Dynamic Cholesky decomposition

CHAR models

Estimation

Asymptotics of Cholesky GARCH Models and Time-Varying Conditional Betas Serge Darolles, Christian Francq∗ and Sébastien Laurent CFE 2018 ∗ CREST

and university of Lille

Pisa · 14 December 2018 Supported by the ANR via the Project MultiRisk (ANR-16-CE26-0015-02)

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Motivation Problem: Given some information set Ft−1 , it is often of interest to regress yt on the components of x t . Solution: yt − E (yt | Ft−1 ) = β 0y x,t {x t − E (x t | Ft−1 )} + ηt , with the dynamic conditional beta (DCB) β y x,t = Σ−1 xx,t Σxy ,t . Practical implementation: An ARCH-type   model for the conditional variance 

Σxx,t

Σxy ,t

 of Σy x,t Σyy ,t   x t − E (x t | Ft−1 )  is needed. t =  yt − E (yt | Ft−1 ) A Cholesky GARCH model directly specifies the DCB.

Dynamic Cholesky decomposition

CHAR models

Estimation

Notation

Let t = (1t , . . . , mt )0 be a vector of m ≥ 2 log-returns satisfying 1/2

t = Σt

(ϑ0 )η t ,

where (η t ) is iid (0, In ), Σt = Σt (ϑ0 ) = Σ(t−1 , t−2 , . . . ; ϑ0 ) > 0, and ϑ0 is a d × 1 vector.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Engle (2002) DCC
√ Σt = D t R t D t = ρijt σiit σjjt , 1/2

1/2

where D t = diag(σ11t , . . . , σmmt ) contains the volatilities of the individual returns, and R t = (ρijt ) the conditional correlations. The time series model needs to incorporate the complicated constraints of a correlation matrix. One often takes R t = (diag Q t )−1/2 Q t (diag Q t )−1/2 where Q t = (1 − θ1 − θ2 )S + θ1 u t−1 u 0t−1 + θ2 Q t−1 , √ with u t = (u1t . . . umt )0 , uit = it / σiit , θ1 + θ2 < 1.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Engle (2016) DCB

Assuming  

xt yt

  | Ft−1

    µ Σ xt  ,  xx,t ∼N   µy Σy x,t t

Σxy ,t Σyy ,t

   

we have   −1 yt | x t ∼ N µyt + Σy x,t Σ−1 xx,t (x t − µx t ), Σyy ,t − Σy x,t Σxx,t Σxy ,t ⇒ β y x,t = Σ−1 xx,t Σxy ,t can be obtained by first estimating a DCC GARCH model.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Drawbacks of DCC-based DCB

1) The stationarity and ergodicity conditions of the DCC are not well known. 2) The correlation constraints are complicated. 3) The asymptotic properties of the QMLE are unknown. 4) The effects of the DCC parameters on β t are hardly interpretable. We now introduce a class of Cholesky GARCH (CHAR) models that avoids all these drawbacks.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Cholesky Decomposition of Σ = Var () Letting v1 := 1 , we have 2 = `21 v1 + v2 = β21 1 + v2 , where β21 = `21 is the beta in the regression of 2 on 1 , and v2 is orthogonal to 1 . Recursively, we have

i =

i−1 X j=1

`ij vj + vi =

i−1 X

βij j + vi ,

for i = 2, . . . , m,

j=1

where vi is uncorrelated to v1 , . . . , vi−1 , and thus uncorrelated to 1 , . . . , i−1 . In general the order of the series matters: Order of the series

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Cholesky Decomposition of Σ = Var () In matrix form,  = Lv

and

v = B,

where L and B = L−1 are lower unitriangular and G := var(v) is diagonal. We obtain the Cholesky decomposition Σ = LGL0 (see Pourahmadi, 1999). Conditioning on Ft−1 , Σt = Lt G t L0t and Σ−1 = B 0t G −1 t B t . We thus need t - a diagonal ARCH-type model for the factors vector v t - a time series model for Lt (or B t ), without particular constraint.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Example: Σ = LGL0 = DRD, m = 2 Letting g1 = σ12 = var(1 ),

2 = `21 1 +v2 ,

σ22 = var(2 )

g2 = varv2 ,

and ρ = cor(1 , 2 ), we have 

1

L= `21

 0 , 1

 g1 G= 0 

g1

Σ= `21 g1

0 g2

 ,

`21 g1 `221 g1 + g2

 σ1 D= 0 



σ12

= ρσ1 σ2

0 σ2

 ,

  1 ρ , R= ρ 1

ρσ1 σ2 σ22

 .

