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lecture3 - EC3062 ECONOMETRICS LINEAR STOCHASTIC MODELS Let...

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EC3062 ECONOMETRICS LINEAR STOCHASTIC MODELS Let { x τ +1 , x τ +2 , . . . , x τ + n } denote n consecutive elements from a stochas- tic process. If their joint distribution does not depend on τ , regardless of the size of n , then the process is strictly stationary. Any two segments of equal length will have the same distribution with (1) E ( x t ) = μ < for all t and C ( x τ + t , x τ + s ) = γ | t s | . The condition on the covariances implies that the dispersion matrix of the vector [ x 1 , x 2 , . . . , x n ] is a bisymmetric Laurent matrix of the form (2) Γ = γ 0 γ 1 γ 2 . . . γ n 1 γ 1 γ 0 γ 1 . . . γ n 2 γ 2 γ 1 γ 0 . . . γ n 3 . . . . . . . . . . . . . . . γ n 1 γ n 2 γ n 3 . . . γ 0 , wherein the generic element in the ( i, j )th position is γ | i j | = C ( x i , x j ). 1
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EC3062 ECONOMETRICS Moving-Average Processes The q th-order moving average MA( q ) process, is defined by (3) y ( t ) = μ 0 ε ( t ) + μ 1 ε ( t 1) + · · · + μ q ε ( t q ) , where ε ( t ) = { ε t ; t = 0 , ± 1 , ± 2 , . . . } is a sequence of i.i.d. random variables with E { ε ( t ) } = 0 and V ( ε t ) = σ 2 ε , defined on a doubly-infinite set of integers. We set can μ 0 = 1. The equation can also written as y ( t ) = μ ( L ) ε ( t ) , where μ ( L ) = μ 0 + μ 1 L + · · · + μ q L q is a polynomial in the lag operator L , for which L j x ( t ) = x ( t j ). This process is stationary, since any two elements y t and y s are the same function of [ ε t , ε t 1 , . . . , ε t q ] and [ ε s , ε s 1 , . . . , ε s q ], which are identically distributed. If the roots of the polynomial equation μ ( z ) = μ 0 + μ 1 z + · · · + μ q z q = 0 lie outside the unit circle, then the process is invertible such that μ 1 ( L ) y ( t ) = ε ( t ) , which is an infinite-order autoregressive representation. 2
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EC3062 ECONOMETRICS Example. Consider the first-order MA(1) moving-average process (4) y ( t ) = ε ( t ) θε ( t 1) = (1 θL ) ε ( t ) . Provided that | θ | < 1, this can be written in autoregressive form as ε ( t ) = 1 (1 θL ) y ( t ) = y ( t ) + θy ( t 1) + θ 2 y ( t 2) + · · · . Imagine that | θ | > 1 instead. Then, to obtain a convergent series, we have to write y ( t + 1) = ε ( t + 1) θε ( t ) = θ (1 L 1 ) ε ( t ) , where L 1 ε ( t ) = ε ( t + 1). This gives (7) ε ( t ) = θ 1 (1 L 1 ) y ( t + 1) = θ 1 y ( t + 1) θ + y ( t + 2) θ 2 + · · · . Normally, this would have no reasonable meaning. 3
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EC3062 ECONOMETRICS The Autocovariances of a Moving-Average Process Consider (8) γ τ = E ( y t y t τ ) = E i μ i ε t i j μ j ε t τ j = i j μ i μ j E ( ε t i ε t τ j ) . Since ε ( t ) is a sequence of independently and identically distributed random variables with zero expectations, it follows that (9) E ( ε t i ε t τ j ) = 0 , if i = τ + j ; σ 2 ε , if i = τ + j . Therefore (10) γ τ = σ 2 ε j μ j μ j + τ . 4
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EC3062 ECONOMETRICS Now let τ = 0 , 1 , . . . , q . This gives (11) γ 0 = σ 2 ε ( μ 2 0 + μ 2 1 + · · · + μ 2 q ) , γ 1 = σ 2 ε ( μ 0 μ 1 + μ 1 μ 2 + · · · + μ q 1 μ q ) , .
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