Regularized Estimation in High-dimensional Time Series Models - - PowerPoint PPT Presentation

Regularized Estimation in High-dimensional Time Series Models

Sumanta Basu

Cornell University

IMA Workshop on Forecasting from Complexity

April 27, 2018

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 1 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 1 / 33

Why High-dimensional Time Series?

Economics and Finance: Macroeconomic policy making (hundreds of macroeconomic series), risk management and monitoring (hundreds of firm health characteristics) Neuroimaging: Functional and effective connectivity analysis from EEG/MEG/fMRI data (hundreds to thousands of brain regions (ROI)) Genomics: Regulatory networks among thousands of genes from short time course (tens of samples) Central questions: structure learning and forecasting of a large, dynamical system Penalized/Regularized estimation/inference methods can be useful Formal theory (beyond i.i.d. data) can guide method development

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 1 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 1 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 1 / 33

Motivation

Systemic risk: widespread failure of the entire financial system Requires understanding of connectivity among financial firms Develop macro-prudential policy: from “too-big-to-fail” to “too-connected-to-fail” Goals: Monitor system-wide risk of financial firms Identify systemically risky institutions How should we measure “systemic risk”? No well defined theoretical agreement so far Extant models lean heavily on “sensible measures” of systemic risk

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 2 / 33

“too-big-to-fail” or “too-central-to-fail”?

Clinton: “... both the governor and the senator have focused only on the big banks. Lehman Brothers, AIG, the shadow banking sector were as big a problem in what caused the Great Recession, I go after them. And I can tell you that the hedge fund billionaires ... ” 1

1https://www.washingtonpost.com/news/the-fix/wp/2016/01/17/the-4th-democratic-debate-

transcript-annotated-who-said-what-and-what-it-meant/

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 3 / 33

Econometric Measures of Systemic Risk

Use publicly available data on firm health (e.g., return, volatility, leverage), study commonality, co-movement, lead-lag relationships: CoVar (Adrian and Brunnermeier, 2011) Marginal and Systemic Expected Shortfall (Acharya et al., 2012) Pairwise vector autoregression (VAR) or Granger causality Network (Billio et al., 2012) Our key point: Co-movements/associations measured in a firm-firm or firm-system basis (pairwise) may lead to incorrect capital requirement policy A system-wide, joint modeling strategy can be more useful

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 4 / 33

Learning Financial Networks: pairwise vs. system-wide

Aug’06 - J ul’09

GOLDMAN S ACHS GOLDMAN S ACHS

P airwis e G ranger C aus ality

B ANK OF AME R ICA COR P

GOLDMAN S ACHS

AIG

B ANK OF AME R ICA COR P

AIG

GOLDMAN S ACHS

Network G ranger C aus ality

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 5 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 5 / 33

Granger Causality

Time series X Granger causal for time series Y 2

X

10 20 30 40 50 60 70 80 90 100

• 3
• 2
• 1

1 2 3 Time

Y

10 20 30 40 50 60 70 80 90 100

• 2

2 4

Network modeling from multivariate systems

◮ independent samples: Correlation, Partial Correlation ◮ time series data: Granger causality, Network Granger causality (NGC)

Vector Autoregression (VAR): flexible modeling framework to capture lead-lag patterns in multivariate systems

2https://en.wikipedia.org/wiki/Granger_causality

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 6 / 33

NGC and VAR

p-dimensional, discrete time, stationary process Xt = {Xt

1,...,Xt p}

Xt = A1Xt−1 +...+AdXt−d +εt, εt i.i.d ∼ N(0,Σε) (1) A1,...,Ad : p×p transition matrices (solid, directed edges) XT−t

j

is Granger-causal for XT

i if At i,j = 0.

Θε = Σ−1

ε : contemporaneous dependence (dotted, undirected edges)

Stability: Eigenvalues of A (z) := Ip −∑d

t=1 Atzt outside {z ∈ C,|z| ≤ 1}

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 7 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 7 / 33

Autoregressive Design

Based on data X0,X1,...,XT, conduct autoregression      (XT)′ (XT−1)′ . . . (Xd)′     

• Y

=      (XT−1)′ (XT−2)′ ··· (XT−d)′ (XT−2)′ (XT−3)′ ··· (XT−1−d)′ . . . ... . . . . . . (Xd−1)′ (Xd−2)′ ··· (X0)′     

• X

   A′

1

. . . A′

d

  

• B∗

+      (εT)′ (εT−1)′ . . . (εd)′     

• E

vec(Y ) = vec(X B∗)+vec(E) = (I ⊗X )vec(B∗)+vec(E) Y

• Np×1

= Z

• Np×q

β ∗

• q×1

+vec(E)

Np×1

vec(E) ∼ N (0,Σε ⊗I) N = (T −d +1), q = dp2

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 8 / 33

Regularized Estimation of VAR

Minimize empirical risk (loss: least squares, negative log likelihood ) with structure inducing penalty Example: Sparse VAR (ℓ1-LS) argmin

A1,...,Ad

L

• Xt,

d

∑

j=1

AjXt−j

• +λ1

d

∑

j=1

Aj1 (ℓ1-ML) argmin

A1,...,Ad,Θε

L

• Xt,

d

∑

j=1

AjXt−j;Θε

• +λ1

d

∑

j=1

Aj1 +λ2∑

i=j

|Θij| Structured sparsity incorporates prior knowledge - help reduce dimension Example of Structure: Group Sparsity, Latent structures

