Quantile regression

[Klaudia Lennerling]

Hello!

I know this might not be stata related but I need help interpreting my results. I performed a quantile regression in stata with three quantiles, 0.25, 0.50 and 0.75. Now I'm having a hard time interpreting my estimates. My dependent variable was logged annual sales and my independent variable was a dummy and then I have added control variables. What does my results mean? Should I interpret my coefficient in the 25th quantile as the 25% lowest sales and the 75th quantile as the 75% lowest sales? I'm a bit confused.

Best regards, Klaudia

[Fei Wang]

The coefficient of x in the 25 quantile equation means the partial effect of x on the 25th percentile of y. Similar for other quantiles.

[Klaudia Lennerling]

Thank you! I mean that I do not understand the distribution within my quantiles. What does the firms in the 25th respectively 75th quantile account for in terms of sales?

[Jackson Monroe]

Klaudia, normal regression gives us estimates of the effect of our regressors on the conditional mean of our dependent variable, E[Y|X=x]. How does the average of Y change when I change x? Quantile regression is the same thing for the conditional quantiles of Y, how does the 25th percentile of Y change as I change x?

[Klaudia Lennerling]

Thank you, I understand the interpretation of the estimate however what I don't understand is how the quantiles are separated and what they mean? Does the 25th quantile mean the firms with the 25% of the lowest sales?

[Jackson Monroe]

Yes, if 100 firms were ranked by sales the 25th percentile would be the firm with 24 firms with lower sales and 75 with higher, give or take a firm.

[Joao Santos Silva]

Dear Klaudia Lennerling,

What Jackson said above needs to be slightly qualified: that is true for firms with the same value of the regressors because you are estimating conditional quantiles. Note that a firm in the first quartile for a given set of regressors may have sales that are in the range of sales in the top quartile for firms with a different set of regressors. So, the coefficient of x in q25 tells you how changes in x affect the first quartile of the distribution of sales for firms with the same regressors. If this is not clear, I suggest you check a suitable textbook or a survey on quantile regression.

Best wishes,

Joao

[Chen Samulsion]

Here is one simple explanation from idre in UCLA: https://stats.idre.ucla.edu/stata/fa...-coefficients/

[Joao Santos Silva]

The helpful link posted by Chen does not cover the case with multiple regressors and that is the most interesting and subtle case.

[Klaudia Lennerling]

Thank you everyone for your help!

[Chen Samulsion]

Actually Fei Wang Joao Santos Silva Jackson Monroe have provided answers. And I quite agree with Joao that better action is to check a suitable textbook or a survey on quantile regression. So let me cite something from textbook. The following quote is from Quantile Regression written by Lingxin Hao & Daniel Q. Naiman, Sage Publications, 2007.

For the LRM (linear-regression model), fitted coefficients can be interpreted as estimated effects, that is, estimates of the change in the mean of the response distribution that results from a one-unit increase in a continuous covariate or the change of the value from 0 to 1 of a dummy covariate. Each of these changes can be interpreted as an estimated difference in means between a reference group and a comparison group. The analog for the QRM (quantile-regression model) is an estimated difference in a particular quantile between a reference group and a comparison group, resulting from a one-unit increase in a continuous covariate or the change of the value from 0 to 1 of a dummy covariate, with other covariates held constant.

The income LRM estimate is $6,314 for ED and $11,452 for WHITE. One more year of schooling is associated with a $6,314 increase in mean income at any fixed level of education. ...... The mean income of blacks is $11,452 lower than that of whites. (P56)

In passing from the LRM to the QRM and focusing on the special case of median regression, the key modification to keep in mind is that we model the conditional median rather than the conditional mean.

In the case of a continuous covariate, the coefficient estimate is interpreted as the change in the median of the response variable corresponding to a unit change in the predictor. (P57)

The coefficient for ED in the conditional-median model is $4,794, which is lower than the coefficient in the conditional-mean model. This suggests that while an increase of one year of education gives rise to an average increase of $6,314 in income, the increase would not be as substantial for most of the population. (P59)

Sometimes, researchers are more interested in the lower or upper tails of a distribution than in the central location. ...... (At the lower tail,) We see that one more year of education can increase income by $1,782 at the .10th quantile and $1,130 at the .05th quantile. (At the right tail,) The coefficient for the .95th quantile is $9,575, much larger than that at the .90th quantile ($8,279), suggesting the contribution of prestigious higher education to income disparity. (P59-60)