Positivity constraints: g1 > 0 and g2 > 0 or σ12 > 0, σ12 > 0 and ρ2 < 1.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Example: Σ = LGL0 , Σ−1 = B 0 G −1 B, m = 3



1

0

 L= `21

1

`31

`32

 0  0 , 1



g1

 Σ= `21 g1 `31 g1



1

 B= −β21 −β31

0 1 −β32

 0  0 , 1

`21 g1 `221 g1 + g2 `21 `31 g1 + `32 g2

 g1  G= 0 0

`31 g1

0

g2

 0 ,

0

g3



 `21 l31 g1 + `32 g2  . 2 2 `31 g1 + `32 g2 + g3

Remark: In Σ = DRD the constraints on the elements of R are ρ212 + ρ213 + ρ223 − 2ρ12 ρ13 ρ23 ≤ 1. In Σ = LGL0 there is no constraint on the `ij ’s.



0

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

A general model for the factors Assume 1/2

v t = Gt

ηt ,

(η t ) iid (0, In ),

where G t = diag(g t ) follows a GJR-like equation g t = ω0 +

q n p o X X 2− A0i,+ v 2+ + A v + B 0j g t−j , 0i,− t−i t−i i=1

j=1

with positive coefficients and v 2+ = t

  + 2 0 + 2 , v1t , · · · , vmt

v 2− = t

  − 2 0 − 2 . v1t , · · · , vmt

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Markovian representation of the factors  0 0 2−0 0 Letting z t = v 2+ , v , g t:(t−p+1) , t:(t−q+1) t:(t−q+1)  0 0 0 0 Υ−0 , 00 0 , 00 ht = ω 00 Υ+ , 0 , ω , ω t 0 t 0 (p−1)m , with m(q−1)    (q−1)m  2+ 2− Υ+ Υ− and obvious notations, we t = diag η t t = diag η t rewrite the model as z t = ht + H t z t−1 , where, in the case p = q = 1, 

+ + Υ+ t A01,+ Υt A01,− Υt B 01



  − − − . Ht =  Υ A Υ A Υ B 01,+ 01,− 01  t  t t A01,+ A01,− B 01

Dynamic Cholesky decomposition

CHAR models

Estimation

Stationarity of the factors

In view of z t = ht + H t z t−1 , there exists a stationary and ergodic sequence (v t ) satisfying 1/2

v t = Gt

η t if and only if γ0 = inf

t≥1

1 E(log kH t H t−1 . . . H 1 k) < 0. t

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Stationarity of β t := −vech0 B t If (v t ) is stationary and ergodic (γ0 < 0), and ( det I m0 −

s X

) C 0i z

i

6= 0 for all |z| ≤ 1,

i=1

then βt = c0



1/2 1/2 v t−1 , . . . , v t−r , g t−1 , . . . , g t−r



+

s X

C 0j β t−j .

j=1

defines a stationary and ergodic sequence (and thus the existence of a stationary CHAR model).

Dynamic Cholesky decomposition

CHAR models

Estimation

Existence of moments If in addition Ekη 1 k2k1 < ∞

and

1 %(EH ⊗k 1 ) < 1,

for some integer k1 > 0, and kc 0 (x) − c 0 (y)k ≤ K kx − yka for some constants K > 0 and a ∈ (0, 1], then the CHAR model satisfies E k1 k2k1 < ∞.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

A simpler triangular parameterization A tractable submodel is i  2 X  2 (k ) − α0i vk2,t−1 +b0i gi,t−1 + +γ  git = ω0i +γ0i+ + 0i− 1,t−1 1,t−1 k =2

with positivity coefficients, and − βij,t = $0ij + ς0ij+ + 1,t−1 + ς0ij− 1,t−1 +

i X

(k )

τ0ij vk ,t−1 + c0ij βij,t−1

k =2

without positivity constraints. Notice the triangular structure and note that the asymmetry is introduced via the first (observed) factor only.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Stationarity for the previous specification There exists a strictly stationary, non anticipative and ergodic solution to the CHAR model when   2   2 − + 1) E log ω01 + γ01+ η1,t−1 + γ01− η1,t−1 + b01 < 0, n o (i) 2) E log α0i ηit2 + b0i < 0 for i = 2, . . . , m, 3) |c0ij | < 1 for all (i, j). Moreover, the stationary solution satisfies Ek1 k2s0 < ∞, Ekg 1 ks0 < ∞, Ekv 1 ks0 < ∞, Ekβ 1 ks0 < ∞ and EkΣ1 ks0 < ∞ for some s0 > 0.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Invertibility of the CHAR

Under the stationarity conditions g t = g(η u , u < t),

β t = β(η u , u < t).

For practical use, we need (uniform) invertibility: g t (ϑ) = g(ϑ; u , u < t), with some abuse of notation.

