Ch9_slides.ppt chemistry lectures for UG
DrOjoBolanleGlory
13 views
85 slides
May 16, 2024
Slide 1 of 85
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
About This Presentation
Chemistry class 305
Slide Content
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 1
Chapter 9
Modelling volatility and correlation
Chapter 9
Modelling volatility and correlation
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 2
An Excursion into Non-linearity Land
•Motivation: the linear structural (and time series) models cannot
explain a number of important features common to much financial data
-leptokurtosis
-volatility clustering or volatility pooling
-leverage effects
•Our “traditional” structural model could be something like:
y
t=
1+
2x
2t+ ... +
kx
kt+ u
t,or more compactly y = X+ u
•We also assumed that u
t
N(0,
2
).
An Excursion into Non-linearity Land
•Motivation: the linear structural (and time series) models cannot
explain a number of important features common to much financial data
-leptokurtosis
-volatility clustering or volatility pooling
-leverage effects
•Our “traditional” structural model could be something like:
y
t=
1+
2x
2t+ ... +
kx
kt+ u
t,or more compactly y = X+ u
•We also assumed that u
t
N(0,
2
).
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 3
A Sample Financial Asset Returns Time Series
Daily S&P 500 Returns for August 2003 –August 2013
A Sample Financial Asset Returns Time Series
Daily S&P 500 Returns for August 2003 –August 2013
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 4
Non-linear Models: A Definition
•Campbell, Lo and MacKinlay (1997) define a non-linear data
generating process as one that can be written
y
t= f(u
t, u
t-1, u
t-2, …)
where u
tis an iid error term and fis a non-linear function.
•They also give a slightly more specific definition as
y
t= g(u
t-1, u
t-2, …)+ u
t
2
(u
t-1, u
t-2, …)
where gis a function of past error terms only and
2
is a variance
term.
•Models with nonlinear g(•) are “non-linear in mean”, while those with
nonlinear
2
(•) are “non-linear in variance”.
Non-linear Models: A Definition
•Campbell, Lo and MacKinlay (1997) define a non-linear data
generating process as one that can be written
y
t= f(u
t, u
t-1, u
t-2, …)
where u
tis an iid error term and fis a non-linear function.
•They also give a slightly more specific definition as
y
t= g(u
t-1, u
t-2, …)+ u
t
2
(u
t-1, u
t-2, …)
where gis a function of past error terms only and
2
is a variance
term.
•Models with nonlinear g(•) are “non-linear in mean”, while those with
nonlinear
2
(•) are “non-linear in variance”.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 5
Types of non-linear models
•The linear paradigm is a useful one.
•Many apparently non-linear relationships can be made linear by a
suitable transformation.
•On the other hand, it is likely that many relationships in finance are
intrinsically non-linear.
•Therearemanytypesofnon-linearmodels,e.g.
-ARCH/GARCH
-switchingmodels
-bilinearmodels
Types of non-linear models
•The linear paradigm is a useful one.
•Many apparently non-linear relationships can be made linear by a
suitable transformation.
•On the other hand, it is likely that many relationships in finance are
intrinsically non-linear.
•Therearemanytypesofnon-linearmodels,e.g.
-ARCH/GARCH
-switchingmodels
-bilinearmodels
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 6
Testing for Non-linearity –The RESET Test
•The“traditional”toolsoftimeseriesanalysis(acf’s,spectralanalysis)
mayfindnoevidencethatwecouldusealinearmodel,butthedata
maystillnotbeindependent.
•Portmanteautestsfornon-lineardependencehavebeendeveloped.
•ThesimplestisRamsey’sRESETtest,whichtooktheform:
•Here the dependent variable is the residual series and the independent
variables are the squares, cubes, …, of the fitted values. ... u y y yv
t t t pt
p
t
0 1
2
2
3
1
Testing for Non-linearity –The RESET Test
•The“traditional”toolsoftimeseriesanalysis(acf’s,spectralanalysis)
mayfindnoevidencethatwecouldusealinearmodel,butthedata
maystillnotbeindependent.
•Portmanteautestsfornon-lineardependencehavebeendeveloped.
•ThesimplestisRamsey’sRESETtest,whichtooktheform:
•Here the dependent variable is the residual series and the independent
variables are the squares, cubes, …, of the fitted values. ... u y y yv
t t t pt
p
t
0 1
2
2
3
1
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 7
Testing for Non-linearity –The BDS Test
•Manyothernon-linearitytestsareavailable-e.g.,theBDSand
bispectrumtest
•BDSisapurehypothesistest.Thatis,ithasasitsnullhypothesisthat
thedataarepurenoise(completelyrandom)
•Ithasbeenarguedtohavepowertodetectavarietyofdeparturesfrom
randomness–linearornon-linearstochasticprocesses,deterministic
chaos,etc)
•TheBDStestfollowsastandardnormaldistributionunderthenull
•Thetestcanalsobeusedasamodeldiagnosticontheresidualsto‘see
whatisleft’
•Iftheproposedmodelisadequate,thestandardisedresidualsshouldbe
whitenoise.
Testing for Non-linearity –The BDS Test
•Manyothernon-linearitytestsareavailable-e.g.,theBDSand
bispectrumtest
•BDSisapurehypothesistest.Thatis,ithasasitsnullhypothesisthat
thedataarepurenoise(completelyrandom)
•Ithasbeenarguedtohavepowertodetectavarietyofdeparturesfrom
randomness–linearornon-linearstochasticprocesses,deterministic
chaos,etc)
•TheBDStestfollowsastandardnormaldistributionunderthenull
•Thetestcanalsobeusedasamodeldiagnosticontheresidualsto‘see
whatisleft’
•Iftheproposedmodelisadequate,thestandardisedresidualsshouldbe
whitenoise.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 8
Chaos Theory
•Chaos theory is a notion taken from the physical sciences
•It suggests that there could be a deterministic, non-linear set of
equations underlying the behaviour of financial series or markets
•Such behaviour will appear completely random to the standard
statistical tests
•A positive sighting of chaos implies that while, by definition, long-
term forecasting would be futile, short-term forecastability and
controllability are possible, at least in theory, since there is some
deterministic structure underlying the data
•Varying definitions of what actually constitutes chaos can be found in
the literature.
Chaos Theory
•Chaos theory is a notion taken from the physical sciences
•It suggests that there could be a deterministic, non-linear set of
equations underlying the behaviour of financial series or markets
•Such behaviour will appear completely random to the standard
statistical tests
•A positive sighting of chaos implies that while, by definition, long-
term forecasting would be futile, short-term forecastability and
controllability are possible, at least in theory, since there is some
deterministic structure underlying the data
•Varying definitions of what actually constitutes chaos can be found in
the literature.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 9
Detecting Chaos
•A system is chaotic if it exhibits sensitive dependence on initial
conditions (SDIC)
•So an infinitesimal change is made to the initial conditions (the initial
state of the system), then the corresponding change iterated through the
system for some arbitrary length of time will grow exponentially
•The largest Lyapunov exponent is a test for chaos
•It measures the rate at which information is lost from a system
•A positive largest Lyapunov exponent implies sensitive dependence,
and therefore that evidence of chaos has been obtained
•Almost without exception, applications of chaos theory to financial
markets have been unsuccessful
•This is probably because financial and economic data are usually far
noisier and ‘more random’ than data from other disciplines
Detecting Chaos
•A system is chaotic if it exhibits sensitive dependence on initial
conditions (SDIC)
•So an infinitesimal change is made to the initial conditions (the initial
state of the system), then the corresponding change iterated through the
system for some arbitrary length of time will grow exponentially
•The largest Lyapunov exponent is a test for chaos
•It measures the rate at which information is lost from a system
•A positive largest Lyapunov exponent implies sensitive dependence,
and therefore that evidence of chaos has been obtained
•Almost without exception, applications of chaos theory to financial
markets have been unsuccessful
•This is probably because financial and economic data are usually far
noisier and ‘more random’ than data from other disciplines
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 10
Neural Networks
•Artificial neural networks (ANNs) are a class of models whose
structure is broadly motivated by the way that the brain performs
computation
•ANNs have been widely employed in finance for tackling time series
and classification problems
•Applications have included forecasting financial asset returns,
volatility, bankruptcy and takeover prediction
•Neural networks have virtually no theoretical motivation in finance
(they are often termed a ‘black box’)
•They can fit any functional relationship in the data to an arbitrary
degree of accuracy.
Neural Networks
•Artificial neural networks (ANNs) are a class of models whose
structure is broadly motivated by the way that the brain performs
computation
•ANNs have been widely employed in finance for tackling time series
and classification problems
•Applications have included forecasting financial asset returns,
volatility, bankruptcy and takeover prediction
•Neural networks have virtually no theoretical motivation in finance
(they are often termed a ‘black box’)
•They can fit any functional relationship in the data to an arbitrary
degree of accuracy.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 11
Feedforward Neural Networks
•The most common class of ANN models in finance are known as
feedforward network models
•These have a set of inputs (akin to regressors) linked to one or more
outputs (akin to the regressand) via one or more ‘hidden’ or
intermediate layers
•The size and number of hidden layers can be modified to give a closer
or less close fit to the data sample
•A feedforward network with no hidden layers is simply a standard
linear regression model
•Neural network models work best where financial theory has virtually
nothing to say about the likely functional form for the relationship
between a set of variables.
Feedforward Neural Networks
•The most common class of ANN models in finance are known as
feedforward network models
•These have a set of inputs (akin to regressors) linked to one or more
outputs (akin to the regressand) via one or more ‘hidden’ or
intermediate layers
•The size and number of hidden layers can be modified to give a closer
or less close fit to the data sample
•A feedforward network with no hidden layers is simply a standard
linear regression model
•Neural network models work best where financial theory has virtually
nothing to say about the likely functional form for the relationship
between a set of variables.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 12
Neural Networks –Some Disadvantages
•Neural networks are not very popular in finance and suffer from
several problems:
–The coefficient estimates from neural networks do not have any real
theoretical interpretation
–Virtually no diagnostic or specification tests are available for estimated
models
–They can provide excellent fits in-sample to a given set of ‘training’ data,
but typically provide poor out-of-sample forecast accuracy
–This usually arises from the tendency of neural networks to fit closely to
sample-specific data features and ‘noise’, and so they cannot ‘generalise’
–The non-linear estimation of neural network models can be cumbersome
and computationally time-intensive, particularly, for example, if the model
must be estimated repeatedly when rolling through a sample.
Neural Networks –Some Disadvantages
•Neural networks are not very popular in finance and suffer from
several problems:
–The coefficient estimates from neural networks do not have any real
theoretical interpretation
–Virtually no diagnostic or specification tests are available for estimated
models
–They can provide excellent fits in-sample to a given set of ‘training’ data,
but typically provide poor out-of-sample forecast accuracy
–This usually arises from the tendency of neural networks to fit closely to
sample-specific data features and ‘noise’, and so they cannot ‘generalise’
–The non-linear estimation of neural network models can be cumbersome
and computationally time-intensive, particularly, for example, if the model
must be estimated repeatedly when rolling through a sample.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 13
Models for Volatility
•Modelling and forecasting stock market volatility has been the subject
of vast empirical and theoretical investigation
•There are a number of motivations for this line of inquiry:
–Volatility is one of the most important concepts in finance
–Volatility, as measured by the standard deviation or variance of returns, is
often used as a crude measure of the total risk of financial assets
–Many value-at-risk models for measuring market risk require the
estimation or forecast of a volatility parameter
–The volatility of stock market prices also enters directly into the Black–
Scholes formula for deriving the prices of traded options
•We will now examine several volatility models.
Models for Volatility
•Modelling and forecasting stock market volatility has been the subject
of vast empirical and theoretical investigation
•There are a number of motivations for this line of inquiry:
–Volatility is one of the most important concepts in finance
–Volatility, as measured by the standard deviation or variance of returns, is
often used as a crude measure of the total risk of financial assets
–Many value-at-risk models for measuring market risk require the
estimation or forecast of a volatility parameter
–The volatility of stock market prices also enters directly into the Black–
Scholes formula for deriving the prices of traded options
•We will now examine several volatility models.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 14
Historical Volatility
•The simplest model for volatility is the historical estimate
•Historical volatility simply involves calculating the variance (or
standard deviation) of returns in the usual way over some historical
period
•This then becomes the volatility forecast for all future periods
•Evidence suggests that the use of volatility predicted from more
sophisticated time series models will lead to more accurate forecasts
and option valuations
•Historical volatility is still useful as a benchmark for comparing the
forecasting ability of more complex time models
Historical Volatility
•The simplest model for volatility is the historical estimate
•Historical volatility simply involves calculating the variance (or
standard deviation) of returns in the usual way over some historical
period
•This then becomes the volatility forecast for all future periods
•Evidence suggests that the use of volatility predicted from more
sophisticated time series models will lead to more accurate forecasts
and option valuations
•Historical volatility is still useful as a benchmark for comparing the
forecasting ability of more complex time models
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 15
Heteroscedasticity Revisited
•Anexampleofastructuralmodelis
withu
t
N(0,).
