004_Modelling Volatility_Arch and Garch Models

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Modelling Volatility: ARCH and GARCH Models

January 24, 2012

() Applied Economoetrics: Topic 8 January 24, 2012 1 / 31


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ARCH and GARCH are time series topics

Notation: Yt fort=1, ..,

T are our observations(e.g. on a stockreturn for each ofT days)

Univariate time series econometric methods werediscussed in 3rd yearcourse.

You may wish to review this material (although Iwill try and briey

explain relevant background below).

Review of basic univariate time series: my textbook (Koop,Introduction to Econometrics) pages 173-196.

Note: material on autoregressive (AR) model is of particular use

The material in this set of slides is taken from my textbook (pages197-205) and Gujarati chapter 15.



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Volatility in Asset Prices

Recall the random walk (withdrift) model:

Yt=+Yt1+t

orYt=+t

where Yt is the rst dierence or change in Yt

Note: ifYt is the log of a variable (e.g. Yt=log (Pt)where Pt is the

price of an asset) then Ytis the percentage change in the variableE.g. Ytcould be stock return (exclusive of dividends)



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Financial assets like stock prices often found to follow random walkwith drift.

Stock prices, on average, increase by per period, but are otherwise

unpredictable.

Stock returns (exclusive of dividends) are on average but areotherwise unpredictable.

Hard to predict stock returns(if it were easy to do, many

econometricians would be rich).However, it is easier to predictvariance (volatility) of stock returns

Why is this interesting?

Volatilities appear in many places in nance relating to risk: portfolio

management, option pricing, pricing of nancial derivatives, VIX,Black-Scholes, etc.

This is not a nance course, so I will just say: having estimates of thevolatilities of asset prices is very useful for nance people



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Assume asset price does followa random walk

To eliminate worrying about the drift, work with

yt= Yt Y

where Y= YtT

Taking deviations from the mean implies that there is no intercept inthe model.

So assume the model is:yt=t



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Before getting to ARCH and GARCH, let me introduce some ideasusing simpler methods

Remember some basic statistics:If you have a random sample, X1, .., XN then an estimate of thevariance is:

Ni=1

XiX

2

N

This result assumes all ofXi have same variance

What if each Xihas a dierent variance? Cannotaverage them alltogether.

Think ofXias a random sample of size N=1 and assume its mean

is zero and you get X2i as an estimate of the variance.Applying these ideas to yt:

y2t is an estimate of volatility of the asset priceat time t.



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Example: Volatility in Stock Prices

Figures on next 3 slides illustrate common pattern with asset prices.

First gure plots of weekly observations of the (logged) stock price ofcertain company.

Second gure plots Y, the percentage change in Y.Third gure plots y2t - an estimate of volatility

Note clustering in volatility: there are times stockprice is much morevolatile than other times

Often happens with nancial assets



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0 50 100 150 200 2503.15

3.2

3.25

3.3

3.35

3.4

3.45

Figure 6.7: Time Series Plot of Log of Stock Price

Week



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0 50 100 150 200 250-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 6.8: Time Series Plot of Change in Stock Price

Week



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0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3x 10

-4 Figure 6.9: Time Series Plot of Volatility

Week



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Properties of AR(1) model:

IfY is high at time t1 it will also tend to be high at time t (if > 0)

Clustering in time of high values ofY

Assume jj < 1 (this is stationarity restrictions, if=1 we have unit

root)corr(Yt, Yt1)=

corr(Yt, Yts)=s

We will use AR models (or similar) but not for Yt, instead for

volatilities



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Example: Volatility in Stock Prices (continued)

Use data in Figure 6.9 and estimate AR(p) model using y2t asdependent variable

Remember: to select lag length can use sequential testing procedure

E.g. estimate AR(4) and if coecient on 4th laginsignicant, movethe AR(3) model, etc.

This strategy yields the AR(1)model for y2

t

.

Table 6.6: AR(1) Model for y2tVariable OLS Estimate t-statistic P-value

Intercept 0.024 1.624 0.106

y

2

t1 0.737 15.552 0.000

Last weeks volatility has strong explanatory power for this weeksvolatility

Clustering in volatility is found


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Autoregressive Conditional Heteroskedasticity (ARCH)

ARCH models (including extensions of them) arethe most popularmodels for nancial volatility.

To allow for generality and conform with how econometrics packageswork context of regression model:

Yt=+1X1t+..+kXkt+t

Note ifX1t=Yt1 then this is an AR model.

If no explanatory variables atall (i.e. = 1 = ..=k=0) then

modelling volatility in dependent variableCommon for dependent variable is a stock return

Common to include at least an intercept



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ARCH model relates to the variance (or volatility) of the error, t.

Use the following notation:

2t=var(t)

2t is volatility at time t.

Error variances are not constant: thus we have heteroskedasticitywhich accounts for the H in ARCH.



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The ARCH model with p lagsis denoted by ARCH(p)

Todays volatility is an averageof past errors squared:

2t=0+12t1+..+p

2tp

0, 1, .., pare coecients that can be estimated in Gretl

We will not discuss estimationof ARCH models:usually maximumlikelihood is used (but other estimators exist)



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If no explanatory variables anddependent variable is the de-meaned

stock return, yt, then t= yt1In this case the ARCH model becomes:

2t=0+1y2t1+..+py

2tp

and volatility depends on recent values ofy2t the metric forvolatility we were using in the preceding section.

