
 Start

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of York, York
YO10 5DD, UK, paola.zerilli@york.ac.uk and lc844.york@gmail.com
1 Introduction
The study of volatility in crude oil and natural gas markets and its interaction with
returns (leverage) has a broad range of Önancial impacts both from an hedging point
of view and also for forecasting purposes.
The main limitation of using daily data is that volatility is not observable (see
 Exact

Bollerslev and Zhou (2002) and Baum and Zerilli (2016)).
 Suffix

As suggested by Zhou
(1996), the availability of highfrequency data has opened up new possibilities in
estimating volatility. Tickbytick data provide an almost continuous observation of
the return series, making the daily volatility observable so that it can be studied
in great detail.
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 Prefix

york@gmail.com
1 Introduction
The study of volatility in crude oil and natural gas markets and its interaction with
returns (leverage) has a broad range of Önancial impacts both from an hedging point
of view and also for forecasting purposes.
The main limitation of using daily data is that volatility is not observable (see
Bollerslev and Zhou (2002) and Baum and Zerilli (2016)). As suggested by
 Exact

Zhou (1996),
 Suffix

the availability of highfrequency data has opened up new possibilities in
estimating volatility. Tickbytick data provide an almost continuous observation of
the return series, making the daily volatility observable so that it can be studied
in great detail.
 (check this in PDF content)

 Start

3229
 Prefix

From an econometric point of view, the employment of
intraday data leads to the estimation of the structural parameters of the stochastic volatility models using simple moment conditions tailored to Öt all the relevant
empirical features of energy and stock index returns.
In terms of hedging strategies in the crude oil market, it has been shown (see
 Exact

Chen, Zerilli and Baum (2018))
 Suffix

that the introduction of the leverage e§ect in the
traditional SV model with normally distributed errors is capable of adequately estimating risk in a VaR and CVaR sense for conservative oil suppliers in both the WTI
and Brent spot markets while it tends to overestimate risk for more speculative oil
suppliers.
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 Prefix

We also show that introducing
jumps improves only the ability of modeling the behaviour of the volatility for the
crude oil futures market.
2
2 Literature review
Traditionally, the termleverage e§ectindicates the negative correlation between asset
returns and changes in their volatility (see
 Exact

AitSahalia et al. (2013)
 Suffix

for an extensive
literature review). The interpretation of this e§ect is very intuitive if we think that
events that have a negative impact on Önancial markets would eventually cause an
increase in volatility.
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There are two possible economic interpretations for this empirical property:
1) when asset returns fall, the Örm becomes more levered as the market value of their
debt increases compared to the market value of their equity, making the stock riskier
and therefore increasing its volatility (see
 Exact

Black (1976));
 Suffix

2) when volatility increases, future prices fall (see French et al. (1987)).
Bekaert and Wu (2000) and AitSahalia et al. (2013) study di§erent possible
interpretations. The latter authors Önd that the leverage e§ect in high frequency data
is not statistically signiÖcant over short periods, but become negative and statistically
signiÖcant over long periods.
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 Prefix

There are two possible economic interpretations for this empirical property:
1) when asset returns fall, the Örm becomes more levered as the market value of their
debt increases compared to the market value of their equity, making the stock riskier
and therefore increasing its volatility (see Black (1976)); 2) when volatility increases, future prices fall (see
 Exact

French et al. (1987)). Bekaert and Wu (2000) and AitSahalia et al. (2013)
 Suffix

study di§erent possible
interpretations. The latter authors Önd that the leverage e§ect in high frequency data
is not statistically signiÖcant over short periods, but become negative and statistically
signiÖcant over long periods.
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 Prefix

The latter authors Önd that the leverage e§ect in high frequency data
is not statistically signiÖcant over short periods, but become negative and statistically
signiÖcant over long periods.
In recent years, the commodity price literature has shown that there is evidence
of leverage e§ects in various energy markets. More speciÖcally,
 Exact

Chan and Grant (2016),
 Suffix

considering lower frequency (weekly) commodity returns conclude that SV
models (with an MA component) are able to replicate the main features of the data
more e¢ ciently than GARCH models. At the same time, they Önd a signiÖcant
negative leverage e§ect in crude oil spot markets.
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 Prefix

More speciÖcally, Chan and Grant (2016), considering lower frequency (weekly) commodity returns conclude that SV
models (with an MA component) are able to replicate the main features of the data
more e¢ ciently than GARCH models. At the same time, they Önd a signiÖcant
negative leverage e§ect in crude oil spot markets.
 Exact

