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Centre for Central Banking Studies
Bank of England
Time Series Modelling in Central Banks: An
application of ARCH and GARCH models
Ibrahim Stevens
Centre for Central Banking Studies
Bank of England
May 2004
Introduction
• Objective
– Overview of time series modelling techniques
– Modelling of volatility clustering
• Motivation
– Understanding the properties of macroeconomic
time series used in central banks
• Information extraction to support monetary and
financial stability policy decisions
– Option pricing, extreme value theory, markets/asset
correlations, event studies, risk calibration
Structure
• Review of key results in econometric time
series
– Stationary and nonstationary time series processes
– Spurious regressions and cointegration and Vector
Autoregressions (VARs)
• Models of changing volatility
– Autoregressive conditional heteroscedasticity
(ARCH)
– Generalised ARCH (GARCH)
Stationary and nonstationary time
series processes
• Economic time series
– Data obtained from observations of an economic variable
over time
• E.g. Interest rates, stock prices, price indices, GDP,
inflation…..
• A time series is a stochastic process
– A sequence of random variables (RVs):
 y1, y0 , y1, y2
• We use means, variances and covariances to capture
most of the information about the probabilistic
structure of time series
• A time series is (weakly) or covariance
stationary if its probability distribution does
not change with time
– Constant mean:   yt     y 
– Constant variance:
Var  yt      0
2
t
– Covariances: Cov  yt , yt k   Cov  yt , yt k    k k
• Stationary time series fluctuate around a constant level
• Two types of stationary variables
– Trend stationary
– Differenced stationary
• Difficult to distinguish between the two
• Variables that needs to be ‘de-trended’ or
‘differenced’ to achieve stationarity are
nonstationary
– To de-trend a series, subtract if from a time trend
– If a time series is integrated or order d, I(d); it has
to be differenced at least d times to make it
stationary
• Integrated series are called unit root processes
S&P 500 weekly prices
(nonstationary/unit root)
S&P 500 weekly return
(differenced-stationary)
Index
1,800
Continously compounded return
0.10
1,600
0.05
1,400
1,200
0.00
1,000
800
-0.05
600
-0.10
400
200
0
93 94 95 96 97 98 99 00 01 02 03
Source: Bloomberg.
-0.15
93 94 95 96 97 98 99 00 01 02 03
Sources: Bloomberg and Bank calculations.
Basic Stochastic Processes
• Autoregressive (AR) models:
yt  0  1 yt 1  2 yt 2 
  p yt  p   t
• Moving average (MA) models:
yt  0  1 t 1  2 t 2 
 q t q
• Autoregressive moving average (ARMA) models:
yt  0  1 yt 1  2 yt 2 
1 t 1   2 t 2 
  q t q
  p yt  p   t 
• White noise (WN) process: yt  0   t
– A zero-mean, constant variance and no autocorrelation :  t
is said to be iid (identically and independently distributed) –
stationary process
• Random walks (RW) (widely used in econometric time
series: y  y  
t
t 1
t
• In a RW, the changes in the levels of the series is
determined by the addition of a random error term
• RWs may contain a drift term:
yt  yt 1     t
• Stochastic processes are estimated in EViews using the AR and MA
functions in the equation window
• Two important properties of RWs
– Markov property: Only current information is
relevant in determining the conditional probability of
the future value of the RV
– Martingale: the conditional expectation of the
future value of a RV is the current value. However,
a martingale need not have a constant variance of
independent errors.
• A RW plus a drift is therefore a martingale
– A positive drift term = sub-martingale
– A negative drift term is a super-martingales
– Practical applications
• Are stock prices a RW?
• RW and the efficient markets hypothesis (rational
expectations)
Mean, variance and autocorrelations of a weekly
stationary variable
• Consider a simple AR(1) model: yt  0  1 yt 1   t
– The mean:   yt     y  
0
1  1
2

– The variance : Var  yt    0
1  1
– The autocorrelation function (ACF):
cov( yt , yt k )  t k 1 ( yt  y )( yt k  y )  k
rk 


