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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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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,
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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).
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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).
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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)
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Time Varying Maximum Likelihood Estimation
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)
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Time Varying Maximum Likelihood Estimation
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
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
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
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
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
1ACF-Var1 PACF-Var1 ACF-Var2 PACF-Var2
0 5 10 15 20
0
1ACF-Var2 PACF-Var2
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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 .
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Time Varying Estimation of Large Dimensional Covariances
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)
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Time Varying Estimation of Large Dimensional Covariances
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)
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Time Varying Estimation of Large Dimensional Covariances
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Time Varying Estimation of Large Dimensional Covariances
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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)
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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)
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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.
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 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 ).
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 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.
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
||Σt − Σtruet ||
TV-n-h=.8-CAI-h 0.345TV-n-h=.7-CAI-h 0.363TV-n-h=.9-CAI-h 0.412
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
||Σt − Σtruet ||
TV-n-h=.8-CAI-h 0.184TV-n-h=.9-CAI-h 0.219TV-n-h=.7-CAI-h 0.225
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Insample Simulations
T=200 N=10 db=.75
||Σt − Σtruet ||
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
LW-oc 0.996sample-estimate 1
T=200 N=40 db=0.75
||Σt − Σtruet ||
TV-n-h=.7-CAI-h 0.584TV-n-h=.8-CAI-h 0.603
TV-n-h=.8-LW 0.695TV-n-h=.7-LW 0.717
TV-n-h=.6-CAI-h 0.727TV-n-h=.9-CAI-h 0.735
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
||Σt − Σtruet ||
TV-n-h=.7-CAI-h 0.584TV-n-h=.8-CAI-h 0.603TV-n-h=.9-CAI-h 0.72
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
LW-oc 0.927sample-estimate 1
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Insample Simulations
T=200 N=10 db=1
||Σt − Σtruet ||
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
||Σt − Σtruet ||
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
||Σt − Σtruet ||
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Insample Simulations
T=400 N=10 db=0.5
||Σt − Σtruet ||
TV-n-h=.7-CAI-h 0.378TV-n-h=.8-CAI-h 0.394
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=.9-CAI-h 0.594TV-n-h=.6-CAI-h 0.604
TV-n-h=.7 0.62CAI-h 0.839
CAI-h-oc 0.89LW 0.967
LW-oc 0.967sample-estimate 1
T=400 N=40 db=0.5
||Σt − Σtruet ||
TV-n-h=.8-CAI-h 0.27TV-n-h=.7-CAI-h 0.344TV-n-h=.9-CAI-h 0.385
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
||Σt − Σtruet ||
TV-n-h=.8-CAI-h 0.17TV-n-h=.9-CAI-h 0.236TV-n-h=.7-CAI-h 0.26
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Insample Simulations
T=400 N=10 db=0.75
||Σt − Σtruet ||
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
||Σt − Σtruet ||
TV-n-h=.7-CAI-h 0.373TV-n-h=.8-CAI-h 0.393
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
LW-oc 0.983sample-estimate 1
400.0000 N=100 db=0.75
||Σt − Σtruet ||
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Insample Simulations
T=400 N=10 db=1
||Σt − Σtruet ||
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
||Σt − Σtruet ||
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
||Σt − Σtruet ||
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Simulations
T=200 N=10 db=0.5
||Σt − Σtruet ||
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
||Σt − Σtruet ||
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
sample estimate 1LW-oc 1.06
TV-e-h=.95 1.11TV-f-h=.9 1.15TV-n-h=.8 1.16
T=200 N=100 db=0.5
||Σt − Σtruet ||
TV-n-CAI-h-oc 0.31CAI-h 0.43CAI-al 0.55
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Simulations
T=200 N=10 db=0.75
||Σt − Σtruet ||
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
