# What does ${\beta} \in \mathbb{R}^{p}$ mean exactly?

Studying different approaches of estimating linear regression models, I recently came across quantile regression.

In the paper I was reading (http://www.econ.uiuc.edu/~roger/research/rq/rq.pdf) the author uses the following notation:

${\beta} \in \mathbb{R}^{p}$

What does the ${p}$ stand for in this context? I have not come across this notation before.

Thank you.

• $\beta$ is a $p$-dimensional vector or tuple taking real values. For example if $p=3$, you would be talking about an ordered triple of real numbers such as $(-1,4,-1)$ – Henry Feb 19 '18 at 10:32
• If $p \in \mathbb{N}$ then ${\beta}$ is a $p$-vector. If not, then ${\beta}$ is function from a set $p$ to the set $\mathbb{R}$. – Bernard Massé Feb 19 '18 at 10:35
• Thank you for the quick responses. – shenflow Feb 19 '18 at 10:41

In quantile regression and more generally in statistics, $p$ often denotes that number of dependent variables (or covariates) in the data. The covariate vector $x_i$ in the paper you read is also $p$-dimensional with component $x_{ij}$ being the value of the $j$th covariate from the $i$th observation, where $i=1,\ldots,n$ and $j=1,\ldots,p$. Here, $n$ often means the number of observations in the data.