**ARCH and GARCH Models**

*Background*
Most of the statistical tools are designed to model the conditional mean
of a random variable.

ARMA and ARIMA time series model assume that there lies stationarity in
the data.

Now there are some situations where variance of error terms varies with
time, so in this situation we can not apply these models.

Autoregressive Conditional Heteroskedasticity (ARCH) models are
specifically designed to model and forecast conditional variances.

ARCH models were introduced by Engle (1982) and generalized as GARCH
(Generalized ARCH) by Bollerslev (1986).

The variance of the dependent variable is modeled as
a function of past values of the dependent variable and independent, or
exogenous variables.

These models are widely used in various branches of econometrics,
especially in financial time series analysis.

**WHY ARCH/GRACH MODELS?**

There are several reasons that you may want to model and forecast
volatility.

First, you may need to analyze the risk of holding an asset or the value
of an option.

Second, forecast confidence intervals may be time-varying, so that more
accurate intervals can be obtained by modeling the variance of the errors.

Third, more efficient estimators can be obtained if
heteroskedasticity in the errors is handled properly.

*Basic Assumptions*
The basic version of the least squares model assumes
that the expected value of all error terms, when squared, is the same at any
given point. This assumption is called homoskedasticity, and it is this
assumption that is the focus of ARCH/ GARCH models.

Data in which the variances of the error terms are
not equal, in which the error terms may reasonably be expected to be larger for
some points or ranges of the data than for others, are said to suffer from
heteroskedasticity.

Instead of considering this as a problem to be
corrected, ARCH and GARCH models treat heteroskedasticity as a variance to be
modeled.

*How To Start Modeling*
Stationarity of time series

·
Unit Root Test

The Box - Jenkins ARIMA Methodology

·
Identification

·
Estimation

·
Diagnostic Checking

·
Forecasting

*Stationarity Checks*

*The ARCH specification*
In developing an ARCH model, you will have to
provide two distinct specifications-one for the conditional mean and one for
the conditional variance.

The ARCH(1) Model

GARCH (1,1) Model

NOTE:

*An ordinary ARCH model is a special case of a GARCH specification in which there are no lagged forecast variances in the conditional variance equation.***GARCH (**

*p, q*) Model
Higher order GARCH models, denoted GARCH (

**), can be estimated by choosing either***p, q***or***p***are greater than 1. The representation of the GARCH(***q, both***) variance is***p, q*
where

**is the order of the GARCH terms and***p***is the order of the ARCH term.***q*

*Model Checking*
1. The
Ljung-Box Q Statistic

2. Jarque-Bera
Statistic

3. Histogram
Normality test

4. ARCH–LM
Test

5. Correlogram
of Standardized Residuals and Correlogram of squared residuals

**A Value-at-Risk Example**

·
Applications of the ARCH/GARCH approach
are widespread in situations where the volatility of returns is a central
issue. Many banks and other financial institutions use the concept of “value at
risk” as a way to measure the risks faced by their portfolios.

·
The 1 percent value at risk is defined
as the number of dollars that one can be 99 percent certain exceeds any losses
for the next day. Statisticians call this a 1 percent quantile, because 1
percent of the outcomes are worse and 99 percent are better.

·
Let’s use the GARCH(1,1) tools to
estimate the 1 percent value at risk of a $1,000,000 portfolio on March 23,
2000. This portfolio consists of 50 percent Nasdaq, 30 percent Dow Jones and 20
percent long bonds. The long bond is a ten-year constant maturity Treasury
bond.1 This date is chosen to be just before the big market slide at the end of
March and April. It is a time of high volatility and great anxiety.

**Hypothetical Portfolio Example**

**ARCH – LM Test**

**Descriptive**

**Table 2**

**Results from table 2**

The portfolio shows substantial evidence of ARCH
effects as judged by the autocorrelations of the squared residuals in Table 2.

The first order autocorrelation is .210, and they
gradually decline to .083 after 15 lags. These autocorrelations are not large,
but they are very significant. They are also all positive, which is uncommon in
most economic time series and yet is an implication of the GARCH(1,1) model.

**Table 3 showing GARCH (1,1)**

**Results from table 3**

The basic GARCH(1,1) results are given in Table 3.

Under this table it lists the dependent variable,
PORT, and the sample period, indicates that it took the algorithm 16 iterations
to maximize the likelihood function and computed standard errors using the
robust method of Bollerslev-Wooldridge.

The three coefficients in the variance equation are
listed as C, the intercept; ARCH(1), the first lag of the squared return; and
GARCH(1), the first lag of the conditional variance. Notice that the
coefficients sum up to a number less than one, which is required to have a mean
reverting variance process.

**Normality of Residuals**

**Residuals Generated from the Model**

Plots the value at risk estimated each day using
this methodology within the sample period and the losses that occurred the next
day. There are about 1 percent of times the value at risk is exceeded, as is expected,
since this is in-sample.

**Conclusion**

ARCH and GARCH models have been applied to a wide
range of time series analyses, but applications in finance have been
particularly successful and have been the focus of this introduction.

Financial decisions are generally based upon the
tradeoff between risk and return; the econometric analysis of risk is therefore
an integral part of asset pricing, portfolio optimization, option pricing and
risk management.

The analysis of ARCH and GARCH models and their many
extensions provides a statistical stage on which many theories of asset pricing
and portfolio analysis can be exhibited and tested.

Reference

Bollerslev, Tim. 1986. “Generalized Autoregressive
Conditional Heteroskedasticity.”

*Journal of Econometrics.*April, 31:3, pp. 307–27.
Bollerslev, Tim and Jeffrey M. Wooldridge. 1992.
“Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time-
Varying Covariances.”

*Econometric Reviews.*11:2, pp. 143–72.
Engle, Robert F. 1982. “Autoregressive Conditional
Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.”

*Econometrica.*50:4, pp. 987–1007.
Engle, Robert
and Gary G. J. Lee. 1999. “A Permanent and Transitory Component Model of Stock
Return Volatility,” in

*Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive W. J. Granger.*Robert F. Engle and Halbert