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 9 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 9 / 33

Challenges in high-dimensional, dependent setting

Random design matrix X , correlated with error matrix E      (XT)′ (XT−1)′ . . . (Xd)′     

• Y

=      (XT−1)′ (XT−2)′ ··· (XT−d)′ (XT−2)′ (XT−3)′ ··· (XT−1−d)′ . . . ... . . . . . . (Xd−1)′ (Xd−2)′ ··· (X0)′     

• X

   A′

1

. . . A′

d

  

• B∗

+      (εT)′ (εT−1)′ . . . (εd)′     

• E

Low-dimension (p, d fixed, N → ∞) X ′X N → Σ,non-singular, X ′E N → 0 High-dimension (p, d grows as N → ∞ ), no limiting distributions Non-asymptotic theory: need Concentration Inequality for dependent data P

• X ′X

N −Σ

• > η
• <?

Requires measuring dependence in the VAR process

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 10 / 33

Quantifying Dependence is Important in High-dimension

500 1000 1500 1 2 3 4 5 6

n ||β ^−β||2 α 0.1 0.3 0.5 0.6 0.7 0.8 0.9 0.95

Estimation error of Lasso with p = 500 predictors, each from a VAR(2) process Xt = 2αXt−1 −α2Xt−2 +error (Basu and Michailidis, 2015)

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 10 / 33

Quantifying Dependence in VAR: Existing Works

For univariate AR(1) process, use strength of autocorrelation p = 1 : Xt = ρXt−1 +εt; |ρ| < 1 For VAR(1), (Negahban and Wainwright, 2011; Loh and Wainwright, 2012) assumes A =

• Λmax(A′A) < 1

p > 1,d = 1 : Xt = AXt−1 +εt; A < 1 (Basu and Michailidis, 2015): A < 1 does not hold for all stable VAR(1), also does not generalize to VAR(2), VAR(3), ... Measuring dependence with A is restrictive

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 11 / 33

Quantifying Dependence via Spectral Density

For univariate processes, “peak" of the spectral density measures stability of the process - (sharper peak = less stable)

−10 −5 5 10 0.0 0.2 0.4 0.6 0.8 1.0 lag (h) Autocovariance Γ(h)

• ρ=0.1

ρ=0.5 ρ=0.7

(a) Autocovariance of AR(1)

−3 −2 −1 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 θ f(θ) ρ=0.1 ρ=0.5 ρ=0.7

(b) Spectral Density of AR(1) For multivariate processes, similar role is played by the maximum eigenvalue of the (matrix-valued) spectral density

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 12 / 33

Quantifying Dependence via Spectral Density

Spectral density function of a covariance stationary process {Xt}, fX(θ) = 1 2π

∞

∑

l=−∞

ΓX(l)e−ilθ, θ ∈ [−π,π] ΓX(l) = E

• Xt(Xt+l)′

, autocovariance matrix of order l Closed form for stable VAR fX(θ) = 1 2π

• A (e−iθ)

−1 Σε

• A ∗(e−iθ)

−1 A (z) = Ip −

d

∑

t=1

Atzt

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 13 / 33

Quantifying Dependence via Spectral Density

Maximum eigenvalue of fX M (fX) = max

θ∈[−π,π]Λmax (fX(θ))

Minimum eigenvalue of fX m(fX) = min

θ∈[−π,π]Λmin (fX(θ))

Can be viewed as measures of “narrowness” of the spectrum Measure stability of subprocesses M (fX,k) := max

J⊂{1,...,p},|J|=kM

• fX(J)
• M (fX,1) ≤ M (fX,2) ≤ ··· ≤ M (fX,p) = M (fX)

For stable VAR, fX has closed form - can be related to network edges Aj, Θε

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 14 / 33

Quantifying Dependence with Spectral Density

For a stable VAR(d) process {Xt}, the maximum eigenvalue of its spectral density captures its stability M (fX) = max

θ∈[−π,π]Λmax (fX(θ))

The minimum eigenvalue of the spectral density captures dependence among its components m(fX) = min

θ∈[−π,π]Λmin (fX(θ))

M (fX) and m(fX) scale with µmin(A ) := min

|z|=1Λmin(A ∗(z)A (z))

µmax(A ) := max

|z|=1Λmax(A ∗(z)A (z))

For stable VAR(1) processes, µmin(A ) scales with (1−ρ(A1))−2, ρ(A1) is the spectral radius of A1 µmax(A ) scales with the capacity (maximum incoming + outgoing effect at a node) of the underlying graph

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 15 / 33

Concentration Inequality for Dependent Random Variables

Proposition (Basu & Michailidis, 2015)

{Xt}t∈Z p-dimensional, stable Gaussian with continuous fX. u,v ∈ Rp unit vectors, S := 1

n ∑n t=1 Xt(Xt)′.

