## 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

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)
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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)
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where f(w) is the marginal density function of w, which is given by
(B.3)
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Adding and subtracting ρ2 u 2 to the exponent term gives
(B.4)
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(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)
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Substituting (B.1) and (B.5) into (B.2), and after a few algebraic manipulations, we can write the conditional density as
(B.6)
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The conditional density function in terms of the means and variances of w and y is given by
(B.7)
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Equation (B.7) shows that the conditional density function of y for fixed value of w is also normally distributed with
(B.8)
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(B.9)
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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)
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Since $ɛ ~ N ( 0 , σ ɛ 2 )$ and is independent from θ, the mean and variance of x i are given by
(B.11)
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(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)
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(p.194)