Financial Econometrics
Dr. Kashif Saleem
Associate Professor (Finance)
Lappeenranta School of Business
Financial Econometrics – Dr. Kashif Saleem
1
Modelling volatility
and correlation
•Autoregressive volatility models
–Autoregressive conditionally heteroscedastic (ARCH) model
–Generalised ARCH (GARCH) model
–Estimation of ARCH/GARCH models
•Extensions to the basic GARCH model
–The GJR model
–The EGARCH model
–GARCH-in-mean
–Forecasting: using GARCH Models
•Multivariate GARCH models
–The VECH model
–The diagonal VECH model
–The BEKK model
Financial Econometrics – Dr. Kashif Saleem
2
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
• Our “traditional” structural model could be something like:
yt = 1 + 2x2t + ... + kxkt + ut,
We also assumed ut  N(0,2).
Financial Econometrics – Dr. Kashif Saleem
3
A Sample Financial Asset Returns Time Series
Daily S&P 500 Returns for January 1990 – December 1999
Return
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
1/01/90
11/01/93
Date
Financial Econometrics – Dr. Kashif Saleem
9/01/97
4
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.
– Ramsey’s RESET Test
– BDS Test
• There are many types of non-linear models, e.g.
- ARCH / GARCH
- switching models
Financial Econometrics – Dr. Kashif Saleem
5
Heteroscedasticity Revisited
• An example of a structural model is
yt = 1 + 2x2t + 3x3t + 4x4t + u t
with ut  N(0, u2 ).
• The assumption that the variance of the errors is constant is known as
homoscedasticity, i.e. Var (ut) =  u2 .
• 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 financial data.
Financial Econometrics – Dr. Kashif Saleem
6
Models for volatility
•
•
•
•
Historical volatility
Implied Volatility
EWMA
AR volatility models
–
–
–
–
–
–
ARCH
GARCH
GARCH-M
EGARCH, GJR
VECH
BEKK
Financial Econometrics – Dr. Kashif Saleem
7
Autoregressive Conditionally Heteroscedastic
(ARCH) Models
• use a model which does not assume that the variance is constant.
• Recall the definition of the variance of ut:
t2 = Var(ut ut-1, ut-2,...) = E[(ut-E(ut))2 ut-1, ut-2,...]
We usually assume that E(ut) = 0
so t2 = Var(ut  ut-1, ut-2,...) = E[ut2 ut-1, ut-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:
t2 = 0 + 1 ut21
• This is known as an ARCH(1) model.
Financial Econometrics – Dr. Kashif Saleem
8
Autoregressive Conditionally Heteroscedastic
(ARCH) Models (cont’d)
• The full model would be
yt = 1 + 2x2t + ... + kxkt + ut , ut  N(0, t2)
where t2 = 0 + 1 ut21
• We can easily extend this to the general case where the error variance
depends on q lags of squared errors:
2
2
2
t2 = 0 + 1 ut 1 +2 ut  2 +...+qut  q
• This is an ARCH(q) model.
• Instead of calling the variance t2, in the literature it is usually called ht,
so the model is
yt = 1 + 2x2t + ... + kxkt + ut , ut  N(0,ht)
where ht = 0 + 1 ut21 +2 ut2 2 +...+q ut2 q
Financial Econometrics – Dr. Kashif Saleem
9
Testing for “ARCH Effects”
1. First, run any postulated linear regression of the form given in the equation
above, e.g.
yt = 1 + 2x2t + ... + kxkt + ut
saving the residuals, ût .
2. Then square the residuals, and regress them on q own lags to test for ARCH
of order q, i.e. run the regression
uˆt2   0   1uˆt21   2uˆt2 2  ...   quˆt2 q  vt
where vt is iid.
Obtain R2 from this regression
3. The test statistic is defined as TR2 (the number of observations multiplied
by the coefficient of multiple correlation) from the last regression, and is
distributed as a 2(q).
Financial Econometrics – Dr. Kashif Saleem
10
Testing for “ARCH Effects” (cont’d)
4. The null and alternative hypotheses are
H0 : 1 = 0 and 2 = 0 and 3 = 0 and ... and q = 0
H1 : 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.
• Residual Tests: Correlogram-Q statistics, Correlogram Squared
Residuals, Histogram-Normality Test
Financial Econometrics – Dr. Kashif Saleem
11
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 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.
