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Unformatted text preview: Data Analysis for Financial Engineers Professor S. Kou Department of IEOR, Columbia University Lecture 5. The second attempt to model the dependent structures: MA and ARMA models 1 The Moving Average Model 1.1 Introduction to the moving average model In the previous lecture, we studied the AR(p) model, which is used to study the dependent structures of stock returns: r t = & + & 1 r t & 1 + & & & + & p r t & p + a t : More generally, we can think of an in&nite order AR model r t = & + & 1 r t & 1 + & 2 r t & 2 + & & & + a t : However, such a model is impossible to built, as it has in&nite number of parameters. One way to reduce the in&nite number of parameters is to introduce some structures among & i , i 1 . In particular, letting & i = i 1 yields r t = & 1 r t & 1 2 1 r t & 2 & & & + a t ; or more precisely an AR( 1 ) representation: r t + 1 r t & 1 + 2 1 r t & 2 + & & & = & + a t : (1) Of course, to make sense of the above in&nite expansion, we need j 1 j < 1 . This is called the invertibility condition. In other words, if a time series is invertible, then the current value ( r t ) is a weighted average of all past values ( r t & 1 ;r t & 2 ;::: ) and the current stochastic shock ( a t ), such as the one in the above AR( 1 ) representation. We will talk more about the invertibility later. From the above model we also have at time t 1 ; r t & 1 + 1 r t & 2 + 2 1 r t & 3 + & & & = & + a t & 1 ; or, multiplying it by 1 ; 1 r t & 1 + 2 1 r t & 2 + 3 1 r t & 3 + & & & = 1 & + 1 a t & 1 : 1 Subtracting it from the original equation yields r t = & + a t & 1 & & 1 a t & 1 ; and r t = (1 & 1 ) & + a t & 1 a t & 1 : This is called the moving average model or order one (MA(1)). More precisely, the de&nition of MA(1) is r t = c + a t & 1 a t & 1 : Similarly MA(q) is de&ned as r t = c + a t & 1 a t & 1 & & q a t & q : 1.2 Properties of the MA(q) 1.2.1 Mean and Variance For simplicity, assume c = 0 . In the MA(1) model, taking expectation we have E [ r t ] = E [ a t ] & 1 E [ a t & 1 ] = c : In other words, the E ( r t ) is a &xed constant independent of t . Taking variance we have, via independence, V ar [ r t ] = V ar [ a t ] + 2 1 V ar [ a t & 1 ] = (1 + 2 1 ) 2 a : This is again a &xed constant independent of t . In general, in the MA(q) model E ( r t ) = 0 ; V ar ( r t ) = (1 + 2 1 + + 2 q ) 2 a : 1.2.2 Autocorrelation Function Consider the MA(1) case &rst. Assume for simplicity that c = 0 . Then multiplying the model by r t & l yields r t & l r t = r t & l a t & 1 r t & l a t & 1 : Therefore, E ( r t & l r t ) = 0 ; if l > 1; E ( r t & 1 r t ) = & 1 2 a : From this, we get = 1 ; 1 = & 1 1 + 2 1 ; l = 0 ; l 2 : 2 Therefore, the ACF of MA(1) simply cuts o/ at lag 1....
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This note was uploaded on 10/16/2010 for the course IEOR 4709 taught by Professor Stevenkou during the Fall '10 term at Columbia.
 Fall '10
 StevenKou
 Financial Engineering

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