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### hayashi_chap5c

Course: EC 571, Winter 2008
School: Portland
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Word Count: 355

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Econometrics Lecture Notes Hayashi, Chapter 5c Unbalanced Panels Missing Observations Assuming common coefficients, for each equation m (=1,...,M) and for each observation i (=1,...,n), yim = zim + i + im Typically, there are missing observations. Assumption: No Selection Bias Whether an observation stays in the sample does not depend on the error term. Missing Observations Let dim = 1 if observation m is in the...

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Econometrics Lecture Notes Hayashi, Chapter 5c Unbalanced Panels Missing Observations Assuming common coefficients, for each equation m (=1,...,M) and for each observation i (=1,...,n), yim = zim + i + im Typically, there are missing observations. Assumption: No Selection Bias Whether an observation stays in the sample does not depend on the error term. Missing Observations Let dim = 1 if observation m is in the sample; dim = 0 otherwise. Let Mi be the number of observations from i. d i1 M di = M M i = di' di = , d im m =1 iM d The Model yi = Fi + di(bi' ) + dii + i (i=1,...,n) ' i1fi1 d d d i1y i1 i1i1 M F = M yi = , , i = M (M i# ) (M i n1) (M i 1) f' iM y iM i iM iM d d d iM iM The Transformed Model Let Qi = IMi di(di'di)-1di' = IMi didi'/Mi Qiyi = QiFi + Qidi(bi' ) + Qidii + Qi i Qiyi = QiFi + Qi i since Qidi = 0 yi* = Fi* + i* y* = F* + * With &quot;zeroing out&quot; missing observation, the transformed model remains the same as without missing observations. Therefore, the formula for the fixed-effects estimator remains the same. Fixed-Effects Estimator The fixed-effects estimator of is the pooled OLS estimator, which is consistent and asymptotically normal: FE = (F*'F*) 1F*'y FE- = (F*'F*) 1F*' * Est(Avar( FE)) = 1/n (F*'F*) 1(F*'VF*) (F*'F*) 1 where V = (y-F* FE)(y-F* FE)' Fixed-Effects Estimator If the assumption of spherical distribution is assumed: E( i i) = 2IMi, so that E( i* i*) = 2Qi Est(Avar( FE)) = n s2 (F*'F*) 1 s2 is the estimator of 2: (y* - F*FE )' (y* - F*FE ) s2 = n Mi - n - # i =1 No Selection Bias Assumption Recall the Orthogonality Assumption for the transf...

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Portland - EC - 570
EconometricsLecture Notes Hayashi, Chapter 3d GMM: Hypotheses TestingHypotheses Testing Suppose A.1-A.5 hold, n ( GMMW - ) d N(0,Avar( GMMW) as n. Suppose there is a consistent estimate of S (=E(gigi'). Let Est(Avar( GMMW) = (Sxz'WSxz)-1Sxz'W
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EconometricsLecture Notes Hayashi, Chapter 1fGeneralized Least Squares Assumption 4 Revisited: E( ' |X) = Var( |X) = 2In Assumption 4 Relaxed (Assumption 4'): E( ' |X) = Var( |X) = 2V(X), with nonsingular and known V(X). OLS estimator of , b
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EconometricsLecture Notes Hayashi, Chapter 2aConvergence: Definitions A sequence of random scalars {zn} = (z1,z2, ) converges in probability to z (a constant or a random variable) if, for any &gt;0, limn Prob(|zn-z|&gt; ) = 0. z is the probability lim
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EconometricsLecture Notes Hayashi, Chapter 1aLinear Regression Notations y: Dependent Variable (Regressand) yi, i = 1,2,n X: Explanatory Variables (Regressors) xi, i = 1,2,n xij, i = 1,2,n; j = 1,2,KLinear Regression Assumption 1: Linear
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EconometricsLecture Notes Hayashi, Chapter 1dHypothesis Testing under Normality Assumption 5: |X ~ N(0, 2In) Implications of Normality Assumption (b- )|X ~ N(0, 2(XX)-1) (bk- k)|X ~ N(0, 2([XX)-1]kk) zk ~ N(0,1) b k k zk = 2 [( X' X) 1 ]kk
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EconometricsLecture Notes Hayashi, Chapter 2c Large Sample TheoryAssumption DGP (Data Generating Process): A stochastic process that generated the finite sample (y,X) must satisfies: Assumption 1: Linearity yi = xi' + i (i=1,2,.,n)Assumption
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EconometricsLecture Notes Hayashi, Chapter 1bMethod-of-Moments Estimator From the implication of strict exgeneity assumption, E(xjk i) = 0 for i = 1,2,.,n and k = 1,2,.,K. That is, Moment Conditions: E(X' ) = 0 X'(y-X ) = 0 or X'y=X'X The meth
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Year Month LEI9619590159.219590259.919590360.519590460.419590560.519590660.519590760.319590859.919590959.919591059.519591158.919591260.019600159.719600259.219600358.619600458.719600558.819600658.9
Portland - EC - 572
21.67521.71621.75621.79621.83721.87821.91821.9592222.04122.08222.12322.16422.20522.24722.28822.32922.37122.41222.45422.49622.53822.5822.62222.66422.70622.74822.7922.83322.87522.91822.9623.00323.04623.08923.13223.17
Portland - EC - 595
Year GNP Inv Price Rate 1963 596.7 90.0 0.7167 3.23 1964 637.7 97.4 0.7277 3.55 1965 691.1 113.5 0.7436 4.04 1966 756.0 125.7 0.7676 4.50 1967 799.6 122.8 0.7906 4.19 1968 873.4 133.3 0.8254 5.16 196
Portland - EC - 572
YEAR QT DPI2000 PCE20001947 1 1096.01017.21947 2 1072.81034.01947 3 1102.81037.51947 4 1089.71037.71948 1 1107.31042.61948 2 1145.31054.31948 3 1168.41056.11948 4 1171.91064.81949 1 1147.61066.11949 2 1151.410
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