Financial Econometrics – Dr. Kashif Saleem
12
Generalised ARCH (GARCH) Models
• Due to Bollerslev (1986). Allow the conditional variance to be dependent
upon previous own lags
• The variance equation is now
t2 = 0 + 1 ut21 +t-12
(1)
• This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the
variance equation.
• We could also write
t-12 = 0 + 1ut22 +t-22
t-22 = 0 + 1 ut23 +t-32
• Substituting into (1) for t-12 :
 t2 = 0 + 1 ut21 +(0 + 1ut22 + t-22)
= 0 + 1 ut21 +0 + 1ut22 + t-22
Financial Econometrics – Dr. Kashif Saleem
13
Generalised ARCH (GARCH) Models (cont’d)
• Now substituting into (2) for t-22
t2 = 0 + 1 ut21 +0 + 1ut22 +2(0 + 1ut23 +t-32)
t2 = 0 + 1 ut21 +0 + 1ut22 +02 + 12ut23 +3t-32
t2 = 0 (1++2) + 1 ut21 (1+L+2L2 ) + 3t-32
• An infinite number of successive substitutions would yield
t2 = 0 (1++2+...) + 1 ut21 (1+L+2L2+...) + 02
• 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):
t2 = 0+1 ut21 +2ut2 2 +...+qut2q +1t-12+2t-22+...+pt-p2
t2
q
=  0   i u
i 1
p
2
t i
   j t  j
2
j 1
Financial Econometrics – Dr. Kashif Saleem
14
Generalised ARCH (GARCH) Models (cont’d)
• 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
Financial Econometrics – Dr. Kashif Saleem
15
The Unconditional Variance under the
GARCH Specification
• The unconditional variance of ut is given by
Var(ut) =
0
1  (1   )
when 1   < 1
•
1    1 is termed “non-stationarity” in variance
•
1   = 1 is termed intergrated GARCH
• For non-stationarity in variance, the conditional variance forecasts will
not converge on their unconditional value as the horizon increases.
Financial Econometrics – Dr. Kashif Saleem
16
Estimation of ARCH / GARCH Models
• Since the model is no longer of the usual linear form, we cannot use
OLS.
• We use another technique known as maximum likelihood.
• 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.
Financial Econometrics – Dr. Kashif Saleem
17
Estimation of ARCH / GARCH Models (cont’d)
•
The steps involved in actually estimating an ARCH or GARCH model
are as follows
1.
Specify the appropriate equations for the mean and the variance - e.g. an
AR(1)- GARCH(1,1) model:
yt =  + yt-1 + ut , ut  N(0,t2)
t2 = 0 + 1 ut21 +t-12
2.
Specify the log-likelihood function to maximise:
T
1 T
1 T
2
2
L   log( 2 )   log(  t )   ( yt    yt 1 ) 2 /  t
2
2 t 1
2 t 1
3. The computer will maximise the function and give parameter values and
their standard errors
Financial Econometrics – Dr. Kashif Saleem
18
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.
Financial Econometrics – Dr. Kashif Saleem
19
The EGARCH Model
• Suggested by Nelson (1991). The variance equation is given by
log(  t )     log(  t 1 )  
2
2
u t 1
 t 1 2
 u
2
t 1



2

  t 1


• Advantages of the model
- Since we model the log(t2), then even if the parameters are negative, t2
will be positive.
- We can account for the leverage effect: if the relationship between
volatility and returns is negative, , will be negative.
Financial Econometrics – Dr. Kashif Saleem
20
The GJR Model
• Due to Glosten, Jaganathan and Runkle
t2 = 0 + 1 ut21 +t-12+ut-12It-1
where It-1 = 1 if ut-1 < 0
= 0 otherwise
• For a leverage effect, we would see  > 0.
• We require 1 +   0 and 1  0 for non-negativity.
Financial Econometrics – Dr. Kashif Saleem
21
An Example of the use of a GJR Model
• Using monthly S&P 500 returns
• Estimating a GJR model, we obtain the following results.
yt  0.172
(3.198)
 t 2  1.243  0.015ut21  0.498 t 1 2  0.604ut21 I t 1
(16.372) (0.437) (14.999) (5.772)
• how volatility rises more after a same magnitude of negative shock
than a positive one?????
