Supplementary material for "Real-time GARCH: Does Current Information Matter?" Ekaterina Smetanina University of Cambridge January 30, 2017 Abstract
In this supplement, we present derivations for extended model with leverage and feedback eect as well as further empirical evidence on the performance of the basic and extended models.
1 Real-time GARCH model with leverage eect. We use the same numeration of the theorems as in the main paper and show how the proofs should be modi ed for the extended vesions of the model.
Proof of Theorem 1.
Now consider the model with leverage given by: rt = λt t 2 λ2t = α + βλ2t−1 + γrt−1 + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0)
where {t } is i.i.d. random variables such that E (t ) = 0, E (2t ) = 1 with the density f . 1
To compute the probability of P (rt ≤ c) note that the rst equation can be rewritten as q 2 + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0) t rt = α + βλ2t−1 + γrt−1
such that q 2 + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0) t ≤ c) P (rt ≤ c) = P ( α + βλ2t−1 + γrt−1
Since the scaling factor of t is positive there is one unique value of d such that q
2 α + βλ2t−1 + γrt−1 + ϕ1 e2 1(e>0) + ϕ2 e2 1(e≤0) e ≤ c
for all e ≤ d. To obtain d let's rst square the above equation such that 2 (α + βλ2t−1 + γrt−1 + ϕ1 d2 1(d>0) + ϕ2 d2 1(d≤0) )d2 = c2 2 (α + βλ2t−1 + γrt−1 )d2 + ϕ1 d4 1(d>0) + ϕ2 d4 1(d≤0) = c2
This quadratic equation has four solutions which are given by: x1,2
x3,4
sp b2t−1 + 4c2 (ϕ1 1(d>0) + ϕ2 1(d≤0) ) − bt−1 =± 2(ϕ1 1(d>0) + ϕ2 1(d≤0) )
s p b2t−1 + 4c2 (ϕ1 1(d>0) + ϕ2 1(d≤0) ) + bt−1 =± − 2(ϕ1 1(d>0) + ϕ2 1(d≤0) )
2 with bt−1 = α + βλ2t−1 + γrt−1 . We disregard x3,4 since we are only interested in the real
valued solutions, such that we have: sp b2t−1 + 4c2 (ϕ1 1(d>0) + ϕ2 1(d≤0) ) − bt−1 d(c) = sign(c) 2(ϕ1 1(d>0) + ϕ2 1(d≤0) )
Since the sign of d(c) is determined by the sign of "c" then corresponding conditions d > 0
2
and d < 0 can be substituted with c > 0 and c < 0. Then eq. 1 becomes sp b2t−1 + 4c2 (ϕ1 1(c>0) + ϕ2 1(c≤0) ) − bt−1 d(c) = sign(c) 2(ϕ1 1(c>0) + ϕ2 1(c≤0) )
Recall that the CDF of rt is given by ˆ
d(c)
P (rt ≤ c) =
f (x)dx −∞
To obtain the density we use Leibniz integral rule with variable limits such that:
∂P (rt ≤ c) ∂d(c) = |c=r f (d(r)) = ∂c ∂c s 2(ϕ1 1(r>0) + ϕ2 1(r≤0) ) p = sign(r)r × 2 2 (bt−1 + 4r (ϕ1 1(r>0) + ϕ2 1(r≤0) ))( b2t−1 + 4r2 (ϕ1 1(r>0) + ϕ2 1(r≤0) ) − bt−1 ) s p −bt−1 + b2t−1 + 4r2 (ϕ1 1(r>0) + ϕ2 1(r≤0) ) )= f (sign(r) 2(ϕ1 1(r>0) + ϕ2 1(r≤0) ) r p 2 = f (d(r)) 2 d(r) bt−1 + 4r (ϕ1 1(r>0) + ϕ2 1(r≤0) ) fr (r) =
Remark: Note that
∂sign(c) ∂c
= 2δ(c), where δ(·) is a Dirac delta function which is zero
everywhere except at 0, where δ(0) = ∞. Recall that we solve the problem of zero values by setting those values to a very small (but not zero) value, thus in our case δ(c) = 0 ∀c .
Proof of Theorem 2. The general model is given by: rt = λt t
(1)
2 + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0) λ2t = α + βλ2t−1 + γrt−1
(2)
3
Since the error term t is i.i.d, it is then obvious that the error process (t )t∈Z is always strictly stationary and ergodic. Thus, (rt )t∈Z is a strictly stationary process if and only if (λt )t∈Z is strictly stationary. Therefore, the task of deriving the strict stationarity conditions
for the whole process (rt , λt )t∈Z can be reduced to deriving strict stationarity conditions for (λ2t )t∈Z , given by eq.(2).
Let's now express (λ2t )t∈Z solely in terms of the error process (t )t∈Z . Repeatedly subsituting for λ2t−1 in eq.(2) , we have: ! t t i−1 Y X Y λ2t = λ0 (β + γ2t−i ) + (β + γ2t−j−1 ) (α + ϕ1 2t−i 1(t >0) + ϕ2 2t−i 1(t ≤0) ), t ≥ 2 (3) i=1
i=0
j=0
In order for eq.(3) to be well de ned we need either to assume the trivial σ -algebra F0 (and a probability measure µ0 ) for the starting value λ20 or to assume that the system extends in nitely far into the past. We proceed by implementing the former approach, de ning:
P[λ20 ∈ Γ] = µ0(Γ)
∀Γ ∈ B
and
(4)
µ0 ((0, ∞)) = 1,
where B denotes the Borel sets on [0, ∞). In order to nd strict stationarity conditions of σt2 we next rewrite eq.2 in the form of the stochastic dierence equation Yt+1 = At Yt + Bt , where Yt , At and Bt are given by: At = β + γ2t ,
Bt = α + ϕ1 2t+1 1(t >0) + ϕ2 2t+1 1(t ≤0)
and
Yt = λ2t
(5)
Since sequences (At )t∈N and (Bt )t∈N are measurable transformations of the stictly stationary and ergodic process ()t∈N we can make use of the Theorem 3.5.8 of Stout (1974) to claim that these sequences are strictly stationary and ergodic as well as the sequence Ψ = (At , Bt )t∈N . If we rewrite eq.3 in terms of eq.5, it follows that (Yt )t∈N = (λ2t )t∈N is the
solution of the stochastic dierence equation Yt+1 = At Yt + Bt . Every such solution then
4
should satisfy the following representation: Yt+1 = At Yt +Bt = At At−1 Yt−1 +At Bt−1 +Bt = At At−1 At−2 Yt−2 +At At−1 Bt−2 +At Bt−1 +Bt = ! ! t t i−1 Y X Y = ··· = At−i Y0 + At−j Bt−i , (6) i=0
with the usual convention that
i=0
Q−1
j=0
j=0
At−j = 1 for the product over an empty index set.
Let's denote by Y an arbitrary R-valued random variable, which is de ned on the same probability space as Ψ. The solution yt (Y, Ψ) of eq.(6) is then given by: 1
yt (Y, Ψ) =
t−1 Y
! Ai
Y0 +
i=0
t−1 X
t−1 Y
i=0
j=t−i
! Aj
Bt−i−1
We have shown earliethat the sequence Ψ = (At , Bt ) is strictly stationary and ergodic, we can now apply Theorem 1 of Brandt (1986) to deduce that yt (Ψ) =
P∞ Qt−1 i=0
j=n−i
Aj Bt−i−1 ,
t ∈ N is strictly stationary solution if and only if the following conditions are satis ed:
P(A0 = 0) > 0 or −∞ 5 E log |A0 | < 0
E ( log |B0 |)+ < ∞,
where x+ = max(0, x) for x ∈ R. Plugging in the expressions for A0 and B0 , given by eq.(5) we get the following strict stationarity conditions:
−∞ 5 E log β + γ20 < 0
+ E log α + ϕ1 20 1(0 >0) + ϕ2 20 1(0 ≤0) < ∞,
This completes the proof.
1 Note
that
Y
and
Ψ
should not necessarily be independent.