β t (ϑ) = β(ϑ; u , u < t) Invertibility conditions

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Full QMLE of the general CHAR A QMLE of the CHAR parameter ϑ0 is e n (ϑ), b n = arg min O ϑ

e n (ϑ) = n−1 O

ϑ∈Θ

n X

et (ϑ), q

t=1

e t (ϑ) = Σ (t−1 , . . . , 1 , e where Σ 0 , e −1 , . . . ; ϑ) and 0

−1

e (ϑ)B e (ϑ)G e t (ϑ)t + et (ϑ) = 0t B q t t

m X

eit (ϑ). log g

i=1

• Does not require matrix inversion. • CAN under general regularity conditions.

Regularity conditions

Dynamic Cholesky decomposition

CHAR models

Estimation

Equation-by-Equation (EbE) estimator Consider the triangular model. In a first step, the parameter (1)

ϑ0 = (ω01 , γ01+ , γ01− , b01 ) is estimated by b (1) = arg min ϑ n (1)

ϑ

∈Θ(1)

n X

e1t (ϑ(1) ), q

t=1

where e1t (ϑ(1) ) = q

21t e1t (ϑ(1) ) g

e1t (ϑ(1) ) = ω1 + γ1+ + and g 1t


2

e1t (ϑ(1) ), + log g

+ γ1− − 1t


2

e1,t−1 (ϑ(1) ). + b1 g

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

EbE second step (2) (2) (2) (2) e2t = g e2t (θ 0(2) ). Let ϑ0 = (ϕ0 , θ 0 ), where βe21,t = βe21,t (ϕ0 ) and g (1)

(2)

Independently or in parallel to ϑ0 , one can estimate ϑ0 by b (2) = arg min ϑ n ϑ(2) ∈Θ(2)

n X

e2t (ϑ(2) ), q

t=1

where, for t = 1, . . . , n, ve2t2 (ϕ(2) ) e2t (ϑ(2) ), + log g (2) e g2t (ϑ )

e2t (ϑ(2) ) q

=

e2t (ϑ(2) ) g

(2) 2 e2,t−1 (ϕ(2) ), = ω2,t−1 + α2 ve2,t−1 (ϕ(2) ) + b2 g

ve2t (ϕ(2) )

= 2t − βe21,t (ϕ(2) )1t ,

βe21,t (ϕ(2) )

(2) = ω21,t−1 + τ21 ve2,t−1 (ϕ(2) ) + c21 βe21,t−1 (ϕ(2) ).

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

EbE remaining steps (+i)

For i ≥ 3, βeij,t depends on ϕ0

  (i) (−i) (−i) = ϕ0 , ϕ0 , where ϕ0 has

eit depends on been estimated in the previous steps. The volatility g (+i)

ϑ0

(i)

(+i)

= (θ 0 , ϕ0

b (i) = arg min ϑ n

n X

ϑ(i) ∈Θ(i) t=1

(i)

(i)

(i)

), and ϑ0 = (θ 0 , ϕ0 ) can be estimated by

eit (ϑ(i) , ϕ b (−i) q ), n

eit (ϑ(+i) ) = ωi,t−1 + g

i X

eit (ϑ(+i) ) = q

veit2 (ϕ(+i) ) eit (ϑ(+i) ), + log g (+i) e git (ϑ )

(k ) ei,t−1 (ϑ(+i) ), αi vek2,t−1 (ϕ(+k ) ) + bi g

k =2

vekt (ϕ(+k ) ) = kt −

k −1 X

βekj,t (ϕ(+k ) )jt ,

j=1

βeij,t (ϕ(+i) ) = ωij,t−1 +

i X k =2

(k ) τij vek ,t−1 (ϕ(+k ) ) + cij βeij,t−1 (ϕ(+i) ),

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

QML vs. EbE

1) If m = 2, the one-step full QMLE and the two-step EbEE are exactly the same. 2) For m ≥ 3, the two estimators are generally different. 3) The QML and EbE estimators are CAN under similar assumptions.

CAN of the EbEE

4) The EbEE is simpler, but is not always less efficient than the full QMLE.

Example

Similar behaviour on simulations

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Asset Pricing for Industry Portfolios We consider the 12 industry portfolios used by Engle (2016), examined in the context of the Fama French 3 factor model. The three factors are: MKT (Market factor = excess log-returns of the SP500), SMB (small minus big size factor) and HML (high minus low value factor) Data are from Ken French’s web site and cover the period 1994-2016. We follow Patton and Verardo (2012) in building hedged portfolios to offset some unwanted exposures to predetermined factors.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Competing models Let t =

(x0t , yt )0

with xt = (MKTt , SMBt , HMLt )0 and yt = rkt .

