gmmexamples - Example stylized consumption...

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Example: stylized consumption function (Campbell and Mankiw (1990) c t = δ 0 + δ 1 y t + δ 2 r t + ε t , t = 1 , . . . , T = δ 0 z t + ε t L = 3 where c t = the log of real per capita consumption (excluding durables), y t = the log of real disposable income, and r t = the ex post real interest rate (T-bill rate - in fl ation rate). Note: See Zivot and Wang (2005), Chapter 21 for S-PLUS code to replicate this example.
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Assumptions { c t , y t , r t } are stationary and ergodic { ε t , I t } is a stationary and ergodic martingale di ff erence sequence (MDS) where I t = { c s , y s , r s } t s =1 denotes the observed information set at time t. Endogeneity and Instruments The variables y t and r t are likely to be contemporaneously correlated with ε t Because { ε t , I t } is a stationary and ergodic MDS, E [ ε t | I t 1 ] = 0 which implies that any variable in I t 1 is a potential instrument. For any variable x t 1 I t 1 , { x t 1 ε t } is an uncorrelated sequence. Data: Annual data over the period 1960 to 1995 taken from Wooldridge (2002)
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Example : Testing the Permanent Income Hypothesis The pure permanent income hypothesis (PIH) due to Hall (1978) states that c t is a martingale so that c t = ε t is a MDS. Hence, the PIH implies the linear restrictions H 0 : δ 1 = δ 2 = 0 which are of the form R δ = r with R = Ã 0 1 0 0 0 1 ! , r = Ã 0 0 ! rank( R ) = 2 If there are temporary income consumers, then δ 1 > 0 .
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c t = δ 1 + δ 2 y t + δ 3 r t + ε t x t = (1 , c t 1 , y t 1 , r t 1 ) 0 , E [ x t ε t ] = 0 , E [ x t x 0 t ε 2 t ] = S
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