Description
This is a PPT on Arch & Garch Model. It talks about modelling volatility and goes on to talk about non linearity.
Modeling Volatility
Non-linearity
? 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
? A “traditional” model could be:
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t,
? We also assumed u
t
~ N(0,o
2
).
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
t
is an iid error term and f is a non-linear function.
? They also give a slightly more specific definition as
y
t
= g(u
t-1
, u
t-2
, …)+ u
t
o
2
(u
t-1
, u
t-2
, …)
where g is a function of past error terms only and o
2
is a variance
term.
? Models with nonlinear g(•) are “non-linear in mean”, while those
with nonlinear o
2
(•) are “non-linear in variance”.
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.
? There are many types of non-linear models, e.g.
- ARCH / GARCH
- switching models
- bilinear models
Testing for Non-linearity
? Portmanteau tests for non-linear dependence have been
developed. The simplest is Ramsey’s RESET test, which took
the form:
? Many other non-linearity tests are available, e.g. the “BDS
test” and the bispectrum test.
? One particular non-linear model that has proved very useful in
finance is the ARCH model due to Engle (1982).
? ? ? ... ? u y y y v
t t t p t
p
t
= + + + + +
÷
| | | |
0 1
2
2
3
1
Heteroscedasticity (Reminder)
? An example of a structural model is
with u
t
~ N(0, ).
? The assumption that the variance of the errors is constant is known as
homoscedasticity, i.e. Var (u
t
) = .
? What if the variance of the errors is not constant?
- heteroscedasticity
- would imply that standard error estimates could be wrong.
? Is the variance of the errors likely to be constant over time? Not for
economic data.
o
u
2
o
u
2
y
t
= |
1
+ |
2
x
2t
+ |
3
x
3t
+ |
4
x
4t
+ u
t
Autoregressive Conditionally Heteroscedastic
(ARCH) Models
? So use a model which does not assume that the variance is constant.
? Recall the definition of the variance of u
t
:
= Var(u
t
| u
t-1
, u
t-2
,...) = E[(u
t
-E(u
t
))
2
| u
t-1
, u
t-2
,...]
We usually assume that E(u
t
) = 0
so = Var(u
t
| u
t-1
, u
t-2
,...) = E[u
t
2
| u
t-1
, u
t-2
,...].
? What could the current value of the variance of the errors plausibly
depend upon?
? Previous squared error terms.
? This leads to the autoregressive conditionally heteroscedastic model for
the variance of the errors:
= o
0
+ o
1
? This is known as an ARCH(1) model.
o
t
2
o
t
2
o
t
2
u
t ÷1
2
Autoregressive Conditionally Heteroscedastic
(ARCH) Models
? The full model would be
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
, u
t
~ N(0, )
where = o
0
+ o
1
? We can easily extend this to the general case where the error variance
depends on q lags of squared errors:
= o
0
+ o
1
+o
2
+...+o
q
? This is an ARCH(q) model.
? Instead of calling the variance , in the literature it is usually called h
t
,
so the model is
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
, u
t
~ N(0,h
t
)
where h
t
= o
0
+ o
1
+o
2
+...+o
q
o
t
2
o
t
2
o
t
2
u
t ÷1
2
u
t q ÷
2
u
t q ÷
2
o
t
2
2
1 ÷ t
u
2
2 ÷ t
u
2
1 ÷ t
u
2
2 ÷ t
u
Another Way of Writing ARCH Models
? For illustration, consider an ARCH(1). Instead of the above, we
can write
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
, u
t
= v
t
o
t
,v
t
~ N(0,1)
? The two are different ways of expressing exactly the same
model. The first form is easier to understand while the second
form is required for simulating from an ARCH model, for
example.
o o o
t t
u = +
÷ 0 1 1
2
Testing for “ARCH Effects”
1. First, run any postulated linear regression of the form given in the equation
above, e.g. y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
saving the residuals, .
2. Then square the residuals, and regress them on q own lags to test for ARCH
of order q, i.e. run the regression
where v
t
is iid.
