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 yi = X ′

i β + ǫ

Xs are plausibly exogenous ǫi 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 yi = X ′

i β + ǫ

X is of full rank X ′X/n converges to a finite matrix (Grenander conditions) ǫi|X is iid(0, σ2) What is not covered? Poorly behaved Xs 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, σ2) Let u ∼ U[0, 1] Probability density function (PDF) φ(ǫ) = 1 (2πσ2)1/2 exp

• − 1

2 ǫ − µ σ 2 Cumulative distribution function (CDF) u = Φ ǫ − µ σ

• Link (quantile) function

ǫ = G(u) = µ + σΦ−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 ǫ = G(u) = µ + σ

• uλ − (1 − u)λ

, 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

f (ǫ) = 1 G ′(G −1(ǫ)) G ′(u) = σλ

• uλ−1 + (1 − u)λ−1

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Tukey Lambda distribution and its extensions Tukey Lambda distribution: Tukey (1960) ǫ = G(u) = µ + σ

• uλ − (1 − u)λ

Generalized Lambda Distribution GLD (GLD-RS: Ramberg and Schmeiser, 1974) ǫ = G(u) = µ + σ

• uλ3 − (1 − u)λ4
• Generalized Tukey Lambda Distribution GTL (GLD-FMKL:

Freimer, Mudholkar, Kollia, Lin, 1978) ǫ = Q(u) = µ + σ uα−δ − 1 α − δ − (1 − u)α+δ − 1 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 ǫ = G(u) = µ + σ

• uλ3 − (1 − u)λ4
• 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 ǫ = G(u) = µ + σ uα−δ − 1 α − δ − (1 − u)α+δ − 1 α + δ

• ,

In general, its CDF has no analytical solution F(ǫ) = u = G −1(ǫ) Given u = G −1(ǫ), its PDF is

f (ǫ) = σ−1 uα−δ−1 + (1 − u)α+δ−1−1

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: ǫ = G(u) = µ + σ uα−δ − 1 α − δ − (1 − u)α+δ − 1 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. Standardized GTL: ˜ ǫ = ǫ−µǫ

σǫ

General GTL: r = τ1 + τ2ǫ

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Flexibility of the GTL distribution The parameter α controls thickness of tails.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 3
• 2
• 1

1 2 3

α=0.35, δ=0.00 α=0.00, δ=0.00 α=-0.35, δ=0.00

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

• 3
• 2
• 1

1 2 3

α=0.00, δ=-0.33 α=0.00, δ=0.00 α=0.00, δ=0.33

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.

0.00 0.10 0.20 0.30 0.40

• 4
• 2

2 4

Normal GTL α = 0.1436 δ = 0.0 α = -0.20 δ = -0.15 α = 0.25 δ = -0.22 α = 1.50 δ = 0.40

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 χ2, Weibull, etc.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

GTL Approximation of Well-known Distributions Normal Distribution

−4 −3 −2 −1 1 2 3 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 GTL Normal

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)

−5 5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 GTL t(5)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

GTL Approximation of Well-known Distributions χ2 Distribution (degrees of freedom equal to 10)

5 10 15 20 25 30 35 40 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 GTL χ 2(10)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

GTL Approximation of Well-known Distributions Weibull Distribution (1,2)

0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 GTL Weibull(1,2)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Characteristics of the GTL distribution GTL link function: ǫ = G(u) = µ + σ uα−δ − 1 α − δ − (1 − u)α+δ − 1 α + δ

• Range: ǫL ≤ ǫ ≤ ǫU

Lower bound:

ǫL = −∞ if α − δ ≤ 0 ǫL = −

1 α+δ if α − δ > 0

Upper bound:

