Econometric take home APPS_Part_23

Econometric take home APPS_Part_23 - 1 "known"...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
γ 1 "known" (identified), the only remaining unknown is γ 2 , which is therefore identified. With γ 1 and γ 2 in hand, β may be deduced from π 2 . With γ 2 and β in hand, σ 22 is the residual variance in the equation ( y 2 - β x - γ 2 y 1 ) = ε 2 , which is directly estimable, therefore, identified. ± 2. Following the method in Example 13.6, for identification of the investment equation, we require that the matrix have rank 5. Columns (1), (4), (6), (7), and (8) each have one element in a different row, so they are linearly independent. Therefore, the matrix has rank five. For the third equation, the required matrix is . Columns (4), (6), (7), (9), and (10) are linearly independent. ± () () () () () () () () () 123456789 1 00 0000 01 0 0 0 0 1 001000 1 0 0 0 0 000100000 33 1 −− αα γγ 3 2 γ 3 β 12 41 42 21 52 γ β β 10 12 31 32 33 52 γ βββ β () () () () () () () () () ( ) 1234567891 0 0 0 0 0 0 0 0 0 0 0 11000 0 1 000 0 0 0 10 0 0 10 0 0 010 1 00000 1 13 2 12 α ββ 3. We find [ A 3 , A 5 ] for each equation. (1) (2) (3) (4) β 32 34 12 13 14 43 4 32 1 0 , [] , , 0 43 44 1 0 Identification requires that the rank of each matrix be M-1 = 3. The second is obviously not identified. In (1), none of the three columns can be written as a linear combination of the other two, so it has rank 3. (Although the second and last columns have nonzero elements in the same positions, for the matrix to have short rank, we would require that the third column be a multiple of the second, since the first cannot appear in the linear combination which is to replicate the second column.) By the same logic, (3) and (4) are identified. ± 4. Obtain the reduced form for the model in Exercise 1 under each of the assumptions made in parts (a) and (b1), (b6), and (b9). (1). The model is y 1 = γ 1 y 2 + β 11 x 1 + β 21 x 2 + β 31 x 3 + ε 1 y 2 = γ 2 y 1 + β 12 x 1 + β 22 x 2 + β 32 x 3 + ε 2 . Therefore, Γ = and B = and Σ is unrestricted. The reduced form is 1 1 2 1 γ γ β β 11 12 22 31 0 0 Π = 1 1 11 1 21 2 11 12 2 2 2 31 2 31 ++ βγ γβ β β and 91
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Ω = ( Γ -1 ) ′Σ ( Γ -1 ) = 1 1 2 2 12 2 11 1 2 22 11 2 21 1 2 1 2 1 2 1 2 2 2 11 22 2 () + + + ++ + + + γγ σγ σ γσ σ γγσ σ (6) The model is y 1 = β 11 x 1 +
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

Econometric take home APPS_Part_23 - 1 "known"...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online