Bootstrap Event Study Tests

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Bootstrap Event Study Tests

• Peter Westfall
• ISQS Dept.
• Joint work with Scott Hein, Finance

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An Example of an Event
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Event (Outlier) Detection

• Main Idea y0 is an outlier if it is unusual
  with respect to typical circumstances.
• Definitions
• Critical value The threshold c that y0 must
  exceed to be called an outlier
• a level The probability that Y0 exceeds c
  under typical circumstances
• p-value The probability that Y0 exceeds the
  particular observed value y0 under typical
  circumstances

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Case 1 Normal distribution, known mean (m),
known variance (s2).

• Let

Y0 is associated with an event if Z is
large. Critical and p-values are from Z
distribution. Ex y0 -7.13, m-.15, s1.0
Þ Z-6.98. a.05 critical value Za/2
1.96. p-value 2P(Zlt-6.98) 3E-12
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Case 2 Normal distribution, unknown m, known s2.

• Let Y1,,Yn denote an i.i.d. sample under
  typical circumstances (excluding Y0). Then

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Case 3 Normal distribution, unknown m, unknown
s2.

• Let Y1,,Yn denote an i.i.d. sample under
  typical circumstances (excluding Y0). Then

Critical and p-values are from tn-1
distribution. Example n87, y0 -7.13,
-.14, s1.013 Þ T-6.86. a.05 critical
value t87-1,a/2 1.99. p-value
2P(T87lt-6.86) 1E-9
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Notes

• The method is essentially asking, how far into
  the tail of the typical distribution is y0?
• (Estimation of the mean just gives a minor
  correction (1 1/n) in the variance formula
• Estimation of the variance gives another minor
  correction Tn-1 instead of Z critical and
  p-values)
• The central limit theorem does not apply since we
  are concerned with the distribution of Y0, not
  the distribution of

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The Distribution of (Y0-m)/s
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Case 1A Known Distribution

• Exact critical values for Z are
• cL a/2 quantile of distribution of Z
• cU 1-a/2 quantile of distribution of Z
• Exact P-Value
• p-value 2 min P(Z z), P(Z ³ z)

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A Simulation-Based Approach

• Simulate many (1,000s) of Zs at random from
  the pdf
• Critical values
• cL is the 100(a/2) percentile of the simulated
  data
• cU is the 100(1-a/2) percentile of the simulated
  data
• P-value
• pL proportion of simulated Zs that are
  smaller than z.
• pU proportion of simulated Zs that are larger
  than z.
• P-value 2min(pL, pU).

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Case 1B Unknown Distribution

• Let Y1,,Yn denote an i.i.d. sample under
  typical circumstances (excluding Y0). Then the
  empirical pdf approximates the true pdf if n is
  large (Glivenko-Cantelli Theorem).
• Thus, approximate critical and p-values can be
  obtained by using the empirical distribution.
• This is the essential nature of the bootstrap.

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Case 1B.i Simulation-Based Approach with known
m, s

• Simulate 1000s of values of Z (Y0 m)/s as
  follows
• Select a value Y01 at random from the observed
  data Y1,,Yn let Z1 (Y01 m)/s
• Select a value Y02 at random from the observed
  data Y1,,Yn let Z2 (Y02 m)/s
• B. Select a value Y0B at random from the observed
  data Y1,,Yn let ZB (Y0B m)/s
• Use the simulated data Z1,,ZB to determine
  critical and p-values.

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Case 1B.ii Unknown m, s

• Use the statistic
• The distribution of the statistic depends on the
  randomness inherent in

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Case 1B.ii Simulation-Based Approach
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Extension Market Model
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Extension Multivariate Market Model
The MVRM models may be expressed as
Ri Xbi Dgi ei, for i
1,,g (firms or portfolios).
Observations within a row of e e1 eg
are correlated this is called cross-sectional
correlation. Observations on e e1 eg
between rows 1,,n are assumed to be independent
in the classical MVRM model. Null hypothesis
H0 g1 gg 0 0
This multivariate test
is computed easily and automatically using
standard statistical software packages, using
exact (under normality) F-tests. The test is
based on Wilks Lambda likelihood ratio criterion.
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Hein, Westfall, Zhang Bootstrap Method

• Fit the MVRM model. Obtain the F-statistic for
  testing H0 using the traditional method (assuming
  normality). Obtain also the ((n1)g) sample
  residual matrix e e1 eg.
• Exclude the row corresponding to event from e,
  leaving the (ng) matrix e-.
• Sample (n1) row vectors, one at a time and with
  replacement, from e-. This gives a ((n1)g)
  matrix R1 Rg .
• Fit the model Ri Xbi Dgi ei, i 1, , g,
  and obtain the test statistic F using the same
  technique used to obtain the F-statistic from the
  original sample.
• Repeat 3 and 4 NBOOT times. The bootstrap
  p-value of the test is the proportion of the
  NBOOT samples yielding an F-statistic that is
  greater than or equal to the original F-statistic
  from step 1.

