The University of Queensland
(School of Economics) Applied Econometrics for Macro and Finance Week 8 1 / 26
ECON7350 Modelling Volatility – I
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Features of Financial Data
Linear models cannot explain several important features common to financial data: leptokurtosis or “fat tails”: high negative returns that have higher-than-expected
probability in historic time-series;
volatility clustering or volatility “pooling”: big shocks tend to follow big shocks (either direction) and small shocks tend to follow small shocks;
leverage effects or volatility “asymmetry”: for example, volatility of a stock price tends to increase when its price drops (Black, 1976).
(School of Economics) Applied Econometrics for Macro and Finance Week 8 2 / 26
Typical Example of Returns
.08 .06 .04 .02 .00
-.02 -.04 -.06 -.08
1970 1975 1980 1985 1990 1995 2000 2005
Figure: Returns to dividend yield S&P 500 index: 1996-2005 Monthly.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 3 / 26
Volatility
In the (invertible) ARMA model:
yt =a0 +a1yt−j +bqεt−j +εt,
we assumed Var(εt | yt−1, yt−2, . . . ) = σ2: this is a linear model.
In financial time-series, we observe large variations in volatility. Volatility is related to the variance of a process, not the mean.
Fluctuations in volatility is a form of heteroscedasticity; in this case, variance is modelled as a function of time.
The most basic approach to modelling volatility is by specifying a conditional variance
function: this results in a non-linear model.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 4 / 26
Non-linear Models
Consider the general non-linear stochastic process (Campbell, et al., 1997): 1
yt = gt(εt−1,εt−2,…)+νt ht2 (εt−1,εt−2,…),
conditional mean conditional variance
where εt ≡ νtht2 (·) and {νt} is a process of independent and identically distributed (iid) errors
with E(νt) = 0 and Var(νt) = 1.
Models with non-linear gt(·) are non-linear in mean. Models with non-constant ht(·) are non-linear in variance. We will focus on linear gt(·), but non-constant ht(·).
(School of Economics) Applied Econometrics for Macro and Finance Week 8 5 / 26
Autoregressive Conditional Heteroscedasticity (ARCH)
Simplest example was proposed by Engle (1982):
ht ≡E(ε2t |εt−1,εt−2,…)=α0 +α1ε2t−1.
This is the Auto-Regressive Conditional Heteroscedasticity of order 1 model, ARCH(1). To ensure that ht ≥ 0 regardless of ε2t−1, we impose α0 ≥ 0 and α1 ≥ 0.
ARCH(1) captures the idea that when a large shock happens in t − 1, there is a greater probability of larger shocks in t.
For larger εt−1, the next shock εt is still mean-zero but has larger variance (proportional to ε2t−1).
(School of Economics) Applied Econometrics for Macro and Finance Week 8 6 / 26
The ARCH(1) specification indicates ε2t and ε2t−1 are correlated. The unconditional variance of εt:
E(ε2t ) = α0 + α1E(ε2t−1). For 0 ≤ α1 < 1, the stable solution is
E(ε2t)= α0 . 1−α1
Therefore, if 0 ≤ α1 < 1 holds, {εt} is unconditionally homoscedastic.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 7 / 26
The ARCH(1) can be easily extended to an ARCH(q) process:
ht ≡E(ε2t |εt−1,εt−2,...)=α0 +α1ε2t−1 +α2ε2t−2 +···+αqε2t−q,
= α 0 + α ( L ) ε 2t − 1 , whereα(L)=α1+α2L+···+αqLq−1 isalagpolynomialoforderq−1.
All coefficients are non-negative, i.e. αj ≥ 0 for j = 0, . . . , q.
To ensure stability, impose α(1) < 1.
In an ARCH(q), only the past q shocks affect the current volatility.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 8 / 26
Combining Linear Models with ARCH Errors
We can specify a regression with ARCH(1) errors as:
yt =β0 +β1x1,t +···+βkxk,t +εt,
ε=να+αε2 , t t 0 1t−1
where νt ∼ N(0,1) and E(εtνt−s) = 0 for all t,s.
