Applications of Random Matrix Theory to Economics, Finance and Political Science
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1 Outline Applications of Random Matrix Theory to Economics, Finance and Political Science Matthew C. 1 1 Department of Economics, MIT Institute for Quantitative Social Science, Harvard University SEA 06 MIT : July 12, 2006
2 Outline Outline 1 Portfolio Selection 2 3
3 Outline Outline 1 Portfolio Selection 2 3
4 Outline Outline 1 Portfolio Selection 2 3
5 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 w i. Variance of portfolio returns is given by V = ij w iσ i C ij σ j w j. Minimize risk for a given expected return or hedge assets against each other - choose weights w for assets in portfolio. Idiosyncratic Volatility Improve portfolio choice by removing idiosyncratic noise correlation matrix using RMT Methods: eigenvalues cut-offs, shrinkage estimators, fixed point covariances (Frahm 05)
6 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 w i. Variance of portfolio returns is given by V = ij w iσ i C ij σ j w j. Minimize risk for a given expected return or hedge assets against each other - choose weights w for assets in portfolio. Idiosyncratic Volatility Improve portfolio choice by removing idiosyncratic noise correlation matrix using RMT Methods: eigenvalues cut-offs, shrinkage estimators, fixed point covariances (Frahm 05)
7 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 w i. Variance of portfolio returns is given by V = ij w iσ i C ij σ j w j. Minimize risk for a given expected return or hedge assets against each other - choose weights w for assets in portfolio. Idiosyncratic Volatility Improve portfolio choice by removing idiosyncratic noise correlation matrix using RMT Methods: eigenvalues cut-offs, shrinkage estimators, fixed point covariances (Frahm 05)
8 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 w i. Variance of portfolio returns is given by V = ij w iσ i C ij σ j w j. Minimize risk for a given expected return or hedge assets against each other - choose weights w for assets in portfolio. Idiosyncratic Volatility Improve portfolio choice by removing idiosyncratic noise correlation matrix using RMT Methods: eigenvalues cut-offs, shrinkage estimators, fixed point covariances (Frahm 05)
9 Cleaning correlations by removing idiosyncratic noise Success? Leads to better risk estimates... But, focuses exclusively on the noise, no real model of asset pricing Sensitive to assumptions on idiosyncratic noise - is there a role for power laws?
10 Cleaning correlations by removing idiosyncratic noise Success? Leads to better risk estimates... But, focuses exclusively on the noise, no real model of asset pricing Sensitive to assumptions on idiosyncratic noise - is there a role for power laws?
11 Cleaning correlations by removing idiosyncratic noise Success? Leads to better risk estimates... But, focuses exclusively on the noise, no real model of asset pricing Sensitive to assumptions on idiosyncratic noise - is there a role for power laws?
12 Cleaning correlations by removing idiosyncratic noise Success? Leads to better risk estimates... But, focuses exclusively on the noise, no real model of asset pricing Sensitive to assumptions on idiosyncratic noise - is there a role for power laws?
13 Cleaning correlations by removing idiosyncratic noise Success? Leads to better risk estimates... But, focuses exclusively on the noise, no real model of asset pricing Sensitive to assumptions on idiosyncratic noise - is there a role for power laws?
14 New Chairman of the Fed believes in! Argues in policy speeches that we need to understand the effect of unobserved latent (international) factors on the US macroeconomy (especially interest rates).
15 New Chairman of the Fed believes in! Argues in policy speeches that we need to understand the effect of unobserved latent (international) factors on the US macroeconomy (especially interest rates).
