GTL Regression: A Linear Model with Skewed and Thick-Tailed - PowerPoint PPT Presentation

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion GTL Regression: A Linear Model with Skewed and Thick-Tailed Disturbances Wim Vijverberg & Takuya Hasebe CUNY-Graduate Center & Sophia University 24 February 2015

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Linear model seems reasonable y i = X ′ i β + ǫ X s are plausibly exogenous ǫ i is plausibly iid : So ... what are you waiting for? OLS is BLUE B stands for BEST L stands for LINEAR U stands for UNBIASED : So ... why write a paper on just such a model? OLS is not always BUE

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion The Classical Linear Regression Model assumes y i = X ′ i β + ǫ X is of full rank X ′ X / n converges to a finite matrix (Grenander conditions) ǫ i | X is iid(0 , σ 2 ) What is not covered? Poorly behaved X s Higher moments of ǫ What does this paper offer? A flexible distributional assumption for ǫ An estimator that is efficient relative to OLS A test for normality that, if normality is rejected, points to this flexible distribution

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Agenda Define this flexible distribution: Generalized Tukey Lambda (GTL) Define the GTL regression model Describe the GTL estimator of the linear model Discuss alternative approaches Monte Carlo results Applications Extensions

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Representing statistical distribution: N ( µ, σ 2 ) Let u ∼ U [0 , 1] Probability density function (PDF) 1 − 1 � ǫ − µ � 2 � � φ ( ǫ ) = (2 πσ 2 ) 1 / 2 exp 2 σ Cumulative distribution function (CDF) � ǫ − µ � u = Φ σ Link (quantile) function ǫ = G ( u ) = µ + σ Φ − 1 ( u ) The CDF has no analytical representation.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Tukey Lambda distribution and its extensions Tukey lambda distribution: defined by a link (quantile) function � u λ − (1 − u ) λ � ǫ = G ( u ) = µ + σ , Its CDF is F ( ǫ ) = u = G − 1 ( ǫ ) In general, the CDF has no analytical solution Given u = G − 1 ( ǫ ), its PDF is 1 f ( ǫ ) = G ′ ( G − 1 ( ǫ )) u λ − 1 + (1 − u ) λ − 1 � G ′ ( u ) � = σλ

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Tukey Lambda distribution and its extensions Tukey Lambda distribution: Tukey (1960) � u λ − (1 − u ) λ � ǫ = G ( u ) = µ + σ Generalized Lambda Distribution GLD (GLD-RS: Ramberg and Schmeiser, 1974) � � u λ 3 − (1 − u ) λ 4 ǫ = G ( u ) = µ + σ Generalized Tukey Lambda Distribution GTL (GLD-FMKL: Freimer, Mudholkar, Kollia, Lin, 1978) � u α − δ − 1 − (1 − u ) α + δ − 1 � ǫ = Q ( u ) = µ + σ α − δ α + δ

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion The impracticality of GLD The parameter space of the Generalized Lambda Distribution GLD has gaps � � u λ 3 − (1 − u ) λ 4 ǫ = G ( u ) = µ + σ Source: Karian et al. (1996)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Generalized Tukey Lambda distribution GTL is defined by a link (quantile) function � u α − δ − 1 − (1 − u ) α + δ − 1 � ǫ = G ( u ) = µ + σ , α − δ α + δ In general, its CDF has no analytical solution F ( ǫ ) = u = G − 1 ( ǫ ) Given u = G − 1 ( ǫ ), its PDF is u α − δ − 1 + (1 − u ) α + δ − 1 � − 1 f ( ǫ ) = σ − 1 � GTL has no gaps in its parameter space

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Flexibility of the GTL distribution GTL link function: � u α − δ − 1 − (1 − u ) α + δ − 1 � ǫ = G ( u ) = µ + σ α − δ α + δ where µ is a location parameter (but is not the mean) where σ is a scale parameter (but is not the standard deviation) where α is a shape parameter controlling tail thickness where δ is a shape parameter controlling skewness GTL in canonical form: set µ = 0 and σ = 1. ǫ = ǫ − µ ǫ Standardized GTL: ˜ σ ǫ General GTL: r = τ 1 + τ 2 ǫ