•Theassumptionthatthevarianceoftheerrorsisconstantisknownas
homoscedasticity,i.e.Var(u
t
)=.
•Whatifthevarianceoftheerrorsisnotconstant?
-heteroscedasticity
-wouldimplythatstandarderrorestimatescouldbewrong.
•Isthevarianceoftheerrorslikelytobeconstantovertime?Notfor
financialdata.
u
2
u
2 yt =
1
+
2
x
2t
+
3
x
3t
+
4
x
4t
+ u t
Heteroscedasticity Revisited
•Anexampleofastructuralmodelis
withu
t
N(0,).
•Theassumptionthatthevarianceoftheerrorsisconstantisknownas
homoscedasticity,i.e.Var(u
t
)=.
•Whatifthevarianceoftheerrorsisnotconstant?
-heteroscedasticity
-wouldimplythatstandarderrorestimatescouldbewrong.
•Isthevarianceoftheerrorslikelytobeconstantovertime?Notfor
financialdata.
u
2
u
2 yt =
1
+
2
x
2t
+
3
x
3t
+
4
x
4t
+ u t
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 16
Autoregressive Conditionally Heteroscedastic
(ARCH) Models
•Souseamodelwhichdoesnotassumethatthevarianceisconstant.
•Recallthedefinitionofthevarianceofu
t
:
=Var(u
t
u
t-1
,u
t-2
,...)=E[(u
t
-E(u
t
))
2
u
t-1
,u
t-2
,...]
WeusuallyassumethatE(u
t
)=0
so=Var(u
t
u
t-1
,u
t-2
,...)=E[u
t
2
u
t-1
,u
t-2
,...].
Whatcouldthecurrentvalueofthevarianceoftheerrorsplausibly
dependupon?
–Previoussquarederrorterms.
•Thisleadstotheautoregressiveconditionallyheteroscedasticmodel
forthevarianceoftheerrors:
=
0
+
1
•ThisisknownasanARCH(1)model
•TheARCHmodelduetoEngle(1982)hasprovedveryusefulin
finance.t
2 t
2 t
2 u
t1
2
Autoregressive Conditionally Heteroscedastic
(ARCH) Models
•Souseamodelwhichdoesnotassumethatthevarianceisconstant.
•Recallthedefinitionofthevarianceofu
t
:
=Var(u
t
u
t-1
,u
t-2
,...)=E[(u
t
-E(u
t
))
2
u
t-1
,u
t-2
,...]
WeusuallyassumethatE(u
t
)=0
so=Var(u
t
u
t-1
,u
t-2
,...)=E[u
t
2
u
t-1
,u
t-2
,...].
Whatcouldthecurrentvalueofthevarianceoftheerrorsplausibly
dependupon?
–Previoussquarederrorterms.
•Thisleadstotheautoregressiveconditionallyheteroscedasticmodel
forthevarianceoftheerrors:
=
0
+
1
•ThisisknownasanARCH(1)model
•TheARCHmodelduetoEngle(1982)hasprovedveryusefulin
finance.t
2 t
2 t
2 u
t1
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 17
Autoregressive Conditionally Heteroscedastic
(ARCH) Models (cont’d)
•Thefullmodelwouldbe
y
t=
1+
2x
2t+...+
kx
kt+u
t,u
t
N(0,)
where=
0
+
1
•Wecaneasilyextendthistothegeneralcasewheretheerrorvariance
dependsonqlagsofsquarederrors:
=
0
+
1
+
2
+...+
q
•ThisisanARCH(q)model.
•Insteadofcallingthevariance,intheliteratureitisusuallycalledh
t
,
sothemodelis
y
t=
1+
2x
2t+...+
kx
kt+u
t,u
t
N(0,h
t
)
whereh
t
=
0
+
1
+
2
+...+
qt
2 t
2 t
2 u
t1
2 u
tq
2 u
tq
2 t
2 2
1tu 2
2tu 2
1tu 2
2tu
Autoregressive Conditionally Heteroscedastic
(ARCH) Models (cont’d)
•Thefullmodelwouldbe
y
t=
1+
2x
2t+...+
kx
kt+u
t,u
t
N(0,)
where=
0
+
1
•Wecaneasilyextendthistothegeneralcasewheretheerrorvariance
dependsonqlagsofsquarederrors:
=
0
+
1
+
2
+...+
q
•ThisisanARCH(q)model.
•Insteadofcallingthevariance,intheliteratureitisusuallycalledh
t
,
sothemodelis
y
t=
1+
2x
2t+...+
kx
kt+u
t,u
t
N(0,h
t
)
whereh
t
=
0
+
1
+
2
+...+
qt
2 t
2 t
2 u
t1
2 u
tq
2 u
tq
2 t
2 2
1tu 2
2tu 2
1tu 2
2tu
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 18
Another Way of Writing ARCH Models
•Forillustration,consideranARCH(1).Insteadoftheabove,wecan
write
y
t=
1+
2x
2t+...+
kx
kt+u
t,u
t
=v
t
t
, v
t
N(0,1)
•Thetwoaredifferentwaysofexpressingexactlythesamemodel.The
firstformiseasiertounderstandwhilethesecondformisrequiredfor
simulatingfromanARCHmodel,forexample.
t t
u
0 11
2
Another Way of Writing ARCH Models
•Forillustration,consideranARCH(1).Insteadoftheabove,wecan
write
y
t=
1+
2x
2t+...+
kx
kt+u
t,u
t
=v
t
t
, v
t
N(0,1)
•Thetwoaredifferentwaysofexpressingexactlythesamemodel.The
firstformiseasiertounderstandwhilethesecondformisrequiredfor
simulatingfromanARCHmodel,forexample.
t t
u
0 11
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 19
Testing for “ARCH Effects”
1.First,runanypostulatedlinearregressionoftheformgivenintheequation
above,e.g.y
t=
1+
2x
2t+...+
kx
kt+u
t
savingtheresiduals,.
2.Thensquaretheresiduals,andregressthemonqownlagstotestforARCH
oforderq,i.e.runtheregression
wherev
t
isiid.
ObtainR
2
fromthisregression
3.TheteststatisticisdefinedasTR
2
(thenumberofobservationsmultiplied
bythecoefficientofmultiplecorrelation)fromthelastregression,andis
distributedasa
2
(q).tuˆ tqtqttt
vuuuu
22
22
2
110
2
ˆ...ˆˆˆ
Testing for “ARCH Effects”
1.First,runanypostulatedlinearregressionoftheformgivenintheequation
above,e.g.y
t=
1+
2x
2t+...+
kx
kt+u
t
savingtheresiduals,.
2.Thensquaretheresiduals,andregressthemonqownlagstotestforARCH
oforderq,i.e.runtheregression
wherev
t
isiid.
ObtainR
2
fromthisregression
3.TheteststatisticisdefinedasTR
2
(thenumberofobservationsmultiplied
bythecoefficientofmultiplecorrelation)fromthelastregression,andis
distributedasa
2
(q).tuˆ tqtqttt
vuuuu
22
22
2
110
2
ˆ...ˆˆˆ
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 20
Testing for “ARCH Effects” (cont’d)
4.Thenullandalternativehypothesesare
H
0
:
1
=0and
2
=0and
3
=0and...and
q
=0
H
1
:
1
0or
2
0or
3
0or...or
q
0.
Ifthevalueoftheteststatisticisgreaterthanthecriticalvaluefromthe
2
distribution,thenrejectthenullhypothesis.
•NotethattheARCHtestisalsosometimesapplieddirectlytoreturns
insteadoftheresidualsfromStage1above.
Testing for “ARCH Effects” (cont’d)
4.Thenullandalternativehypothesesare
H
0
:
1
=0and
2
=0and
3
=0and...and
q
=0
H
1
:
1
0or
2
0or
3
0or...or
q
0.
Ifthevalueoftheteststatisticisgreaterthanthecriticalvaluefromthe
2
distribution,thenrejectthenullhypothesis.
•NotethattheARCHtestisalsosometimesapplieddirectlytoreturns
insteadoftheresidualsfromStage1above.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 21
Problems with ARCH(q) Models
•Howdowedecideonq?
•Therequiredvalueofqmightbeverylarge
•Non-negativityconstraintsmightbeviolated.
–WhenweestimateanARCHmodel,werequire
i
>0i=1,2,...,q
(sincevariancecannotbenegative)
•AnaturalextensionofanARCH(q)modelwhichgetsaroundsomeof
theseproblemsisaGARCHmodel.
Problems with ARCH(q) Models
•Howdowedecideonq?
•Therequiredvalueofqmightbeverylarge
•Non-negativityconstraintsmightbeviolated.
–WhenweestimateanARCHmodel,werequire
i
>0i=1,2,...,q
(sincevariancecannotbenegative)
•AnaturalextensionofanARCH(q)modelwhichgetsaroundsomeof
theseproblemsisaGARCHmodel.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 22
Generalised ARCH (GARCH) Models
•DuetoBollerslev(1986).Allowtheconditionalvariancetobedependent
uponpreviousownlags
•Thevarianceequationisnow
(1)
•ThisisaGARCH(1,1)model,whichislikeanARMA(1,1)modelforthe
varianceequation.
•Wecouldalsowrite
•Substitutinginto(1)for
t-1
2
:t
2
= 0 + 1
2
1tu+t-1
2
t-1
2
= 0 + 1
2
2tu+t-2
2
t-2
2
= 0 + 1
2
3tu+t-3
2
t
2
= 0 + 1
2
1tu+(0 + 1
2
2tu+t-2
2
) = 0 + 1
2
1tu+0 + 1
2
2tu+t-2
2
Generalised ARCH (GARCH) Models
•DuetoBollerslev(1986).Allowtheconditionalvariancetobedependent
uponpreviousownlags
•Thevarianceequationisnow
(1)
•ThisisaGARCH(1,1)model,whichislikeanARMA(1,1)modelforthe
varianceequation.
•Wecouldalsowrite
•Substitutinginto(1)for
t-1
2
:t
2
= 0 + 1
2
1tu+t-1
2
t-1
2
= 0 + 1
2
2tu+t-2
2
t-2
2
= 0 + 1
2
3tu+t-3
2
t
2
= 0 + 1
2
1tu+(0 + 1
2
2tu+t-2
2
) = 0 + 1
2
1tu+0 + 1
2
2tu+t-2
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 23
Generalised ARCH (GARCH) Models (cont’d)
•Nowsubstitutinginto(2)for
t-2
2
•Aninfinitenumberofsuccessivesubstitutionswouldyield
•SotheGARCH(1,1)modelcanbewrittenasaninfiniteorderARCHmodel.
•WecanagainextendtheGARCH(1,1)modeltoaGARCH(p,q):t
2
=0 + 1
2
1tu+0 + 1
2
2tu+
2
(0 + 1
2
3tu+t-3
2
) t
2
= 0 + 1
2
1tu+0 + 1
2
2tu+0
2
+ 1
22
3tu+
3
t-3
2
t
2
= 0 (1++
2
) + 1
2
1tu(1+L+
2
L
2
) +
3
t-3
2
t
2
= 0 (1++
2
+...) + 1
2
1tu(1+L+
2
L
2
+...) +
0
2
t
2
= 0+1
2
1tu+2
2
2tu+...+q
2
qt
u
+1t-1
2
+2t-2
2
+...+pt-p
2
t
2
=
q
i
p
j
jtjiti
u
1 1
22
0
Generalised ARCH (GARCH) Models (cont’d)
•Nowsubstitutinginto(2)for
t-2
2
•Aninfinitenumberofsuccessivesubstitutionswouldyield
•SotheGARCH(1,1)modelcanbewrittenasaninfiniteorderARCHmodel.
•WecanagainextendtheGARCH(1,1)modeltoaGARCH(p,q):t
2
=0 + 1
2
1tu+0 + 1
2
2tu+
2
(0 + 1
2
3tu+t-3
2
) t
2
= 0 + 1
2
1tu+0 + 1
2
2tu+0
2
+ 1
22
3tu+
3
t-3
2
t
2
= 0 (1++
2
) + 1
2
1tu(1+L+
2
L
2
) +
3
t-3
2
t
2
= 0 (1++
2
+...) + 1
2
1tu(1+L+
2
L
2
+...) +
0
2
t
2
= 0+1
2
1tu+2
2
2tu+...+q
2
qt
u
+1t-1
2
+2t-2
2
+...+pt-p
2
t
2
=
q
i
p
j
jtjiti
u
1 1
22
0
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 24
Generalised ARCH (GARCH) Models (cont’d)
•ButingeneralaGARCH(1,1)modelwillbesufficienttocapturethe
volatilityclusteringinthedata.
