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Private Sector Involvement and International Financial CrisesAn Analytical Perspective$

Michael Chui and Prasanna Gai

Print publication date: 2005

Print ISBN-13: 9780199267750

Published to Oxford Scholarship Online: July 2005

DOI: 10.1093/0199267758.001.0001

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(p.191) Appendix B Conditional Distributions

(p.191) Appendix B Conditional Distributions

Source:
Private Sector Involvement and International Financial Crises
Publisher:
Oxford University Press

Let f(w,y) be the joint density function of two normally distributed variables with means μw and μy, and variances σ w 2 and σ y 2 respectively. Then the joint density function is given as

(B.1)
f ( w , y ) = 1 2 π σ w σ y 1 ρ 2 exp [ 1 2 ( 1 ρ 2 ) ( u 2 2 ρ u υ + υ 2 ) ] ,
where u = (w − μw)/σw, υ = (y − μy)/σy, and ρ = cov (w,y)/(σwσy) is the correlation coefficient.

By definition, the conditional density function of y for fixed value of w is

(B.2)
f ( y w ) = f ( w , y ) f ( w ) ,
where f(w) is the marginal density function of w, which is given by
(B.3)
f ( w ) = f ( w , y ) d y = { exp [ 1 / ( 2 ( 1 ρ 2 ) ) ( u 2 2 ρ u υ + υ 2 ) ] 2 π σ w σ y 1 ρ 2 } d y = 1 2 π σ w 1 ρ 2 exp [ 1 2 ( 1 ρ 2 ) ( u 2 2 ρ u υ + υ 2 ) ] d υ .
Adding and subtracting ρ2 u 2 to the exponent term gives
(B.4)
f ( w ) = exp ( u 2 / 2 ) 2 π σ w 1 ρ 2 exp [ 1 2 ( 1 ρ 2 ) ( υ ρ u ) 2 ] d υ .
(p.192) And let z = ( ν ρ u ) / 1 ρ 2 such that d ν = 1 ρ 2 d z , so (B.4) can be rewritten as
(B.5)
f ( x ) = exp ( u 2 / 2 ) 2 π σ w 1 2 π exp ( z 2 ) d z = 1 2 π σ w exp ( u 2 2 ) .
Substituting (B.1) and (B.5) into (B.2), and after a few algebraic manipulations, we can write the conditional density as
(B.6)
f ( y w ) = 1 2 π 1 ρ 2 ) σ y exp { 1 2 [ υ ρ u 1 ρ 2 ] 2 } .
The conditional density function in terms of the means and variances of w and y is given by
(B.7)
f ( y w ) = 1 2 π 1 ρ 2 σ y exp [ 1 2 ( y [ μ y + ρ ( σ y / σ w ) ( w μ w ) ] 1 ρ 2 σ y ) ] .
Equation (B.7) shows that the conditional density function of y for fixed value of w is also normally distributed with
(B.8)
mean = μ y + ρ σ y σ w ( w μ w ) ,
(B.9)
variance = ( 1 ρ 2 ) σ y 2 .
Note that (B.8) corresponds to a linear regression of y on w, with ρσyw being the estimate of the slope parameter.

Suppose that the fundamental θ is normally distributed with mean μθ and variance σ θ 2 . Investor i only observes a private signal x i about θ such that

(B.10)
x i = θ + ɛ i .
Since ɛ ~ N ( 0 , σ ɛ 2 ) and is independent from θ, the mean and variance of x i are given by
(B.11)
E ( x i ) = μ θ , var ( x i ) = σ θ 2 + σ ɛ 2 .
(p.193) Notice that the correlation coefficient ρ = σ θ 2 / σ θ 2 ( σ θ 2 + σ ɛ 2 ) . To derive the conditional mean and variance of f(θ|x), we make use of (B.8) and (B.9) by substituting var ( x i ) , μθ, and σθ for μw, σw, μy, and σy respectively. After some algebraic manipulations, we get,
(B.12)
mean = μ θ σ ɛ 2 + x σ θ 2 σ θ 2 + σ ɛ 2 , variance = σ θ 2 σ ɛ 2 σ θ 2 + σ ɛ 2 .
(p.194)