Chapter 6: Time Series Analysis

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ATM 552 Notes: Time Series Analysis - Section 6a Page 106AmplitudeAmplitudeAmplitudeAmplitudeAmplitudeAmplitude210-1Periodic-20 0.5 1 1.5 2 2.5 32Time10-1Two Periods-20 0.5 1 1.5 2 2.5 32Time10-1White Noise-20 0.5 1 1.5 2 2.5 32Time10-1-20 0.5 1 1.5 2 2.5 3Time21Red Noise (0.8)+ Period0-1-20 0.5 1 1.5 2 2.5 32Time10-1Red NoiseSawtooth WaveCorrelationCorrelationCorrelationCorrelationCorrelationCorrelation1-20 -15 -10 -5 0 5 10 15 20Time Lag1-210 0.5 1 1.5 2 2.5 3 -20 -15 -10 -5 0 5 10 15 20TimeTime LagFigure. Comparison of time series and autocorrelation functions for various cases.0.5Copyright 2005 Dennis L. Hartmann 1/25/05 5:04 PM 10610.500.500.5PeriodicTwo Periods1-20 -15 -10 -5 0 5 10 15 20Time Lag10.500.5-0.50.5White Noise1-20 -15 -10 -5 0 5 10 15 20Time Lag10.500.50-0.5-110.50Red Noise (0.8)-1-20 -15 -10 -5 0 5 10 15 20Time Lag1Red Noise (0.8)+ Period-20 -15 -10 -5 0 5 10 15 20Time LagSawtooth Wave
ATM 552 Notes: Time Series Analysis - Section 6a Page 1076.1.3 Statistical Prediction and Red NoiseConsider a prediction equation of the formx ˆ ( t + t)= a' xt ()+ b' xtt ( ) (6.7)where x = 0 . a' and b' are chosen to minimize the rms error on dependent data. Recall fromour discussion of multiple regression that for two predictors x 1 and x 2 used to predict yrx ( 2 , y) rx ( 1 ,y)rx ( 1 ,x 2 )In the case where the equality holds, r(x 2 ,y) is equal to the “minimum useful correlation”discussed in Chapter 3 and will not improve the forecasting skill beyond the level possibleby using x 1 alone. In the case of trying to predict future values from prior times, r(x 2 ,y) =r(2t), and r(x 1 ,y) = r(x 1 ,x 2 ) = r(t) so that we must haver( 2t) > r( t) 2in order to justify using a second predictor at two time steps in the past. Note that for rednoiser( 2t) = r( t) 2so that the value at two lags previous to now always contributes exactly the minimum useful,and nearly automatic, correlation, and there is no point in using a second predictor if thevariable we are trying to predict is red noise. All we can use productively is the presentvalue and the autocorrelation function,xt+ ( t)= xt () with an R 2 = a 2 = r( t) 2This is just what is called a persistence forecast, we assume tomorrow will be like today.6.1.4 White NoiseIn the special case r(t) = a = 0, our time series is a series of random numbers,uncorrelated in time so that r() = (0) a delta function. For such a “white noise” timeseries, even the present value is of no help in projecting into the future. The probabilitydensity function we use is generally normally distributed about zero mean, and this isgenerated by the ‘randn’ function in Matlab.Copyright 2005 Dennis L. Hartmann 1/25/05 5:04 PM 107
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Chapter 6: Time Series Analysis

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