16 are Back! BY t + ΓZ t + ΛF t + U t = 0, (1) where t = 1..T. Assumption 1: The number of endogenous variables increases with the sample size T. Thus, n, T and n/t c (0, ). Definitions Y t : n 1dimensional vector of endogenous variables Z t : k 1vector of exogenous variables F t : p 1 vector of unobserved factors Λ : n p matrix of factor loadings
17 are Back! BY t + ΓZ t + ΛF t + U t = 0, (1) where t = 1..T. Assumption 1: The number of endogenous variables increases with the sample size T. Thus, n, T and n/t c (0, ). Definitions Y t : n 1dimensional vector of endogenous variables Z t : k 1vector of exogenous variables F t : p 1 vector of unobserved factors Λ : n p matrix of factor loadings
18 are Back! BY t + ΓZ t + ΛF t + U t = 0, (1) where t = 1..T. Assumption 1: The number of endogenous variables increases with the sample size T. Thus, n, T and n/t c (0, ). Definitions Y t : n 1dimensional vector of endogenous variables Z t : k 1vector of exogenous variables F t : p 1 vector of unobserved factors Λ : n p matrix of factor loadings
19 are Back! BY t + ΓZ t + ΛF t + U t = 0, (1) where t = 1..T. Assumption 1: The number of endogenous variables increases with the sample size T. Thus, n, T and n/t c (0, ). Definitions Y t : n 1dimensional vector of endogenous variables Z t : k 1vector of exogenous variables F t : p 1 vector of unobserved factors Λ : n p matrix of factor loadings
20 are Back! BY t + ΓZ t + ΛF t + U t = 0, (1) where t = 1..T. Assumption 1: The number of endogenous variables increases with the sample size T. Thus, n, T and n/t c (0, ). Definitions Y t : n 1dimensional vector of endogenous variables Z t : k 1vector of exogenous variables F t : p 1 vector of unobserved factors Λ : n p matrix of factor loadings
21 are Back! BY t + ΓZ t + ΛF t + U t = 0, (1) where t = 1..T. Assumption 1: The number of endogenous variables increases with the sample size T. Thus, n, T and n/t c (0, ). Definitions Y t : n 1dimensional vector of endogenous variables Z t : k 1vector of exogenous variables F t : p 1 vector of unobserved factors Λ : n p matrix of factor loadings
22 Special case Y t = ΛF t + U t, (2) where t = 1..T. Familiar to signal processing world: k unobserved signals F and noise U. Follows from No Arbitrage restrictions in asset pricing models. Y usually corresponds to asset returns (stocks, bonds etc.) Identification of factors, estimation of number of factors, estimation of loadings, tests on estimated factors...
23 Special case Y t = ΛF t + U t, (2) where t = 1..T. Familiar to signal processing world: k unobserved signals F and noise U. Follows from No Arbitrage restrictions in asset pricing models. Y usually corresponds to asset returns (stocks, bonds etc.) Identification of factors, estimation of number of factors, estimation of loadings, tests on estimated factors...
24 Brown (1989) Puzzle An economy with K factors, each of which is priced and contributes equally to the returns and calibrated to actual data from the NYSE. Nevertheless, he finds evidence that estimations are biased towards a single factor model. Answer: Phase transition in spiked model (Paul 05) { } { } a.s. b) ˆλ i NσF 2 σ2 β + σ2 ɛ σɛ 2 T, for i = 2...K σ 2 F σ2 β a.s. c) ˆλ i σɛ 2 (1 + N/T ) 2, for i = 2...K ( ) 2 depending on N <> 1 σɛ 2 T ; ( 06 N > 101, 396) σ 2 F σ2 β
25 Brown (1989) Puzzle An economy with K factors, each of which is priced and contributes equally to the returns and calibrated to actual data from the NYSE. Nevertheless, he finds evidence that estimations are biased towards a single factor model. Answer: Phase transition in spiked model (Paul 05) { } { } a.s. b) ˆλ i NσF 2 σ2 β + σ2 ɛ σɛ 2 T, for i = 2...K σ 2 F σ2 β a.s. c) ˆλ i σɛ 2 (1 + N/T ) 2, for i = 2...K ( ) 2 depending on N <> 1 σɛ 2 T ; ( 06 N > 101, 396) σ 2 F σ2 β
26 Brown (1989) Puzzle An economy with K factors, each of which is priced and contributes equally to the returns and calibrated to actual data from the NYSE. Nevertheless, he finds evidence that estimations are biased towards a single factor model. Answer: Phase transition in spiked model (Paul 05) { } { } a.s. b) ˆλ i NσF 2 σ2 β + σ2 ɛ σɛ 2 T, for i = 2...K σ 2 F σ2 β a.s. c) ˆλ i σɛ 2 (1 + N/T ) 2, for i = 2...K ( ) 2 depending on N <> 1 σɛ 2 T ; ( 06 N > 101, 396) σ 2 F σ2 β
27 Brown (1989) Puzzle An economy with K factors, each of which is priced and contributes equally to the returns and calibrated to actual data from the NYSE. Nevertheless, he finds evidence that estimations are biased towards a single factor model. Answer: Phase transition in spiked model (Paul 05) { } { } a.s. b) ˆλ i NσF 2 σ2 β + σ2 ɛ σɛ 2 T, for i = 2...K σ 2 F σ2 β a.s. c) ˆλ i σɛ 2 (1 + N/T ) 2, for i = 2...K ( ) 2 depending on N <> 1 σɛ 2 T ; ( 06 N > 101, 396) σ 2 F σ2 β
