Slides-04 Time Series Analysis using ARMA models: Part 2
Univariate Time Series Analysis: ARIMA models
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Building ARIMA models
An ARMA Process
Examples ARMA(p, q) process
Consider the following ARMA(1,1) process:
yt = 2 + 0.5yt−1 + 0.9εt−1 + εt , εt ∼ N (0, 1) , T = 100
This is a stationary series as 0.5 < 1. Properties I The expected value is given by E (yt) = 2 /(1− 0.5) = 4 I The variance is given by 1 + β21 + 2α1β1 1 + 0.92 + 2× 0.5× 0.9 Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process I The ACF is given by (1 + α1β1) (α1 + β1) 1 + β21 + 2α1β1 (1 + 0.5× 0.9) (0.5 + 0.9) 1 + 0.92 + 2× 0.5× 0.9 ρ2 = α1ρ1 = 0.5× 0.7491 = 0.3745 ρ3 = α1ρ2 = 0.5× 0.3745 = 0.1873 ρ4 = α1ρ3 = 0.5× 0.1873 = 0.0936 I The PACF is given by τ11 = ρ1 = 0.7491 ρ3 − τ21ρ2 − τ22ρ1 1− τ21ρ1 − τ22ρ2 where τ21 = τ11 − τ22 τ11 = 1.0675 Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Figure 46 : Theoretical ACF and PACF of generated ARMA(1,1) process 1 2 3 4 5 6 7 8 9 10 Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Figure 47 : Dynamic impact of a shock εt on y t-5 t t+5 t+10 t+15 t+20 t+25 Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Figure 48 : A generated ARMA(1,1) process 0 10 20 30 40 50 60 70 80 90 100 Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Figure 49 : Sample ACF and PACF of generated ARMA(1,1) process 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com