1

Chapter 5 : Volatility ModelsChapter 5 : Volatility Models

• Similar to linear regression analysis, many time series exhibit a non-constant variance (heteroscedasticity). In a regression model, suppose that yt = 0 + 1x1t + 2x2t + … + t; var(t) = 2

t

then instead of using the ordinary least squares (OLS) procedure, one should use a generalized least squares (GLS) method to account for the heterogeneity of t.

• With financial time series, it is often observed that variations of the time series are quite small for a number of successive periods, then large for a while, then smaller again. It would be desirable if these changes in volatility can be incorporated into the model.

2

• This plot shows the weekly dollar/sterling exchange rate from January 1980 to December 1988 (470 observations)

3

• This first difference of the series is shown here

4

• The levels exhibit wandering movement of a random walk, and consistent with this, the differences are stationary about zero and show no discernable pattern, except that the differences tend to be clustered (large changes tend to be followed by large changes and small changes tend to be followed by small changes)

• An examination of the series’ ACF and PACF reveals some of the cited characteristics

5

The ARIMA ProcedureThe ARIMA Procedure

Name of Variable = ratesName of Variable = rates

Period(s) of Differencing 1Period(s) of Differencing 1 Mean of Working Series -0.00092Mean of Working Series -0.00092 Standard Deviation 0.02754Standard Deviation 0.02754 Number of Observations 469Number of Observations 469

Observation(s) eliminated by differencing 1Observation(s) eliminated by differencing 1

AutocorrelationsAutocorrelations

Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std ErrorLag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error

0 0.00075843 1.00000 | |********************| 00 0.00075843 1.00000 | |********************| 0 1 -0.0000487 -.06416 | .*| . | 0.0461761 -0.0000487 -.06416 | .*| . | 0.046176

2 6.52075E-6 0.00860 | . | . | 0.0463652 6.52075E-6 0.00860 | . | . | 0.046365 3 0.00005996 0.07906 | . |** | 0.0463693 0.00005996 0.07906 | . |** | 0.046369 4 0.00004290 0.05657 | . |*. | 0.0466554 0.00004290 0.05657 | . |*. | 0.046655 5 -0.0000173 -.02284 | . | . | 0.0468015 -0.0000173 -.02284 | . | . | 0.046801 6 2.67563E-6 0.00353 | . | . | 0.0468256 2.67563E-6 0.00353 | . | . | 0.046825 7 0.00006114 0.08061 | . |** | 0.0468267 0.00006114 0.08061 | . |** | 0.046826 8 -9.5206E-6 -.01255 | . | . | 0.0471218 -9.5206E-6 -.01255 | . | . | 0.047121 9 6.54731E-6 0.00863 | . | . | 0.0471289 6.54731E-6 0.00863 | . | . | 0.047128

10 0.00003322 0.04380 | . |*. | 0.04713110 0.00003322 0.04380 | . |*. | 0.047131 11 -0.0000507 -.06689 | .*| . | 0.04721811 -0.0000507 -.06689 | .*| . | 0.047218 12 0.00001356 0.01788 | . | . | 0.04741912 0.00001356 0.01788 | . | . | 0.047419 13 0.00001637 0.02158 | . | . | 0.04743413 0.00001637 0.02158 | . | . | 0.047434 14 0.00003604 0.04752 | . |*. | 0.04745514 0.00003604 0.04752 | . |*. | 0.047455 15 1.26289E-6 0.00167 | . | . | 0.04755615 1.26289E-6 0.00167 | . | . | 0.047556 16 0.00002185 0.02881 | . |*. | 0.04755616 0.00002185 0.02881 | . |*. | 0.047556 17 3.2823E-7 0.00043 | . | . | 0.04759317 3.2823E-7 0.00043 | . | . | 0.047593 18 -0.0000340 -.04483 | .*| . | 0.04759318 -0.0000340 -.04483 | .*| . | 0.047593 19 0.00005576 0.07352 | . |*. | 0.04768319 0.00005576 0.07352 | . |*. | 0.047683 20 5.5947E-6 0.00738 | . | . | 0.04792420 5.5947E-6 0.00738 | . | . | 0.047924 21 -3.8865E-6 -.00512 | . | . | 0.04792721 -3.8865E-6 -.00512 | . | . | 0.047927