28 Minimum Distance/GMM Estimation of Model Parameters Assume we observe sample covariance ˆΩ from a model Ω(θ). Let Π(θ) = lim N Etr(Ω s ) (Free Probability) and Π = tr( ˆΩ s ) (Sample). Then, θ = argmin θ (Π(θ) Π) V 1 (Π(θ) Π). Strategy for Identifying the Number of Factors Order the sample eigenvalues in decreasing order λ 1, λ 1,..., λ N. Estimate θ using CMD and compute J-test (over-id test) Hansen (1982) recursively using the smallest N q eigenvalues k = argmin k=0,1,... Ĵ(λ k+1, λ k+2,..., λ N ; θ)
29 Minimum Distance/GMM Estimation of Model Parameters Assume we observe sample covariance ˆΩ from a model Ω(θ). Let Π(θ) = lim N Etr(Ω s ) (Free Probability) and Π = tr( ˆΩ s ) (Sample). Then, θ = argmin θ (Π(θ) Π) V 1 (Π(θ) Π). Strategy for Identifying the Number of Factors Order the sample eigenvalues in decreasing order λ 1, λ 1,..., λ N. Estimate θ using CMD and compute J-test (over-id test) Hansen (1982) recursively using the smallest N q eigenvalues k = argmin k=0,1,... Ĵ(λ k+1, λ k+2,..., λ N ; θ)
30 Minimum Distance/GMM Estimation of Model Parameters Assume we observe sample covariance ˆΩ from a model Ω(θ). Let Π(θ) = lim N Etr(Ω s ) (Free Probability) and Π = tr( ˆΩ s ) (Sample). Then, θ = argmin θ (Π(θ) Π) V 1 (Π(θ) Π). Strategy for Identifying the Number of Factors Order the sample eigenvalues in decreasing order λ 1, λ 1,..., λ N. Estimate θ using CMD and compute J-test (over-id test) Hansen (1982) recursively using the smallest N q eigenvalues k = argmin k=0,1,... Ĵ(λ k+1, λ k+2,..., λ N ; θ)
31 Minimum Distance/GMM Estimation of Model Parameters Assume we observe sample covariance ˆΩ from a model Ω(θ). Let Π(θ) = lim N Etr(Ω s ) (Free Probability) and Π = tr( ˆΩ s ) (Sample). Then, θ = argmin θ (Π(θ) Π) V 1 (Π(θ) Π). Strategy for Identifying the Number of Factors Order the sample eigenvalues in decreasing order λ 1, λ 1,..., λ N. Estimate θ using CMD and compute J-test (over-id test) Hansen (1982) recursively using the smallest N q eigenvalues k = argmin k=0,1,... Ĵ(λ k+1, λ k+2,..., λ N ; θ)
32 Minimum Distance/GMM Estimation of Model Parameters Assume we observe sample covariance ˆΩ from a model Ω(θ). Let Π(θ) = lim N Etr(Ω s ) (Free Probability) and Π = tr( ˆΩ s ) (Sample). Then, θ = argmin θ (Π(θ) Π) V 1 (Π(θ) Π). Strategy for Identifying the Number of Factors Order the sample eigenvalues in decreasing order λ 1, λ 1,..., λ N. Estimate θ using CMD and compute J-test (over-id test) Hansen (1982) recursively using the smallest N q eigenvalues k = argmin k=0,1,... Ĵ(λ k+1, λ k+2,..., λ N ; θ)
33 Example: Can estimate much weaker factors which appear in international APT models e.g. exchange rate risks.
34 Example: Can estimate much weaker factors which appear in international APT models e.g. exchange rate risks.
35 Extending factor reasoning Can we extend factor approach beyond covariance matrices? Testing for Delay correlations Application to financial contagion: estimate a factor model based on stock market returns for large number of firms across a region (e.g. South America) Estimate numerous factors F j for j = 1..p (country, industries etc) Let τ be a lag Is F j (0) correlated to F k (τ)?
36 Extending factor reasoning Can we extend factor approach beyond covariance matrices? Testing for Delay correlations Application to financial contagion: estimate a factor model based on stock market returns for large number of firms across a region (e.g. South America) Estimate numerous factors F j for j = 1..p (country, industries etc) Let τ be a lag Is F j (0) correlated to F k (τ)?
37 Extending factor reasoning Can we extend factor approach beyond covariance matrices? Testing for Delay correlations Application to financial contagion: estimate a factor model based on stock market returns for large number of firms across a region (e.g. South America) Estimate numerous factors F j for j = 1..p (country, industries etc) Let τ be a lag Is F j (0) correlated to F k (τ)?
38 Extending factor reasoning Can we extend factor approach beyond covariance matrices? Testing for Delay correlations Application to financial contagion: estimate a factor model based on stock market returns for large number of firms across a region (e.g. South America) Estimate numerous factors F j for j = 1..p (country, industries etc) Let τ be a lag Is F j (0) correlated to F k (τ)?
