# Category A COMPANION TO Theoretical Econometrics

## Hypothesis Testing

An obvious question is how to carry out various diagnostic tests done in the parametric econometrics within the nonparametric and semiparametric models. Several papers have appeared in the recent literature which deal with this issue. We present them here and show their links.

First consider the problem of testing a specified parametric model against a nonparametric alternative, H0 : /(p, x) = E(yi | xi) against H1 : m(x) = E(yi | xi). The idea behind the Ullah (1985) test statistic is to compare the parametric RSS (PRSS) XЙ2, x = Vi – /(S, xi) with the nonparametric RSS (NPRSS), XB2, where щ = – m(xi). His test statistic is

or simply T * = (PRSS – NPRSS), and reject the null hypothesis when T1 is large. л/П T1 has a degenerage distribution under H0...

## Bayesian Analysis. of Stochastic. Frontier Models

Gary Koop and Mark F. J. Steel

1 Introduction

Stochastic frontier models are commonly used in the empirical study of firm1 efficiency and productivity. The seminal papers in the field are Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977), while a recent survey is provided in Bauer (1990). The ideas underlying this class of models can be demonstrated using a simple production model2 where output of firm i, Yi, is produced using a vector of inputs, Xi, (i = 1 … N). The best practice technology for turning inputs into output depends on a vector of unknown parameters, p, and is given by:

Y = f (Xi; P). (24.1)

This so-called production frontier captures the maximum amount of output that can be obtained from a given level of inputs...

## Time series: a brief historical introduction

Time series data have been used since the dawn of empirical analysis in the mid-seventeenth century. In the "Bills of Mortality" John Graunt compared data on births and deaths over the period 1604-60 and across regions (parishes); see Stigler (1986). The time dimension of such data, however, was not properly understood during the early stages of empirical analysis. Indeed, it can be argued that the time dimension continued to bedevil statistical analysis for the next three centuries before it could be tamed in the context of proper statistical models for time series data.

The descriptive statistics period: 1665-1926

Up until the last quarter of the nineteenth century the time dimension of ob­served data and what that entails was not apparent to the descriptive statistics literature which c...

## 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...

## 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...

## 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

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...