•WhyisGARCHBetterthanARCH?
-moreparsimonious-avoidsoverfitting
-lesslikelytobreechnon-negativityconstraints
Generalised ARCH (GARCH) Models (cont’d)
•ButingeneralaGARCH(1,1)modelwillbesufficienttocapturethe
volatilityclusteringinthedata.
•WhyisGARCHBetterthanARCH?
-moreparsimonious-avoidsoverfitting
-lesslikelytobreechnon-negativityconstraints
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 25
The Unconditional Variance under the GARCH
Specification
•The unconditional variance of u
tis given by
when
• is termed “non-stationarity” in variance
• is termed intergrated GARCH
•For non-stationarity in variance, the conditional variance forecasts will
not converge on their unconditional value as the horizon increases.Var(ut) =
)(1
1
0
1
< 1
1
1
1
= 1
The Unconditional Variance under the GARCH
Specification
•The unconditional variance of u
tis given by
when
• is termed “non-stationarity” in variance
• is termed intergrated GARCH
•For non-stationarity in variance, the conditional variance forecasts will
not converge on their unconditional value as the horizon increases.Var(ut) =
)(1
1
0
1
< 1
1
1
1
= 1
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 26
Estimation of ARCH / GARCH Models
•Sincethemodelisnolongeroftheusuallinearform,wecannotuse
OLS.
•Weuseanothertechniqueknownasmaximumlikelihood.
•Themethodworksbyfindingthemostlikelyvaluesoftheparameters
giventheactualdata.
•Morespecifically,weformalog-likelihoodfunctionandmaximiseit.
Estimation of ARCH / GARCH Models
•Sincethemodelisnolongeroftheusuallinearform,wecannotuse
OLS.
•Weuseanothertechniqueknownasmaximumlikelihood.
•Themethodworksbyfindingthemostlikelyvaluesoftheparameters
giventheactualdata.
•Morespecifically,weformalog-likelihoodfunctionandmaximiseit.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 27
Estimation of ARCH / GARCH Models (cont’d)
•ThestepsinvolvedinactuallyestimatinganARCHorGARCHmodel
areasfollows
1.Specifytheappropriateequationsforthemeanandthevariance-e.g.an
AR(1)-GARCH(1,1)model:
2.Specifythelog-likelihoodfunctiontomaximise:
3.Thecomputerwillmaximisethefunctionandgiveparametervaluesand
theirstandarderrorsyt = + yt-1 + ut , ut N(0,t
2
) t
2
= 0 + 1
2
1tu+t-1
2
T
t
ttt
T
t
t
yy
T
L
1
22
1
1
2
/)(
2
1
)log(
2
1
)2log(
2
Estimation of ARCH / GARCH Models (cont’d)
•ThestepsinvolvedinactuallyestimatinganARCHorGARCHmodel
areasfollows
1.Specifytheappropriateequationsforthemeanandthevariance-e.g.an
AR(1)-GARCH(1,1)model:
2.Specifythelog-likelihoodfunctiontomaximise:
3.Thecomputerwillmaximisethefunctionandgiveparametervaluesand
theirstandarderrorsyt = + yt-1 + ut , ut N(0,t
2
) t
2
= 0 + 1
2
1tu+t-1
2
T
t
ttt
T
t
t
yy
T
L
1
22
1
1
2
/)(
2
1
)log(
2
1
)2log(
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 28
Parameter Estimation using Maximum Likelihood
•Considerthebivariateregressioncasewithhomoscedasticerrorsfor
simplicity:
•Assumingthatu
t
N(0,
2
),theny
tN( ,
2
)sothatthe
probabilitydensityfunctionforanormallydistributedrandomvariable
withthismeanandvarianceisgivenby
(1)
•Successivevaluesofy
t
wouldtraceoutthefamiliarbell-shapedcurve.
•Assumingthatu
t
areiid,theny
t
willalsobeiid.ttt uxy
21 tx
21
2
2
212
21
)(
2
1
exp
2
1
),(
tt
tt
xy
xyf
Parameter Estimation using Maximum Likelihood
•Considerthebivariateregressioncasewithhomoscedasticerrorsfor
simplicity:
•Assumingthatu
t
N(0,
2
),theny
tN( ,
2
)sothatthe
probabilitydensityfunctionforanormallydistributedrandomvariable
withthismeanandvarianceisgivenby
(1)
•Successivevaluesofy
t
wouldtraceoutthefamiliarbell-shapedcurve.
•Assumingthatu
t
areiid,theny
t
willalsobeiid.ttt uxy
21 tx
21
2
2
212
21
)(
2
1
exp
2
1
),(
tt
tt
xy
xyf
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 29
Parameter Estimation using Maximum Likelihood
(cont’d)
•Thenthejointpdfforallthey’scanbeexpressedasaproductofthe
individualdensityfunctions
(2)
•Substitutingintoequation(2)foreveryy
t
fromequation(1),
(3)
T
t
tt
TT
tT
xy
xyyyf
1
2
2
212
2121
)(
2
1
exp
)2(
1
),,...,,(
T
t
tt
T
tT
Xyf
Xyf
XyfXyfXyyyf
1
2
21
2
421
2
2212
2
1211
2
2121
),(
),(
)...,(),(),,...,,(
Parameter Estimation using Maximum Likelihood
(cont’d)
•Thenthejointpdfforallthey’scanbeexpressedasaproductofthe
individualdensityfunctions
(2)
•Substitutingintoequation(2)foreveryy
t
fromequation(1),
(3)
T
t
tt
TT
tT
xy
xyyyf
1
2
2
212
2121
)(
2
1
exp
)2(
1
),,...,,(
T
t
tt
T
tT
Xyf
Xyf
XyfXyfXyyyf
1
2
21
2
421
2
2212
2
1211
2
2121
),(
),(
)...,(),(),,...,,(
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 30
Parameter Estimation using Maximum Likelihood
(cont’d)
•Thetypicalsituationwehaveisthatthex
t
andy
t
aregivenandwewantto
estimate
1
,
2
,
2
.Ifthisisthecase,thenf()isknownasthelikelihood
function,denotedLF(
1
,
2
,
2
),sowewrite
(4)
•Maximumlikelihoodestimationinvolveschoosingparametervalues(
1
,
2
,
2
)thatmaximisethisfunction.
•Wewanttodifferentiate(4)w.r.t.
1
,
2
,
2
,but(4)isaproductcontaining
Tterms.
T
t
tt
TT
xy
LF
1
2
2
212
21
)(
2
1
exp
)2(
1
),,(
Parameter Estimation using Maximum Likelihood
(cont’d)
•Thetypicalsituationwehaveisthatthex
t
andy
t
aregivenandwewantto
estimate
1
,
2
,
2
.Ifthisisthecase,thenf()isknownasthelikelihood
function,denotedLF(
1
,
2
,
2
),sowewrite
(4)
•Maximumlikelihoodestimationinvolveschoosingparametervalues(
1
,
2
,
2
)thatmaximisethisfunction.
•Wewanttodifferentiate(4)w.r.t.
1
,
2
,
2
,but(4)isaproductcontaining
Tterms.
T
t
tt
TT
xy
LF
1
2
2
212
21
)(
2
1
exp
)2(
1
),,(
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 31
•Since ,wecantakelogsof(4).
•Then,usingthevariouslawsfortransformingfunctionscontaining
logarithms,weobtainthelog-likelihoodfunction,LLF:
•whichisequivalentto
(5)
•Differentiating(5)w.r.t.
1
,
2
,
2
,weobtain
(6)max()maxlog(())
x x
fx fx
Parameter Estimation using Maximum Likelihood
(cont’d)
T
t
tt
xyT
TLLF
1
2
2
21
)(
2
1
)2log(
2
log
T
t
tt
xyTT
LLF
1
2
2
212 )(
2
1
)2log(
2
log
2
2
21
1
1.2).(
2
1
tt
xyLLF
•Since ,wecantakelogsof(4).
•Then,usingthevariouslawsfortransformingfunctionscontaining
logarithms,weobtainthelog-likelihoodfunction,LLF:
•whichisequivalentto
(5)
•Differentiating(5)w.r.t.
1
,
2
,
2
,weobtain
(6)max()maxlog(())
x x
fx fx
Parameter Estimation using Maximum Likelihood
(cont’d)
T
t
tt
xyT
TLLF
1
2
2
21
)(
2
1
)2log(
2
log
T
t
tt
xyTT
LLF
1
2
2
212 )(
2
1
)2log(
2
log
2
2
21
1
1.2).(
2
1
tt
xyLLF
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 32
(7)
(8)
•Setting(6)-(8)tozerotominimisethefunctions,andputtinghatsabove
theparameterstodenotethemaximumlikelihoodestimators,
•From(6),
(9)
Parameter Estimation using Maximum Likelihood
(cont’d)
4
2
21
22
)(
2
11
2
tt xyTLLF
2
21
2
.2).(
2
1
ttt xxyLLF 0)
ˆˆ
(
21 tt xy 0
ˆˆ
21 tt xy 0
ˆˆ
21 tt xTy 0
1
ˆˆ
1
21 tt x
T
y
T
xy
21
ˆˆ
(7)
(8)
•Setting(6)-(8)tozerotominimisethefunctions,andputtinghatsabove
theparameterstodenotethemaximumlikelihoodestimators,
•From(6),
(9)
Parameter Estimation using Maximum Likelihood
(cont’d)
4
2
21
22
)(
2
11
2
tt xyTLLF
2
21
2
.2).(
2
1
ttt xxyLLF 0)
ˆˆ
(
21 tt xy 0
ˆˆ
21 tt xy 0
ˆˆ
21 tt xTy 0
1
ˆˆ
1
21 tt x
T
y
T
xy
21
ˆˆ
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 33
•From(7),
(10)
•From(8),
Parameter Estimation using Maximum Likelihood
(cont’d) 0)
ˆˆ
(
21 ttt xxy 0
ˆˆ
2
21 tttt xxxy 0
ˆˆ
2
21 tttt xxxy
tttt xxyxyx )
ˆ
(
ˆ
2
2
2
2
2
2
2
ˆˆ
xTyxTxyx
ttt yxTxyxTx
ttt )(
ˆ
22
2 )(
ˆ
222
xTx
yxTxy
t
tt
2
2142
)
ˆˆ
(
ˆ
1
ˆ
tt xy
T
•From(7),
(10)
•From(8),
Parameter Estimation using Maximum Likelihood
(cont’d) 0)
ˆˆ
(
21 ttt xxy 0
ˆˆ
2
21 tttt xxxy 0
ˆˆ
2
21 tttt xxxy
tttt xxyxyx )
ˆ
(
ˆ
2
2
2
2
2
2
2
ˆˆ
xTyxTxyx
ttt yxTxyxTx
ttt )(
ˆ
22
2 )(
ˆ
222
xTx
yxTxy
t
tt
2
2142
)
ˆˆ
(
ˆ
1
ˆ
tt xy
T
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 34
•Rearranging,
(11)
•How do these formulae compare with the OLS estimators?
(9)&(10)areidenticaltoOLS
(11)isdifferent.TheOLSestimatorwas
•ThereforetheMLestimatorofthevarianceofthedisturbancesisbiased,
althoughitisconsistent.
•Buthowdoesthishelpusinestimatingheteroscedasticmodels?
2 21
T
u
t
2 21
Tk
u
t
Parameter Estimation using Maximum Likelihood
(cont’d)
2
21
2
)
ˆˆ
(
1
ˆ
tt xy
T
•Rearranging,
(11)
•How do these formulae compare with the OLS estimators?
(9)&(10)areidenticaltoOLS
(11)isdifferent.TheOLSestimatorwas
•ThereforetheMLestimatorofthevarianceofthedisturbancesisbiased,
althoughitisconsistent.
•Buthowdoesthishelpusinestimatingheteroscedasticmodels?
2 21
T
u
t
2 21
Tk
u
t
Parameter Estimation using Maximum Likelihood
(cont’d)
2
21
2
)
ˆˆ
(
1
ˆ
tt xy
T
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 35
Estimation of GARCH Models Using
Maximum Likelihood
•Nowwehavey
t
=+y
t-1
+u
t
,u
t
N(0,)
•Unfortunately,theLLFforamodelwithtime-varyingvariancescannotbe
maximisedanalytically,exceptinthesimplestofcases.Soanumerical
procedureisusedtomaximisethelog-likelihoodfunction.Apotential
problem:localoptimaormultimodalitiesinthelikelihoodsurface.
