# Interpret Intercept Plot from Series of Quantile Regressions in R

Essentially, I am looking at a set of housing data from 2000-2017 and I am examining 'affordability' by year through quantile regression. The Y variable (affordability) is a ratio of what an affordable mortgage payment would be (based on median income that year) to an estimate of what an actual mortgage payment for an individual house would be (based on interest rates and that home's actual sale price). The X is year, and the data is binned by year because this is a class project and I was told to avoid trying to do a real time series. So I use this code in R to run the regression:

QR <- rq(data$$Ratio ~ data$$Year, tau=seq(0.1, 0.9, by=0.1))
sumQR <- summary(QR)
sumQR
plot(sumQR)


And I get a series of plots that look like this (sorry they're each so small):

What I (think) I have is a plot for each year that shows the effect of being in that year, relative to the baseline (which I think is the year 2000), on the 'affordability' of houses at every 10th quantile of house price. So, for example, in 2006, the effect of being in 2006 vs 2000 is greater for cheap homes than expensive ones. In 2016, the effect of being in 2016 vs 2000 is greater for expensive homes than cheap ones. So my questions:

1. Am I interpreting the plots for individual years correctly?
2. How do I interpret the plot for Intercept? Is it just the plot for the year 2000, because that's being treated as the baseline?

## 1 Answer

You would interpret the results from quantile regression more or less in the same way you would ordinary least squares regression. It looks like in your data, year is a categorical factor variable and so you are getting a different plot for each level of that variable. The intercept is the estimated value of the response at the first value (or reference level) of your factor variable at each quantile. As expected, the plot of the intercept estimate is increasing steadily with increasing quantiles.

• Thank you for answering. If I understand right, the first plot (the intercept plot) is the expected level of the affordability index at each quantile of affordability in the year 2000, the baseline year. So is it reasonable to say that the intercept plot is the distribution of the affordability index in 2000, the distribution to which we compare the distributions of other years when we run this regression? – hemera Mar 12 at 17:15
• The first part of your statement is reasonable."If I understand right, the first plot (the intercept plot) is the expected level of the affordability index at each quantile of affordability in the year 2000, the baseline year." The second part less so but not too far off. – JustGettinStarted Mar 12 at 17:23
• The intercept is an estimate. It is the estimate of the affordability index in the year 2000. Note that this is estimate is in the context of the other parameters of your model. It is not a stand alone estimate of the distribution of affordability – JustGettinStarted Mar 12 at 17:25