Financial Econometrics – Dr. Kashif Saleem
22
Tests for asymmetries in volatility
•
Engle and Ng (1993) have proposed a set of tests for asymmetry in volatility,
known as: sign and size bias tests.
•
determine whether an asymmetric model is required for a given series, or
whether the symmetric GARCH model can be deemed adequate.
•
Define S−t−1 as an indicator dummy that takes the value 1 if ˆut−1 < 0 and zero
otherwise.
•
The test for sign bias is based on the significance or otherwise of φ1 in
•
ˆu2t = φ0 + φ1S−t−1 + υt
•
If positive and negative shocks to ˆut−1 impact differently upon the conditional
variance, then φ1 will be statistically significant.
23
Financial Econometrics – Dr. Kashif Saleem
Tests for asymmetries in volatility
•
It could also be the case that the magnitude or size of the shock will affect
whether the response of volatility to shocks is symmetric or not.
•
In this case, a negative size bias test would be conducted, based on a
regression where S−t−1 is now used as a slope dummy variable.
•
Negative size bias is argued to be present if φ1 is statistically significant in the
regression
•
ˆu2t = φ0 + φ1S− t−1ut−1 + υt
•
Finally, defining S+t−1= 1 − S−t−1, so that S+t−1 picks out the observations with
positive innovations.
Financial Econometrics – Dr. Kashif Saleem
24
News Impact Curves
The news impact curve plots the next period volatility (ht) that would arise from various
positive and negative values of ut-1, given an estimated model.
News Impact Curves for S&P 500 Returns using Coefficients from GARCH and GJR Model Estimates
0.14
GARCH
GJR
Value of Conditional Variance
0.12
0.1
0.08
0.06
0.04
0.02
0
-1
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Value of Lagged Shock
Financial Econometrics – Dr. Kashif Saleem
25
GARCH-in Mean
• We expect a risk to be compensated by a higher return. So why not let
the return of a security be partly determined by its risk?
• Engle, Lilien and Robins (1987) suggested the ARCH-M specification.
A GARCH-M model would be
yt =  + t-1+ ut , ut  N(0,t2)
t2 = 0 + 1 ut21 +t-12
•  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.
Financial Econometrics – Dr. Kashif Saleem
26
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.
• Consider the following GARCH(1,1) model:
yt    ut , ut  N(0,t2),  t 2   0   1u t21   t 1 2
• What is needed is to generate are forecasts of T+12 T, T+22 T, ...,
T+s2 T where T denotes all information available up to and
including observation T.
• Adding one to each of the time subscripts of the above conditional
variance equation, and then two, and then three would yield the
following equations
T+12 = 0 + 1 uT2 +T2 , T+22 = 0 + 1 uT+12 +T+12 , T+32 = 0 +
1 uT+22 +T+22
Financial Econometrics – Dr. Kashif Saleem
27
Forecasting Variances
using GARCH Models (Cont’d)
• Let 1f,T 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.
•  1f,T 2 would be obtained by taking the conditional expectation of the
first equation at the bottom of slide 28:
2
 1f,T = 0 + 1 uT2 +T2
2
• Given,  1f,T how is 2f,T , 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 28:
2
2
2
 2f,T = 0 + 1E( uT 1 T) +  1f,T
2
2
• where E( uT 1 T) is the expectation, made at time T, of uT 1, which is
the squared disturbance term.
2
2
Financial Econometrics – Dr. Kashif Saleem
28
Forecasting Variances
using GARCH Models (Cont’d)
• We can write
E(uT+12  t) = T+12
• But T+12 is not known at time T, so it is replaced with the forecast for
it,  f 2 , so that the 2-step ahead forecast is given by
1,T
2
2
2
 2f,T = 0 + 1 1f,T + 1f,T
2
f 2 =  + ( +) f
 2,T
1,T
0
1
• By similar arguments, the 3-step ahead forecast will be given by
2
 3f,T = ET(0 + 1 + T+22)
= 0 + (1+) 2f,T 2
2
= 0 + (1+)[ 0 + (1+) 1f,T ]
2
= 0 + 0(1+) + (1+)2 1f,T
• Any s-step ahead forecast (s  2) would be produced by
s 1
f
s ,T
h
  0  ( 1   ) i 1  ( 1   ) s 1 h1f,T
i 1
Financial Econometrics – Dr. Kashif Saleem
29
Testing Non-linear Restrictions or
Testing Hypotheses about Non-linear Models
• Usual t- and F-tests are still valid in non-linear models, but they are
not flexible enough.
