Written by 43161,

On 4/2, we have discussed about outliers and quantile regression. You said that quantile regression is one of the robust regression methods, models quantile of the distribution of Y, and is very needed and helpful to resolve none-normality and heteroskedasticity. According to Wiki's definition, quantile regression, also, will be more robust in response to large outliers. However, I am still confusing with this regression model and so have spent much time to better understand it.

At 00:39:10, you showed some quantile regression models such as Q _.5(Y) = b₀^(.5) + b₁^(.5)x and Q _.2(Y) = b₀^(.2) + b₁^(.2)x, and explained the reason why there is no error term in a quantile regression model. But, I don't understand your explanation that because of quantile, there is no error term. Thus, I would like you to explain more details about this.  Furthermore, in the case that we use (.2) or (.8) quantile regression model, what does .2or .8 quantile exactly mean? Does it mean that in order to estimate parameters, such quantile model is going to use 20 percentiles' or 80 percentiles' values of Y on fixed X? This is very unclear to me.

In addition, at 00:39:30, you said that we can have a different model at every quantiles and such different models have different shapes, intercepts, and slopes. At oil example, 00:46:18, we could see that depending upon different percentiles, some quantile models were significant, but others were not significant. If so, can we just pick one of significant models after running different quantile models as many as we can? Otherwise, is there any way to choose the best one among them?  

 Thank you for your time.

Anonymous:

Written by 43161,

On 4/2, we have discussed about outliers and quantile regression. You said that quantile regression is one of the robust regression methods, models quantile of the distribution of Y, and is very needed and helpful to resolve none-normality and heteroskedasticity. According to Wiki's definition, quantile regression, also, will be more robust in response to large outliers. However, I am still confusing with this regression model and so have spent much time to better understand it.

At 00:39:10, you showed some quantile regression models such as Q _.5(Y) = b₀^(.5) + b₁^(.5)x and Q _.2(Y) = b₀^(.2) + b₁^(.2)x, and explained the reason why there is no error term in a quantile regression model. But, I don't understand your explanation that because of quantile, there is no error term. Thus, I would like you to explain more details about this.  Furthermore, in the case that we use (.2) or (.8) quantile regression model, what does .2or .8 quantile exactly mean? Does it mean that in order to estimate parameters, such quantile model is going to use 20 percentiles' or 80 percentiles' values of Y on fixed X? This is very unclear to me.

In addition, at 00:39:30, you said that we can have a different model at every quantiles and such different models have different shapes, intercepts, and slopes. At oil example, 00:46:18, we could see that depending upon different percentiles, some quantile models were significant, but others were not significant. If so, can we just pick one of significant models after running different quantile models as many as we can? Otherwise, is there any way to choose the best one among them?  

 Thank you for your time.

You said, "that because of quantile, there is no error term. Thus, I would like you to explain more details about this. "

It's just like our usual model E(Y|X=x) = beta0 + beta1*x.   That model states that the means of the pdfs of Y for different X=x fall on a straight line, and there is no error term there either.    Similarly, we could model a quantile (like the median) of the pdfs of Y for different X=x as falling on a straight line.

The definition of the .20 quantile of the pdf of Y for a given X=x is the value q(.20,x) such that 20% of the Y values that are observable when X=x are less than q(.20,x), and 80% are larger.  Please review the concept "probability distribution."

As far as different models for different quantiles goes, your question about "which one to pick" kind of misses the point.  Pick them all.  Pick one that is relevant.  They all provide different information.  For example, VaR in finance is a .05 quantile.   Or just use the median model, as a natural alterntive to the usual OLS mean-based model.

Also recall that regression model is a model for the distribution of Y for a given x.  The quantile models (note plural) give you a way to estimate these distributions.

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