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 …
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 …
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 …
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 …
Forecast comparison and evaluation
The most reliable way to evaluate a forecast or to compare forecasting methods is by examining out of sample performance. To evaluate the forecasting performance of a single model or …
The Johansen’s method
Johansen (1995) develops a maximum likelihood estimation procedure based on the so-called reduced rank regression method that, as the other methods to be later discussed, presents some advantages over the …
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, …
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 …
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 …
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. …
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 …
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 …
The Gaussian AR(1) Case without Intercept: Part 1
2.1 Introduction Consider the AR(1) model without intercept, rewritten as3 Ayt = a0yt-1 + ut, where ut is iid N(0, a2), (29.2) and y t is observed for t = …
Uses of Vector Autoregressive Models
When an adequate model for the DGP of a system of variables has been found it may be used for forecasting and economic analysis. Different tools have been proposed for …
Instrumental regressions
Consider the limited information (LI) structural regression model: y = Ye + Xayі + u = Z5 + u, (23.3) Y = Xana + X2n2 + V, (23.4) where Y …
Salient Features of US Macroeconomic Time Series Data
The methods discussed in this chapter will be illustrated by application to five monthly economic time series for the US macroeconomy: inflation, as measured by the annual percentage change in …
Common trends representation
As mentioned above, there is a dual relationship between the number of cointegrating vectors (r) and the number of common trends (n - r) in an n - dimensional system. …
Duration Variables
In this section we introduce basic concepts in duration analysis and present the commonly used duration distributions. 2.1 Survivor and hazard functions Let us consider a continuous duration variable Y …
Bayesian inference
In order to define the sampling model,5 we make the following assumptions about and zi for i = 1 ... N: 1. p(vi | h_1) = f N(vi |0, h_1) …
Autoregressive Models: Univariate
The objective of this section is to provide a brief overview of the most commonly used time series model, the AR(1), from both, the traditional and the probabilistic reduction (PR) …
Testing complex unit roots
Before proceeding to the examination of the procedure proposed by Hylleberg et al. (1990) it will be useful to consider some of the issues related to testing complex unit roots, …
The ACD model
This model was introduced by Engle and Russell (1998) to represent the dynamics of durations between trades on stock or exchange rate markets. Typically, intertrade durations are generated by a …
Tests for stochastic dominance
In the area of income distributions and tax analysis, it is important to look at Lorenz curves and similar comparisons. In practice, a finite number of ordinates of the desired …
Weak convergence of random functions
In order to establish the limiting distribution of (29.14), and other asymptotic results, we need to extend the well known concept of convergence in distribution of random variables to convergence …
Forecasting VAR processes
Neglecting deterministic terms and exogenous variables the levels VAR form (32.1) is particularly convenient to use in forecasting the variables yt. Suppose the ut are generated by an independent rather …
Identification in Parametric Models
Paul Bekker and Tom Wansbeek * 1 Introduction Identification is a notion of essential importance in quantitative empirical branches of science like economics and the social sciences. To the extent …
The Population Moment Condition and Identification
In this section, we consider the conditions under which the population moment provides sufficient information to determine uniquely 0 0 from all other elements in the parameter space 0 C …
Motivation for nonnested statistics
From a statistical view point the main difference between the nested and nonnested hypothesis testing lies in the fact that the usual loglikelihood ratio or Wald statistics used in the …
Least squares estimation
When attention is focused on modeling just the conditional mean, least squares methods are inferior to the approach of the previous subsection. Linear least squares regression of V on x …
Criteria for Choosing Concomitants in RCMs
Equations (19.9)-(19.12) incorporate in a consistent way all the prior information that is usually available about these equations. The most difficult step arises in the form of equation (19.13). Not …
Sampling Theory Estimation and Inference with Unknown Covariance Matrix
Consider again the linear model y = Xp + e with error-covariance matrix V = о2Л. In this section we relax the assumption that Л is known. As we saw …
Instrumental variables
To introduce the notion of instrumental variables, we start from (8.3): y = Хв + u, where consistent estimation was hampered by the correlation between X and u. If there …
Collinearity in the linear regression model
Denote the linear regression model as y = Xp + e, (12.4) where y is a T x 1 vector of observations on the dependent variable, X is a T …
Direct representation
A second commonly used approach to the formal specification of spatial autocorrelation is to express the elements of the variance-covariance matrix in a parsimonious fashion as a "direct" function of …
Binary and Multinomial Response Models
In this section we present the basic analysis of models with a single explanatory variable that is observed as a dichotomous (binary or binomial) or polychotomous (multinomial) variable. Both binary …
