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• Statistical DistributionsBYUJames B. McDonald

• Statistical Distributions James B. McDonaldBrigham Young UniversityMay 2013The research assistance of Brad Larsen, Patrick Turley, and Sean Kerman is gratefully acknowledged as are comments from Richard Michelfelder and Panayiotis Theodossiou.

• Statistical Distributions

IntroductionSome families of statistical distributionsRegression applicationsCensored regressionQualitative response modelsOption pricing VaR (value at risk)Conclusion

• Statistical Distributions

Introduction Some families of statistical distributionsFamilies Regression applicationsCensored regressionQualitative response modelsOption pricing VaR (value at risk)Conclusion

• Some families of statistical distributionsFamilies f(y;), = vector of parameters GB: GB1, GB2, GG (0
• GB distribution tree

• Probability Density Functions

• Probability Density Functions

• Probability Density FunctionsGB2 PDF evaluated at different parameter values:

• Some families of statistical distributionsFamilies GB: GB1, GB2, GGEGB: EGB1, EGB2, EGG (Y is real valued)

• EGB distribution tree

• Probability Density Functions

• Probability Density Functions

• Probability Density FunctionsEGB2 PDF evaluated at different parameter values:

• Some families of statistical distributionsFamilies GB: GB1, GB2, GGEGB: EGB1, EGB2, EGG SGT (Skewed generalized t): SGED, GT, ST, t, normal (Y is real valued)

• SGT distribution tree

• Probability Density Functions

• Probability Density FunctionsSGT PDF evaluated at different parameter values:

• Some families of statistical distributionsFamilies GB: GB1, GB2, GGEGB: EGB1, EGB2, EGG SGT (Skewed generalized t): SGED, GT, ST, t, normalIHS

• Probability Density Functionswhere IHS

• Probability Density FunctionsIHS PDF evaluated at different parameter values:

• Some families of statistical distributionsFamilies GB: GB1, GB2, GGEGB: EGB1, EGB2, EGG SGT (Skewed generalized t): SGED, GT, ST, t, normalIHSg-and-h distribution (Y is real valued)

• g-and-h distributionDefinition:

where Z ~ N[0,1]

h>0h

• g-and-h distributionIs known as the g distribution where the parameter g allows for skewness.Is known as the h distribution Symmetric Allows for thick tails

• Probability Density Functionsg-and-h PDF evaluated at different parameter values with h>0:

• Probability Density Functionsg-and-h PDF evaluated at different parameter values with h
• Some families of statistical distributionsFamilies f(y;)GB: GB1, GB2, GGEGB: EGB1, EGB2, EGG SGT (Skewed generalized t): SGED, GT, ST, t, normalIHSg-and-h distributionOther distributions: extreme value, Pearson family,

• Some families of statistical distributionsFamilies f(y;)GB: GB1, GB2, GGEGB: EGB1, EGB2, EGG SGT (Skewed generalized t): SGED, GT, ST, t, normalIHSg- and h-distributionOther distributions: extreme value, Pearson family, Extensions:

, 2. Multivariate

• Statistical Distributions

Introduction Some families of statistical distributionsFamilies PropertiesRegression applicationsCensored regressionQualitative response modelsOption pricing VaR (value at risk)Conclusion

• Some families of statistical distributionsPropertiesMomentsGB family

for h < aq with c=1

• Some families of statistical distributionsPropertiesMomentsGB familyGB1

• Some families of statistical distributionsPropertiesMomentsGB familyGB1GB2

• Some families of statistical distributionsPropertiesMomentsGB familyGB1GB2GG

• Some families of statistical distributionsPropertiesMomentsGB familyEGB family

• EGB moments

EGGEGB1EGB2MeanVarianceSkewnessExcess kurtosis

• EGB2 moment space

• Some families of statistical distributionsPropertiesMomentsGB familyEGB familySGT family

• SGT familyfor h < pq=d.f.