We can specify an AR(1) model with ARCH(2) errors as:
yt = a0 + a1yt−1 + εt, ε=να+αε2 +αε2 ,
t t 0 1t−1 2t−2 where νt ∼ N(0,1) and E(εtνt−s) = 0 for all t,s.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 9 / 26
Examples of Models with ARCH Errors (Enders, 2015)
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Consequence of ARCH Errors
The presence of ARCH errors does not invalidate OLS, but there exist more efficient non-linear estimators to estimate conditional-mean coefficients.
Breusch-Pagan test for heteroscedasticity: H0 : α1 = · · · = αq = 0, H1 : ARCH(q). 1 Estimate the conditional mean using OLS and save residuals in {ε }.
2 Regressε =α +αε +···+αε andcomputeR. t01t−1qt−q ε
3 Test using LMARCH ≡ TRε2 ∼ χ2q.
Testing can provide a diagnostic tool, which is useful in some contexts.
If the objective is to forecast volatility, then it is always relevant to extend a given class of models (e.g. ARMA or ARDL) to specifications with heteroscedastic errors.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 11 / 26
Generalized ARCH (GARCH)
The Generalized ARCH model (proposed by Bollerslev, 1986) is: ε2t =νt2ht,
h t = α 0 + α j ε 2t − j + β j h t − j ,
= α0 + α(L)ε2t−1 + β(L)ht−1,
known as GARCH(p, q).
Again, we require αj ≥ 0 for j = 0, . . . , q and βj ≥ 0 for j = 1, . . . , p to ensure ht ≥ 0. In practice, the most commonly used is the GARCH(1, 1):
ht = α0 + α1ε2t−1 + β1ht−1.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 12 / 26
ARMA(1,1)Representationforε2t inaGARCH(1,1)
Define the surprise in the squared innovation ηt = ε2t − ht, and re-write the GARCH(1, 1) as an ARMA(1, 1) for ε2t :
ε2t =νt2ht,
ht = α0 + α1ε2t−1 + β1ht−1,
and adding / subtracting β1ε2t−1, ε2t yields
ht = α0 + α1ε2t−1 + β1ht−1+(ε2t − ε2t )+(β1ε2t−1 − β1ε2t−1), ε2t = α0 + (α1 + β1)ε2t−1 − β1(ε2t−1 − ht−1) + (ε2t − ht),
ε 2t = α 0 + ( α 1 + β 1 ) ε 2t − 1 + η t − β 1 η t − 1 .
The “residual” ηt is uncorrelated over time, but it is heteroscedastic.
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Stability in a GARCH(1, 1)
The autoregressive coefficient is α1 + β1, so stability requires α1 + β1 < 1; values of α1 + β1 close to 1 indicate highly persistent volatility.
In a stable GARCH(1, 1), the unconditional variance of εt is: E ( ε 2t ) = α 0 .
Similar to AR and MA processes, The GARCH(1, 1) can be written as an infinite-order ARCH
model with geometrically declining coefficients:
ht = α0(1 + β1 + β12 + · · · ) + α1(ε2t−1 + β1ε2t−2 + β12ε2t−3 + · · · ),
= 0 + α 1 β j − 1 ε 2t − j . 1−β1 1
The GARCH(1, 1) is a parsimonious alternative to a high-order ARCH.
(School of Economics) Applied Econometrics for Macro and Finance Week 8 14 / 26
Linear regression to forecast S&P 500 excess returns (dependent variable EXRET).
