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BUSINESS
QBUS6830
Financial Time Series and Forecasting
Practice questions for final exam
Q1
(a) If we can observe repeated finite values of a real data series, like a price series,
explain why and how it could possibly have an infinite mean, infinite variance and
infinite 4th moments.
It is not just possible, but quite usual, for a variable to have infinite variance but
still yield finite values of itself. A variance is a mathematical formula that involves
a sum over an infinite number of values over the real line, and effectively averages
(X )2. As an example, a Student-t distribution with 2 degrees of freedom has
infinite variance, yet will only generate finite values from its distribution. Variance
being infinite simply means that the tails of the distribution are fat or thick or long
enough so that the weighted average of (X )2 , weighted by the probability
density function, over the whole real line is infinite. It implies we should expect to
see outliers in the data from this distribution.
Another example is an asset price series. The price can never be infinite, but as the
series progresses over time, it does not have to have, and indeed usually does not
have, a long-run or any mean reversion, and if it follows a random walk it has no
long-run variance either.
(b) What is the purpose of a factor model?
The purpose of a factor model is to find a small number of underlying components
in multiple series of data, so as to learn what might drive their variation. It applies
to situations where a number of variables have been observed at the same times,
over the same time period. The factor model tries to find a set of nonlinear
combinations of these series that can explain most of the variation in these series.
(c) Explain why the CAPM is not usually used for forecasting purposes. Explain one
method you could use to forecast with this model.
The CAPM relates asset premiums to market premiums, both at the same time. To
forecast what will happen to the asset premium in the next period (e.g. tomorrow),
using the CAPM would require knowing the market premium for that same time,
the next period (e.g. tomorrow). Of course this is not possible. The only way to
forecast with the CAPM is to make an assumption about what will happen
tomorrow in the market. E.g. we can do a stress test by seeing what would happen
if the market premium was a certain value, or a list of values.
(d) Why are likelihood methods most favoured, in general, for estimation in
volatility models like GARCH, but not favoured when estimating CAPM models
(where least squares is favoured)?
GARCH models are time series regressions in the squares of the data: i.e. they are
models for squared returns. For LS estimation to have good properties in this case, we
need the 4th moments of the squared returns, i.e. the 8th moment of returns to be finite.
Now, returns certainly do seem prone to outliers and extreme observations, and have
much fatter tails than a Gaussian, making even a finite 4th moment questionable.
Further, since the 8th moment involves the average of (rt – )8 , and return data has
outliers, it seems somewhat unlikely that this 8th moment would be finite for return data.
Thus, LS methods are not preferred for GARCH models in general.
When estimating a CAPM model, we only need the 4th moment of returns to be finite
(not the 8th moment) for LS estimation to have good properties. In this case, LS
estimation is favoured for regression since under the LS assumptions the estimates
found have desirable properties: like being unbiased, consistent and relatively efficient
among estimators for regression parameters. Further, likelihood methods force us to
make an assumption about the conditional distribution of Y given X, while LS
estimation does not. Also, LS estimators give the conditional mean of Y given X, which
is important in asset pricing models.
(e) Explain the leverage effect and how it is potentially captured by the GJR-
GARCH model.
The leverage effect is a theory that suggests that as asset prices fall, the volatility of that
asset’s returns increases. Naturally as prices fall a firm’s equity is decreased. If at the
same time the level of debt for the company is unchanged the drop in equity will
increase the debt/equity ratio and thus leverage increases. The leverage effect associates
this with a subsequent increase in volatility due to the increased risk the firm faces. The
GJR-GARCH model
2 2 20 1 1 1 1 1
1
1
1
0, 0
1, 0
t t t t
t
t
t
I a
a
I
a
includes a dummy variable that is 1 whenever a return shock is negative. This dummy
variable allows the ARCH affect to be different for negative, than for positive, return
shocks. If the variable is positive, then volatility would be higher following negative
shocks (since at-1
2 is also positive) than following positive return shocks. Note that this
is not exactly the leverage effect, since here volatility would increase only when the
return was higher than it estimated mean (i.e. at = rt - t). However, the GJR would
capture the leverage effect if the mean was set to zero.
(f) Compare the Value at Risk and Expected Shortfall risk measures, listing and
discussing at least one advantage and one disadvantage of each, compared to
each other.
Value at Risk is the maximum amount that an asset return would realise in a fixed
period of time at a given probability level. VaR at 1% is then the 1% percentile or
quantile of a return distribution over a fixed period of time.
Expected shortfall (ES) is the average amount that an asset return would realise in a
fixed period of time if it was more extreme than a return quantile at a given probability
level. ES at 1% is then the average return for returns below the 1% percentile or quantile
of a return distribution, i.e. below the 1% VaR, over a fixed period of time.
An advantage of ES is that it represents the average loss in a specific part of the return
distribution, which is perhaps more representative of such losses than the VaR, which
represents the minimum loss in that part of the distribution. A disadvantage of ES is
that it is often a point that is very far out in the tails of the returns distribution, and is
thus estimated with a high level of uncertainty or standard error (since hardly any actual
observations are this far out in the tails in real data sets), and is very sensitive to outliers.
A disadvantage of VaR is that it represents a minimum loss in one part of the return
distribution, which is not really representative of the range or typical losses in that part
of the distribution. An advantage of VaR is that it is not as extreme as ES and thus can
be estimated with more certainty and with less standard error.
(g) A GARCH model specifies the one-step-ahead forecast return distribution.
Explain why it does not specify the two-step-ahead return distribution and why this
distribution is not the same as the one-step-ahead forecast return distribution.
A GARCH model can be written as:
; t t t t t tr a a
2 2 2
0 1 1 1 1t t ta
~ (0,1)
where 0 and 1
t
t t
D
E Var
This setting implies that the one-step-ahead distribution for returns is:
21 1 1| ~ ,t t t tr D where t+1 and t+1 are known at time t, so only t+1 varies here,
and it varies according to D.
For the 2-step-ahead distribution 2 2 2 2
| |t t t t t tr
Here 2 2 2 2 2
2 0 1 1 1 1 0 1 1 1 1 1t t t t t ta . Since this depends on
2
1t , which is not known at time t, this two-step-ahead standard deviation is a random
variable. Thus
2 2 2 2| |t t t t t tr is a random variable that involves the
distribution of the multiplication of the rv t+2 and the rv t+2. We do not, in general
know what the distribution of this new rv is, except that it is not the same as D.
(h) Why is forecasting important in finance, especially in the context of investment?
Investment involves making a decision and seeing and realising the subsequent result
of that decision. In the simplest case if we buy an asset, we make money if the asset
price subsequently increases and lose money if that price subsequently decreases. Thus,
our investment decisions are based on what we think we will happen after we make our
investment decision. Thus forecasting is important, since we are at least implicitly
forecasting what will occur. A little thought will help us realise that all investments
require forecasts of what will happen to prices or other financial instruments.
Q2
We consider the daily prices for the asset NAB (National Australia Bank) from January
2003 until June 2012 with 2434 observations.
Percentage log returns for NAB appear in the bottom plot above.