ǫU = ∞ if α + δ ≤ 0 ǫU =

1 α+δ if α + δ > 0

The kth 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: ǫL ≤ ǫ ≤ ǫU The kth moment exists if min(α − δ, α + δ) > − 1

k

1 1.25 0.5 0.75 1 1.25 Left bound on μ Left bound on σ Left bound on κ4

• 1.25
• 1
• 0.75
• 0.5
• 0.25

0.25 0.5 0.75 1 1.25

• 1.25
• 1
• 0.75
• 0.5
• 0.25

0.25 0.5 0.75 1 1.25 Left bound on εL finite Left bound on εR finite

• 1.25
• 1
• 0.75
• 0.5
• 0.25

0.25 0.5 0.75 1 1.25

• 1.25
• 1
• 0.75
• 0.5
• 0.25

0.25 0.5 0.75 1 1.25 Left bound on μ Left bound on σ Left bound on κ4

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

The GTL regression model Model: yi = X ′

i β + σǫi

ǫi ∼ iid GTL(α, δ) The GTL distribution is stated in its canonical form. We allow (α, δ) ∈ R2 We do not impose E[ǫi] = 0 since E[ǫi] does not exist for all (α, δ). σ is merely a scaling parameter. If min(α − δ, α + δ) > − 1

2, the variance of σǫi exists.

If min(α − δ, α + δ) ≤ − 1

2, this variance does not exist.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

MLE estimation Define θ = (β′, α, δ)′. Define ui = G −1( 1

σ(yi − x′ i β))

The GTL estimator (GTLE) maximizes the following likelihood function: L(y, x, θ) = −n ln σ −

n

• i=1

ln G ′(ui) = −n ln σ −

n

• i=1

ln

• uλ1−1

i

+ (1 − ui)λ2−1 . The function G −1 is numerically evaluated with a hybrid algorithm that mixes the bisection and Newton-Raphson algorithms (Press et al., 1986, Numerical Recipes).

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Consistency and Asymptotic Normality Define θ = (β′, α, δ)′. Let θ0 be the true parameter vector. Let Θ be the parameter space. Assumptions A.1 Θ is compact with σ > 0. A.2 (α, δ) are such that α − δ < 1

2 and α + δ < 1 2.

A.3 θ0 ∈ int(Θ). A.4 (i) x ∈ X ⊂ Rk. (ii) x is weakly exogenous. (iii) x has finite moments up to the fourth order. (iv) E[xx′] has full rank. A.5 (i) ǫ is independently and identically GTL(α, δ)-distributed, with the GTL distribution stated in canonical form. (ii) ǫ is independent of x.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Consistency and Asymptotic Normality Theorem Given Assumptions 1, 3, 4, and 5, ˆ θ is a consistent estimator of θ0 for all θ0 ∈ Θ except for those θ0 for which (α0, δ0) = (1, 0) or (α0, δ0) = (2, 0). Theorem Given the linear regression model yi = X ′

i β + σǫi and Assumptions

1-5, ˆ θ is asymptotically normally distributed N(θ0, V0), where V0 = n−1A0 is estimated as V (ˆ θ) = −

• n
• i=1

∇θθℓi(ˆ θ) −1 where ℓi(θ) = ln gy(y|xi; θ) and gy(y|xi; θ) is the conditional density of y.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Consistency and Asymptotic Normality The additional assumption imposed to obtain asymptotic normality is Assumption 2: A.2 (α, δ) are such that α − δ < 1

2 and α + δ < 1 2.

• 1.25
• 1
• 0.75
• 0.5
• 0.25

0.25 0.5 0.75 1

• 1.25
• 1
• 0.75
• 0.5
• 0.25

0.25 0.5 Left bound on μ Left bound on σ Right bound on Asy.N.

δ α

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

OLS vs GTL OLS is equivalent to quasi-MLE assuming iid N(0, σ2) disturbances FOC under QMLE-normal yield

n

• i=1

˜ ǫixi = 0 FOC under MLE-GTL yield

n

• i=1

1 σ G ′′(ui) (G ′(ui))2 xi = 0 ≡

n

• i=1

w(˜ ǫi)˜ ǫixi ≡

n

• i=1

ψ(˜ ǫi)xi where ui = G −1(σǫ˜ ǫi + µǫ).