22 0.00001112 0.01466 | . | . | 0.04792822 0.00001112 0.01466 | . | . | 0.047928 23 -0.0000168 -.02212 | . | . | 0.04793823 -0.0000168 -.02212 | . | . | 0.047938

24 0.00003914 0.05161 | . |*. | 0.04795924 0.00003914 0.05161 | . |*. | 0.047959

"." marks two standard errors"." marks two standard errors

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Partial Autocorrelations

Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

1 -0.06416 | .*| . | 2 0.00450 | . | . | 3 0.08023 | . |** | 4 0.06742 | . |*. | 5 -0.01626 | . | . | 6 -0.00704 | . | . | 7 0.07182 | . |*. | 8 -0.00271 | . | . | 9 0.00843 | . | . | 10 0.03316 | . |*. | 11 -0.07116 | .*| . | 12 0.01058 | . | . | 13 0.01856 | . | . | 14 0.05192 | . |*. | 15 0.01636 | . | . | 16 0.02016 | . | . | 17 -0.01202 | . | . | 18 -0.04319 | .*| . | 19 0.06369 | . |*. | 20 0.01375 | . | . | 21 0.00007 | . | . | 22 0.00120 | . | . | 23 -0.03788 | .*| . | 24 0.05154 | . |*. |

7

Engle (1982, Econometrica) called this form of heteroscedasticity, where 2

t depends on 2t1, 2

t2, 2t3, etc.

“autoregressive conditional heteroscedasticity (ARCH)”. More formally, the model is

20 1 1 1

2 20

1

; ~ 0, t t k kt t t t t

q

t i t ii

y x x N

,x,x,y tttt 2111 where represents the past realized values of the series. Alternatively we may write the error process as

102

1

1

20 ,N~; t

q

iititt

8

This equation is called an ARCH(q) model. We require that 0 > 0 and i ≥ 0 to ensure that the conditional variance is positive. Stationarity of the series requires that

q

ii .

1

1

9

Typical stylized facts about the ARCH(q) process include:

1. {t} is heavy tailed, much more so than the Gaussian White noise process.

2. Although not much structure is revealed in the correlation function of {t}, the series {t

2} is highly correlated.

3. Changes in {t} tends to be clustered.

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As far as testing is concerned, there are many methods. Three simple approaches are as follows:

1. Time series test. Since an ARCH(p) process implies that {t

2} follows an AR(p), one can use the Box-Jenkins approach to study the correlation structure of t

2 to identify the AR properties

2. Ljung-Box-Pierce test

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3. Lagrange multipler test

H0: 1 = 2 = … q = 0

H1: 1 ≥ 0, i = 1, …, q (with at least one inequality)

To conduct the test,i) Regress et

2 on its lags depends on the assumed order of the ARCH process. For an ARCH(q) process, we regress et

2 on e2t1 … e2

tq.

ii) The LM statistic is under H0, where R2 is the coefficient of determination from the auxiliary regression.

2 2~a

qLM n q R

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• The following SAS program estimates an ARCH model for the monthly stock returns of Intel Corporation from January 1973 to December 1977

• data intel;• infile 'd:\teaching\ms6217\m-intc.txt';• input r t;• r2=r*r;• lr2=lag(r2);• proc reg;• model r2=lr2;• proc arima;• identify var=r nlag=10;• run;• proc arima;• identify var=r2 nlag=10;• run;• proc autoreg;• model r= /garch =(q=4);• run;• proc autoreg;• model r= /garch =(q=1);• output out=out1 r=e;• run;• proc print data=out1;• var e;• run;

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• The REG Procedure• Model: MODEL1• Dependent Variable: r2

• Analysis of Variance

• Sum of Mean• Source DF Squares Square F Value Pr > F

• Model 1 0.01577 0.01577 9.53 0.0022• Error 297 0.49180 0.00166• Corrected Total 298 0.50757

