Efficient Estimation of a Multivariate Multiplicative Volatility Model · 2015. 3. 2. · Efficient...

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Efficient Estimation of a Multivariate Multiplicative Volatility Model Christian Hafner and Oliver Linton UCL and LSE Cass Business School, December 7th, 2007 Christian Hafner and Oliver Linton UCL and LSE London School of Economics Multivariate Spline GARCH

Transcript of Efficient Estimation of a Multivariate Multiplicative Volatility Model · 2015. 3. 2. · Efficient...

Page 1: Efficient Estimation of a Multivariate Multiplicative Volatility Model · 2015. 3. 2. · Efficient Estimation of a Multivariate Multiplicative Volatility Model Christian Hafner and

Efficient Estimation of a Multivariate

Multiplicative Volatility Model

Christian Hafner and Oliver LintonUCL and LSE

Cass Business School, December 7th, 2007

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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SPLINE GARCH MODEL

Engle and Rangel (2006)

yt = τ1/2t g1/2

t εt

with (for example) unit GARCH(1,1) short run dynamics

gt = 1− β − γ + βgt−1 + γy2t−1

τt−1

Slowly varying component to volatility

τt = c exp

(w0t +

k

∑i=1

wi (t − ti−1)2+ + ztδ

)

εt i.i.d. with mean zero and variance one

Estimation method based on fitting spline. Flexible choice ofk .

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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We consider models without covariates.

Interpretτt = m(t/T ),

where m(u), u ∈ [0, 1] is an unknown smooth function onrescaled time

Semiparametric model with parameters (β, γ) and unknownfunction m(.). Error distribution standard normal forcompleteness.

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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The local autocorrelation function (LACF) of squared returnsin the ER model is time invariant, i.e.,

ρy2(t , j) =cov(y2

t , y2t−j )√

var(y2t )var(y2

t−j )

=v (t/T )v (t − j/T )cov(gtε2t , gt−j ε

2t−j )√

v2(t/T )v2(t − j/T )var(gt ε2t)var(gt−j ε2t−j )

=cov(gt ε2t , gt−j ε

2t−j )

var(gt ε2t)

= ρy ∗2(t , j) = ρy ∗2(j),

where y ∗2t = gt ε2t , and ρy ∗2(t , j) is time invariant because of

the stationarity of gt ε2t .

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Nice featureEy2

t = m(t/T )

makes estimation simple. Local in time kernel method forexample

m(u) =∑

Tt=1 Kh(u − t/T )y2

t

∑Tt=1 Kh(u − t/T )

,

where K is a kernel, h is a bandwidth, and Kh(.) = K (./h)/h.

Given consistent estimate of m(.) one can estimate (β, γ).

One issue is standard error construction and efficiency.

m(u)−m(u) =1

T

T

∑t=1

Kh(u − t/T )(y2t −m(t/T )) + bias

=1

T

T

∑t=1

Kh(u − t/T )m(t/T )(gt ε2t − 1) + bias

Error is not mds :(

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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If one knew gt , then

m(u) =∑

Tt=1 Kh(u − t/T )(y2

t /gt)

∑Tt=1 Kh(u − t/T )

,

m(u)−m(u) =1

T

T

∑t=1

Kh(u − t/T )(y2t /gt −m(t/T )) + bias

=1

T

T

∑t=1

Kh(u − t/T )m(t/T )(ε2t − 1) + bias

Error is mds :)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Semiparametric Efficiency. Assuming εt is normally distributed, thelog likelihood function

ℓT (φ, m) = −1

2

T

∑t=1

log σ2t (φ, m)− 1

2

T

∑t=1

y2t /σ2

t (φ, m)

σ2t (φ, m) = m(t/T )gt(φ)

for some unknown function m(.).We suppose that

log m(t/T ) =∞

∑j=0

θj ψj (t/T )

for some orthonormal basis ψj∞j=0 with ψ0(u) = 1, and

1

T

T

∑t=1

ψj (t/T )ψk(t/T ) → δjk ,

where δjk = 1 if j = k and 0 if j 6= k .Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Then consider some finite order approximation mθ(t/T ) withJ < ∞. We have:

∂ log mθ(t/T )

∂θj

= ψj (t/T )

Iθθ = limT→∞

E

[∂ℓT (φ0, θ0)

∂θ

∂ℓT (φ0, θ0)

∂θ⊤

]= E [(ε2t − 1)2]IJ

Iφθ = E [(ε2t − 1)2]E

[∂ log gt

∂φ

]lim

T→∞

1

T

T

∑t=1

[ψj (t/T )]j

= E [(ε2t − 1)2]E

[∂ log gt

∂φ

](1, 0, . . . , 0)⊤

√T

∂ℓT (φ0, θ0)

∂θj

= − 1√T

T

∑t=1

(ε2t − 1)ψj (t/T ),

because∫ 1

0ψj (u)du = 0 for all j ≥ 1.