T
var( yt )
0
 t k 1 ( yt  y )
T
• Where k is the is the number of lags and y is the sample mean
• The ACF is simple a regression of the variable on a constant and
its k-period lagged variables
• The partial autocorrelation function (PACF):
– Usually calculated by fitting autoregressive models of
increasing order and taking the last coefficient in each model
as the sample PACF. A cut-off is determined by 2/ T
• The Ljung-Box Q-test (Q-stats) is the standard test of the
significance of ACF with the null of zero autocorrelation
(available in EViews):
Q  T (T  2) k 1 rk2 /(T  k )
s
 ( m)
Identifying difference stationary process:
unit roots tests
• Recall
– A time series, y, is integrated of the order d, denoted as (y
~ I (d)) if it has to be differenced d times to make it
stationary, (Δd yt )
• A non stationary (differenced) time series contains at
least on unit root
• Several methods are available to test for unit roots:
Augmented Dickey-Fuller, Phillips-Perron,
Kwiatkowski-Phillips-Schmidt-Shin (KPSS)
– Other issues : structured breaks, ARCH effects
• Stock prices are integrated (unit roots) but stock
returns are stationary
Spurious regression and cointegration
• Spurious regressions
– Regressing two non-stationary variables against each other
contravenes the assumptions of the classical regression
model (stationary variables plus zero-mean and constantvariance innovation term:
– …..produces biased standard errors…..
– …....and thus biased hypothesis testing
• Likely to reject a false null hypothesis and thus accept
an incorrect relationship – without any economic
meaning
– Models with high R2 and low Durbin Watson statistics
• Cointegration
– Stationarity means that a variable is in statistical equilibrium
– The link between nonstationary variables and long-run
economic equilibrium is know as cointegration
– Formalised in Engle and Granger (1987)
• Two variables, are said to be cointegrated of order d, b, denoted
CI(d,b), if :
– They are both I(d)
– And there exist some  (cointegrating parameter) such that :
et  yt   xt  
– When b=1 and d=1, et ~1(0)
I (d  b), b  0
• Cointegration and error-correction model
– Cointegrated processes embody and error-correction
mechanism
• Engle and Granger (1987) suggest the following
procedure:
• Stage 1: estimate a long-run cointegrating regression and
test for cointegration:
yt   0  1 xt  ut
• Residual-based cointegration test (ut ~ I(0))
use MacKinnon (1991) critical values instead (available
in EViews):
ut  ut 1  et
• Stage 2 (if two variables are cointegrated, they have a valid
error correction model) : Construct the following error
correction model:
n
n
i 1
i 1
n
n
i 1
i 1
xt  1uˆt 1   B1yt 1   C1xt 1   xt
yt   2uˆt 1   B2 xt 1   C2 yt 1   yt
• with 1 and 2 ≠ 0 (the speed of adjustment parameters)
• The error correction is due to what is known as the Granger
representation theorem: a cointegrated system can be
represented in autoregressive moving average, or error
correction form, but not as a VAR in the differenced variable
• Due mainly to the fact that vector MA (VMA) models
are noninvertible (the MA terms contain a unit root).
– However, this is a problem rectified by the inclusion of the
long-run error correction term in the error correction model
above.
• Multivariate extensions
– Johansen’s cointegrating VAR model (The default
cointegration method in EViews) – Johansen (1988) and
Johansen and Juselius (1990)
• Vector error correction models (VECM)
– Combine short-run (dynamic model) information
with long-run (static model ) information
• Practical issues in cointegration and ECM’s:
– On average, in each period, a proportion of disequilibrium
from the previous period is corrected otherwise, the process
will explode to ± - would not return to its equilibrium
– This implies that the two variables have a long-run
equilibrium
• The variables may diverge from their equilibrium
relation in the short-run but are forced back to
equilibrium
– Examples
• PPP
• Interest rate differentials
• Capital market integration
Unrestricted VARs, near-VARs and
Structural VAR (SVAR) models
• We are often interested in the dynamic relationships
between variables
– But have no strong theory about how they interact
• An unrestricted VAR model expresses a random
variable as a function of its own lag and lags of other
exogenous variables
– One equation is estimated for each variables with equal
number lags on the right-hand side variables
– When there are unequal number of lags of the variable, the
system becomes a near-VAR model
• VAR models are easy to use and often informative
• and are good for making short-term forecasts
• By imposing suitable theoretical restrictions
(especially on the error structure) we can recover the
restricted (or structural) VAR
• The unrestricted VAR is a statistical description of the
data, the SVAR adds some economics
• Innovation accounting methods are used to assess the
dynamic response of variables to fundamental shocks
– Known as Impulse response functions and variance
decompositions
• E.g Impulse response functions are used to determine
the lags of a monetary transmission mechanism
Modelling Changing Volatility
• Why changing volatility
– Volatility greater during some periods than others
• Time-varying risk (conditional variance) characteristics
– Implications for asset pricing dynamics
– Persistence: volatile periods tends to linger
• Positive autocorrelations
– Implications for forecasting asset returns
– Constant unconditional variance
• Due to Engle (1982) and Bollerslev (1986)
Conditional and unconditional moments
• For the general (zero-mean, constant-variance and iid
innovations) AR(1) process:
xt  a0  a1 xt 1   t
– The conditional mean forecast for the next period is:
t xt 1  a0  a1 xt
– The conditional forecast variance is:
2
2
2