• SGT moment space

• SGT family moment space

• Some families of statistical distributionsFamiliesPropertiesMomentsGB familyEGB familySGT familyIHS

• IHS moment space

• Some families of statistical distributionsFamiliesPropertiesMomentsGB familyEGB familySGT familyIHS g-and-h family

• g- and h-familyMoments exist up to order 1/h (0
• g-and-h moment space (h>0)(visually equivalent to the IHS)

• Moment space for g-and-h (h>0) and g-and-h (h real)

• Moment space of SGT, EGB2, IHS, and g-and-h

• Some families of statistical distributionsPropertiesMomentsCumulative distribution functions (see appendix)Involve the incomplete gamma and beta functions

• Some families of statistical distributionsPropertiesMomentsCumulative distribution functions (see appendix)Involve the incomplete gamma and beta functionsGini coefficients (G)

• Gini Coefficients (G)Definition:

(Dorfman, 1979, RESTAT)

• Gini CoefficientsInterpretation:G = 2A

• Gini CoefficientsApplication: Stochastic DominanceMeasures of income and wealth inequality

• Some families of statistical distributionsPropertiesMomentsCumulative distribution functions (see appendix)Gini coefficients (G)Incomplete moments

• Incomplete momentsDefinition:

Applications: Option pricing formulasLorenz Curves

• Incomplete momentsConvenient theoretical results:

DistributionLNGGGB2

• Some families of statistical distributionsPropertiesMomentsCumulative distribution functions (see appendix)Gini coefficients (G)Incomplete momentsMixture models

• Mixture ModelsLet denote a structural or conditional density of the random variable Y where and denote vectors of distributional parameters. Let the density of be given by the mixing distribution . The observed or mixed distribution can be written as

• Mixture Models

Observed modelStructural modelMixing distribution

• Some families of statistical distributionsPropertiesMomentsCumulative distribution functions (see appendix)Gini coefficients (G)Incomplete momentsMixture modelsHazard functions (Duration dependence)

• Hazard functionsDefinition:

Let denote the pdf of a spell (S) or duration of an event. is the probability that that S>s.The corresponding hazard function is defined by

which can be thought of as representing the rate or likelihood that a spell will be completed after surviving s periods.

• Hazard functionsApplications:

Does the probability of ending a strike, unemployment spell, expansion, or stock run depend on the length of the strike, unemployment spell, or of the run? With unemployment,A job seeker might lower their reservation wage and become more likely to find a job Increasing hazard functionHowever, if being out of work is a signal of damaged goods, the longer they are out of work might decrease employment opportunities Decreasing hazard function.An alternative example might deal with attempts to model the time between stock trades. Engle and Russell (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66: 1127-1162Hazard function of time between trades is decreasing as t increases or the longer the time between trades the less likely the next trade will occur.

• Hazard functionsApplications:

BubblesMcQueen and Thorley (1994) Bubbles, stock returns, and duration dependence. Journal of Financial and Quantitative Analysis, 29:379-401Efficient markets hypothesis, stock runs should not exhibit duration dependence (constant hazard function)McQueen and Thorley argue that asset prices may contain bubbles which grow each period until they burst causing the stock market to crash. Hence, bubbles cause runs of positive stock returns to exhibit duration dependencethe longer the run the less likely it will end (decreasing hazard function), but runs of negative stock returns exhibit no duration dependenceGrimshaw, McDonald, McQueen, and Thorley. 2005, Communications in StatisticsSimulation and Computation, 34: 451-463.What model should we use to characterize duration dependence?Exponentialconstant Gammathe hazard function can increase, decrease, or be constantWeibullthe hazard function can increase, decrease, or be constantGeneralized Gamma: the hazard function can be increasing, decreasing, constant, -shaped, or -shaped

• Hazard functions Possible shapes for the GG hazard functions

• Statistical Distributions

Introduction Some families of statistical distributionsFamilies PropertiesModel selectionRegression applicationsCensored regressionQualitative response modelsOption pricing VaR (value at risk)Conclusion

• Some families of statistical distributionsModel selectionGoodness of fit statisticsLog-likelihood values

for individual data

for grouped data

Partition the data into g groups,

Empirical frequency:

Theoretical frequency:

• Model Selection

Goodness of fit statisticsLog-likelihood valuesPossible Measures

• Model Selection

Goodness of fit statisticsLog-likelihood valuesPossible MeasuresAkaike Information Criterion (AIC)

A tool for model selectionAttaches a penalty to over-fitting a model

• Model Selection

Goodness of fit statisticsTes