Then there exists c > 0 such that for any η > 0, P

• v′ (S−Γ(0))v
• > 2πM (fX)η
• ≤ 2exp
• −cnmin{η2,η}
• P
• u′ (S−Γ(0))v
• > 6πM (fX)η
• ≤ 6exp
• −cnmin{η2,η}
• In particular, for any i,j ∈ {1,...,p},

P

• Sij −Γij(0)
• > 6πM (fX)η
• ≤ 6exp
• −cnmin{η2,η}
• (2)

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 16 / 33

General Outline of deriving estimation error bound

For a fixed realization of X and E, non-asymptotic upper bounds on estimation error derived under

◮ Restricted Eigenvalue or Strong Convexity: Nullspace of S = X′X/N "stays

away" from a small set of approximately sparse vectors

◮ Deviation Condition: ||X′E/n|| is small under suitable norms

For i.i.d. data, these assumptions are known to hold with high probability, proofs use classical concentration inequalities General Strategy: Integrate the new concentration inequalities for dependent data with existing analyses for i.i.d. data, verify that these assumptions hold with high probability when data comes from stable processes

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 17 / 33

Example: Stochastic Regression with Lasso

Linear Model: exogenous predictors, serially correlated noise yt = β ∗,Xt+εt, t = 1,2,...,n Estimation error: If n

• M 2(fX)/m2(fX)
• klogp and λ ≍
• logp/n, then with

probability at least 1−c1 exp[−c2 logp], ˆ βlasso −β ∗ [(M (fX)+M (fε))/m(fX)]

• klogp/n

500 1000 1500 2000 2500 3000 0.5 1.0 1.5 2.0 2.5 3.0 3.5 n ||β ^−β||2

• 128

256 512 1024

(a) ˆ β −β ∗ vs. n

10 20 30 0.5 1.0 1.5 2.0 2.5 3.0 3.5 n/(k logp) ||β ^−β||2

• 128

256 512 1024

(b) ˆ β −β ∗ vs. n/klogp Figure : ℓ2-estimation error (over 50 replicates): {Xt} ∼AR(2), {εt} ∼ MA(2)

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 18 / 33

Consistency of sparse VAR estimates

Proposition (Basu & Michailidis, 2015)

For stable, Gaussian VAR(d), Λmin(Σε) > 0, there exist φ(At,Σε) > 0 and constants ci > 0 such that for N ≥ φ2(At,Σε)klogdp2, the Lasso estimate with λN ≍

• (logdp2)/N

satisfies, with probability at least 1−c1 exp

• −c2(logdp2)
• ,

d

∑

h=1

• ˆ

Ah −Ah

• ≤ φ(At,Σε)
• k(logdp2)/N
• Consistency in high-dimension: Even if d, p = O(N2), klogdp2/N → 0 as long as

k = o(N) Error has two components:

1

φ(At,Σε) large ⇔ M (fX) large, m(fX) small

2

• k logdp2/N: Estimation error for independent data

Estimation errors same as i.i.d. data, modulo a price of dependence

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 19 / 33

Beyond Sparse VAR Estimation

The concentration bounds can be used to prove estimation consistency for other regularized methods with high-dimensional, Gaussian time series (Basu and Michailidis, 2015; Basu et al., 2015)

◮ Regression with Lasso; non-convex penalty (SCAD, MCP) ◮ Sparse covariance estimation with time series data ◮ Regression / VAR with Group Lasso ◮ Low rank and Low rank+Sparse VAR ◮ Tensor Regression with dependent data (Raskutti and Yuan, 2015) ◮ Time series with local dependence (Schweinberger et al., 2015) ◮ De-biasing Lasso: Confidence intervals, p-values of regression/VAR

coefficients [ongoing]

◮ VARMA, Impulse Response, Spectral Density [ongoing]

The results have a common theme estimation error for dependent data

• Measure of narrowness
• f spectrum

× estimation error for i.i.d. data

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 20 / 33

Beyond Sparse VAR Estimation

The concentration bounds can be used to prove estimation consistency for other regularized methods with high-dimensional, Gaussian time series (Basu and Michailidis, 2015; Basu et al., 2015)

◮ Regression with Lasso; non-convex penalty (SCAD, MCP) ◮ Sparse covariance estimation with time series data ◮ Regression / VAR with Group Lasso ◮ Low rank and Low rank+Sparse VAR ◮ Tensor Regression with dependent data (Raskutti and Yuan, 2015) ◮ Time series with local dependence (Schweinberger et al., 2015) ◮ De-biasing Lasso type estimates: Confidence intervals, p-values of

regression/VAR coefficients [ongoing]

◮ VARMA, Impulse Response, Spectral Density [ongoing]

The results have a common theme estimation error for dependent data

• Measure of narrowness
• f spectrum

× estimation error for i.i.d. data

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 20 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 20 / 33

Inference in High-dimensional Regression

De-biasing Lasso: Zhang and Zhang (2013), van de Geer et al (2014), Javanmard and Montanari (2014) High-level roadmap:

◮ Linear Model Y = Xβ ∗ +ε with random design ◮ One-step bias correction of Lasso estimate ˆ

β ˜ β = ˆ β + 1 nMX′(Y −X ˆ β), M is an approximation of Θ, the inverse of Σ = E(X1(X1)′). We adopt this for a simple VAR(1): Xt = AXt−1 +εt, Σε = diag(σ2

1 ,...,σ2 p )

