Applications of Random Matrix Theory to Economics, Finance and

Outline

Applications of Random Matrix Theory toEconomics, Finance and Political Science

Matthew C. Harding1

1Department of Economics, MITInstitute for Quantitative Social Science, Harvard University

SEA’06 MIT : July 12, 2006

Harding RMT Applications

Outline

Outline

1 Portfolio Selection

2 Factor Models

3 Beyond Covariances


Outline

Outline


2 Factor Models



Outline

Outline


2 Factor Models



Portfolio SelectionFactor Models

Beyond Covariances

Econophysics and Portfolio Section

First attempt at applying RMT in Finance - cleaning data(Potters and Bouchaud)

Aim: choose a portfolio of N assets with weights wi . Varianceof portfolio returns is given by V =

∑ij wiσiCijσjwj .

Minimize risk for a given expected return or hedge assetsagainst each other - choose weights w for assets in portfolio.

Idiosyncratic Volatility

Improve portfolio choice by removing idiosyncratic noisecorrelation matrix using RMT

Methods: eigenvalues cut-offs, shrinkage estimators, fixedpoint covariances (Frahm ’05)



Beyond Covariances











Beyond Covariances











Beyond Covariances











Beyond Covariances

Cleaning correlations by removing idiosyncratic noise

Success?

Leads to better risk estimates ...

But, focuses exclusively on the noise, no real model ofasset pricing

Sensitive to assumptions on idiosyncratic noise - is there arole for power laws?



Beyond Covariances


Success?






Beyond Covariances


Success?






Beyond Covariances


Success?






Beyond Covariances


Success?






Beyond Covariances

New Chairman of the Fed ...

... believes in Factor Models!

Argues in policy speeches that we need to understand theeffect of unobserved latent (international) factors on the USmacroeconomy (especially interest rates).



Beyond Covariances

New Chairman of the Fed ...

... believes in Factor Models!

Argues in policy speeches that we need to understand theeffect of unobserved latent (international) factors on the USmacroeconomy (especially interest rates).



Beyond Covariances

Factor Models are Back !

BYt + ΓZt + ΛFt + Ut = 0, (1)

where t = 1..T .Assumption 1: The number of endogenous variablesincreases with the sample size T . Thus, n →∞, T →∞ andn/T → c ∈ (0,∞).

Definitions

Yt : n × 1dimensional vector of endogenous variables

Zt : k × 1vector of exogenous variables

Ft : p × 1 vector of unobserved factors

Λ : n × p matrix of factor loadings



Beyond Covariances


BYt + ΓZt + ΛFt + Ut = 0, (1)


Definitions







Beyond Covariances


BYt + ΓZt + ΛFt + Ut = 0, (1)


Definitions







Beyond Covariances


BYt + ΓZt + ΛFt + Ut = 0, (1)


Definitions







Beyond Covariances


BYt + ΓZt + ΛFt + Ut = 0, (1)


Definitions







Beyond Covariances


BYt + ΓZt + ΛFt + Ut = 0, (1)


Definitions







Beyond Covariances

Special case

Yt = ΛFt + Ut , (2)

where t = 1..T .

Familiar to signal processing world: k unobserved signals Fand noise U.

Follows from No Arbitrage restrictions in asset pricing models.

Y usually corresponds to asset returns (stocks, bonds etc.)

RMT Applications

Identification of factors, estimation of number of factors,estimation of loadings, tests on estimated factors ...



Beyond Covariances

Special case

Yt = ΛFt + Ut , (2)

where t = 1..T .

Familiar to signal processing world: k unobserved signals Fand noise U.

Follows from No Arbitrage restrictions in asset pricing models.

Y usually corresponds to asset returns (stocks, bonds etc.)

RMT Applications

Identification of factors, estimation of number of factors,estimation of loadings, tests on estimated factors ...



Beyond Covariances

Brown (1989) Puzzle

An economy with K factors, each of which is priced andcontributes equally to the returns and calibrated to actualdata from the NYSE.

Nevertheless, he finds evidence that estimations arebiased towards a single factor model.

