# Corrected LM goodness-of-fit tests with applicaton to stock returns

Abstract (Summary)

Many commonly used econometric procedures develop efficient estimates of
model parameters by making explicit assumptions about data distributions, often
assuming that the distribution is normal. Nonparametric econometric procedures give
less efficient estimates while often assuming that the variances of unknown
distributions are finite. Importantly, the shape and scale of the data often have a crucial
effect on the estimates of unknown model parameters. The usual Central Limit
Theorems assume that as the number of unknown random variables in a sum increases
without limit, the sum of the variables converges asymptotically to a normal
distribution. However, if even one of those variables has an infinite variance, then the
limiting distribution of the resulting sum belongs to the set of non-Gaussian Pareto-
Lévy stable distributions. If the data conforms to a leptokurtic stable distribution,
econometric procedures that rely on either normal errors or finite variance are
invalidated, hencecastingdoubtonastudy’seconomicfindings.Theprocedures
described herein fill gaps in the current methods based on the possibilities that many
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distributions are not normal and some may even have theoretical infinite variances
despite all sample variances necessarily being finite.
All data sets and series of residuals from regressions, including those from a
model with error terms from an underlying distribution with infinite second moments,
have finite sample variances and appear to more closely conform to any assumed error
distribution than do the actual errors. Many well-known goodness-of-fit tests rely on
theempiricaldistributionofresidualsbeingarbitrarilyclosetothe“true”underlying
error distribution; or, equivalently, that model parameter estimates are actually equal to
theparameter’s“true,”typicalyunknownvalues.Whilethisasumptionmaybe
approximately correct for a very large sample size, such tests are biased towards
acceptance with finite sample sizes.
Certainly it is difficult to visually distinguish between finite and infinite second
moments by assessing histograms. Standard goodness-of-fit tests are biased towards
acceptance of any hypothesized distribution if the test statistics do not contain explicit
corrections for the fact that estimates of model parameters are used rather than unknown
true values. Goodness-of-fit tests that use only the most extreme distributional
deviation are not as efficient as those that use all the entire distribution.
Whether or not the true distribution has infinite variance, the bias can be avoided
by Lagrange Multiplier goodness-of-fit tests proposed herein. If a sample is
independent and identically distributed according to a distribution F (with time series
data a transformation can be applied to estimate an IID series), then the distribution
transform of the data produces a histogram that is approximately uniform over the unit
interval. Large deviations from uniformity provide evidence against F. The
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construction of an alternative hypothesis space surrounding the null hypothesis ensures
that deviations in any direction can be detected.
Such tests can be constructed so that they have more power against alternative
hypotheses and less size distortion than standard tests. They achieve these
improvements by correcting for the presence of unknown model parameters. The test
statistic is asymptotically chi-squared. Exact finite sample sizes are calculated
employing Monte Carlo simulations; however, for samples with as few as 30
observations, size distortion is quite low.
Unknown model parameters can be estimated by the maximum likelihood
principle without asymptotically biasing the test. Furthermore, the test meets the
optimality conditions of the Neyman-Pearson lemma against any simple alternative
hypothesis in its parameter space. It is an omnibus test with the null hypothesis nested
in the space of alternatives.
Tests against many non-standard distributions are conducted including
symmetric stable distributions, generalized Student-t distributions, generalized error
distributions (GED), and mixtures of Gaussian distributions.
These econometric tests are not restricted to economic or financial studies, but
can be applied in any discipline employing econometric or statistical techniques. With
these tests, economists and other researchers will have a new tool yielding better results
on more data sets, with or without obvious outliers.
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Bibliographical Information:

Advisor:

School:The Ohio State University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:lagrange problem goodness of fit tests statistical hypothesis testing distribution probability theory gaussian t test statistics

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