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Time Varying Estimation and Inference withApplication to Large Dimensional Covariance

Estimation and Portfolio Management

G. Kapetanios (with L. Giraitis, Y. Dendramis et al)

14th January 2015

Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management

Introduction

I Modelling structural change is crucial for most econometricanalyses and especially forecasting

I There are a variety of different approaches to such modelling.These include:

I Structural breaks: Change in parameters is rare and abrupt.

I Smooth deterministic change over time with no abruptchanges.

I Random Coefficient (RC) models: These are the major focusof the current presentation.


The usual approach

I RC models have become increasingly popular recently.I They are estimated by being cast in state space form and then

using filters, such as the Kalman filter, usually as part of aBayesian estimation framework.

I Work has ranged across topics such as accounting for theGreat Moderation, documenting changes in the effect ofmonetary policy shocks and documenting changes in thedegree of exchange rate pass-through.

I A selection of papers that make use of such models includeCogley and Sargent (2001), Cogley and Sargent (2005),Cogley, Sargent, and Primiceri (2010), Benati (2010), Benatiand Surico (2008), Mumtaz and Surico (2009), Pesaran,Pettenuzzo, and Timmermann (2006) and Koop and Potter(2007).

I Problems with these models: Unclear theoretical properties,heavy computational costs.


A new approach

I It is clear that easier, robust and less costly estimationmethods with clear theoretical properties would be welcome.

I Kernel estimation of coefficient processes in models of smoothdeterministic change is well established and fully analysed inthe statistical literature. In the context of structural changethis approach is simply a refinement of estimating modelswith rolling windows.

I Giraitis, Kapetanios and Yates (JoE, 2014) propose applyingkernel estimation to RC models.

I They formalise a kernel estimator for the unobservedcoefficient process and derive its theoretical properties.

I These estimator properties are very attractive: Consistencyand asymptotic normality. Also great small sample properties.

I A number of follow-up papers take this approach to morecomplex and realistic settings.


Presentation Roadmap

I We will present the basic idea within the context of a simpleAR model and discuss theoretical properties.

I Then, we move on to more realistic models that allow for timevarying variances.

I Third, we discuss time varying Maximum Likelihoodestimation and its theoretical properties coupled with anapplication to the modelling of the sterling money market.

I We consider the issue of selecting relevant tuning parametersusing data

I The above allow us to develop time varying estimation forlarge dimensional covariance matrices and present a basicresult that can be used for a variety of different covarianceestimators

I Finally we will present some Monte Carlo evidence on theseestimators which can feed directly into the construction ofportfolios of assets.


Some preliminaries

I For simplicity of analysis we begin with a univariate dynamicmodel. We then extend in many directions.

I We consider the AR(1) model:

yt = ρt−1yt−1 + ut , t = 1, 2, · · · , n, ut ∼ IID(0, σ2u)

I AR is a workhorse class of models. Many possibilitiesdepending on how ρn,t−1 is specified.

I The most closely related specification relates to Locallystationary models: Priestley (1965), Dahlhaus (1997)

ρn,t−1 = µ(t/n), 1 ≤ t ≤ n deterministic, smooth


Random ρt−1

I Random coefficient (RC) case: ρt random process andbounded between -1 and 1. Many ways to bound. This issueis not discussed in the macro/finance literature.

I We choose a straightforward standartization

ρt = ρat

max0≤k≤t |ak |, t = 1, 2, · · · , n,

{at} determines random drift. ρ restricts ρt awayfrom −1 and 1. Both {at}, ρ unknown. Observe:ρk ∈ [−ρ, ρ] ⊂ (−1, 1), for all k = 1, · · · , n.

I at evolves as I (d), d > 1/2 process: {vt} stationary

at = at−1 + vt , t = 1, · · · , n,

I Popular choice: vt i.i.d., at driftless random walk,


Estimation and Inference

I We wish to estimate the coefficients ρ1, · · · , ρnI We suggest as an estimator a weighted sample autocorrelation

at lag 1 given by

ρn,t :=

∑nk=1 K ( t−kH )ykyk−1∑nk=1 K ( t−kH )y 2

k−1

, (1)

where K (x) ≥ 0, x ∈ R is a continuous bounded kernelfunction.

I This is simply a generalisation of a rolling window estimatorgiven by

ρn,t :=

∑t+Hk=t−H ykyk−1∑t+Hk=t−H y 2

k−1

,

which is a local sample correlation of yt ’s at lag 1, based on2H + 1 observations yt−H , · · · , yt+H .

I For the bandwidth we assume that H →∞ and H = o(n).


Estimator properties

I This estimator is consistent and asymptotic normal assummarised belowTheorem 3

ρn,t − ρt = ξn,t + OP((H/n)γ) = OP(1/√

H) + OP((H/n)γ),

TH,t

(1− ρ2t )1/2

ξn,t →D N(0, 1).

(ii) If H = o(nγ/(0.5+γ)), then

TH,t√1− ρ2

n,t

(ρn,t − ρt

)→D N(0, 1).

where γ = d − 1/2 and

btk := K (t − k

H),TH,t :=

∑nk=1 btk(∑n

k=1 b2tk

)1/2, ξn,t :=

∑nk=1 btkukyk−1∑nk=1 btky 2

k−1

.


Some comments

I For γ ≥ 1/2, we can take H = o(n1/2) and then no knowledgeof γ is needed. Using the above the estimation of standarderrors is easily feasible.

I Estimator requires persistence of ρn,tI 0 < γ < 1 defines the magnitude of error term in normal

approximation:

I Larger γ → stronger persistence → better approximation

I {vj} in at = at−1 + vt can have short, long or negativememory.

I Bandwidth H = o(n1/2) yields negligible error for shortmemory {vj} (γ = 1/2) and long memory vj (1/2 < γ < 1).

I When γ → 0, trending of at and the quality of approximationdeteriorate.

I For stationary {at}, estimation is not consistent.


Further Extensions

I The above focuses on a simple univariate homoscedasticmodel.

I Good for proving the concept but not for actual empirical work

I Giratis, Kapetanios and Yates (2014) provide furtherextensions that enable analysing more realistic models

I The first extension is to heteroscedastic vector autoregressivemodels (VARs)

I We can show that kernel estimates are consistent underconditions that bound the norm of the coefficient matrices.

I The second crucial extension enables estimation of possiblevariation in the unconditional variances of the model shocks.


The setup

I We consider the m-dimensional VAR(1) model:

yt = Ψt−1yt−1 + ut , t = 1, 2, · · · , n, ut ∼ IID(0,Σu)

I VAR is a workhorse class of models. Many possibilitiesdepending on how Ψt−1 is specified.

I The most closely related specification again relates to Locallystationary models: Priestley (1965), Dahlhaus (1997)

Ψn,t−1 = µ(t/n), 1 ≤ t ≤ n deterministic, smooth


Random Ψt−1

I Random coefficient (RC) case: Ψt random process witheigenvalues bounded between -1 and 1. Many ways to bound.Let Ψt−1 = [ψt−1,ij ]. Sufficient bounding can be implementedby defining

Ψt−1 = [ψt−1,ij ], ψt,ij = ψt−1,ij+vψt,ij , t = 1, · · · , n; i , j = 1, · · · ,mwhere vψt,ij is a zero mean i.i.d. sequence with finite variance.Then,

ψt−1,ij = ψψt−1,ij

max0≤k≤t∑m

j=1 |ψt−1,ij |, t = 1, 2, · · · , n,

and 0 < ψ < 1. This ensures that the maximum eigenvalue ofΨn,t−1 is bounded above by one in absolute value.

