Ρ σ ρ σ ρ ε ε ε ρε ε ρ ε ρ ε 2 2 t u 2

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ρ σ ρ σ = - ρ ε = ε ε = ρε + ε = ρ ε + + ρ ε 2 2 t u 2 u 2 Var[ ]            1 ρ ε + σ σ = - ρ
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Part 14: Generalized Regression Autocovariances ™  25/44 t t 1 t 1 t t 1 t 1 t 1 t t 1 t-1 t 2 u 2 t t 2 t 1 t t 2 Continuing... Cov[ , ] =  Cov[ u , ]                  =   Cov[ , ] Cov[u , ]                  =   Var[ ] Var[ ]                  = (1 ) Cov[ , ] =  Cov[ u , ]          - - - - - - - - - ε ε ρε + ε ρ ε ε + ε ρ ε = ρ ε ρσ - ρ ε ε ρε + ε t 1 t 2 t t 2 t t 1 2 2 u 2          =   Cov[ , ] Cov[u , ]                  =   Cov[ , ]                  =  and so on. (1 ) - - - - ρ ε ε + ε ρ ε ε ρ σ - ρ
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Part 14: Generalized Regression Autocorrelation Matrix ™  26/44 2 1 2 2 2 2 3 2 1 2 3 1 1 1 1 1 (Note, trace   = n as required.) - - - - - - ρ ρ ρ ρ ρ ρ σ σ = ρ ρ ρ ÷ ρ ρ ρ Ω Ω L L L M M M O M L T T u T T T T
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Part 14: Generalized Regression Generalized Least Squares ™  27/44 ( 29 - - - - ρ = - ρ ÷ ÷ - ρ ÷ ÷ - ρ ÷ ÷ ÷ - ρ 2 1/ 2 2 1 2 1 1/ 2 3 2 T T 1 1 0 0 ... 0 1 0 ... 0 0 1 ... 0 ... ... ... ... ... 0 0 0 0 1 y y y y y ... y Ω y = Ω
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Part 14: Generalized Regression The Autoregressive Transformation ™  28/44 t t t t 1 t t 1 t 1 t t 1 t t 1 t t 1 t    y                       u    y y y ( )  +  ( ) y y ( )  +  u (Where did the first observation go?) - - - - - - = + ε ε = ρε + ρ = ρ + ρε - - - - - - - - - - - - ρ = - ρ ε - ρε - ρ = - ρ t t-1 t t-1 t t-1 x ' β x ' β x x ' β x x ' β
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Part 14: Generalized Regression Unknown p The problem (of course), is unknown. For now, we will consider two methods of estimation: n Two step, or feasible estimation. Estimate first, then do GLS. Emphasize - same logic as White and Newey-West. We don’t need to estimate . We need to find a matrix that behaves the same as (1/n) X-1X . n Properties of the feasible GLS estimator p Maximum likelihood estimation of , 2, and all at the same time. n Joint estimation of all parameters. Fairly rare. Some generalities… n We will examine two applications: Harvey’s model of heteroscedasticity and Beach-MacKinnon on the first order autocorrelation model ™  29/44
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Part 14: Generalized Regression Specification p must be specified first. p A full unrestricted contains n(n+1)/2 - 1 parameters. (Why minus 1? Remember, tr( ) = n, so one element is determined.) p is generally specified in terms of a few parameters. Thus, = ( ) for some small parameter vector . It becomes a question of estimating . p Examples: ™  30/44
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Part 14: Generalized Regression Harvey’s Model of Heteroscedasticity p Var[i | X ] = 2 exp( z i) p Cov[i,j | X ] = 0 e.g.: zi = firm size e.g.: z i = a set of dummy variables (e.g., countries) (The groupwise heteroscedasticity model.) p [2 ] = diagonal [exp(  + zi )],  = log(2) ™  31/44
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Part 14: Generalized Regression AR(1) Model of Autocorrelation ™  32/44 2 1 2 2 2 2 3
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