跨度回归 跨度回归，偏度回归 偏度回归 与峰度回归及 Stata 应用 与峰度回归及 Stata 应用 Spread Regression Skewness Regression and Spread Regression, Skewness Regression and Kurtosis Regression with Applications in Stata 陈强 山东大学经济学院 qiang2chen2@126.com 公众号 / 网站： econometrics ‐ stata 公众号 / 网站： econometrics stata 网易云课堂： http://study.163.com/u/metrics 2020/8/11 1
Abstract Abstract • Quantile regression provides a powerful tool to • Quantile regression provides a powerful tool to study the effects of covariates on key quantiles of conditional distribution Yet we often lack a conditional distribution. Yet we often lack a general picture about how covariates affect the overall shape of conditional distribution overall shape of conditional distribution. • We propose quantile ‐ based spread regression, W til b d d i skewness regression and kurtosis regression to quantify the effects of covariates on the spread, tif th ff t f i t th d skewness and kurtosis of conditional distribution. 2020/8/11 2
Abstract (cont ) Abstract (cont.) • This methodology is then applied to U.S. census data with substantive findings. g • We demonstrate the implementation of W d h i l i f spread, skewness and kurtosis regressions with official Stata command iqreg , and two user ‐ written commands skewreg and user written commands skewreg and kurtosisreg . 2020/8/11 3
Outline Outline 1. Introduction d i 2. Quantile ‐ based Measures of Conditional Spread, Skewness and Kurtosis 3 The Spread Regression 3. The Spread Regression 4. The Skewness Regression 5. The Kurtosis Regression h 6. An Application to the U.S. Wage Data pp g 7. Stata Application 2020/8/11 4
1. Introduction 1 Introduction • Quantile regression provides a powerful tool to Q il i id f l l study the effects of covariates on key quantiles of conditional distribution of dependent variable diti l di t ib ti f d d t i bl given covariates. • But there are (too) many regression quantiles… • How do covariates affect the overall shape of How do covariates affect the overall shape of conditional distribution? 2020/8/11 5
How to Characterize Distribution How to Characterize Distribution • A simple way to characterize conditional distribution by looking at summary statistics: y g y • Location (mean, median) L i ( di ) • Scale (variance, spread, or interquartile range) ( , p , q g ) • Asymmetry (skewness) • Fat tails or tail risk (kurtosis) 2020/8/11 6
Quantile based Measures Quantile ‐ based Measures • Median • Spread (e.g. Interquartile Range) • Skewness (defined by quantiles) Skewness (defined by quantiles) • Kurtosis (defined by quantiles) 2020/8/11 7
Advantages of Quantile ‐ based Measures • Moment ‐ based measures may not exist, whereas b d i h quantile ‐ based measures are always well defined • Moment ‐ based measures are sensitive to outliers, whereas quantile ‐ based measures are robust to outliers • Quantile ‐ based measures can easily connect with Quantile based measures can easily connect with quantile regression. 2020/8/11 8
A Motivating Example A Motivating Example • Take a look at the classic Engel (1857) dataset • Food expenditure is regressed on household income 2020/8/11 9
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How Much Can We See How Much Can We See • The spread increases with the only covariate household income. But by how much, and is it y statistically significant? • How about the effect of household income on the conditional skewness and conditional kurtosis of food expenditure? kurtosis of food expenditure? 2020/8/11 11
Our Contributions Our Contributions • We propose a model that examines the impact of covariates on important properties p p p p conditional distribution, such as quantile ‐ based measures of spread skewness and based measures of spread, skewness, and kurtosis. • Estimated conditional spread, skewness and Estimated conditional spread, skewness and kurtosis functions are of additional interests 2020/8/11 12