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Simulation Study True Type I error rates
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Simulation Study True Type I error rates
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Alternative Method (Kramer,2001)

• Test statistic is Z S ti/(g1/2st), where
  ti is the t-statistic from the univariate
  dummy-variable-based regression model for firm i,
  and st is the sample standard deviation of the g
  t-statistics.
• Algorithm
• (i) create a pseudo-population of
  t-statistics ti ti - reflecting the null
  hypothesis case where the true mean of the
  t-statistics is zero,
• (ii) sample g values with replacement from
  the pseudo-population and compute Z from these
  pseudo-values,
• (iii) repeat (ii) NBOOT times, obtaining
  Z1, , Zb. The p-value for the test is then
  2min(pU, pL), where pL is the proportion of the
  NBOOT bootstrap samples yielding Zi Z, and
  where pU is the proportion of the NBOOT samples
  yielding Zi ³ Z.
• Assumption The statistics are
  cross-sectionally independent

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Modified Kramer Method

• Model-Based bootstrap Kramer Bootstrap Kramers
  Z S ti/(g1/2st), but by resampling MVRM
  residual vectors as in HWZ.
• Model-based sum t Bootstrap St Sti by
  resampling MVRM residual vectors as in HWZ.

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Table 1. Simulated Type I error rates as a
function of cross-sectional correlation.
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/------------------------------------------------
--------------/ / Name bootevnt
/ / Title
Macro to calculate bootstrap p-values for event
/ / studies
/ / Author Peter H.
Westfall, westfall_at_ttu.edu / /
Release SAS Version 6.12 or higher, requires
SAS/IML / /--------------------------------
------------------------------/ / Inputs

/ /
/ / DATASET Data set to be
analyzed (required) / /

/ / YVARS List of y variables used in
the multivariate / / regression
model, separated by blanks (required) / /

/ / XVARS List of x variables used
in the multivariate / /
regression model, separated by blanks (required)
/ /
/ / EVENT Name of dummy
variable indicating event / /
observation (e.g., day). This is required.
/ /
/ / EXCLUDE Name of
dummy variable indicating days that / /
should be excluded from the resampling.
If there / / are multiple event
days in the model, then all / /
those days should be excluded because the
/ / residuals are mathematically
zero. If there are / / not
multiple eventdays, then the EXCLUDE
/ / variable should be identical to
the EVENT / / variable.
/ /

/ / NBOOT Number of bootstrap samples.
This input is / / required.
Pick a number as large as possible / /
subject to time constraints. Start with
100 / / and work your way up,
noting the accuracy as / /
given by the confidence interval in the output.
/ /
/ / MODELBOOT 1 for
requesting model-based bootstrap tests, / /
0 to exclude them.
/ /
/ / NPBOOT 1
to request Kramer's nonparametric bootstrap
/ / tests, 0 to exclude them.
/ /
/ / SEED
Seed value for random numbers (0 default)
/ /
/ /---------------------------
-----------------------------------/ / Output
This macro computes normality-assuming exact p-
/ / values and bootstrap approximate p-values
that do not / / require the normality
assumption. A 95 confidence interval / / for
the true bootstrap p-value (which itself is
approximate / / because it uses the empirical,
not the true, residual / / distribution)
also is given.
/ /---------------------------------------------
-----------------/
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Invocation of Macro
libname fin "c\research\coba" data sinkey
set fin.sinkey run bootevnt(datasetsinkey,
yvarspr1 pr2 pr3 pr4, xvarsds m1 m2 m3
dsm d2 d3 d4 d5 d6, eventd1,
excludeexclude, nboot1000, modelboot1,
npboot1, seed182161)
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Normality-Assuming
Tests for Event
TSQ F NDF DDF PVAL
15.025505 3.6957895 4
183 0.0064153
NBOOT
Model-based bootstrap Binder p-value,
using 20000 samples
with 95 confidence limits on the true bootstrap
p-value
BOOTP LCL UCL
0.01115 0.0096947 0.0126053
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Model-based bootstrap Kramer
p-value, using 20000 samples
with 95 confidence limits on the true
bootstrap p-value
BOOTKP LCLK UCLK
0.0609 0.0561373
0.0656627
NBOOT
Model-based bootstrap Sum t p-value, using
20000 samples with 95
confidence limits on the true bootstrap
p-value
BOOTTSUMP LCLSUMT UCLSUMT
0.0001 -0.000096 0.000296
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1.55 of the
bootstrap samples had 0 variance

NBOOT Nonparametric bootstrap
Kramer p-value, using 20000 samples
with 95 confidence limits on the
true bootstrap p-value
BOOTTNP LCLNP UCLNP
0.1404 0.1333184
0.1452147
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Robustness of Bootstrap to Serial Correlation

• Recall that the method is essentially a
  comparison of Y0 to the distribution of Y1,,Yn.
• If the empirical distribution of Y1,,Yn
  converges to F, then the unconditional null
  probability of an event also converges to a
  F(ca/2) (1-F(c1-a/2)).
• Such convergence occurs for typical stationary
  time series processes.

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Conclusions

• We use t, not z even when n is large. Why?
  Because t is generally more accurate.
• We should use bootstrap tests instead of
  traditional tests for precisely the same reason.
• We must account for cross-sectional correlation
  in the analysis.
• The recommended method is our bootstrap with a
  modification of Kramers Z (The model-based sum t
  method)Software is available from
  westfall_at_ba.ttu.edu