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Lecture 2: ARMA Models Bus 41910, Autumn Quarter 2008, Mr. Ruey S. Tsay Autoregressive Moving-Average (ARMA) models form a class of linear time series models which are widely applicable and parsimonious in parameterization. By allowing the order of an ARMA model to increase, one can approximate any linear time series model with desirable accuracy. [This is similar to using rational polynomials to approximate a general polynomial; see the impluse response function.] In what follows, we assume that is a sequence of independent and identically distrib- uted random variables with mean zero and ﬁnite variance . That is, is a white noise series. Sometimes, we use to denote the variance of A. Autoregressive (AR) Processes 1. AR( ) Model: or , where is a constant. If is stationary with mean ) = , then the model can be written as )( ) = . The latter is the parameterization implemented in the R package. 2. Stationarity: All zeros of the polynomial ) lie outside the unit circle. (Why?) 3. Moments: (Assume stationarity) Mean: Taking expectation of both sides of the model equation, we have ) = (1 This relation between and is important for stationary time series. Autocovariance function: For simplicity, assume = 0. Multiplying both sides of the AR model by and taking expectations, we have [( ] = For = 0, For `> 0, = 0

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Autocorrelation function (ACF): . From autocovariance function, we have for = 0 0 for `> 0. Since the ACFs satisfy the -th order diﬀerence equation, namely = 0 for `> 0, they decay exponentially to zero as 4. Yule-Walker equation: Consider the above equations jointly for = 1 ,p , we have which is the -order Yule-Walker equation. For given ’s, the equation can be used to solve for . Of course, we can obtain ’s for given ’s and via the moment generating function or the -weight representation to be discussed shortly. 5. MA representation: =0 where the ’s are referred to as the -weights of the model. These -weights are also called the impulse response function and can be obtained from the ’s by equating the coeﬃcients of = 1 + 1 = (1 )(1 + Thus, we have =1 for `>p. Again, the -weights satisfy the diﬀerence equation = 0 for ` > p so that they also decay exponentially to zero as goes to inﬁnite.

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6. The moment generating function: Γ( ) = 7. A simple example: the AR(1) case Stationarity condition: 1. Mean: ACF: Yule-Walker equation: MA representation: so that the -weights are The variance of : Var( ) = 8. An AR(2) model: (1 Stationarity condition: Zeros of ) are outside the unit circle. Mean: ACF: = 1, (1 ), and , for j > 1. Why? Yule-Walker equation: or equivalently, #" MA representation: where 1 +

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The variance of can be obtained from the moment equations as Var( ) = 1 + [(1 B. Moving-Average (MA) Processes 1. MA( ) Model: or 2. Stationarity: Finite order MA models are stationary. 3. Mean: By taking expectation on both sides, we obtain ) = Thus, for MA models, the constant term is the mean of the process. 4. Invertibility: Can we write an MA model as an AR model? Consider ) = Let ) = 1 . We call the ’s the weights of the process Similar to , for the above AR representation to be meaningful we require that =1 The necessary and suﬃcient condition for such a convergent -weight sequence is that all the zeros of ) lie outside the unit circle. 5. Autocovariance function: Again for simplicity, assume = 0. Multiple both sides by and take expectation. We have (1 + for = 0 =0 for = 1 ,q 0 for `>q where 1. This shows that for an MA( ) process, the autocovariance function is zero for ` > q . This is a special feature of MA processes and it provides a convenient way to identify an MA model in practice.

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6. Autocorrelation function: . Again, from the autocovariance function we have = 0 for 0 for `>q In other words, the ACF has only a ﬁnite number of non-zero lags. Thus, for an MA( ) model, and are uncorrelated provided that `>q . For this reason, MA models are referred to as short memory time series. 7. The moment generating function: Γ( ) = 8. A simple example: the MA(1) case θa Invertibility condition: 1. Mean: ) = ACF: 1 for = 0 1+ for = 1 0 for `> 1. The above result shows that 5 for an MA(1) model. Variance: = (1 + Moment generating fucntion: Γ( ) = [(1 + )] From the MGF, we have = (1 + , and = 0 for 2.