2. Quantile ‐ based Measures of Conditional Spread, Skewness and Kurtosis x y • Consider a random variable and covariates ( p ‐ dim vector), denote the distribution ( p dim vector), denote the distribution Y x y F y x function of conditional on as ( | ) y y and the quantile function of and the quantile function of conditional on conditional on x Q x ( | ) is Y • We want to study how the distributional y properties of (spread, skewness and kurtosis) vary with x 2020/8/11 13
The Spread Regression The Spread Regression SP SP • Let be a measure of the spread of y y x x given , then is varying with SP , and y y suppose this relationship is captured by the functional relationship functional relationship x ( ) ( ) SP m y y • We call this relationship as the “spread We call this relationship as the spread regression” relationship. 2020/8/11 14
The Skewness Regression The Skewness Regression • Similarly, let be a measure of the SK y x y y skewness of given , then is varying g y g SK y y x with , and suppose this relationship is captured by the functional relationship captured by the functional relationship x ( ) ( ) SK s y y • We call this relationship as the “skewness We call this relationship as the skewness regression” relationship. 2020/8/11 15
The Kurtosis Regression The Kurtosis Regression KUR KUR y • Let be a measure of the kurtosis of y x x KUR given , then is varying with , and g y g y y suppose this relationship is captured by the functional relationship functional relationship x ( ) ( ) KUR k y y • We call this relationship as the “kurtosis We call this relationship as the kurtosis regression” relationship. 2020/8/11 16
Quantile based Measurements Quantile ‐ based Measurements m x k x s x ( ) ( ) • The properties of , and are ( ) dependent on how we measure the spread, p p skewness and kurtosis. • We consider quantile ‐ based measures for the spread, skewness and kurtosis, and study the relationship between spread, skewness, relationship between spread, skewness, x kurtosis and useful covariates . 2020/8/11 17
Measurement of Spread Measurement of Spread • A widely used robust measure of the spread is A id l d b t f th d i the Interquartile Range (IQR), then x x x ( ) (0.75| ) (0.25| ) SP m Q Q y Y Y • In general, for appropriate chosen , we may x x y y measure the spread of th d f given i by b x x x x x x ( , ) ( , ) (1 (1 | ) | ) ( | ) ( | ) SP SP m m Q Q Q Q y Y Y Y Y • For example, = 0.25 or 0.1 2020/8/11 18
Measurement of Skewness Measurement of Skewness • Consider the following robust measure of skewness based on quantiles (Bowley,1920) : q ( y ) x x x x x x x x (0.75| ) (0.75| ) (0.5| ) (0.5| ) (0.5| ) (0.5| ) (0.25| ) (0.25| ) Q Q Q Q Q Q Q Q Y Y Y Y y y x ( ) SK s x x (0.75| ) (0.25| ) Q Q y Y y • In general, for appropriate chosen , we may x x y y measure the skewness of th k f given i b by x x x x (1 | ) (0.5| ) (0.5| ) ( | ) Q Q Q Q y y y y y y y y x ( ( , ) ) SK SK s x x (1 | ) ( | ) Q Q y y y 2020/8/11 19
Intuition of Skewness Measure Intuition of Skewness Measure 2020/8/11 20
Measurement of Kurtosis Measurement of Kurtosis • Consider the following robust measure of kurtosis C id th f ll i b t f k t i based on quantiles (Moors, 1988): x x x x (7 / 8| ) (5 / 8| ) (3/ 8| ) (1/ 8| ) Q Q Q Q y y y y x ( ) KUR k y y x x (6 / 8| ) (6 / 8| ) (2 / 8| ) (2 / 8| ) Q Q Q Q y y • Moors (1988) shows that the conventional moment ‐ based measure of kurtosis can be interpreted as a measure of the dispersion of a interpreted as a measure of the dispersion of a distribution around the two values . 2020/8/11 21
Quantile Regression Quantile Regression • We consider the linear quantile regression model, which assumes that the conditional z x quantile functions of y given are (1 ) linear in covariates: linear in covariates: Q Q x x θ θ z z ( | ) ( | ) ( ) ( ) y • Extensions to other quantile regression • Extensions to other quantile regression models can also be analyzed 2020/8/11 22
Estimation of Quantile Regression Estimation of Quantile Regression • The quantile regression estimator solves n ˆ( ) θ z θ argmin ( ) y t t 1 p θ θ 1 1 t t • is the check function is the check function ( ) ( ) ( ( 0) 0) u u u u I u I u • The estimated conditional quantile function: θ ˆ ˆ z z θ | | ( ) ( ) Q Q y t t t 2020/8/11 23
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