Obtain R
2
from this regression
3. The test statistic is defined as TR
2
(the number of observations multiplied
by the coefficient of multiple correlation) from the last regression, and is
distributed as a _
2
(q).
t
uˆ
t q t q t t t
v u u u u + + + + + =
÷ ÷ ÷
2 2
2 2
2
1 1 0
2
ˆ ... ˆ ˆ ˆ ¸ ¸ ¸ ¸
Testing for “ARCH Effects”
4. The null and alternative hypotheses are
H
0
: ¸
1
= 0 and ¸
2
= 0 and ¸
3
= 0 and ... and ¸
q
= 0
H
1
: ¸
1
= 0 or ¸
2
= 0 or ¸
3
= 0 or ... or ¸
q
= 0.
If the value of the test statistic is greater than the critical value from the _
2
distribution, then reject the null hypothesis.
? Note that the ARCH test is also sometimes applied directly to returns instead of the
residuals from Stage 1 above.
Problems with ARCH(q) Models
? How do we decide on q?
? The required value of q might be very large
? Non-negativity constraints might be violated.
? When we estimate an ARCH model, we require o
i
>0 ¬ i=1,2,...,q (since
variance cannot be negative)
? A natural extension of an ARCH(q) model which gets around some of these
problems is a GARCH model.
Generalised ARCH (GARCH) Models
? Due to Bollerslev (1986). Allow the conditional variance to be dependent
upon previous own lags
? The variance equation is now
(1)
? This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the
variance equation.
? We could also write
? Substituting into (1) for o
t-1
2
:
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +|o
t-1
2
o
t-1
2
= o
0
+ o
1
2
2 ÷ t
u +|o
t-2
2
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +|(o
0
+ o
1
2
2 ÷ t
u +|o
t-2
2
)
= o
0
+ o
1
2
1 ÷ t
u +o
0
| + o
1
|
2
2 ÷ t
u +|o
t-2
2
Generalised ARCH (GARCH) Models
? Now substituting into (2) for o
t-2
2
? An infinite number of successive substitutions would yield
? So the GARCH(1,1) model can be written as an infinite order ARCH
model.
? We can again extend the GARCH(1,1) model to a GARCH(p,q):
o
t
2
=o
0
+ o
1
2
1 ÷ t
u +o
0
| + o
1
|
2
2 ÷ t
u +|
2
(o
0
+ o
1
2
3 ÷ t
u +|o
t-3
2
)
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +o
0
| + o
1
|
2
2 ÷ t
u +o
0
|
2
+ o
1
|
2 2
3 ÷ t
u +|
3
o
t-3
2
o
t
2
= o
0
(1+|+|
2
) + o
1
2
1 ÷ t
u (1+|L+|
2
L
2
) + |
3
o
t-3
2
o
t
2
= o
0
(1+|+|
2
+...) + o
1
2
1 ÷ t
u (1+|L+|
2
L
2
+...) + |
·
o
0
2
o
t
2
= o
0
+o
1
2
1 ÷ t
u +o
2
2
2 ÷ t
u +...+o
q
2
q t
u
÷
+|
1
o
t-1
2
+|
2
o
t-2
2
+...+|
p
o
t-p
2
o
t
2
=
¿ ¿
= =
÷ ÷
+ +
q
i
p
j
j t j i t i
u
1 1
2
2
0
o | o o
Generalised ARCH (GARCH) Models
? But in general a GARCH(1,1) model will be sufficient to
capture the volatility clustering in the data.
? Why is GARCH Better than ARCH?
- more parsimonious - avoids overfitting
- less likely to breech non-negativity constraints
The Unconditional Variance under the
GARCH Specification
? The unconditional variance of u
t
is 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(u
t
) =
) ( 1
1
0
| o
o
+ ÷
| o +
1
< 1
| o +
1
> 1
| o +
1
= 1
Estimation of ARCH / GARCH Models
? Since the model is no longer of the usual linear form, we cannot use OLS.
? We use MLE.
? The method works by finding the most likely values of the parameters given
the actual data.
? More specifically, we form a log-likelihood function and maximise it.