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Therefore, the semiparametric efficient score function is

∂ℓ∗T (φ0)

∂φ=

1

T

T

∑t=1

(ε2t − 1)

[∂ log gt

∂φ− E

(∂ log gt

∂φ

)]

The asymptotic variance of the efficient estimator is

avar(T 1/2φ) =E[(ε2t − 1)2

]

var[

∂ log gt

∂φ

]

Efficient estimator of m has variance

avar(T 1/2h1/2m(u)) = m(u)2E[(ε2t − 1)2

]

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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MULTIVARIATE CASE

We suppose that yt ∈ RN satisfies the model

yt = Σ(t/T )1/2G 1/2t εt ,

where:

εt is i.i.d. (or mds)Σ(u), u ∈ [0, 1] is a deterministic covariance matrix ofunknown functional form, smooth or cadlagGt ∈ Ft−1 is a parametric stochastic covariance matrix processwith EGt = IN . For example, Gt could be a unit BEKK process

Gt(φ) = IN − AA⊤ − BB⊤ +

AΣ(t − 1/T )−1/2yt−1y⊤t−1Σ(t − 1/T )−1/2A⊤

+BGt−1B⊤

in which case φ = (vec(A)⊤, vec(B)⊤)⊤ denote the freeparameters of Gt .

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Conditional variance

var(yt |Ft−1) = Σ(t/T )1/2GtΣ(t/T )1/2

Unconditional Variance

var(yt) = Σ(t/T )

Let W (t , j) = Σ(t − j/T )1/2 ⊗ Σ(t − j/T )1/2,zt = vec(G 1/2

t εt ε⊤t G 1/2t − I ), and Mj = E [ztz

⊤t−j ]. The local

autocorrelation matrix is

Ψ(t , j) = diag[Γ(t , 0)]−1/2Γ(t , j)diag[Γ(t , 0)]−1/2.

Γ(t , 0) = D+W (t , 0)MW (t , 0)D⊤+

Γ(t , j) = D+W (t , 0)MjW (t , j)D⊤+ .

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Step 1 Initial Estimation of Σ

Under the model assumptions,

E [yty⊤t ] = Σ(t/T )

for all t with t = 1, . . . , T . Therefore, one can estimate Σ(u) bythe Poo and Linton (2001) estimator

Σ(u) =∑

Tt=1 Kh(u − t/T )yty

⊤t

∑Tt=1 Kh(u − t/T )

,

where K is a kernel, h is a bandwidth, and Kh(.) = K (./h)/h.This can be interpreted as the minimizer of the local log-likelihoodcriterion

lT (Ω; u) = −1

2

T

∑t=1

Kh(u − t/T )l(Ω; yt),

l(Ω; yt) = log det(Ω) + y⊤t Ω−1yt = log det(Ω) + tr(yty

⊤t Ω−1)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Step 2 Initial Estimation of φFirst, one computes the profiled G , i.e., for each φ, let

Gt(φ) = I − AA⊤ − BB⊤

+AΣ(t − 1/T )−1/2yt−1y⊤t−1Σ(t − 1/T )−1/2A⊤

+BGt−1(φ)B⊤,

and then computes the profiled global likelihood function

ℓT (φ) =T

∑t=1

log det Ωt(φ) +T

∑t=1

y⊤t Ω−1

t (φ)yt

Ωt(φ) = Σ(t/T )1/2Gt(φ)Σ(t/T )1/2

Maximize ℓT (φ) with respect to φ to give φ.

Since Σ(t/T ) does not depend on φ, can replace ℓT (φ) by

ℓT (φ) =T

∑t=1

log det Gt(φ) +T

∑t=1

y⊤t G−1

t (φ)yt

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Step 3 Improved Estimation

Suppose one knew the random variable Gt , how to improvethe estimate of Σ(t/T ) and hence of φ? Unlike in scalar caseone cant just ”divide through” by G 1/2

t , since

G−1/2t yt = G−1/2

t Σ(t/T )1/2G 1/2t εt 6= Σ(t/T )1/2εt .

Our approach is to treat Gt as fixed known numbers inside thelocal likelihood. Let θ = vech(Θ) ∈ R

N(N+1)/2 be the uniqueelements of Θ.

lT (θ; u) = −1

2

T

∑t=1

Kh(u − t/T )l(Ωt(θ); yt),

Ωt(θ) = ΘGtΘ

Then maximize lT (θ; u) with respect to θ. In practice weapply this with the estimated G

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Next one computes a new profiled G , i.e., for each φ, let

Gt(φ) = I − AA⊤ − BB⊤

+AΣ(t − 1/T )−1/2yt−1y⊤t−1Σ(t − 1/T )−1/2A⊤

+BGt−1(φ)B⊤,

where some initialization G0(φ) is chosen, and then compute theprofiled global likelihood function

ℓT (φ) =T

∑t=1

log det Ωt(φ) +T

∑t=1

y⊤t Ω−1

t (φ)yt

Ωt(φ) = Θφ(t/T )Gt(φ)Θφ(t/T ),

and maximize it with respect to φ to give φ and θφ and hence

Σ(u) = Θ2φ(u).

