Transcript Building Econometric Models

Chapter 10
Simulation Methods
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Simulation Methods in Econometrics and Finance
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The Monte Carlo Method
This technique is often used in econometrics when the properties of a particular
estimation method are not known.
Examples from econometrics include:
1. Quantifying the simultaneous equations bias induced by treating an
endogenous variable as exogenous.
2. Determining the appropriate critical values for a Dickey-Fuller test.
3. Determining what effect heteroscedasticity has upon the size and power of a
test for autocorrelation.
Simulations are also often extremely useful tools in finance, in situations such as:
1. The pricing of exotic options, where an analytical pricing formula is unavailable.
2. Determining the effect on financial markets of substantial changes in the
macroeconomic environment.
3. “Stress-testing” risk management models to determine whether they generate
capital requirements sufficient to cover losses in all situations.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Conducting Simulations Experiments
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In all of the above examples, the basic way that such a study would be conducted
(with additional steps and modifications where necessary) is as follows.
1.
Generate the data according to the desired data generating process
(DGP), with the errors being drawn from some given distribution.
2.
Do the regression and calculate the test statistic.
3.
Save the test statistic or whatever parameter is of interest.
4.
Go back to stage 1 and repeat N times.
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N is the number of replications, and should be as large as is feasible.
The central idea behind Monte Carlo is that of random sampling from a given
distribution.
Therefore, if the number of replications is set too small, the results will be
sensitive to “odd” combinations of random number draws.
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‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Random Number Generation
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We generate numbers that are a continuous uniform(0,1) according to the
following recursion:
yi+1 = (a yi + c) modulo m
then
Ri+1 = yi+1 / m for i = 0, 1,...,T
for T random draws, where y0 is the seed (the initial value of y), a is a
multiplier and c is an increment.
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An example of a discrete uniform number generator would be a die or a
roulette wheel. Computers generate continuous uniform random number
draws.
These are pseudo-random numbers
The U(0,1) draws can be transformed into other standard distributions (e.g
normal, t, etc).
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Variance Reduction Techniques
• The sampling variation in a Monte Carlo study is measured by the
standard error estimate, denoted Sx:
var(x)
Sx 
N
where var(x) is the variance of the estimates of the quantity of interest
over the N replications
• In order to achieve acceptable accuracy, the number of replications
may have to be set at an infeasibly high level.
• An alternative way to reduce Monte Carlo sampling error is to use a
variance reduction technique
• Two of the intuitively simplest and most widely used methods are
antithetic variates and control variates.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Antithetic Variates
• One reason that a lot of replications are typically required of a Monte
Carlo study is that it may take many, repeated sets of sampling before
the entire probability space is adequately covered.
• What is really required is for successive replications to cover different
parts of the probability space.
• The antithetic variate technique involves taking the complement of a
set of random numbers and running a parallel simulation on those.
• For each replication conducted using random draws ut, an additional
replication with errors given by - ut is also used.
• Suppose that the average value of the parameter of interest across 2
sets of Monte Carlo replications is given by
x  ( x1  x 2 ) / 2
where x1 and x2 are the average parameter values for replications set 1
and 2 respectively.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Antithetic Variates (cont’d)
• The variance of will be given by:
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var( x )  var( x1 )  var( x2 )  2 cov(x1 , x2 ).
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• If no antithetic variates are used:
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var( x )  var( x1 )  var( x 2 ) .
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• However, the use of antithetic variates would lead the covariance to be
negative, and therefore the Monte Carlo sampling error to be reduced.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Control Variates
• The application of control variates involves employing a variable
similar to that used in the simulation, but whose properties are known
prior to the simulation.
• Denote the variable whose properties are known by y, and that whose
properties are under simulation by x.
• The simulation is conducted on x and also on y, with the same sets of
random number draws being employed in both cases.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Control Variates (cont’d)
• Denoting the simulation estimates of x and y using hats, a new estimate
of x can be derived from:
x*  y  ( xˆ  yˆ )
• Again, it can be shown that the Monte Carlo sampling error of this
quantity, x*, will typically be lower than that of x.
