Transcript Analysis of Financial Data Spring 2012 Lecture: Introduction

Analysis of Financial Data
Spring 2012
Lecture 10: ARCH and GARCH Models – 2
Priyantha Wijayatunga
Department of Statistics, Umeå University
[email protected]
Course homepage:
http://www8.stat.umu.se/kursweb/vt012/staa2st017mom2/
AR/ARCH Models
• When the residuals of an AR (or a linear regression) model
are heteroscedastic (residuals have nonconstant conditional
variance) ARCH and GARCH models can model this
variation,
• The AR (or the linear regression) model is only used to model
the conditional mean of the (stationary) time series (or data)
• How do you find that the residuals are heteroscedastic? Do
residual diagnostics
• Residuals of the AR (the linear regression) model may have
their conditional mean zero implying that residuals are not
autocorrelated
• But they may not be independent (because noncorrelation
does not imply independence) – violation of a modeling
assumption
AR/ARCH Models
• Look at the squared residuals to see this dependence
through caculation of autocorrelation function ACF of squared
residuals (Hint: time–point–wise sample conditional variance
of the residuals is just the squared residual value now )
• If squared residuals are autocorelated fit another AR model to
them. It will become the ARCH model for nonconstant
conditional variance of the residuals of former AR
(regression) model (for conditional mean)
• Finally combine two models:
time series value at time t = AR + ARCH
Example AR/ARCH Model
Example AR/ARCH Model
AR(1) model could be fine
Example AR/ARCH Model
AR(1) model fit to the time series
Model Statistics
Model Fit statistics
Model
Number of Predictors
Xt-Model_1
Ljung-Box Q(18)
Stationary R-squared
1
Statistics
,516
21,821
DF
Sig.
17
Number of Outliers
,192
0
ARIMA Model Parameters
Estimate
Xt-Model_1
Xt
DAY, not periodic
No Transformation
No Transformation
Constant
SE
t
Sig.
49,890
,127
393,026
,000
AR
Lag 1
,717
,022
32,510
,000
Numerator
Lag 0
,000
,000
,547
,585
Example AR/ARCH Model
Look at the residuals: they are not autocorrelated
Example AR/ARCH Model
Look at the residuals: their heteroscedasticity
Example AR/ARCH Model
Look at the residuals: their heteroscedasticity
Statistics
Noise residual from Xt-Model_1
N
Valid
Missing
1000
0
Mean
-,0001
Std. Deviation
,57065
Skewness
,359
Std. Error of Skewness
,077
Kurtosis
3,569
Std. Error of Kurtosis
Percentiles
,155
25
-,3352
50
-,0055
75
,3157
Example AR/ARCH Model
Look at the nonconstant conditional variance of residuals with
squared residuals
Example AR/ARCH Model
Fit an AR model for squared residuals
Model Statistics
Model Fit statistics
Model
Number of Predictors
ResSquares-Model_1
Stationary R-squared
1
Ljung-Box Q(18)
Statistics
,175
14,452
DF
Sig.
16
Number of Outliers
,565
0
ARIMA Model Parameters
Estimate
ResSquares-Model_1
ResSquares
No Transformation
Constant
AR
DAY, not periodic
No Transformation
Numerator
SE
t
Sig.
,213
,088
2,412
,016
Lag 1
,252
,031
8,213
,000
Lag 2
,251
,031
8,189
,000
Lag 0
,000
,000
1,454
,146
Final Model AR(1)/ARCH(2)
X t  49.89  0.717X t 1   0.213 0.2522t 1  0.2522t  2
 ~ WhiteNoise(0,1)
ARCH(q) Model
The process at is an ARCH(1) models if
at   t  0  1at21
at2  ( 0  1at21  ...   q at2q ) t2
where  t ~ WhiteNoise(0,1) for all t
GARCH(p,q) Model
• ARCH models: conditional variance depends on short–
term history of volatility in the time series
• But if we want more persistent volatility we use GARCH
model
• Past conditional variances σ2t-1, σ2t-2,.. too are determining
the present σt2:
The process at is an GARCH(p,q) models if
at   t  0  1at21  ...   q at2q  1 t21  ...   p t2 p
where  t ~ WhiteNoise(0,1) for all t
Process at is uncorrelated has stationary mean and variance
Process has at2 ACF like an ARMA process
ARIMA(p,d,q)/GARCH(l,m) Model
• Generally the first model you may fit can be a
ARIMA(p,d,q) model
• Then look at the residuals
• If they are heteroscedastic fit a GARCH(l,m) model to
them: fit an ARMA(l,m) to squared residuals
• Researchers have noticed that often time series have
outliers: errors/residuals are not normally distributed and
they needed to be modeled with heavy–tailed distributions
• The time series may often be conditional heteroscedastic
• GARCH processes can model both heavy–tailed
behaviour and any conditional heteroscedasticity
Fitting GARCH(p,q) Model
• As in the case of ARCH(p) model fitting we get the
squared residuals
• To fit a GARCH(p,q) model, fit a ARMA(p,q) model to the
sqaured residuals
• Fitting GARCH(1,1) is fitting ARMA(1,1) to squared
residuals
2
2
2
 t   0  1at 1  1 t 1
at2   0  1at21   t
where  t ~ WhiteNoise(0,1) for all t
 t2  at2   t
at2   0  (1  1 )at21   t  1 t 1