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C. Mixed ARMA Processes 1. ARMA( p,q ) Model: or . Again, for a stationary process, we can rewrite the model as )( ) = 2. Stationarity: All zeros of ) lie outside the unit circle. 3. Invertibility: All zeros of ) lie outside the unit circle. 4. AR representation: (1) , where ) = The -weight can be obtained by equating the coeﬃcients of in ) = 5. MA representation: (1) , where ) = Again, the -weights can be obtained by equating coeﬃcients. Note that ) = 1 for all for an ARMA model. This identity has many applications. 6. Moments: (Assume stationarity) Mean: Autocovariance function: (Assume = 0) Using the result ) = for = 0 for `> 0 for `< 0, and the same technique as before, we have (1 for = 0 +1 for = 1 ,q 0 for + 1 where = 1 and = 0 for j >q

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Autocorrelation function: satisﬁes = 0 for `>q. You may think that the ACF satisfy the diﬀerence equation = 0 for + 1 with , as initial conditions. 7. Generalized Yule-Walker Equation: Consider the above equations of ACF for + 1 ,q , we have +1 +2 +2 +1 +1 +3 +2 +1 which is referred to as a -th order generalized Yule-Walker equation for the ARMA( p,q process. It can be used to solve for ’s given the ACF ’s. 8. Moment generating function: Γ( ) = 9. A simple example: The ARMA(1,1) case φZ θa Stationarity condition: 1. Invertibility condition: 1. Mean: Variance: (1+ ACF: (1 )( 1 + for `> 10. AR representation: where ). 11. MA representation: where ).

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1 Illurative Examples Some examples of ARMA models are given below with demonstration. U.S. quarterly growth rate of GDP. Daily returns of the US-EU exchange rate. Chemical concentration readings, Series A, of Box and Jenkins (1976).

Ruey S Tsay Autoregressive MovingAverage ARMA models form a class of linear time series models which are widely applicable and parsimonious in parameterization By allowing the order of an ARMA model to increase one can approximate any linear time se ID: 22266

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Page 1

Lecture 2: ARMA Models Bus 41910, Autumn Quarter 2008, Mr. Ruey S. Tsay Autoregressive Moving-Average (ARMA) models form a class of linear time series models which are widely applicable and parsimonious in parameterization. By allowing the order of an ARMA model to increase, one can approximate any linear time series model with desirable accuracy. [This is similar to using rational polynomials to approximate a general polynomial; see the impluse response function.] In what follows, we assume that is a sequence of independent and identically distrib- uted random variables with mean zero and ﬁnite variance . That is, is a white noise series. Sometimes, we use to denote the variance of A. Autoregressive (AR) Processes 1. AR( ) Model: or , where is a constant. If is stationary with mean ) = , then the model can be written as )( ) = . The latter is the parameterization implemented in the R package. 2. Stationarity: All zeros of the polynomial ) lie outside the unit circle. (Why?) 3. Moments: (Assume stationarity) Mean: Taking expectation of both sides of the model equation, we have ) = (1 This relation between and is important for stationary time series. Autocovariance function: For simplicity, assume = 0. Multiplying both sides of the AR model by and taking expectations, we have [( ] = For = 0, For `> 0, = 0

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Autocorrelation function (ACF): . From autocovariance function, we have for = 0 0 for `> 0. Since the ACFs satisfy the -th order diﬀerence equation, namely = 0 for `> 0, they decay exponentially to zero as 4. Yule-Walker equation: Consider the above equations jointly for = 1 ,p , we have which is the -order Yule-Walker equation. For given ’s, the equation can be used to solve for . Of course, we can obtain ’s for given ’s and via the moment generating function or the -weight representation to be discussed shortly. 5. MA representation: =0 where the ’s are referred to as the -weights of the model. These -weights are also called the impulse response function and can be obtained from the ’s by equating the coeﬃcients of = 1 + 1 = (1 )(1 + Thus, we have =1 for `>p. Again, the -weights satisfy the diﬀerence equation = 0 for ` > p so that they also decay exponentially to zero as goes to inﬁnite.