5
Proof of Theorem 3. Consider the general model: rt = λt t
(7)
2 λ2t = α + βλ2t−1 + γrt−1 + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0)
(8)
Substituting eq.(8) into the squared eq.(7) yields: 2 rt2 = α2t + βλ2t−1 2t + γrt−1 2t + ϕ1 4t 1(t >0) + ϕ2 4t 1(t ≤0)
(9)
The rst moments of rt2 and λ2t are given by:
2 E[rt2 ] = α + βE[λ2t−1 ] + γE[rt−1 ] + ϕ1 E 4t |t > 0 + ϕ2 E 4t |t ≤ 0 = 2 = α + βE[λ2t−1 ] + γE[rt−1 ] + E ε4t |εt > 0 ϕ1 + E ε4t |εt ≤ 0 ϕ2
(10)
2 E[λ2t ] = α + βE[λ2t−1 ] + γE[rt−1 ] + ϕ1 E 2t |t > 0 + ϕ2 E 2t |t ≤ 0 = 2 ] + ϕ1 + ϕ2 = α + βE[λ2t−1 ] + γE[rt−1
(11)
Combining eq.(10)-(11) then yields: E[rt2 ] = E[λ2t ] + (E ε4t − 1)(ϕ1 + ϕ2 )
Recursively subsituting eq.(12) into eq.(11) we get:
6
(12)
2 E[λ2t ] = α + βE[λ2t−1 ] + γE[rt−1 ] + ϕ1 + ϕ2 = α + βE[λ2t−1 ]+ + γ E[λ2t−1 ] + (E ε4t − 1)(ϕ1 + ϕ2 ) + ϕ1 + ϕ2
(13)
Rearranging the above equation and denoting by η = E [ε4t ] − 1 we get: E[λ2t ] = α + (ϕ1 + ϕ2 )(ηγ + 1) + (β + γ)E[λ2t−1 ] = = (α + (ϕ1 + ϕ2 )(ηγ + 1)) + (β + γ) α + (ϕ1 + ϕ2 )(ηγ + 1) + (β + γ)E[λ2t−2 ] = · · · t−2 X = · · · = (α + (ϕ1 + ϕ2 )(ηγ + 1)) (β + γ)i + (β + γ)t−1 E[λ21 ] i=0
Assuming weak stationarity, the E[λ21 ] is then given by: 4 α + (ϕ1 + ϕ2 ) (E [εt ] − 1)γ + 1 E[λ21 ] =
1 − (β + γ)
In order for E[λ21 ] to be positive and nite the following conditions of one of these two cases must be satis ed: β+γ <1 α + (ϕ + ϕ )((E [ε4 ] − 1)γ + 1) > 0 1 2 t
β+γ >1 α + (ϕ + ϕ )((E [ε4 ] − 1)γ + 1) < 0 1 2 t
Proof of Theorem 4. Recall that E[rt2 ] = E[λ2t ] + η(ϕ1 + ϕ2 ), where η = E [ε4t ] − 1. Then the conditions for weak
7
stationarity of E[r12 ] become β+γ <1 α + (ϕ + ϕ )(1 + η − βη) > 0 1 2
or
β+γ >1 α + (ϕ + ϕ )(1 + η − βη) < 0 1 2
and the unconditional variance of rt2 , E[r12 ], is given by E[r12 ] =
Proof of Theorem 5.
α + (ϕ1 + ϕ2 )(1 + η − βη) 1 − (β + γ)
The proof is very similar to the one for the baseline model, we
therefore omit it.
Proof of Theorem 6. 2 λ4t = α2 + 2αβλ2t−1 + 2αγrt−1 + 2α2t (ϕ1 1(t >0) + ϕ2 1(t ≤0) ) + β 2 λ4t−1 + 2 4 + 2βγλ2t−1 rt−1 + +γ 2 rt−1 + 2βλ2t−1 2t (ϕ1 1(t >0) + ϕ2 1(t ≤0) )+ 2 + 2γrt−1 2t (ϕ1 1(t >0) + ϕ2 1(t ≤0) ) + +4t (ϕ21 1(t >0) + ϕ22 1(t ≤0) )
(14)
Let's make use of the following notation, µj := E[jt (ϕ1 1(t >0) + ϕ2 1(t ≤0) )] (moments of t (ϕ1 1(t >0) +ϕ2 1(t <0) ) are derived in Appendix 2) and k := ϕ1 1(t >0) +ϕ2 1(t ≤0) . Multiplying
eq. (14) by 4t from both sides and taking expectations we get E[rt4 ] = E[λ4t 4t ] which is then given by E[rt4 ] = E[λ4t rt4 ] = α2 µ4 + 2αβµ4 E[λ4t−1 ] + 2αγ E[λt−1 ]2 + 2(ϕ1 + ϕ2 )) + β 2 µ4 E[λ2t−1 ]+ 4 + γ 2 µ4 E[rt−1 ] + 2βγµ4 E[λ2t−1 ] E[λ2t−1 ] + 2(ϕ1 + ϕ2 )) + 2αµ6 k + 2βµ6 kE[λ2t−1 ]+ + 2γµ6 E[λ2t−1 ] + 2(ϕ1 + ϕ2 ) + µ8 k 2
8
Rearranging the above equation and assuming that rt is fourth order stationary we get E[r14 ] 1 − γ 2 µ4 = α2 µ4 + µ8 k 2 + 2αµ6 k + 4(ϕ1 + ϕ2 )αγµ4 + 4βγµ4 (ϕ1 + ϕ2 )+ 2 + 4γµ6 (ϕ1 + ϕ2 ) + E[λ2t−1 ] 2αβµ4 + β 2 µ4 + 2βµ6 k + 2αγµ4 + 2γµ6 + 2βγµ4 E[λ2t−1 ]
which can be compactly written in the following form 2
E[r14 ] =
ξ1 + E[λ21 ]ξ2 + 2βγµ4 [E[λ21 ]] 1 − γ 2 µ4
where ξ1 = (α2 µ4 + µ8 k2 + 2αµ6 k + 4(ϕ1 + ϕ2 )αγµ4 + 4βγµ4 (ϕ1 + ϕ2 ) + 4γµ6 (ϕ1 + ϕ2 )) > 0 and ξ2 = (2αβµ4 + β 2 µ4 + 2βµ6 k + 2αγµ4 + 2γµ6 ) > 0. Since E[r4 ] must be positive, γ 2
