Ch. 3 Estimation 1 The Nature of statistical Inference It is argued that it is important to develop a mathematical model purporting to provide a generalized description of the data generating process. A prob bility model in the form of the parametric family of the density functions p=f(:0),0E e and its various ramifications formulated in last chapter

Ch. 2 Probability Theory 1 Descriptive Study of Data 1.1 Histograms and Their Numerical Characteristics By descriptive study of data we refer to the summarization and exposition(tab- ulation, grouping, graphical representation) of observed data as well as the derivation of numerical characteristics such as measures of location, dispersion and shape

Ch. 24 Johansen's mle for Cointegration We have so far considered only single-equation estimation and testing for cointe- gration. While the estimation of single equation is convenient and often consis- tent, for some purpose only estimation of a system provides sufficient information This is true, for example, when we consider the estimation of multiple cointe- grating vectors, and inference about the number of such vectors. This chapter examines methods of finding the cointegrating rank and derive the asymptotic

Ch. 23 Cointegration 1 Introduction An important property of (1) variables is that there can be linear combinations of theses variables that are I(O). If this is so then these variables are said to be cointegrated. Suppose that we consider two variables Yt and Xt that are I(1) (For example, Yt= Yt-1+ St and Xt= Xi-1+nt.)Then, Yt and Xt are said to be cointegrated if there exists a B such

Ch. 22 Unit root in Vector Time series 1 Multivariate Wiener Processes and multivari- ate FCLT Section 2.1 of Chapter 21 described univariate standard Brownian motion W(r) as a scalar continuous-time process(W: rE0, 1-R). The variable W(r) has a N(O, r)distribution across realization, and for any given realization, w(r) is continuous function of the date r with independent increments. If a set of k such independent processes, denoted

Ch. 21 Univariate Unit Root process 1 Introduction Consider OLS estimation of a AR(1)process, Yt= pYt-1+ut where ut w ii d (0, 0), and Yo=0. The OLS estimator of p is given by and we also have

Ch. 20 Processes with Deterministic Trends 1 Traditional Asymptotic Results of OlS Suppose a linear regression model with stochastic regressor given by Y=x!3+e,t=1,2,…,T,;B∈R or in matrix form y=xB+E We are interested in the asymptotic properties such as consistency and limiting

where a subscribed element of a matrix is always read as arou, column. Here we confine the element to be real number a vector is a matrix with one row or one column. Therefore a row vector is Alxk and a column vector is AixI and commonly denoted as ak and ai,respec- tively. In the followings of this course, we follow conventional custom to say that a vector is a columnvector except for

Ch. 19 Models of Nonstationary Time Series In time series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are nonstationary. How to deal with the nonstationary data and use what we have learned from stationary model are the main subjects of this chapter 1 Integrated Process

Ch. 18 Vector Time series 1 Introduction In dealing with economic variables often the value of one variables is not only related to its predecessors in time but, in addition, it depends on past values of other variables. This naturally extends the concept of univariate stochastic process to vector time series analysis. This chapter describes the dynamic in