I But this is just an example. What matters is that

||Ψt −Ψt+h|| = Op(h/t)

This kind of condition seems necessary for consistency.


Estimation

I We wish to estimate the coefficients Ψ1, · · · ,Ψn

I We suggest as an estimator a weighted sample autocorrelationat lag 1 given by

Ψn,t :=

(n∑

k=1

K (t − k

Hψ)yky ′k−1

)(n∑

k=1

K (t − k

HΨ)yk−1y ′k−1

)−1

,

(2)

where K (x) ≥ 0, x ∈ R is a continuous bounded function.I This is simply a generalisation of a rolling window estimator

given by

Ψn,t :=

t+Hψ∑k=t−Hψ

yky ′k−1

t+Hψ∑k=t−Hψ

yk−1y ′k−1

−1

,

which is a local sample correlation of yt ’s at lag 1, based on2Hψ + 1 observations yt−Hψ , · · · , yt+Hψ .

I For the bandwidth we assume that Hψ →∞ and Hψ = o(n).Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management

Estimation of time varying variance

I We wish to also allow a time varying variance for ut andestimate it

I

ut = Ht−1εt , E [ut |Ft−1] = 0 (3)

with respect to some filtration Ft , where Ht = {ht,ij} is am ×m time varying random volatility process, and εt is avector-valued standardized i.i.d. noise, Eεt = 0, Eεtε

′t = I.

Denote by Σt = Ht−1H′t−1 = E [utu′t |Ft−1] the conditional

variance-covariance matrix.

I We then obtain the residuals, ut from the conditional meanmodel, and fit a time varying simple location model to everyelement of ut u

′t . We denote the bandwidth parameter by Hh.


Estimator properties

Let κn,ψ := (Hψ/n)1/2 + H−1/2ψ , κn,h := (Hh/n)1/2 + H

−1/2h ,ktj :=

K((t − j)/Hψ

),Kt =

∑nj=1 ktj ,K2,t =

∑nj=1 k2

tj . ForHψ = o(n/ log n), Hh = o(n/ log n),

Ψt −Ψt = Op(κn,ψ), (4)

Σuu,t −Σt = Op

(κ2n,ψ + κn,h

). (5)

In addition, if HψHψ = o(n), then for any real m × 1- vector asuch that ||a|| = 1,

(Kt/K2,t)1/2H−1

t−1(Ψt −Ψt)( n∑j=1

ktjyj−1y′j−1

)1/2a→D N (0, I) (6)

In addition, if HhHh = o(n) and H1/2h << Hψ << n/(Hh log n)1/2,

then

(Lt/L1/22,t )H−1

t−1(Σuu,t −Σt)H′t−1−1 →D Z (7)

where the elements of Z are independent normal variables.Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management

Time Varying Maximum Likelihood Estimation and anapplication to Bank Lending

I Giraitis, Kapetanios, Wetherilt and Zikes (2013) have recentlyextended the above method to handle time varying MaximumLikelihood estimation with random coefficients.

I They use a set of interdependent bivariate Tobit models tomodel interactions between bank lending of a set of banksover time in the sterling money market. Their model canhandle parsimoniously a large set of variables while allowingfor structural change.

I They consider the Tobit model

yt =

{β′0,txt + ut , if β′0,txt + ut > 0

0, otherwise(8)

where the latent variable β′0,txt + ut is defined by a vectorβ0,t , a k × 1 vector of known random/deterministic regressorsxt and the noise ut = σ0,tεt , εt ∼ NID(0, 1).


Time Varying Maximum Likelihood Estimation

I They assume that β0,t and σ0,t are bounded (truncated)random/deterministic processes independent of εt satisfyingthe following smoothness condition: for 1 ≤ h ≤ t, as h→∞,

supj :|j−t|≤h

||θ0,t − θ0,j ||2 = Op (h/t) . (9)



I To accommodate estimation of time varying parameter θ0,t ,they use the weighted likelihood function

Lθ,t,T :=∏j

′(1− Fθ,j)

ktj∏j

′′(gθ,t(yj))ktj ,

with the weights ktj = ktj/(∑T

j=1 ktj

), ktj := K

((t − j)/H

),

where K (x) ≥ 0, x ∈ R is a continuous bounded function andthe bandwidth parameter H →∞, H = o(T/ log T ).

I They define the MLE estimate θt of θ0,t as the maximiser ofthe weighted log-likelihood:

Qθ,t,T := log Lθ,t,T =∑j

′ktj log(1− Fθ,j) +

∑j

′′ktj log gθ,t(yj),

θt := argmaxθQθ,t,T . (10)



TheoremLet y1, ..., yT be a sample of the Tobit model and t = [τT ] where0 < τ < 1 is fixed. Denote κH,T := (H/T )1/2 + H−1/2. Then the

MLE estimate θt of the parameter θ0,t has the following properties.

(i) (Consistency). There exist an open neighborhood Bt of θ0,t

such that

θt := argminθ∈BtQθ,t,T →P θ0,t , and θt − θ0,t = Op(κH,T ).

(ii) (Asymptotic normality). In addition, if HH = o(T ), then

Σ−1/2t (θt − θ0,t)→D N (0, I ), Σt := Kt,T

(−∂2Qθ0,t ,t

∂θ∂θ′)−1

where Kt,T :=∑T

j=1 k2tj ∼ const H−1, and

(−

∂2Qθ0,t ,t

∂θ∂θ′

)is a

positive definite matrix.


Simulation

yt = ct + ρtyt−1 + βtxt + εt

True Estimated 95% CI

0 100 200 300 400 500 600 700 800 900 1000

-1

0

1 c True Estimated 95% CI

0 100 200 300 400 500 600 700 800 900 1000

0.0

0.5

1.0 β

0 100 200 300 400 500 600 700 800 900 1000-1

0

1ρ


Summaries of VAR coefficient matrix

Largest eigenvalue

0.5

0.6

0.7

0.8

11/03

11/03

05/05 12/06 07/08 02/10 09/11

(1) Reserves avg. (2) Crisis starts (3) Lehman (4) QE

Largest eigenvalue

Diagonals Off-diagonals Abs. off-diagonals

0.0

2.5

5.0

7.5

05/05 12/06 07/08 02/10 09/11

Diagonals Off-diagonals Abs. off-diagonals


Diagnostics - generalized residuals

ACF-Var1 PACF-Var1

0 5 10 15 20

0

1ACF-Var1 PACF-Var1 ACF-Var2 PACF-Var2

0 5 10 15 20

0

1ACF-Var2 PACF-Var2

ACF-Var1 PACF-Var1

0 5 10 15 20

0


0 5 10 15 20

0

1ACF-Var2 PACF-Var2

ACF-Var1 PACF-Var1

0 5 10 15 20

0


0 5 10 15 20

0

1ACF-Var2 PACF-Var2

ACF-Var1 PACF-Var1

0 5 10 15 20

0


0 5 10 15 20

0

1ACF-Var2 PACF-Var2

ACF-Var1 PACF-Var1

0 5 10 15 20

0


0 5 10 15 20

0

1ACF-Var2 PACF-Var2

ACF-Var1 PACF-Var1

0 5 10 15 20

0


0 5 10 15 20

0

1ACF-Var2 PACF-Var2


A diversion: How to get tuning parameters

I In all the above the bandwidth was taken as given and achoice based on random coefficients and MSE optimality wassuggested.

I But one may wish to get a data dependent choice.

I Further, there may be other tuning parameters. For example,tuning parameters associated with the estimation of largecovariance matrices.

