dynamicpanelslides - Introduction to Dynamic Panel Data...

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Introduction to Dynamic Panel Data: Autoregressive Models with Fixed E ff ects Eric Zivot December 2, 2009
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Dynamic Panel Data y im = ρy i,m 1 + α i + η im i = 1 , . . . , n (individuals) m = 1 , . . . , M (time periods) Typical assumptions 1. Stationarity: | ρ | < 1 2. E [ η im | y i 0 , . . . , y i,m 1 , α i ] = 0 3a. Homoskedasticity: η im iid (0 , σ 2 η ) 3b. Homeskedasticity: α i iid (0 , σ 2 α )
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Example: International Di ff erence in Growth Rates (Hayashi, Section 5.4) y im = ρy i,m 1 + α i + η im y im = ( ρ 1) y i,m 1 + α i + η im 1. y im = ln ( Y ( t m ) /L ( t m )) = log per capita output at time t m 2. Y ( t m ) = aggregate output 3. L ( t m ) = aggregate hours worked 4. α i = (1 ρ ) { ln( q i ) ln( A i (0)) 5. q = steady state level of ouput per e ff ective labor 6. q ( t ) = Y ( t ) / ( A ( t ) L ( t )) output per e ff ective labor input at time t
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7. A ( t ) = level of labor augmenting technical progress 8. ln( q ( t m )) = (1 ρ ) ln( q ) + ρ ln( q ( t m 1 )) 9. ρ = exp( λ ( t m t m 1 )) , λ = speed of convergence ( > 0)
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Stationary Model Representation By recursive substitution y i 1 = ρy i 0 + α i + η i 1 y i 2 = ρy i 2 + α i + η i 2 = ρ [ ρy i 0 + α i + η i 1 ] + α i + η i 2 = ρ 2 y i 0 + (1 + ρ ) α i + ρη i 1 + η i 2 . . . y im = ρ m y i 0 + α i m 1 X s =0 ρ s + m 1 X s =0 ρ s η i,m s
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Now E [ y im | α i ] = ρ m E [ y i 0 | α i ] + α i m 1 X s =0 ρ s + m 1 X s =0 ρ s E [ η i,m s | α i ] = ρ m E [ y i 0 | α i ] + α i m 1 X s =0 ρ s For large m ρ m 0 m 1 X s =0 ρ s 1 1 ρ
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so that E [ y im | α i ] α i 1 ρ var[ y im | α i ] X s =0 var[ ρ s η i,m s | α i ] = σ 2 η 1 ρ 2
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Estimation Can’t use RE estimation because E [ y i,m 1 · α i ] = E ρ m 1 y i 0 + α i m 2 X s =0 ρ s + m 2 X s =0 ρ s η i,m s α i 6 = 0 What about FE estimation?
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Write the model in FE matrix notation as y i = ρ y i, 1 + α i 1 M + η i y i M × 1 = y i 1 . . . y iM , y i, 1 M × 1 = y i 0 .
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