Hedging strategy: Et−1 (rkt | x t ) = βk ,MKT ,t MKTt + βk ,SMB,t SMBt + βk ,HML,t HMLt . Competing models: 1) CCC-GARCH(1,1) 2) DCC-GARCH(1,1) 3) CHAR with constant betas 4) CHAR with time varying betas βij,t = $ij + τij vi,t−1 vj,t−1 + cij βij,t−1 CHAR model on (MKTt , SMBt , HMLt , rkt )0 or (MKTt , HMLt , SMBt , rkt )0 by minimizing AIC (or equivalently BIC).

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

BusEq-Mkt 1 2

Buseq: Business Equipment – Computers, Software, and Electronic Equipment

2002

2004

2006

2008

2010

2012

2014

2016

2002

2004

2006

2008

2010

2012

2014

2016

BusEq-SMB -0.5 0 0.5

1

2000

BusEq-HML -2 -1 0

2000

2000

C-CHAR CCC

2002

2004

2006

2008

2010

2012

2014

CHAR DCC

2016

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Model Confidence Set test (Hansen, Lunde and Nason, 2011) C-CHAR

CHAR

CCC

DCC

BusEq

X

Chems

X

Durbl

X

Enrgy

X

X

Hlth

X

X

Manuf

X

Money

X

NoDur

X

Other

X

Shops

X

Telcm

X

Utils

X

X

Models included in the MCS in the beta hedging exercise. Models highlighted with the symbol X are contained in the model confidence set using a MSE loss function. The significance level for the MCS is set to 20%, and 10,000 bootstrap samples (with a block length of 5 observations).

Dynamic Cholesky decomposition

CHAR models

Estimation

Transaction costs : MKT

SMB

HML

BusEq

0.356

0.380

0.341

Chems

0.310

0.263

0.376

Durbl

0.419

0.464

0.693

Enrgy

0.373

0.337

0.456

Hlth

0.461

0.667

0.397

Manuf

0.442

0.402

0.430

Money

0.390

0.397

0.366

NoDur

0.414

0.383

0.296

Other

0.273

0.343

0.335

Shops

0.344

0.297

0.395

Telcm

0.334

0.414

0.640

Utils

0.465 P1,678

0.408

0.431

∆β

k ,j

=

t=2

∆βCHAR ∆βDCC

|βk ,j,t+1|t − βk ,j,t|t−1 |.

For each column, the figures correspond to the ratio between the value of ∆β

k ,j

obtained for

the CHAR and the DCC-DCB models.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Conclusion Compare to other multivariate GARCH (in particular BEKK and DCC), the Cholesky-GARCH models introduced here have several advantages. 1) Precise stationarity and moment conditions exist. 2) The parameters are directly interpretable in terms of DCB. 3) There is no complicated correlation constraint. 4) The estimation can be done without matrix invertion. 5) The asymptotic theory of the QMLE is available. 6) EbE estimation is possible for triangular models. 7) The model works nicely in practice, in particuilar for beta hedging.

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Conclusion Compare to other multivariate GARCH (in particular BEKK and DCC), the Cholesky-GARCH models introduced here have several advantages. 1) Precise stationarity and moment conditions exist. 2) The parameters are directly interpretable in terms of DCB. 3) There is no complicated correlation constraint. 4) The estimation can be done without matrix invertion. 5) The asymptotic theory of the QMLE is available. 6) EbE estimation is possible for triangular models. 7) The model works nicely in practice, in particuilar for beta hedging. Thanks for your attention!

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

CAN of the EbE estimator Theorem (CAN of the EbEE)  bn = Under B1-B5 the EbEE ϑ b n → ϑ0 , ϑ

0

b (1) , . . . , ϑ b (m) ϑ n n

0

0 satisfies

almost surely as n → ∞.

Under the additional assumption B6, as n → ∞, √

  (i) L (i) b n ϑn − ϑ0 →N

 0, Σ

(i)



:= J

(i)

−1

I

(i)



J

for i = 1 and i = 2. (i)

Jn

=

e (i) (ϑ b (+i) ) ∂2O n n 0 , ∂ϑ(i) ∂ϑ(i)

(i)

In =

n b (+i) ) ∂ e b (+i) ) 1 X ∂e qit (ϑ qit (ϑ n n 0 ∂ϑ(i) ∂ϑ(i)

n t=1

(i)

−1 

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

CAN of the EbE estimator (2)

β32,t = $032 + . . . + τ032 v2,t−1 + c032 β32,t−1 , where v2,t−1 = ε2,t−1 − β21,t−1 ε1,t−1 .

