Slides-04 Time 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 < 1 i=1 |αi | = 0.5 + 0.3 = 0.8 < 1 Properties I The expected value of the series is given by E (yt) = 2 /(1 + 0.5− 0.3) = 1.67 I The variance is given by (1 + 0.3) (1 + 0.5− 0.3) (1− 0.5− 0.3) Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process I The ACF is given by ρ1 = −0.5 /(1− 0.3) = −0.7143 2 /(1− 0.3) + 0.3 = 0.6571 ρ3 = 0.5× 0.6571 + 0.3× 0.7143 = −0.5429 ρ4 = 0.5× 0.5429 + 0.3× 0.6571 = 0.4686 I The PACF is given by τ11 = −0.7143 τkk = 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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