• Root MSE 0.04069 R-Square 0.0311• Dependent Mean 0.01766 Adj R-Sq 0.0278• Coeff Var 230.46618

• Parameter Estimates

• Parameter Standard• Variable DF Estimate Error t Value Pr > |t|

• Intercept 1 0.01455 0.00256 5.68 <.0001• lr2 1 0.17624 0.05710 3.09 0.0022

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H0: 1 = 0

H1: otherwise

LM = 299(0.0311)

= 9.2989 > 21, 0.05 = 3.84

Therefore, we reject H0

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• The ARIMA Procedure

• Name of Variable = r

• Mean of Working Series 0.028556• Standard Deviation 0.129548• Number of Observations 300

• Autocorrelations

• Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 1 Std Error

• 0 0.016783 1.00000 | |********************| 0• 1 0.00095235 0.05675 | . |*. | 0.057735• 2 -0.0000497 -.00296 | . | . | 0.057921• 3 0.00098544 0.0587 | . |*. | 0.057921• 4 -0.0005629 -.03354 | .*| . | 0.058119• 5 -0.0007545 -.04496 | .*| . | 0.058184• 6 0.00038362 0.0228 | . | . | 0.058299• 7 -0.0002817 -.00678 .*| . | 0.058329• 8 -0.0006309 -.03759 | .*| . | 0.059918• 9 -0.0009289 -.05535 | .*| . | 0.059996• 10 0.00097606 0.05816 | . |*. | 0.060166

• "." marks two standard errors

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Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.05675 | . |*. | 2 -0.00620 | . | . | 3 0.05943 | . |*. | 4 -0.04059 | .*| . | 5 -0.04022 | .*| . | 6 0.02403 | . | . | 7 -0.16854 | ***| . | 8 -0.01354 | . | . | 9 -0.06333 | .*| . | 10 0.08700 | . |** |

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The ARIMA Procedure Name of Variable = r2 Mean of Working Series 0.017598 Standard Deviation 0.041145 Number of Observations 300 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 0.0016929 1.00000 | |********************| 0 1 0.00029832 0.17622 | . |**** | 0.057735 2 0.00018962 0.11201 | . |** | 0.059501 3 0.00037532 0.22169 | . |**** | 0.060200 4 0.00033045 0.19519 | . |**** | 0.062862 5 0.00019604 0.11580 | . |**. | 0.064851 6 0.00016872 0.09966 | . |**. | 0.065537 7 0.00016590 0.09799 | . |**. | 0.066040 8 0.00005835 0.03447 | . |* . | 0.066523 9 0.00011312 0.06682 | . |* . | 0.066582 10 0.00008283 0.04893 | . |* . | 0.066805 "." marks two standard errors

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Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.17622 | . |**** | 2 0.08355 | . |** | 3 0.19631 | . |**** | 4 0.13255 | . |*** | 5 0.04335 | . |*. | 6 0.02042 | . | . | 7 0.01435 | . | . | 8 -0.04129 | .*| . | 9 0.02083 | . | . | 10 -0.00074 | . | . |

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The AUTOREG Procedure Dependent Variable r Ordinary Least Squares Estimates SSE 5.03481522 DFE 299 MSE 0.01684 Root MSE 0.12976 SBC -369.15479 AIC -372.85857 Regress R-Square 0.0000 Total R-Square 0.0000 Durbin-Watson 1.8834 Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.0286 0.007492 3.81 0.0002 Algorithm converged. GARCH Estimates SSE 5.03595361 Observations 300 MSE 0.01679 Uncond Var 0.01661204 Log Likelihood 205.372347 Total R-Square . SBC -376.522 AIC -398.74469 Normality Test 83.9101 Pr > ChiSq <.0001 The AUTOREG Procedure Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.0266 0.006782 3.92 <.0001 ARCH0 1 0.009459 0.001418 6.67 <.0001 ARCH1 1 0.2474 0.1107 2.24 0.0254 ARCH2 1 0.0737 0.0686 1.08 0.2822 ARCH3 1 0.0421 0.0714 0.59 0.5552 ARCH4 1 0.0673 0.0622 1.08 0.2792