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Flexibility of the GTL distribution The parameter α controls thickness of tails. 1 0.9 0.8 0.7 α=0.35, δ=0.00 0.6 α=0.00, δ=0.00 0.5 α= - 0.35, δ=0.00 0.4 0.3 0.2 0.1 0 -3 -2 -1 0 1 2 3 The smaller α is, the thicker tails are. ( δ is fixed at 0.)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Flexibility of the GTL distribution The parameter δ controls direction of skewness. 0.9 0.8 0.7 α=0.00, δ= -0.33 0.6 α=0.00, δ=0.00 α=0.00, δ=0.33 0.5 0.4 0.3 0.2 0.1 0 -3 -2 -1 0 1 2 3 δ = 0 ( black ) ⇒ symmetric δ > 0 ( red ) ⇒ left-skewed δ < 0 ( blue ) ⇒ right-skewed ( α is fixed at 0.)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Flexibility of the GTL distribution A GTL distribution exhibits a variety of shapes. 0.40 α = 0.1436 α = -0.20 δ = 0.0 δ = -0.15 0.30 Normal 0.20 GTL 0.10 0.00 -4 -2 0 2 4 α = 0.25 α = 1.50 δ = 0.40 δ = -0.22

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Flexibility of GTL distribution Relative to other distributions, GTL nests: Logistic distribution α = 0 , δ = 0 Exponential distribution α − δ → ∞ and α + δ = 0 Uniform distribution α = 1 , δ = 0 ” α = 2 , δ = 0 ” ( α, δ ) = ( α, α − 1) for α → ∞ ” ( α, δ ) = ( α, − α + 1) for α → ∞ and GTL approximately nests Normal distribution α = 0 . 1436 , δ = 0 Student’s t ( r ) for r ≥ 1 − 0 . 8416 ≤ α ≤ 0 . 1436 , δ = 0 Gumbel α = 0 . 1422 , δ = − 0 . 2290 χ 2 , Weibull, etc.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion GTL Approximation of Well-known Distributions Normal Distribution 0.4 GTL Normal 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −4 −3 −2 −1 0 1 2 3 4

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion GTL Approximation of Well-known Distributions Student’s t Distribution (degrees of freedom equal to 5) 0.4 GTL t(5) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −5 0 5

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion GTL Approximation of Well-known Distributions χ 2 Distribution (degrees of freedom equal to 10) 0.1 GTL 0.09 χ 2 (10) 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 5 10 15 20 25 30 35 40

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion GTL Approximation of Well-known Distributions Weibull Distribution (1,2) 0.9 GTL Weibull(1,2) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Characteristics of the GTL distribution GTL link function: � u α − δ − 1 − (1 − u ) α + δ − 1 � ǫ = G ( u ) = µ + σ α − δ α + δ Range: ǫ L ≤ ǫ ≤ ǫ U Lower bound: ǫ L = −∞ if α − δ ≤ 0 1 ǫ L = − α + δ if α − δ > 0 Upper bound: ǫ U = ∞ if α + δ ≤ 0 1 ǫ U = α + δ if α + δ > 0 The k th moment exists if min ( α − δ, α + δ ) > − 1 k

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion Characteristics of the GTL distribution Range: ǫ L ≤ ǫ ≤ ǫ U The k th moment exists if min ( α − δ, α + δ ) > − 1 k 1.25 1.25 1 1 0.75 0.75 0.5 0.5 0.25 0.25 Left bound on μ Left bound on μ Left bound on ε L finite 0 0 Left bound on σ Left bound on σ Left bound on ε R finite -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 0.5 0.75 1 1.25 Left bound on κ4 Left bound on κ4 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 1.25 -1 -1 1 -1.25 -1.25

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion The GTL regression model Model: X ′ y i = i β + σǫ i ǫ i ∼ iid GTL ( α, δ ) The GTL distribution is stated in its canonical form. We allow ( α, δ ) ∈ R 2 We do not impose E [ ǫ i ] = 0 since E [ ǫ i ] does not exist for all ( α, δ ). σ is merely a scaling parameter. If min ( α − δ, α + δ ) > − 1 2 , the variance of σǫ i exists. If min ( α − δ, α + δ ) ≤ − 1 2 , this variance does not exist.