•Thewaywedotheoptimisationis:
1. Set up LLF.
2. Use regression to get initial guesses for the mean parameters.
3. Choose some initial guesses for the conditional variance parameters.
4. Specify a convergence criterion -either by criterion or by value.
t
2 t
2
= 0 + 1
2
1tu+t-1
2
T
t
ttt
T
t
t
yy
T
L
1
22
1
1
2
/)(
2
1
)log(
2
1
)2log(
2
Estimation of GARCH Models Using
Maximum Likelihood
•Nowwehavey
t
=+y
t-1
+u
t
,u
t
N(0,)
•Unfortunately,theLLFforamodelwithtime-varyingvariancescannotbe
maximisedanalytically,exceptinthesimplestofcases.Soanumerical
procedureisusedtomaximisethelog-likelihoodfunction.Apotential
problem:localoptimaormultimodalitiesinthelikelihoodsurface.
•Thewaywedotheoptimisationis:
1. Set up LLF.
2. Use regression to get initial guesses for the mean parameters.
3. Choose some initial guesses for the conditional variance parameters.
4. Specify a convergence criterion -either by criterion or by value.
t
2 t
2
= 0 + 1
2
1tu+t-1
2
T
t
ttt
T
t
t
yy
T
L
1
22
1
1
2
/)(
2
1
)log(
2
1
)2log(
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 36
Non-Normality and Maximum Likelihood
•Recallthattheconditionalnormalityassumptionforu
t
isessential.
•Wecantestfornormalityusingthefollowingrepresentation
u
t
=v
t
t
v
t
N(0,1)
•Thesamplecounterpartis
•Arethenormal?Typicallyarestillleptokurtic,althoughlesssothan
the.Isthisaproblem?Notreally,aswecanusetheMLwitharobust
variance/covarianceestimator.MLwithrobuststandarderrorsiscalledQuasi-
MaximumLikelihoodorQML.
t t t
u
0 11
2
21
2 v
u
t
t
t
t
t
t
u
v
ˆ
ˆ
ˆ tvˆ tvˆ tuˆ
Non-Normality and Maximum Likelihood
•Recallthattheconditionalnormalityassumptionforu
t
isessential.
•Wecantestfornormalityusingthefollowingrepresentation
u
t
=v
t
t
v
t
N(0,1)
•Thesamplecounterpartis
•Arethenormal?Typicallyarestillleptokurtic,althoughlesssothan
the.Isthisaproblem?Notreally,aswecanusetheMLwitharobust
variance/covarianceestimator.MLwithrobuststandarderrorsiscalledQuasi-
MaximumLikelihoodorQML.
t t t
u
0 11
2
21
2 v
u
t
t
t
t
t
t
u
v
ˆ
ˆ
ˆ tvˆ tvˆ tuˆ
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 37
Extensions to the Basic GARCH Model
•SincetheGARCHmodelwasdeveloped,ahugenumberofextensions
andvariantshavebeenproposed.Threeofthemostimportant
examplesareEGARCH,GJR,andGARCH-Mmodels.
•ProblemswithGARCH(p,q)Models:
-Non-negativityconstraintsmaystillbeviolated
-GARCHmodelscannotaccountforleverageeffects
•Possiblesolutions:theexponentialGARCH(EGARCH)modelorthe
GJRmodel,whichareasymmetricGARCHmodels.
Extensions to the Basic GARCH Model
•SincetheGARCHmodelwasdeveloped,ahugenumberofextensions
andvariantshavebeenproposed.Threeofthemostimportant
examplesareEGARCH,GJR,andGARCH-Mmodels.
•ProblemswithGARCH(p,q)Models:
-Non-negativityconstraintsmaystillbeviolated
-GARCHmodelscannotaccountforleverageeffects
•Possiblesolutions:theexponentialGARCH(EGARCH)modelorthe
GJRmodel,whichareasymmetricGARCHmodels.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 38
The EGARCH Model
•SuggestedbyNelson(1991).Thevarianceequationisgivenby
•Advantagesofthemodel
-Sincewemodelthelog(
t
2
),theneveniftheparametersarenegative,
t
2
willbepositive.
-Wecanaccountfortheleverageeffect:iftherelationshipbetween
volatilityandreturnsisnegative,,willbenegative.
2
)log()log(
2
1
1
2
1
12
1
2
t
t
t
t
tt
uu
The EGARCH Model
•SuggestedbyNelson(1991).Thevarianceequationisgivenby
•Advantagesofthemodel
-Sincewemodelthelog(
t
2
),theneveniftheparametersarenegative,
t
2
willbepositive.
-Wecanaccountfortheleverageeffect:iftherelationshipbetween
volatilityandreturnsisnegative,,willbenegative.
2
)log()log(
2
1
1
2
1
12
1
2
t
t
t
t
tt
uu
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 39
The GJR Model
•DuetoGlosten,JaganathanandRunkle
whereI
t-1
=1ifu
t-1
<0
=0otherwise
•Foraleverageeffect,wewouldsee>0.
•We require
1
+ 0 and
1
0 for non-negativity.t
2
= 0 + 1
2
1tu+t-1
2
+ut-1
2
It-1
The GJR Model
•DuetoGlosten,JaganathanandRunkle
whereI
t-1
=1ifu
t-1
<0
=0otherwise
•Foraleverageeffect,wewouldsee>0.
•We require
1
+ 0 and
1
0 for non-negativity.t
2
= 0 + 1
2
1tu+t-1
2
+ut-1
2
It-1
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 40
An Example of the use of a GJR Model
•UsingmonthlyS&P500returns,December1979-June1998
•EstimatingaGJRmodel,weobtainthefollowingresults.)198.3(
172.0
ty )772.5()999.14()437.0()372.16(
604.0498.0015.0243.1
1
2
1
2
1
2
1
2
ttttt
Iuu
An Example of the use of a GJR Model
•UsingmonthlyS&P500returns,December1979-June1998
•EstimatingaGJRmodel,weobtainthefollowingresults.)198.3(
172.0
ty )772.5()999.14()437.0()372.16(
604.0498.0015.0243.1
1
2
1
2
1
2
1
2
ttttt
Iuu
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 41
News Impact Curves
The news impact curve plots the next period volatility (h
t) that would arise from various
positive and negative values of u
t-1, given an estimated model.
News Impact Curves for S&P 500 Returns using Coefficients from GARCH and GJR
Model Estimates:0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91
Value of Lagged Shock
Value of Conditional Variance
GARCH
GJR
News Impact Curves
The news impact curve plots the next period volatility (h
t) that would arise from various
positive and negative values of u
t-1, given an estimated model.
News Impact Curves for S&P 500 Returns using Coefficients from GARCH and GJR
Model Estimates:0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.91
Value of Lagged Shock
Value of Conditional Variance
GARCH
GJR
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 42
GARCH-in Mean
•Weexpectarisktobecompensatedbyahigherreturn.Sowhynotlet
thereturnofasecuritybepartlydeterminedbyitsrisk?
•Engle,LilienandRobins(1987)suggestedtheARCH-Mspecification.
AGARCH-Mmodelwouldbe
•canbeinterpretedasasortofriskpremium.
•Itispossibletocombineallorsomeofthesemodelstogethertoget
morecomplex“hybrid”models-e.g.anARMA-EGARCH(1,1)-M
model.yt = + t-1+ ut , ut N(0,t
2
) t
2
= 0 + 1
2
1tu+t-1
2
GARCH-in Mean
•Weexpectarisktobecompensatedbyahigherreturn.Sowhynotlet
thereturnofasecuritybepartlydeterminedbyitsrisk?
•Engle,LilienandRobins(1987)suggestedtheARCH-Mspecification.
AGARCH-Mmodelwouldbe
•canbeinterpretedasasortofriskpremium.
•Itispossibletocombineallorsomeofthesemodelstogethertoget
morecomplex“hybrid”models-e.g.anARMA-EGARCH(1,1)-M
model.yt = + t-1+ ut , ut N(0,t
2
) t
2
= 0 + 1
2
1tu+t-1
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 43
What Use Are GARCH-type Models?
•GARCHcanmodelthevolatilityclusteringeffectsincetheconditional
varianceisautoregressive.Suchmodelscanbeusedtoforecastvolatility.
•Wecouldshowthat
Var(y
t
y
t-1
,y
t-2
,...)=Var(u
t
u
t-1
,u
t-2
,...)
•Somodelling
t
2
willgiveusmodelsandforecastsfory
t
aswell.
•Variance forecasts are additive over time.
What Use Are GARCH-type Models?
•GARCHcanmodelthevolatilityclusteringeffectsincetheconditional
varianceisautoregressive.Suchmodelscanbeusedtoforecastvolatility.
•Wecouldshowthat
Var(y
t
y
t-1
,y
t-2
,...)=Var(u
t
u
t-1
,u
t-2
,...)
•Somodelling
t
2
willgiveusmodelsandforecastsfory
t
aswell.
•Variance forecasts are additive over time.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 44
Forecasting Variances using GARCH Models
•Producing conditional variance forecasts from GARCH models uses a
very similar approach to producing forecasts from ARMA models.
•It is again an exercise in iterating with the conditional expectations
operator.
•ConsiderthefollowingGARCH(1,1)model:
,u
t
N(0,
t
2
),
•Whatisneededistogenerateforecastsof
T+1
2
T
,
T+2
2
T
,...,
T+s
2
T
where
T
denotesallinformationavailableuptoand
includingobservationT.
•Addingonetoeachofthetimesubscriptsoftheaboveconditional
varianceequation,andthentwo,andthenthreewouldyieldthe
followingequations
T+1
2
=
0
+
1
u
T
2
+
T
2
,
T+2
2
=
0
+
1
u
T+1
2
+
T+1
2
,
T+3
2
=
0
+
1
u
T+2+
T+2
2tt uy 2
1
2
110
2
ttt
u
Forecasting Variances using GARCH Models
•Producing conditional variance forecasts from GARCH models uses a
very similar approach to producing forecasts from ARMA models.
•It is again an exercise in iterating with the conditional expectations
operator.
•ConsiderthefollowingGARCH(1,1)model:
,u
t
N(0,
t
2
),
•Whatisneededistogenerateforecastsof
T+1
2
T
,
T+2
2
T
,...,
T+s
2
T
where
T
denotesallinformationavailableuptoand
includingobservationT.
•Addingonetoeachofthetimesubscriptsoftheaboveconditional
varianceequation,andthentwo,andthenthreewouldyieldthe
followingequations
T+1
2
=
0
+
1
u
T
2
+
T
2
,
T+2
2
=
0
+
1
u
T+1
2
+
T+1
2
,
T+3
2
=
0
+
1
u
T+2+
T+2
2tt uy 2
1
2
110
2
ttt
u
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 45
Forecasting Variances
using GARCH Models (Cont’d)
•Let be the one step ahead forecast for
2
made at time T. This is
easy to calculate since, at time T, the values of all the terms on the
RHS are known.
• would be obtained by taking the conditional expectation of the
first equation at the bottom of slide 36:
•Given, how is , the 2-step ahead forecast for
2
made at time T,
calculated?Taking the conditional expectation of the second equation
at the bottom of slide 36:
=
0
+
1
E(
T
) +
•where E(
T
) is the expectation, made at time T, of , which is
the squared disturbance term.2
,1
f
T
2
,1
f
T
2
,1
f
T
= 0 + 1
2
Tu +T
2
2
,1
f
T 2
,2
f
T
2
,2
f
T
2
1Tu 2
,1
f
T
2
1Tu 2
1Tu
Forecasting Variances
using GARCH Models (Cont’d)
•Let be the one step ahead forecast for
2
made at time T. This is
easy to calculate since, at time T, the values of all the terms on the
RHS are known.
• would be obtained by taking the conditional expectation of the
first equation at the bottom of slide 36:
•Given, how is , the 2-step ahead forecast for
2
made at time T,
calculated?Taking the conditional expectation of the second equation
at the bottom of slide 36:
=
0
+
1
E(
T
) +
•where E(
T
) is the expectation, made at time T, of , which is
the squared disturbance term.2
,1
f
T
2
,1
f
T
2
,1
f
T
= 0 + 1
2
Tu +T
2
2
,1
f
T 2
,2
f
T
2
,2
f
T
2
1Tu 2
,1
f
T
2
1Tu 2
1Tu
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 46
Forecasting Variances
using GARCH Models (Cont’d)
•We can write
E(u
T+1
2
t
) =
T+1
2
•But
T+1
2
is not known at time T, so it is replaced with the forecast for
it, , so that the 2-step ahead forecast is given by
=
0
+
1
+
=
0
+ (
1
+)
•By similar arguments, the 3-step ahead forecast will be given by
= E
T
(
0
+
1
u
T+2
2
+
T+2
2
)
=
0
+ (
1
+)
=
0
+ (
1
+)[
0
+ (
1
+) ]
=
0
+
0
(
1
+) + (
1
+)
2
•Any s-step ahead forecast (s2)would be produced by 2
,1
f
T
2
,2
f
T
2
,1
f
T
2
,1
f
T
2
,2
f
T
2
,1
f
T
2
,3
f
T
2
,2
f
T
2
,1
f
T
2
,1
f
T
2
,1
1
1
1
1
1
10
2
,
)()(
f
T
s
s
i
if
Ts
Forecasting Variances
using GARCH Models (Cont’d)
•We can write
E(u
T+1
2
t
) =
T+1
2
•But
T+1
2
is not known at time T, so it is replaced with the forecast for
it, , so that the 2-step ahead forecast is given by
=
0
+
1
+
=
0
+ (
1
+)
•By similar arguments, the 3-step ahead forecast will be given by
= E
T
(
0
+
1
u
T+2
2
+
T+2
2
)
=
0
+ (
1
+)
=
0
+ (
1
+)[
0
+ (
1
+) ]
=
0
+
0
(
1
+) + (
1
+)
2
•Any s-step ahead forecast (s2)would be produced by 2
,1
f
T
2
,2
f
T
2
,1
f
T
2
,1
f
T
2
,2
f
T
2
,1
f
T
2
,3
f
T
2
,2
f
T
2
,1
f
T
2
,1
f
T
2
,1
1
1
1
1
1
10
2
,
)()(
f
T
s
s
i
if
Ts
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 47
What Use Are Volatility Forecasts?