Extensions to the Basic GARCH Model
? Since the GARCH model was developed, a huge number of extensions and
variants have been proposed. Three of the most important examples are
EGARCH, GJR, and GARCH-M models.
? Problems with GARCH(p,q) Models:
- Non-negativity constraints may still be violated
- GARCH models cannot account for leverage effects
? Possible solutions: the exponential GARCH (EGARCH) model or the GJR
model, which are asymmetric GARCH models.
The EGARCH Model
? Suggested by Nelson (1991). The variance equation is given by
? Advantages of the model
- Since we model the log(o
t
2
), then even if the parameters are negative, o
t
2
will be positive.
- We can account for the leverage effect: if the relationship between
volatility and returns is negative, ¸, will be negative.
(
(
¸
(
¸
÷ + + + =
÷
÷
÷
÷
÷
t
o
o
o
¸ o | e o
2
) log( ) log(
2
1
1
2
1
1
2
1
2
t
t
t
t
t t
u
u
GARCH-in Mean
? We expect a risk to be compensated by a higher return. So why not let the
return be partly determined by its risk?
? Engle, Lilien and Robins (1987) suggested the ARCH-M specification. A
GARCH-M model would be
? o can be interpreted as a sort of risk premium.
? It is possible to combine all or some of these models together to get more
complex “hybrid” models - e.g. an ARMA-EGARCH(1,1)-M model.
y
t
= µ + oo
t-1
+ u
t
, u
t
~ N(0,o
t
2
)
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +|o
t-1
2
What Use Are GARCH-type Models?
? GARCH can model the volatility clustering effect since the conditional
variance is autoregressive. Such models can be used to forecast volatility.
? We could show that
Var (y
t
| y
t-1
, y
t-2
, ...) = Var (u
t
| u
t-1
, u
t-2
, ...)
? So modelling o
t
2
will give us models and forecasts for y
t
as well.
? Variance forecasts are additive over time.
doc_305818598.ppt
This is a PPT on Arch & Garch Model. It talks about modelling volatility and goes on to talk about non linearity.
Modeling Volatility
Non-linearity
? 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
? A “traditional” model could be:
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t,
? We also assumed u
t
~ N(0,o
2
).
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
t
is an iid error term and f is a non-linear function.
? They also give a slightly more specific definition as
y
t
= g(u
t-1
, u
t-2
, …)+ u
t
o
2
(u
t-1
, u
t-2
, …)
where g is a function of past error terms only and o
2
is a variance
term.
? Models with nonlinear g(•) are “non-linear in mean”, while those
with nonlinear o
2
(•) are “non-linear in variance”.
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.
? There are many types of non-linear models, e.g.
- ARCH / GARCH
- switching models
- bilinear models
Testing for Non-linearity
? Portmanteau tests for non-linear dependence have been
developed. The simplest is Ramsey’s RESET test, which took
the form:
? Many other non-linearity tests are available, e.g. the “BDS
test” and the bispectrum test.
? One particular non-linear model that has proved very useful in
finance is the ARCH model due to Engle (1982).
? ? ? ... ? u y y y v
t t t p t
p
t
= + + + + +
÷
| | | |
0 1
2
2
3
1
Heteroscedasticity (Reminder)
? An example of a structural model is
with u
t
~ N(0, ).
? The assumption that the variance of the errors is constant is known as
homoscedasticity, i.e. Var (u
t
) = .
? What if the variance of the errors is not constant?
- heteroscedasticity
- would imply that standard error estimates could be wrong.
? Is the variance of the errors likely to be constant over time? Not for
economic data.
o
u
2
o
u
2
y
t
= |
1
+ |
2
x
2t
+ |
3
x
3t
+ |
4
x
4t
+ u
t
Autoregressive Conditionally Heteroscedastic
(ARCH) Models
? So use a model which does not assume that the variance is constant.
? Recall the definition of the variance of u
t
:
= Var(u
t
| u
t-1
, u
t-2
,...) = E[(u
t
-E(u
t
))
2
| u
t-1
, u
t-2
,...]