39 Extending factor reasoning Can we extend factor approach beyond covariance matrices? Testing for Delay correlations Application to financial contagion: estimate a factor model based on stock market returns for large number of firms across a region (e.g. South America) Estimate numerous factors F j for j = 1..p (country, industries etc) Let τ be a lag Is F j (0) correlated to F k (τ)?
40 Extending factor reasoning Can we extend factor approach beyond covariance matrices? Testing for Delay correlations Application to financial contagion: estimate a factor model based on stock market returns for large number of firms across a region (e.g. South America) Estimate numerous factors F j for j = 1..p (country, industries etc) Let τ be a lag Is F j (0) correlated to F k (τ)?
41 Symmetrized delay correlation matrix Under Null F ij are iid. Φ = F (0)F(τ) + F (τ)f(0) 2T Compute the empirical eigenvalue distribution F Φ (λ) See what happens when there are delay correlations...
42 Symmetrized delay correlation matrix Under Null F ij are iid. Φ = F (0)F(τ) + F (τ)f(0) 2T Compute the empirical eigenvalue distribution F Φ (λ) See what happens when there are delay correlations...
43 Symmetrized delay correlation matrix Under Null F ij are iid. Φ = F (0)F(τ) + F (τ)f(0) 2T Compute the empirical eigenvalue distribution F Φ (λ) See what happens when there are delay correlations...
44 Symmetrized delay correlation matrix Under Null F ij are iid. Φ = F (0)F(τ) + F (τ)f(0) 2T Compute the empirical eigenvalue distribution F Φ (λ) See what happens when there are delay correlations...
45 Symmetrized delay correlation matrix Under Null F ij are iid. Φ = F (0)F(τ) + F (τ)f(0) 2T Compute the empirical eigenvalue distribution F Φ (λ) See what happens when there are delay correlations...
46
47 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
48 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
49 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
50 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
51 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
52 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
53 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
54 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
55 Can obtain moments of the eigenvalue distribution from the Cauchy transform and construct J-test... Moments of the eigenvalue distribution mφ 1 = 0 mφ 2 = c/2 mφ 3 = 0 mφ 4 = (1/2)c2 + (3/8)c 3 mφ 5 = 0 mφ 6 = (5/8)c3 + (5/16)c 5 + (9/8)c 4 Or look at the largest eigenvalue...
56
57 Random Distance Matrices Spectral measures for large random Euclidean matrices (Bordenave 06) Entries are functions of positions of n random points in a compact set R d, A = (D(X i X j )) i,j=1..n. Special case D(X) = D( X) (0, 1), adjacency matrix for a random graph. More general distance in issue space agreement beyond polarization study of US congress Also risk sharing between companies in developing countries.
58 Random Distance Matrices Spectral measures for large random Euclidean matrices (Bordenave 06) Entries are functions of positions of n random points in a compact set R d, A = (D(X i X j )) i,j=1..n. Special case D(X) = D( X) (0, 1), adjacency matrix for a random graph. More general distance in issue space agreement beyond polarization study of US congress Also risk sharing between companies in developing countries.
59 Random Distance Matrices Spectral measures for large random Euclidean matrices (Bordenave 06) Entries are functions of positions of n random points in a compact set R d, A = (D(X i X j )) i,j=1..n. Special case D(X) = D( X) (0, 1), adjacency matrix for a random graph. More general distance in issue space agreement beyond polarization study of US congress Also risk sharing between companies in developing countries.
60 Random Distance Matrices Spectral measures for large random Euclidean matrices (Bordenave 06) Entries are functions of positions of n random points in a compact set R d, A = (D(X i X j )) i,j=1..n. Special case D(X) = D( X) (0, 1), adjacency matrix for a random graph. More general distance in issue space agreement beyond polarization study of US congress Also risk sharing between companies in developing countries.
61 Random Distance Matrices Spectral measures for large random Euclidean matrices (Bordenave 06) Entries are functions of positions of n random points in a compact set R d, A = (D(X i X j )) i,j=1..n. Special case D(X) = D( X) (0, 1), adjacency matrix for a random graph. More general distance in issue space agreement beyond polarization study of US congress Also risk sharing between companies in developing countries.
62 Random Distance Matrices Spectral measures for large random Euclidean matrices (Bordenave 06) Entries are functions of positions of n random points in a compact set R d, A = (D(X i X j )) i,j=1..n. Special case D(X) = D( X) (0, 1), adjacency matrix for a random graph. More general distance in issue space agreement beyond polarization study of US congress Also risk sharing between companies in developing countries.
63
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