||Σt − Σtruet ||
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-h=.7-LW 0.77TV-n-LW-oc 0.78
TV-n-LW(I)-oc 0.78TV-n-h=.7 0.79
CAI-h 0.89LW 0.97
sample estimate 1LW-oc 1.66
T=200 N=100 db=0.75
||Σt − Σtruet ||
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-h=.9-LW 0.88TV-n-LW-oc 0.89
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Simulations
T=200 N=10 db=1
||Σt − Σtruet ||
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
||Σt − Σtruet ||
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
TV-n-LW-oc 0.57TV-n-LW(I)-oc 0.57
LW 0.89sample estimate 1
CAI-h 1.04LW-oc 1.77
T=200 N=100 db=1
||Σt − Σtruet ||
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Simulations
T=400 N=10 db=0.5
||Σt − Σtruet ||
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
sample estimate 1LW-oc 1.29
T=400 N=40 db=0.5
||Σt − Σtruet ||
TV-n-CAI-h-oc 0.35CAI-h 0.69CAI-al 0.76
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
||Σt − Σtruet ||
TV-n-CAI-h-oc 0.29CAI-h 0.53CAI-al 0.61
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Simulations
T=400 N=10 db=0.75
||Σt − Σtruet ||
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
sample estimate 1LW-oc 1.78
T=400 N=40 db=0.75
||Σt − Σtruet ||
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
sample estimate 1LW-oc 1.87
T=400 N=100 db=0.75
||Σt − Σtruet ||
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Simulations
T=400 N=10 db=1
||Σt − Σtruet ||
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
TV-n-oc 0.4TV-n-CAI-h-oc 0.41
TV-n-LW-oc 0.42TV-n-LW(I)-oc 0.42
CAI-h 0.91LW 0.92
sample estimate 1LW-oc 1.8
T=400 N=40 db=1
||Σt − Σtruet ||
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-n-oc 0.38TV-n-CAI-h-oc 0.38
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
sample estimate 1LW-oc 2.2
T=400 N=100 db=1
||Σt − Σtruet ||
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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)
The 20 S&P500 stocks with the highestaverage capitalization
03/93 to 12/13 (219 obs) from 01/09 to 12/13 (60 obs)
The 30 S&P500 stocks with the highestaverage capitalization
03/93 to 12/13 (219 obs) from 01/09 to 12/13 (60 obs)
The 40 S&P500 stocks with the highestaverage capitalization
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
A summary of the empirical results
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 methods 7 9 9 6 3 5 10 9 10 10 10 10 10 10Fixed methods 3 1 1 4 7 5 0 1 0 0 0 0 0 0
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 sample covariance 1 0 0 0 1 0 3 0 3 3 0 1 0 0TV method with h=.5 1 0 1 1 0 0 2 2 6 4 4 3 1 4TV method with h=.6 1 1 1 0 0 0 0 0 3 2 2 2 3 2TV method with h=.7 2 3 0 1 0 0 4 1 0 1 2 2 2 2TV method with h=.8 0 3 6 4 0 2 3 5 0 0 1 1 0 0TV method with h=.9 0 0 0 0 3 1 0 0 0 0 1 2 3 1
TV CV methods 3 2 1 0 0 2 1 1 1 3 0 0 1 1TV shrink the inverse 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TV LW 1 factor 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
A summary of the empirical results
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 methods 3 3 4 4 5 6 1 4 5 3 5 5 7 7Fixed methods 7 7 6 6 5 4 9 6 5 7 5 5 3 3
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 sample covariance 0 0 0 0 0 1 0 0 1 0 0 0 0 0TV method with h=.5 0 0 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 0 0 0TV method with h=.7 0 0 0 0 0 0 0 0 0 0 0 0 0 0TV method with h=.8 0 0 0 0 0 1 0 0 0 0 1 1 1 1TV method with h=.9 2 2 3 3 4 4 0 3 4 2 3 3 5 5
TV CV methods 0 0 0 0 0 0 0 0 0 0 0 0 0 0TV shrink the inverse 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TV LW 1 factor 1 1 1 1 1 1 0 1 1 1 1 1 1 1
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
A summary of the empirical results
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 methods 10 10 10 10 10 7 10 9 10 10 10 7 9 9Fixed methods 0 0 0 0 0 3 0 1 0 0 0 3 1 1
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 sample covariance 4 5 5 4 3 1 2 0 5 6 1 2 1 3TV method with h=.5 1 0 0 0 0 0 2 0 2 0 0 0 0 0TV method with h=.6 3 3 1 3 1 0 0 3 1 1 2 0 1 0TV method with h=.7 1 2 3 4 3 0 3 5 0 2 1 0 1 0TV method with h=.8 0 3 5 2 6 1 0 1 1 2 4 4 3 1TV method with h=.9 0 0 0 0 0 4 0 0 0 0 1 2 3 3
TV CV methods 5 2 1 1 0 2 5 0 6 5 2 1 1 5TV shrink the inverse 2 0 0 0 0 0 2 0 0 2 0 0 0 1
TV LW 1 factor 1 0 1 3 3 1 1 3 3 1 0 2 2 4
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
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
N=10, Industry Portfolios
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
CAI-h 1 1sample estimate 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
N=17, Industry Portfolios
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Empirical Results
N=30, Industry Portfolios
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 1 1sample estimate 1 1
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Empirical Results
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management
Out of Sample Empirical Results
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
Kapetanios et al Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management