Kristoufek (2014)
 Suffix

focuses on the
leverage e§ect in commodity futures markets and provides an extensive literature
review in this area. Fan et al. (2008) estimate VaR of crude oil prices using a
GEDGARCH approach with daily WTI and Brent prices spanning from 1987 to
2006.
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7513
 Prefix

At the same time, they Önd a signiÖcant
negative leverage e§ect in crude oil spot markets. Kristoufek (2014) focuses on the
leverage e§ect in commodity futures markets and provides an extensive literature
review in this area.
 Exact

Fan et al. (2008)
 Suffix

estimate VaR of crude oil prices using a
GEDGARCH approach with daily WTI and Brent prices spanning from 1987 to
2006. They Önd that this type of model speciÖcation does as well as the standard
normal distribution at a 95% conÖdence level.
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They Önd that this type of model speciÖcation does as well as the standard
normal distribution at a 95% conÖdence level. They also test and Önd evidence for
asymmetric leverage e§ects without modelling them directly.
 Exact

Youssef et al. (2015)
 Suffix

3
evaluate VaR and CVar for crude oil and gasoline markets using a long memory
GARCHEVT approach. Their Öndings and backtesting exercise show that crude
oil markets are characterized by asymmetry, fat tails and long range memory.
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Youssef et al. (2015) 3
evaluate VaR and CVar for crude oil and gasoline markets using a long memory
GARCHEVT approach. Their Öndings and backtesting exercise show that crude
oil markets are characterized by asymmetry, fat tails and long range memory. In
the commodity price literature,
 Exact

Kristoufek (2014) and Nomikos and Andriosopoulos (2012),
 Suffix

using daily data, Önd aninverse leverage e§ectin the natural gas market:
a positive correlation coe¢ cient. Larsson and Nossman (2011) Önd evidence for
stochastic volatility and jumps in both returns and volatility daily spot prices of
WTI crude oil from 1989 to 2009.
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In
the commodity price literature, Kristoufek (2014) and Nomikos and Andriosopoulos (2012), using daily data, Önd aninverse leverage e§ectin the natural gas market:
a positive correlation coe¢ cient.
 Exact

Larsson and Nossman (2011)
 Suffix

Önd evidence for
stochastic volatility and jumps in both returns and volatility daily spot prices of
WTI crude oil from 1989 to 2009. See Kristoufek (2014) for an extensive literature
review on the leverage e§ect in commodity markets.
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 Prefix

In
the commodity price literature, Kristoufek (2014) and Nomikos and Andriosopoulos (2012), using daily data, Önd aninverse leverage e§ectin the natural gas market:
a positive correlation coe¢ cient. Larsson and Nossman (2011) Önd evidence for
stochastic volatility and jumps in both returns and volatility daily spot prices of
WTI crude oil from 1989 to 2009. See
 Exact

Kristoufek (2014)
 Suffix

for an extensive literature
review on the leverage e§ect in commodity markets.
We contribute to the current debate by testing for the existence of the leverage
e§ect when considering a nearcontinuous observation of the processes with the ability
to study their volatility in great detail because of the use of high frequency futures
returns in the S&P500, natural gas and crude oil markets and by st
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 Prefix

In this paper, we employ a stochastic volatility model in order to
4
analyze the uncertainty of S&P500, natural gas and crude oil futures returns. The
empirical analysis makes use of highfrequency (tickbytick) data from the futures
markets, aggregated to 10minute intervals during the trading day.
 Exact

Schwartz (1997), Schwartz and Smith (2000), and Casassus and CollinDufresne (2005)
 Suffix

propose multifactor models for energy prices where returns are only a§ected
by Gaussian shocks, but they constrain volatility to be constant. Pindyck (2004)
examines the volatility of energy spot and futures prices, estimating the standard
deviation of their Örst di§erences.
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 Prefix

Schwartz (1997), Schwartz and Smith (2000), and Casassus and CollinDufresne (2005) propose multifactor models for energy prices where returns are only a§ected
by Gaussian shocks, but they constrain volatility to be constant.
 Exact

Pindyck (2004)
 Suffix

examines the volatility of energy spot and futures prices, estimating the standard
deviation of their Örst di§erences.
Mason and Wilmot (2014) investigate the potential presence of jumps in two
key daily natural gas prices: the spot price at the Henry Hub in the US, and the
spot price for natural gas at the National Balancing Point in the UK.
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Schwartz (1997), Schwartz and Smith (2000), and Casassus and CollinDufresne (2005) propose multifactor models for energy prices where returns are only a§ected
by Gaussian shocks, but they constrain volatility to be constant. Pindyck (2004) examines the volatility of energy spot and futures prices, estimating the standard
deviation of their Örst di§erences.
 Exact