must also satisfy: 1 − γ 2 E[4t ] > 0
⇔
γ2 <
1 E[4t ]
2 Real-time GARCH model with leverage and feedback.
Proof of Theorem 1. Now consider the model with leverage and feedback eects which is given by: rt = λt t 2 2 1(rt >0) + γ2 rt−1 1(rt ≤0) + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0) λ2t = α + βλ2t−1 + γ1 rt−1
where {t } is i.i.d. random variables such that E (t ) = 0, E (2t ) = 1 with the density f . To compute the probability of P (rt ≤ c) note that the rst equation can be rewritten as rt =
q 2 2 α + βλ2t−1 + γ1 rt−1 1(rt >0) + γ2 rt−1 1(rt ≤0) + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0) t
such that
9
q 2 2 1(rt ≤0) + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0) t ≤ c) 1(rt >0) + γ2 rt−1 P (rt ≤ c) = P ( α + βλ2t−1 + γ1 rt−1
Since the scaling factor of t is positive there is one unique value of d such that q
2 2 1(rt ≤0) + ϕ1 e2 1(e>0) + ϕ2 e2 1(e≤0) e ≤ c 1(rt >0) + γ2 rt−1 α + βλ2t−1 + γ1 rt−1
for all e ≤ d. To obtain d let's rst square the above equation such that 2 2 (α + βλ2t−1 + γ1 rt−1 1(rt >0) + γ2 rt−1 1(rt ≤0) + ϕ1 d2 1(d>0) + ϕ2 d2 1(d≤0) )d2 = c2 2 2 (α + βλ2t−1 + γ1 rt−1 1(rt >0) + γ2 rt−1 1(rt ≤0) )d2 + ϕ1 d4 1(d>0) + ϕ2 d4 1(d≤0) = c2
This quadratic equation has four solutions which are given by: x1,2
x3,4
sp b2t−1 + 4c2 (ϕ1 1(d>0) + ϕ2 1(d≤0) ) − bt−1 =± 2(ϕ1 1(d>0) + ϕ2 1(d≤0) )
s p b2t−1 + 4c2 (ϕ1 1(d>0) + ϕ2 1(d≤0) ) + bt−1 =± − 2(ϕ1 1(d>0) + ϕ2 1(d≤0) )
2 2 with bt−1 = α + βλ2t−1 + γ1 rt−1 1(rt >0) + γ2 rt−1 1(rt ≤0) . We disregard x3,4 since we are only
interested in the real valued solutions, such that we have: sp b2t−1 + 4c2 (ϕ1 1(d>0) + ϕ2 1(d≤0) ) − bt−1 d(c) = sign(c) 2(ϕ1 1(d>0) + ϕ2 1(d≤0) )
Since the sign of d(c) is determined by the sign of "c" then corresponding conditions d > 0 and d < 0 can be substituted with c > 0 and c ≤ 0. Then eq. 2 becomes sp b2t−1 + 4c2 (ϕ1 1(c>0) + ϕ2 1(c≤0) ) − bt−1 d(c) = sign(c) 2(ϕ1 1(c>0) + ϕ2 1(c≤0) )
10
Recall that the CDF of rt is given by ˆ
d(c)
P (rt ≤ c) =
f (x)dx −∞
To obtain the density we use Leibniz integral rule with variable limits such that:
∂P (rt ≤ c) ∂d(c) = |c=r f (d(r)) = ∂c s∂c 2(ϕ1 1(r>0) + ϕ2 1(r≤0) ) p = sign(r)r × 2 (bt−1 + 4r2 (ϕ1 1(r>0) + ϕ2 1(r≤0) ))( b2t−1 + 4r2 (ϕ1 1(r>0) + ϕ2 1(r≤0) ) − bt−1 ) s p −bt−1 + b2t−1 + 4r2 (ϕ1 1(r>0) + ϕ2 1(r≤0) ) )= f (sign(r) 2(ϕ1 1(r>0) + ϕ2 1(r≤0) ) r p 2 = f (d(r)) d(r) bt−1 + 4r2 (ϕ1 1(r>0) + ϕ2 1(r≤0) ) fr (r) =
Remark: Note that
∂sign(c) ∂c
= 2δ(c), where δ(·) is a Dirac delta function which is zero
everywhere except at 0, where δ(0) = ∞. Recall that we solve the problem of zero values by setting those values to a very small (but not zero) value, thus in our case δ(c) = 0 ∀c .
Proof of Theorem 2. The general model is given by: rt = λt t
(15)
2 2 1(rt >0) + γ2 rt−1 1(rt ≤0) + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0) λ2t = α + βλ2t−1 + γ1 rt−1
(16)
Since the error term t is i.i.d, it is then obvious that the error process (t )t∈Z is always strictly stationary and ergodic. Thus, (rt )t∈Z is a strictly stationary process if and only if (λt )t∈Z is strictly stationary. Therefore, the task of deriving the strict stationarity conditions 11
for the whole process (rt , λt )t∈Z can be reduced to deriving strict stationarity conditions for (λ2t )t∈Z , given by eq.(16).
Let's now express (λ2t )t∈Z solely in terms of the error process (t )t∈Z . Repeatedly subsituting for λ2t−1 in eq.(16) , we have: ! t t i−1 Y X Y λ2t = λ0 (β+(γ1 +γ2 )2t−i )+ (β + (γ1 + γ2 )2t−j−1 ) (α+ϕ1 2t−i 1(t >0) +ϕ2 2t−i 1(t ≤0) ), t ≥ 2 i=1
i=0
j=0
(17)
In order for eq.(17) to be well de ned we need either to assume the trivial σ -algebra F0 (and a probability measure µ0 ) for the starting value λ20 or to assume that the system extends in nitely far into the past. We proceed by implementing the former approach, de ning:
P[λ20 ∈ Γ] = µ0(Γ)
∀Γ ∈ B
and
(18)
µ0 ((0, ∞)) = 1,
where B denotes the Borel sets on [0, ∞). In order to nd strict stationarity conditions of σt2 we next rewrite eq.(16) in the form of the stochastic dierence equation Yt+1 = At Yt + Bt ,
where Yt , At and Bt are given by:
At = β + (γ1 + γ2 )2t ,
Bt = α + ϕ1 2t+1 1(t >0) + ϕ2 2t+1 1(t ≤0)
and
Yt = λ2t
(19)
Since sequences (At )t∈N and (Bt )t∈N are measurable transformations of the stictly stationary and ergodic process ()t∈N we can make use of the Theorem 3.5.8 of Stout (1974) to claim that these sequences are strictly stationary and ergodic as well as the sequence Ψ = (At , Bt )t∈N . If we rewrite eq.17 in terms of eq.(19), it follows that (Yt )t∈N = (λ2t )t∈N is
the solution of the stochastic dierence equation Yt+1 = At Yt + Bt . Every such solution then
12
should satisfy the following representation: Yt+1 = At Yt +Bt = At At−1 Yt−1 +At Bt−1 +Bt = At At−1 At−2 Yt−2 +At At−1 Bt−2 +At Bt−1 +Bt = ! ! t t i−1 Y X Y = ··· = At−i Y0 + At−j Bt−i , (20) i=0
with the usual convention that
i=0
Q−1
j=0
j=0
At−j = 1 for the product over an empty index set.
Let's denote by Y an arbitrary R-valued random variable, which is de ned on the same probability space as Ψ. The solution yt (Y, Ψ) of eq.(20) is then given by: 2
yt (Y, Ψ) =
t−1 Y
! Ai
Y0 +
i=0
t−1 X
t−1 Y
i=0
j=t−i
! Aj
(21)
Bt−i−1
We have shown earlier that the sequence Ψ = (At , Bt ) is strictly stationary and ergodic, we can now apply Theorem 1 of Brandt (1986) to deduce that yt (Ψ) =
P∞ Qt−1 i=0
j=n−i
Aj Bt−i−1 ,
t ∈ N is strictly stationary solution if and only if the following conditions are satis ed:
P(A0 = 0) > 0 or −∞ 5 E log |A0 | < 0
E ( log |B0 |)+ < ∞,
where x+ = max(0, x) for x ∈ R. Plugging in the expressions for A0 and B0 , given by eq.19 we get the following strict stationarity conditions:
−∞ 5 E log β + (γ1 + γ2 )20 < 0
+ E log α + ϕ1 20 1(0 >0) + ϕ2 20 1(0 ≤0) < ∞,
This completes the proof.
2 Note
that
Y
and
Ψ
should not necessarily be independent.
13
Proof of Theorem 3. Consider the general model: rt = λt t
(22)
2 2 λ2t = α + βλ2t−1 + γ1 rt−1 1(rt >0) + γ2 rt−1 1(rt ≤0) + ϕ1 2t 1(t >0) + ϕ2 2t 1(t ≤0)
(23)
Substituting eq.(23) into the squared eq.(22) yields:
2 2 rt2 = α2t + βλ2t−1 2t + (γ1 rt−1 1(rt >0) + γ2 rt−1 1(rt ≤0) )2t + ϕ1 4t 1(t >0) + ϕ2 4t 1(t ≤0)
(24)
The rst moments of rt2 and λ2t are given by:
2 2 E[rt2 ] = α + βE[λ2t−1 ] + γ1 E[rt−1 |rt > 0] + γ2 E[rt−1 |rt ≤ 0] + ϕ1 E 4t |t > 0 + ϕ2 E 4t |t ≤ 0
(25) and 2 2 E[λ2t ] = α + βE[λ2t−1 ] + γ1 E[rt−1 |rt > 0] + γ2 E[rt−1 |rt ≤ 0] + ϕ1 E 2t |t > 0 + ϕ2 E 2t |t ≤ 0
(26) Combining eq.(25)-(26) then yields: E[rt2 ]
=
E[λ2t ]
4 + (ϕ1 + ϕ2 ) E εt − 1
14
(27)
Denoting by η = E [ε4t ] − 1 and recursively subsituting eq.(27) into eq.(26) we get: 2 2 E[λ2t ] = α+βE[λ2t−1 ]+γ1 E[rt−1 |rt > 0]+γ2 E[rt−1 |rt ≤ 0]]+ϕ1 +ϕ2 = α+βE[λ2t−1 ]+ + (γ1 + γ2 ) E[λ2t−1 ] + η(ϕ1 + ϕ2 ) + ϕ1 + ϕ2 (28)
Rearranging the the above equation we get E[λ2t ] = α + (ϕ1 + ϕ2 )(η(γ1 + γ2 ) + 1) + (β + (γ1 + γ2 ))E[λ2t−1 ] = = (α+(ϕ1 +ϕ2 )(η(γ1 +γ2 )+1))+(β+(γ1 +γ2 )) α + (ϕ1 + ϕ2 )(η(γ1 + γ2 ) + 1) + (β + (γ1 + γ2 ))E[λ2t−2 ] = · = · · · = (α + (ϕ1 + ϕ2 )(η(γ1 + γ2 ) + 1))
t−2 X
(β + (γ1 + γ2 ))i + (β + (γ1 + γ2 ))t−1 E[λ21 ]
i=0
Assuming weak stationarity, the E[λ21 ] is then given by: E[λ21 ] =
α + (ϕ1 + ϕ2 ) [η(γ1 + γ2 ) + 1] 1 − (β + γ1 + γ2 )
In order for E[λ21 ] to be positive and nite the following conditions of one of these two cases must be satis ed: β + γ1 + γ2 < 1 α + (ϕ + ϕ ) [η(γ + γ ) + 1] > 0 1 2 1 2
β + γ1 + γ2 > 1 α + (ϕ + ϕ ) [η(γ + γ ) + 1] < 0 1 2 1 2
Proof of Theorem 4. Recall that E[rt2 ] = E[λ2t ] + η(ϕ1 + ϕ2 ), thus the conditions for weak stationarity of E[r12 ]
15
become
β + γ1 + γ2 < 1 α + (ϕ + ϕ )(1 + η − βη) > 0 1
or
2
β + γ1 + γ2 > 1 α + (ϕ + ϕ )(1 + η − βη) < 0 1
2
and the unconditional variance of rt2 , E[r12 ], is given by E[r12 ] =
Proof of Theorem 5.
