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