I Giraitis, Kapetanios and Price (JoE, 2013) have addressedthis in the context of forecasting under structural change.


A diversion: How to get tuning parameters

I They derive cross-validation approaches to determine the rateat which data should be downweighted when forecasting inthe presence of structural change.

I They prove that one can retrieve the optimal bandwidth thatminimises the MSE of forecasts under a variety of structuralchange settings.

I And show that such a strategy can be very effective forforecasting US spreads and price indices.

I We conjecture that such a strategy would work fordetermining more than one tuning parameters simultaneously.


Time Varying Estimation of Large Dimensional Covariances

I We can use the above methods to investigate the estimationof large dimensional covariance matrices in the presence ofstructural change.

I Letyt = Ht−1εt , E [ut |Ft−1] = 0 (11)

with respect to some filtration Ft , where Ht = {ht,ij} is anp × p time varying volatility process, and εt = (ε1,t , ..., εp,t)

′

is a vector-valued standardized α-mixing process, such thatEεt = 0, Eεtε

′t = I.

I Denote by Σt = [σij ,t ] = Ht−1H′t−1 = E [utu′t |Ft−1] the

conditional variance-covariance matrix, where Ht isFt-measurable with respect to some filtration Ft .



I We assumeh4t,ij ≤ C , for1 ≤ k ≤ t/2 (12)

Further, for 1 ≤ k ≤ t/2 and some ϑ > 0,

max1≤i ,j≤N

max1≤s≤k

|σij ,t − σij ,t+s |2 = Oa.s.(d(N) (k/t)ϑ). (13)

I Assume further that

supi

Pr [|εi ,t | > a] ≤ C1e−C2aq , q > 1 (14)



I Define

Σt = [σij ,t ] = L−1t

n∑j=1

ltjyjy′j , ltj := L(

t − j

H), Lt :=

n∑j=1

ltj ,

(15)

I Then we have the following theorem

TheoremUnder (12)-(14), and for any 0 < δ < 1, we have that

maxij|σij ,t − σij ,t | = Op

((log p)(q+1)/q

H1−(1+δ)/2

)(16)



I (16) is obtained under weaker than usual assumptions. Weneed to allow for heterogeneity over time and we also allow formixing whereas usually an iid assumption is made. As a resultof our weaker assumptions as well as time variation a slowerrate that usual is obtained. See Theorems 3.3 and 3.4 ofWhite and Wooldridge (1991).

I The full sample version of (16) forms the core of a variety ofresults on estimation of Σ in the case of full sampleestimation. So (16) can be used to replicate all these resultsfor the time varying case with minimal further modifications.



I Examples of full sample estimation include the work of Ledoitand Wolf on shrinkage estimators of Σ, and the work of Bickeland Levina and Cai and Liu on threshold estimators of Σ.

I It is worth mentioning that none of the above allow for non-iiddata in their derivations. However, the mixing assumption wemake is not novel for the full sample estimation since Fan,Liao and Micheva (2013) make it, although they base theirwork on an exponential inequality in a recent working paperand do not seem to be aware of either White and Wooldridge(1991) or the work on exponential inequalities cited therein.


Ledoit and Wolf (2003, 2004)

I Relies on an asymptotically optimal shrinkage approach

ΣLW = ρ1Σt arg et + ρ2Σ (17)

Σt arg et :variance target estimator, Σ :full sample covarianceestimator, ρ1, ρ2, µ are positive constants. When the Σt arg et

is the identity (Σt arg et = I ) the optimal weights are given

by:ρ1 = mTb2T/d2

T , ρ2 = a2T/d2

T , mT = N−1tr(

Σ)

,

d2T = N−1tr

(Σ2)−m2

T , a2T = d2

T − b2T , b2

T = min(

b2, d2

T

)and b

2T = 1

NT 2

∑Tt=1

(∑Ni=1 (xit − x i )

2)− 1

NT tr(

Σ2)

I One could use several candidates for the Σt arg et , such as the

diagonal of Σ

ΣLW = ρ1diag(

Σ)

+ ρ2Σ (18)


Cai and Liu (2013)

The adaptive thresholding estimator:

ΣCL (δ) = (σij)N×N with σij = sλij (σij) , (19)

σij is the i , j element of the full sample covariance matrix estimate

Σsλij is a thresholding rule (hard, soft, adaptive lasso)

λij := λij (δ) = δ

√θij log N

T

and

θij =1

T

T∑t=1

[(xit − x i ) (xjt − x j)− σij ]2 (20)

δ is a regularization parameter that can be fixed at δ = 2 or chosenthrough cross validation


A theorem on the time varying Cai-Liu estimator

TheoremLet {Σt}∞t=1 belong to U (q, c0(N),M) and let our Assumptions

hold. Let λij ,t = κθ1/2ij ,t

(log Hκ)1/2(log N)(q+1)/q

H1−(1+δ)/2 , for any 0 < δ < 1,

some finite κ, where H = H (log Hκ)1/2 for some κ > 1. Let

H = o(

d(N)−1ϑ+1 T

ϑϑ+1

). Then,

∥∥∥Tλij

(Σt

)− Σt

∥∥∥ = Op

(c0(N)

(log Hκ)1/2 (log N)(q+1)/q

H1−(1+δ)/2

), for all t.

(21)Further, let {Σt}∞t=1 belong to U (q, c0(N),M, ε), for some ε > 0.Then, for any 0 < δ < 1,∥∥∥∥Tλij

(Σt

)−1− Σ−1

t

∥∥∥∥ = Op

(c0(N)

(log Hκ)1/2 (log N)(q+1)/q

H1−(1+δ)/2

), for all t.

(22)Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management

Relaxing our assumptions

The Theorem holds under restrictive conditions. We impose bothmixing and exponentially declining tails. We can relax both at thecost of lower rates. The following Corollary gives this result.

Corollary

Let {Σt}∞t=1 belong to U (q, c0(N),M) and let our Assumptions

hold. Let εt have finite eighth moments. Let λij ,t = κθ1/2ij ,t NH−η,

for any 0 < η < 1/2, and some finite κ. Let H = H (log Hκ)1/2 for

some κ > 1, and H = o(

d(N)−1

2η+ϑTϑ

2η+ϑN2

2η+ϑ

).Then, for all

η < 1/2, ∥∥∥Tλij

(Σt

)− Σt

∥∥∥ = op(c0(N)NH−η

), for all t. (23)


Evaluation Exercises

I We carry out a Monte Carlo and an empirical exercise

I The Monte Carlo considers the estimation of the time varyingcovariance matrix both within the sample and at the end ofthe sample.

I The empirical exercise focuses on the construction ofminimum variance portfolios and considers their out-of-sampleperformance.


Minimum Variance portfolio

I The global minimum variance (GMV) portfolio is the portfoliowhich is designed to minimize investors exposure to risk

wt := w(

Σt

)=

Σ−1t 1N

1′NΣ−1t 1N

I Cross Validation is designed to minimize the out of samplevariance of the portfolio.