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 21 / 33

De-biasing Lasso VAR

How to find M? vdGeer et al (2014): node-wise regression, J-M (2014): convex

• ptimization

Set ˆ Σ = X′X/n and µ =

• logp/n. Construct M, an approximate inverse of ˆ

Σ, in high-dimension:

1

for i = 1,...,p do:

⋆ Set mi: a solution of the convex program:

minimize m′ ˆ Σm subject to

• ˆ

Σm−ei

• ∞ ≤ µ

where e′

i = (0,0,...,1,...,0)1×p

2

Set M = (m1,...,mp)′. If any of the problems is not feasible, set M = I. Correct bias of Lasso: (Javanmard and Montanari, 2014) ˜ Ai: := ˆ Ai: + 1 nMX′ Yi: −Xˆ Ai:

• Sumanta Basu (Cornell)

High-dimensional Time Series April 27, 2018 22 / 33

Bias Corrected Lasso VAR Estimates

Set δ = ˆ Ai: −Ai:, the bias of Lasso. The debiased Lasso estimates take the form ˜ Ai: = Ai: −(M ˆ Σ−I)δ + 1 nMX′E:i Concentration: Bias term controlled by algorithm, variance term tractable (holds for Lasso VAR estimates) Asymptotics (work in progress): Set M = Θ. Denote the jth row of M by m′

j.

Assume slogp/√n → 0, where s = maxi=1,...,p |{j : Aij = 0}|. Then, lim

n→∞

√n(˜ Aij −Aij) ˆ σi

• m′

j ˆ

Σmj 1/2

d

→ N(0,1) Key ingredients: generalize scaled lasso (Sun and Zhang (2012)) for time series, martingale CLT under high-dimensional scaling Plugging in M for Θ, can operationalize confidence interval construction for Aij and testing hypotheses H0 : Aij = 0 vs. H1 : Aij = 0

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 23 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 23 / 33

Data and Method Description

Monthly stock returns (Rit) of p = 25 largest (by market capitalization) firms in each of the three sectors - Banks (BA), Broker-Dealers (PB) and Insurance (INS) T = 36 month rolling windows from Jan′90−Dec′12 Pre-processing: GARCH(1,1) filter applied on each univariate return series, Rit Method: 75 De-biased Lasso regressions + FDR control at 20% using Benjamini and Hochberg (1995) Firm j connected to firm i if Aij or Aji is statistically significant

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 24 / 33

NGC learns a sparse network

Aug’06 - J ul’09

GOLDMAN S ACHS GOLDMAN S ACHS

P airwis e G ranger C aus ality

B ANK OF AME R ICA COR P

GOLDMAN S ACHS

AIG

B ANK OF AME R ICA COR P

AIG

GOLDMAN S ACHS

Network G ranger C aus ality

Compared to pairwise Granger causality (Bilio et al, 2012), NGC provides a more interpretable structure

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 24 / 33

Top Central Firms: AIG and GS

Firms with high degree: AIG emerged as central 3 months before Lehman bankruptcy on Sep ’08 Goldman Sachs also became central during crisis period 2007-2009

1 2 3 4 5 6 7 Degree May'07 Jul'07 Sep'07 Nov'07 Jan'08 Mar'08 May'08 Jul'08

Sep'08

Nov'08 Jan'09 Mar'09 May'09 Jul'09 Sep'09 Nov'09 Jan'10 Mar'10 May'10

HIG WB AMTD JNS NCC GS BEN UNH MFC MMC AXP TD MS LEH LM TROW GNW BMO COF CME ET AB FII RJF JNC ALL PRU ACE PFG CNA UNH NCC MS BEN WB AXP COF STT JNS GS CME TROW MET HIG AET CI GNW BNS TD STI BMO BCM CFC MER LEH SCH LM AMTD AB AGE FII JNC JEF AMG AIG MFC PRU SLF AFL MMC ACE PFG MBI HUM COF BEN NCC CFC MS GS CME UNH GNW BAC WFC WB RY BBT STT MER LEH CIT AB JNS ENH AIG AET AFL ACE MMC CI BSC AMTD ET BLK AGE EV FII MFC MET ALL SLF CB PFG LNC MBI CVH COF MBI TROW BAC WB GS JNS FII CFC MS LEH BEN CME SLF CI RY FRE TD BBT NCC SLM STT MER BSC AMTD ET EV RJF IDC WDR AINV UNH MFC HIG AFL CB MMC XL HUM HUM FRE CI MBI BEN LEH TROW FII BNS BK COF GS CME ET EV CVH MBI FRE CI BEN HUM LEH FII FRE CI AB BK BMO NCC LM JNS IVZ WB MS LEH BEN CME BSC CB MMC IVZ TD STI CI BAC AB UBS BMO PNC MS PGR CI IVZ TD BMO MS HIG CB CI WB AB MS ACE AB CB WB AXP MS ACE BEN CB AB FII CB BEN AB NDAQ BEN BAC BNS AB NDAQ CB AOC HUM BEN HUM NDAQ AB AOC XL BEN TROW NDAQ WDR AOC AXP MS UNH HUM XL BEN AOC WDR

GS AIG

WDR AXP BEN TROW XL BNS AOC BX AMTD BNS AOC TROW

AIG AIG AIG AIG AIG AIG GS GS GS GS GS AIG AIG AIG AIG AIG AIG AIG AIG GS GS

Network Granger Causality

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 25 / 33

Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 25 / 33

Beyond Sparse VAR Estimation

The concentration bounds can be used to prove estimation consistency for other regularized methods with high-dimensional, Gaussian time series (Basu and Michailidis, 2015; Basu et al., 2015)