MB 2 PE 1 WINTER
Explanatory Variables
credit spread (yield on Moody’s Aaa minus BBa debt), lagged one month
dividend yield S&P 500 index, lagged one month (in % per month) 12-month interest rate, lagged one month
12-month interest rate, lagged two months
1-month interest rate, lagged one month
1-month interest rate, lagged two months inflation, lagged two months
change industrial production, lagged two months change in monetary base, lagged two months price earnings ratio (S&P500), lagged one month dummy, 1 in November to April, 0 otherwise
(School of Economics) Applied Econometrics for Macro and Finance Week 8 15 / 26
Estimation Results
Dependent Variable: EXRET Method: Least Squares Date: 04/14/16 Time: 16:10 Sample: 1966M01 2005M12 Included observations: 480
Variable Coefficient C 3.300976
Std. Error
2.325907 6.450444 6.011935 6.866371 5.694212 12.11589 11.96867 12.08628 12.55039 6.159389 9.921162 0.391515
t-Statistic
1.419221 −1.767067 0.996253 −2.037178 −0.370747 1.583666 −1.625224 −3.420670 2.859809 −2.086729 1.844602 2.057941
0.1565 0.0779 0.3196 0.0422 0.7110 0.1139 0.1048 0.0007 0.0044 0.0375 0.0657 0.0401
0.432739 4.382373 5.715573 5.819917 5.756588 2.076803
−11.39836 5.989411 −13.98802 −2.111113 19.18752 −19.45177 −41.34318 35.89172 −12.85298 18.30060 0.805715
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 5.684130 Prob(F-statistic) 0.000000
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0.117856 0.097122 4.164128 8115.100
Mean dependent var S.D. dependent var Akaike info criterion Bayesian info criterion Hannan-Quinn criter. Durbin-Watson stat
(School of Economics) Applied Econometrics for Macro and Finance
Estimation Results
Dependent Variable: EXRET Method: Least Squares Date: 04/14/16 Time: 16:10 Sample: 1966M01 2005M12 Included observations: 480
Variable Coefficient C 3.300976
Std. Error
2.325907 6.450444 6.011935 6.866371 5.694212 12.11589 11.96867 12.08628 12.55039 6.159389 9.921162 0.391515
t-Statistic
1.419221 −1.767067 0.996253 −2.037178 −0.370747 1.583666 −1.625224 −3.420670 2.859809 −2.086729 1.844602 2.057941
0.1565 0.0779 0.3196 0.0422 0.7110 0.1139 0.1048 0.0007 0.0044 0.0375 0.0657 0.0401
0.432739 4.382373 5.715573 5.819917 5.756588 2.076803
−11.39836 5.989411 −13.98802 −2.111113 19.18752 −19.45177 −41.34318 35.89172 −12.85298 18.30060 0.805715
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 5.684130 Prob(F-statistic) 0.000000
PE 1 DY 1 INF 2 IP 2
I3 2 I12 1 I12 2 MB 2 CS 1 WINTER
0.117856 0.097122 4.164128 8115.100
Mean dependent var S.D. dependent var Akaike info criterion Bayesian info criterion Hannan-Quinn criter. Durbin-Watson stat
(School of Economics) Applied Econometrics for Macro and Finance
Residuals ACF/PACF
Sample: 1966M01 2006M12 Included observations: 480
Autocorrelation
Partial Correlation
AC PAC Q-Stat
1 -0.039 -0.039 0.7182 2 -0.016 -0.018 0.8473 3 0.039 0.037 1.5679 4 -0.041 -0.038 2.3705 5 0.096 0.094 6.8308 6 -0.000 0.004 6.8308 7 0.042 0.049 7.6779 8 -0.000 -0.006 7.6780 9 0.024 0.033 7.9529
10 0.058 0.048 9.6071 11 0.000 0.009 9.6071 12 0.037 0.030 10.293 13 0.006 0.007 10.309 14 -0.045 -0.048 11.328 15 -0.005 -0.020 11.339 16 0.012 0.008 11.413 17 0.080 0.075 14.645 18 0.014 0.015 14.736
0.397 0.655 0.667 0.668 0.234 0.337 0.362 0.466 0.539 0.476 0.566 0.590 0.668 0.660 0.728 0.783 0.621 0.680
(School of Economics)
Applied Econometrics for Macro and Finance
Squared Residuals ACF/PACF
Sample: 1966M01 2006M12 Included observations: 480
Autocorrelation
Partial Correlation
AC PAC Q-Stat
