Conditional Quantile Regressions

Because no-one is average

Fernando Rios-Avila

Random coefficents

One approach to both understanding, and simulating QREG is by also understanding the intuition behind the data generating process.

$\begin{aligned} y &= b_0(\tau)+b_1(\tau)x_1 + +b_2(\tau)x_2+...+b_k(\tau) x_k \\ \tau &\sim runiform(0,1) \end{aligned}$

where all coefficients are a function (preferably monotonically increasing or decreasing) of $\tau$ .

We want them to be monotonically increasing or decreasing because we want that $X \beta(\tau_1 ) \geq X \beta(\tau_2 ) \ \text{ if } \ \tau_1 > \tau_2$

SVC model with a latent running variable

Another way of thinking about Qreg is to align it to the semiparametric method we introduced ealier. SVC model.
In SVC, there is an observed running variable $z$ , and we focus on analyzing how the “local” effects of $X$ on $Y$ change as a function of $z$ .
The difference with Qreg is that the running variable is unknown $\tau$ .
- Given the outcome, and characteristics we can identify something like a “latent” component.

There are a few (recent) papers that focus on estimation and identification of these models. The general intuition is that the qreg model is identified by the following moment condition:

$E\Big( 1[x\beta(\tau) - y > 0 ] - \tau \Big) = 0$

but substitute the indicator function with a smooth function. CDF

$E\Big( F(x\beta(\tau) - y) - \tau \Big) = 0$

Being differentiable, this problem is relatively easier to solve (given good initial values)

Scale and Location Model

Another approach that can be used to understand Quantile regressions (and elaborate the interpretation) is to assume that the coefficients are in fact capturing two components:

$y = Xb + Xg(\tau)$

Location: $Xb$ which indicates what is the average/typical relationship between X and Y.
Scale: $Xg(\tau)$ which indicates how far one could be from the average effect, given a relative rank $\tau$

Estimation of this model is not standard. But can be manually implemented:

Estimate OLS and get residuals
Estimate QREG using those residuals

Requires additional care for the estimation of SE

Scale and Location 2: Heteroskedasticity

A second approach that is useful to understand and interpret CQreg is to consider a parametric version of the LS model:

$y = Xb + \gamma (X) * e \text{ with } \gamma(X)>>0$

This shows the relationship between a quantile regressions and heteroskedasticity in the error term.
If we assume Heteroskedasticity is parametric ( $\gamma(x) = X \gamma$ ), it constrains the relationship across all quantile coefficients:

$y = X(b+\gamma F^{-1}(\tau)) \rightarrow b(\tau)=b+\gamma\times q(\tau)$

Making it more efficient, albeit imposing constrains of the relationship.

Estimation and Statistical Inference

As hinted previously, there are many approaches that can be used for the estimation of Conditional Quantile regressions.

Official: qreg, sqreg, bsqreg, iqreg
CContributed: qreg2, qrprocess, mmqreg, smqreg, sivqr
For Standard errors, however, there are main 3 options. Under the assumption of iid error. Non iid error (robust), and assuming clustered standard errors.

$\begin{aligned} iid: \Sigma_\beta &=\frac{\tau(1-\tau)}{N f^2_y(F^{-1}(\tau))}(X'X)^{-1} \\ niid: \Sigma_\beta &= \tau(1-\tau) (X'f(0|x)X)^{-1} \ (X'X) \ (X'f(0|x)X)^{-1} \\ alt: \Sigma_\beta &= (IF_\beta \ ' IF_\beta) N^{-2} \end{aligned}$

Or simply Bootstrap

Solution 1: Correlated Random Effects

Idea: Include “PAnel average” of all indep variables as regressors.

They should control (at least partially) for the unobserved effects.

$Q_p(y|X) = X\beta + \alpha \bar X + \epsilon$

Code

bysort id: egen x1p = mean(x1)
bysort id: egen x2p = mean(x2)
qui:qreg y x1 x2 x1p x2p , q(10)
est sto m1
qui:qreg y x1 x2  x1p x2p , q(50)
est sto m2
qui:qreg y x1 x2  x1p x2p , q(90)
est sto m3
esttab m1 m2 m3 m10 m20 m30, se nogaps mtitle(q10 q50 q90)