Cross Validation Objective Functions

To determine to bandwidth (h), shrinkage (ρ), and thresholding(δ) coefficients, we rely on cross validation.Given a sample of size T , of N returns {Rt}Tt=1, we select theparameter of interest (θ = (h, ρ, δ)) by numerically minimizing theobjective function, in the sample. The objective function could bethe portfolio variance

QovT ,θ :=

1

Tn

T∑τ=T0

wθ,τ |τ−1Rτ −1

Tn

T∑τ=T0

wθ,τ |τ−1Rτ

2

(24)

where wθ,τ |τ−1 is the GMV portfolio computed using the dataavailable up to time τ -1, T0 = o (T ) and Tn := T − T0 + 1. Or itcould be the MSE of the covariance matrix estimator

QocT ,θ :=

1

Tn

T∑τ=T0

||Σθ,−τ − (Rτ − µτ )′ (Rτ − µτ ) || (25)

where Σθ,−τ obtained by dropping data at time τ .Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management

Time Varying Estimation of Large DimensionalCovariances: Monte Carlo

I To generate Σt we do the following

I First, we draw T N × 1 vectors bt= (b1t , b2t , ..., bNt)′ where

for i ≤ Nb (< N), Nb =[Ndb

],

bit =

(2.5bit

max1≤j≤t bij

)+ 2.5

where bit = bit−1 + ξit and ξit ∼ niid(0.1, 1). For i > Nb, weset bit = 0.



I Second, let

hit =

10hit

max1≤j≤t

∣∣∣hit

∣∣∣+ 10, i = 1, ..,N

where hit = hit−1 + ηit and ηit ∼ niid(0.1, 1). This ensuresthat hit , is bounded between 0 and 20.

I Third, sample an N × 1 vector d = (d1, d2, .., dN), di˜χ22.

Then, define T N × 1 vectors et = (e1t , e2t , .., eNt),eit = hitdi . Then, compute the N × N matrix Dt = diag (et).Set

Σt = Dt + b′tbt

I Finally, denote Σt = [σijt ], set Σt = [σijt√σii1σjj1

] and generate

data as yt = Σ1/2t εt where εt ∼ N(0, I ).



I We consider db = 0.5, 0.75, 1, N = 10, 40, 100 andT = 100, 200, 400.

I We use two cross-validation exercises.

I One focuses on in-sample estimation and uses a leave-one out

approach in minimising ||Σ(−1)t − yty

′t ||, where ||.|| denotes

the Frobenius norm, to determine H and any other tuningparameters, using the whole sample. The notation .(−1)

indicates the leave-one out estimator.

I The other focuses on pseudo out-of-sample estimation, morerelevant for portfolio optimisation, and minimises||Σt − yt+1y

′t+1|| over the last 20 periods of the sample.

I The above define estimation and forecasting strategies thatare then evaluated using ||Σt −Σt || and ||Σt −Σt+1||,respectively as the evaluation criterion.


A summary of the Monte Carlo results

Whole sample

N 10 10 10 50 50 50 10 10 10 50 50 50T 200 200 200 200 200 200 400 400 400 400 400 400db 1 0.75 0.5 1 0.75 0.5 1 0.75 0.5 1 0.75 0.5

TV methods 10 10 10 10 10 8 10 10 10 10 10 10Whole Sample methods 0 0 0 0 0 2 0 0 0 0 0 0

TV CL 0 6 8 0 10 8 2 7 9 1 9 10TV FAN 0 0 0 0 0 0 0 0 0 0 0 0TV LW 5 3 2 5 0 0 5 2 1 5 1 0

TV sample covariance 5 1 0 5 0 0 3 1 0 4 0 0TV method with h=.5 0 0 0 0 0 0 0 0 0 0 0 0TV method with h=.6 1 0 0 0 0 0 0 0 1 0 0 1TV method with h=.7 2 2 1 2 1 1 2 3 1 2 1 1TV method with h=.8 4 4 4 4 3 2 6 3 3 4 3 2TV method with h=.9 2 3 2 2 3 2 1 2 2 1 3 3

TV CV methods 1 1 3 2 3 3 1 2 3 3 3 3


A summary of the Monte Carlo results

End of sample

N 10 10 10 50 50 50 10 10 10 50 50 50T 200 200 200 200 200 200 400 400 400 400 400 400db 1 0.75 0.5 1 0.75 0.5 1 0.75 0.5 1 0.75 0.5

TV methods 10 10 10 10 10 6 10 10 10 10 10 7Fixed methods 0 0 0 0 0 4 0 0 0 0 0 3

TV CL 2 5 8 1 9 6 3 5 8 3 9 7TV FAN 0 0 0 0 0 0 0 0 0 0 0 0TV LW 4 2 2 4 0 0 3 1 1 3 0 0

TV sample covariance 4 3 0 5 1 0 4 4 1 4 1 0TV method with h=.5 0 0 0 0 0 0 0 0 0 0 0 0TV method with h=.6 0 0 0 0 0 0 0 0 0 0 0 0TV method with h=.7 2 3 1 2 1 1 4 3 1 3 1 1TV method with h=.8 5 3 2 5 3 1 6 5 2 6 3 1TV method with h=.9 3 4 4 3 3 1 0 2 4 1 3 2