◮ Regression with Lasso; non-convex penalty (SCAD, MCP) ◮ Sparse covariance estimation with time series data ◮ Regression / VAR with Group Lasso ◮ Low rank and Low rank+Sparse VAR ◮ Tensor Regression with dependent data (Raskutti and Yuan, 2015) ◮ Time series with local dependence (Schweinberger et al., 2015) ◮ De-biasing Lasso estimates: Confidence intervals, p-values of

regression/VAR coefficients [ongoing]

◮ VARMA, Impulse Response, Spectral Density [ongoing]

The results have a common theme estimation error for dependent data

• Measure of narrowness
• f spectrum

× estimation error for i.i.d. data

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 25 / 33

Realized Volatility Network with Lasso VAR

As before, constructed Lasso VAR networks on daily realized volatility during crisis period Fails to provide sparse, interpretable structures - modeling latent factors may be helpful

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Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 26 / 33

Example: Low-Rank VAR

p-dimensional stationary process {Xt} entire dynamics driven by a r-dimensional (r ≪ p) unobserved process of factors {Ft} Xt = ΛFt +ξ t, ξ t ∼ N(0,Σξ ), Cov(ξ t,ξ s) = 0 if t = s Ft = HFt−1 +ηt, ηt ∼ N(0,Ση), Cov(ηt,ηs) = 0, if t = s If the matrix of factor loadings Λp×r is full rank r with a left inverse Λ−, then Xt = Λ

• HFt−1 +ηt

+ξ t = Λ

• HΛ−(Xt−1 −ξ t−1)+ηt

+ξ t = ΛHΛ− Xt−1 +

• Ληt +ξ t −ΛHΛ−ξ t−1

= LXt−1 +εt L = ΛHΛ− has rank at most r and the new error process εt = Ληt +ξ t −Lξ t−1 has a MA component

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 27 / 33

Example: LowRank+Sparse VAR

Dynamics of {Xt} governed by

1

r-dimensional unobserved process of factors {Ft}

2

intereaction among its components, captured by a sparse transition matrix S Xt = ΛFt +SXt−1 +ξ t, S sparse Ft = HFt−1 +ηt A similar calculation shows Xt = ΛHΛ− Xt−1 −SXt−2 −ξ t−1 +SXt−1 +ξ t +Ληt = (L+S)Xt−1 −LSXt−2 +εt ≈ (L+S)Xt−1 +εt, assuming the second order effects in LS are small Can be generalized to group sparse structures G, G+S We model L+S VAR as a first step towards L+S VARMA

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Model

Xt = AXt−1 +εt,εt i.i.d. N(0,Σε) (3) A = L0 +S0, rank(L0) = r, S00 = s, r ≪ p, s ≪ p2 (4) L0 captures the effects of latent variables Data: {X0,...,XT}, we form autoregressive design    (XT)′ . . . (X1)′   

• Y

=    (XT−1)′ . . . (X0)   

• X

A′ +    (εT)′ . . . (ε1)′   

• E

(5) Linear regression problem with N = T samples and q = p2 variables Goal: estimate L0 and S0 with high accuracy when N ≪ p2.

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Estimation Procedure

We use the convex program proposed in (Agarwal et al., 2012): (ˆ L′, ˆ S′) = argmin

L,S∈Rp×p:Lmax≤α/p

1 2 Y −X (L+S)2

F +λNL∗ + µNS1

(6) λN, µN are non-negative tuning parameters controlling the regularization of sparse and low-rank component Identifiability: The parameter α controls for degree of non-identifiable matrices allowed in the model class In practice, we choose α in [1, p]

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Estimation Error

Proposition

There exist universal positive constants ci > 0 such that for N pM 2(fX)/m2(fX), for any S0 with S0max ≤ α, any solution (ˆ L, ˆ S) of the program (6) with appropriately chosen λN and µN satisfies, with probability at least 1−c1 exp[−c2 logp], ˆ S−S02

F +ˆ

L−L02

F ≤ c0φ2(A,Σε)µ2 max(A )

Λ2

min(Σε)

(rp+slogp) N + 32Λ2

min(Σε)

µ2

max(A )

sα2 p2 (7) where φ(A,Σε) = Λmax(Σε)[1+(1+ µmax(A ))/µmin(A )]. error emanating from randomness in the data and limited sample capacity, this error goes to zero as N → ∞ error due to unidentifiability of the problem, does not change with sample size The first term scales as O(rp/N), which is much smaller than p2/N if r ≪ p The effect of dependence is stronger when µmax(A ),Λmax(Σε) are larger and µmin(A ),Λmin(Σε) are smaller, i.e., when the spectrum is more spiky

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Data Description and Main Findings