Answer: Phase transition in spiked model (Paul ’05)

b) λia.s.→

Nσ2

F σ2β + σ2

ε

1 + 1

Tσ2

ε

σ2F σ2

β

, for i = 2...K

c) λia.s.→ σ2

ε (1 +√

N/T )2, for i = 2...K

depending onN <> 1T

(σ2

ε

σ2F σ2

β

)2

; (Harding ’06 N > 101, 396)



Beyond Covariances

Brown (1989) Puzzle




b) λia.s.→

Nσ2

F σ2β + σ2

ε

1 + 1

Tσ2

ε

σ2F σ2

β

, for i = 2...K

c) λia.s.→ σ2

ε (1 +√

N/T )2, for i = 2...K

depending onN <> 1T

(σ2

ε

σ2F σ2

β

)2

; (Harding ’06 N > 101, 396)



Beyond Covariances

Brown (1989) Puzzle




b) λia.s.→

Nσ2

F σ2β + σ2

ε

1 + 1

Tσ2

ε

σ2F σ2

β

, for i = 2...K

c) λia.s.→ σ2

ε (1 +√

N/T )2, for i = 2...K

depending onN <> 1T

(σ2

ε

σ2F σ2

β

)2

; (Harding ’06 N > 101, 396)



Beyond Covariances

Brown (1989) Puzzle




b) λia.s.→

Nσ2

F σ2β + σ2

ε

1 + 1

Tσ2

ε

σ2F σ2

β

, for i = 2...K

c) λia.s.→ σ2

ε (1 +√

N/T )2, for i = 2...K

depending onN <> 1T

(σ2

ε

σ2F σ2

β

)2

; (Harding ’06 N > 101, 396)



Beyond Covariances

Minimum Distance/GMM Estimation of Model Parameters

Assume we observe sample covariance Ω from a model Ω(θ).Let Π(θ) = limN→∞Etr(Ωs) (Free Probability) and Π = tr(Ωs)

(Sample). Then, θ = argminθ(Π(θ)− Π)′ V−1 (Π(θ)− Π).

Strategy for Identifying the Number of Factors

Order the sample eigenvalues in decreasing orderλ1, λ1, ..., λN .

Estimate θ using CMD and compute J-test (over-id test)Hansen (1982) recursively using the smallest N − qeigenvalues

k = argmink=0,1,...J(λk+1, λk+2, ..., λN ; θ)



Beyond Covariances










Beyond Covariances










Beyond Covariances










Beyond Covariances










Beyond Covariances

Example:

Can estimate much weaker factors which appear ininternational APT models e.g. exchange rate risks.



Beyond Covariances

Example:

Can estimate much weaker factors which appear ininternational APT models e.g. exchange rate risks.



Beyond Covariances

Extending factor reasoning

Can we extend factor approach beyond covariance matrices?

Testing for Delay correlations

Application to financial contagion: estimate a factor modelbased on stock market returns for large number of firmsacross a region (e.g. South America)

Estimate numerous factors Fj for j = 1..p (country,industries etc)

Let τ be a lag

Is Fj(0) correlated to Fk (τ)?



Beyond Covariances






Let τ be a lag




Beyond Covariances






Let τ be a lag




Beyond Covariances






Let τ be a lag




Beyond Covariances






Let τ be a lag




Beyond Covariances






Let τ be a lag




Beyond Covariances

Symmetrized delay correlation matrix

Φ =F ′(0)F (τ) + F ′(τ)F (0)

2T

Under Null Fij are iid.

Compute the empirical eigenvalue distribution FΦ(λ)

See what happens when there are delay correlations ...



Beyond Covariances


Φ =F ′(0)F (τ) + F ′(τ)F (0)

2T






Beyond Covariances


Φ =F ′(0)F (τ) + F ′(τ)F (0)

2T






Beyond Covariances


Φ =F ′(0)F (τ) + F ′(τ)F (0)

2T






Beyond Covariances


Φ =F ′(0)F (τ) + F ′(τ)F (0)

2T






Beyond Covariances



Beyond Covariances

Can obtain moments of the eigenvalue distribution from theCauchy transform and construct J-test ...

Moments of the eigenvalue distribution

m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4

Or look at the largest eigenvalue ...



Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



m1Φ = 0

m2Φ = c/2

m3Φ = 0

m4Φ = (1/2)c2 + (3/8)c3

m5Φ = 0

m6Φ = (5/8)c3 + (5/16)c5 + (9/8)c4




Beyond Covariances



Beyond Covariances

Random Distance Matrices

Spectral measures for large random Euclidean matrices(Bordenave ’06)

Entries are functions of positions of n random points in acompact set Rd , A = (D(Xi − Xj))i,j=1..n.

Special case D(X ) = D(−X ) ∈ (0, 1), adjacency matrix fora random graph.

More general distance in issue space ”agreement beyondpolarization” study of US congress

Also risk sharing between companies in developingcountries.



Beyond Covariances









Beyond Covariances









Beyond Covariances









Beyond Covariances









Beyond Covariances









Beyond Covariances


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Applications of Random Matrix Theory to Economics, Finance and