TV CV methods 0 0 3 0 3 3 0 0 3 0 3 3


Insample Simulations

T=200 N=10 db=0.5

||Σt − Σtruet ||

TV-n-h=.7-CAI-h 0.48TV-n-h=.8-CAI-h 0.486TV-n-h=.9-CAI-h 0.62

TV-n-h=.8-LW 0.634TV-n-h=.7-LW 0.638

TV-n-h=.6-CAI-h 0.671TV-n-h=.95-CAI-h 0.681

TV-n-h=.8 0.738TV-n-oc 0.739

TV-e-h=.9 0.745CAI-h 0.794

CAI-h-oc 0.85LW 0.927

LW-oc 0.927sample-estimate 1

T=200 N=40 db=0.5



TV-n-h=.95-CAI-h 0.441CAI-h 0.495

CAI-h-oc 0.499CAI-al 0.537

TV-n-h=.6-CAI-h 0.611TV-n-h=.8-LW 0.647TV-n-h=.9-LW 0.666

TV-n-h=.95-LW 0.687LW 0.738

LW-oc 0.739TV-n-oc 0.908

sample-estimate 1

T=200 N=100 db=0.5



TV-n-h=.95-CAI-h 0.235CAI-h-oc 0.258

CAI-h 0.268CAI-al 0.301

TV-n-h=.9-LW 0.46TV-n-h=.95-LW 0.467

TV-n-h=.6-CAI-h 0.477TV-n-h=.8-LW 0.493

LW 0.495LW-oc 0.495

TV-n-oc 0.983sample-estimate 1



T=200 N=10 db=.75


TV-n-oc 0.394TV-n-h=.5-LW 0.442

TV-n-h=.5-CAI-h 0.444TV-n-h=.5 0.456

TV-n-h=.6-CAI-h 0.469TV-n-h=.6-LW 0.473

TV-e-h=.6 0.477TV-n-h=.6 0.485TV-e-h=.7 0.507

TV-n-h=.7-CAI-h 0.531CAI-h 0.991

CAI-h-oc 0.993LW 0.996


T=200 N=40 db=0.75



TV-n-h=.8-LW 0.695TV-n-h=.7-LW 0.717


TV-n-oc 0.753TV-n-h=.8 0.758TV-e-h=.9 0.768

TV-e-h=.95 0.787CAI-h-oc 0.839

CAI-h 0.909LW-oc 0.959

LW 0.96sample-estimate 1

T=200 N=100 db=0.75



CAI-h-oc 0.72TV-n-h=.6-CAI-h 0.732

TV-n-h=.8-LW 0.737TV-n-h=.95-CAI-h 0.771

TV-n-h=.9-LW 0.801TV-n-h=.7-LW 0.812

TV-n-h=.95-LW 0.841TV-n-oc 0.848

CAI-h 0.87LW 0.927




T=200 N=10 db=1


TV-n-oc 0.446TV-n-h=.5 0.47

TV-n-h=.5-LW 0.476TV-e-h=.6 0.48TV-n-h=.6 0.505

TV-n-h=.6-LW 0.508TV-e-h=.7 0.51

TV-n-h=.5-CAI-h 0.514TV-e-h=.5 0.524TV-f-h=.6 0.528

sample-estimate 1CAI-h-oc 1

LW 1.002LW-oc 1.004CAI-h 1.025

T=200 N=40 db=1


TV-n-h=.6-LW 0.46TV-n-oc 0.461

TV-n-h=.6 0.464TV-e-h=.8 0.479

TV-n-h=.7-LW 0.48TV-n-h=.7 0.486TV-e-h=.7 0.497TV-f-h=.7 0.509

TV-n-h=.5-LW 0.552TV-n-h=.5 0.558

LW-oc 0.995LW 0.996

sample-estimate 1CAI-h-oc 1.001

CAI-h 1.203

T=200 N=100 db=1


TV-n-h=.6 0.542TV-e-h=.8 0.548TV-n-h=.7 0.553

TV-n-oc 0.555TV-n-h=.6-LW 0.562TV-n-h=.7-LW 0.574

TV-e-h=.7 0.576TV-f-h=.7 0.583TV-f-h=.8 0.617TV-e-h=.9 0.639

sample-estimate 1CAI-h-oc 1.001

LW 1.025LW-oc 1.03CAI-h 1.55



T=400 N=10 db=0.5



TV-n-h=.8-LW 0.532TV-n-h=.7-LW 0.535

TV-n-oc 0.567TV-e-h=.9 0.584TV-n-h=.8 0.589


TV-n-h=.7 0.62CAI-h 0.839

CAI-h-oc 0.89LW 0.967


T=400 N=40 db=0.5



TV-n-h=.95-CAI-h 0.445CAI-h 0.544CAI-al 0.558

CAI-h-oc 0.559TV-n-h=.8-LW 0.639TV-n-h=.9-LW 0.683

TV-n-h=.95-LW 0.733TV-n-h=.6-CAI-h 0.753

LW 0.832LW-oc 0.832

TV-n-oc 0.848sample-estimate 1

T=400 N=100 db=0.5



TV-n-h=.95-CAI-h 0.271CAI-h-oc 0.328

CAI-h 0.329CAI-al 0.346

TV-n-h=.9-LW 0.569TV-n-h=.95-LW 0.592TV-n-h=.8-LW 0.602

LW 0.649LW-oc 0.649

TV-n-h=.6-CAI-h 0.706TV-n-oc 0.95

sample-estimate 1



T=400 N=10 db=0.75


TV-n-h=.7-CAI-h 0.355TV-n-h=.7-LW 0.398

TV-n-oc 0.402TV-n-h=.7 0.406TV-e-h=.8 0.423

TV-n-h=.8-CAI-h 0.426TV-f-h=.8 0.446

TV-n-h=.6-CAI-h 0.451TV-e-h=.9 0.456

TV-n-h=.8-LW 0.467CAI-h 0.966

CAI-h-oc 0.975sample-estimate 1

LW 1.001LW-oc 1.002

T=400 N=40 db=0.75



TV-n-h=.8-LW 0.526TV-n-oc 0.544

TV-n-h=.7-LW 0.552TV-e-h=.9 0.553

TV-n-h=.6-CAI-h 0.559TV-n-h=.8 0.559TV-n-h=.7 0.593TV-f-h=.8 0.613CAI-h-oc 0.875

CAI-h 0.886LW 0.983


400.0000 N=100 db=0.75


TV-n-h=.7-CAI-h 0.399TV-n-h=.8-CAI-h 0.402TV-n-h=.9-CAI-h 0.58TV-n-h=.6-CAI-h 0.628

TV-n-h=.8-LW 0.636TV-n-h=.95-CAI-h 0.665

TV-n-h=.8 0.681TV-n-oc 0.682

TV-e-h=.9 0.691TV-e-h=.95 0.715

CAI-h-oc 0.764CAI-h 0.81

LW 0.982LW-oc 0.984

sample-estimate 1



T=400 N=10 db=1


TV-n-h=.7-LW 0.293TV-n-h=.7 0.297TV-e-h=.8 0.299

TV-n-oc 0.304TV-n-h=.6-LW 0.305

TV-n-h=.6 0.307TV-f-h=.7 0.324

TV-n-h=.7-CAI-h 0.328TV-e-h=.7 0.331TV-f-h=.8 0.351

LW 1sample-estimate 1

LW-oc 1CAI-h-oc 1.001

CAI-h 1.018

T=400 N=40 db=1


TV-n-h=.7-LW 0.329TV-n-h=.7 0.332

TV-n-oc 0.335TV-e-h=.8 0.337

TV-n-h=.6-LW 0.356TV-n-h=.6 0.357TV-f-h=.7 0.378TV-f-h=.8 0.381TV-e-h=.7 0.387TV-e-h=.9 0.422

LW 1sample-estimate 1

LW-oc 1CAI-h-oc 1

CAI-h 1.069

T=400 N=100 db=1


TV-n-h=.7-LW 0.345TV-n-h=.7 0.346

TV-n-oc 0.348TV-e-h=.8 0.35

TV-n-h=.6-LW 0.369TV-n-h=.6 0.369TV-f-h=.7 0.388TV-e-h=.7 0.399TV-f-h=.8 0.4TV-e-h=.9 0.44

LW-oc 1sample-estimate 1

CAI-h-oc 1LW 1.001

CAI-h 1.127


Out of Sample Simulations

T=200 N=10 db=0.5


TV-n-CAI-h-oc 0.59TV-n-h=.8-LW 0.67

TV-e-h=.95 0.69TV-n-h=.8 0.69TV-f-h=.9 0.72TV-e-h=.9 0.72TV-n-h=.9 0.74TV-f-h=.95 0.75