Daily stock prices on 75 large US firms - 25 each from three sectors: banking (BA), insurance (INS) and broker-dealers (PB) [Data source: CRSP] Daily realized volatility estimated using log(hi−low) Three different 36 month windows: (i) Stable Period (Sep ’02-Aug ’05), (ii) Leading up to Crisis (Sep ’05 - Aug ’08), (iii) Post-crisis (Jan ’10 - Dec ’12) Lasso and L+S VAR fitted on each dataset Tuning parameters chosen by AIC L+S VAR extracted 7−9 rank low-rank components pre-crisis, GC networks have similar edge density In crisis period, GC network estimated by L+S VAR much sparser than Lasso post-crisis, L+S extracts strong connectivity among broker-dealers in low-rank component

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Financial Networks during financial crisis (2005-08)

Figure : (Left) Lasso estimates a denser network than (Right) L+S VAR

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Outline

1

Introduction

2

Sparse Vector Autoregression (VAR) Motivation: Measuring Systemic Risk Network Granger Causality and VAR Modeling and Implementation Estimation in Sparse VAR Inference in Sparse VAR Back to Measuring Systemic Risk

3

Incorporating Latent Structure in VAR

4

Extension to Other Time Series Models

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Beyond Sparse VAR Estimation

The concentration bounds can be used to prove estimation consistency for other regularized methods with high-dimensional, Gaussian time series (Basu and Michailidis, 2015; Basu et al., 2015)

◮ Regression with Lasso; non-convex penalty (SCAD, MCP) ◮ Sparse covariance estimation with time series data ◮ Regression / VAR with Group Lasso ◮ Low rank and Low rank+Sparse VAR ◮ Tensor Regression with dependent data (Raskutti and Yuan, 2015) ◮ Time series with local dependence (Schweinberger et al., 2015) ◮ De-biasing Lasso estimates: Confidence intervals, p-values of

regression/VAR coefficients [ongoing]

◮ VARMA, Impulse Response, Spectral Density [ongoing]

The results have a common theme estimation error for dependent data

• Measure of narrowness
• f spectrum

× estimation error for i.i.d. data

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

VARMA (... Ines’ talk on Monday)

Model: {at} Gaussian white noise: yt = Φ1yt−1 +Φ2yt−2 +...+Φpyt−p +Θ1at−1 +...+Θqat−q +at Two-stage estimation:

1

Phase-I: ˆ Π1:...:˜

p =

argmin

Π1,Π2,...,Π[˜

p]∑

t

yt −∑

j

Πjyt−j2 +P(Π1,...,Π[˜

p])

2

Phase-II: ˆ Φj, ˆ Θk

• 1≤j≤p,1≤k≤q = argmin

Φ1:p,Θ1:q∑ t

yt −∑

j

Φjyt−j −∑

k

Θk ˆ εt−k2 +P(Φ1:p,Θ1:q) ˆ εt estimated residuals from phase-I Phase-I requires extension to VAR(∞), generalized concentration bounds Phase-II requires measuring stability of the joint process {yt−1,...,yt−p,εt−1,...,εt−q} Both accomplished using spectral density based concentration bounds

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Spectral Density Estimation in High-dimension

Spectral density captures both lead-lag and contemporaneous dependence amongst time series Coherence and Partial coherence commonly used in functional connectivity studies in neuroscience Classical Approach: Smoothed Periodogram Shrinkage Methods (Bohm and von Sachs (2009)): Take weighted average with diagonal matrix No sparsity assumption, theory works in a regime p2/n → 0 Thresholding methods used in covariance methods can be potentially helpful Spectral density based measures show that thresholding works if slogp/n → 0

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Collaborators

George Michailidis Ali Shojaie Xianqi Li Sreyoshi Das Amiyatosh Purnanandam Ines Wilms Jacob Bien David S. Matteson Yige Li Yiming Sun

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Sumanta Basu and George Michailidis (2015), Regularized Estimation in Sparse High-dimensional Time Series Models, Annals of Statistics. Sumanta Basu, Xianqi Li and George Michailidis (2018), Low-Rank and Structured Modeling of High-dimensional Vector Autoregressions, submitted. Sumanta Basu, Sreyoshi Das, George Michailidis and Amiyatosh Purnanandam (2017), A System-wide Approach to Measure Connectivity in the Financial Sector. Available on SSRN http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2816137. Ines Wilms, Sumanta Basu, Jacob Bien and David S. Matteson (2017), Sparse Identification and Estimation of High-Dimensional Vector AutoRegressive Moving

• Averages. Available at https://arxiv.org/abs/1707.09208.

Sumanta Basu and David S. Matteson, Estimation in Sparse High-dimensional Time Series Models, in revision at WIRES Computational Statistics.

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Acharya, V., Engle, R., and Richardson, M. (2012), “Capital shortfall: A new approach to ranking and regulating systemic risks,” The American Economic Review, 59–64. Adrian, T. and Brunnermeier, M. K. (2011), “CoVaR,” Tech. rep., National Bureau of Economic Research. Agarwal, A., Negahban, S., and Wainwright, M. J. (2012), “Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions,” The Annals of Statistics, 40, 1171–1197. Basu, S. and Michailidis, G. (2015), “Regularized estimation in sparse high-dimensional time series models,” The Annals of Statistics, 43, 1535–1567. Basu, S., Shojaie, A., and Michailidis, G. (2015), “Network Granger Causality with Inherent Grouping Structure,” Journal of Machine Learning Research, 16, 417–453. Benjamini, Y. and Hochberg, Y. (1995), “Controlling the false discovery rate: a practical and powerful approach to multiple testing,” Journal of the Royal Statistical Society. Series B (Methodological), 289–300. Bernanke, B. S., Boivin, J., and Eliasz, P . (2005), “Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach,” The Quarterly Journal of Economics, 120, 387–422.