Theorem (CAN of the EbEE) (+i)

(+(i−1))

Denoting by Σϕ− (or by Σϕ+ (+i)

) the bottom-right sub-matrix of

(+(i−1))

Σ (or of Σ ) corresponding to the asymptotic variance of b n(−i) (which is equal to ϕ b (+(i−1)) ϕ ), and using the convention n (+2) (2) Σ = Σ , for i = 3, . . . , m we also have  √  (+i) (+i) b n ϑ − ϑ0 n

with

 Σ

(+i)

 =

(i)

Σϑ

 0 −1 (i)0 (+(i−1)) − J (i) K Σ + ϕ

(i)

where Σϑ =

L

→N



0, Σ

(+i)



  −1 (+(i−1)) − J (i) K (i) Σ +  ϕ  (+(i−1)) Σ + ϕ

 −1  −1  e (i) (ϑ b (+i) ) ∂2 O (i) (+(i−1)) (i)0 n n J (i) and K n = J (i) I (i) + K (i) Σ + K 0 ϕ

∂ϑ(i) ∂ϕ(−i)

Return

Dynamic Cholesky decomposition

CHAR models

Estimation

Invertibility of the CHAR For practical use, we need uniform invertibility: β t (ϑ) = β(ϑ; u , u < t). More precisely, we need



e sup β t (ϑ) − β t (ϑ) ≤ K ρt ,

ϑ∈Θ

e (ϑ) = β(ϑ; t−1 , . . . , 1 , e where β 0 , e −1 , . . . ) for fixed initial t values e 0 , e −1 , . . . , and β t (ϑ0 ) = β t .

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Invertibility conditions for the "triangular" model In vector form, the model

vekt (ϑ) = kt −

kX −1

βekj,t (ϑ)jt ,

j=1

βeij,t (ϑ) = ωij,t−1 +

i X

(k ) τij vek ,t−1 (ϑ) + cij βeij,t−1 (ϑ),

k =2 − with ωijt = $ij + ςij+ + 1t + ςij− 1t , writes

e t−1 (ϑ)t−1 + C β e (ϑ) =w t−1 + T B e (ϑ) β t t−1 n o 0 e (ϑ) =w t−1 + T t−1 + C − (t−1 ⊗ T )D0m β t−1 e (ϑ). :=w ∗t−1 + S t−1 β t−1

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Invertibility conditions for the "triangular" model By the Cauchy rule, the triangular model e (ϑ) = w ∗ + S t−1 β e (ϑ). β t t−1 t−1 is uniformly invertible under the conditions E log+ sup kw 1 + T 1 k < ∞, ϑ∈Θ

n

Y

1

γS := lim sup log sup S t−i < 0.

n→∞ n ϑ∈Θ i=1

Return

Dynamic Cholesky decomposition

CHAR models

Estimation

Regularity conditions for the triangular model Let ωit = ωi + γi+ + 1t


2

+ γi− − 1t


2

. The regularity conditions

for CAN of the QMLE of the triangular model

git (ϑ) = ωi,t−1 +

βij,t (ϑ) = ωij,t−1 +

i X k =2 i X

(k )

αi vk2,t−1 (ϑ) + bi gi,t−1 (ϑ) (k )

τij vk ,t−1 (ϑ) + cij βij,t−1 (ϑ)

k =2

are B1 |c0ij | < 1 and other stationarity conditions.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Regularity conditions for the triangular model Let ωit = ωi + γi+ + 1t


2

+ γi− − 1t


2

. The regularity conditions

for CAN of the QMLE of the triangular model

git (ϑ) = ωi,t−1 +

βij,t (ϑ) = ωij,t−1 +

i X k =2 i X

(k )

αi vk2,t−1 (ϑ) + bi gi,t−1 (ϑ) (k )

τij vk ,t−1 (ϑ) + cij βij,t−1 (ϑ)

k =2

are B2 For i = 2, . . . , m, the distribution of ηit2 conditionally on {ηjt , j 6= i} is non-degenerate. The support of η1t contains at least two positive points and two negative points.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Regularity conditions for the triangular model Let ωit = ωi + γi+ + 1t


2

+ γi− − 1t


2

. The regularity conditions

for CAN of the QMLE of the triangular model

git (ϑ) = ωi,t−1 +

βij,t (ϑ) = ωij,t−1 +

i X k =2 i X

(k )

αi vk2,t−1 (ϑ) + bi gi,t−1 (ϑ) (k )

τij vk ,t−1 (ϑ) + cij βij,t−1 (ϑ)

k =2

are B3 Positivity and invertibility conditions: for all ϑ, (k )

γi+ , γi− , αi

≥ 0 ωi > 0, |bi | < 1 and |cij | < 1.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Regularity conditions for the triangular model Let ωit = ωi + γi+ + 1t