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The AUTOREG Procedure Dependent Variable r Ordinary Least Squares Estimates SSE 5.03481522 DFE 299 MSE 0.01684 Root MSE 0.12976 SBC -369.15479 AIC -372.85857 Regress R-Square 0.0000 Total R-Square 0.0000 Durbin-Watson 1.8834 Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.0286 0.007492 3.81 0.0002 Algorithm converged. GARCH Estimates SSE 5.0357148 Observations 300 MSE 0.01679 Uncond Var 0.01855167 Log Likelihood 202.39693 Total R-Square . SBC -387.68251 AIC -398.79386 Normality Test 54.6463 Pr > ChiSq <.0001 The AUTOREG Procedure Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.0268 0.006102 4.40 <.0001 ARCH0 1 0.0105 0.001351 7.75 <.0001 ARCH1 1 0.4355 0.1127 3.86 0.0001

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Obs e 292 0.07416 293 -0.03744 294 -0.09077 295 0.26844 296 -0.02342 297 -0.02479 298 -0.19238 299 -0.01871 300 -0.12183

22

22301

22302

ˆ 0.0105 0.4355 0.12183

0.01696

ˆ 0.0105 0.4355 0.06196

0.01789

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In general, the -step ahead forecast is

22

22

2110

22

21

212

210

21

qtqttt

qtqttt

eˆeˆˆˆˆˆ

eˆeˆeˆˆˆ

q

iitit ˆˆˆˆ

1

20

2

24

Generalized Autoregressive Conditional Heteroscedasticity (GARCH)

The first empirical application of ARCH models was done by Engle (1982, Econometrica) to investigate the relationship between the level and volatility of inflation. It was found that a large number of lags was required in the variance functions. This would necessitate the estimation of a large number of parameters subject to inequality constraints. Using the concept of an ARMA process. Bollerslev (1986, Journal of Econometrics) generalized Engle’s ARCH model and introduced the GARCH model.

25

Specifically, a GARCH model is defined as

2

11

20

2

21110 0

jt

p

jj

q

iitit

tttttt ,N~ ;xy

with 0 > 0, i ≥ 0, i =1, … q, j ≥ 0, j = 1, … p imposed to ensure that the conditional variances are positive.

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Usually, we only consider lower order GARCH processes such as GARCH (1, 1), GARCH (1, 2), GARCH (2, 1) and GARCH (2, 2) processes

For a GARCH (1, 1) process, for example the forecasts are

1for 21110

2

21

210

21

ˆˆˆˆˆ

ˆˆeˆˆˆ

tt

ttt

27

Other diagnostic checks:• AIC, SBC

• Note that t = tt. So we should consider “standardized” residuals

and conduct Ljung-Box-Pierce test for

ttt ˆeˆ

.ˆ ˆtt2 and

28

Consider the monthly excess return of the S&P500 index from 1926 for 792 observations:

data sp500;infile 'd:\teaching\ms4221\sp500.txt';input r;proc autoreg;model r=/garch = (q=1);run;proc autoreg;model r=/garch = (q=2);run;proc autoreg;model r=/garch = (q=4);run;proc autoreg;model r=/garch =(p=1, q=1);run;proc autoreg;model r=/garch =(p=1, q=2);run;

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The AUTOREG Procedure Dependent Variable r Ordinary Least Squares Estimates SSE 2.70318776 DFE 791 MSE 0.00342 Root MSE 0.05846 SBC -2244.3895 AIC -2249.0641 Regress R-Square 0.0000 Total R-Square 0.0000 Durbin-Watson 1.8161 Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.006143 0.002077 2.96 0.0032 Algorithm converged. GARCH Estimates SSE 2.70465846 Observations 792 MSE 0.00341 Uncond Var 0.00332216 Log Likelihood 1156.49078 Total R-Square . SBC -2292.9579 AIC -2306.9816 Normality Test 3455.8233 Pr > ChiSq <.0001 The AUTOREG Procedure Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.007506 0.001928 3.89 <.0001 ARCH0 1 0.002742 0.0000651 42.10 <.0001 ARCH1 1 0.1748 0.0397 4.40 <.0001