1.Optionpricing
C=f(S,X,
2
,T,r
f
)
2.Conditionalbetas
3.Dynamichedgeratios
TheHedgeRatio-thesizeofthefuturespositiontothesizeofthe
underlyingexposure,i.e.thenumberoffuturescontractstobuyorsellper
unitofthespotgood.
it
imt
mt
,
,
,
2
What Use Are Volatility Forecasts?
1.Optionpricing
C=f(S,X,
2
,T,r
f
)
2.Conditionalbetas
3.Dynamichedgeratios
TheHedgeRatio-thesizeofthefuturespositiontothesizeofthe
underlyingexposure,i.e.thenumberoffuturescontractstobuyorsellper
unitofthespotgood.
it
imt
mt
,
,
,
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 48
What Use Are Volatility Forecasts? (Cont’d)
•Whatistheoptimalvalueofthehedgeratio?
•Assumingthattheobjectiveofhedgingistominimisethevarianceofthe
hedgedportfolio,theoptimalhedgeratiowillbegivenby
whereh=hedgeratio
p=correlationcoefficientbetweenchangeinspotprice(S)and
changeinfuturesprice(F)
S=standarddeviationofS
F=standarddeviationofF
•Whatifthestandarddeviationsandcorrelationarechangingovertime?
Usehp
s
F
tF
ts
tt
ph
,
,
What Use Are Volatility Forecasts? (Cont’d)
•Whatistheoptimalvalueofthehedgeratio?
•Assumingthattheobjectiveofhedgingistominimisethevarianceofthe
hedgedportfolio,theoptimalhedgeratiowillbegivenby
whereh=hedgeratio
p=correlationcoefficientbetweenchangeinspotprice(S)and
changeinfuturesprice(F)
S=standarddeviationofS
F=standarddeviationofF
•Whatifthestandarddeviationsandcorrelationarechangingovertime?
Usehp
s
F
tF
ts
tt
ph
,
,
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 49
Testing Non-linear Restrictions or
Testing Hypotheses about Non-linear Models
•Usualt-andF-testsarestillvalidinnon-linearmodels,buttheyare
notflexibleenough.
•Therearethreehypothesistestingproceduresbasedonmaximum
likelihoodprinciples:Wald,LikelihoodRatio,LagrangeMultiplier.
•Considerasingleparameter,tobeestimated,DenotetheMLEas
andarestrictedestimateas.~
ˆ
Testing Non-linear Restrictions or
Testing Hypotheses about Non-linear Models
•Usualt-andF-testsarestillvalidinnon-linearmodels,buttheyare
notflexibleenough.
•Therearethreehypothesistestingproceduresbasedonmaximum
likelihoodprinciples:Wald,LikelihoodRatio,LagrangeMultiplier.
•Considerasingleparameter,tobeestimated,DenotetheMLEas
andarestrictedestimateas.~
ˆ
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 50
Likelihood Ratio Tests
•Estimateunderthenullhypothesisandunderthealternative.
•ThencomparethemaximisedvaluesoftheLLF.
•Soweestimatetheunconstrainedmodelandachieveagivenmaximised
valueoftheLLF,denotedL
u
•Thenestimatethemodelimposingtheconstraint(s)andgetanewvalueof
theLLFdenotedL
r
.
•Whichwillbebigger?
•L
r
L
u
comparabletoRRSSURSS
•TheLRteststatisticisgivenby
LR=-2(L
r
-L
u
)
2
(m)
wherem=numberofrestrictions
Likelihood Ratio Tests
•Estimateunderthenullhypothesisandunderthealternative.
•ThencomparethemaximisedvaluesoftheLLF.
•Soweestimatetheunconstrainedmodelandachieveagivenmaximised
valueoftheLLF,denotedL
u
•Thenestimatethemodelimposingtheconstraint(s)andgetanewvalueof
theLLFdenotedL
r
.
•Whichwillbebigger?
•L
r
L
u
comparabletoRRSSURSS
•TheLRteststatisticisgivenby
LR=-2(L
r
-L
u
)
2
(m)
wherem=numberofrestrictions
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 51
Likelihood Ratio Tests (cont’d)
•Example:WeestimateaGARCHmodelandobtainamaximisedLLFof
66.85.Weareinterestedintestingwhether=0inthefollowingequation.
y
t
=+y
t-1
+u
t
,u
t
N(0,)
=
0
+
1
+
•Weestimatethemodelimposingtherestrictionandobservethemaximised
LLFfallsto64.54.Canweaccepttherestriction?
•LR=-2(64.54-66.85)=4.62.
•Thetestfollowsa
2
(1)=3.84at5%,sorejectthenull.
•DenotingthemaximisedvalueoftheLLFbyunconstrainedMLasL()
andtheconstrainedoptimumas .Thenwecanillustratethe3testing
proceduresinthefollowingdiagram:
t
2
t
2 u
t1
2 L(
~
) 2
1t
ˆ
Likelihood Ratio Tests (cont’d)
•Example:WeestimateaGARCHmodelandobtainamaximisedLLFof
66.85.Weareinterestedintestingwhether=0inthefollowingequation.
y
t
=+y
t-1
+u
t
,u
t
N(0,)
=
0
+
1
+
•Weestimatethemodelimposingtherestrictionandobservethemaximised
LLFfallsto64.54.Canweaccepttherestriction?
•LR=-2(64.54-66.85)=4.62.
•Thetestfollowsa
2
(1)=3.84at5%,sorejectthenull.
•DenotingthemaximisedvalueoftheLLFbyunconstrainedMLasL()
andtheconstrainedoptimumas .Thenwecanillustratethe3testing
proceduresinthefollowingdiagram:
t
2
t
2 u
t1
2 L(
~
) 2
1t
ˆ
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 52
Comparison of Testing Procedures under Maximum
Likelihood: Diagramatic Representation
L
A
ˆ
L
B
~
L
~
ˆ
Comparison of Testing Procedures under Maximum
Likelihood: Diagramatic Representation
L
A
ˆ
L
B
~
L
~
ˆ
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 53
Hypothesis Testing under Maximum Likelihood
•TheverticaldistanceformsthebasisoftheLRtest.
•TheWaldtestisbasedonacomparisonofthehorizontaldistance.
•TheLMtestcomparestheslopesofthecurveatAandB.
•WeknowattheunrestrictedMLE,L(),theslopeofthecurveiszero.
•Butisit“significantlysteep”at ?
•Thisformulationofthetestisusuallyeasiesttoestimate.L(
~
)
ˆ
Hypothesis Testing under Maximum Likelihood
•TheverticaldistanceformsthebasisoftheLRtest.
•TheWaldtestisbasedonacomparisonofthehorizontaldistance.
•TheLMtestcomparestheslopesofthecurveatAandB.
•WeknowattheunrestrictedMLE,L(),theslopeofthecurveiszero.
•Butisit“significantlysteep”at ?
•Thisformulationofthetestisusuallyeasiesttoestimate.L(
~
)
ˆ
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 54
An Example of the Application of GARCH Models
-Day & Lewis (1992)
•Purpose
•ToconsidertheoutofsampleforecastingperformanceofGARCHand
EGARCHModelsforpredictingstockindexvolatility.
•Impliedvolatilityisthemarketsexpectationofthe“average”levelof
volatilityofanoption:
•Whichisbetter,GARCHorimpliedvolatility?
•Data
•Weeklyclosingprices(WednesdaytoWednesday,andFridaytoFriday)
fortheS&P100Indexoptionandtheunderlying11March83-31Dec.89
•Impliedvolatilityiscalculatedusinganon-lineariterativeprocedure.
An Example of the Application of GARCH Models
-Day & Lewis (1992)
•Purpose
•ToconsidertheoutofsampleforecastingperformanceofGARCHand
EGARCHModelsforpredictingstockindexvolatility.
•Impliedvolatilityisthemarketsexpectationofthe“average”levelof
volatilityofanoption:
•Whichisbetter,GARCHorimpliedvolatility?
•Data
•Weeklyclosingprices(WednesdaytoWednesday,andFridaytoFriday)
fortheS&P100Indexoptionandtheunderlying11March83-31Dec.89
•Impliedvolatilityiscalculatedusinganon-lineariterativeprocedure.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 55
The Models
•The“Base”Models
Fortheconditionalmean
(1)
Andforthevariance (2)
or (3)
where
R
Mt
denotesthereturnonthemarketportfolio
R
Ft
denotestherisk-freerate
h
tdenotestheconditionalvariancefromtheGARCH-typemodelswhile
t
2
denotestheimpliedvariancefromoptionprices.ttFtMt uhRR
10 11
2
110
ttt huh )
2
()ln()ln(
2/1
1
1
1
1
1110
t
t
t
t
tt
h
u
h
u
hh
The Models
•The“Base”Models
Fortheconditionalmean
(1)
Andforthevariance (2)
or (3)
where
R
Mt
denotesthereturnonthemarketportfolio
R
Ft
denotestherisk-freerate
h
tdenotestheconditionalvariancefromtheGARCH-typemodelswhile
t
2
denotestheimpliedvariancefromoptionprices.ttFtMt uhRR
10 11
2
110
ttt huh )
2
()ln()ln(
2/1
1
1
1
1
1110
t
t
t
t
tt
h
u
h
u
hh
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 56
The Models (cont’d)
•Addinalaggedvalueoftheimpliedvolatilityparametertoequations(2)
and(3).
(2)becomes
(4)
and(3)becomes
(5)
•WeareinterestedintestingH
0
:=0in(4)or(5).
•Also,wewanttotestH
0
:
1
=0and
1
=0in(4),
•andH
0
:
1
=0and
1
=0and=0and=0in(5).2
111
2
110
tttt huh )ln()
2
()ln()ln(
2
1
2/1
1
1
1
1
1110
t
t
t
t
t
tt
h
u
h
u
hh
The Models (cont’d)
•Addinalaggedvalueoftheimpliedvolatilityparametertoequations(2)
and(3).
(2)becomes
(4)
and(3)becomes
(5)
•WeareinterestedintestingH
0
:=0in(4)or(5).
•Also,wewanttotestH
0
:
1
=0and
1
=0in(4),
•andH
0
:
1
=0and
1
=0and=0and=0in(5).2
111
2
110
tttt huh )ln()
2
()ln()ln(
2
1
2/1
1
1
1
1
1110
t
t
t
t
t
tt
h
u
h
u
hh
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 57
The Models (cont’d)
•Ifthissecondsetofrestrictionsholds,then(4)&(5)collapseto
(4’)
•and(3)becomes
(5’)
•Wecantestalloftheserestrictionsusingalikelihoodratiotest.2
10
2
tth )ln()ln(
2
10
2
tth
The Models (cont’d)
•Ifthissecondsetofrestrictionsholds,then(4)&(5)collapseto
(4’)
•and(3)becomes
(5’)
•Wecantestalloftheserestrictionsusingalikelihoodratiotest.2
10
2
tth )ln()ln(
2
10
2
tth
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 58
In-sample Likelihood Ratio Test Results:
GARCH Versus Implied Volatility
ttFtMt uhRR
10 (8.78)
11
2
110
ttt
huh (8.79)
2
111
2
110
tttt huh (8.81)
2
10
2
tth (8.81)
Equation for
Variance
specification
0 1 010
-4
1 1 Log-L
2
(8.79) 0.0072
(0.005)
0.071
(0.01)
5.428
(1.65)
0.093
(0.84)
0.854
(8.17)
- 767.321 17.77
(8.81) 0.0015
(0.028)
0.043
(0.02)
2.065
(2.98)
0.266
(1.17)
-0.068
(-0.59)
0.318
(3.00)
776.204 -
(8.81) 0.0056
(0.001)
-0.184
(-0.001)
0.993
(1.50)
- - 0.581
(2.94)
764.394 23.62
Notes: t-ratios in parentheses, Log-L denotes the maximised value of the log-likelihood function in
each case.
