程序代写代做代考 C ECON3350/7350 Volatility Models – I

ECON3350/7350 Volatility Models – I
Eric Eisenstat
The University of Queensland
Lecture 7
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 8
1 / 24

Volatility Models
Recommended readings
Author
Title
Chapter
Call No
Enders Verbeek
Applied Econometric Time Series, 4e
A Guide to Modern Econometrics
3 8.11
HB139 .E55 2015 HB139 .V465 2012
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 2 / 24

Features of Financial Data
Linear models cannot explain a number of important features common to financial data:
leprokurtosis 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).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 3 / 24

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.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8
4 / 24

Volatility
In the classical regression model:
yi =β1 +β2×2,i +···+βkxk,i +εi, we assumed Var(εi) = σ2.
In financial time-series, we observe large variations in volatility. Volatility is related to the variance of a process, not the mean.
Volatility is a form of heteroskedasticity; in this case, conditional variance is modeled as a function of time.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 5 / 24

Non-linear Models
Consider the non-linear stochastic process (Campbell, Lo and MacKinlay, 1997):
1
yt = g(εt−1,εt−2,…)+νt h2 (εt−1,εt−2,…).
􏰎 􏰍􏰌 􏰏 􏰎 􏰍􏰌 􏰏
conditional mean volatility
Models with non-linear g(·) are non-linear in mean. Models with non-constant h(·) are non-linear in variance. We will focus on linear g(·), but non-constant h(·).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 6 / 24

Autoregressive Conditional Heteroskedasticity (ARCH)
Simplest example was proposed by Engle (1982):
ht ≡E(ε2t |εt−1,εt−2,…)=α0 +α1ε2t−1.
This is called the Auto-Regressive Conditional Heteroskedasticity of order 1 model, or 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).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8
7 / 24

ARCH(1)
The ARCH(1) specification indicates ε2t and ε2t−1 are correlated. However, the unconditional variance of εt is homosckedastic:
E(ε2t ) = α0 + α1E(ε2t−1). For 0 ≤ α1 < 1, the stationary solution is E(ε2t)= α0 . 1−α1 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 8 / 24 ARCH(q) 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 stationarity, impose α(1) < 1. In an ARCH(q), only the past q shocks affect the current volatility. Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 9 / 24 Combining Linear Models with ARCH Errors We can specify a regression with ARCH(1) errors as: yt =β1 +β2x2,t +···+βkxk,t +εt, ε=ν􏰠α+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 + a2yt−1 + εt, ε=ν􏰠α+αε2 +αε2 , t t 0 1t−1 2t−2 where νt ∼ N(0,1) and E(εtνt−s) = 0 for all t,s. Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 10 / 24 Examples of Models with ARCH Errors Source: Enders (2015) Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 11 / 24 Consequence of ARCH Errors The presences of ARCH errors does not invalidate OLS, but there exist more efficient non-linear estimators to estimate conditional-mean coefficients. If the objective is to forecast volatility, then it is relevant to test for the presence of ARCH errors and estimate an ARCH model. Breusch-Pagan type test for heteroskedasticity: H0 : α1 = · · · = αq = 0 versus H1 : ARCH(q). 1 Estimate the conditional mean using OLS and save residuals in {ε􏰭 }. t 2222 2 Regressε􏰭 =α +αε􏰭 +···+αε􏰭 andcomputeR. t01t−1qt−q e 3 Test using LMARCH ≡ Re2 ∼ χ2q . Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 12 / 24 Generalized ARCH (GARCH) The Generalized ARCH model (proposed by Bollerslev, 1986) is: ε2t =νt2ht, qp h t = α 0 + 􏰁 α j ε 2t − j + 􏰁 β j h t − j , j=1 j=1 = α0 + α(L)ε2t−1 + β(L)ht−1, known as GARCH(p, q). Again, we require αj ≥ 0 for j = 0, . . . , q and β1 ≥ 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. Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 13 / 24 GARCH(1, 1) Define the surprise in the squared innovation ηt = ε2t − ht, and re-write the GARCH(1, 1) as an ARMA(1, 1): ε2t =νt2ht, ht = α0 + α1ε2t−1 + β1ht−1, and adding / subracting 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 “error” ηt is uncorrelated over time, but heteroskedastic. Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 14 / 24 GARCH(1, 1) The autoregressive coefficient α1 + β1, so stationarity requires α1 + β1 < 1; values of α1 + β1 close to 1 indicate highly persistent volatility. Under stationarity, the unconditional variance of εt is: E ( ε 2t ) = α 0 . 1−α1 −β1 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 + · · · ), α∞ = 0 + α 1 􏰁 β j − 1 ε 2t − j . 1−β1 1 j=1 The GARCH(1, 1) is a parsimonious alternative to a high-order ARCH. Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 15 / 24 Example Linear regression to forecast S&P 500 excess returns (dependent variable EXRET). CS 1 DY 1 I12 1 I12 2 I3 1 I3 2 INF 2 IP 2 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 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 16 / 24 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 Prob. 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 R-squared 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 1 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 Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat −1359.737 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 17 / 24 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 Prob 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 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 18 / 24 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 Prob 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 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 19 / 24 GARCH(1, 1) Estimation Results Dependent Variable: EXRET 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 Prob. 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 1 I3 2 I12 1 I12 2 MB 2 CS 1 WINTER C RESID(-1)ˆ2 GARCH(-1) R-squared 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 Schwarz criterion Hannan-Quinn criter. Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 20 / 24 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 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 21 / 24 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 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 22 / 24 Forecasting Volatility from GARCH(p, q) 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 given by: ht+1 = E(ht+1 | εt, εt−1, . . . ) = α0 + α1ε2t + β1ht Assuming the parameters are known, the j-step-ahead forecast for j > 1 is
given by:
E ( h t + j | ε t , ε t − 1 , . . . ) = α 0 + α 1 E ( ε 2t + j − 1 | ε t , ε t − 1 , . . . )
+ β1E(ht+j−1 | εt, εt−1, . . . ),
=α0 +(α1 +β1)E(ht+j−1|εt,εt−1,…), since E(ε2t+j−1 |εt,εt−1,…) = E(ht+j−1 |εt,εt−1,…).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 23 / 24

Forecasting Volatility from GARCH(p, q)
We can write this more compactly as
Et(ht+j ) = (α1 + β1)Et(ht+j−1),
=α0􏰗1+(α1 +β1)+(α1 +β1)2 +···+(α1 +β1)j−1􏰘
+(α1 +β1)jht.
If α1 + β1 < 1, the conditional forecast of ht+j will converge as j −→ ∞ to the long-run value: lim Et(ht+j) = E(ht) = j→∞ α0 . 1 − α1 − β1 Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 24 / 24