α + (ϕ1 + ϕ2 )(1 + η − βη) 1 − (β + γ1 + γ2 )
The proof is very similar to the one for the baseline model, we
therefore omit it.
Proof of Theorem 6. 2 2 λ4t = α2 +2αβλ2t−1 +2α(γ1 rt−1 1(rt >0) +γ2 rt−1 1(rt ≤0) )+2α2t (ϕ1 1(t >0) +ϕ2 1(t ≤0) )+β 2 λ4t−1 +
2 2 4 4 +2βλ2t−1 (γ1 rt−1 1(rt >0) +γ2 rt−1 1(rt ≤0) )++γ12 rt−1 1(rt >0) +γ22 rt−1 1(rt ≤0) +2βλ2t−1 2t (ϕ1 1(t >0) +ϕ2 1(t ≤0) )+ 2 2 +2(γ1 rt−1 1(rt >0) +γ2 rt−1 1(rt ≤0) )2t (ϕ1 1(t >0) +ϕ2 1(t ≤0) )+4t (ϕ21 1(t >0) +ϕ22 1(t ≤0) )
(29) Let's make use of the following notation, µj := E[jt (ϕ1 1(t >0) + ϕ2 1(t ≤0) )] (moments of t (ϕ1 1(t >0) + ϕ2 1(t <0) ) are derived in Appendix 2), k1 := γ1 1(rt >0) + γ2 1(rt <0) and k2 := ϕ1 1(t >0) + ϕ2 1(t <0) . Multiplying eq. (29) by 4t from both sides and taking expectations we
get E[rt4 ] = E[λ4t 4t ] which is then given by E[rt4 ] = E[λ4t rt4 ] = α2 µ4 +2αβµ4 E[λ4t−1 ]+2α(γ1 +γ2 ) E[λt−1 ]2 + 2(ϕ1 + ϕ2 )) +β 2 µ4 E[λ2t−1 ]+ 4 + k12 µ4 E[rt−1 ] + 2β(γ1 + γ2 )µ4 E[λ2t−1 ] E[λ2t−1 ] + 2(ϕ1 + ϕ2 )) + 2αµ6 k2 + 2βµ6 k2 E[λ2t−1 ]+ + 2(γ1 + γ2 )µ6 E[λ2t−1 ] + 2(ϕ1 + ϕ2 ) + µ8 k22
16
Rearranging the above equation and assuming that rt is fourth order stationary we get E[r14 ] 1 − k12 µ4 = α2 µ4 +µ8 k22 +2αµ6 k2 +4(ϕ1 +ϕ2 )α(γ1 +γ2 )µ4 +4β(γ1 +γ2 )µ4 (ϕ1 +ϕ2 )+ +4(γ1 +γ2 )µ6 (ϕ1 +ϕ2 )+E[λ2t−1 ] 2αβµ4 + β 2 µ4 + 2βµ6 k2 + 2α(γ1 + γ2 )µ4 + 2(γ1 + γ2 )µ6 + 2 + 2β(γ1 + γ2 )µ4 E[λ2t−1 ]
which can be compactly written in the following form E[r14 ]
ξ1 + E[λ21 ]ξ2 + 2β(γ1 + γ2 )µ4 [E[λ21 ]] = 1 − k12 µ4
2
where ξ1 = α2 µ4 + µ8 k22 + 2αµ6 k + 4(ϕ1 + ϕ2 )α(γ1 + γ2 )µ4 + 4β(γ1 + γ2 )µ4 (ϕ1 + ϕ2 ) + 4(γ1 + γ2 )µ6 (ϕ1 + ϕ2 ) > 0 and ξ2 = (2αβµ4 + β 2 µ4 + 2βµ6 k + 2α(γ1 + γ2 )µ4 + 2(γ1 + γ2 )µ6 ) > 0.
Since E[r4 ] must be positive, k12 = (γ12 + γ22 ) must also satisfy: 1 − k12 E[4t ] > 0
⇔
(γ12 + γ22 ) <
1 E[4t ]
3 Further empirical results
3.1 Data The original data for application was bought from Kibot and constitutes DJ30 1-minute high-frequency return data, of which we picked IBM and GE for our empirical application. The data was then aggregated to 5-minutes for calculating the 5-minute realized volatility to serve as a proxy of the end-of-the day volatility. We also use the S&P500 index historical returns which is freely available from the realized library at Oxford-Man Institute of Quantitative Finance. This data includes the open-to-close daily returns and the 5-minute realized volatility.
17
3.2 Empirical results We start by providing the descriptive statistics of all the stocks in the applications. We then present additional results for the forecast evaluations for a dierent (from the main paper) splitting point. In particular, for IBM and GE stocks we use 3500 and 1000 observations for estimation and evaluation respectively. For S&P500 index we use 2500 and 500 observations for estimation and evaluation respectively. In addition, we also present the results of Hansen's (2005) Test for Superior Predictive Ability (SPA) for all stocks and all forecasting horizons.
18
19
3477
4772
4762
Sample size
Min
Max
0.0146 -0.0874 0.1291
St.dev.
2.7 ∗ 10−4
0.0112 -0.0935 0.1022
−4.11 ∗ 10−4 0.0167 -0.1367 0.1322
2.14 ∗ 10−4
Mean
-0.2780
0.0619
0.1623
13.7925
11.9920
7.5856
16919
16080
4192
Skewness Kurtosis Normality Test
The sample period is January 2, 2002 through August 29, 2008. The Normality Test is the Jarque-Bera test which has χ2 distribution with 2 degrees of
S&P 500 28/01/2003-1/12/2016
2/01/1998-1/12/2016
2/01/1998 - 1/12/2016
Dates (DD/MM/YY)
freedom under the null hypothesis of normally distributed errors. The 5% critical value is, therefore, 5.99.
Note:
3
GE
IBM
1
2
Returns
Dataset #
Table 1: Descriptive statistics of the datasets.
20
MSE 7.7463 7.5221 7.6131 7.6200 7.5321 7.5321 7.4786 7.4709 10.6315 10.6642 0.0180 0.0180
pM CS 0.0860? 0.9800? 0.6450? 0.2310? 0.7930? 0.7930? 0.9800? 1?
MSE 8.1501 7.9944 7.7985 7.8180 8.2112 8.2112 8.2394 8.5474 12.0087 12.3391 0 0
pM CS 0.2260? 0.4950? 1? 0.9220? 0.0407 0.0407 0.0020 0.0020
GE MSE 3.0635 2.3407 2.3647 3.2466 3.0900 3.0900 3.5230 2.9446 3.3888 3.6002 0.0040 0.0030
pM CS 0.0200 0.7410? 1? 0.0076 0.0250 0.0250 0.0050 0.1410?