1 0.144 0.144 9.9571 2 0.101 0.083 14.941 3 0.122 0.100 22.179 4 0.018 -0.019 22.345 5 0.026 0.007 22.662 6 0.012 -0.004 22.738 7 -0.055 -0.060 24.210 8 -0.026 -0.016 24.543 9 0.080 0.097 27.651
10 0.083 0.081 31.002 11 0.018 -0.012 31.156 12 0.007 -0.026 31.183 13 0.079 0.067 34.268 14 0.041 0.019 35.083 15 0.095 0.075 39.565 16 0.025 -0.006 39.883 17 -0.012 -0.019 39.949 18 0.004 -0.017 39.958
0.002 0.001 0.000 0.000 0.000 0.001 0.001 0.002 0.001 0.001 0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.002
(School of Economics)
Applied Econometrics for Macro and Finance
GARCH(1, 1) Estimation Results
Variable Coefficient C 2.117416
Std. Error
2.480921 7.102047 5.394256 6.239101 5.526029 9.239231 9.851971 10.65276 11.28990 4.579737 10.58376 0.375668
z-Statistic
0.853480 −1.022685 1.725337 −1.890206 −0.839156 2.891035 −2.854826 −4.664927 3.998238 −3.387797 1.468809 2.111698
1.870863 2.633798 14.42264
0.3934 0.3065 0.0845 0.0587 0.4014 0.0038 0.0043 0.0000 0.0001 0.0007 0.1419 0.0347
0.0614 0.0084 0.0000
0.432739 4.382373 5.671065 5.801495 5.722334
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C RESID(-1)ˆ2 GARCH(-1)
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
−7.263156 9.306910 −11.79318 −4.637199 26.71094 −28.12567 −49.69434 45.13972 −15.51522 15.54553 0.793297
Variance Equation
1.078377 0.108658 0.832238
0.114786 0.093979 4.171367 8143.342
−1346.055 2.064821
0.576406 0.041255 0.057704
Mean dependent var S.D. dependent var Akaike info criterion Bayesian info criterion Hannan-Quinn criter.
(School of Economics) Applied Econometrics for Macro and Finance
GARCH(1, 1) Estimation Results
Variable Coefficient C 2.117416
Std. Error
2.480921 7.102047 5.394256 6.239101 5.526029 9.239231 9.851971 10.65276 11.28990 4.579737 10.58376 0.375668
z-Statistic
0.853480 −1.022685 1.725337 −1.890206 −0.839156 2.891035 −2.854826 −4.664927 3.998238 −3.387797 1.468809 2.111698
1.870863 2.633798 14.42264
0.3934 0.3065 0.0845 0.0587 0.4014 0.0038 0.0043 0.0000 0.0001 0.0007 0.1419 0.0347
0.0614 0.0084 0.0000
0.432739 4.382373 5.671065 5.801495 5.722334
PE 1 DY 1 INF 2 IP 2
I3 2 I12 1 I12 2 MB 2 CS 1 WINTER
C RESID(-1)ˆ2 GARCH(-1)
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
−7.263156 9.306910 −11.79318 −4.637199 26.71094 −28.12567 −49.69434 45.13972 −15.51522 15.54553 0.793297
Variance Equation
1.078377 0.108658 0.832238
0.114786 0.093979 4.171367 8143.342
−1346.055 2.064821
0.576406 0.041255 0.057704
Mean dependent var S.D. dependent var Akaike info criterion Bayesian info criterion Hannan-Quinn criter.
(School of Economics) Applied Econometrics for Macro and Finance
GARCH(1, 1) Estimation Results
Variable Coefficient C 2.117416
Std. Error
2.480921 7.102047 5.394256 6.239101 5.526029 9.239231 9.851971 10.65276 11.28990 4.579737 10.58376 0.375668
z-Statistic
0.853480 −1.022685 1.725337 −1.890206 −0.839156 2.891035 −2.854826 −4.664927 3.998238 −3.387797 1.468809 2.111698
1.870863 2.633798 14.42264
0.3934 0.3065 0.0845 0.0587 0.4014 0.0038 0.0043 0.0000 0.0001 0.0007 0.1419 0.0347
0.0614 0.0084 0.0000
0.432739 4.382373 5.671065 5.801495 5.722334
PE 1 DY 1 INF 2 IP 2
I3 2 I12 1 I12 2 MB 2 CS 1 WINTER
C RESID(-1)ˆ2 GARCH(-1)
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
−7.263156 9.306910 −11.79318 −4.637199 26.71094 −28.12567 −49.69434 45.13972 −15.51522 15.54553 0.793297
Variance Equation
1.078377 0.108658 0.832238
0.114786 0.093979 4.171367 8143.342
−1346.055 2.064821
0.576406 0.041255 0.057704
Mean dependent var S.D. dependent var Akaike info criterion Bayesian info criterion Hannan-Quinn criter.