Mason and Wilmot (2014)
 Suffix

investigate the potential presence of jumps in two
key daily natural gas prices: the spot price at the Henry Hub in the US, and the
spot price for natural gas at the National Balancing Point in the UK.
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We focus on the role
of volatility as a measure of variability and uncertainty of futures returns and we
examine the impact of the leverage e§ect between futures returns and their variance.
In our investigation, we use the moment conditions derived from the stochastic
variance as proposed by
 Exact

Bollerslev and Zhou (2002) and Garcia et al. (2011).
 Suffix

3 Data
The raw data used in this study are 10minute aggregations of natural gas, crude
oil and S&P500 futures contract transactionslevel data provided by TickData, Inc.
Industry analysts have noted that to avoid market disruptions, major participants
5
in the natural gas futures market roll over their positions from the near contract
to the nextnear contract over several days before the near c
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ret WTI 0.00 0.01 1.64 1.73 0.23 0.09 7.33
Table 2:Descriptive Statistics for the daily futures realised variance 20012016
mean median min max stdev skew kurt
fut rv SP500 0.00 0.00 0.00 0.32 0.01 12.75 258.67
fut rv NG0.06 0.02 0.00 5.86 0.19 19.15 495.16
fut rv WTI 0.03 0.01 0.00 2.68 0.08 20.95 648.12
4 Estimation method
Following
 Exact

Bollerslev and Zhou (2002),
 Suffix

who use continuously observed futures prices
on oil, we build a conditional moment estimator for stochastic variance models based
on matching the sample moments ofRealized Variancewith population moments of
10
Table 3: Test Statistics and Pvalues for daily futures returns
KSmirnov pval SFrancia pvalQpval
fut ret SP5000.0870.000 14.078 0.000 156.575 0.000
fut ret NG0.0560.000 11.640
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X
tkst
(x(s)dN(s))
2
12
Barndor§Nielsen and Shephard (2004) proposed the Realized Bipower Variation
as a consistent estimate of integrated volatility component in the presence of jumps:
BV(t;k;n) =2
Pnk
i=2
\f
\fr
tk+ikn;1n
\f
\f
\f
\f
\fr
tk+
(i1)k
n;
1
n
<\f
\f
\f
RV(t;k;n)BV(t;k;n)!QV(t;k)IV(t;k)
QV(t;k)IV(t;k) =
X
tkst
(x(s)dN(s))
2
asn!1
A similar approach was used in
 Exact

Baum and Zerilli (2016).
 Suffix

In this section, we are going to introduce the three models that are going to be
estimated and compared in terms of ability to Öt the data, risk measurement and
out of sample performance: the Stochastic Volatility model (SV), the Stochastic
Volatility model with jumps (SVJ) and the Stochastic Volatility model with leverage
(SVL).
13
4.1 Stochastic Volatility model (SV)
We model the returns on cru
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models that are going to be
estimated and compared in terms of ability to Öt the data, risk measurement and
out of sample performance: the Stochastic Volatility model (SV), the Stochastic
Volatility model with jumps (SVJ) and the Stochastic Volatility model with leverage
(SVL).
13
4.1 Stochastic Volatility model (SV)
We model the returns on crude oil, natural gas and S&P500 futures using the
 Exact

Heston (1993)
 Suffix

model, setting the leverage e§ect equal to zero at the beginning.
dpt=dln(Ft)
=
p
VtdW1t
dVt=(>Vt)dt+p
VtdW2t
dW1tdW2t= 0
In this model, there are two orthogonal Wiener processes,dW1tanddW2t, driving
the evolution of returns and volatility.
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It is the parameter of the Poisson counting process that takes values:
1when an extreme event happens
0otherwise
The main moment conditions used for the SV speciÖcation, are augumented using
two additional moment conditions.
From
 Exact

Bollerslev and Zhou (2002)
 Suffix

Appendix Bpage 62(B:14)a§ects all the moment conditions (impact of jumps)
At time(t;t+ 1)
e1Jt;t+1=E[RVt;t+1jGt]RVt;t+1
=E[BPt;t+1jGt] +2xdtRVt;t+1
since
E[RVt;t+1jGt] =E[BPt;t+1jGt] +2xdt
18
and
e2Jt;t+1=E
RV2t;t+1
\f
\fGt
RV2t;t+1
=E
BP2t;t+1
\f
\fGt
+ 22xE
RV2t;t+1
\f
\fGt
dt4xdtRV2t;t+1
They are then augmented using appropriate functions of the past values of the
realised variance
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This main moment condition is then augmented using past
realizations of the returns and the variance as instruments (see the Appendix for
more details). Additional details about the moment conditions and more speciÖcally
about the equations for the leverage e§ect can be found in
 Exact