S&P 500
MSE 7.2190 7.1344 8.2480 7.3727 6.9187 6.9187 6.7465 6.7475 10.6647 10.7480 pM CS 0.3350? 0.3350? 0.0030 0.0190 0.3350? 0.3350? 1? 0.9000?
IBM
MSE 7.6631 7.4285 7.4972 7.5034 7.4990 7.4990 7.5338 7.1448 12.5353 12.8422
GE
5-step ahead volatility forecasts
S&P 500
pM CS MSE pM CS 0.0010 2.4406 0.8980? 0.4250? 2.2203 1? ? 0.4250 2.9394 0.0030 0.0020 2.6726 0.0840? 0.0650? 2.5065 0.0840? 0.0650? 2.5065 0.0840? 0.0005 3.0442 0.0010 ? 1 2.5014 0.4510? 0.0030 0 3.6407 0.0010 0.0030 0 3.8242 0.0010 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
1-step ahead volatility forecasts
Table 2: Forecasts evaluation based on MSE loss (full sample)
21
MSE 7.7152 8.1210 9.6969 7.3950 7.3143 7.3143 7.1370 6.7755 10.8180 10.8315
pM CS 0.0600? 0.0220 0.0040 0.1200? 0.2100? 0.2100? 0.4400? 1?
MSE 8.1713 7.8414 8.1607 7.7765 7.9002 7.9002 7.9059 7.7423 13.0564 13.1357
pM CS 0.0020 0.3370? 0 0.8810? 0.0330 0.0330 0.0070 1?
GE
MSE 2.3666 2.7384 3.2950 2.6656 2.8557 2.8557 3.0443 2.6921 4.9484 4.9833
S&P 500
pM CS 0.2280? 0 0.0010 0.0540? 0.0230 0.0230 0.0210 1?
IBM
MSE 7.2884 8.8373 8.4532 6.6619 7.5387 7.5387 7.5980 6.4964 9.5910 9.5793
MSE 8.7735 8.4173 1.8763 8.1922 8.5938 8.5938 8.5702 8.5495 13.4800 13.4706
pM CS 0 0.4340? 0 1? 0.2320? 0.2320? 0 0.4340?
GE
S&P 500
MSE 2.3117 3.0292 4.2312 2.6328 3.1654 3.1654 3.1615 2.7804 4.3769 4.0984
pM CS 1? 0.0500? 0 0.0860? 0.0220 0.0220 0.0280 0.0580? 0 0 0 0 0 0 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
15-step ahead volatility forecasts
pM CS 1? 0.0080 0.0080 0.2370? 0.0080 0.0080 0.0080 0.0080 0.0030 0 0 0.0030 0 0 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
10-step ahead volatility forecasts
Table 3: Forecasts evaluation based on MSE loss (full sample)
22
QLIKE 1.6699 1.6346 1.6463 1.7975 1.7491 1.7491 1.7680 1.7675 4.4698 4.5641 0 0
pM CS 0.2070? 1? 0.4190? 0 0.0300 0.0300 0.0110 0.0110
QLIKE 1.8956 1.8712 1.8469 1.9312 1.9110 1.9110 1.9126 1.9303 3.6475 3.5951 0.0020 0 0 0
pM CS 0.0050 0.0590? 1? 0.0050 0.0050 0.0050
GE QLIKE 1.1945 1.1731 1.2018 1.1964 1.1992 1.1992 1.2121 1.2190 1.7107 1.8226 0 0
0.0020 0.0010
0.0030 0.0030
pM CS 0.3730? 1? 0.1870? 0.2120?
S&P 500
QLIKE 1.5883 1.6342 1.7512 1.6917 1.5965 1.5965 1.5925 1.5892 4.8455 4.7744 0 0
pM CS 1? 0.0020 0 0.0020 0.0240 0.0240 0.0640? 0.9980?
IBM
QLIKE 1.8344 1.8209 1.8626 1.8733 1.8233 1.8233 1.8242 1.8214 3.8940 3.8821
0.0080 0.0080 0.0080
pM CS 0.0080 1? 0.0030 0.0030
0 0
0.9240?
GE
5-step ahead volatility forecasts
0.0020 0.0020
pM CS 1? 0.0040 0.0040 0.0130 0.0130 0.0130 0.0040 0.0960?
S&P 500 QLIKE 1.1494 1.2149 1.3692 1.1588 1.1586 1.1586 1.1885 1.1503 1.9021 2.0884
Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
1-step ahead volatility forecasts
Table 4: Forecasts evaluation based on QLIKE loss (full sample)
23
QLIKE 1.6380 1.7283 1.9339 1.6631 1.6477 1.6477 1.6564 1.6534 5.1174 5.2332 0 0
pM CS 1? 0.0170 0.0170 0.5300? 0.8600? 0.8600? 0.7000? 0.7500?
QLIKE 1.8422 1.8413 1.9578 1.8657 1.8539 1.8539 1.8523 1.8270 4.2593 3.9802 0 0
0.1380? 1?
0.0310 0.0310
pM CS 0.2140? 0.2140? 0.0050 0.0010
GE QLIKE 1.1708 1.3044 1.5653 1.1779 1.2177 1.2177 1.2309 1.2051 2.2480 2.4818 0 0
pM CS 1? 0.0020 0.0010 0.1420? 0.0020 0.0020 0.0020 0.0030
S&P 500
QLIKE 1.5951 1.7442 1.9423 1.5507 1.6301 1.6301 1.6765 1.5594 4.9485 4.9358 0 0
pM CS 0.1780? 0.0160 0.0010 1? 0.0300 0.0300 0.0170 0.7960?
IBM
QLIKE 1.8787 1.8815 2.0533 1.8766 1.9022 1.9022 1.8994 1.8750 4.3935 4.0953
0 0
pM CS 0.9800? 0.9730? 0.0005 0.9800? 0.0415 0.0415 0.4490? 1?
GE
15-step ahead volatility forecasts
0 0
pM CS 1? 0.0030 0 0.9760? 0.0320 0.0320 0.0320 0.0410
S&P 500 QLIKE 1.1889 1.3582 1.7091 1.1957 1.2735 1.2735 1.2662 1.2475 2.6542 2.8627
Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
10-step ahead volatility forecasts
Table 5: Forecasts evaluation based on QLIKE loss (full sample)
24
MSE 6.7167 6.3892 6.5041 6.7018 6.7627 6.7627 6.8142 6.9739 9.7043 9.8900 0.0010 0.0010
pM CS 0.2030? 1? 0.3580? 0.2610? 0.0710? 0.0710? 0.0480 0.0300
MSE 8.5147 8.2735 7.8039 8.9624 8.8659 8.8659 9.0494 9.1617 11.7697 11.8497
GE
0 0
pM CS 0.0010 0.0090 1? 0.0010 0.0010 0.0010 0.0010 0.0010
MSE 0.9792 0.9170 0.7468 0.8677 0.9356 0.9356 0.9556 1.0148 1.2867 1.2867 pM CS 0.0050 0.0080 1? 0.0080 0.0080 0.0080 0.0080 0.0050
0.0050 0.0050
S&P 500
MSE 5.5722 5.3574 6.9006 5.9271 5.4976 5.4976 5.6150 5.4914 9.3924 9.5864 pM CS 0.7020? 1? 0.0330 0.0910? 0.7410? 0.7410? 0.0910? 0.7410?
IBM
MSE 8.0393 7.7015 7.5956 8.7449 8.0786 8.0786 8.4446 8.1116 12.2431 12.3219
GE
5-step ahead volatility forecasts
S&P 500
pM CS MSE pM CS 0.0470 0.7123 0.0050 ? 0.7470 0.6629 0.1980? ? 1 0.6963 0.1570? 0.0010 0.7232 0.0050 0.0220 0.7242 0.0050 0.0220 0.7242 0.0050 0.0030 0.7215 0.0050 0.0030 0.5777 1? 0.0010 0 1.4434 0 0.0010 0 1.4242 0 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
1-step ahead volatility forecasts
Table 6: Forecasts evaluation based on MSE loss (pre-crisis period)
25
MSE 6.0329 6.3490 7.4060 6.1188 5.7626 5.7626 5.8161 5.8161 9.7543 9.8677
pM CS 0.5740? 0.1010? 0.0030 0.1010? 1? 1? 0.8670? 0.8670?