CAI-h 0.76TV-n-oc 0.76

TV-n-h=.7-LW 0.76TV-n-h=.9-LW 0.77

TV-n-h=.95 0.78TV-n-LW(I)-oc 0.78

TV-n-LW-oc 0.79LW 0.94

sample estimate 1LW-oc 1.21

T=200 N=40 db=0.5


TV-n-CAI-h-oc 0.46CAI-h 0.61CAI-al 0.73

TV-n-h=.9-LW 0.82TV-n-LW-oc 0.82

TV-n-LW(I)-oc 0.85TV-n-h=.95-LW 0.85TV-n-h=.8-LW 0.87

TV-n-h=.95 0.93TV-n-oc 0.93

LW 0.93TV-n-h=.9 0.95TV-f-h=.95 0.99


TV-e-h=.95 1.11TV-f-h=.9 1.15TV-n-h=.8 1.16

T=200 N=100 db=0.5



TV-n-LW-oc 0.7TV-n-h=.9-LW 0.7

TV-n-h=.95-LW 0.71TV-n-LW(I)-oc 0.72

LW 0.76TV-n-h=.8-LW 0.89

LW-oc 0.9TV-n-oc 1

sample estimate 1TV-n-h=.95 1.03TV-n-h=.9 1.1TV-f-h=.95 1.17

CAI-s 1.25TV-e-h=.95 1.43TV-f-h=.9 1.48



T=200 N=10 db=0.75


TV-e-h=.9 0.41TV-n-h=.7 0.42

TV-n-h=.7-LW 0.43TV-n-h=.8 0.43TV-f-h=.8 0.44

TV-e-h=.95 0.45TV-e-h=.8 0.46

TV-n-h=.8-LW 0.46TV-n-oc 0.46

TV-n-CAI-h-oc 0.46TV-f-h=.9 0.46

TV-n-LW-oc 0.5TV-n-LW(I)-oc 0.51

TV-n-h=.9 0.52CAI-h 0.68

LW 0.73sample estimate 1

LW-oc 1.18

T=200 N=40 db=0.75


TV-n-CAI-h-oc 0.59TV-n-h=.8 0.67TV-e-h=.95 0.68TV-e-h=.9 0.69

TV-n-h=.8-LW 0.71TV-f-h=.9 0.71TV-n-oc 0.72

TV-n-h=.9 0.72TV-f-h=.95 0.75TV-n-h=.95 0.76


TV-n-LW(I)-oc 0.78TV-n-h=.7 0.79

CAI-h 0.89LW 0.97


T=200 N=100 db=0.75


TV-n-CAI-h-oc 0.61TV-n-h=.9 0.79

TV-n-h=.95 0.8TV-e-h=.95 0.81TV-n-h=.8 0.81

TV-n-oc 0.82TV-f-h=.95 0.82

TV-n-h=.8-LW 0.83TV-f-h=.9 0.85TV-e-h=.9 0.88


TV-n-LW(I)-oc 0.89TV-n-h=.95-LW 0.93

CAI-h 0.95sample estimate 1

LW 1.02LW-oc 1.62



T=200 N=10 db=1


TV-n-h=.7 0.47TV-n-h=.7-LW 0.48

TV-e-h=.8 0.49TV-e-h=.9 0.51TV-f-h=.8 0.51TV-n-h=.6 0.53

TV-n-h=.6-LW 0.53TV-n-h=.8 0.54TV-f-h=.7 0.56

TV-n-h=.8-LW 0.57TV-n-oc 0.58

TV-n-CAI-h-oc 0.62TV-n-LW-oc 0.63

TV-n-LW(I)-oc 0.64LW 0.96

CAI-h 1sample estimate 1

LW-oc 1.89

T=200 N=40 db=1


TV-n-h=.7 0.45TV-n-h=.7-LW 0.45

TV-e-h=.8 0.45TV-f-h=.8 0.48TV-n-h=.6 0.49

TV-n-h=.6-LW 0.49TV-e-h=.9 0.5TV-f-h=.7 0.52TV-n-oc 0.53

TV-n-h=.8 0.53TV-n-CAI-h-oc 0.53TV-n-h=.8-LW 0.54


LW 0.89sample estimate 1

CAI-h 1.04LW-oc 1.77

T=200 N=100 db=1


TV-e-h=.8 0.54TV-n-h=.6 0.54TV-n-h=.7 0.54

TV-n-h=.6-LW 0.55TV-n-h=.7-LW 0.56

TV-f-h=.7 0.57TV-e-h=.7 0.58TV-f-h=.8 0.58TV-e-h=.9 0.6

TV-n-oc 0.62TV-n-CAI-h-oc 0.63

TV-n-h=.8 0.64TV-n-LW-oc 0.67

TV-n-LW(I)-oc 0.67LW 0.98

sample estimate 1CAI-h 1.34LW-oc 1.79



T=400 N=10 db=0.5


TV-n-CAI-h-oc 0.41TV-e-h=.9 0.52TV-n-h=.8 0.52

TV-n-h=.8-LW 0.53TV-e-h=.95 0.55

TV-n-h=.7-LW 0.57TV-f-h=.9 0.57TV-n-oc 0.6

TV-f-h=.8 0.6TV-n-h=.7 0.6

TV-n-LW(I)-oc 0.61TV-n-LW-oc 0.62TV-e-h=.8 0.66TV-n-h=.9 0.68

CAI-h 0.85LW 0.99


T=400 N=40 db=0.5



TV-n-h=.8-LW 0.81TV-n-h=.9 0.82

TV-n-h=.9-LW 0.83TV-n-oc 0.84

TV-n-h=.95 0.84TV-n-LW(I)-oc 0.84

TV-f-h=.95 0.84TV-n-LW-oc 0.84TV-e-h=.95 0.85

TV-n-h=.95-LW 0.9TV-f-h=.9 0.9TV-n-h=.8 0.92

sample estimate 1LW 1.03

LW-oc 1.36

T=400 N=100 db=0.5



TV-n-h=.9-LW 0.86TV-n-h=.95-LW 0.89TV-n-LW(I)-oc 0.89

TV-n-LW-oc 0.89TV-n-oc 0.96

TV-n-h=.95 0.97LW 0.99

sample estimate 1TV-n-h=.9 1.01

TV-n-h=.8-LW 1.04TV-f-h=.95 1.06

LW-oc 1.21TV-e-h=.95 1.23TV-f-h=.9 1.31TV-n-h=.8 1.42



T=400 N=10 db=0.75


TV-n-h=.7 0.36TV-n-h=.7-LW 0.36

TV-f-h=.8 0.37TV-e-h=.9 0.37TV-e-h=.8 0.37TV-n-h=.8 0.4

TV-n-CAI-h-oc 0.41TV-n-h=.8-LW 0.42

TV-n-oc 0.43TV-e-h=.95 0.45TV-f-h=.9 0.46

TV-n-LW-oc 0.47TV-n-LW(I)-oc 0.48TV-n-h=.6-LW 0.48

CAI-h 0.81LW 0.87


T=400 N=40 db=0.75


TV-n-CAI-h-oc 0.45TV-e-h=.9 0.52TV-n-h=.8 0.53

TV-n-h=.8-LW 0.56TV-e-h=.95 0.56TV-f-h=.9 0.58TV-f-h=.8 0.59

TV-n-h=.7-LW 0.59TV-n-h=.7 0.59

TV-n-oc 0.61TV-n-LW-oc 0.64

TV-n-LW(I)-oc 0.65TV-e-h=.8 0.65TV-n-h=.9 0.67

CAI-h 0.86LW 0.98


T=400 N=100 db=0.75


TV-n-CAI-h-oc 0.5TV-n-h=.8 0.66TV-e-h=.95 0.67TV-e-h=.9 0.68

TV-n-h=.8-LW 0.69TV-f-h=.9 0.71TV-n-h=.9 0.75

TV-n-oc 0.75TV-f-h=.95 0.77

TV-n-LW(I)-oc 0.8TV-n-LW-oc 0.8TV-f-h=.8 0.81

TV-n-h=.95 0.81TV-n-h=.7-LW 0.82

CAI-h 0.9sample estimate 1

LW 1.07LW-oc 2.16



T=400 N=10 db=1


TV-n-h=.7 0.3TV-e-h=.8 0.3

TV-n-h=.7-LW 0.3TV-f-h=.8 0.32TV-f-h=.7 0.34TV-n-h=.6 0.34

TV-n-h=.6-LW 0.35TV-e-h=.9 0.36TV-e-h=.7 0.37TV-n-h=.8 0.39



CAI-h 0.91LW 0.92


T=400 N=40 db=1


TV-e-h=.8 0.3TV-n-h=.7 0.31

TV-n-h=.7-LW 0.31TV-n-h=.6 0.33

TV-n-h=.6-LW 0.33TV-f-h=.7 0.34TV-f-h=.8 0.34TV-e-h=.7 0.36


TV-e-h=.9 0.38TV-n-LW-oc 0.4

TV-n-LW(I)-oc 0.4TV-n-h=.8 0.42

LW 0.96CAI-h 0.98


T=400 N=100 db=1


TV-n-h=.7 0.34TV-e-h=.8 0.35

TV-n-h=.7-LW 0.35TV-f-h=.8 0.38TV-n-h=.6 0.39TV-f-h=.7 0.39