------------------------------------------------------------------------------------------------------------
                      (1)             (2)             (3)             (4)             (5)             (6)   
                      q10             q50             q90             m10             m20             m30   
------------------------------------------------------------------------------------------------------------
x1                  0.825***        0.972***        1.195***        0.780***        0.988***        1.219***
                 (0.0250)        (0.0246)        (0.0460)        (0.0190)        (0.0139)        (0.0224)   
x2                  1.176***        1.023***        0.774***        1.219***        1.000***        0.763***
                 (0.0252)        (0.0248)        (0.0463)        (0.0189)        (0.0138)        (0.0223)   
x1p                 0.310***        0.413***        0.663***                                                
                 (0.0575)        (0.0565)         (0.106)                                                   
x2p                 0.283***        0.406***        0.551***                                                
                 (0.0564)        (0.0555)         (0.104)                                                   
ui                                                                  0.654***        1.000***        1.352***
                                                                 (0.0180)        (0.0132)        (0.0212)   
_cons              -0.951***        0.861***        3.146***       -0.458***        0.970***        2.420***
                 (0.0278)        (0.0273)        (0.0511)        (0.0236)        (0.0173)        (0.0279)   
------------------------------------------------------------------------------------------------------------
N                    6019            6019            6019            6019            6019            6019   
------------------------------------------------------------------------------------------------------------
Standard errors in parentheses
* p<0.05, ** p<0.01, *** p<0.001

Solution 2: FE are Fixed

Canay (2011) proposes make the “simplifying” that “Fixed effects” are constant across quantiles.
Thus a two step procedure is proposed:
- First: Estimate the fixed effects using OLS
- Second: Estimate the quantile regression using outcome after taking FE “off”

$\begin{aligned} y &= X\beta + \alpha_i + \epsilon \\ Q_\tau(y - \hat \alpha_i |X) &= X\beta(\tau) + \epsilon_\tau \\ \end{aligned}$

Code

qui:reghdfe y x1 x2, absorb(fe = id)
gen y_fe = y - fe
qui:qreg y_fe x1 x2   , q(10)
est sto m1
qui:qreg y_fe x1 x2   , q(50)
est sto m2
qui:qreg y_fe x1 x2 , q(90)
est sto m3
esttab m1 m2 m3 m10 m20 m30, se nogaps mtitle(q10 q50 q90)

(4 missing values generated)

------------------------------------------------------------------------------------------------------------
                      (1)             (2)             (3)             (4)             (5)             (6)   
                      q10             q50             q90             m10             m20             m30   
------------------------------------------------------------------------------------------------------------
x1                  0.727***        0.992***        1.277***        0.780***        0.988***        1.219***
                 (0.0213)        (0.0120)        (0.0218)        (0.0190)        (0.0139)        (0.0224)   
x2                  1.117***        1.004***        0.853***        1.219***        1.000***        0.763***
                 (0.0212)        (0.0119)        (0.0216)        (0.0189)        (0.0138)        (0.0223)   
ui                                                                  0.654***        1.000***        1.352***
                                                                 (0.0180)        (0.0132)        (0.0212)   
_cons              -0.296***        1.021***        2.336***       -0.458***        0.970***        2.420***
                 (0.0272)        (0.0153)        (0.0277)        (0.0236)        (0.0173)        (0.0279)   
------------------------------------------------------------------------------------------------------------
N                    6015            6015            6015            6019            6019            6019   
------------------------------------------------------------------------------------------------------------
Standard errors in parentheses
* p<0.05, ** p<0.01, *** p<0.001

Solution 3: Modified Canay(2011)

Same as before, but rather than “removing” fixed effects, we control for them in the model:

$\begin{aligned} y &= X\beta + \alpha_i + \epsilon \\ Q_\tau(y |X) &= X\beta(\tau) + \gamma \hat \alpha_i + \epsilon_\tau \\ \end{aligned}$

Code

qui:qreg y x1 x2 fe  , q(10)
est sto m1
qui:qreg y x1 x2 fe  , q(50)
est sto m2
qui:qreg y x1 x2 fe, q(90)
est sto m3
esttab m1 m2 m3 m10 m20 m30, se nogaps mtitle(q10 q50 q90)


------------------------------------------------------------------------------------------------------------
                      (1)             (2)             (3)             (4)             (5)             (6)   
                      q10             q50             q90             m10             m20             m30   
------------------------------------------------------------------------------------------------------------
x1                  0.800***        0.993***        1.171***        0.780***        0.988***        1.219***
                 (0.0182)        (0.0137)        (0.0206)        (0.0190)        (0.0139)        (0.0224)   
x2                  1.193***        1.006***        0.831***        1.219***        1.000***        0.763***
                 (0.0181)        (0.0137)        (0.0205)        (0.0189)        (0.0138)        (0.0223)   
fe                  0.691***        0.991***        1.279***                                                
                 (0.0159)        (0.0120)        (0.0180)                                                   
ui                                                                  0.654***        1.000***        1.352***
                                                                 (0.0180)        (0.0132)        (0.0212)   
_cons              -0.334***        1.017***        2.354***       -0.458***        0.970***        2.420***
                 (0.0227)        (0.0171)        (0.0257)        (0.0236)        (0.0173)        (0.0279)   
------------------------------------------------------------------------------------------------------------
N                    6015            6015            6015            6019            6019            6019   
------------------------------------------------------------------------------------------------------------
Standard errors in parentheses
* p<0.05, ** p<0.01, *** p<0.001