MSE 8.1277 7.7152 9.2257 8.7023 7.9221 7.9121 8.2775 8.0458 12.5970 12.6200
GE
pM CS 0.0150 1? 0.0010 0.0030 0.0200 0.0200 0.0150 0.0160
MSE 0.6861 0.6749 0.8790 0.7219 0.8677 0.8677 0.7932 0.7206 1.5362 1.4465
S&P 500
pM CS 0.3230? 0 0 0.8000? 0.8000? 0.8000? 0.8000? 1?
IBM
MSE 6.3493 7.3264 7.3221 6.2503 6.2536 6.2536 6.1988 6.1269 9.9399 9.9348
MSE 8.2510 7.8282 9.7927 8.6674 8.0534 8.0534 8.3260 8.1217 12.7783 12.7865
pM CS 0.0670? 1? 0.0010 0.0100 0.0670? 0.0670? 0.0670? 0.0670?
GE
S&P 500
MSE 0.7336 0.7507 1.1247 0.7722 1.0698 1.0698 0.9628 0.9649 1.5680 1.4721
pM CS 1? 0.3330? 0.0060 0.3330? 0.0430 0.0430 0.0970? 0.0510? 0 0.0010 0.0060 0 0.0010 0.0060 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
15-step ahead volatility forecasts
pM CS 0.4630? 1? 0.0200 0.4630? 0.0460 0.0460 0.2720? 0.4630? 0 0 0.0020 0 0 0.0020 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
10-step ahead volatility forecasts
Table 7: Forecasts evaluation based on MSE loss (pre-crisis period)
26
QLIKE 1.8275 1.7611 1.7647 1.8899 1.8626 1.8626 1.8591 1.8782 4.4555 4.5577 0 0
pM CS 0.0470 1? 0.8270? 0.0470 0.0470 0.0470 0.0470 0.0470
QLIKE 1.8366 1.7946 1.7524 1.9925 1.9136 1.9136 1.9427 1.9661 4.3848 4.2149 0.0005 0.0005 0 0
pM CS 0.0440 0.0900? 1? 0.0010 0.0010 0.0010
GE QLIKE 1.1107 1.0841 1.0343 1.0823 1.1038 1.1038 1.1110 1.1251 1.7249 1.7753 pM CS 0.0070 0.0070 1? 0.0110
0 0
0.0010 0.0010
0.0070 0.0070
S&P 500
QLIKE 1.6874 1.7003 1.8277 1.7682 1.6994 1.6994 1.6925 1.6949 4.7130 4.8067 0 0
pM CS 1? 0.0360 0.0100 0.0100 0.5180? 0.5180? 0.9420? 0.9420?
IBM
QLIKE 1.7618 1.7257 1.7840 1.9373 1.7534 1.7534 1.7892 1.8197 4.6344 4.6010
GE
5-step ahead volatility forecasts
0.0400 0.0160 0.0050 0.0050 0 0.0030
pM CS 0.0400 1? 0.0050 0.0400
0 0
pM CS 0.0130 0.2000? 0.0130 0.0130 0.0130 0.0130 0.0130 1?
S&P 500 QLIKE 1.0443 1.0250 1.0417 1.0424 1.0444 1.0444 1.0468 1.0061 2.0014 1.9824
Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
1-step ahead volatility forecasts
Table 8: Forecasts evaluation based on QLIKE loss (pre-crisis period)
27
QLIKE 1.7187 1.7784 1.9869 1.7761 1.7214 1.7214 1.7203 1.7063 5.1874 5.2865 0 0
pM CS 0.3460? 0.0120 0.0500? 0.0120 0.2200? 0.2200? 0.2200? 1?
QLIKE 1.7532 1.7281 1.8860 1.9117 1.7382 1.7382 1.7548 1.7705 5.0168 4.9003 0 0
pM CS 0.1470? 1? 0 0 0.1470? 0.1470? 0.1470? 0.1220?
GE QLIKE 1.0285 1.0240 1.1043 1.0370 1.0837 1.0837 1.0551 1.0279 2.1537 2.0969 pM CS 0.5910? 1? 0.0330 0.2250? 0.0040 0.0040 0.1620? 0.7240?
0 0
S&P 500
QLIKE 1.7429 1.8350 1.9367 1.7804 1.7607 1.7607 1.7525 1.7298 5.4927 5.0503 0 0.0030
pM CS 0.1970? 0.0050 0.0010 0.0650? 0.0650? 0.0650? 0.1970? 1?
IBM
QLIKE 1.7497 1.7416 1.9780 1.8851 1.7514 1.7514 1.7532 1.7512 5.2806 5.0949
0 0
pM CS 0.8630? 1? 0.0030 0.0030 0.2020? 0.2020? 0.2020? 0.8630?
GE
15-step ahead volatility forecasts
0.0020 0.0020
pM CS 1? 0.2750? 0.0040 0.0570? 0.0040 0.0040 0.0130 0.0380
S&P 500 QLIKE 1.0369 1.0438 1.1648 1.0489 1.1760 1.1760 1.1139 1.0965 2.3087 2.2251
Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
10-step ahead volatility forecasts
Table 9: Forecasts evaluation based on QLIKE loss (pre-crisis period)
28
MSE 6.2327 5.9920 5.9683 6.0514 6.1056 6.1056 6.1244 6.3716 7.5156 7.5863 0 0
pM CS 0.0120 0.1290? 1? 0.1290? 0.0485 0.0485 0.0300 0.0100
MSE 18.7194 18.4969 18.9934 18.8168 18.9128 18.9128 18.9858 18.9847 21.0603 21.1826 0 0
pM CS 0.2430? 1? 0 0.0530? 0.0010 0.0010 0 0
GE MSE 2.9334 2.5160 2.0691 3.7069 2.7454 2.7454 3.1765 3.9746 3.0103 3.1124 0.0020 0.0020
pM CS 0.0030 0.2220? 1? 0 0.0450 0.0450 0.0020 0
S&P 500
MSE 5.6666 5.4313 5.5147 5.3890 5.7514 5.7514 5.5878 5.5871 7.7353 7.7649 pM CS 0.0860? 0.5590? 0.5590? 1? 0 0 0.3490? 0.3490?
IBM
MSE 17.0593 16.7183 18.4381 17.5382 17.5499 17.5499 17.5415 17.5423 21.5986 21.6292
GE
5-step ahead volatility forecasts
S&P 500
pM CS MSE pM CS ? 0.5030 1.4610 1? 1? 1.5339 0.3320? 0 2.1450 0.0010 0.0410 2.5506 0.0010 0.0010 2.4824 0.0010 0.0010 2.4824 0.0010 0.0010 2.4313 0.0010 0.0010 2.3337 0.0010 0 0 3.4718 0 0 0 3.4438 0 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
1-step ahead volatility forecasts
Table 10: Forecasts evaluation based on MSE loss (crisis and post-crisis period)
29
MSE 6.2682 6.1328 7.1830 6.0047 6.3446 6.3446 6.3124 6.3195 7.9433 7.9682
pM CS 0.3810? 0.4890? 0 1? 0.0200 0.0200 0.1340? 0.1340?