• In fact, it can be shown that control variates will reduce the Monte
Carlo variance if
Corr ( xˆ, yˆ ) 
1 Var ( yˆ )
2 Var ( xˆ )
• For example, if the topic of interest is the pricing of an Asian option,
the control variate approach would be given by
PA* = ( PˆA  PˆBS ) + PBS
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the use of Monte Carlo Simulation in
Econometrics: Deriving a Set of Critical Values for a
Dickey-Fuller Test
• Recall, that the equation for a Dickey Fuller test applied to some series
yt is the regression
yt =  yt-1 + ut
• The test is one of H0:  = 1 against H1:   1. The relevant test statistic
is given by
ˆ  1

SE (ˆ)
• Under the null hypothesis of a unit root, the test statistic does not
follow a standard distribution, and therefore a simulation would be
required to obtain the relevant critical values.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the use of Monte Carlo Simulation in
Econometrics: Deriving a Set of Critical Values for a
Dickey-Fuller Test (cont’d)
The simulation would be conducted in the following steps:
• Construct the data generating process under the null hypothesis – that is,
obtain a series for y that follows a unit root process. This would be done by
– Draw a series of length T, the required number of observations, from a
normal distribution. This will be the error series, so that ut  N(0,1).
– Assume a first value for y, i.e. a value for y at time t = 0.
– Construct the series for y recursively, starting with y1, y2, and so on:
» y1 = y0 + u1
» y2 = y1 + u2
» y3 = y2 + u3
»
…
» yT = yT-1 + uT
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the use of Monte Carlo Simulation in
Econometrics: Deriving a Set of Critical Values for a
Dickey-Fuller Test (cont’d)
– Calculate the test statistic, .
– Repeat steps 1 and 2 N times to obtain N replications of the experiment.
A distribution of values for  will be obtained across the replications.
– Order the set of N values of  from the lowest to the highest. The
relevant 5% critical value will be the 5th percentile of this distribution.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of how to simulate the Price of a
Financial Option
• Although the example presented here is for a simple plain vanilla call option,
the same basic approach would be used to value an exotic option. The steps
are:
– Specify a data generating process for the underlying asset. A random
walk with drift model is usually assumed. Specify also the assumed size
of the drift component and the assumed size of the volatility parameter.
Specify also a strike price K, and a time to maturity, T.
– Draw a series of length T, the required number of observations for the
life of the option, from a normal distribution. This will be the error
series, so that ut  N(0,1).
– Form a series of observations of length T on the underlying asset.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of how to simulate the Price of a
Financial Option (cont’d)
– Observe the price of the underlying asset at maturity observation T. For a
call option, if the value of the underlying asset on maturity date, PT < K,
the option expires worthless for this replication. If the value of the
underlying asset on maturity date, PT > K, the option expires in the
money, and has value on that date equal to PT - K, which should be
discounted back to the present day using the risk free rate.
– Repeat steps 1 to 4 N times, and take the average value of the option over
the N replications. This average will be the price of the option.
• The data for the underlying can be generated according to a more empirically
realistic process.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Bootstrapping
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In finance, the bootstrap is often used instead of a pure simulation.
This is mainly because financial asset returns do not follow the standard statistical
distributions that are used in simulations.
Bootstrapping provides an alternative where by definition, the properties of the
artificially constructed series will be similar to those of the actual series.
Bootstrapping is used to obtain a description of the properties of empirical
estimators by using the sample data points themselves.
Bootstrapping is similar to pure simulation but the former involves sampling from
real data rather than creating new data.
Suppose we have a sample of data, y = y1, y2, ..., yT and we want to estimate some
parameter .
We can get an approximation to the statistical properties of T by studying a sample
of bootstrap estimators.
We do this by taking N samples of size T with replacement from y and recalculating  with each new sample.
We then get a series of  estimates.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Bootstrapping (cont’d)
• Advantage of bootstrapping: it allows the researcher to make inferences
without making strong distributional assumptions.