2

+ γi− − 1t


2

. The regularity conditions

for CAN of the QMLE of the triangular model

git (ϑ) = ωi,t−1 +

βij,t (ϑ) = ωij,t−1 +

i X k =2 i X

(k )

αi vk2,t−1 (ϑ) + bi gi,t−1 (ϑ) (k )

τij vk ,t−1 (ϑ) + cij βij,t−1 (ϑ)

k =2

are (2) (i) B4 Identifiability conditions: (γ0i+ , γ0i− , α0i , . . . , α0i ) 6= 0 and (2)

(i)

(ς0ij+ , ς0ij− , τ0ij , . . . , τ0ij ) 6= 0.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Regularity conditions for the triangular model Let ωit = ωi + γi+ + 1t


2

+ γi− − 1t


2

. The regularity conditions

for CAN of the QMLE of the triangular model

git (ϑ) = ωi,t−1 +

βij,t (ϑ) = ωij,t−1 +

i X k =2 i X

(k )

αi vk2,t−1 (ϑ) + bi gi,t−1 (ϑ) (k )

τij vk ,t−1 (ϑ) + cij βij,t−1 (ϑ)

k =2

are B5 Uniform invertibility condition.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Regularity conditions for the triangular model Let ωit = ωi + γi+ + 1t


2

+ γi− − 1t


2

. The regularity conditions

for CAN of the QMLE of the triangular model

git (ϑ) = ωi,t−1 +

βij,t (ϑ) = ωij,t−1 +

i X k =2 i X

(k )

αi vk2,t−1 (ϑ) + bi gi,t−1 (ϑ) (k )

τij vk ,t−1 (ϑ) + cij βij,t−1 (ϑ)

k =2

are B6 Moments conditions (larger than 6) on kt k ,

sup ϑ∈V (ϑ0 )

kβ t (ϑ)k ,



∂β t (ϑ)

sup 0 . ∂ϑ ϑ∈V (ϑ0 )

Return

Dynamic Cholesky decomposition

CHAR models

Estimation

Example Consider a static model with git = 1 for i ∈ {1, 2, 3} and 1t

= v1t

2t

= l21 v1t + v2t

3t

=

l31 v1t + l32 v2t + v3t = l32 v2t + v3t . |{z} 0

In terms of the β’s: 2t

= β21 1t + v2t

3t

= −β21 β32 1t + β32 2t + v3t | {z } β31

→ ϑ = (β21 , β32 ).

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Example Assume for instance that the variable v2t is independent of the vector (v1t , v3t )0 and that this vector is distributed as the product ηu, where the random variable η and the vector u are p independent, e.g., u ∼ N (0, I 2 ) and η ∼ (ν − 2)/νStν , ν > 4.     √  L b QMLE: n ϑn − ϑ0 → N 0,   EbEE:

 √  n βb21 − β0,21 =

2 Eη 4 1+β32 2 )2 (1+β32

0

P n−1/2 nt=1 η2t 1t L P → n−1 nt=1 21t

   . Eη 4  0

N (0, 1) .

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

10

20

(1+β32 2Eη 4)/(1+β32 2)2

30

Illustration

8 7 ν (d

2

egre e

1

6 of fr eedo m

)

5

0 -1

β 32

Figure: Ratio between the asymptotic variance of the QML estimator of β21 and its EbE counterpart as a function of the degree of freedom ν and β32 .

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Illustration Full QML

EbE

3.5

n × MSE of β^21

3.0

2.5

2.0

1.5

1.0 5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

25.0

27.5

30.0

Return

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Example: Σ = LGL0 , m = 3



1

0

 L= l21

1

l31

l32

 0  0 , 1 

g1

 Σ= l21 g1 l31 g1



1

 B= −β21 −β31

0 1 −β32

 0  0 , 1

l21 g1 2 l21 g1 + g2

l21 l31 g1 + l32 g2

 g1  G= 0 0

l31 g1

0

g2

 0 ,

0

g3



 l21 l31 g1 + l32 g2  . 2 2 l31 g1 + l32 g2 + g3

Remark: In Σ = DRD the constraints on the elements of R are ρ212 + ρ213 + ρ223 − 2ρ12 ρ13 ρ23 ≤ 1. In Σ = LGL0 there is no constraint on the `ij ’s.



0

Dynamic Cholesky decomposition

CHAR models

Estimation

Sequential construction of Σt = Lt G t L0t 1/2

Let the factors v t = G t 1/2

Taking Σt

η t , η t iid (0, I m ) (vit =

1/2

1/2

= Lt G t , we have t = Σt Step 1: v1t = 1t

Step 2:

Step 3:

√

git ηit ).