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GARCH Estimates SSE 2.70325396 Observations 792 MSE 0.00341 Uncond Var 0.00322879 Log Likelihood 1216.10915 Total R-Square . SBC -2405.52 AIC -2424.2183 Normality Test 653.3418 Pr > ChiSq <.0001 The AUTOREG Procedure Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.006432 0.001747 3.68 0.0002 ARCH0 1 0.001793 0.0000827 21.69 <.0001 ARCH1 1 0.1286 0.0297 4.34 <.0001 ARCH2 1 0.3160 0.0335 9.42 <.0001

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GARCH Estimates SSE 2.70432935 Observations 792 MSE 0.00341 Uncond Var 0.0037328 Log Likelihood 1247.06474 Total R-Square . SBC -2454.0821 AIC -2482.1295 Normality Test 387.4606 Pr > ChiSq <.0001 The AUTOREG Procedure Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.007344 0.001452 5.06 <.0001 ARCH0 1 0.001095 0.0000950 11.53 <.0001 ARCH1 1 0.0586 0.0321 1.83 0.0678 ARCH2 1 0.1925 0.0231 8.31 <.0001 ARCH3 1 0.2217 0.0482 4.60 <.0001 ARCH4 1 0.2339 0.0432 5.42 <.0001

32

GARCH Estimates SSE 2.70454696 Observations 792 MSE 0.00341 Uncond Var 0.00324989 Log Likelihood 1269.46195 Total R-Square . SBC -2512.2257 AIC -2530.9239 Normality Test 95.0051 Pr > ChiSq <.0001 The AUTOREG Procedure Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.007453 0.001547 4.82 <.0001 ARCH0 1 0.0000818 0.0000238 3.44 0.0006 ARCH1 1 0.1203 0.0197 6.12 <.0001 GARCH1 1 0.8545 0.0189 45.15 <.0001

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GARCH Estimates SSE 2.70373438 Observations 792 MSE 0.00341 Uncond Var 0.00355913 Log Likelihood 1271.02525 Total R-Square . SBC -2508.6777 AIC -2532.0505 Normality Test 81.4625 Pr > ChiSq <.0001 The AUTOREG Procedure Standard Approx Variable DF Estimate Error t Value Pr > |t| Intercept 1 0.006974 0.001510 4.62 <.0001 ARCH0 1 0.0000913 0.0000284 3.21 0.0013 ARCH1 1 0.0578 0.0374 1.54 0.1226 ARCH2 1 0.0883 0.0440 2.01 0.0446

GARCH1 1 0.8282 0.0228 36.26 <.0001

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• proc autoreg;• model r=/garch =(p=1, q=2);• output out=out1 r=e cev=vhat;• run;• data out1;• set out1;• shat=sqrt(vhat);• s=e/shat;• ss=s*s;• proc arima;• identify var=ss nlag=10;• run;

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The ARIMA Procedure Name of Variable = ss Mean of Working Series 1.000064 Standard Deviation 1.80578 Number of Observations 792 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 3.260843 1.00000 | |********************| 0 1 0.030967 0.00950 | .|. | 0.035533 2 -0.115510 -.03542 | *|. | 0.035537 3 0.055883 0.01714 | .|. | 0.035581 4 -0.084938 -.02605 | *|. | 0.035592 5 -0.109266 -.03351 | *|. | 0.035616 6 -0.029573 -.00907 | .|. | 0.035655 7 -0.150360 -.04611 | *|. | 0.035658 8 0.165407 0.05073 | .|* | 0.035734 9 0.177608 0.05447 | .|* | 0.035824 10 0.015212 0.00466 | .|. | 0.035929 "." marks two standard errors

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Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.00950 | .|. | 2 -0.03552 | *|. | 3 0.01785 | .|. | 4 -0.02771 | *|. | 5 -0.03177 | *|. | 6 -0.01067 | .|. | 7 -0.04751 | *|. | 8 0.05160 | .|* | 9 0.04893 | .|* | 10 0.00743 | .|. | Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations--------------------

6 2.81 6 0.8324 0.009 -0.035 0.017 -0.026 -0.034 -0.009