Category A COMPANION TO Theoretical Econometrics

Durations

Christian Gourieroux and Joann Jasiak

1 Introduction

Duration data represent times elapsed between random arrivals of events. They play an important role in many areas of science such as engineering, manage­ment, physics, economics, and operational research. In economics, duration data frequently appear in labor and health studies, insurance analysis, and finance. For example, a commonly used duration-based statistic is the average individual lifetime, called the life expectancy, provided yearly by national surveys. It serves a variety of purposes. The macroeconomists quote it as an indicator of the level of development and welfare of the society, while applied microeconomists con­sider it implicitly in designing and pricing contracts such as life insurances...

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The Stochastic Frontier Model with Cross-Sectional Data

2.1 Introduction and notation

The model given in equation (24.2) implicitly assumes that all deviations from the frontier are due to inefficiency. This assumption is also typically made in the DEA approach. However, following standard econometric practice, we add a random error to the model, Z, to capture measurement (or specification) error,4 resulting in:

The addition of measurement error makes the frontier stochastic, hence the term "stochastic frontier models". We assume that data for i = 1 … N firms is available and that the production frontier,/(•), is log-linear (e. g. Cobb-Douglas or translog). We define Xi as a 1 x (k + 1) vector (e. g. Xi = (1 Li K) in the case of a Cobb – Douglas frontier with two inputs, L and K) and, hence, (24.3) can be written as:

Vi = X-P + vi – zu (24...

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A PROBABILISTIC FRAMEWORK FOR TIME SERIES

A time series is defined as a finite sequence of observed data {y1, y2,…, yT} where the index t = 1, 2,…, T, denotes time. The probabilistic concept which corresponds to this series is that of a real stochastic process: {Yt, t Є T}, defined on the probability space (S, %, P(-)) : Y(-,-) : {S x T} ^ RY, where S denotes the outcomes set, % is the relevant о-field, P(-) : % ^ [0, 1], T denotes the relevant index set (often T := {1, 2,…, T,…}) and RY denotes a subset of the real line.

The probabilistic foundation of stochastic processes comes in the form of the finite joint distribution of the process {Yt, t Є T} as formulated by Kolmogorov (1933) in the form of Kolmogorov’s extension theorem...

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Testing the Seasonal Unit Root Null Hypothesis

In this section we discuss the test procedures proposed by Dickey et al. (1984) and Hylleberg, Engle, Granger, and Yoo (HEGY) (1990) to test the null hypo­thesis of seasonal integration. It should be noted that while there are a large number of seasonal unit root tests available (see, for example, Rodrigues (1998) for an extensive survey), casual observation of the literature shows that the HEGY test is the most frequently used procedure in empirical work. For simplicity of presentation, throughout this section we assume that augmentation of the test regression to account for autocorrelation is unnecessary and that presample starting values for the DGP are equal to zero.

3.1 The Dickey-Hasza-Fuller test

The first test of the null hypothesis yt ~ SI(1) was proposed by Dickey, Hasza, and Fu...

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The Poisson process

There exist two alternative ways to study a sequence of event arrivals. First we can consider the sequence of arrival dates or equivalently the sequence of durations

Y1r…, Yn,… between consecutive events. Secondly, we can introduce the counting process [N(t), t varying], which counts the number of events observed between 0 and t. The counting process is a jump process, whose jumps of unitary size occur at each arrival date. It is equivalent to know the sequence of durations or the path of the counting process.

The Poisson process is obtained by imposing the following two conditions:

1. The counting process has independent increments, i. e. N(tn) – N(tn-1), N(tn-1) – N(tn-2),…, N(t1) – N(t0) are independent for any t0 < t1 < t2 < … < tn, and any n.

2...

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Nonparametric Tests of Inequality Restrictions

All of the above models and hypotheses were concerned with comparing means and/or variance parameters of either known or asymptotically normal distribu­tions. We may not know the distributions and/or be interested in comparing more general characteristics than the first few moments, and the distributions being compared may not be from the same family. All of these situations require a nonparametric development that can also deal with ordered hypotheses.

Order relations between distributions present one of the most important and exciting areas of development in economics and finance. These include stochastic dominance relations which in turn include Lorenz dominance, and such others as likelihood and uniform orders...

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