We usually assume that E(u
t
) = 0
so = Var(u
t
| u
t-1
, u
t-2
,...) = E[u
t
2
| u
t-1
, u
t-2
,...].
? What could the current value of the variance of the errors plausibly
depend upon?
? Previous squared error terms.
? This leads to the autoregressive conditionally heteroscedastic model for
the variance of the errors:
= o
0
+ o
1
? This is known as an ARCH(1) model.
o
t
2
o
t
2
o
t
2
u
t ÷1
2
Autoregressive Conditionally Heteroscedastic
(ARCH) Models
? The full model would be
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
, u
t
~ N(0, )
where = o
0
+ o
1
? We can easily extend this to the general case where the error variance
depends on q lags of squared errors:
= o
0
+ o
1
+o
2
+...+o
q
? This is an ARCH(q) model.
? Instead of calling the variance , in the literature it is usually called h
t
,
so the model is
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
, u
t
~ N(0,h
t
)
where h
t
= o
0
+ o
1
+o
2
+...+o
q
o
t
2
o
t
2
o
t
2
u
t ÷1
2
u
t q ÷
2
u
t q ÷
2
o
t
2
2
1 ÷ t
u
2
2 ÷ t
u
2
1 ÷ t
u
2
2 ÷ t
u
Another Way of Writing ARCH Models
? For illustration, consider an ARCH(1). Instead of the above, we
can write
y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
, u
t
= v
t
o
t
,v
t
~ N(0,1)
? The two are different ways of expressing exactly the same
model. The first form is easier to understand while the second
form is required for simulating from an ARCH model, for
example.
o o o
t t
u = +
÷ 0 1 1
2
Testing for “ARCH Effects”
1. First, run any postulated linear regression of the form given in the equation
above, e.g. y
t
= |
1
+ |
2
x
2t
+ ... + |
k
x
kt
+ u
t
saving the residuals, .
2. Then square the residuals, and regress them on q own lags to test for ARCH
of order q, i.e. run the regression
where v
t
is iid.
Obtain R
2
from this regression
3. The test statistic is defined as TR
2
(the number of observations multiplied
by the coefficient of multiple correlation) from the last regression, and is
distributed as a _
2
(q).
t
uˆ
t q t q t t t
v u u u u + + + + + =
÷ ÷ ÷
2 2
2 2
2
1 1 0
2
ˆ ... ˆ ˆ ˆ ¸ ¸ ¸ ¸
Testing for “ARCH Effects”
4. The null and alternative hypotheses are
H
0
: ¸
1
= 0 and ¸
2
= 0 and ¸
3
= 0 and ... and ¸
q
= 0
H
1
: ¸
1
= 0 or ¸
2
= 0 or ¸
3
= 0 or ... or ¸
q
= 0.
If the value of the test statistic is greater than the critical value from the _
2
distribution, then reject the null hypothesis.
? Note that the ARCH test is also sometimes applied directly to returns instead of the
residuals from Stage 1 above.
Problems with ARCH(q) Models
? How do we decide on q?
? The required value of q might be very large
? Non-negativity constraints might be violated.
? When we estimate an ARCH model, we require o
i
>0 ¬ i=1,2,...,q (since
variance cannot be negative)
? A natural extension of an ARCH(q) model which gets around some of these
problems is a GARCH model.
Generalised ARCH (GARCH) Models
? Due to Bollerslev (1986). Allow the conditional variance to be dependent
upon previous own lags
? The variance equation is now
(1)
? This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the
variance equation.
? We could also write
? Substituting into (1) for o
t-1
2
:
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +|o
t-1
2
o
t-1
2
= o
0
+ o
1
2
2 ÷ t
u +|o
t-2
2
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +|(o
0
+ o
1
2
2 ÷ t
u +|o
t-2
2
)
= o
0
+ o
1
2
1 ÷ t
u +o
0
| + o
1
|
2
2 ÷ t
u +|o
t-2
2
Generalised ARCH (GARCH) Models
? Now substituting into (2) for o
t-2
2
? An infinite number of successive substitutions would yield
? So the GARCH(1,1) model can be written as an infinite order ARCH
model.