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Billio, M., Getmansky, M., Lo, A. W., and Pelizzon, L. (2012), “Econometric measures of connectedness and systemic risk in the finance and insurance sectors,” Journal of Financial Economics, 104, 535–559. Blanchard, O. and Perotti, R. (2002), “An empirical characterization of the dynamic effects of changes in government spending and taxes on output,” the Quarterly Journal of economics, 117, 1329–1368. Hiemstra, C. and Jones, J. (1994), “Testing for linear and nonlinear Granger causality in the stock price-volume relation,” Journal of Finance, 1639–1664. Javanmard, A. and Montanari, A. (2014), “Confidence intervals and hypothesis testing for high-dimensional regression,” The Journal of Machine Learning Research, 15, 2869–2909. Loh, P .-L. and Wainwright, M. J. (2012), “High-dimensional regression with noisy and missing data: provable guarantees with nonconvexity.” Ann. Stat., 40, 1637–1664. Negahban, S. and Wainwright, M. J. (2011), “Estimation of (near) low-rank matrices with noise and high-dimensional scaling,” Ann. Statist., 39, 1069–1097. Raskutti, G. and Yuan, M. (2015), “Convex Regularization for High-Dimensional Tensor Regression,” arXiv preprint arXiv:1512.01215.

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Schweinberger, M., Babkin, S., and Ensor, K. B. (2015), “High-Dimensional Multivariate Time Series With Local Dependence,” arXiv preprint arXiv:1510.02159. Sims, C. (1972), “Money, income, and causality,” The American Economic Review, 62, 540–552.

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VAR in Macroeconomics

Testing relationship between money and income (Sims, 1972) Stock price-volume relation (Hiemstra and Jones, 1994) Dynamic effect of government spending and taxes on output (Blanchard and Perotti, 2002) Measure effect of monetary policy on economy (Bernanke et al., 2005)

• 6
• 4
• 2

2 4 6 Feb-60 Aug-60 Feb-61 Aug-61 Feb-62 Aug-62 Feb-63 Aug-63 Feb-64 Aug-64 Feb-65 Aug-65 Feb-66 Aug-66 Feb-67 Aug-67 Feb-68 Aug-68 Feb-69 Aug-69 Feb-70 Aug-70 Feb-71 Aug-71 Feb-72 Aug-72 Feb-73 Aug-73 Feb-74 Aug-74

Employment Federal Funds Rate Consumer Price Index

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VAR in Genomics

Learn directed regulatory network of interactions from time course gene expression data

0.2 0.4 0.6 0.8 1 1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

lysR lysA alaS tpr fis mtlR marR marA rob putA marB zwf nfo inaA ygiC

Gene Expression level Time Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

VAR models in Neuroscience

Identify connectivity among brain regions from time course fMRI data Connectivity of VAR generative model (Seth et al., 2013)

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Data

Two types of data structures :

1

Short Panels: p variables observed on n > 1 replicates, T small to moderate (genomics, microeconomics)

2

Long Time Series: n = 1 replicate, T moderate or large, p large (macroeconomics, finance) Estimation methods identical Theory more challenging for long time series due to dependence

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Maximum Likelihood Estimation (ℓ1-ML)

p regressions coupled through Θε Novel, parallelizable block relaxation algorithm: Start with ℓ1-LS at k = 0 For k ≥ 1, ˆ A(k)

1:d

= argmin

A1,...,Ad

L

• Xt,

d

∑

j=1

AjXt−j; ˆ Θ(k−1)

ε

• +λ1

d

∑

j=1

Aj1 ˆ Θ(k) = argmin

Θε≥0

L

• Xt,

d

∑

j=1

ˆ A(k)

j

Xt−j;Θε

• +λ1

d

∑

j=1

Aj1 +λ2∑

i=j

|Θij| Objective function non-convex, but convex w.r.t. {Aj} and Θε Algorithm converges to stationary point near truth with high probability under high-dimensional scaling3

3Lin*, Basu*, Banerjee and Michailidis (2016), preprint.

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Concentration Inequality for Dependent Data

Proposition (Basu & Michailidis, 2015)

For a stationary, centered Gaussian time series {Xt}t∈Z with continuous spectral density fX, there exists a constant c > 0 such that for any k-sparse vectors u,v ∈ Rp with u ≤ 1, v ≤ 1, k ≥ 1, and any η ≥ 0, the sample Gram matrix S = X ′X /N concentrates around ΓX(0) as P

• v′ (S−ΓX(0))v
• > 2πM (fX,k)η
• ≤ 2exp
• −cnmin{η2,η}
• (8)

P

• u′ (S−ΓX(0))v
• > 6πM (fX,2k)η
• ≤ 6exp
• −cnmin{η2,η}
• (9)

In particular, for any i,j ∈ {1,...,p}, we have P

• Sij −Γij(0)
• > 6πM (fX,2)η
• ≤ 6exp
• −cnmin{η2,η}
• (10)

Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Network Heatmaps during financial crisis (2005-08)

Figure : Increased connectivity in banking and insurance sectors captured in low-rank component of L+S VAR

