Econometric theory and methods, Russell Davidson - Chapter 2 doc

... X 1 ˆ β 1 + X 2 ˆ β 2 + M X y. (2. 43) Premultiplying the leftmost and rightmost expressions in (2. 43) by X 2  M 1 , we obtain X 2  M 1 y = X 2  M 1 X 2 ˆ β 2 . (2. 44) Copyright c  1999, Russell Davidson ... triangles. Copyright c  1999, Russell Davidson and James G. MacKinnon 2. 8 Exercises 85 P X y = X 2 β 2 + u;e) M 1 y = X 2 β 2 + u;f) M 1 y = M 1...

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... matrices:  A 11 A 12 A 21 A 22  B 11 B 12 B 21 B 22  =  A 11 B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22  . Check all the expressions on the right-hand side, ... =  k 1 k 2 X 11 X 12 n 1 X 21 X 22 n 2  with the submatrix X 11 of dimensions n 1 × k 1 , X 12 of dimensions n 1 × k 2 , X 21 of dimen...

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... (6.91) and (6. 92) . 6 .21 The original HRGNR proposed by Davidson and MacKinnon (1985a) is ι = ´ U ´ M 1 ´ X 2 b 2 + residuals, (6.96) where ´ U, ´ X 1 , and ´ X 2 are as defined in Section 6.8, b 2 is ... asymptotically equivalent to n 1 /2 ˆ β 2 , where ˆ β 2 is the NLS estimator of β 2 from (6.69). Copyright c  1999, Russell Davidson and James G. MacKinnon 6.3...

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... expectation and variance of u 1 are respectively 0 and σ 2 ε /(1 − ρ 2 ), then we see at once that E(u 2 ) = E(ρu 1 ) + E(ε 2 ) = 0, and that Var(u 2 ) = Var(ρu 1 + ε 2 ) = σ 2 ε  ρ 2 1 − ρ 2 + 1  = σ 2 ε 1 ... of u t is σ 2 u ≡ E  (ε t + α 1 ε t−1 ) 2  = σ 2 ε + α 2 1 σ 2 ε = (1 + α 2 1 )σ 2 ε , the covariance of u t and u t−1 is E  (ε t + α 1 ε...

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... ˆσ 2 a ˆσ 2 2  + ˆu 2 2t ˆσ 2 2 − 1, or log  ˆσ 2 1 + ˆσ 2 a ˆσ 2 2  + ˆu 2 2t ˆσ 2 2 − ˆσ 2 1 + ( Z X ) 2 t ˆσ 2 1 + σ 2 a . (15.83) 15 .23 The file nonnested.data contains 40 observations on artificially generated vari- ables 1 , 2 , 2 , 3 , ... vari- ables 1 , 2 , 2 , 3 , and 4 . Consider the nonnested linear regression models H 1 : = α 1 + β 1 1 + β 2...

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... Section 2. 4, M 2 ≡ I − X 2 (X 2  X 2 ) −1 X 2  . This estimate is ˆ β 1 = x 1  M 2 y x 1  M 2 x 1 , and, by a calculation similar to that leading to (3 .28 ), its variance is σ 2 0  x 1  M 2 x 1  −1 = σ 2 0 x 1  M 2 x 1 . ... E  n  t=1 ˆu 2 t  = E   ˆ u 2  ≤ E  u 2  = E  n  t=1 u 2 t  = n  t=1 E(u 2 t ) = nσ 2 0 . This suggests that,...

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... noncentrality parameter given by Λ ≡ 1 σ 2 β 2  X 2  M 1 X 2 (X 2  M 1 X 2 ) −1 X 2  M 1 X 2 β 2 = 1 σ 2 β 2  X 2  M 1 X 2 β 2 . Copyright c  1999, Russell Davidson and James G. MacKinnon ... ≡  y 1 y 2  , and X ≡  X 1 X 2  , where y 1 and y 2 are, respectively, an n 1 vector and an n 2 vector, while X 1 and X 2 are n 1 × k and...

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... β 2 . The left-hand side of the equation is equal to Pr  s 2 t α /2 ≤ ˆ β 2 − β 20 ≤ s 2 t 1−(α /2)  = Pr  −s 2 t α /2 ≥ β 20 − ˆ β 2 ≥ −s 2 t 1−(α /2)  = Pr  ˆ β 2 − s 2 t α /2 ≥ β 20 ≥ ˆ β 2 − ... so Pr  t α /2 ≤ ˆ β 2 − β 20 s 2 ≤ t 1−(α /2)  = 1 − α, (5. 12) where t α /2 and t 1−(α /2) denote the α /2 and 1 − (α /2) quantiles of the t(...

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... Blomquist and Dahlberg (1999), Donald and Newey (20 01), Hahn and Hausman (20 02) , Kleibergen (20 02) , and Stock, Wright, and Yogo (20 02) . There remain many unsolved problems. Copyright c  1999, Russell ... 1999, Russell Davidson and James G. MacKinnon 8 .2 Correlation Between Error Terms and Regressors 311 where u t ≡ u ◦ t + v 2t − β 2 v 1t . Thus Var(u t ) is e...

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... becomes  µ (j+1) σ 2 (j+1)  =  µ (j) σ 2 (j)  −     ∂ ¯m 1 ∂µ ∂ ¯m 1 ∂σ 2 ∂ ¯m 2 ∂µ ∂ ¯m 2 ∂σ 2      ¯m 1 (µ (j) , σ 2 (j) ) − ¯z ¯m 2 (µ (j) , σ 2 (j) ) − ¯y  , Copyright c  1999, Russell Davidson ... as ¯m 1 (µ, σ 2 ) = ¯z and ¯m 2 (µ, σ 2 ) = ¯y, (9.111) where ¯z and ¯y are the sample averages of the z t and the y t , respectively, and ¯m...

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