In this chapter, you will learn how to: Explain the defining characteristics of various types of stochastic processes, identify the appropriate time series model for a given data series, produce forecasts for ARMA and exponential smoothing models, evaluate the accuracy of predictions using various metrics, estimate time series models and produce forecasts from them in EViews. | Chapter 6 Univariate time series modelling and forecasting ‘Introductory Econometrics for Finance’ c Chris Brooks 2013 1 Univariate Time Series Models • Where we attempt to predict returns using only information contained in their past values. Some Notation and Concepts • A Strictly Stationary Process A strictly stationary process is one where P{yt1 ≤ b1 , . . . , ytn ≤ bn } = P{yt1 +m ≤ b1 , . . . , ytn +m ≤ bn } • A Weakly Stationary Process ‘Introductory Econometrics for Finance’ c Chris Brooks 2013 2 Univariate Time Series Models (Cont’d) If a series satisfies the next three equations, it is said to be weakly or covariance stationary (1) E (yt ) = µ t = 1, 2, . . . , ∞ (2) E (yt − µ)(yt − µ) = σ 2 < ∞ (3) E (yt1 − µ)(yt2 − µ) = γt2 −t1 ∀ t1 , t2 • So if the process is covariance stationary, all the variances are the same and all the covariances depend on the difference between t1 and t2 . The moments E (yt − E (yt ))(yt−s − E (yt−s )) = γs , s = 0, 1, 2, . . . are known as the covariance function. • The covariances, γs , are known as autocovariances. ‘Introductory Econometrics for Finance’ c Chris Brooks 2013 3 Univariate Time Series Models (Cont’d) • However, the value of the autocovariances depend on the units of measurement of yt . • It is thus more convenient to use the autocorrelations which are the autocovariances normalised by dividing by the variance: τs = γs , γ0 s = 0, 1, 2, . . . • If we plot τs against s=0,1,2,. then we obtain the autocorrelation function or correlogram. ‘Introductory Econometrics for Finance’ c Chris Brooks 2013 4 A White Noise Process • A white noise process is one with (virtually) no discernible structure. A definition of a white noise process is E (yt ) = µ var(yt ) = σ 2 γt−r = σ2 0 if t = r otherwise • Thus the autocorrelation function will be zero apart from a single peak of 1 at s=0. τs ∼ approx. N(0, 1/T ) where T = ˆ sample size • We can use this to do significance tests for the .