2
denotes the value of the test statistic, which follows a
2
(1) in the case of (8.81) restricted
to (8.79), and a
2
(2) in the case of (8.81) restricted to (8.81). Source: Day and Lewis (1992).
Reprinted with the permission of Elsevier Science.
In-sample Likelihood Ratio Test Results:
GARCH Versus Implied Volatility
ttFtMt uhRR
10 (8.78)
11
2
110
ttt
huh (8.79)
2
111
2
110
tttt huh (8.81)
2
10
2
tth (8.81)
Equation for
Variance
specification
0 1 010
-4
1 1 Log-L
2
(8.79) 0.0072
(0.005)
0.071
(0.01)
5.428
(1.65)
0.093
(0.84)
0.854
(8.17)
- 767.321 17.77
(8.81) 0.0015
(0.028)
0.043
(0.02)
2.065
(2.98)
0.266
(1.17)
-0.068
(-0.59)
0.318
(3.00)
776.204 -
(8.81) 0.0056
(0.001)
-0.184
(-0.001)
0.993
(1.50)
- - 0.581
(2.94)
764.394 23.62
Notes: t-ratios in parentheses, Log-L denotes the maximised value of the log-likelihood function in
each case.
2
denotes the value of the test statistic, which follows a
2
(1) in the case of (8.81) restricted
to (8.79), and a
2
(2) in the case of (8.81) restricted to (8.81). Source: Day and Lewis (1992).
Reprinted with the permission of Elsevier Science.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 59
In-sample Likelihood Ratio Test Results:
EGARCH Versus Implied Volatility
ttFtMt uhRR
10 (8.78)
)
2
()ln()ln(
2/1
1
1
1
1
1110
t
t
t
t
tt
h
u
h
u
hh (8.80)
)ln()
2
()ln()ln(
2
1
2/1
1
1
1
1
1110
t
t
t
t
t
tt
h
u
h
u
hh
(8.82)
)ln()ln(
2
10
2
tth (8.82)
Equation for
Variance
specification
0 1 010
-4
1 Log-L
2
(c) -0.0026
(-0.03)
0.094
(0.25)
-3.62
(-2.90)
0.529
(3.26)
-0.273
(-4.13)
0.357
(3.17)
- 776.436 8.09
(e) 0.0035
(0.56)
-0.076
(-0.24)
-2.28
(-1.82)
0.373
(1.48)
-0.282
(-4.34)
0.210
(1.89)
0.351
(1.82)
780.480 -
(e) 0.0047
(0.71)
-0.139
(-0.43)
-2.76
(-2.30)
- - - 0.667
(4.01)
765.034 30.89
Notes: t-ratios in parentheses, Log-L denotes the maximised value of the log-likelihood function in
each case.
2
denotes the value of the test statistic, which follows a
2
(1) in the case of (8.82) restricted
to (8.80), and a
2
(2) in the case of (8.82) restricted to (8.82). Source: Day and Lewis (1992).
Reprinted with the permission of Elsevier Science.
In-sample Likelihood Ratio Test Results:
EGARCH Versus Implied Volatility
ttFtMt uhRR
10 (8.78)
)
2
()ln()ln(
2/1
1
1
1
1
1110
t
t
t
t
tt
h
u
h
u
hh (8.80)
)ln()
2
()ln()ln(
2
1
2/1
1
1
1
1
1110
t
t
t
t
t
tt
h
u
h
u
hh
(8.82)
)ln()ln(
2
10
2
tth (8.82)
Equation for
Variance
specification
0 1 010
-4
1 Log-L
2
(c) -0.0026
(-0.03)
0.094
(0.25)
-3.62
(-2.90)
0.529
(3.26)
-0.273
(-4.13)
0.357
(3.17)
- 776.436 8.09
(e) 0.0035
(0.56)
-0.076
(-0.24)
-2.28
(-1.82)
0.373
(1.48)
-0.282
(-4.34)
0.210
(1.89)
0.351
(1.82)
780.480 -
(e) 0.0047
(0.71)
-0.139
(-0.43)
-2.76
(-2.30)
- - - 0.667
(4.01)
765.034 30.89
Notes: t-ratios in parentheses, Log-L denotes the maximised value of the log-likelihood function in
each case.
2
denotes the value of the test statistic, which follows a
2
(1) in the case of (8.82) restricted
to (8.80), and a
2
(2) in the case of (8.82) restricted to (8.82). Source: Day and Lewis (1992).
Reprinted with the permission of Elsevier Science.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 60
Conclusions for In-sample Model Comparisons &
Out-of-Sample Procedure
•IVhasextraincrementalpowerformodellingstockvolatilitybeyond
GARCH.
•Butthemodelsdonotrepresentatruetestofthepredictiveabilityof
IV.
•Sotheauthorsconductanoutofsampleforecastingtest.
•Thereare729datapoints.Theyusethefirst410toestimatethe
models,andthenmakea1-stepaheadforecastofthefollowingweek’s
volatility.
•Thentheyrollthesampleforwardoneobservationatatime,
constructinganewonestepaheadforecastateachstep.
Conclusions for In-sample Model Comparisons &
Out-of-Sample Procedure
•IVhasextraincrementalpowerformodellingstockvolatilitybeyond
GARCH.
•Butthemodelsdonotrepresentatruetestofthepredictiveabilityof
IV.
•Sotheauthorsconductanoutofsampleforecastingtest.
•Thereare729datapoints.Theyusethefirst410toestimatethe
models,andthenmakea1-stepaheadforecastofthefollowingweek’s
volatility.
•Thentheyrollthesampleforwardoneobservationatatime,
constructinganewonestepaheadforecastateachstep.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 61
Out-of-Sample Forecast Evaluation
•Theyevaluatetheforecastsintwoways:
•Thefirstisbyregressingtherealisedvolatilityseriesontheforecastsplus
aconstant:
(7)
where isthe“actual”valueofvolatility,andisthevalueforecasted
foritduringperiodt.
•Perfectlyaccurateforecastsimplyb
0
=0andb
1
=1.
•Butwhatisthe“true”valueofvolatilityattimet?
Day&Lewisuse2measures
1.Thesquareoftheweeklyreturnontheindex,whichtheycallSR.
2.Thevarianceoftheweek’sdailyreturnsmultipliedbythenumber
oftradingdaysinthatweek.
t ftt
bb
1
2
0 1
2
1
t1
2
ft
2
Out-of-Sample Forecast Evaluation
•Theyevaluatetheforecastsintwoways:
•Thefirstisbyregressingtherealisedvolatilityseriesontheforecastsplus
aconstant:
(7)
where isthe“actual”valueofvolatility,andisthevalueforecasted
foritduringperiodt.
•Perfectlyaccurateforecastsimplyb
0
=0andb
1
=1.
•Butwhatisthe“true”valueofvolatilityattimet?
Day&Lewisuse2measures
1.Thesquareoftheweeklyreturnontheindex,whichtheycallSR.
2.Thevarianceoftheweek’sdailyreturnsmultipliedbythenumber
oftradingdaysinthatweek.
t ftt
bb
1
2
0 1
2
1
t1
2
ft
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 62
Out-of Sample Model Comparisons
1
2
10
2
1
tftt bb (8.83)
Forecasting Model Proxy for ex
post volatility
b0 b1 R
2
Historic SR 0.0004
(5.60)
0.129
(21.18)
0.094
Historic WV 0.0005
(2.90)
0.154
(7.58)
0.024
GARCH SR 0.0002
(1.02)
0.671
(2.10)
0.039
GARCH WV 0.0002
(1.07)
1.074
(3.34)
0.018
EGARCH SR 0.0000
(0.05)
1.075
(2.06)
0.022
EGARCH WV -0.0001
(-0.48)
1.529
(2.58)
0.008
Implied Volatility SR 0.0022
(2.22)
0.357
(1.82)
0.037
Implied Volatility WV 0.0005
(0.389)
0.718
(1.95)
0.026
Notes: Historic refers to the use of a simple historical average of the squared returns to forecast
volatility; t-ratios in parentheses; SR and WV refer to the square of the weekly return on the S&P 100,
and the variance of the week’s daily returns multiplied by the number of trading days in that week,
respectively. Source: Day and Lewis (1992). Reprinted with the permission of Elsevier Science.
Out-of Sample Model Comparisons
1
2
10
2
1
tftt bb (8.83)
Forecasting Model Proxy for ex
post volatility
b0 b1 R
2
Historic SR 0.0004
(5.60)
0.129
(21.18)
0.094
Historic WV 0.0005
(2.90)
0.154
(7.58)
0.024
GARCH SR 0.0002
(1.02)
0.671
(2.10)
0.039
GARCH WV 0.0002
(1.07)
1.074
(3.34)
0.018
EGARCH SR 0.0000
(0.05)
1.075
(2.06)
0.022
EGARCH WV -0.0001
(-0.48)
1.529
(2.58)
0.008
Implied Volatility SR 0.0022
(2.22)
0.357
(1.82)
0.037
Implied Volatility WV 0.0005
(0.389)
0.718
(1.95)
0.026
Notes: Historic refers to the use of a simple historical average of the squared returns to forecast
volatility; t-ratios in parentheses; SR and WV refer to the square of the weekly return on the S&P 100,
and the variance of the week’s daily returns multiplied by the number of trading days in that week,
respectively. Source: Day and Lewis (1992). Reprinted with the permission of Elsevier Science.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 63
Encompassing Test Results: Do the IV Forecasts
Encompass those of the GARCH Models?
1
2
4
2
3
2
2
2
10
2
1
tHtEtGtItt bbbbb (8.86)
Forecast comparison b0 b1 b2 b3 b4 R
2
Implied vs. GARCH
-0.00010
(-0.09)
0.601
(1.03)
0.298
(0.42)
- - 0.027
Implied vs. GARCH
vs. Historical
0.00018
(1.15)
0.632
(1.02)
-0.243
(-0.28)
- 0.123
(7.01)
0.038
Implied vs. EGARCH
-0.00001
(-0.07)
0.695
(1.62)
- 0.176
(0.27)
- 0.026
Implied vs. EGARCH
vs. Historical
0.00026
(1.37)
0.590
(1.45)
-0.374
(-0.57)
- 0.118
(7.74)
0.038
GARCH vs. EGARCH
0.00005
(0.37)
- 1.070
(2.78)
-0.001
(-0.00)
- 0.018
Notes: t-ratios in parentheses; the ex post measure used in this table is the variance of the week’s daily
returns multiplied by the number of trading days in that week. Source: Day and Lewis (1992).
Reprinted with the permission of Elsevier Science.
Encompassing Test Results: Do the IV Forecasts
Encompass those of the GARCH Models?
1
2
4
2
3
2
2
2
10
2
1
tHtEtGtItt bbbbb (8.86)
Forecast comparison b0 b1 b2 b3 b4 R
2
Implied vs. GARCH
-0.00010
(-0.09)
0.601
(1.03)
0.298
(0.42)
- - 0.027
Implied vs. GARCH
vs. Historical
0.00018
(1.15)
0.632
(1.02)
-0.243
(-0.28)
- 0.123
(7.01)
0.038
Implied vs. EGARCH
-0.00001
(-0.07)
0.695
(1.62)
- 0.176
(0.27)
- 0.026
Implied vs. EGARCH
vs. Historical
0.00026
(1.37)
0.590
(1.45)
-0.374
(-0.57)
- 0.118
(7.74)
0.038
GARCH vs. EGARCH
0.00005
(0.37)
- 1.070
(2.78)
-0.001
(-0.00)
- 0.018
Notes: t-ratios in parentheses; the ex post measure used in this table is the variance of the week’s daily
returns multiplied by the number of trading days in that week. Source: Day and Lewis (1992).
Reprinted with the permission of Elsevier Science.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 64
Conclusions of Paper
•WithinsampleresultssuggestthatIVcontainsextrainformationnot
containedintheGARCH/EGARCHspecifications.
•Outofsampleresultssuggestthatnothingcanaccuratelypredict
volatility!
Conclusions of Paper
•WithinsampleresultssuggestthatIVcontainsextrainformationnot
containedintheGARCH/EGARCHspecifications.
•Outofsampleresultssuggestthatnothingcanaccuratelypredict
volatility!
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 65
Stochastic Volatility Models
•ItisacommonmisconceptionthatGARCH-typespecificationsare
stochasticvolatilitymodels
•However,asthenamesuggests,stochasticvolatilitymodelsdiffer
fromGARCHprincipallyinthattheconditionalvarianceequationofa
GARCH specificationiscompletelydeterministicgivenall
informationavailableuptothatofthepreviousperiod
•ThereisnoerrorterminthevarianceequationofaGARCHmodel,
onlyinthemeanequation
•Stochasticvolatilitymodelscontainaseconderrorterm,whichenters
intotheconditionalvarianceequation.