MSE 18.0992 18.0221 18.2768 18.4066 18.5479 18.5479 18.5323 18.5323 22.0640 22.0184
pM CS 0.7830? 1? 0.5500? 0.4100? 0.0040 0.0040 0.0040 0.0040
GE
MSE 1.5839 1.8795 3.2326 2.5872 2.0659 2.0659 3.3424 3.8959 3.8312 3.7899
S&P 500
pM CS 0.0375 0.8260? 0 1? 0.0420 0.0420 0.0400 0.0400
IBM
MSE 6.4385 6.4249 8.9399 6.2829 6.4109 6.4109 6.4271 6.4271 7.8991 7.8436
MSE 18.9034 19.0633 18.9831 19.0384 19.2771 19.2771 19.3308 19.4422 22.3056 22.3674
pM CS 1? 0.6960? 0.8150? 0.4860? 0.0010 0.0010 0.0010 0.0010
GE
S&P 500
MSE 2.7712 3.0348 5.1513 4.6473 3.4050 3.4050 5.9951 6.4489 3.3435 3.3017
pM CS 1? 0.2970? 0 0.0010 0.0240 0.0240 0 0 0 0 0.0190 0 0 0.0190 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
15-step ahead volatility forecasts
pM CS 1? 0.0920? 0.0005 0.0030 0.0030 0.0030 0.0005 0.0005 0 0 0 0 0 0 Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
10-step ahead volatility forecasts
Table 11: Forecasts evaluation based on MSE loss (crisis and post-crisis period)
30
QLIKE 1.6420 1.5585 1.5547 1.6014 1.6324 1.6324 1.6385 1.7007 3.6716 3.7560 0 0
pM CS 0.0010 0.0170 1? 0.0020 0.0010 0.0010 0.0010 0
QLIKE 1.7096 1.6412 1.7169 1.7256 1.7634 1.7634 1.7388 1.7861 6.1544 6.3239 0.0070 0.0070 0 0
pM CS 0.1100? 1? 0.2430? 0.1010? 0.0070 0.0070
GE QLIKE 0.8979 0.8974 0.9357 0.8839 0.8989 0.8989 0.9002 0.9118 1.5393 1.5286 0 0
pM CS 0.7740? 0.7740? 0.1460? 1? 0.6520? 0.6520? 0.1590? 0.1460?
S&P 500
QLIKE 1.4865 1.4819 1.5107 1.4573 1.5021 1.5021 1.4982 1.4982 4.2910 4.2648 0 0
pM CS 0.0280 0.3680? 0.0130 1? 0.0280 0.0280 0.0320 0.0320
IBM
QLIKE 1.4975 1.4834 1.5652 1.5516 1.5094 1.5094 1.5960 1.5960 6.7186 7.0263
0.0370 0.0370 0 0 0 0
pM CS 0.3950? 1? 0.0020 0.0020
GE
5-step ahead volatility forecasts
0 0
pM CS 1? 0.0130 0.0010 0.3430? 0.0140 0.0140 0.0010 0.02200
S&P 500 QLIKE 0.8930 0.9429 1.0983 0.8710 0.9117 0.9117 0.9870 0.8838 1.8969 2.0189
Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
1-step ahead volatility forecasts
Table 12: Forecasts evaluation based on QLIKE loss (crisis and post-crisis period)
31
QLIKE 1.5782 1.5455 1.6307 1.5081 1.5930 1.5930 1.5592 1.5696 4.3180 4.3838 0 0
pM CS 0.0790? 0.2720? 0.0110 1? 0.0390 0.0390 0.1210? 0.1210?
QLIKE 1.5290 1.5483 1.5562 1.5757 1.5592 1.5592 1.5625 1.5566 8.3275 6.9157 0 0
0.0960?
0.0450
pM CS 1? 0.4230? 0.4230? 0.0450 0.1000? 0.1000?
GE QLIKE 0.7839 0.9516 1.2066 0.8058 0.9034 0.9034 1.0620 0.9089 1.9029 1.9891 0 0
pM CS 1? 0.0030 0.0030 0.1770? 0.0040 0.0040 0.0010 0.0040
S&P 500
QLIKE 1.6010 1.5656 1.7161 1.5384 1.5854 1.5854 1.5755 1.5750 3.9585 3.3071 0 0
pM CS 0.0270 0.5610? 0.0100 1? 0.0330 0.0330 0.5610? 0.5610?
IBM
QLIKE 1.5587 1.6058 1.5732 1.5944 1.6096 1.6096 1.6089 1.6014 6.7221 7.0731
0 0
pM CS 1? 0.2000? 0.4700? 0.2890? 0.0400 0.0400 0.0400 0.2000?
GE
15-step ahead volatility forecasts
0 0
pM CS 1? 0.0220 0.0100 0.2700? 0.0220 0.0220 0.0100 0.0220
S&P 500 QLIKE 0.9650 1.1436 1.4201 0.9916 1.1129 1.1129 1.2943 1.1286 2.4365 2.4325
Note: pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
Note:
pM CS are the p-values from Model Con dence Set test of Hansen et al.(2011). The p-values that are c? . marked with a ? are those in the model con dence set M 95%
Model RT-GARCH RT-GARCH-L RT-GARCH-LF A-PARCH(2,2)-St.t distr. GARCH(1,1)-N (0, 1) GARCH(1,2)-N (0, 1) GARCH(1,1)-St.t distr. GARCH(1,2)-St.t distr. Simple NoVaS Exponential NoVaS
IBM
10-step ahead volatility forecasts
Table 13: Forecasts evaluation based on QLIKE loss (crisis and post-crisis period)
32
0.0450 0.0580 0.0650 0.0300 0.0300 0.0550 0.1170 0.2570 0.3940 0.0260 0.0390 0.1150 0.0260 0.0390 0.1150 0.1180 0.1260 0.1450 1 1 1 0 0 0 0 0 0
SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au
Real-time GARCH
Real-time GARCH-L
Real-time GARCH-LF
A-PARCH(2,2) St.t distr.
GARCH(1,1)-N (0, 1)
GARCH(1,2)-N (0, 1)
GARCH(1,1)-St.t distr.
GARCH(1,2)-St.t distr.
Simple NoVaS
Exponential NoVaS
0 0 0
0 0 0
0.0020 0.0020 0.0020
0.0060 0.0060 0.0060
0.0050 0.0130 0.0270
0.0200 0.0200 0.0200
0.5450 0.6710 0.7860
0.0250 0.0250 0.0670
1 1 1
QLIKE 0.5390 0.5990 0.9470
p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0
0 0 0
1 1 1
0.1010 0.1010 0.1580
0.0010 0.0010 0.0010
0.0010 0.0010 0.0010
0.0010 0.0020 0.0030
0.0005 0.0005 0.0005
0.0010 0.0010 0.0050
MSE 0.1760 0.1760 0.5210
h=5
0 0 0
0 0 0
0.0070 0.0070 0.0070
0 0 0
0.0010 0.0010 0.0010
0.0010 0.0010 0.0010
0.0160 0.0160 0.0160
0.0010 0.0010 0.0010
0.0010 0.0010 0.0010
QLIKE 1 1 1
p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0
0 0 0
0.0950 0.0950 0.1130
0.0950 0.0950 0.1130
0.0020 0.0200 0.0840
0.0010 0.0200 0.0840
1 1 1
0 0 0
0.0020 0.0020 0.0020
MSE 0.1000 0.3000 0.5040
0 0 0
0 0 0
0.0110 0.0200 0.0220
0.0110 0.0200 0.0200
0.0050 0.0050 0.0060
0.0050 0.0050 0.0060
1 1 1
0.0015 0.0015 0.0050
0.0020 0.0020 0.0040
QLIKE 0.1100 0.1140 0.2000
h=10 p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0
0 0 0
0.0010 0.0020 0.0140
0.0020 0.0020 0.0030
0.0020 0.0020 0.0040
0.0020 0.0020 0.0040
1 1 1
0.0010 0.0010 0.0040
0.0410 0.0410 0.0580
MSE 0.2090 0.3830 0.5910
h=15
0 0 0
0 0 0
0.0020 0.0030 0.0180
0.0020 0.0020 0.0020
0.0020 0.0020 0.0040
0.0020 0.0020 0.0040
1 1 1
0.0010 0.0010 0.0050
0.0380 0.0380 0.0660
QLIKE 0.1500 0.1500 0.3050
Note: Table reports the p-value of Hansen's (2005) Superior Predictive Ability test statistics (SP Ac ), described by eq. (??),as well as its lower (SP Al ) and upper (SP Au ) bounds for dierent forecast horizons.
MSE 0.0290 0.0290 0.0440
p
h=1
SP Al SP Ac SP Au
Benchmark
Forecast horizon h
Table 14: Test for Superior Predictive ability (SPA) for IBM data.
33
0.7030 0.9080 0.9080 1 1 1 0.4610 0.6860 0.7230 0.0770 0.0770 0.0800 0.0770 0.0770 0.0800 0.0530 0.0530 0.1000 0.0670 0.0670 0.0690 0.0010 0.0010 0.0010 0 0 0
SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au
Real-time GARCH
Real-time GARCH-L
Real-time GARCH-LF
A-PARCH(2,2) St.t distr.
GARCH(1,1)-N (0, 1)
GARCH(1,2)-N (0, 1)
GARCH(1,1)-St.t distr.
GARCH(1,2)-St.t distr.