ARCH model is closely relatedto AR

ARCH models have similar properties to AR models except that

these properties relate to the volatility of the series.


E l V l tilit i St k P i ( ti d)


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With ARCH models do not need to subtract themean o of stock

returns as we did previouslyBy simply including an intercept in regression model we are allowingfor a random walk with drift

Use logged stock price data and take rst dierence to create Y

Estimate ARCH(1) model with Y as dependentvariable and anintercept in the regression equation I get Table 6.7 (next slide)

The upper part of Table 6.7 refers to the coecients in the regressionequation (here only an intercept)

The lower part of the table refers to the ARCH equation.I have specied ARCH(1) model, so have an intercept (withcoecient labeled 0 in the ARCH equation) and one lag of theerrors squared (labeled 1 in the ARCH equation).



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Table 6.7: ARCH(1) Model using Stock Return Data

Variable Coecient

Estimate P-value

95% CondenceInterval

Regression Equation with Yas Dependent Variable

Intercept 0.105 0.000 [0.081, 0.129]ARCH Equation

Intercept 0.024 0.000 [0.016, 0.032]2t1 0.660 0.000 [0.302, 1.018]




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Numbers labeled CoecientEstimate are estimates of thecoecients

Numbers labeled P-value are P-values for testing whethercorresponding coecient equals zero.

Here all P-values are all less than 0.05 so all coecients aresignicant at the 5% level.

Estimate of1 is 0.660, indicating that volatilitythis week dependsstrongly on the errors squared last week.

Persistence in volatility of similar degree to whatwe found before

Remember we previously found the AR(1) coecient for the variabley2t to be 0.737.




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Lag length selection in ARCHmodels can be done in the samemanner as with any time series model.

Can use an information criterion to select a model

Or look at P-values for whether coecients equal zero (and, if they

do seem to be zero, the accompanying variables can be dropped).E.g. if we estimate an ARCH(2) model table onnext slide.

Results similar to ARCH(1) model.

But coecient on 2t2 is not signicant, sinceP-value is greater

than 0.05.

Thus, ARCH(1) model is adequate and the second lag added byARCH(2) model does not add signicant explanatory power



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Table 6.8: ARCH(2) Model using Stock Return DataVariable

CoecientEstimate

P-value 95% Condence

Interval


Intercept 0.109 0.000 [0.087, 0.131]

ARCH EquationIntercept 0.025 0.000 [0.016, 0.033]2t1 0.717 0.000 [0.328, 1.107]2t2 0.043 0.487 [0.165, 0.079]


Extensions of ARCH


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Extensions of ARCH

Many extensions of ARCH model are used by nancialeconometricians.

Generalized ARCH (or GARCH) is most popularand we will discuss itbelow

Gretl has a menu of GARCHvariants with seven variants of thewith acronyms like TARCH, EGARCH, NARCH, etc.

Will not discuss these in this course, but if you are interested innancial volatility, you can learn more about these models.

Here focus on GARCH



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ARCH had volatility depending on lagged errors squared

GARCH adds to this lags of volatility itself

Properties of GARCH similar to ARCH

But GARCH is much more exible, much more capable of matching awide variety of patterns of nancial volatility

Financial time series often have fat tails: moreextreme outcomes inthe tails of the distribution than a Normal distribution would allow for

Important property of GARCH: allows for fat tails

GARCH model with (p, q) lags is denoted by GARCH(p, q) and has avolatility equation of:

2t=0+1

2t1+..+p

2tp+1

2t1+..+p

2tp

Maximum likelihood estimation can be done by Gretl (and otherpackages)




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Table contains GARCH(1,1) results, we obtain the results in the

following table.Looks like ARCH(1) was goodenough for this data set sincecoecient on 2t1 is insignicant

Table 6.9: GARCH(1, 1) Model using Stock Return Data

Variable Coecient

Estimate P-value

95% CondenceInterval


Intercept 0.109 0.000 [0.087, 0.131]

ARCH EquationIntercept 0.026 0.000 [0.015, 0.038]2t1 0.714 0.000 [0.327, 1.101]2t1 0.063 0.457 [0.231, 0.104]


GARCH: Summary


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GARCH: Summary

So far our most general modelis a regression with GARCH errors.

To summarize, this means:

Yt=+1X1t+..+kXkt+t

t is N

0, 2t

and

2t=0+12t1+..+p

2tp+1

2t1+..+p

2tp

But it is useful to introduce one more extension


GARCH in mean (GARCH-M)


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GARCH in mean (GARCH M)

What if the volatility directly inuences the dependent variable?

E.g. volatility in stock markets causes people toworry about riskwhich impacts on stock returns

Volatility is an explanatory variable in the regression:

Yt=+1X1t+..+kXkt+t+t

t is N

0, 2t

and

2

t

=0

+1

2

t1+..+

p

2

tp+1

2

t1+..+p

2

tp

Exactly like GARCH model except we have added tas explanatoryvariable in the regression.

This is the GARCH-M model



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GARCH-M model can be estimated using maximum likelihood in Gretl

These models will be investigated in the computer tutorial



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004_Modelling Volatility_Arch and Garch Models

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