Solution 4: LS model

Machado and Silva (2019) propose a different approach. They suggest modeling the quantile regression using linear models for a scale and location model.
This simplifies the task of estimating multiple equations:

$\begin{aligned} y &= X\beta + \epsilon \\ \hat\epsilon^2 &= X\gamma + \nu \\ Q\left( \frac{\hat\epsilon}{X\gamma}\right) &= q(\tau) \\ \beta(\tau) &= \beta + \gamma \times q(\tau) \end {aligned}$

qui:mmqreg y x1 x2   , q(10)  abs(id) robust
est sto m1
qui:mmqreg y x1 x2   , q(50)  abs(id) robust
est sto m2
qui:mmqreg y x1 x2 , q(90) abs(id) robust
est sto m3
esttab m1 m2 m3 m10 m20 m30, se nogaps mtitle(q10 q50 q90)


------------------------------------------------------------------------------------------------------------
                      (1)             (2)             (3)             (4)             (5)             (6)   
                      q10             q50             q90             m10             m20             m30   
------------------------------------------------------------------------------------------------------------
main                                                                                                        
x1                  0.780***        0.993***        1.214***        0.780***        0.988***        1.219***
                 (0.0183)        (0.0147)        (0.0203)        (0.0190)        (0.0139)        (0.0224)   
x2                  1.231***        0.992***        0.745***        1.219***        1.000***        0.763***
                 (0.0197)        (0.0162)        (0.0233)        (0.0189)        (0.0138)        (0.0223)   
ui                                                                  0.654***        1.000***        1.352***
                                                                 (0.0180)        (0.0132)        (0.0212)   
_cons              -0.346***        1.017***        2.431***       -0.458***        0.970***        2.420***
                 (0.0220)        (0.0207)        (0.0251)        (0.0236)        (0.0173)        (0.0279)   
------------------------------------------------------------------------------------------------------------
N                    6019            6019            6019            6019            6019            6019   
------------------------------------------------------------------------------------------------------------
Standard errors in parentheses
* p<0.05, ** p<0.01, *** p<0.001

1 / 42

Conditional Quantile Regressions Because no-one is average Fernando Rios-Avila

Conditional Quantile Regressions
Introduction
$Q(\theta) = F^{-1}(\theta)$
Other advantages? Yes!
And are also very...
Technical Note
Statistical Inference
So we have: \[ \begin{aligned}...
Quantile SE \[ Var(\hat...
Example Using Bootstrap...
From $Q_Y$ to $Q_{Y|X}$
Slide 12
Q and Loss functions...
From $\beta$ to...
Interpretation: Why is it so different from OLS?
But CQreg?
what does it mean?...
Visualizing Differences...
Example: Wages…
Example: Wages…
Other Interpretations of Qreg
Random coefficents
In this specification...
SVC model with a latent running variable
There are a few (recent)...
Example (with sivqr)
Scale and Location Model
Scale and Location 2: Heteroskedasticity
Example (with mmqreg)...
Estimation and Statistical Inference
Problems and Considerations
Quantile Regressions with Fixed Effects
The problem
Simulating some data
Accounting for Fixed effects
Benchmark
Ignoring Fixed Effects
Solution 1: Correlated Random Effects
Solution 2: FE are Fixed
Solution 3: Modified Canay(2011)
Solution 4: LS model
Next topic…Unconditional Quantiles

Conditional Quantile Regressions

Introduction

Question: What are quantiles? and why do we care??

$Q(\theta) = F^{-1}(\theta)$

Other advantages? Yes!

Technical Note

Statistical Inference

Quantile SE

Example

Using Bootstrap

Analytical SE

From $Q_Y$ to $Q_{Y|X}$

Q and Loss functions

From $\beta$ to $X\beta$

Interpretation: Why is it so different from OLS?

But CQreg?

what does it mean?

Visualizing Differences in Interpretation

Example: Wages…

Example: Wages…

Other Interpretations of Qreg

Random coefficents

SVC model with a latent running variable

Example (with `sivqr`)

Scale and Location Model

Scale and Location 2: Heteroskedasticity

Example (with `mmqreg`)

Estimation and Statistical Inference

Problems and Considerations

Quantile Regressions with Fixed Effects

The problem

Simulating some data

Accounting for Fixed effects

Benchmark

Ignoring Fixed Effects

Solution 1: Correlated Random Effects

Solution 2: FE are Fixed

Solution 3: Modified Canay(2011)

Solution 4: LS model

Next topic…Unconditional Quantiles