Time Series Analysis Chapter 02 (4up).pdf - 2 Outline 1 2 3...

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TIME SERIES ANALYSIS Chapter 2, Stationary processes (omit §2.6) Outline 2 1. Basic properties 2. Linear processes 3. Introduction to ARMA processes 4. Properties of sample mean and ACF 5. Forecasting stationary time series 1. Basic properties 3 Let ° ± be a stationary time series with mean ² ACVF ³ & ACF ´ , ℎ = 0, ±1, ±2, ⋯ Basic properties of ACVF & ACF Basic properties of ACVF ³ 0 ≥ 0 ³ ≤ ³ 0 ³ = ³ −ℎ Basic properties of ACF ´ 0 = 1 ´ ≤ 1 ´ = ´ −ℎ 4
Nonnegative definite function A real-valued function µ defined on the integers is nonnegative definite if ¶ ¶ · ¸ µ ¹ − º · » ¼ »½¾ ¼ ¸½¾ ≥ 0 for all positive integers ¿ and vectors À = · ¾ , ⋯ , · ¼ Á with real-valued components · ¸ . 5 Theorem µ · is the ACVF of a stationary time series if & only if (i) µ · is an even function, & (ii) µ · is nonnegative definite. Note, (ii) is hard to verify, so the theorem is usually used for disproof. 6 Remark To show that µ · is the ACVF of a stationary process, it is often simpler to find the process that has µ · as its ACVF than to verify (ii). Example: which of the following functions are ACVF? µ = −1  µ = 1 + !"# $ℎ/2 + !"# $ℎ/4 µ = ’ 1 ¹( ℎ = 0 0.4 ¹( ℎ = ±1 0 "+ℎ,-.¹#, 7 Role of ACVF & ACF in time series forecasting ACVF & ACF provide useful measure of dependence among time series data. Hence, they play an important role in time series forecasting. 8
Illustration example Suppose ° ± is a stationary Gaussian time series with mean ² , ACVF ³ · , and ACF ´ Suppose we have observed ° ¼ We want to forecast ° ¼/Â , ℎ ≥ 1, based on ° ¼ 9 Best MSE predictor Criterion of the best prediction: Find the predictor that minimizes 0 ° ¼/Â − ( ° ¼ 1 over all possible functions ( Such predictor, say 2 ° ¼ , is called the best MSE predictor . What is the best MSE predictor of ° ¼/Â based on ° ¼ 10 · . . . . ? . 1
Best linear predictor (BLP) Criterion of the best prediction: Find the predictor that minimizes 0 ° ¼/Â − ( ° ¼ 1 over all linear functions ( of the form ·° ¼ + 6 . Such predictor, say ° ¼ , is called the best linear predictor . Finding BLP is equivalent to finding · & 6 to minimize 9 ·, 6 = 0 ° ¼/Â − ·° ¼ − 6 1 . 13 Example (cont’d) 14 Remarks For Gaussian time series, best MSE predictor = BLP. In general, best MSE predictor gives smaller MSE than BLP.

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