<p>I cannot say how helpful this is for you but if your model is</p>

<p>$$y_i = a(u_i) + b(u_i)x_i, \;\; u_i|x_i \sim U[0,1] \;\;\forall i$$</p>

<p>Then </p>

<p>$$\text{Var}(y_i\mid x_i) = \text{Var}[a(u_i)\mid x_i] + x_i^2\text{Var}[b(u_i)\mid x_i] +2x_i\text{Cov}[a(u_i),b(u_i)\mid x_i]$$</p>

<p>which tells you that $y_i$ is conditionally heteroskedastic.
This can acquire a complicated integral expression if you use the "Law of unconscious statistician" and the fact the the denisty function of a $U[0,1]$ random variable is just unity:</p>

<p>$$\text{Var}(y_i\mid x_i) = \int_0^1[a(u)]^2du - \left(\int_0^1a(u)du \right)^2 \\ + x_i^2\cdot \left[\int_0^1[b(u)]^2du - \left(\int_0^1b(u)du\right)^2\right] \\
+2x_i\cdot \left[\int_0^1a(u)b(u)du - \int_0^1a(u)du\int_0^1b(u)du \right]$$</p>

CC BY-SA 3.0
2016-03-23T22:57:23.600Z
Thanks! This is definitely a start, so I'll see if I can work something out and get a less messy result
DornerA
CC BY-SA 3.0
2016-03-23T15:24:13.817Z
2016-03-23T15:24:13.817Z
Alecos Papadopoulos
<p>I have been looking at quantile regression (since it is a much better method when trying to quantify welfare effects), and I am struggling with the following, standard model:</p>

<p>$\textbf{Model:}$</p>

<p>$y=x'\beta(u)$ where $u|x\text{~}Uniform\,[0,1]$ and for any $x,\, x'\beta(\tau)$ is a strictly increasing function in $\tau$. </p>

<p>For simplicity, assume $x=(1,x_1)$ meaning that we have one regressor and an intercept </p>

<p>$\textbf{Question:}$</p>

<p>What is $Var(y|x)$?</p>

CC BY-SA 3.0
2016-03-23T14:33:27.263Z
The way you've written it, your coefficient vector is a random variable, as in a Bayesian setting. Is this indeed the case?
Alecos Papadopoulos
CC BY-SA 3.0
2016-03-23T14:39:00.230Z
@AlecosPapadopoulos yes, the coefficient vector is a function of u which is a random variable with a uniform conditional distribution
DornerA
<econometrics><variance>
Variance of Quantile Regression
DornerA
# Variance of Quantile Regression

(Asked by: username_0 on 2016-03-21)

I have been looking at quantile regression (since it is a much better method when trying to quantify welfare effects), and I am struggling with the following, standard model:

$\textbf{Model:}$

$y=x'\beta(u)$ where $u|x\text{~}Uniform\,[0,1]$ and for any $x,\, x'\beta(\tau)$ is a strictly increasing function in $\tau$.

For simplicity, assume $x=(1,x\_1)$ meaning that we have one regressor and an intercept

$\textbf{Question:}$

What is $Var(y|x)$?

**Tags:** econometrics, variance

> **Comments:**
> **username_1**: The way you've written it, your coefficient vector is a random variable, as in a Bayesian setting. Is this indeed the case?  
> **username_0**: @username_1 yes, the coefficient vector is a function of u which is a random variable with a uniform conditional distribution  

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## 1 answers

### Answer by username_1 on 2016-03-23. Score: 2 | Accepted answer

I cannot say how helpful this is for you but if your model is

$$y\_i = a(u\_i) + b(u\_i)x\_i, \;\; u\_i|x\_i \sim U[0,1] \;\;\forall i$$

Then

$$\text{Var}(y\_i\mid x\_i) = \text{Var}[a(u\_i)\mid x\_i] + x\_i^2\text{Var}[b(u\_i)\mid x\_i] +2x\_i\text{Cov}[a(u\_i),b(u\_i)\mid x\_i]$$

which tells you that $y\_i$ is conditionally heteroskedastic.
This can acquire a complicated integral expression if you use the "Law of unconscious statistician" and the fact the the denisty function of a $U[0,1]$ random variable is just unity:

$$\text{Var}(y\_i\mid x\_i) = \int\_0^1[a(u)]^2du - \left(\int\_0^1a(u)du \right)^2 \\ + x\_i^2\cdot \left[\int\_0^1[b(u)]^2du - \left(\int\_0^1b(u)du\right)^2\right] \\
+2x\_i\cdot \left[\int\_0^1a(u)b(u)du - \int\_0^1a(u)du\int\_0^1b(u)du \right]$$

> **Comments:**
> **username_0**: Thanks! This is definitely a start, so I'll see if I can work something out and get a less messy result  

---