t  xt 1  a0  a1xt   t t 1  


• The unconditional mean converges to the long-run mean:
a0 / 1  a1 
• The unconditional variance is:

  xt 1  a0 / 1  a1  
2

   t 1  a1 t  a12 t 1  a13 t 1 

    / 1  a 
2
2
2
1
• The unconditional mean forecast has the greater variance
– The conditional mean forecast is preferable
• A more accurate measure of asset returns
ARCH Processes
• Conditional mean is zero: t 1 t  0
• Conditional variance is autoregressive:
  0      
2
t
2
1 t 1
2
2 t 2

 
2
q t p
• The unconditional mean is zero and
unconditional variance is constant
Modelling ARCH processes
• For a conditional mean-zero  t
ARCH ( p )
• The ARCH innovations are:
 t2  0  1 t21   2 t22 
  p t2 p  vt
• Where vt is white noise
• A valid ARCH requires at least one non-zero coefficient in the
ARCH model
-The number of significant coefficients determine the order of
the ARCH process
• To guarantee this,
below by -0
  0, vt must be bounded from
2
t
– which, implies that the error term of the ARCH sequence cannot be
Gaussian or normal
– Use a square-root process instead:
 t   0  1 t21   2 t22 
–
–
  q t2q  wt
wt and  t 1 are independent
2
wt is iid with wt  0 and wt  1
– The expected values of the squared innovations is therefore equal
to the conditional variance
GARCH Processes
• For similar reasons as the ARCH process write as a
1/ 2
square-root process:  t  wi ( t )
or
 t   0  1 t21 
  p t2 p  1 t21 
  q t2q  wt
• The conditional variance is:
 t2  0  1 t21 
  p t2 p  1 t21 
 q t2q
Identification and estimation of
ARCH/GARCH models
• Estimate mean equation assuming constant variance
• Test for ARCH effects and evidence of
autocorrelation
– Use LM-test and Q stats in EViews
• If ARCH effects are significant
– Re-estimate the model with ARCH innovations
• Estimate the mean and variance equation jointly (recursively) using
maximum likelihood (LM)
• Check for any remaining ARCH effects or residual autocorrelations
in standardised and squared-standardised residuals
• Test the distributional properties of the innovations
The ARCH LM-test
• The test statistics is
 2 (q )
– Estimate a mean equation with constant variance
using ordinary least squares (OLS) and save the
innovations (residuals)
T  R2
– Estimate the squared residuals on a constant and q
lags and get the R2
– Test for the presence of ARCH against the null of
iid innovations
GARCH Simulations
• Model 1
– AR(1)-GARCH(1,1) model
yt  0.0005  0.6 yt 1   t
  0.00008  0.9628
2
2
t 1
 0.0318
2
t 1
Innovations
Innovation
0.15
0.1
0.05
0
-0.05
0
100
200
300
400
500
600
Conditional Standard Deviations
0
100
200
300
400
500
Returns
0
100
200
300
400
500
700
800
900
1000
600
700
800
900
1000
600
700
800
900
1000
Standard Deviation
0.02
0.015
0.01
0.005
0.1
Return
0.05
0
-0.05
-0.1
GARCH Simulation 2
• Model 2
– ARMA(1,1)-GARCH(1,1) model
yt  0  0.6 yt 1   t  0.4 t 1
  0.00001  0.8
2
2
t 1
 0.1
2
t 1
Innovations
Innovation
0.04
0.02
0
-0.02
-0.04
0
100
200
300
400
500
600
Conditional Standard Deviations
0
100
200
300
400
500
Returns
0
100
200
300
400
500
700
800
900
1000
600
700
800
900
1000
600
700
800
900
1000
Standard Deviation
0.014
0.012
0.01
0.008
0.006
Return
0.05
0
-0.05
Some GARCH Extensions
• Asymmetric GARCH models (available in EViews)
– Exponential GARCH (EGARCH) and
– Threshold GARCH (TGARCH) models
– GJR-GARCH
• Leverage effects in conditional time seriesallowing for the asymmetric effects of positive and negative returns
– For example negative shocks to stock returns may have a larger
impact on volatility than positive shocks:
» The so-called “good news” and “bad news” asymmetric impact
on stock returns – implications for monetary policy
– Models based on t-density
– Long memory GARCH models
– Models with changes in regimes
Multivariate (MVGARCH) models
• A useful tool for computing time-varying (conditional)
correlations
• Conditional asset pricing models
• Studying volatility spillovers between asset markets –
feedback effects
• The general k-dimensional process is given as:
 t  zt H
1/ 2
t
• zt is a k-dimensional mean-zero iid process with covariance
matrix equivalent to Ik
• The mean of the process is zero:   t t 1   0
• The covariance matrix is:
  t t t 1   H t