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

OLS vs GTL ψ(˜ ǫ) for selected values of α and δ

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10

• 5
• 3
• 1

1 3 5 OLS α = 0.1436, δ = 0

Comparing N(0, 1) with GTL(0.1436, 0)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

OLS vs GTL ψ(˜ ǫ) for selected values of α and δ

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5

• 5
• 3
• 1

1 3 5 OLS α = 0 α = -0.15 α = -0.30 α = -0.45

Increasingly thick tails: δ = 0

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

OLS vs GTL ψ(˜ ǫ) for selected values of α and δ

• 10
• 6
• 2

2 6 10

• 5
• 3
• 1

1 3 5 OLS δ = 0.15 δ = 0 δ = -0.15

Thin tails with skewness: α = 0.30

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

OLS vs GTL ψ(˜ ǫ) for selected values of α and δ

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5

• 5
• 3
• 1

1 3 5 OLS δ = 0.15 δ = 0 δ = -0.15

Thick tails with skewness: α = −0.30

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Test for normality – even when running OLS Wald test: general GTL(α, δ) model Likelihood ratio test: general GTL(α, δ) + restricted GTL(0.1436, 0) models Lagrange multiplier test: restricted GTL(0.1436, 0) model Vuong test: general GTL(α, δ) + “general” N(0, σ2) models

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Test for normality – even when running OLS Lagrange multiplier test: restricted GTL(0.1436, 0) model Could we insert OLS residuals into the LM formula instead of estimating a restricted GTL(0.1436, 0) model? Range violations may occur: p(|ǫ| > 4.819|ǫ ∼ N(0, 1)) = 1.44 × 10−6. Thus, the probability of any violation in sample of n = 5000 equals 0.0072. (α, δ) = (0.1436, 0) vs N(0, σ2): power = 0.074 (α, δ) = (0.1436, 0) vs (α, δ) = (0.10, 0): power = 0.849 (α, δ) = (0.1436, 0) vs (α, δ) = (0.20, 0): power = 0.995 (α, δ) = (0.1436, 0) vs (α, δ) = (0.1436, 0.02): power = 0.770

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Prediction Model for i = 1, . . . , n: yi = X ′

i β + ǫi ≡ µi + ǫi where we impose E[ǫj] = 0

Forecast type 1 for j > n: ˜ µj = ˜ X ′

j ˆ

βGTL ˜ µj ∼ AN( ˜ X ′

j β, ˜

X ′

j Var( ˆ

βGTL) ˜ Xj) Forecast type 2 for j > n: ˜ yj = ˜ X ′

j ˆ

βGTL ˜ ej ≡ ˜ yj − yj = ˜ X ′

j (ˆ

βGTL − β) − ǫj ˜ ej ∼ AN( ˜ X ′

j β, ˜

X ′

j Var(ˆ

βGTL) ˜ Xj) + GTL(ˆ α, ˆ δ)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Monte Carlo results Data generating process: yi = β1 + β2x2i + β3x3i + σǫi for i = 1, . . . , n n = 250 or 5000 x2i is a standard normal random variate x3i is a χ2(5) random variate that is standardized to have mean 0 and variance x2i and x3i are independent ǫi is a canonical GTL(α, δ) variate for various combinations of α and δ. β1 = β2 = β3 = 1 Scaling parameter σ is chosen such that the range of the GTL distribution from the 0.1% quantile to the 99.9% quantile has the same length as that of a normal N(0, 2) distribution.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Monte Carlo results

DGP RMSE of OLS RMSE of GTLE α δ σ β2 β3 β1 β2 β3 β1 A: GTL as an approximation of the standard normal distribution, N = 250 0.1436 0.00 1.188 0.107 0.110 0.110 0.108 0.111 0.126 B: Various GTL distributions, small sample, N = 250 0.33

• 0.10

1.477 0.106 0.108 0.199 0.089 0.094 0.132 0.33 0.00 1.508 0.107 0.109 0.109 0.097 0.101 0.133 0.33 0.10 1.477 0.106 0.108 0.200 0.091 0.093 0.132