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Distribution theory

Can obtain asymptotic normality for estimators of Σ(u) and φ.

For the inefficient estimators the asymptotic variances are verycomplicated because errors are not martingale differences andthe smoothing variable is t/T

In the multivariate case, I conjecture that the semiparametricefficient score function is

∂ℓ∗T (φ0)

∂φi

=1√T

T

∑t=1

vec

(G−1/2

t

∂Gt

∂φi

G−1/2t − E

[G−1/2

t

∂Gt

∂φi

G−1/2t

])⊤

×vec(

εt ε⊤t − IN

)

The efficient estimator has a simple asymptotic variance

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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APPLICATIONWe apply the proposed estimator to the bivariate series of dailyDow Jones and NASDAQ index returns, March 26, 1990 to March23, 2000.

Parameter First stage Efficient

a11 0.103716 0.111952a12 0.031615 0.047084a21 0.061927 0.062203a22 0.319424 0.330660b11 0.922463 0.929655b12 0.011324 0.006116b21 -0.219697 -0.181059b22 0.863677 0.865463

Table: Estimated Parameters of Gt using the first stage and the efficientestimator.

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Looking at the estimated conditional and unconditional standarddeviation and correlation plots in Figures 1 to 6, no majordifferences can be detected visually between the first stage andefficient estimator.Note that the decline in correlations towards the end of thesample, due to the decoupling of technology and brick and mortarstocks during the New Economy boom, is more pronounced in ourcase than it is using DCC or OGARCH models.While the conditional correlation attains 95 percent at times, theunconditional correlation never goes beyond 90 percent.

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Dow Jones volatility (first stage)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Dow Jones volatility (efficient)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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NASDAQ volatility (first stage)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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NASDAQ volatility (efficient)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Correlation (first stage)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Correlation (efficient)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

Page 25: Efficient Estimation of a Multivariate Multiplicative Volatility Model · 2015. 3. 2. · Efficient Estimation of a Multivariate Multiplicative Volatility Model Christian Hafner and

One can allow Σ to have a finite number of discontinuities by usingonly one sided kernels. Suppose that our model is that for someknown union of intervals U = ∪L

ℓ=1[uℓ−, uℓ

+] ⊂ [0, 1],

Σ(u) = Σc(u) + Σd1(u ∈ U),

where Σc(·) is a smooth unknown function and Σd is an unknownmatrix. This model is potentially useful for studying the effect ofbusiness cycles on volatility in which case U might correspond torecession periods. Σc(u) is estimated by the standard two-sidedkernel estimator. We now show how to estimate Σd .

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

Page 26: Efficient Estimation of a Multivariate Multiplicative Volatility Model · 2015. 3. 2. · Efficient Estimation of a Multivariate Multiplicative Volatility Model Christian Hafner and

Let

Σ−(u) =∑

Tt=1 K−

h (u − t/T )yty⊤t

∑Tt=1 K−

h (u − t/T )

Σ+(u) =∑

Tt=1 K+

h(u − t/T )yty

⊤t

∑Tt=1 K+

h(u − t/T )

,

where K−, K+ are respectively left and right sided kernels definedon [−1, 0] and [0, 1] respectively, say. Then let

Σd =L

∑ℓ=1

wℓ−(

Σ+(u−)− Σ−(u−))

+ wℓ+

(Σ−(u+)− Σ+(u+)

)

for some weighting sequence wℓ−, wℓ+Lℓ=1 with

∑Lℓ=1 wℓ− + wℓ+ = 1.

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

Page 27: Efficient Estimation of a Multivariate Multiplicative Volatility Model · 2015. 3. 2. · Efficient Estimation of a Multivariate Multiplicative Volatility Model Christian Hafner and

Volatility of DJ with structural break during NBER recession(July 1990 to March 1991)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Volatility of NASDAQ with structural break during NBER recession(July 1990 to March 1991)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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Correlation with structural break during NBER recession(July 1990 to March 1991)

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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CONCLUDING REMARKS

Multivariate multiplicative volatility model

Some properties are similar to univariate case, others not so

Establish consistency and asymptotic normality of estimators.Efficiency almost.

Computation ok.

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH

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THE END

Christian Hafner and Oliver Linton UCL and LSE London School of Economics

Multivariate Spline GARCH