Stochastic Volatility Models
•ItisacommonmisconceptionthatGARCH-typespecificationsare
stochasticvolatilitymodels
•However,asthenamesuggests,stochasticvolatilitymodelsdiffer
fromGARCHprincipallyinthattheconditionalvarianceequationofa
GARCH specificationiscompletelydeterministicgivenall
informationavailableuptothatofthepreviousperiod
•ThereisnoerrorterminthevarianceequationofaGARCHmodel,
onlyinthemeanequation
•Stochasticvolatilitymodelscontainaseconderrorterm,whichenters
intotheconditionalvarianceequation.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 66
Autoregressive Volatility Models
•Asimpleexampleofastochasticvolatilitymodelistheautoregressive
volatilityspecification
•Thismodelissimpletounderstandandsimpletoestimate,becauseit
requiresthatwehaveanobservablemeasureofvolatilitywhichisthen
simplyusedasanyothervariableinanautoregressivemodel
•ThestandardBox-Jenkins-typeproceduresforestimating
autoregressive(orARMA)modelscanthenbeappliedtothisseries
•Forexample,ifthequantityofinterestisadailyvolatilityestimate,we
couldusesquareddailyreturns,whichtriviallyinvolvestakinga
columnofobservedreturnsandsquaringeachobservation
•Themodelestimatedforvolatility,
t
2
,isthen
Autoregressive Volatility Models
•Asimpleexampleofastochasticvolatilitymodelistheautoregressive
volatilityspecification
•Thismodelissimpletounderstandandsimpletoestimate,becauseit
requiresthatwehaveanobservablemeasureofvolatilitywhichisthen
simplyusedasanyothervariableinanautoregressivemodel
•ThestandardBox-Jenkins-typeproceduresforestimating
autoregressive(orARMA)modelscanthenbeappliedtothisseries
•Forexample,ifthequantityofinterestisadailyvolatilityestimate,we
couldusesquareddailyreturns,whichtriviallyinvolvestakinga
columnofobservedreturnsandsquaringeachobservation
•Themodelestimatedforvolatility,
t
2
,isthen
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 67
A Stochastic Volatility Model Specification
•Theterm‘stochasticvolatility’isusuallyassociatedwithadifferent
formulationtotheautoregressivevolatilitymodel,apossibleexample
ofwhichwouldbe
whereη
tisanotherN(0,1)randomvariablethatisindependentofu
t
•Thevolatilityislatentratherthanobserved,andsoismodelled
indirectly
•Stochasticvolatilitymodelsaresuperiorintheorycomparedwith
GARCH-typemodels,buttheformeraremuchmorecomplexto
estimate.
A Stochastic Volatility Model Specification
•Theterm‘stochasticvolatility’isusuallyassociatedwithadifferent
formulationtotheautoregressivevolatilitymodel,apossibleexample
ofwhichwouldbe
whereη
tisanotherN(0,1)randomvariablethatisindependentofu
t
•Thevolatilityislatentratherthanobserved,andsoismodelled
indirectly
•Stochasticvolatilitymodelsaresuperiorintheorycomparedwith
GARCH-typemodels,buttheformeraremuchmorecomplexto
estimate.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 68
Covariance Modelling: Motivation
•Alimitationofunivariatevolatilitymodelsisthatthefittedconditional
varianceofeachseriesisentirelyindependentofallothers
•Thisispotentiallyanimportantlimitationfortworeasons:
–Ifthereare‘volatilityspillovers’betweenmarketsorassets,theunivariate
modelwillbemis-specified
–Itisoftenthecasethatthecovariancesbetweenseriesareofinteresttoo
–Thecalculationofhedgeratios,portfoliovalueatriskestimates,CAPM
betas,andsoon,allrequirecovariancesasinputs
•MultivariateGARCHmodelscanbeusedforestimationof:
–ConditionalCAPMbetas
–Dynamichedgeratios
–Portfoliovariances
Covariance Modelling: Motivation
•Alimitationofunivariatevolatilitymodelsisthatthefittedconditional
varianceofeachseriesisentirelyindependentofallothers
•Thisispotentiallyanimportantlimitationfortworeasons:
–Ifthereare‘volatilityspillovers’betweenmarketsorassets,theunivariate
modelwillbemis-specified
–Itisoftenthecasethatthecovariancesbetweenseriesareofinteresttoo
–Thecalculationofhedgeratios,portfoliovalueatriskestimates,CAPM
betas,andsoon,allrequirecovariancesasinputs
•MultivariateGARCHmodelscanbeusedforestimationof:
–ConditionalCAPMbetas
–Dynamichedgeratios
–Portfoliovariances
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 69
Simple Covariance Models:
Historical and Implied
•Inexactlythesamefashionasforvolatility,thehistoricalcovariance
orcorrelationbetweentwoseriescanbecalculatedfromasetof
historicaldata
•Impliedcovariancescanbecalculatedusingoptionswhosepayoffsare
dependentonmorethanoneunderlyingasset
•Therelativelysmallnumberofsuchoptionsthatexistlimitsthe
circumstancesinwhichimpliedcovariancescanbecalculated
•Examplesincluderainbowoptions,‘crackspread’optionsfordifferent
gradesofoil,andcurrencyoptions.
Simple Covariance Models:
Historical and Implied
•Inexactlythesamefashionasforvolatility,thehistoricalcovariance
orcorrelationbetweentwoseriescanbecalculatedfromasetof
historicaldata
•Impliedcovariancescanbecalculatedusingoptionswhosepayoffsare
dependentonmorethanoneunderlyingasset
•Therelativelysmallnumberofsuchoptionsthatexistlimitsthe
circumstancesinwhichimpliedcovariancescanbecalculated
•Examplesincluderainbowoptions,‘crackspread’optionsfordifferent
gradesofoil,andcurrencyoptions.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 70
Implied Covariance Models
•Togiveanillustrationforcurrencyoptions,theimpliedvarianceofthe
cross-currencyreturnsisgivenby
where and aretheimpliedvariancesofthexandy
returnsrespectively,and istheimpliedcovariancebetween
xandy
•SoiftheimpliedcovariancebetweenUSD/DEMandUSD/JPYisof
interest,thentheimpliedvariancesofthereturnsofUSD/DEMand
USD/JPYandthereturnsofthecross-currencyDEM/JPYarerequired.
Implied Covariance Models
•Togiveanillustrationforcurrencyoptions,theimpliedvarianceofthe
cross-currencyreturnsisgivenby
where and aretheimpliedvariancesofthexandy
returnsrespectively,and istheimpliedcovariancebetween
xandy
•SoiftheimpliedcovariancebetweenUSD/DEMandUSD/JPYisof
interest,thentheimpliedvariancesofthereturnsofUSD/DEMand
USD/JPYandthereturnsofthecross-currencyDEM/JPYarerequired.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 71
EWMA Covariance Models
•AEWMAspecificationgivesmoreweightinthecovariancetorecent
observationsthananestimatebasedonthesimpleaverage
•TheEWMAmodelestimatesforvariancesandcovariancesattimetin
thebivariatesetupwithtworeturnsseriesxandymaybewrittenas
h
ij,t=λh
ij,t−1+(1−λ)x
t−1y
t−1
whereijforthecovariancesandi=j;x=yforthevariances
•Thefittedvaluesforhalsobecometheforecastsforsubsequent
periods
•λ(0<λ<1)denotesthedecayfactordeterminingtherelativeweights
attachedtorecentversuslessrecentobservations
•Thisparametercouldbeestimatedbutisoftensetarbitrarily(e.g.,
Riskmetricsuseadecayfactorof0.97formonthlydatabut0.94for
daily).
EWMA Covariance Models
•AEWMAspecificationgivesmoreweightinthecovariancetorecent
observationsthananestimatebasedonthesimpleaverage
•TheEWMAmodelestimatesforvariancesandcovariancesattimetin
thebivariatesetupwithtworeturnsseriesxandymaybewrittenas
h
ij,t=λh
ij,t−1+(1−λ)x
t−1y
t−1
whereijforthecovariancesandi=j;x=yforthevariances
•Thefittedvaluesforhalsobecometheforecastsforsubsequent
periods
•λ(0<λ<1)denotesthedecayfactordeterminingtherelativeweights
attachedtorecentversuslessrecentobservations
•Thisparametercouldbeestimatedbutisoftensetarbitrarily(e.g.,
Riskmetricsuseadecayfactorof0.97formonthlydatabut0.94for
daily).
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 72
EWMA Covariance Models -Limitations
•Thisequationcanberewrittenasaninfiniteorderfunctionofonlythe
returnsbysuccessivelysubstitutingoutthecovariances:
•TheEWMAmodelisarestrictedversionofanintegratedGARCH
(IGARCH)specification,anditdoesnotguaranteethefittedvariance-
covariancematrixtobepositivedefinite
•EWMAmodelsalsocannotallowfortheobservedmeanreversionin
thevolatilitiesorcovariancesofassetreturnsthatisparticularly
prevalentatlowerfrequencies.
EWMA Covariance Models -Limitations
•Thisequationcanberewrittenasaninfiniteorderfunctionofonlythe
returnsbysuccessivelysubstitutingoutthecovariances:
•TheEWMAmodelisarestrictedversionofanintegratedGARCH
(IGARCH)specification,anditdoesnotguaranteethefittedvariance-
covariancematrixtobepositivedefinite
•EWMAmodelsalsocannotallowfortheobservedmeanreversionin
thevolatilitiesorcovariancesofassetreturnsthatisparticularly
prevalentatlowerfrequencies.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 73
Multivariate GARCH Models
•Multivariate GARCH models are used to estimate and to forecast
covariances and correlations.
•The basic formulation is similar to that of the GARCH model, but where the
covariances as well as the variances are permitted to be time-varying.
•There are 3 main classes of multivariate GARCH formulation that are widely
used: VECH, diagonal VECH and BEKK.
VECH and Diagonal VECH
•e.g. suppose that there are two variables used in the model. The conditional
covariance matrix is denoted H
t, and would be 2 2. H
tand VECH(H
t) are
t
t
t
t
h
h
h
HVECH
12
22
11
)(
tt
tt
t
hh
hh
H
2221
1211
Multivariate GARCH Models
•Multivariate GARCH models are used to estimate and to forecast
covariances and correlations.
•The basic formulation is similar to that of the GARCH model, but where the
covariances as well as the variances are permitted to be time-varying.
•There are 3 main classes of multivariate GARCH formulation that are widely
used: VECH, diagonal VECH and BEKK.
VECH and Diagonal VECH
•e.g. suppose that there are two variables used in the model. The conditional
covariance matrix is denoted H
t, and would be 2 2. H
tand VECH(H
t) are
t
t
t
t
h
h
h
HVECH
12
22
11
)(
tt
tt
t
hh
hh
H
2221
1211
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 74
VECH and Diagonal VECH
•In the case of the VECH, the conditional variances and covariances would
each depend upon lagged values of all of the variances and covariances
and on lags of the squares of both error terms and their cross products.
•In matrix form, it would be written
•Writing out all of the elements gives the 3 equations as
•Such a model would be hard to estimate. The diagonal VECH is much
simpler and is specified, in the 2 variable case, as follows:112212111012
1222
2
121022
1112
2
111011
tttt
ttt
ttt
huuh
huh
huh
111
tttt HVECHBVECHACHVECH
ttt
HN,0~
1
1123312232111312133
2
232
2
1313112
1122312222111212123
2
222
2
1212122
1121312212111112113
2
212
2
1111111
tttttttt
tttttttt
tttttttt
hbhbhbuuauauach
hbhbhbuuauauach
hbhbhbuuauauach
VECH and Diagonal VECH
•In the case of the VECH, the conditional variances and covariances would
each depend upon lagged values of all of the variances and covariances
and on lags of the squares of both error terms and their cross products.
•In matrix form, it would be written
•Writing out all of the elements gives the 3 equations as
•Such a model would be hard to estimate. The diagonal VECH is much
simpler and is specified, in the 2 variable case, as follows:112212111012
1222
2
121022
1112
2
111011
tttt
ttt
ttt
huuh
huh
huh
111
tttt HVECHBVECHACHVECH
ttt
HN,0~
1
1123312232111312133
2
232
2
1313112
1122312222111212123
2
222
2
1212122
1121312212111112113
2
212
2
1111111
tttttttt
tttttttt
tttttttt
hbhbhbuuauauach
hbhbhbuuauauach
hbhbhbuuauauach
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 75
BEKK and Model Estimation for M-GARCH
•Neither the VECH nor the diagonal VECH ensure a positive definite
variance-covariance matrix.
•An alternative approach is the BEKK model (Engle & Kroner, 1995).
•The BEKK Model uses a Quadratic form for the parameter matrices to
ensure a positive definite variance / covariance matrix H
t.