• 0.33
• 0.10

0.454 0.133 0.144 0.255 0.066 0.068 0.066

• 0.33

0.00 0.482 0.118 0.122 0.124 0.071 0.073 0.071

• 0.33

0.10 0.454 0.127 0.126 0.239 0.067 0.069 0.067

• 0.67
• 0.25

0.136 1.737 1.667 1.932 0.025 0.026 0.025

• 0.67

0.00 0.202 0.336 0.337 0.341 0.038 0.040 0.037

• 0.67

0.25 0.136 1.046 0.823 1.126 0.025 0.026 0.025

• 1.00
• 0.50

0.024 112.585 89.227 103.243 0.005 0.005 0.005

• 1.00

0.00 0.079 2.503 2.217 2.408 0.018 0.019 0.018

• 1.00

0.50 0.024 39.742 27.930 35.099 0.005 0.005 0.005

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Monte Carlo results QQ plots of OLS estimators of β2 and β3, n = 250

0.4 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 OLS estimate of β2 Normal quantile 95% upper 95% lower β2 45 degree 1 0.4 0.6 0.8 1 1.2 1.4 1.6 0.6 0.8 1 1.2 1.4 OLS estimate of β3 Normal quantile 95% upper 95% lower β3 45 degree 1

ˆ β2 for (α, δ) = (0.1436, 0) ˆ β3 for (α, δ) = (0.1436, 0)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Monte Carlo results QQ plots of OLS estimators of β2 and β3, n = 250

• 0.2

0.2 0.6 1 1.4 1.8 0.5 0.7 0.9 1.1 1.3 1.5 OLS estimate of β2 Normal quantile 95% upper 95% lower β2 45 degree 1 0.5 1 1.5 2 2.5 0.5 0.7 0.9 1.1 1.3 1.5 OLS estimate of β3 Normal quantile 95% upper 95% lower β3 45 degree 1

ˆ β2 for (α, δ) = (−0.33, 0) ˆ β3 for (α, δ) = (−0.33, 0)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Monte Carlo results QQ plots of OLS estimators of β2 and β3, n = 250

• 35
• 25
• 15
• 5

5 15 25

• 5
• 3
• 1

1 3 5 7 OLS estimate of β2 Normal quantile 95% upper 95% lower β2 45 degree 1

• 25
• 15
• 5

5 15 25

• 5
• 3
• 1

1 3 5 7 OLS estimate of β3 Normal quantile 95% upper 95% lower β3 45 degree 1

ˆ β2 for (α, δ) = (−0.67, 0) ˆ β3 for (α, δ) = (−0.67, 0)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Monte Carlo results QQ plots of GTL estimators of β2 and β3, n = 250

0.85 0.9 0.95 1 1.05 1.1 1.15 0.9 0.95 1 1.05 1.1 GTL estimate of β2 Normal quantile 95% upper 95% lower β2 45 degree 1 0.85 0.9 0.95 1 1.05 1.1 1.15 0.9 0.95 1 1.05 1.1 GTL estimate of β3 Normal quantile 95% upper 95% lower β3 45 degree 1

ˆ β2 for (α, δ) = (−0.67, 0) ˆ β3 for (α, δ) = (−0.67, 0)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Monte Carlo results Ratio of average estimated variance to Monte Carlo variance

DGP OLS GTL α δ σ β2 β3 β1 β2 β3 β1 A: Various GTL distributions, small sample, N = 250

• 0.33

0.00 0.482 1.073 1.004 0.960 1.005 0.945 1.024

• 0.33

0.10 0.454 1.037 1.071 1.018 0.989 0.933 1.016

• 0.67

0.00 0.202 0.960 0.994 0.931 0.997 0.922 1.045

• 0.67

0.25 0.136 0.794 1.368 1.042 0.987 0.912 1.032

• 1.00

0.00 0.079 0.830 1.132 0.893 0.978 0.896 1.054

• 1.00

0.50 0.024 0.777 1.707 0.996 0.991 0.903 1.033 B: Various GTL distributions, large sample, N = 5000

• 0.33

0.00 0.482 1.035 1.006 0.897 1.034 1.038 0.997

• 0.33

0.10 0.454 1.076 0.999 0.921 1.042 1.035 1.003

• 0.67

0.00 0.202 1.307 1.446 0.980 1.019 1.066 1.005

• 0.67

0.25 0.136 1.529 2.627 1.020 1.035 1.034 1.011

• 1.00

0.00 0.079 2.727 0.966 0.998 1.008 1.080 1.017

• 1.00

0.50 0.024 20.456 0.382 1.005 1.024 0.998 1.014

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Alternatives Choose another distribution that permits skewness and thick tails.

