[SOLVED] CS代考 Slides-04 Time Series Analysis using ARMA models: Part 2

Slides-04Time Series Analysis using ARMA models: Part 2

Univariate Time Series Analysis: ARIMA models

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Building ARIMA models

Autoregressive Process

Consider the following AR(2) process:

yt = 2− 0.5yt−1 + 0.3yt−2 + εt , εt ∼ N (0, 1) , T = 100

The characteristic equation is given by

1 + 0.5z − 0.3z2 = 0

The characteristic roots are

0.52 + 4× 0.3

0.52 + 4× 0.3

This is a stationary series as the characteristic roots are larger than
1 in absolute value.

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Note that stationarity could also be concluded from

i=1 αi = −0.5 + 0.3 = −0.2 < 1i=1 |αi | = 0.5 + 0.3 = 0.8 < 1PropertiesI The expected value of the series is given byE (yt) = 2 /(1 + 0.5− 0.3) = 1.67I The variance is given by(1 + 0.3) (1 + 0.5− 0.3) (1− 0.5− 0.3)Univariate Time Series Analysis: ARIMA modelsBuilding ARIMA modelsAutoregressive ProcessI The ACF is given byρ1 = −0.5 /(1− 0.3) = −0.71432 /(1− 0.3) + 0.3 = 0.6571ρ3 = 0.5× 0.6571 + 0.3× 0.7143 = −0.5429ρ4 = 0.5× 0.5429 + 0.3× 0.6571 = 0.4686I The PACF is given byτ11 = −0.7143τkk = 0 ∀k > 2

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Figure 37 : Theoretical ACF and PACF of generated AR(2) process

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Figure 38 : Dynamic impact of a shock εt on y

t-5 t t+5 t+10 t+15 t+20 t+25

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