takes the value 1 when t falls in season s and et ~ iid(0, о2). Typically, yt is a first difference series in order to account for the zero frequency unit root commonly found in economic time series. When a model like (31...

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Duration Time Series

In this section we focus our attention on duration time series, i. e. sequences of random durations, indexed by their successive numbers in the sequence and possibly featuring temporal dependence. In practice these data are generated, for example, by randomly occurring transactions on credit cards, by claims ran­domly submitted to insurance agencies at unequal intervals, or by assets traded at a time varying rate on stock markets. According to the traditional time series analysis the ultimate purpose of our study is to model and estimate the dynamics of these stochastic duration processes.

There are two major characteristics which account for the distinct character of duration time series...

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Nonlinear Models and Nonlinear Inequality Restrictions

Wolak (1989, 1991) gives a general account of this topic. He considers the general formulation in (25.18) with nonlinear restrictions. Specifically, consider the fol­lowing problem:

S = P + v h(p) > 0

v ~ N(0, Y) (25.22)

where h( ) is a smooth vector function of dimension p with a derivative matrix denoted by H(). We wish to test

Подпись:H0 : h(P) > 0, vs. H1 : p Є RK.

This is very general since model classes that allow for estimation results given in (25.22) are very broad indeed. As the results in Potscher and Prucha (1991a, 1991b) indicate, many nonlinear dynamic processes in econometrics permit con­sistent and asymptotically normal estimators under regularity conditions.

In general an asymptotically exact size test of the null in (25...

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Unit Roots

Herman J. Bierens*

1 Introduction

In this chapter I will explain the two most frequently applied types of unit root tests, namely the Augmented Dickey-Fuller tests (see Fuller, 1996; Dickey and Fuller, 1979, 1981), and the Phillips-Perron tests (see Phillips, 1987; Phillips and Perron, 1988). The statistics and econometrics levels required for understanding the material below are Hogg and Craig (1978) or a similar level for statistics, and Green (1997) or a similar level for econometrics. The functional central limit theorem (see Billingsley, 1968), which plays a key role in the derivations in­volved, will be explained in this chapter by showing its analogy with the concept of convergence in distribution of random variables, and by confining the discus­sion to Gaussian unit root processes...

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Model checking

Once a model has been specified and estimated its adequacy is usually checked with a range of tests and other statistical procedures. Many of these model check­ing tools are based on the residuals of the final model. Some of them are applied to the residuals of individual equations and others are based on the full residual vectors. Examples of specification checking tools are visual inspection of the plots of the residuals and their autocorrelations. In addition, autocorrelations of squared residuals may be considered to check for possible autoregressive condi­tional heteroskedasticity (ARCH). Although it may be quite insightful to inspect the autocorrelations visually, formal statistical tests for remaining residual auto­correlation should also be applied...

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Statistical Issues: A Practical Approach to Core Questions

The hypothesis testing problem is often presented as one of deciding between two hypotheses: the hypothesis of interest (the null H0) and its complement (the alternative HA). For the purpose of the exposition, consider a test problem pertain­ing to a parametric model (Y P0), i. e. the case where the data generating process (DGP) is determined up to a finite number of unknown real parameters 0 £ 0, where 0 refers to the parameter space (usually a vector space), Y is the sample space, P0 is the family of probability distributions on Y Furthermore, let Y denote the observations, and 00 the subspace of 0 compatible with H0.

A statistical test partitions the sample space into two subsets: a set consistent with H0 (the acceptance region), and its complements whose elements are viewed as inconsi...

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