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Unformatted text preview: Financial Data Analysis Professor S. Kou, Department of IEOR, Columbia University Lecture 4. Autoregression (AR) Models 1 AR(1) Model The autoregression model with order one, or AR(1) model is formulated as r t = & + & 1 r t & 1 + a t ; where a t is a sequence of white noise with mean zero and variance 2 a , & and & 1 are constants. 1.1 Mean and Variance The conditional mean and variance of the AR(1) model is given by E ( r t j r t & 1 ) = & + & 1 r t & 1 ; V ar ( r t j r t & 1 ) = V ar ( a t ) = 2 a : The unconditional mean satis&es the following recursion E ( r t ) = E f E ( r t j r t & 1 ) g = E f & + & 1 r t & 1 g = & + & 1 E f r t & 1 g : From this we have E ( r t ) = & + & 1 E f r t & 1 g = & f 1 + & 1 g + & 2 1 E f r t & 2 g = & f 1 + & 1 + & 2 1 g + & 3 1 E f r t & 3 g = & & & = & f 1 + & 1 + & 2 1 + & & & + & t & 2 1 g + & t & 1 1 E f r 1 g For the series to converge, we need to assume that j & 1 j < 1 . In fact, it can be shown that the necessary and su cient condition for the r t to be weakly stationary is that j & 1 j < 1 : If j & 1 j < 1 , then the mean is a constant, E ( r t ) = , and the above recursion becomes = & + & 1 ; which implies that E ( r t ) = = & 1 & 1 : Similarly, if j & 1 j < 1 , then the unconditional variance is given by V ar ( r t ) = E [ V ar ( r t j r t & 1 )] + V ar [ E ( r t j r t & 1 )] = 2 a + V ar [ & + & 1 r t & 1 ] = 2 a + & 2 1 V ar [ r t & 1 ] ; and therefore V ar ( r t ) = 2 a 1 & 2 1 1 1.2 ACF Since r t & & = 1 ( r t & 1 & & ) + a t ; and for any l 1 , we have Cov ( r t ;r t & l ) = E f ( r t & & )( r t & l & & ) g = 1 E f ( r t & 1 & & )( r t & l & & ) g + E f a t ( r t & l & & ) g : Note that the expectation in the second term is zero as E f a t ( r t & l & & ) g = E f a t g E ( r t & l & & ) = 0 ; via independence. Thus, for l 1 ; Cov ( r t ;r t & l ) = 1 E f ( r t & 1 & & )( r t & l & & ) g = 1 Cov ( r t & 1 ;r t & l ) : Therefore, for l 1 ; l = Corr ( r t ;r t & l ) = 1 Cov ( r t & 1 ;r t & l ) p V ar ( r t ) V ar ( r t & l ) = 1 Cov ( r t & 1 ;r t & l ) p V ar ( r t & 1 ) V ar ( r t & l ) = 1 Corr ( r t & 1 ;r t & l ) = 1 l & 1 : Since = 1 we have l = l 1 : In summary, the autocorrelation function of AR(1) model must have an exponential decay. 2 AR(2) Model The AR(2) model is formulated as r t = + 1 r t & 1 + 2 r t & 2 + a t ; where a t is a sequence of white noise with mean zero and variance 2 a , and ; 1 ; 2 are constants. 2.1 Mean and Variance The conditional mean and variance are E ( r t j r t & 1 ;r t & 2 ) = + 1 r t & 1 + 2 r t & 2 ; V ar ( r t j r t & 1 ;r t & 2 ) = V ar ( a t ) = 2 a : Thus we have the following recursion for the (unconditional) mean E ( r t ) = E f E ( r t j r t & 1 ;r t & 2 ) g = E f + 1 r t & 1 + 2 r t & 2 g = + 1 E f r t & 1 g + 2 E f r t & 2 g : 1.2 ACF Since r t & & = 1 ( r t & 1 & & ) + a t ; and for any l 1 , we have Cov ( r t ;r...
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 Fall '10
 StevenKou
 Financial Engineering

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