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# Chapter Part III Mixed AutoregressiveMoving Average Models Even though the ARp and MAq models are somewhat unrealistic by themselves we can mi hem to form the extremely useful ARMApq models PDF document - DocSlides

briana-ranney | 2014-12-18 | General

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### Presentations text content in Chapter Part III Mixed AutoregressiveMoving Average Models Even though the ARp and MAq models are somewhat unrealistic by themselves we can mi hem to form the extremely useful ARMApq models

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Chapter 3, Part III: Mixed Autoregressive-Moving Average Models Even though the AR(p) and MA(q) models are somewhat unrealistic by themselves, we can mi hem to form the extremely useful ARMA(p,q) models. The ARMA(p,q) series { } is generated by 12 pt pt 12 qt + ... + ε+ ε+ ε+ ... Thus, { } is essentially the sum of an autoregression (on past values of { }) and a moving average (o tt t white noise process { }). Given { } together with starting values of { }, the whole series { can be formed iteratively. The series will be stationary provided that the AR part is stationary, i.e., pro ided that the AR parameters ,..., satisfy the conditions given in Part II. Note that the AR(p) and MA(q) models are both special cases of the mixed model AR ARMA ,0) , MA ARMA (0 , Forecasting for ARMA models proceeds as usual. In the one-step case, for example, we hav = + ... + +ε+ ε+ ε+ ... 11 pn 11 qn o that the optimal forecast is = + ... + ε+ ε+ ... ,1 1 pn 11 qn and the one-step forecast error i = ,1 ,1 tt To actually form the optimal forecast, we need { } . These can be estimated by startin ith 0 and then forming the recursively by 0,1 ttt 1,1 ε= ft 1,2 ,...,n .

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-2- he practical importance of mixed ARMA models is a consequence of the following theorem: If are independent series with tt 11 22 then their sum is ARMA ), ARMA ARMA ), where typically ttt max( 12 1221 hus, the sum of independent ARMA processes is again ARMA. The theorem implies that mixed models are much more plausible for real-world economic series than the individual AR and MA odels. For example, suppose that the series of interest is AR(p) but can only be observed with a meas- urement error, assumed to be white noise, MA(0). Then the observed series will be ARMA(p,p). As nother example, note that many economic variables are aggregates (sums) of other series. If each component is AR, the aggregate will be ARMA. If some series are AR and others MA then the sum i gain ARMA. The only case where the ARMA model does not arise is if each component is MA. Then the sum will be MA.

The ARMApq series is generated by 12 pt pt 12 qt 949 949 949 Thus is essentially the sum of an autoregression on past values of and a moving average o tt t white noise process Given together with starting values of the whole series ID: 25823

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Page 1

Chapter 3, Part III: Mixed Autoregressive-Moving Average Models Even though the AR(p) and MA(q) models are somewhat unrealistic by themselves, we can mi hem to form the extremely useful ARMA(p,q) models. The ARMA(p,q) series { } is generated by 12 pt pt 12 qt + ... + ε+ ε+ ε+ ... Thus, { } is essentially the sum of an autoregression (on past values of { }) and a moving average (o tt t white noise process { }). Given { } together with starting values of { }, the whole series { can be formed iteratively. The series will be stationary provided that the AR part is stationary, i.e., pro ided that the AR parameters ,..., satisfy the conditions given in Part II. Note that the AR(p) and MA(q) models are both special cases of the mixed model AR ARMA ,0) , MA ARMA (0 , Forecasting for ARMA models proceeds as usual. In the one-step case, for example, we hav = + ... + +ε+ ε+ ε+ ... 11 pn 11 qn o that the optimal forecast is = + ... + ε+ ε+ ... ,1 1 pn 11 qn and the one-step forecast error i = ,1 ,1 tt To actually form the optimal forecast, we need { } . These can be estimated by startin ith 0 and then forming the recursively by 0,1 ttt 1,1 ε= ft 1,2 ,...,n .

Page 2

-2- he practical importance of mixed ARMA models is a consequence of the following theorem: If are independent series with tt 11 22 then their sum is ARMA ), ARMA ARMA ), where typically ttt max( 12 1221 hus, the sum of independent ARMA processes is again ARMA. The theorem implies that mixed models are much more plausible for real-world economic series than the individual AR and MA odels. For example, suppose that the series of interest is AR(p) but can only be observed with a meas- urement error, assumed to be white noise, MA(0). Then the observed series will be ARMA(p,p). As nother example, note that many economic variables are aggregates (sums) of other series. If each component is AR, the aggregate will be ARMA. If some series are AR and others MA then the sum i gain ARMA. The only case where the ARMA model does not arise is if each component is MA. Then the sum will be MA.

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