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13 Introduction toTime-Series Analysis. What is in this Chapter?. This chapter discusses the basic time-series models: autoregressive (AR) and moving average (MA) models, stationary and nonstationary time series, and the Box-Jenkins approach to time-series modeling
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What is in this Chapter? • This chapter discusses • the basic time-series models: autoregressive (AR) and moving average (MA) models, • stationary and nonstationary time series, • and the Box-Jenkins approach to time-series modeling • Some R2 measures applicable in evaluating goodness of fit in time-series models are also discussed.
13.1 Introduction • A time series is a sequence of numerical data in which each item is associated with a particular instant in time • One can quote numerous examples: monthly unemployment, weekly measures of money supply, 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).
13.1 Introduction • 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 multivariatetime-series analysis or, more simply, multiple time-series analysis
13.1 Introduction • 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 a 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
13.1 Introduction • 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.
13.3 Stationary and Nonstationary Time Series • Strict Stationarity • One way of describing a stochastic process is to specify the joint distribution of thevariables Xt. This is quite complicated and not usually attempted in practice. Instead, what is usually done is that we define the first and second moments of the variables Xt These are:
13.3 Stationary and Nonstationary Time Series • Weak Stationarity
13.3 Stationary and Nonstationary Time Series • Properties of Autocorrelation Function
13.4 Some Useful Models for Time Series • In this section we discuss several different types of stochastic processes that are usefulin modeling time series: • (1) a purely random process, • (2) a random walk, • (3) a movingaverage (MA) process, • (4) an autoregressive (AR) process, • (5) an autoregressive movingaverage (ARMA) process, and • (6) an autoregressive integrated moving average (ARIMA)process.
13.5 Estimation of AR, MA, and ARMA Models • Testing Goodness of Fit • When an AR, MA, or ARMA model has been fitted to a given time series, it is advisable to check that the model does really give an adequate description of the data • There are two criteria often used that reflect the closeness of fit and the number of parameters estimated. • One is the Akaike information criterion (AIC), and the other is the Schwartz Bayesiancriterion (SBC) • The latter is also called the Bayesian information criterion (BIC).
13.6 The Box-Jenkins Approach • The Box-Jenkins approach is one of the most widely used methodologies for the analysis of time-series data • It is popular because of its generality; it can handle any series, stationary or not, with or without seasonal elements, and it has well-documented computer programs
13.6 The Box-Jenkins Approach • Although Box and Jenkins have been neither the originators nor the most important contributors in the field of ARMA models • They have popularized these models and made them readily accessible to everyone, so much that ARMA models are sometimes referred to as Box-Jenkins models.
13.6 The Box-Jenkins Approach • The basic steps in the Box-Jenkins methodology are • (1) differencing the series so as toachieve stationarity, • (2) identification of a tentative model, • (3) estimation of the model, • (4) diagnostic checking (if the model is found inadequate, we go back to step 2), and • (5) using the model for forecasting and control. Schematically, we can describe the steps as in Figure 13.3.
13.6 The Box-Jenkins Approach • 1. Differencing to achieve stationarity: How do we conclude whether a time series is stationary or not? • We can do this by studying the graph of the correlogram of the series. • The correlogram of a stationary series drops off as k, the number of lags, becomes large, but this is not usually the case for a nonstationary series. • Thus the common procedure is to plot the correlogram of the given series yt and successive differences Δy, Δy, and so on, and look at the correlograms at each stage. • We keep differencing until the correlogram dampens
13.6 The Box-Jenkins Approach • 2. Once we have used the differencing procedure to get a stationary time series, we examine the correlogram to decide on the appropriate orders of the AR and MA components. • The correlogram of a MA process is zero after a point. • That of an ARprocess declines geometrically. The correlograms of ARMA processes show different patterns (but all dampen after a while). • Based on these, one arrives at a tentative ARMA model. • This step involves more of a judgmental procedure than the use of any clear-cut rules.
13.6 The Box-Jenkins Approach • 3. The next step is the estimation of the tentative ARMA model identified in step 2. We have discussed in the preceding section the estimation of ARMA models. • 4. The next step is diagnostic checking to check the adequacy of the tentative model. We discussed in the preceding section the Q and Q* statistics commonly used in diagnostic checking. As argued there, the (^-statistic is inappropriate in autoregressive models and thus we need to replace it with some LM test statistic. • 5. The final step is forecasting
13.6 The Box-Jenkins Approach • Illustrative Example • As an illustrative example we consider the problem of forecasting hog marketings considered by Leuthold et al. • It is an old study, but the example illustrates how the correlogram can be used to arrive at a model that uses higher than first-order differences. • The data consist of 275 daily observations. • The correlograms for the original data are presented in Figure 13.4. The correlogram does not damp, thus indicating nonstationarity.
13.6 The Box-Jenkins Approach • The peaks at 5, 10, 15, ... indicate a strong 5-day weekly effect • Figure 13.5 shows the correlogram of first differences. It still shows peaks at 5, 10, 15, and so on, and it does not show any sign of damping. • Since the peaks do not damp, it suggests a fifth-order MA component as well • We next try fifth differences, that is Xt - Xt-5 • The correlogram, shown in Figure 13.6, damps. • But the initial decline and oscillation suggest the use of an ARMA model rather than a pure AR or MA model • Leuthold et al. finally arrive at the model