Skewed-t distribution Skewed Generalized Error distribution “Stable” distribution (has no second moment unless it coincides with N(0, σ2)) Mixture of normal distributions

Parameter-intensive: 3 normals use 6 parameters Allows more than one mode Overfitting

Choose another estimation approach

Compute Var(ˆ βOLS) by sandwich estimator (QMLE) or bootstrap Estimate by Least Absolute Deviation Use the M-estimation approach (QMLE) Estimate by data trimming or winsorizing

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Alternatives Choose a non-parametric estimation approach

This ignores the linearity of the location assumption: yi = X ′

i β + ǫi

A nonparametric (kernel) estimator is essentially a weighted average of y values with Xs in the neighborhood of Xi. The weights account for the distance from Xi, not for the thickness

• f the tails of the distribution of ǫi.

Thus, the nonparametric estimator is even more unstable than OLS.

Choose a semi-parametric estimation approach: single-index models

This makes the linear location assumption: yi = X ′

i β + ǫi

Assumption about the distribution of ǫi: E[ǫi|X] = 0, E[ǫ2

i |X]

must be finite. What else? Anything about higher-order moments?

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Summary Tests for normality in six applications

Application n Wald LR LM LM V Vuong lnWage males 54687 1127.5 1303.4 583.7 42

• 11.16

MORG US lnWage females 46045 1339.4 1618.2 670.4 52

• 11.80

MORG US Housing prices 546 8.5 11.4 8.17

• 1.69

Canada Speeding tickets, 31674 791.1 69015.0 5072.0 8

• 52.67

Massachusetts Trade cr. and div. 37983 1326.8 4374.6 1299.0 24

• 24.96

World CAPM, 25 portf 635 (25) (17) Fama/French, US Note: Critical value for Wald, LR, LM = 9.21 (1%), 5.99 (5%). Note: Critical value for Vuong = -2.33 (1% one-tailed), -1.645 (5% one-tailed)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Summary Estimates of (α, δ) in six applications

OLS residuals ˆ α ˆ δ Application Skewness Kurtosis Est. Sr.Err. Est. St.Err. lnWage males

• 0.23

4.36 0.030 0.004 0.029 0.002 lnWage females

• 0.19

4.89 0.003 0.004 0.013 0.003 Housing prices

• 0.19

3.51 0.010 0.050 0.034 0.025 Speeding tickets,

• 1.20

5.07

• 1.761

0.083 1.308 0.048 Trade cr. and div.

• 0.71

4.90

• 0.078

0.008 0.139 0.005 CAPM, 25 portf. graph graph

Recall: For a normal distribution, (α, δ) = (0.1436, 0). Recall: For a logistic distribution, (α, δ) = (0, 0)

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Summary Difference between normal and logistic (standardized)

Left bound on σ Left bound on κ4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

• 4
• 3
• 2
• 1

1 2 3 4 Normal Logistic

Range Normal Logistic −1000 < −2.375 0.0088 0.0133 −2.375 < −0.682 0.2395 0.2123 −0.682 < 0.682 0.5035 0.5488 0.682 < 2.375 0.2395 0.2123 2.375 < 1000 0.0088 0.0133

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Trade creation and trade diversion, 1960-2005 Trade creation estimates

OLS GTLE Estimate Stan.Err. Estimate Stan.Err.

ˆ βOLS − ˆ βGTL ˆ βGTL SEGTL SEOLS

Trade creation dummy variables tc.nafta 0.371 0.252 0.500 0.220

• 0.257

0.874 tc.eu 0.427 0.115 0.196 0.098 1.182 0.853 tc.efta 0.685 0.129 0.518 0.116 0.323 0.896 tc.eea 0.179 0.095 0.261 0.079

• 0.312

0.837 tc.caricom 2.823 0.513 2.417 0.459 0.168 0.894 tc.ap 0.828 0.186 0.804 0.159 0.030 0.852 tc.mercosur 1.086 0.306 1.035 0.315 0.049 1.030 tc.asean 0.467 0.216 0.492 0.185

