The preceding discussion shows thai there are some drawbacks in choosing between models on the basis of estimated error variance. Hence some alternative criteria in terms of estimated mean-squared error of prediction have been suggested. These criteria, called Cp, PC, Sp, have been discussed in Section 12.7.

Introduction to Time-Series Analysis

13.1 Introduction

13.2 Two Methods of Time-Series Analysis: Frequency Domain and Time Domain

13.3 Stationary and Nonstationary Time Series

13.4 Some Useful Models for Time Series

13.5 Estimation of AR, MA, and ARMA Models

13.6 The Box-Jenkms Approach

13.7 R Measures in Time-Series Models Summary

Exercises Data Sets

13.1 Introduction

A time series is a sequence of numerical data in which each item is associated with a particular mstant in time. One can quote numerous examples: monthly unemployment, weekly measures of money supply, M, and Mj, daily closing prices of stock indices, and so on. In fact with the current progress in computer

technology we have daily series on interest rates, the hourly "telerate" interest rate index, and stock prices by the minute (or even second). A cartoon in the New Yorker magazine shows a digital display at a bank:

Time: 11.23

Temp: 83

Interest rate: 8.62

An analysis of a single sequence of data is called univariate time-series analysis. An analysis of several sets of data for the same sequence of time periods is called multivariate time-series analysis or, more simply, multiple time-series analysis (e.g., an analysis on the basis of monthly data, the relationships among unemployment, price level, money supply, etc., falls under multiple time-series analysis). The purpose of time-series analysis is to study the dynamics or temporal structure of the data.

For a long time there has been very little communication between econometricians and time-series analysts. Econometricians have emphasized economic theory and a study of contemporaneous relationships. Lagged variables were introduced but not in any systematic way, and no serious attempts were made to study the temporal structure of the data. Theories were imposed on the data even when the temporal structure of the data was not in conformity with the theories. The time-series analysts, on the other hand, did not believe in economic theories and thought that they were better off allowing the data to determine the model. Since the mid-1970s these two approaches-the time-series approach and the econometric approach-have been converging. Econometricians now use some of the basic elements of time-series analysis in checking the specification of their econometric models, and some economic theories have influenced the direction of time series work.

13.2 Two Methods of Time-Series Analysis: Frequency Domain and Time Domain

Time-series analysis can be roughly divided into two types of methods: frequency-domain methods and time-domain methods. In this chapter we discuss time-domain methods only. We shall, however, explain what these two methods are.

In models underlying the frequency-domain analysis, the time series X, is expressed as the sum of independently varying cosine and sine curves with random amplitudes. We thus write X, as

X, = H-i-2lYj cos(27r0 + Zj sin(2-ir )]

where Fs and Zs are uncorrelated random variables with zero expectations and variances (rif), and the summation is over all frequencies. The frequencies fu fl f, , are equally spaced and separated by a small interval /. The