Garcia et al. (2011).
 Suffix

As
shown in Tables 20, 22, 24 all the moment conditions are in accordance with the data
and the overall HansenísJstatistic indicates that the overidentifying restrictions are
valid. As shown in Tables 19, 21, 23 all estimated parameters of the model are very
precisely estimated (is signiÖcant at 10% level for WTI) and take on sensible values
from an analytical perspective.
26
We Önd that stochast
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More speciÖcally, we Önd
signiÖcant evidence of aleverage e§ectfor S&P500 and crude oil markets: a negative
shock to returns increases volatility in these markets. In contrast, we Önd evidence
ofinverse leverage e§ectfor the natural gas market (in line with
 Exact

Kristoufek (2014)).
 Suffix

27
Table 19:GMM estimates for SVL model for the S&P500 futures:
09/2001ñ06/2016
0.0424(2.92)
>
0.00649(5.55)
0.249(17.96)
0.379(11.29)
N3704
t statistics in parentheses
p<0:10;p<0:05;p<0:01
28
29
Table 21:GMM estimates for SVL model for the Natural Gas futures:
09/2001ñ06/2016
0.760(3.45)
>
0.0460(5.60)
0.925(3.49)
0.201(4.57)
N3708
t statistics in parentheses
p<0:10;p<0
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Despite its
popularity, an obvious and distinctive limitation of theV aRapproach is that it only
speciÖes the maximum one can lose at a given risk level, but provides no indication
for how much more thanV aRone can lose if extreme tail events happen. This may
lead to an equivalentV aRestimate for two di§erent positions, though they have
completely di§erent risk exposures.
 Exact

Artzner et al. (1999)
 Suffix

proposed the concept of
coherent risk measure, which has become the paradigm of risk measurement. A good
alternative is conditional ValueatRisk (CV aR), which is a coherent risk measure
and retains the beneÖts ofV aRin terms of the capability to deÖne quantiles of the
loss distribution.
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Although theCV aRapproach has been widely used for risk analysis, the implementation of backtesting forCV aRmodels is much harder than forV aRmodels.
Nevertheless, formal backtesting methods can be found in literature, such as the
most commonly used approach zeromean residual test by
 Exact

McNeil and Frey (2000)
 Suffix

which rely on bootstrapping or one sampletprinciple, censored Gaussian method
by Berkowitz (2001) and the functional delta approach by Kerkhol and Melenverg
40
(2004)2. However, applying these methods tend to be di¢ cult and overly complex.
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 Prefix

Nevertheless, formal backtesting methods can be found in literature, such as the
most commonly used approach zeromean residual test by McNeil and Frey (2000) which rely on bootstrapping or one sampletprinciple, censored Gaussian method
by
 Exact

Berkowitz (2001) and
 Suffix

the functional delta approach by Kerkhol and Melenverg
40
(2004)2. However, applying these methods tend to be di¢ cult and overly complex.
The application of these methods is based upon the realization of speciÖc conditions,
hence it is possible to backtestCV aRonly under speciÖc circumstances.
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 Prefix

aRs
andFRV aRdas follows:
FRV aRd=
1
T
XT
t=1
It(yt<V aRd;t)
FRV aRu=
1
T
XT
t=1
It(yt> V aRu;t)
whereV aRd;tandV aRu;tare the estimatedV aRsfor downside and upside risk
at time t for a given conÖdence interval,Tis the number of observations andIt()is
2A comprehensive discussion of variousCV aRbacktesting methodologies as well as their implementations at di§erent circumstances can refer to
 Exact

Wimmerstedt (2015).
 Suffix

41
the indicator function which is deÖned as:
Downside:It=
1if yt<V aRd;t
0if ytV aRu;t
Upside:It=
1if yt> V aRu;t
0if ytV aRd;t
Furthermore, there are three formal tests based on the above criteria to backtest
theV aRestimates.
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 Prefix

methodologies as well as their implementations at di§erent circumstances can refer to Wimmerstedt (2015). 41
the indicator function which is deÖned as:
Downside:It=
1if yt<V aRd;t
0if ytV aRu;t
Upside:It=
1if yt> V aRu;t
0if ytV aRd;t
Furthermore, there are three formal tests based on the above criteria to backtest
theV aRestimates. The unconditional coverage test (LRuc), proposed by
 Exact

Kupiec (1995),
 Suffix

is to examine whether the null hypothesisH0:FR=can be satisÖed. A
good performance of theV aRmodel should be accompanied by an accurate unconditional coverage, that is, the failure rate is statistically expected to be equal to the
prescribedV aRlevel.
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 Prefix

