Question

Say I have two variables x1 and x2 that are positively correlated with one another. What is the intuition behind why this correlation is positive while the covariance in the covariance matrix for beta hat is negative?

Answer 1

correlation coefficient between two variables is given by,

its value ranges from -1 to +1. For the positive value of this coefficient we say that the variable has a positive relationship that is they increase or decrease together and for the negative value we say that the variable has a negative relationship that is, if one increases other decreases and vice versa.

Now look at (1), the denominator is always a positive quantity (since variance is a non negative function). So it is the numerator which determines the sign of the correlation co efficient. So, if the numerator i.e. the value of the covariance is negative then the correlation will be negative and if the numerator is positive then positive correlation.

now you are saying that,

two variables x1 and x2 that are positively correlated with one another.

What is the intuition behind why this correlation is positive while the covariance in the covariance matrix for beta hat is negative?

If the variables x1 and x2 are positively correlated with one another then the covariance between them must be positive by law.

The covariance in the covariance matrix for beta hat must not be the covariance between X1 and x2 for they have positive correlation. It must be the covariance for something else, it can possibly for the two different values of beta hat, or there can be other possibilities but it cannot be for x1 and x2 at all.