# BUS 10123 Chapter Notes - Chapter 13: Autoregressive Conditional Heteroskedasticity, Stock Market Index, HeteroscedasticityExam

by OC2511532

School

Kent State UniversityDepartment

Business Administration InterdisciplinaryCourse Code

BUS 10123Professor

Eric Von HendrixStudy Guide

FinalThis

**preview**shows pages 1-2. to view the full**7 pages of the document.**1

1. (a) The scope of possible answers to this part of the question is limited only

by the imagination! Simulations studies are useful in any situation where the

conditions used need to be fully under the control of the researcher (so that

an application to real data will not do) and where an analytical solution to the

problem is also unavailable. In econometrics, simulations are particularly

useful for examining the impact of model mis-specification on the properties

of estimators and forecasts. For example, what is the impact of ignored

structural breaks in a series upon GARCH model estimation and forecasting?

What is the impact of several very large outliers occurring one after another

on tests for ARCH? In finance, an obvious application of simulations, as well

as those discussed in Chapter 11, is to producing â€śscenariosâ€ť for stress-

testing risk measurement models. For example, what would be the impact on

bank portfolio volatility if the correlations between European stock indices

rose to one? What would be the impact on the price discovery process or on

market volatility if the number and size of index funds increased

substantially?

(b) Pure simulation involves the construction of an entirely new dataset made

from artificially constructed data, while bootstrapping involves resampling

with replacement from a set of actual data.

Which technique of the two is the more appropriate would obviously depend

on the situation at hand. Pure simulation is more useful when it is necessary

to work in a completely controlled environment. For example, when

examining the effect of a particular mis-specification on the behaviour of

hypothesis tests, it would be inadvisable to use bootstrapping, because of

course the boostrapped samples could contain other forms of mis-

specification. Consider an examination of the effect of autocorrelation on the

power of the regression F-test. Use of bootstrapped data may be

inappropriate because it violates one or more other assumptions â€“ for

example, the data may be heteroscedastic or non-normal as well. If the

bootstrap were used in this case, the result would be a test of the effect of

several mis-specifications on the F-test!

Bootstrapping is useful, however, when it is desirable to mimic some of the

distributional properties of actual data series, even if we are not sure quite

what they are. For example, when simulating future possible paths for price

series as inputs to risk management models or option prices, bootstrapping is

useful. In such instances, pure simulation would be less appropriate since it

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

2

would bring with it a particular set of assumptions in order to simulate the

data â€“ e.g. that returns are normally distributed. To the extent that these

assumptions are not supported by the real data, the simulated option price

or risk assessment could be inaccurate.

(c) Variance reduction techniques aim to reduce Monte Carlo sampling error.

In other words, they seek to reduce the variability in the estimates of the

quantity of interest across different experiments, rather like reducing the

standard errors in a regression model. This either makes Monte Carlo

simulation more accurate for a given number of replications, making the

answers more robust, or it enables the same level of accuracy to be achieved

using a considerably smaller number of replications. The two techniques that

were discussed in Chapter 11 were antithetic variates and control variates.

Mathematical details were given in the chapter and will therefore not be

repeated here.

Antithetic variates try to ensure that more of the probability space is covered

by taking the opposite (usually the negative) of the selected random draws,

and using those as another set of draws to compute the required statistics.

Control variates use the known analytical solutions to a similar problem to

improve accuracy. Obviously, the success of this latter technique will depend

on how close the analytical problem is to the actual one under study. If the

two are almost unrelated, the reduction in Monte Carlo sampling variation

will be negligible or even negative (i.e. the variance will be higher than if

control variates were not used).

(d) Almost all statistical analysis is based on â€ścentral limit theoremsâ€ť and

â€ślaws of large numbersâ€ť. These are used to analytically determine how an

estimator will behave as the sample tends to infinity, although the behaviour

could be quite different for small samples. If a sample of actual data that is

too small is used, there is a high probability that the sample will not be

representative of the population as a whole. As the sample size is increased,

the probability of obtaining a sample that is unrepresentative of the

population is reduced. Exactly the same logic can be applied to the number of

replications employed in a Monte Carlo study. If too small a number of

replications is used, it is possible that â€śoddâ€ť combinations of random number

draws will lead to results that do not accurately reflect the data generating

process. This is increasingly unlikely to happen as the number of replications

is increased. Put another way, the whole probability space will gradually be

appropriately covered as the number of replications is increased.

###### You're Reading a Preview

Unlock to view full version