•In matrix form, the BEKK model is
•Model estimation for all classes of multivariate GARCH model is
again performed using maximum likelihood with the following LLF:
where Nis the number of variables in the system (assumed 2 above),
is a vector containing all of the parameters, and Tis the number of obs. BBAHAWWH
tttt 111
T
t
tttt
HH
TN
1
1'
log
2
1
2log
2
BEKK and Model Estimation for M-GARCH
•Neither the VECH nor the diagonal VECH ensure a positive definite
variance-covariance matrix.
•An alternative approach is the BEKK model (Engle & Kroner, 1995).
•The BEKK Model uses a Quadratic form for the parameter matrices to
ensure a positive definite variance / covariance matrix H
t.
•In matrix form, the BEKK model is
•Model estimation for all classes of multivariate GARCH model is
again performed using maximum likelihood with the following LLF:
where Nis the number of variables in the system (assumed 2 above),
is a vector containing all of the parameters, and Tis the number of obs. BBAHAWWH
tttt 111
T
t
tttt
HH
TN
1
1'
log
2
1
2log
2
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 76
Correlation Models and the CCC
•The correlations between a pair of series at each point in time can be
constructed by dividing the conditional covariances by the product of
the conditional standard deviations from a VECH or BEKK model
•A subtly different approach would be to model the dynamics for the
correlations directly
•In the constant conditional correlation (CCC) model, the correlations
between the disturbances to be fixed through time
•Thus, although the conditional covariances are not fixed, they are tied
to the variances
•The conditional variances in the fixed correlation model are identical
to those of a set of univariate GARCH specifications (although they
are estimated jointly):
Correlation Models and the CCC
•The correlations between a pair of series at each point in time can be
constructed by dividing the conditional covariances by the product of
the conditional standard deviations from a VECH or BEKK model
•A subtly different approach would be to model the dynamics for the
correlations directly
•In the constant conditional correlation (CCC) model, the correlations
between the disturbances to be fixed through time
•Thus, although the conditional covariances are not fixed, they are tied
to the variances
•The conditional variances in the fixed correlation model are identical
to those of a set of univariate GARCH specifications (although they
are estimated jointly):
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 77
More on the CCC
•The off-diagonal elements of H
t, h
ij,t (ij), are defined indirectly via
the correlations, denoted ρ
ij:
•Is it empirically plausible to assume that the correlations are constant
through time?
•Several tests of this assumption have been developed, including a test
based on the information matrix due and a Lagrange Multiplier test
•There is evidence against constant correlations, particularly in the
context of stock returns.
More on the CCC
•The off-diagonal elements of H
t, h
ij,t (ij), are defined indirectly via
the correlations, denoted ρ
ij:
•Is it empirically plausible to assume that the correlations are constant
through time?
•Several tests of this assumption have been developed, including a test
based on the information matrix due and a Lagrange Multiplier test
•There is evidence against constant correlations, particularly in the
context of stock returns.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 78
The Dynamic Conditional Correlation Model
•Several different formulations of the dynamic conditional correlation
(DCC) model are available, but a popular specification is due to Engle
(2002)
•The model is related to the CCC formulation but where the correlations
are allowed to vary over time.
•Define the variance-covariance matrix, H
t, as H
t= D
tR
tD
t
•D
tis a diagonal matrix containing the conditional standard deviations
(i.e. the square roots of the conditional variances from univariate
GARCH model estimations on each of the Nindividual series) on the
leading diagonal
•R
tis the conditional correlation matrix
•Numerous parameterisations of R
tare possible, including an
exponential smoothing approach
The Dynamic Conditional Correlation Model
•Several different formulations of the dynamic conditional correlation
(DCC) model are available, but a popular specification is due to Engle
(2002)
•The model is related to the CCC formulation but where the correlations
are allowed to vary over time.
•Define the variance-covariance matrix, H
t, as H
t= D
tR
tD
t
•D
tis a diagonal matrix containing the conditional standard deviations
(i.e. the square roots of the conditional variances from univariate
GARCH model estimations on each of the Nindividual series) on the
leading diagonal
•R
tis the conditional correlation matrix
•Numerous parameterisations of R
tare possible, including an
exponential smoothing approach
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 79
The DCC Model –A Possible Specification
•A possible specification is of the MGARCH form:
where:
•S is the unconditional correlation matrix of the vector of standardised
residuals (from the first stage estimation), u
t= D
t
−1
ϵ
t
•ιis a vector of ones
•Q
tis an N×Nsymmetric positive definite variance-covariance matrix
•◦denotes the Hadamard or element-by-element matrix multiplication
procedure
•This specification for the intercept term simplifies estimation and
reduces the number of parameters.
The DCC Model –A Possible Specification
•A possible specification is of the MGARCH form:
where:
•S is the unconditional correlation matrix of the vector of standardised
residuals (from the first stage estimation), u
t= D
t
−1
ϵ
t
•ιis a vector of ones
•Q
tis an N×Nsymmetric positive definite variance-covariance matrix
•◦denotes the Hadamard or element-by-element matrix multiplication
procedure
•This specification for the intercept term simplifies estimation and
reduces the number of parameters.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 80
The DCC Model –A Possible Specification
•Engle (2002) proposes a GARCH-esque formulation for dynamically
modelling Q
twith the conditional correlation matrix, R
tthen constructed
as
where diag(·) denotes a matrix comprising the main diagonal elements
of (·) and Q
∗
is a matrix that takes the square roots of each element in Q
•This operation is effectively taking the covariances in Q
tand dividing
them by the product of the appropriate standard deviations in Q
t
∗
to
create a matrix of correlations.
The DCC Model –A Possible Specification
•Engle (2002) proposes a GARCH-esque formulation for dynamically
modelling Q
twith the conditional correlation matrix, R
tthen constructed
as
where diag(·) denotes a matrix comprising the main diagonal elements
of (·) and Q
∗
is a matrix that takes the square roots of each element in Q
•This operation is effectively taking the covariances in Q
tand dividing
them by the product of the appropriate standard deviations in Q
t
∗
to
create a matrix of correlations.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 81
DCC Model Estimation
•The model may be estimated in a single stage using ML although this will
be difficult. So Engle advocates a two-stage procedure where each variable
in the system is first modelled separately as a univariate GARCH
•A joint log-likelihood function for this stage could be constructed, which
would simply be the sum (over N) of all of the log-likelihoods for the
individual GARCH models
•In the second stage, the conditional likelihood is maximised with respect
to any unknown parameters in the correlation matrix
•The log-likelihood function for the second stage estimation will be of the
form
•where θ
1and θ
2denote the parameters to be estimated in the 1
st
and 2
nd
stages respectively.
DCC Model Estimation
•The model may be estimated in a single stage using ML although this will
be difficult. So Engle advocates a two-stage procedure where each variable
in the system is first modelled separately as a univariate GARCH
•A joint log-likelihood function for this stage could be constructed, which
would simply be the sum (over N) of all of the log-likelihoods for the
individual GARCH models
•In the second stage, the conditional likelihood is maximised with respect
to any unknown parameters in the correlation matrix
•The log-likelihood function for the second stage estimation will be of the
form
•where θ
1and θ
2denote the parameters to be estimated in the 1
st
and 2
nd
stages respectively.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 82
Asymmetric Multivariate GARCH
•Asymmetric models have become very popular in empirical applications,
where the conditional variances and / or covariances are permitted to react
differently to positive and negative innovations of the same magnitude
•In the multivariate context, this is usually achieved in the Glosten et al.
(1993) framework
•Kroner and Ng (1998), for example, suggest the following extension to the
BEKK formulation (with obvious related modifications for the VECH or
diagonal VECH models)
where z
t−1is an N-dimensional column vector with elements taking the
value −ϵ
t−1if the corresponding element of ϵ
t−1is negative and zero
otherwise.
Asymmetric Multivariate GARCH
•Asymmetric models have become very popular in empirical applications,
where the conditional variances and / or covariances are permitted to react
differently to positive and negative innovations of the same magnitude
•In the multivariate context, this is usually achieved in the Glosten et al.
(1993) framework
•Kroner and Ng (1998), for example, suggest the following extension to the
BEKK formulation (with obvious related modifications for the VECH or
diagonal VECH models)
where z
t−1is an N-dimensional column vector with elements taking the
value −ϵ
t−1if the corresponding element of ϵ
t−1is negative and zero
otherwise.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 83
An Example: Estimating a Time-Varying Hedge Ratio
for FTSE Stock Index Returns
(Brooks, Henry and Persand, 2002).
•Data comprises 3580 daily observations on the FTSE 100 stock index and
stock index futures contract spanning the period 1 January 1985 -9 April 1999.
•Several competing models for determining the optimal hedge ratio are
constructed. Define the hedge ratio as .
–No hedge (=0)
–Naïve hedge (=1)
–Multivariate GARCH hedges:
•Symmetric BEKK
•Asymmetric BEKK
In both cases, estimating the OHR involves forming a 1-step ahead
forecast and computingt
tF
tCF
t
h
h
OHR
1,
1,
1
An Example: Estimating a Time-Varying Hedge Ratio
for FTSE Stock Index Returns
(Brooks, Henry and Persand, 2002).
•Data comprises 3580 daily observations on the FTSE 100 stock index and
stock index futures contract spanning the period 1 January 1985 -9 April 1999.
•Several competing models for determining the optimal hedge ratio are
constructed. Define the hedge ratio as .
–No hedge (=0)
–Naïve hedge (=1)
–Multivariate GARCH hedges:
•Symmetric BEKK
•Asymmetric BEKK
In both cases, estimating the OHR involves forming a 1-step ahead
forecast and computingt
tF
tCF
t
h
h
OHR
1,
1,
1
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 84
OHR ResultsIn Sample
Unhedged
= 0
Naïve Hedge
= 1
Symmetric Time
Varying
Hedge
tF
tFC
t
h
h
,
,
Asymmetric
Time Varying
Hedge
tF
tFC
t
h
h
,
,
Return 0.0389
{2.3713}
-0.0003
{-0.0351}
0.0061
{0.9562}
0.0060
{0.9580}
Variance 0.8286 0.1718 0.1240 0.1211
Out of Sample
Unhedged
= 0
Naïve Hedge
= 1
Symmetric Time
Varying
Hedge
tF
tFC
t
h
h
,
,
Asymmetric
Time Varying
Hedge
tF
tFC
t
h
h
,
,
Return 0.0819
{1.4958}
-0.0004
{0.0216}
0.0120
{0.7761}
0.0140
{0.9083}
Variance 1.4972 0.1696 0.1186 0.1188
OHR ResultsIn Sample
Unhedged
= 0
Naïve Hedge
= 1
Symmetric Time
Varying
Hedge
tF
tFC
t
h
h
,
,
Asymmetric
Time Varying
Hedge
tF
tFC
t
h
h
,
,
Return 0.0389
{2.3713}
-0.0003
{-0.0351}
0.0061
{0.9562}
0.0060
{0.9580}
Variance 0.8286 0.1718 0.1240 0.1211
Out of Sample
Unhedged
= 0
Naïve Hedge
= 1
Symmetric Time
Varying
Hedge
tF
tFC
t
h
h
,
,
Asymmetric
Time Varying
Hedge
tF
tFC
t
h
h
,
,
Return 0.0819
{1.4958}
-0.0004
{0.0216}
0.0120
{0.7761}
0.0140
{0.9083}
Variance 1.4972 0.1696 0.1186 0.1188
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 85
Plot of the OHR from Multivariate GARCH
Conclusions
-OHR is time-varying and less
than 1
-M-GARCH OHR provides a
better hedge, both in-sample
and out-of-sample.
-No role in calculating OHR for
asymmetriesSymmetric BEKK
Asymmetric BEKK
Time Varying Hedge Ratios
500 1000 1500 2000 2500 3000
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Plot of the OHR from Multivariate GARCH
Conclusions
-OHR is time-varying and less
than 1
-M-GARCH OHR provides a
better hedge, both in-sample
and out-of-sample.
-No role in calculating OHR for
asymmetriesSymmetric BEKK
Asymmetric BEKK
Time Varying Hedge Ratios
500 1000 1500 2000 2500 3000
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Tags
Categories
ScienceDownload
Download Slideshow Get the original presentation fileQuick Actions
Statistics
- Views 13
- Slides 85
- Age 564 days
Related Slideshows
23
Earthquakes_Type of Faults_Science G8.pptx
OctabellFabila1
31 views
15
Quiz #1 Science 10 in the first quarter for jhs
HendrixAntonniAmante
30 views
9
Astronomy history from long ago till doday
ssuserbd9abe
29 views
9
Great history of astronomy from long ago till today
ssuserbd9abe
27 views
20
EARTHQUAKE-DRILL.powerpoint.............
chalobrido8
30 views
9
History of astronomy from old times to the present times
ssuserbd9abe
30 views