• Instead of imposing a shape on the sampling distribution of the  ’s,
bootstrapping involves empirically estimating the sampling distribution by
looking at the variation of the statistic within sample.
• We draw a set of new samples with replacement from the sample and
calculate the test statistic of interest from each of these.
• Call the test statistics calculated from the new samples *.
• The samples are likely to be quite different from each other.
• We thus get a distribution of *’s.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the Use of Bootstrapping in a
Regression Context
• Consider a standard regression model, y =  X + u .
• The regression model can be bootstrapped in two ways.
1. Resample the Data
• Take the data, and sample the entire rows corresponding to observation i
together.
The steps would then be
1. Generate a sample of size T from the original data by sampling with
replacement from the whole rows taken together.
2. Calculate  *, the coefficient matrix for this bootstrap sample.
3. Go back to stage 1 and generate another sample of size T. Repeat this a
total of N times.
• Problem with this approach is that we are sampling from the regressors.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the Use of Bootstrapping in a
Regression Context (cont’d)
• The only random influence in the regression is the errors, u, so why not just
bootstrap from those?
2. Resampling from the Residuals
The steps are
1. Estimate the model on the actual data, obtain the fitted values y , and
calculate the residuals, u .
2. Take a sample of size T with replacement from these residuals (and
call these u *), and generate a bootstrapped dependent variable:
y*  y  u *
3.
4.
Then regress the original X data on this new dependent variable to get
a bootstrapped coefficient vector,  * .
Go back to stage 2, and repeat a total of N times.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Situations where the Bootstrap will be Ineffective
• If there are extreme outliers in the data, the conclusions of the
bootstrap may be affected.
• Use of the bootstrap implicitly assumes that the data from which the
sampling is done are independent. This would not hold, for example, if
the data were autocorrelated.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the Use of Bootstrapping to Calculate
Capital Risk Requirements (Hsieh, 1993)
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Financial Motivation
An important use of bootstrapping in a financial context is in the calculation
of value at risk. VaR gives the amount in financial terms that is expected to
be lost with a given probability over a given time interval.
E.g. 95%, 10-day VaR = $100m
How do we calculate, say, the amount of capital required to cover 95% of the
daily losses on a particular position?
This is a forecasting problem considered by Hsieh (1993).
Data: daily log returns on foreign currency (against the US dollar) futures
series from 22 February 1985 until 9 March 1990 (1275 observations) for the
British pound (denoted BP), the German mark (GM), the Japanese yen (JY),
and the Swiss franc (SF).
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the Use of Bootstrapping to Calculate
Capital Risk Requirements (Hsieh, 1993)
• The first step in the analysis is to find a model that describes the main features
of the data that can be used to construct the bootstraps.
• Hsieh shows that his fx returns have no linear or non-linear structure in the
mean, but they have evidence of non-linear structure in the variance.
• What models are examples of this type? The GARCH family.
• Hsieh uses EGARCH and Autoregressive Volatility models.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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An Example of the Use of Bootstrapping to Calculate
Capital Risk Requirements (Hsieh, 1993)
• The standardised residuals (e.g. uˆ t / ˆ t for the EGARCH model) are tested
for non-randomness; little evidence of any structure is found.
• We can thus sample with replacement from these standardised residuals.
• From these, we can simulate a set of future paths for  t2 and then yt by
bootstrapping from vt= ( uˆ t / ˆ t ).
• Hsieh simulates the future path of returns for 180 days after the end of the
estimation sample.
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Calculating the Capital Risk Requirements
In each case, the maximum drawdown (loss) can be calculated over a given
holding period by
Q= (P0 - P1 )  number of contracts
where P0 is the initial value of the position, and P1 is the lowest simulated
price (for a long position) or highest simulated price (for a short position)
over the holding period.
• Hsieh repeats this procedure 2,000 times to get 2,000 maximum losses.
The maximum loss is calculated assuming holding periods of 1, 5, 10, 15,
20, 25, 30, 60, 90 and 180 days.
It is assumed that the futures position is opened on the final day of the
sample used to estimate the models, 9 March 1990.