TV-n-h=.6-LW 0.39TV-e-h=.9 0.42TV-e-h=.7 0.42

TV-n-CAI-h-oc 0.44TV-n-oc 0.44

TV-n-h=.8 0.45TV-n-LW(I)-oc 0.47

TV-n-LW-oc 0.47LW 0.96

sample estimate 1CAI-h 1.03LW-oc 2.32


Data Description

Portfolio Time Period out of sample observations5 industry portfolios 03/63 to 12/13 (609 obs) from 01/06 to 12/13 (95 obs)10 industry portfolios 03/63 to 12/13 (609 obs) from 01/06 to 12/13 (95 obs)17 industry portfolios 03/63 to 12/13 (609 obs) from 01/06 to 12/13 (95 obs)30 industry portfolios 03/63 to 12/13 (609 obs) from 01/06 to 12/13 (95 obs)6 size and book to market portfolios 03/63 to 12/13 (609 obs) from 01/06 to 12/13 (95 obs)25 size and book to market portfolios 03/63 to 12/13 (609 obs) from 01/06 to 12/13 (95 obs)The 5 S&P500 stocks with the highestaverage capitalization

03/93 to 12/13 (219 obs) from 01/09 to 12/13 (60 obs)

The 10 S&P500 stocks with the highestaverage capitalization

03/93 to 12/13 (219 obs) from 01/09 to 12/13 (60 obs)


03/93 to 12/13 (219 obs) from 01/09 to 12/13 (60 obs)


03/93 to 12/13 (219 obs) from 01/09 to 12/13 (60 obs)


03/93 to 12/13 (219 obs) from 01/09 to 12/13 (60 obs)

Performance Criteria :

var =1

Tn

T∑τ=T0

wθ,τ|τ−1Rτ −1

Tn

T∑τ=T0

wθ,τ|τ−1Rτ

2

and SR =T∑

τ=T0

wθ,τ|τ−1Rτ

/var

T0 = o (T ) , Tn := T − T0 + 1. Tn = 60 observations for the S&P500 stocks, Tn = 95 for the remainingportfolios


A summary of the empirical results

variance

N 5 10 12 17 30 48 6 25 5 10 20 30 40 50T 960 960 960 960 960 960 963 963 250 250 250 250 250 250

TV methods 10 10 10 10 10 7 10 9 10 10 10 9 9 10Fixed methods 0 0 0 0 0 3 0 1 0 0 0 1 1 0

TV CL 2 3 2 0 0 0 3 2 3 0 0 0 0 0TV FAN 0 2 0 1 0 1 1 0 0 0 8 1 3 0TV LW 2 1 2 3 4 4 1 3 0 1 2 1 3 1

TV sample covariance 3 3 5 3 3 1 2 0 4 6 0 3 1 4TV method with h=.5 1 0 0 0 0 0 3 0 1 0 0 0 0 0TV method with h=.6 4 5 1 3 1 0 0 3 1 1 2 0 0 0TV method with h=.7 0 0 2 6 3 1 2 5 0 2 1 0 1 0TV method with h=.8 0 3 5 0 6 1 0 0 1 2 4 3 3 1TV method with h=.9 0 0 0 0 0 3 0 0 1 0 1 2 3 3

TV CV methods 5 2 2 1 0 2 5 1 6 5 2 4 2 6TV shrink the inverse 2 0 0 0 0 0 2 0 0 2 0 1 0 1

TV LW 1 factor 1 1 1 3 3 1 1 4 3 1 0 3 2 4



sharp ratio

N 5 10 12 17 30 48 6 25 5 10 20 30 40 50T 960 960 960 960 960 960 963 963 250 250 250 250 250 250


TV CL 3 7 4 0 2 1 5 4 5 6 9 9 9 10TV FAN 3 2 5 6 0 4 2 5 2 0 0 0 1 0TV LW 0 0 0 0 0 0 0 0 0 1 1 0 0 0



TV LW 1 factor 0 0 0 0 0 0 0 0 0 0 0 0 0 0



turnover

N 5 10 12 17 30 48 6 25 5 10 20 30 40 50T 960 960 960 960 960 960 963 963 250 250 250 250 250 250


TV CL 0 0 1 1 2 0 0 1 1 0 0 0 0 0TV FAN 0 0 0 0 0 0 0 0 0 0 2 2 4 4TV LW 1 1 1 1 1 3 0 1 1 1 1 1 1 1



TV LW 1 factor 1 1 1 1 1 1 0 1 1 1 1 1 1 1



certainty equivalent

N 5 10 12 17 30 48 6 25 5 10 20 30 40 50T 960 960 960 960 960 960 963 963 250 250 250 250 250 250


TV CL 2 3 2 0 0 1 2 4 2 0 0 0 0 0TV FAN 0 2 0 1 1 2 2 0 0 0 7 2 4 0TV LW 1 0 2 2 3 2 1 2 0 1 2 1 2 1



TV LW 1 factor 1 0 1 3 3 1 1 3 3 1 0 2 2 4


Out of Sample Empirical Results

N=5, Industry Portfolios

var SR

TV-e-h=.6 0.68 1.11TV-f-h=.5 0.681 1.119

TV-e-ov 0.685 1.11TV-n-ov 0.685 1.081

TV-n-h=.5 0.685 1.081TV-n-LW-ov 0.692 1.075

TV-n-LW(I)-ov 0.693 1.068TV-ok-ov 0.694 1.078TV-n-oc 0.728 0.72TV-ok-oc 0.745 0.785

TV-n-LW-oc 0.777 0.643TV-n-LW(I)-oc 0.789 0.636

LW-oc 0.951 0.734LW 0.992 0.972

CAI-h-ov 1 1CAI-h-oc 1 1

CAI-h 1 1sample estimate 1 1

LW-ov 1.071 0.641true 1.249 0.621

TV-n-CAI-h-oc 1.835 0.648TV-n-CAI-ov 2.103 0.404


var SR

TV-f-h=.8 0.705 1.144TV-f-oc 0.705 0.989

TV-e-h=.8 0.708 1.059TV-n-h=.7 0.712 1.031TV-n-h=.6 0.768 0.831

TV-n-CAI-ov 0.777 0.813TV-n-LW-ov 0.828 0.691

TV-n-ov 0.837 0.735TV-n-LW(I)-ov 0.839 0.717

TV-ok-oc 0.844 0.704TV-n-LW-oc 0.844 0.671

TV-n-LW(I)-oc 0.847 0.666TV-n-oc 0.848 0.75LW-oc 0.955 0.777

LW 0.987 0.978CAI-h-ov 1 1


LW-ov 1.072 0.621CAI-h-oc 1.409 0.889

TV-n-CAI-h-oc 1.496 0.5true 1.503 0.63


var SR

TV-e-h=.8 0.686 1.073TV-f-oc 0.696 1.083

TV-n-h=.7 0.697 1.057TV-f-h=.8 0.71 1.134

TV-n-h=.6-LW 0.727 0.918TV-n-LW-ov 0.749 0.876

TV-n-LW(I)-ov 0.763 0.865TV-n-ov 0.768 0.919

TV-n-LW(I)-oc 0.845 0.679TV-n-LW-oc 0.86 0.653

TV-n-oc 0.863 0.84TV-n-CAI-ov 0.876 0.807

TV-ok-oc 0.904 0.767LW-oc 0.908 0.782

LW 0.986 0.985CAI-h 1 1

sample estimate 1 1LW-ov 1.02 0.712

CAI-h-ov 1.051 0.918CAI-h-oc 1.95 0.103

true 2.06 0.57TV-n-CAI-h-oc 8.868 0.177




var SR

TV-f-h=.8 0.749 0.881TV-n-h=.7 0.771 0.743TV-e-h=.8 0.773 0.718TV-e-h=.9 0.784 0.873

