ARMA processes

In order to get a feeling for ACF/PACF and ARMA models, we simulate fiveAR/MA processes:1. yt = 1− 0.8yt−1 + εt

2. yt = 1+ 0.8yt−1 + εt

3. yt = 1+ 1.2yt−1 − 0.5yt−2 + εt

4. yt = 1+ εt − 0.8εt−1

5. yt = yt−1 + εt

Time Series Plot

yt = 1− 0.8yt−1 + εt

ACF and PACF Plots

yt = 1− 0.8yt−1 + εt

ACF PACF

Estimated Spectrum

yt = 1− 0.8yt−1 + εt

Time Series Plot

yt = 1+ 0.8yt−1 + εt

ACF and PACF Plots

yt = 1+ 0.8yt−1 + εt

ACF PACF

Estimated Spectrum

yt = 1+ 0.8yt−1 + εt

Time Series Plot

yt = 1+ 1.2yt−1 − 0.5yt−2 + εt

ACF and PACF Plots

yt = 1+ 1.2yt−1 − 0.5yt−2 + εt

ACF PACF

STATA Regression Output

yt = 1+ 1.2yt−1 − 0.5yt−2 + εt

AIC/BIC Model Comparison Table

yt = 1+ 1.2yt−1 − 0.5yt−2 + εt

AIC BIC

• Both AIC and BIC suggest AR(2) specification (not surprisingly!)

Estimated Spectrum

yt = 1+ 1.2yt−1 − 0.5yt−2 + εt

Time Series Plot

yt = 1+ εt − 0.8εt−1

ACF and PACF Plots

yt = 1+ εt − 0.8εt−1

ACF PACF

Estimated Spectrum

yt = 1+ εt − 0.8εt−1

Time Series Plot

yt = yt−1 + εt

ACF and PACF Plots

yt = yt−1 + εt

ACF PACF

Estimated Spectrum

yt = yt−1 + εt