1	APS 425 – Fall 2012 Time Series Analysis: ARIMA Models Instructor: G. William Schwert 585-275-2470 schwert@schwert.simon.rochester.edu
2	Topics Typical time series plot Pattern recognition in auto and partial autocorrelations Stationarity & invertibility Stochastic Seasonality (seasonal ARIMA)
3	Partial autocorrelations Used to identify pure AR models Estimate a sequence of AR(k) models, and report the last coefficient estimate, fkk, for each lag k: AR(k): Zt = a + f1k Zt-1 + . . . + fkk Zt-k + at Graph the pacf coefficients, fkk, and see where they become zero, which implies that the right model is a (k-1)th order AR process
4	"AR(1):" AR(1): Zt = a + f1 Zt-1 + at f1 = .9
5	"AR(1):" AR(1): Zt = a + f1 Zt-1 + at f1 = .5
6	"AR(1):" AR(1): Zt = a + f1 Zt-1 + at f1 = -.5
7	"AR(1):" AR(1): Zt = a + f1 Zt-1 + at f1 = -.9
8	"AR(1):" AR(1): Zt = a + f1 Zt-1 + at f1 = .9
9	"AR(1):" AR(1): Zt = a + f1 Zt-1 + at f1 = -.9
10	"AR(2):" AR(2): Zt = a + f1 Zt-1 + f2 Zt-2 + at f1 = 1.4 f2 = -.45
11	"AR(2):" AR(2): Zt = a + f1 Zt-1 + f2 Zt-2 + at f1 = .4 f2 = .45
12	"AR(2):" AR(2): Zt = a + f1 Zt-1 + f2 Zt-2 + at f1 = 1.4 f2 = -.45
13	"Autoregressive Models" Autoregressive Models: Summary
14	"MA(1):" MA(1): Zt = a + at - q1 at-1 q1 = .9
15	"MA(1):" MA(1): Zt = a + at - q1 at-1 q1 = .5
16	"MA(1):" MA(1): Zt = a + at - q1 at-1 q1 = -.5
17	"MA(1):" MA(1): Zt = a + at - q1 at-1 q1 = .9
18	"Moving Average Models" Moving Average Models: Summary
19	"Autoregressive Moving Average Models" Autoregressive Moving Average Models
20	"ARMA(1,1):" ARMA(1,1): Zt = a + f1 Zt-1 + at - q1 at-1 f1 = .9, q1 = .5
21	"ARMA(1,1):" ARMA(1,1): Zt = a + f1 Zt-1 + at - q1 at-1 f1 = .9, q1 = .5
22	"Autoregressive Moving Average Models" Autoregressive Moving Average Models: Summary
23	"Autoregressive Integrated Moving Average ARIMA(p,d,q" Autoregressive Integrated Moving Average ARIMA(p,d,q) Models
24	"ARIMA(0,1,1):" ARIMA(0,1,1): (Zt - Zt-1) = at - .8 at-1 [T=150]
25	"ARIMA(0,1,1):" ARIMA(0,1,1): (Zt - Zt-1) = at - .5 at-1 [T=150]
26	"ARIMA(0,1,1" ARIMA(0,1,1) (Zt - Zt-1) = at - .8 at-1
27	"Seasonal ARIMA(P,D,Q)s Models" Seasonal ARIMA(P,D,Q)s Models
28	"Seasonal ARIMA(0,1,1)s Models" Seasonal ARIMA(0,1,1)s Models
29	"Seasonal Exponential" Seasonal Exponential Smoothing Forecasts
30	"Seasonal ARIMA(P,D,Q)_s Models" Seasonal ARIMA(P,D,Q)_s Models Interacting with ARIMA(p,d,q) Models
31	"Seasonal ARIMA(P,D,Q)_s Models" Seasonal ARIMA(P,D,Q)_s Models Interacting with ARIMA(p,d,q) Models
32	"Integrated Moving Average Models" Integrated Moving Average Models: Summary
33	Links Return to APS 425 Home Page: http://schwert.simon.rochester.edu/A425/A425main.htm