• There are three hypothesis testing procedures based on maximum
likelihood principles:
• Wald,
• Likelihood Ratio,
• Lagrange Multiplier.
Financial Econometrics – Dr. Kashif Saleem
30
Likelihood Ratio Tests
• Estimate under the null hypothesis and under the alternative.
• Then compare the maximised values of the LLF.
• So we estimate the unconstrained model and achieve a given maximised
value of the LLF, denoted Lu
• Then estimate the model imposing the constraint(s) and get a new value of
the LLF denoted Lr.
• The LR test statistic is given by
LR = -2(Lr - Lu)  2(m)
where m = number of restrictions
Financial Econometrics – Dr. Kashif Saleem
31
Likelihood Ratio Tests (cont’d)
• Example: We estimate a GARCH model and obtain a maximised LLF of
66.85. We are interested in testing whether  = 0 in the following equation.
yt =  + yt-1 + ut , ut  N(0, t )
2
t2 = 0 + 1 ut21 +   t 1
2
• We estimate the model imposing the restriction and observe the maximised
LLF falls to 64.54. Can we accept the restriction?
• LR = -2(64.54-66.85) = 4.62.
• The test follows a 2(1) = 3.84 at 5%, so reject the null.
Financial Econometrics – Dr. Kashif Saleem
32
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
• BEKK.
Financial Econometrics – Dr. Kashif Saleem
33
VECH- Bollerslev, Engle and Wooldridge
(1988)
• In matrix form, it would be written:
VECH H t   C  A VECH  t 1t 1   B VECH H t 1 
•
•
•
•
•
•
•
 t  t 1 ~ N 0, H t 
suppose that there are two variables used in the model.
The conditional covariance matrix is denoted Ht, and would be 2  2.
a 2 × 1 innovation (disturbance) vector
ψt−1 represents the information set at time t – 1
C is a 3 × 1 parameter vector
A and B are 3 × 3 parameter matrices
VECH(·) denotes the column-stacking operator applied to the upper
portion of the symmetric matrix.
Financial Econometrics – Dr. Kashif Saleem
34
VECH- Bollerslev, Engle and
Wooldridge (1988)
How the VECH model works…..
 h11t 
VECH ( H t )  h22t 
 h12t 
Financial Econometrics – Dr. Kashif Saleem
35
VECH- Bollerslev, Engle and
Wooldridge (1988)
• 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.
Financial Econometrics – Dr. Kashif Saleem
36
Diagonal VECH
• The diagonal VECH is much simpler and is specified, in
the 2 variable case, as follows:
h11t   0   1u12t 1   2 h11t 1
h22t   0   1u 22t 1   2 h22t 1
h12t   0   1u1t 1u 2t 1   2 h12t 1
Financial Econometrics – Dr. Kashif Saleem
37
BEKK and Model Estimation for M-GARCH
•
•
•
Neither the VECH nor the diagonal VECH ensure a positive definite variancecovariance matrix.
An alternative approach is the BEKK model (Engle & Kroner, 1995).
In matrix form, the BEKK model is
H t  W W  AH t 1 A  Bt 1t 1B
•
Model estimation for all classes of multivariate GARCH model is again
performed using maximum likelihood with the following LLF:
TN
1 T
   
log 2   log H t   t' H t1 t
2
2 t 1
where N is the number of variables in the system (assumed 2 above),  is a
vector containing all of the parameters to be estimated, and T is the number of
observations.


Financial Econometrics – Dr. Kashif Saleem
38
What we have learned !!!!
• classical linear regression model
– CAPM, APT, assumptions and diagnostic tests
• time series modelling
– ARMA processes
– VAR MODEL
– SEM
• long-run relationships
– Cointegration Models , VECM
• Modelling volatility and correlation
– Univariate GARCH models
– Multivariate GARCH models
How to examine all these models using E-views
Financial Econometrics – Dr. Kashif Saleem
39