η t = Lt v t .

g1,t

= Var (v1t | Ft−1 )

2t

= l21,t v1t + v2t

v2t

= 2t − l21,t v1t

g2,t

= Var (v2t | Ft−1 )

3t

= l31,t v1t + l32,t v2t + v3t

v3t

= 3t − l31,t v1t − l32,t v2t

g3,t

= Var (v3t | Ft−1 )

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

= B 0t G −1 Alternative construction of Σ−1 t Bt t t = Lt v t and therefore B t t = v t , where the subdiagonal elements of the lower unitriangular matrix B t = L−1 are −βij,t t Step 1: v1t = 1t

Step 2:

Step 3:

g1,t

= Var (v1t | Ft−1 )

2t

= β21,t 1t + v2t

v2t

= 2t − β21,t 1t

g2,t

= Var (v2t | Ft−1 )

3t

= β31,t 1t + β32,t 2t + v3t

v3t

= 3t − β31,t 1t − β32,t 2t

g3,t

= Var (v3t | Ft−1 ) Return

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

The order of the series Replace t = (1t , 2t , 3t )0 by ∗t = (2t , 1t , 3t )0 (the position of the last component is imposed by the problem) ∗ ∗ ∗ Step 1∗ : v1t = 2t , g1,t = Var (v1t | Ft−1 ) Step 2∗ :

Step 3∗ :

1t

∗ = β12,t 2t + v2t

∗ g2,t

∗ = Var (v2t | Ft−1 )

3t

∗ = β32,t 2t + β31,t 1t + v3t

g3,t

= Var (v3t | Ft−1 )

In particular, β12,t = β21,t β 2

g1t

21,t g1t +g2t

specifications, the order matters.

∗ (v2t 6= v2t )

∗ (v3t = v3t )

: for most parametric

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

The order of the series (matrix form)

Σ−1 t

 −1 2 −1 2 g −1 + β21 g2 + β31,t g3,t  1 −1 −1 =  −β21 g2 + β31,t β32,t g3,t −1 −β31,t g3,t

∗t = ∆t , 

−1 −β21 g2−1 + β31,t β32,t g3,t −1 2 β32,t g3,t + g2−1 −1 −β32,t g3,t

−1 −β31,t g3,t



 −1  −β32,t g3,t . −1 g3,t

∆Σ∗−1 ∆= t

−1 2 β31,t g3,t + g2∗−1

 −β12 g ∗−1 + β31,t β32,t g −1 2 3,t  −1 −β31 g3,t

−1 −β12 g2∗−1 + β31,t β32,t g3,t −1 2 ∗−1 2 g3,t g2 + β32,t g1∗−1 + β12 −1 −β32,t g3,t

−1 −β31,t g3,t



 −1  −β32,t g3,t . −1 g3,t

We have ∆Σ∗−1 ∆ = Σ−1 when t t 2 2 2 2 2 β12 = β21 g1 /(β21 g1 + g2 ), g1∗ = β21 g1 + g2 and g2∗ = g1 − β21 g1 /(β21 g1 + g2 )

When the first parameters are time-invariant, the order does not matter. Return

Dynamic Cholesky decomposition

CHAR models

Estimation

Data Generating process

t

1/2

= Σt

(ϑ0 )η t ,

Σt

= Lt G t L0t ,

gi,t

2 + 0.8gi,t−1 , = 0.1 + 0.1vi,t−1

βij,t

= 0.1 + 0.2vi,t−1 + 0.8βij,t−1 ,

where (η t ) is iid (0, Im ), t = 1, . . . , n.

Application

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Results for m = 5, η t ∼ N (0, Im ), 1000 replications FULL QML BIAS

RMSE-STD

EbE

5% CP

95% CP

BIAS

RMSE-STD

5% CP

95% CP 92.251

n=1000 ω

0.0202

0.0158

4.444

91.695

0.0190

0.0100

4.070

α

0.0037

0.0024

4.018

91.111

0.0036

-0.0004

3.659

91.696

β

-0.0245

0.0178

7.834

94.007

-0.0230

0.0092

7.379

94.347 92.559

$

0.0008

0.0011

5.769

91.538

0.0007

0.0001

4.604

τ

0.0012

0.0011

7.306

93.939

0.0011

-0.0001

6.208

94.779

c

-0.0017

0.0022

8.115

93.805

-0.0016

0.0003

7.009

94.913

0.0000

0.0050

6.520

92.820

0.0000

0.0021

5.639

93.644 93.138

ALL n=2000 ω

0.0098

0.0014

3.824

92.745

0.0087

0.0000

3.305

α

0.0019

0.0014

4.412

92.966

0.0017

-0.0004

3.766

93.410

β

-0.0119

0.0034

6.642

94.804

-0.0104

-0.0006

6.130

95.460 93.410

$

0.0002

-0.0009

6.483

91.324

0.0002

-0.0003

4.550

τ

0.0003

0.0001

7.145

93.493

0.0004

-0.0002

5.690

94.812

c

-0.0005

-0.0014

8.039

94.069

-0.0004

-0.0006

6.266

95.690

0.0000

0.0002

6.468

93.143

0.0000

-0.0004

5.135

94.426

ALL

2 gi,t = ωi + αi vi,t−1 + βi gi,t−1 and βij,t = $ij + τij vi,t−1 + cij βij,t−1

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Results for m = 5, η t ∼ t(0, 1, 7), 1000 replications FULL QML BIAS