• Where t1 is the information set in the previous period
• Specifying the conditional covariance matrix, is nontrivial
• We adopt three approaches here
– Diagonal vec model (Bollerslev et. Al. 1988)
– The constant conditional correlation (CCC) model
(Bollerslev 1990)
– The BEKK model (Engle and Kroner 1995)
– The dynamic conditional correlation (DCC) model (Engle
2002, Engle and Sheppard 2001)
• The diagonal-vec model (bivariate):
H t  W  A1  ( t 1 t1 )  B1  H t 1
• Where  is the Hadamard product
• The coefficient matrices (W, A, B) are constrained to
be diagonal 3(k (k  1) / 2)
• The number of parameters estimated in this model is
equal to
– The number of parameters increases with number variables
• In a bivariate MVGARCH diagonal-vec model, there are nine
parameters to be estimated
– The conditional covariance matrix in MVGARCH models
must be positive definite
• Expanding the bivariate diagonal-vec MVGARCH
model yields the following system:
*
*
2




 h11,t  11
11 0
0
1,t 1 



  *  
*
 h12,t    12    0  22 0  1,t 1 2,t 1 
* 
2
 h   *   0

0


33 
2,t 1
 22,t   22  

 11*

 0
 0

0

*
22
0
0   h11,t 1 


0   h12,t 1 
33*   h22,t 1 
• CCC-MVGARCH models
• The parameters of the diagonal-vec model increases
considerably as the MVGRACH dimension increases
• The CCC-MVGARCH model assumes that the conditional
correlations between the elements of  t are time variant
• The conditional covariance matrix is written as:
Ht  diag

h11,t ,

, hNN ,t R diag

h11,t ,
, hNN ,t

• And the time-invariant correlation matrix is written as:
 1
R

 1N
1N 


1 
• Where  ij is the correlation between the variables
• The conditional covariance matrix can be written in compact
form as:
H t  Dt1/ 2 RDt1/ 2
• For the two-variable CCC-MVGARCH model:
 h11,t
Ht  
 0