• 0.051

0.858 tc.anzcerta 0.969 0.141 0.748 0.130 0.295 0.920 tc.apec 1.599 0.095 1.291 0.085 0.238 0.889 tc.laia

• 0.133

0.141

• 0.432

0.134

• 0.691

0.950 tc.cacm 2.314 0.150 1.931 0.139 0.198 0.927 tc.bilateralPTA 0.110 0.128 0.098 0.117 0.122 0.916 Dependent variable: Log of bilateral imports.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Trade creation and trade diversion, 1960-2005 Trade diversion estimates

OLS GTLE Estimate Stan.Err. Estimate Stan.Err.

ˆ βOLS − ˆ βGTL ˆ βGTL SEGTL SEOLS

Trade diversion dummy variables td.nafta 0.151 0.073 0.081 0.061 0.875 0.841 td.eu 0.651 0.051 0.434 0.047 0.500 0.908 td.efta 0.376 0.059 0.202 0.054 0.866 0.921 td.eea

• 0.142

0.048

• 0.101

0.043 0.409 0.894 td.caricom

• 0.577

0.100

• 0.539

0.097 0.071 0.972 td.ap 0.105 0.074 0.115 0.068

• 0.088

0.925 td.mercosur 0.030 0.073

• 0.019

0.063

• 2.554

0.869 td.asean 0.474 0.070 0.395 0.061 0.198 0.869 td.anzcerta

• 0.759

0.098

• 0.657

0.086 0.156 0.879 td.apec 0.439 0.049 0.341 0.042 0.288 0.871 td.laia

• 0.561

0.060

• 0.533

0.054 0.051 0.913 td.cacm

• 0.174

0.078

• 0.120

0.074 0.451 0.945 td.bilateralPTA

• 0.292

0.054

• 0.275

0.045 0.064 0.832 Dependent variable: Log of bilateral imports.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Trade creation and trade diversion, 1960-2005 Distributions estimated for the trade creation and diversion equation

0.05 0.1 0.15 0.2 0.25 0.3

• 6
• 4
• 2

2 4 6 Normal GTL

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Speeding tickets, Massachussets, 2001

OLS GTLE Estimate Stan.Err. Estimate Stan.Err.

ˆ βOLS − ˆ βGTL ˆ βGTL SEGTL SEOLS

ln(Mph over) 0.8649 0.0144 1.2332 0.0057

• 0.299

0.396 OutTown 0.0071 0.0107 0.0001 0.0006 119.524 0.054 OutState 0.0353 0.0070 0.0007 0.0004 52.026 0.062 Afr.American

• 0.0207

0.0095

• 0.0010

0.0004 20.402 0.045 Hispanic 0.0216 0.0101

• 0.0003

0.0007

• 85.838

0.064 Female

• 0.0564

0.0375 0.0003 0.0017

• 204.497

0.046 ln(Age) 0.0002 0.0073 0.0015 0.0007

• 0.878

0.092 Female × ln(Age) 0.0092 0.0105 0.0000 0.0005 946.064 0.049 ln(CourtDist) 0.0158 0.0035

• 0.0002

0.0003

• 81.445

0.074 ln(Pvalue.pc)

• 0.0043

0.0218

• 0.0007

0.0011 5.209 0.050 OR 0.0202 0.0700 0.0058 0.0020 2.502 0.028 OR × OutTown 0.0083 0.0630

• 0.0082

0.0019

• 2.008

0.031 OR × ln(CourtDist) 0.0019 0.0089 0.0003 0.0004 5.139 0.042 SP 0.0299 0.2949

• 0.0016

0.0155

• 19.662

0.053 SP × OutTown 0.0258 0.0187 0.0000 0.0008

• 4569.156

0.041 SP × ln(CourtDist) 0.0059 0.0040 0.0007 0.0004 7.853 0.101 SP × ln(Pvalue.pc)

• 0.0018

0.0265

• 0.0002

0.0014 9.790 0.051 SP × OR

• 0.0164

0.0319

• 0.0018

0.0017 8.270 0.053 Intercept 2.2716 0.2551 1.4817 0.0190 σ 0.0030 0.0005 α

• 1.7612

0.0833 δ 1.3083 0.0483 log Likelihood

• 9440.55

25066.89 (Absolute) Average 360.723 0.075 Dependent variable: Log of amount of fine.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Speeding tickets, Massachussets, 2001 Distributions estimated for the speeding ticket equation

The GTL density is top-truncated; it peaks at 33.23 at ǫ = 0.0075.