RMSE-STD

EbE

5% CP

95% CP

BIAS

RMSE-STD

5% CP

95% CP 90.284

n=1000 ω

0.0236

0.0153

4.468

89.811

0.0225

0.0050

4.227

α

0.0056

0.0015

2.931

89.362

0.0057

-0.0066

2.860

90.074

β

-0.0309

0.0181

9.409

92.719

-0.0293

-0.0013

8.980

93.270 92.650

$

0.0010

-0.0002

5.922

91.052

0.0010

-0.0014

4.532

τ

0.0014

-0.0013

7.080

94.173

0.0012

-0.0016

5.910

95.152

c

-0.0019

0.0005

8.818

93.995

-0.0018

-0.0024

7.308

95.226

ALL

-0.0001

0.0037

6.717

92.259

-0.0001

-0.0015

5.730

93.298 91.734

n=2000 ω

0.0062

0.0018

3.624

92.584

0.0093

0.0005

3.141

α

0.0009

0.0007

3.490

91.678

0.0020

-0.0017

2.748

91.189

β

-0.0079

0.0025

7.282

94.899

-0.0120

-0.0006

7.546

94.984 94.079

$

0.0002

0.0001

8.188

90.336

0.0003

-0.0004

4.308

τ

0.0002

0.0002

7.953

92.953

0.0005

-0.0006

4.984

95.507

c

-0.0004

0.0001

9.463

91.695

-0.0007

-0.0008

5.965

95.725

ALL

-0.0001

0.0006

7.289

92.125

0.0000

-0.0006

4.883

94.281

2 gi,t = ωi + αi vi,t−1 + βi gi,t−1 and βij,t = $ij + τij vi,t−1 + cij βij,t−1

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

Results for m = 10, n = 4000 and 1000 replications

Summary statistics on the 165 parameters Mean biais

-0.000873171

Mean (rmse-Mean STD)

0.00155948

Mean Coverage Prob 5%

5.042389091

Mean Coverage Prob 95%

93.6329697 Return

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

The order of the series Replace t = (1t , 2t , 3t )0 by ∗t = (2t , 1t , 3t )0 (the position of the last component is imposed by the problem) ∗ ∗ ∗ Step 1∗ : v1t = 2t , g1,t = Var (v1t | Ft−1 ) Step 2∗ :

Step 3∗ :

1t

∗ = β12,t 2t + v2t

∗ g2,t

∗ = Var (v2t | Ft−1 )

3t

∗ = β32,t 2t + β31,t 1t + v3t

g3,t

= Var (v3t | Ft−1 )

In particular, β12,t = β21,t β 2

g1t

21,t g1t +g2t

specifications, the order matters.

∗ (v2t 6= v2t )

∗ (v3t = v3t )

: for most parametric

Dynamic Cholesky decomposition

CHAR models

Estimation

Application

The order of the series (matrix form)

Σ−1 t

 −1 2 −1 2 g −1 + β21 g2 + β31,t g3,t  1 −1 −1 =  −β21 g2 + β31,t β32,t g3,t −1 −β31,t g3,t

∗t = ∆t , 

−1 2 β32,t g3,t + g2−1 −1 −β32,t g3,t

−1 −β31,t g3,t



 −1  −β32,t g3,t . −1 g3,t

∆Σ∗−1 ∆= t

−1 2 β31,t g3,t + g2∗−1

−1 −β12 g2∗−1 + β31,t β32,t g3,t

 −β12 g ∗−1 + β31,t β32,t g −1 2 3,t  −1 −β31 g3,t

−1 2 ∗−1 2 g3,t g2 + β32,t g1∗−1 + β12 −1 −β32,t g3,t

We have ∆Σ∗−1 ∆ = Σ−1 when t t β12 =

−1 −β21 g2−1 + β31,t β32,t g3,t

2 β21 g1 /(β21 g1

+ g2 ),

g1∗

=

−1 −β31,t g3,t



 −1  −β32,t g3,t . −1 g3,t

Return

2 β21 g1

2 2 2 + g2 and g2∗ = g1 − β21 g1 /(β21 g1 + g2 )

When the first parameters are time-invariant, the order does not matter.