(School of Economics) Applied Econometrics for Macro and Finance
GARCH(1, 1) Residuals ACF/PACF
Date: 09/05/16 Time: 23:52 Sample: 1966M01 2005M12 Included observations: 480
Autocorrelation
Partial Correlation
AC PAC Q-Stat Prob*
1 -0.045 -0.045 0.9832 0.321 2 0.015 0.013 1.0869 0.581 3 0.048 0.049 2.1947 0.533 4 -0.037 -0.033 2.8459 0.584 5 0.096 0.092 7.2982 0.199 6 -0.019 -0.013 7.4798 0.279 7 0.043 0.043 8.3919 0.299 8 0.008 0.002 8.4262 0.393 9 0.054 0.062 9.8343 0.364
10 0.051 0.043 11.138 0.347 11 0.005 0.014 11.152 0.431 12 0.031 0.018 11.617 0.477 13 -0.006 -0.004 11.632 0.558 14 -0.019 -0.030 11.803 0.622 15 -0.007 -0.019 11.829 0.692 16 -0.008 -0.012 11.859 0.754 17 0.087 0.081 15.643 0.549
18 0.012 0.016 15.712 0.613
(School of Economics)
Applied Econometrics for Macro and Finance
GARCH(1, 1) Squared Residuals ACF/PACF
Date: 09/05/16 Time: 23:53 Sample: 1966M01 2005M12 Included observations: 480
Autocorrelation
Partial Correlation
AC PAC Q-Stat Prob*
1 -0.037 -0.037 0.6589 0.417 2 0.017 0.015 0.7929 0.673 3 0.061 0.062 2.5731 0.462 4 -0.024 -0.020 2.8463 0.584 5 -0.021 -0.025 3.0590 0.691 6 -0.011 -0.016 3.1227 0.793 7 -0.091 -0.089 7.2083 0.408 8 -0.040 -0.045 8.0105 0.432 9 0.051 0.052 9.2635 0.413
10 0.043 0.060 10.168 0.426 11 -0.022 -0.019 10.399 0.495 12 -0.022 -0.038 10.629 0.561 13 0.048 0.038 11.772 0.546 14 0.007 0.009 11.796 0.623 15 0.082 0.081 15.102 0.444 16 0.022 0.032 15.351 0.499 17 -0.031 -0.021 15.825 0.536
18 -0.003 -0.019 15.829 0.604
(School of Economics)
Applied Econometrics for Macro and Finance
Forecasting Volatility with the GARCH(1, 1)
Consider a model with GARCH errors and volatility given by ht+1 = α0 + α1ε2t + β1ht. Assuming the parameters are known, the one-step-ahead forecast is:
E(ht+1|It)≡E(ht+1|εt,εt−1,...)=α0 +α1ε2t +β1ht
For j > 1, the key assumption is that {νt} is an iid process, which implies that νt is
independent of ht. Consequently, for j > 1:
E(ε2 |I ) = E(ν2 h |I ) = E(ν2 |I )E(h |I ) = E(h
t+j−1 t t+j−1 t+j−1 t t+j−1 t t+j−1 t t+j−1 Assuming the parameters are known, the j-step-ahead (j > 1) forecast is:
E(ht+j | It) = α0 + α1E(ε2t+j−1 | It) + β1E(ht+j−1 | It), = α0 + (α1 + β1)E(ht+j−1 | It).
(School of Economics) Applied Econometrics for Macro and Finance Week 8
Forecasting Volatility with the GARCH(1, 1)
Then, successively substituting yields:
E(ht+j | It) = α0 + (α1 + β1)E(ht+j−1 | It),
=α01+(α1 +β1)+(α1 +β1)2 +···+(α1 +β1)j−2
+ (α1 + β1)j−1(α1ε2t + β1ht).
If α1 + β1 < 1, the conditional forecast of ht+j will converge as j −→ ∞ to the long-run
forecast, which is the unconditional mean of ht:
lim E(ht+j | It) = α0 = E(ht).
Note: this is also the unconditional mean obtained from the ARMA(1, 1) representation for ε2t , which is equivalent to the unconditional variance of the process {yt}.
j→∞ 1 − α1 − β1
(School of Economics) Applied Econometrics for Macro and Finance Week 8 26 / 26
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