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Financial Networks in stable period (2002-05)

Figure : (Left) Lasso and (Right) L+S VAR estimate networks of similar sparsity

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Systemic Risk

Three key drivers: Counter-party exposure (Allen and Gale, 2000) Common holdings of assets/liabilities Fire-sale/spillover externalities (Bhattacharyya and Gale, 1987) Network based Models: interbank lending network (Glasserman and Young (2013)) Data not always publicly available primarily captures contagion through counter-party exposure

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AIG and GS in pairwise GC Network

AIG and GS have high centrality, comparable to COF and MBI in 2007

1 2 3 4 5

Degree

May'07 Jul'07 Sep'07 Nov'07 Jan'08 Mar'08 May'08 Jul'08

Sep'08

Nov'08 Jan'09 Mar'09 May'09 Jul'09 Sep'09 Nov'09 Jan'10 Mar'10 May'10

COF

AGE JNC HIG PRU SLF

COF

AGE UNH WM ET WB MER SCH SLF LEH JNS

COF

UNH LEH WM

GS

AGE

MBI COF

FRE

AIG

WB

MBI

FRE WB MS

CI MBI

TD HUM FRE AMG AB

AIG

FRE AB TD CI TD AB BAC WB EV TD WB IVZ AMG BAC AB WB AB MS HIG CI TD WB TD MS HIG TD AB MS DB SLM TD AB MS BEN DB TD SLM BEN AB DB SLM AB UBS BEN MS SLM BEN TROW DB AIG UBS SLM BEN DB UBS SLM BEN AIG DB TROW AB BX UBS AOC DB AB XL

AIG AIG AIG AIG AIG AIG AIG AIG AIG AIG AIG AIG GS GS GS GS GS GS GS GS GS GS

6 7

Pairwise Granger Causality

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Evolution of Network Centrality: Degree

Identifies systemically important events

1 2 3 4 5

Degree

Degree May'90−Apr'93 Oct'90−Sep'93 Mar'91−Feb'94 Aug'91−Jul'94 Jan'92−Dec'94 Jun'92−May'95 Nov'92−Oct'95 Apr'93−Mar'96 Sep'93−Aug'96 Feb'94−Jan'97 Jul'94−Jun'97 Dec'94−Nov'97 May'95−Apr'98 Oct'95−Sep'98 Mar'96−Feb'99 Aug'96−Jul'99 Jan'97−Dec'99 Jun'97−May'00 Nov'97−Oct'00 Apr'98−Mar'01 Sep'98−Aug'01 Feb'99−Jan'02 Jul'99−Jun'02 Dec'99−Nov'02 May'00−Apr'03 Oct'00−Sep'03 Mar'01−Feb'04 Aug'01−Jul'04 Jan'02−Dec'04 Jun'02−May'05 Nov'02−Oct'05 Apr'03−Mar'06 Sep'03−Aug'06 Feb'04−Jan'07 Jul'04−Jun'07 Dec'04−Nov'07 May'05−Apr'08 Oct'05−Sep'08 Mar'06−Feb'09 Aug'06−Jul'09 Jan'07−Dec'09 Jun'07−May'10 Nov'07−Oct'10 Apr'08−Mar'11 Sep'08−Aug'11 Feb'09−Jan'12 Jul'09−Jun'12 Dec'09−Nov'12 Thai baht Devaluation Russian effective default LTCM Dot com bubble Bear Stearns Lehman Brothers l1var Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33

Evolution of Degree: pairwise vs. L1VAR

0.5 1.0 1.5 2.0 2.5

Total Number of Edges (Scaled by Historical Average over 1990−2012)

Degree May'90−Apr'93 Oct'90−Sep'93 Mar'91−Feb'94 Aug'91−Jul'94 Jan'92−Dec'94 Jun'92−May'95 Nov'92−Oct'95 Apr'93−Mar'96 Sep'93−Aug'96 Feb'94−Jan'97 Jul'94−Jun'97 Dec'94−Nov'97 May'95−Apr'98 Oct'95−Sep'98 Mar'96−Feb'99 Aug'96−Jul'99 Jan'97−Dec'99 Jun'97−May'00 Nov'97−Oct'00 Apr'98−Mar'01 Sep'98−Aug'01 Feb'99−Jan'02 Jul'99−Jun'02 Dec'99−Nov'02 May'00−Apr'03 Oct'00−Sep'03 Mar'01−Feb'04 Aug'01−Jul'04 Jan'02−Dec'04 Jun'02−May'05 Nov'02−Oct'05 Apr'03−Mar'06 Sep'03−Aug'06 Feb'04−Jan'07 Jul'04−Jun'07 Dec'04−Nov'07 May'05−Apr'08 Oct'05−Sep'08 Mar'06−Feb'09 Aug'06−Jul'09 Jan'07−Dec'09 Jun'07−May'10 Nov'07−Oct'10 Apr'08−Mar'11 Sep'08−Aug'11 Feb'09−Jan'12 Jul'09−Jun'12 Dec'09−Nov'12 Thai baht Devaluation Russian effective default LTCM Dot com bubble Bear Stearns Lehman Brothers l1var pwvar Sumanta Basu (Cornell) High-dimensional Time Series April 27, 2018 33 / 33