TV-n-h=.8-LW 0.793 0.885TV-n-LW-ov 0.95 0.689

TV-n-LW(I)-ov 0.971 0.696LW 0.994 0.988

sample estimate 1 1CAI-h 1 1

CAI-h-ov 1 1TV-n-ov 1.011 0.599LW-oc 1.109 0.712LW-ov 1.114 0.725

TV-n-LW(I)-oc 1.219 0.496TV-n-LW-oc 1.245 0.503

TV-n-oc 1.302 0.484TV-ok-oc 1.379 0.469

true 2.559 0.58CAI-h-oc 2.902 0.091

TV-n-CAI-h-oc 3.06 0.698TV-n-CAI-ov 8.664 0.61

N=6, size-book to market portfolios

var SR

TV-n-h=.5 0.777 1.009TV-n-h=.6 0.78 1.054TV-e-h=.7 0.782 0.955TV-e-h=.6 0.789 1.04

TV-f-ov 0.79 0.975TV-n-ov 0.792 1.028

TV-n-LW-ov 0.794 1.025TV-n-LW(I)-ov 0.804 1.015

TV-n-oc 0.833 0.855TV-n-LW(I)-oc 0.851 0.729

TV-n-LW-oc 0.851 0.72TV-ok-oc 0.924 0.947

sample estimate 1 1CAI-h-ov 1 1CAI-h-oc 1 1

CAI-h 1 1LW-ov 1.002 1.002

LW 1.049 0.935LW-oc 1.128 0.798

true 1.467 0.793TV-n-CAI-ov 1.624 1.163

TV-n-CAI-h-oc NaN NaN

N=25, size-book to market portfolios

var SR

TV-n-LW(I)-ov 0.946 0.99TV-n-h=.6-LW 0.953 0.933

TV-n-LW-ov 0.977 0.781TV-e-h=.8 0.99 0.834TV-n-h=.7 0.995 0.874CAI-h-oc 0.998 1.194


CAI-h-ov 1 1.093TV-n-CAI-ov 1.006 1.162

TV-e-ov 1.007 0.797LW 1.009 0.953

TV-ok-ov 1.009 0.752TV-n-ov 1.014 0.862LW-ov 1.067 0.78LW-oc 1.211 0.592

TV-n-oc 1.532 0.565TV-n-LW-oc 1.576 0.28

TV-n-LW(I)-oc 1.585 0.285TV-ok-oc 1.647 0.861

true 1.869 0.877TV-n-CAI-h-oc 2.059 0.782



N=5, SP500

var SR

TV-n-h=.6-LW 0.65 1.03TV-n-LW-ov 0.69 0.98

TV-n-h=.5-LW 0.7 1.31TV-e-oc 0.7 0.76

TV-f-h=.7 0.71 0.89TV-n-h=.8 0.71 0.86

TV-n-LW-oc 0.72 0.67TV-n-oc 0.73 0.61TV-n-ov 0.74 0.79

TV-n-CAI-ov 0.74 0.78TV-ok-oc 0.74 0.61

TV-n-LW(I)-ov 0.83 0.8TV-n-LW(I)-oc 0.9 0.49

CAI-h-oc 0.96 1.02sample estimate 1 1

CAI-h-ov 1.2 1.34LW-oc 1.35 0.66LW-ov 2.04 0.6

LW 2.47 0.82true 5.09 0.88

CAI-h 64.97 0.35TV-n-CAI-h-oc NaN NaN

N=10, SP500

var SR

TV-f-h=.7 0.7 1.48TV-f-ov 0.72 1.5

TV-ok-ov 0.72 1.5TV-e-h=.8 0.74 1.13TV-f-h=.8 0.78 0.81

TV-n-LW(I)-ov 0.85 1.09TV-n-LW-ov 0.87 1.05

TV-n-LW(I)-oc 0.94 1.32TV-n-ov 0.96 0.98LW-ov 1 1.01

sample estimate 1 1LW-oc 1 1

CAI-h-oc 1 1LW 1.04 1.48

TV-n-LW-oc 1.16 0.91TV-n-oc 1.4 0.64TV-ok-oc 1.52 0.64

TV-n-CAI-ov 1.76 -0.2true 2.44 2.19

CAI-h-ov 2.76 -0.58CAI-h 40748.46 1.92

TV-n-CAI-h-oc NaN NaN

N=20, SP500

var SR

TV-n-LW(I)-ov 0.82 1.47LW 0.82 1.8

TV-n-h=.95-LW 0.82 1.64TV-n-h=.9-LW 0.83 1.54TV-n-h=.8-LW 0.85 1.28

TV-n-LW-ov 0.88 1.13TV-f-h=.95 0.93 1.06

LW-ov 0.93 1.22TV-n-h=.6-LW 0.93 1.23

TV-n-ov 0.98 1.03LW-oc 1 1

sample estimate 1 1CAI-h-oc 1 0.98CAI-h-ov 1.02 0.97

TV-n-CAI-ov 1.14 0.48TV-n-LW(I)-oc 1.36 1.35

TV-n-LW-oc 1.49 1.02true 1.64 2.59

TV-n-oc 1.84 1.1TV-ok-oc 3.26 0.96

CAI-h 12.39 -0.92TV-n-CAI-h-oc NaN NaN



N=30, SP500

TV-n-h=.9-LW 0.9 1.29TV-n-h=.95-LW 0.91 1.39

LW 0.91 1.56TV-n-h=.8-LW 0.92 0.94

TV-f-h=.95 0.96 0.98LW-ov 0.97 1.14

TV-n-h=.95 0.99 0.92LW-oc 1 1

sample estimate 1 1CAI-h-oc 1.02 0.96CAI-h-ov 1.13 0.82

TV-n-LW-ov 1.36 0.1TV-n-LW(I)-ov 1.37 0.28

TV-n-ov 1.49 0.14true 1.71 2.38

TV-n-CAI-ov 1.94 0.18TV-n-LW-oc 2.16 0

TV-n-LW(I)-oc 2.23 0.06TV-ok-oc 2.48 0.03TV-n-oc 2.99 -0.03

TV-n-CAI-h-oc 38.32 1.96CAI-h 596.25 1.33

N=40, SP500

TV-n-h=.9-LW 0.89 1.31TV-n-h=.95-LW 0.89 1.41

LW 0.92 1.57TV-n-h=.8-LW 0.92 0.92

LW-ov 0.96 1.07TV-n-LW(I)-ov 1 1.23

TV-n-h=.95 1 0.9LW-oc 1 1

sample estimate 1 1TV-n-h=.9 1.01 0.82CAI-h-oc 1.03 1.02TV-n-ov 1.07 1.08CAI-h-ov 1.22 0.82

TV-n-LW-ov 1.23 0.41TV-n-CAI-ov 1.42 0.19

true 1.8 2.57TV-n-LW(I)-oc 2.51 0.4

TV-n-LW-oc 2.68 0.08TV-ok-oc 3.38 0.19TV-n-oc 3.5 0.26

CAI-h 41.51 -1.35TV-n-CAI-h-oc 288.56 -0.84


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