A
good performance of theV aRmodel should be accompanied by an accurate unconditional coverage, that is, the failure rate is statistically expected to be equal to the
prescribedV aRlevel.
The method proposed by
 Exact

Kupiec (1995)
 Suffix

is capable to test the overestimates or
underestimates of aV aRmodel. It does not, however, consider whether the exceptions are scattered or if they appear in clusters3. In order to examine whether the
V aRviolations are serially uncorrelated over time, Christo§ersen (1998) proposes
the independent likelihood ratio test (LRind).
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 Prefix

This test investigates if the failure rate is equal to
the expected prescribed risk level and if the exceptions are independently distributed
over time. The null hypothesis for this test is that the exceptions are independent
and that the expected failure rate is equal to prescribed risk level.
6.1 Outofsample VaR and CVaR estimations
As in
 Exact

Fan et al. (2008) and
 Suffix

Youssef at al. (2015), in this subsection, we test the
forecasting ability of the SV and SVL models by computing the outofsample VaR
3Kupiecís (1995) approach is an unconditional test. On the other hand, we need to conditionally
examine theV aRperformance under the timevarying volatility framework.
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 Prefix

futures and speculative investors who are short futures would be better
o§ not considering the leverage e§ect in order to have more precise forecasts of risk
in a VaR sense.
45
46
47
48
7 Conclusions
In this paper we estimate a Stochastic Volatility model using high frequency data on
crude oil, natural gas and stock index (S&P500) futures.
In terms of hedging strategies, it has been shown (see
 Exact

Chen, Zerilli and Baum 2018),
 Suffix

at lower frequencies and for crude oil spot returns, that the introduction of
the leverage e§ect in the traditional SV model with Normally distributed errors is
capable of adequately estimating risk (in a VaR and CVaR sense) for conservative
oil suppliers in both the WTI and Brent markets while it tends to overestimate risk
for more speculative oil suppliers.
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59536
 Prefix

In this Appendix we also show the expressions of 2 additional moment conditions
that have the main purpose of modelling the leverage e§ect: we have not used them
in our results as, when tested, these moment conditions did not add any relevant
information to the available set and therefore they revelead to be redundant in this
case.
8.1 Residual 1
From
 Exact

Bollerslev and Zhou (2002),
 Suffix

page 56Appendix A.1equation(A:3)
51
e1t+1;t+2=E[Vt+1;t+2jGt]Vt+1;t+2
=E[Vt;t+1jGt] +\fVt+1;t+2
whereVt+1;t+2is therealised varianceandGtis the information set.
dt=Tt=t+ 1t= 1
=t+ 2(t+ 1) = 1
=e\f=>(1)
a=
1
(1) =
\f
>
b=>(1a)
=>\f
8.2 Residual 2
E
V2t+1;t+2
\f
\fGt
=H1E
V2t;t+1
\f
\fGt
+IE[Vt;t+1jGt] +J
e2t+1;t+2=E
V2t+1;t+2
\f
\fGt
V2t+1;t+2
=H1E
V2t;t+1
\f
\fGt
+IE[Vt;t+1jGt]
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60410
 Prefix

21
e2B=
<2
>
>(12) +
>
2(+ 5) (1)
C=2
(1) =a2
D=
2>
2(1)
2
=a22>
52
H1 =2
G1 =\f2
I=
1
a2(C+ 2\f) +
2
(2ab+A)
a
N=
a2
D+\f2
+\f(2ab+A) +
12
b2+B
J=bI+N
8.3 Residual 3
Residuals 3 to 5 are built in order to deal with the leverage aspect as they focus on
the relationship between futures returns and their variance.
This moment condition derives from the paper by
 Exact

Garcia et al. (2011)
 Suffix

page 32
top equation in Box I:
e3t+1;t+2=
E[pt;t+1Vt+1;t+2jGt]b
a
pt;t+1Vt;t+1
Considering the relationship between the population varianceVtand the realised
varianceVt;t+1
E[Vt;t+1jFt] =aVt+b
Vt=
E[Vt;t+1jFt]b
a
Ab+ab2aB(A2ab)E[Vt;t+1jFt] +aE
V2t;t+1
\f
\fFt
V2t=
a3
E[pt;t+1Vt;t+1jGt] =
e(E[Vt;t+1jGt]+>(ek1))
=
e>
E[Vt;t+1jGt]b
a
+>(ek1)
E[pt;t+1Vt+1;t+2jGt]b
a
53
e3t+1;t+2=
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