? We can again extend the GARCH(1,1) model to a GARCH(p,q):
o
t
2
=o
0
+ o
1
2
1 ÷ t
u +o
0
| + o
1
|
2
2 ÷ t
u +|
2
(o
0
+ o
1
2
3 ÷ t
u +|o
t-3
2
)
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +o
0
| + o
1
|
2
2 ÷ t
u +o
0
|
2
+ o
1
|
2 2
3 ÷ t
u +|
3
o
t-3
2
o
t
2
= o
0
(1+|+|
2
) + o
1
2
1 ÷ t
u (1+|L+|
2
L
2
) + |
3
o
t-3
2
o
t
2
= o
0
(1+|+|
2
+...) + o
1
2
1 ÷ t
u (1+|L+|
2
L
2
+...) + |
·
o
0
2
o
t
2
= o
0
+o
1
2
1 ÷ t
u +o
2
2
2 ÷ t
u +...+o
q
2
q t
u
÷
+|
1
o
t-1
2
+|
2
o
t-2
2
+...+|
p
o
t-p
2
o
t
2
=
¿ ¿
= =
÷ ÷
+ +
q
i
p
j
j t j i t i
u
1 1
2
2
0
o | o o
Generalised ARCH (GARCH) Models
? But in general a GARCH(1,1) model will be sufficient to
capture the volatility clustering in the data.
? Why is GARCH Better than ARCH?
- more parsimonious - avoids overfitting
- less likely to breech non-negativity constraints
The Unconditional Variance under the
GARCH Specification
? The unconditional variance of u
t
is 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(u
t
) =
) ( 1
1
0
| o
o
+ ÷
| o +
1
< 1
| o +
1
> 1
| o +
1
= 1
Estimation of ARCH / GARCH Models
? Since the model is no longer of the usual linear form, we cannot use OLS.
? We use MLE.
? The method works by finding the most likely values of the parameters given
the actual data.
? More specifically, we form a log-likelihood function and maximise it.
Extensions to the Basic GARCH Model
? Since the GARCH model was developed, a huge number of extensions and
variants have been proposed. Three of the most important examples are
EGARCH, GJR, and GARCH-M models.
? Problems with GARCH(p,q) Models:
- Non-negativity constraints may still be violated
- GARCH models cannot account for leverage effects
? Possible solutions: the exponential GARCH (EGARCH) model or the GJR
model, which are asymmetric GARCH models.
The EGARCH Model
? Suggested by Nelson (1991). The variance equation is given by
? Advantages of the model
- Since we model the log(o
t
2
), then even if the parameters are negative, o
t
2
will be positive.
- We can account for the leverage effect: if the relationship between
volatility and returns is negative, ¸, will be negative.
(
(
¸
(
¸
÷ + + + =
÷
÷
÷
÷
÷
t
o
o
o
¸ o | e o
2
) log( ) log(
2
1
1
2
1
1
2
1
2
t
t
t
t
t t
u
u
GARCH-in Mean
? We expect a risk to be compensated by a higher return. So why not let the
return be partly determined by its risk?
? Engle, Lilien and Robins (1987) suggested the ARCH-M specification. A
GARCH-M model would be
? o can be interpreted as a sort of risk premium.
? It is possible to combine all or some of these models together to get more
complex “hybrid” models - e.g. an ARMA-EGARCH(1,1)-M model.
y
t
= µ + oo
t-1
+ u
t
, u
t
~ N(0,o
t
2
)
o
t
2
= o
0
+ o
1
2
1 ÷ t
u +|o
t-1
2
What Use Are GARCH-type Models?
? GARCH can model the volatility clustering effect since the conditional
variance is autoregressive. Such models can be used to forecast volatility.
? We could show that
Var (y
t
| y
t-1
, y
t-2
, ...) = Var (u
t
| u
t-1
, u
t-2
, ...)
? So modelling o
t
2
will give us models and forecasts for y
t
as well.
? Variance forecasts are additive over time.
doc_305818598.ppt