1 2 3

• 1.5
• 1
• 0.5

0.5 1 1.5 Normal GTL

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

CAPM, basic and extended, 1960-2012 (monthly) Estimates of (α, δ), basic CAPM

• 0.12
• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

0.02

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

δ α

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

CAPM, basic and extended, 1960-2012 (monthly) Estimates of (α, δ), extended CAPM

• 0.12
• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

0.02

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

δ α

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

CAPM, basic and extended, 1960-2012 (monthly) Normal and GTL densities for CAPM and Extended CAPM models Portfolio ME:1, B/M:1, ˆ α = −0.135, ˆ δ = −0.044

0.05 0.1 0.15 0.2

• 18
• 12
• 6

6 12 18 CAPM: normal CAPM: GTL ECAPM: normal ECAPM: GTL

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Financial risk management and GTL-GARCH

Given the GTL link function ǫ = G(u), it is straightforward to compute Value at Risk of a variable r = τ1 + τ2ǫ: VaRp = τ1 + τ2G(p) Given p, the Expected Shortfall is computed analytically as ESp = τ1 + τ2ESǫ(p) ESǫ(p) = E (ǫ | ǫ ≤ VaRǫ) = − 2δ α2 − δ2 + pα−δ (α − δ) (α − δ + 1)+ (1 − p)α+δ+1 − 1 p(α + δ)(α + δ + 1) : A GARCH(q1, q2) process: rt = σt ˜ ǫt σ2

t

= ω +

q2

• i=1

ηir 2

t−i + q1

• j=1

βjσ2

t−j,

t ∈ Z = {0, ±1, . . .} :

Source: Chu-Ping Vijverberg, Wim Vijverberg, and S¨ uleyman Ta¸ spinar, “Financial Risk Measurement with the Generalized Tukey Lambda Distribution.”

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

GTL and Binary Choice Models With the GTL(α, δ) family of distributions, a family of binary choice models arises: y∗

i

= X ′

i β + ǫi

yi = I(y∗

i ≥ 0) = I(ǫi ≥ −X ′ i β)

ǫi ∼ GTL(α, δ) This “pregibit” family nests exactly or approximately: Probit Logit Gossit or robit Loglog and cloglog Linear probability model

Source: Wim Vijverberg and Chu-Ping Vijverberg, “Pregibit: A Family of Binary Choice Models.” Empirical Economics, forthcoming.

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Extending Heckman’s selection model A GTL(α, δ) density is univariate. With a copula function, two or more univariate densities can be joined to create a multivariate density. Thus, combine two univariate GTL densities to create a flexible generalization of the Heckman selection model:

1 Selection equation: si = 1(Ziγ + νi > 0) 2 Outcome equation: yi = Xi ′β + σǫi if si = 1 3 νi ∼ GTL(αs, δs) 4 ǫi ∼ GTL(α1, δ1) 5 F(νi, ǫi) = C(Fν(νi), Fǫ(ǫi))

This can be expanded with a second outcome equation, where one

• r the other outcome is observed.

:

Source: Wim Vijverberg and Takuya Hasebe, “A Flexible Sample Selection Model: A GTL-Copula Approach.”

Intro GTL distribution GTL regression What about OLS? Alternatives Applications Extensions Conclusion

Conclusion GTL regression: Deals with non-normal disturbances in a flexible way Efficient relative to OLS when disturbances are non-normal Time-efficient relative to semi-parametric estimation methods—and statistically efficient if the GTL distribution captures the data generating process well enough Permits disturbances that don’t even have a mean or variance (or skewness or kurtosis) LM test can be used as a post-estimation test after running OLS Can be extended to include heteroskedasticity, serial correlation, GARCH-type behavior Might also increase efficiency of IV and GMM estimators