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6. The moment generating function: Γ( ) = 7. A simple example: the AR(1) case Stationarity condition: 1. Mean: ACF: Yule-Walker equation: MA representation: so that the -weights are The variance of : Var( ) = 8. An AR(2) model: (1 Stationarity condition: Zeros of ) are outside the unit circle. Mean: ACF: = 1, (1 ), and , for j > 1. Why? Yule-Walker equation: or equivalently, #" MA representation: where 1 +

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The variance of can be obtained from the moment equations as Var( ) = 1 + [(1 B. Moving-Average (MA) Processes 1. MA( ) Model: or 2. Stationarity: Finite order MA models are stationary. 3. Mean: By taking expectation on both sides, we obtain ) = Thus, for MA models, the constant term is the mean of the process. 4. Invertibility: Can we write an MA model as an AR model? Consider ) = Let ) = 1 . We call the ’s the weights of the process Similar to , for the above AR representation to be meaningful we require that =1 The necessary and suﬃcient condition for such a convergent -weight sequence is that all the zeros of ) lie outside the unit circle. 5. Autocovariance function: Again for simplicity, assume = 0. Multiple both sides by and take expectation. We have (1 + for = 0 =0 for = 1 ,q 0 for `>q where 1. This shows that for an MA( ) process, the autocovariance function is zero for ` > q . This is a special feature of MA processes and it provides a convenient way to identify an MA model in practice.

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6. Autocorrelation function: . Again, from the autocovariance function we have = 0 for 0 for `>q In other words, the ACF has only a ﬁnite number of non-zero lags. Thus, for an MA( ) model, and are uncorrelated provided that `>q . For this reason, MA models are referred to as short memory time series. 7. The moment generating function: Γ( ) = 8. A simple example: the MA(1) case θa Invertibility condition: 1. Mean: ) = ACF: 1 for = 0 1+ for = 1 0 for `> 1. The above result shows that 5 for an MA(1) model. Variance: = (1 + Moment generating fucntion: Γ( ) = [(1 + )] From the MGF, we have = (1 + , and = 0 for 2.

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C. Mixed ARMA Processes 1. ARMA( p,q ) Model: or . Again, for a stationary process, we can rewrite the model as )( ) = 2. Stationarity: All zeros of ) lie outside the unit circle. 3. Invertibility: All zeros of ) lie outside the unit circle. 4. AR representation: (1) , where ) = The -weight can be obtained by equating the coeﬃcients of in ) = 5. MA representation: (1) , where ) = Again, the -weights can be obtained by equating coeﬃcients. Note that ) = 1 for all for an ARMA model. This identity has many applications. 6. Moments: (Assume stationarity) Mean: Autocovariance function: (Assume = 0) Using the result ) = for = 0 for `> 0 for `< 0, and the same technique as before, we have (1 for = 0 +1 for = 1 ,q 0 for + 1 where = 1 and = 0 for j >q

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Autocorrelation function: satisﬁes = 0 for `>q. You may think that the ACF satisfy the diﬀerence equation = 0 for + 1 with , as initial conditions. 7. Generalized Yule-Walker Equation: Consider the above equations of ACF for + 1 ,q , we have +1 +2 +2 +1 +1 +3 +2 +1 which is referred to as a -th order generalized Yule-Walker equation for the ARMA( p,q process. It can be used to solve for ’s given the ACF ’s. 8. Moment generating function: Γ( ) = 9. A simple example: The ARMA(1,1) case φZ θa Stationarity condition: 1. Invertibility condition: 1. Mean: Variance: (1+ ACF: (1 )( 1 + for `> 10. AR representation: where ). 11. MA representation: where ).

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1 Illurative Examples Some examples of ARMA models are given below with demonstration. U.S. quarterly growth rate of GDP. Daily returns of the US-EU exchange rate. Chemical concentration readings, Series A, of Box and Jenkins (1976).

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