• Then he takes the 90th percentile of these 2000 maximum losses to get the
amount of capital required to cover losses on 90% of days.
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‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Minimum Capital Risk Requirements for
Currency Futures as a Percentage of the
Initial Value of the Position
No. of days
AR
BP
1
5
10
15
20
25
30
60
90
180
0.73
1.90
2.83
3.54
4.10
4.59
5.02
7.24
8.74
11.38
Long Position
Unconditional
density
0.91
2.30
3.27
3.94
4.61
5.15
5.58
7.44
8.70
10.67
EGARCH
AR
0.93
2.61
4.19
5.72
6.96
8.25
9.08
14.50
17.91
24.25
0.80
2.18
3.38
4.45
5.24
6.20
7.11
11.64
15.45
25.81
Short Position
Unconditional
density
0.98
2.76
4.22
5.48
6.33
7.36
8.33
12.87
16.90
27.36
DM
1
5
10
15
20
25
30
60
90
180
0.72
1.89
2.77
3.52
4.05
4.55
4.93
7.16
8.87
11.38
0.87
2.18
3.14
3.86
4.45
4.90
5.37
7.24
8.39
10.35
0.83
2.34
3.93
5.37
6.54
7.86
8.75
13.14
16.06
21.69
0.89
2.23
3.40
4.36
5.19
6.14
7.02
11.36
14.68
24.25
1.00
2.70
4.12
5.30
6.14
7.21
7.88
12.38
16.16
26.25
0.95
2.91
5.03
6.92
8.91
10.69
12.36
20.86
27.75
45.68
JY
1
5
10
15
20
25
30
60
90
180
0.56
1.61
2.59
3.30
3.95
4.42
4.95
6.99
8.43
10.97
0.74
1.99
2.82
3.46
4.10
4.58
4.92
6.84
8.00
10.27
0.72
2.22
3.46
4.37
5.09
5.78
6.34
8.72
10.51
13.99
0.68
1.92
3.06
4.11
5.13
5.91
6.58
10.53
13.61
21.86
0.87
2.36
3.53
4.60
5.45
6.30
6.85
10.74
14.00
22.21
0.86
2.73
4.41
5.79
6.77
7.98
8.81
13.58
17.63
27.39
SF
1
0.82
0.97
0.89
0.93
1.12
5
1.99
2.51
2.48
2.23
2.93
10
2.87
3.60
4.12
3.37
4.53
15
3.67
4.35
5.60
4.22
5.67
20
4.24
5.10
6.82
5.09
6.69
25
4.81
5.65
8.12
5.90
7.77
30
5.23
6.20
9.12
6.70
8.47
60
7.69
8.41
13.73
10.55
13.10
90
9.23
9.93
16.89
13.60
17.06
180
12.18
12.57
22.92
21.72
27.45
Source: Hsieh (1993). Reprinted with the permission of the University of Washington, School of
Business Administration.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
EGARCH
1.05
3.00
4.88
6.67
8.43
10.46
12.06
20.71
28.03
48.02
0.98
2.98
5.09
7.03
8.86
10.93
12.50
21.27
27.80
45.47
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Minimum Capital Risk Requirements:
Discussion
• The MCRRs derived from bootstrapping the price changes themselves (the
“unconditional approach”) are in most cases higher than those generated
from the other two methods, especially at short investment horizons.
• As the holding period increases from 1 towards 180 days, the MCRR
estimates from the ARV model converge upon those of the unconditional
densities. On the other hand, those of the EGARCH model do not
converge, even after 180 days
• It can also be observed that the MCRRs for short positions are larger than
those of comparative long positions.
‘Introductory Econometrics for Finance’ © Chris Brooks 2002
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Problems with Simulation (including bootstrapping)
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It might be computationally expensive
The results might not be precise
The results are often hard to replicate
Simulation results are experiment-specific.
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To conclude, simulation is an extremely useful tool that can be applied
to an enormous variety of problems. The technique has grown in
popularity over the past decade, and continues to do so. However, like
all tools, it is dangerous in the wrong hands.
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