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# lecture4 - EC3062 ECONOMETRICS IDENTIFICATION OF ARMA...

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EC3062 ECONOMETRICS IDENTIFICATION OF ARMA MODELS A stationary stochastic process can be characterised, equivalently, by its autocovariance function or its partial autocovariance function. It can also be characterised by is spectral density function, which is the Fourier transform of the autocovariances { γ τ ; τ = 0 , ± 1 , ± 2 , . . . } : f ( ω ) = τ = −∞ γ τ cos( ωτ ) = γ 0 + 2 τ =1 γ τ cos( ωτ ) . Here, ω [0 , π ] is an angular velocity, or frequency value, in radians per period. The empirical counterpart of the spectral density function is the periodogram I ( ω j ), which may be defined as 1 2 I ( ω j ) = T 1 τ =1 T c τ cos( ω j τ ) = c 0 + 2 T 1 τ =1 c τ cos( ω j τ ) , where ω j = 2 πj/T ; j = 0 , 1 , . . . , [ T/ 2] are the Fourier frequencies and { c τ ; τ = 0 , ± 1 , . . . , ± ( T 1) } , with c τ = T 1 T 1 t = τ ( y t ¯ y )( y t τ ¯ y ), are the empirical autocovariances. 1

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EC3062 ECONOMETRICS The Periodogram and the Autocovariances We need to show this definition of the peridogram is equivalent to the previous definition, which was based on the following frequency decompo- sition of the sample variance: 1 T T 1 t =0 ( y t ¯ y ) 2 = 1 2 [ T/ 2] j =0 ( α 2 j + β 2 j ) , where α j = 2 T t y t cos( ω j t ) = 2 T t ( y t ¯ y )cos( ω j t ) , β j = 2 T t y t sin( ω j t ) = 2 T t ( y t ¯ y )sin( ω j t ) . Substituting these into the term T ( α 2 j + β 2 j ) / 2 gives the periodogram I ( ω j ) = 2 T T 1 t =0 cos( ω j t )( y t ¯ y ) 2 + T 1 t =0 sin( ω j t )( y t ¯ y ) 2 . 2
EC3062 ECONOMETRICS The quadratic terms may be expanded to give I ( ω j ) = 2 T t s cos( ω j t )cos( ω j s )( y t ¯ y )( y s ¯ y ) + 2 T t s sin( ω j t )sin( ω j s )( y t ¯ y )( y s ¯ y ) , Since cos( A )cos( B ) + sin( A )sin( B ) = cos( A B ), this can be written as I ( ω j ) = 2 T t s cos( ω j [ t s ])( y t ¯ y )( y s ¯ y ) On defining τ = t s and writing c τ = t ( y t ¯ y )( y t τ ¯ y ) /T , we can reduce the latter expression to I ( ω j ) = 2 T 1 τ =1 T cos( ω j τ ) c τ , which is a Fourier transform of the empirical autocovariances. 3

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EC3062 ECONOMETRICS 0 2.5 5 7.5 10 0 π /4 π /2 3 π /4 π Figure 1. The spectral density function of an MA(2) process y ( t ) = (1 + 1 . 250 L + 0 . 800 L 2 ) ε ( t ) . 4
EC3062 ECONOMETRICS 0 20 40 60 0 π /4 π /2 3 π /4 π Figure 2. The graph of a periodogram calculated from 160 observations on a simulated series generated by an MA(2) process y ( t ) = (1 + 1 . 250 L + 0 . 800 L 2 ) ε ( t ). 5

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EC3062 ECONOMETRICS 0 10 20 30 0 π /4 π /2 3 π /4 π Figure 3. The spectral density function of an AR(2) process (1 0 . 273 L + 0 . 810 L 2 ) y ( t ) = ε ( t ). 6
EC3062 ECONOMETRICS 0 25 50 75 100 125 0 π /4 π /2 3 π /4 π Figure 4. The graph of a periodogram calculated from 160 observations on a simulated series generated by an AR(2) process (1 0 . 273 L + 0 . 810 L 2 ) y ( t ) = ε ( t ). 7

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EC3062 ECONOMETRICS 0 20 40 60 0 π /4 π /2 3 π /4 π Figure 5. The spectral density function of an ARMA(2, 1) process (1 0 . 273 L + 0 . 810 L 2 ) y ( t ) = (1 + 0 . 900 L ) ε ( t ).
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lecture4 - EC3062 ECONOMETRICS IDENTIFICATION OF ARMA...

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