0  1

h22,t   12
12   h11,t

  0
0 

h22,t 
• The individual variances in the system are assumed to follow a
GARCH(1,1) process:
hii ,t  ii   ii i2,t 1  ii hii ,t 1
• The bivariate BEKK-MVGARCH model
– A very general class of time-varying correlation
MVGARCH model
– The covariance matrix is written as:
H t  CC  A1 t 1 t1 A1  B1H t 1B1
– Where C, A and B are kxk matrices and C is upper
triangular; C C  W  0 is symmetric positive definite
• The coefficient matrices are written as quadratic forms
to ensure positive definiteness
• Each variance and covariance term is affected by the
covariance between the variables and variance of the
other variable
• Matrix diagonal and vector diagonal versions of the
BEKK-MVGARCH model have also been suggested
• To see the generality of the BEKK-MVGARCH model
consider the following two-variable BEKK model:
 h11,t
h
 12,t
2
h12,t 
1,t 1 2,t 1   11 12 
 11 12   1,t 1
 C C  
 

 

2
h22,t 







 21
22   1,t 1 2,t 1
22 
2,t 1
  21
 11 12   h11,t 1 h12,t 1   11 12 


 h

h




22,t 1   21
 21
22   12,t 1
22 
• Linear expansions shows that each variance and
covariance term is a function lagged variances and
covariances
• DCC-MVGARCH models
• The DCC model is very similar to the CCC-MVGARCH
model
• The conditional covariance matrix is given as:
H t  Dt Rt Dt
• D, is a kxk matrix of time-varying standard deviations from
hit on the ith diagonal
univariate GARCH models with
• The model is estimated in two stages
– Univarate GARCH models are estimated for all the
variables and the residuals are extracted and standardised
– The standardised residuals are modelled as a correlationGARCH model
• The residuals standardised by their standard deviations are:
 it  rit / hit ;  it ~ N  0, Rt 
• The dynamic correlation structure is given as
Qt  1   n   n  Q   n  t 1 t1    nQt 1
Rt  Qt*1Qt Qt*1
• Where Q is the unconditional covariance of the standardised
residuals
• and
 q11

 0
*
Qt  

 0

0
0
q22
0
0
0
• The typical element of
0 

0 


qkk 
Rt is ij ,t 
qij ,t
qii ,t q jj ,t
• Asymmetric versions of the DCC have been estimated by
Cappiello et al 2003
• The model can also be estimated with different n and βn
coefficients for each asset in the DCC structure
Other MVGARCH Models
• The exponentially weighted moving average
(EWMA) covariance matrix
– Widely used by practitioners
• Factor and orthogonal GARCH models
• Flexible MVGARCH model (Ledoit, et al. (2003))
• MVGARCH models based on a multivariate
conditional t-density
Software and computational issues in
GARCH modelling
• Nonlinear estimation can be problematic due to
convergence problems with the optimisation routine
• Univariate GARCH models
– EViews, PC-Give/OX, RATS, S-plus (FinMetrix),
GAUSS (FANPAC), Matlab (GARCH Toolbox)
and Microfit
• Multivariate GARCH models
• RATS, S-plus (FinMetrix), GAUSS (FANPAC),
Matlab
Summary
• Econometric time series modelling
– Stationary and nonstationary processes
• Constant and time-dependent moments
– Spurious regressions, cointegration and VARS
• Modelling volatility clustering
– Univariate ARCH/GARCH models
– Multivariate GARCH models
• Covariance matrix estimation
Additional Reading
• Textbooks
– Enders, W. 2003. Applied Econometric Time Series, second edition.
Wiley
– Engle, R. 1995. ARCH: Selected Readings, Oxford University Press
– Green, W.H. 2000. Econometric Analysis, fourth edition, Prentice
Hall
– Hamilton, J. 1994. Time Series Analysis, Princeton University Press
– Hayashi, F. 2001. Econometrics, Princeton University Press
– Tsay, R. 2002. Analysis of Financial Time Series, Wiley
• Journal Articles
– There are literally hundreds of journal articles on econometric
time series modelling. The Journal of Applied Econometrics,
Journal of Business and Economic Statistics, Econometrica,
Journal of Time Series Analysis, Journal of Econometrics, Journal
of Finance, Journal Financial Economics etc. etc. contains
hundreds of papers in this area
• References for the classic papers can be found in the textbooks
listed above