Simple NoVaS
Exponential NoVaS
0 0 0
0 0 0
0.0070 0.0070 0.0090
0.0070 0.0080 0.0090
0.0050 0.0050 0.0060
0.0050 0.0050 0.0060
0.0170 0.0170 0.0180
0.4010 0.4080 0.6070
1 1 1
QLIKE 0.0030 0.0420 0.0800
p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0
0 0 0
1 1 1
0.0890 0.0890 0.0970
0.0150 0.0150 0.0390
0.0150 0.0150 0.0390
0.0470 0.0500 0.0500
0 0 0
0.0820 0.1280 0.1360
MSE 0.0740 0.0980 0.1090
h=5
0 0 0
0 0 0
0.0120 0.0120 0.0420
0.0020 0.0020 0.0020
0.0040 0.0040 0.0040
0.0040 0.0040 0.0040
0.0340 0.0610 0.0660
0 0 0
0.0400 0.0540 0.0580
QLIKE 1 1 1
p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0.0010 0.0010
0 0 0.0030
1 1 1
0.0020 0.0020 0.0090
0 0 0
0 0 0
0.4450 0.6060 0.8920
0 0 0
0.0010 0.0520 0.0600
MSE 0.1320 0.1470 0.2090
0 0 0
0 0 0
0.2060 0.2600 0.5530
0.0010 0.0020 0.0030
0 0 0
0 0 0
0.2560 0.2560 0.5500
0 0 0
0.0400 0.0510 0.0510
QLIKE 1 1 1
h=10 p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0.0020
0 0 0.0010
0.0020 0.0020 0.0190
0 0 0.0020
0.0010 0.0010 0.0060
0.0010 0.0010 0.0060
1 1 1
0.0010 0.0010 0.0040
0.0010 0.0010 0.0050
MSE 0.0760 0.1130 0.1610
h=15
0 0 0
0 0 0
0.0010 0.0010 0.0030
0 0.0010 0.0020
0.0010 0.0010 0.0060
0.0010 0.0010 0.0060
1 1 1
0.0010 0.0010 0.0040
0.0010 0.0010 0.0050
QLIKE 0.1080 0.1080 0.3750
Note: Table reports the p-value of Hansen's (2005) Superior Predictive Ability test statistics (SP Ac ), described by eq. (??),as well as its lower (SP Al ) and upper (SP Au ) bounds for dierent forecast horizons.
MSE 0.1290 0.1340 0.1340
p
h=1
SP Al SP Ac SP Au
Benchmark
Forecast horizon h
Table 22: Test for Superior Predictive ability (SPA) for GE data.
34
0.2100 0.2410 0.2450 1 1 1 0.0050 0.0450 0.0170 0.0300 0.0450 0.0450 0.0300 0.0450 0.0450 0.0430 0.0460 0.1170 0.0710 0.0860 0.0890 0 0 0 0 0 0
SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au SP Al SP Ac SP Au
Real-time GARCH
Real-time GARCH-L
Real-time GARCH-LF
A-PARCH(2,2) St.t distr.
GARCH(1,1)-N (0, 1)
GARCH(1,2)-N (0, 1)
GARCH(1,1)-St.t distr.
GARCH(1,2)-St.t distr.
Simple NoVaS
Exponential NoVaS
0 0 0
0 0 0
0.7870 0.9750 0.9940
0.0600 0.0650 0.1460
0.0300 0.0450 0.0700
0.0300 0.0450 0.0700
1 1 1
0.0020 0.0020 0.0150
0.1040 0.1230 0.1260
QLIKE 0.1050 0.1150 0.1740
p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0
0 0 0
0.2100 0.3500 0.4360
0.2100 0.2740 0.2750
0.0410 0.0440 0.0490
0.0410 0.0440 0.0490
0.4330 0.7500 0.8590
0.0400 0.0550 0.0628
1 1 1
MSE 0.1050 0.1260 0.3800
h=5
0 0 0
0 0 0
0.0010 0.0010 0.0040
0.0020 0.0020 0.0050
0.0380 0.0390 0.0470
0.0380 0.0390 0.0470
1 1 1
0.0020 0.0040 0.0050
0.0050 0.0050 0.0200
QLIKE 0.0910 0.1090 0.3000
p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0
0 0 0
0.0110 0.0320 0.0380
0.0110 0.0350 0.0380
0.0120 0.0120 0.0150
0.0120 0.0120 0.0150
0.1420 0.2030 0.4380
0.0200 0.0220 0.0350
0.0020 0.0020 0.0020
MSE 1 1 1
0 0 0
0 0 0
0.0020 0.0090 0.0120
0.0010 0.0090 0.0120
0.0130 0.0170 0.0250
0.0130 0.0170 0.0250
1 1 1
0.0010 0.0010 0.0010
0.0050 0.0090 0.0100
QLIKE 0.2500 0.5500 0.6900
h=10 p
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
SP Al SP Ac SP Au
0 0 0
0 0 0
0.0200 0.0220 0.0220
0.0180 0.0220 0.0310
0.0020 0.0220 0.0220
0.0020 0.0220 0.0220
0.1950 0.1950 0.4100
0.0010 0.0050 0.0100
0.0070 0.0070 0.0080
MSE 1 1 1
h=15
0 0 0
0 0 0
0.0450 0.0550 0.0850
0.0450 0.0550 0.0870
0.0300 0.0350 0.0850
0.0300 0.0350 0.0850
1 1 1
0.0156 0.0164 0.0323
0.0010 0.0010 0.0040
QLIKE 0.6380 0.8800 0.9770
Note: Table reports the p-value of Hansen's (2005) Superior Predictive Ability test statistics (SP Ac ), described by eq. (??),as well as its lower (SP Al ) and upper (SP Au ) bounds for dierent forecast horizons.
MSE 0.0010 0.0100 0.0200
p
h=1
SP Al SP Ac SP Au
Benchmark
Forecast horizon h
Table 23: Test for Superior Predictive ability (SPA) for S&P 500 data.
In order to show that the Real-time GARCH model is a better approximation to the data than the GARCH(1,1)-N model, we plot the error density forecasts. The errors fot ˆ γˆ , ϕˆ can the RT-GARCH, similarly to the standard GARCH case, given the estimates αˆ , β,
be obtained recursively as follows: set the initial value for λˆ 1= 1, then calculate ˆb1 = q ˆ 1 + γˆ r2 , use this expression to get ˆ2 = ˆb2 + 4r2 ϕˆ − ˆb1 /2ϕˆ, which in turn can be α ˆ + βˆλ 2 1 1 used to further update λˆ 2 = ˆb1 + ϕˆ ˆ2 and then get the next error term ˆ3 .
Figure 1:
Plot of the error density forecasts for IBM stock.
. The gure displays the error density forecasts. p The histogram corresponds to t+l = rt+l / K(Xσ )t+l , where K(Xσ ) is the realized kernel. The dashed line represents ˆ t+l for the the kernel density estimator of the t+l = rt+l /ˆ σt+l for the standard GARCH(1,1) model and t+l = rt+l /λ Real-time GARCH models.
35
Figure 2:
Plot of the error density forecasts for GE stock.
. The gure displays the error density forecasts. The p histogram corresponds to t+l = rt+l / K(Xσ )t+l , where K(Xσ ) is the realized kernel. The dashed line represents ˆ t+l for the the kernel density estimator of the t+l = rt+l /ˆ σt+l for the standard GARCH(1,1) model and t+l = rt+l /λ Real-time GARCH models.
36
Plot of the error density forecasts for S&P 500 Index.
Figure 3: . The gure displays the error density p forecasts. The histogram corresponds to t+l = rt+l / K(Xσ )t+l , where K(Xσ ) is the realized kernel. The dashed line represents the kernel density estimator of the t+l = rt+l /ˆ σt+l for the standard GARCH(1,1) model and t+l = ˆ t+l for the Real-time GARCH models. rt+l /λ
References [1] Brandt, A., 1986, The stochastic equation Yn+1 = An Yn +Bn with stationary coe cients,
Advances in Applied Probability , 18, pp. 211-220. [2] Hansen, P. R., 2005, A test for superior predictive ability, Journal of Business&Economic
statistics,23(4), 365-380.
[3] Stout, W.F., 1974, Almost sure convergence, probability and mathematical statistics, New York: Academic press.
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