Yes, the covariance and correlation coefficient between two variables will always have the same sign. This is easily seen from the definition of correlation coefficient. Let `X` and `Y` be two random variables, then correlation coefficient `rho_(X,Y)` is given by
where `cov(X,Y)` is covariance and `sigma_x` and `sigma_y` are standard deviations of `X` and `Y` respectively.
` `Therefore, because standard deviation of any random variable is positive it follows that covariance and correlation coefficient must have the same sign.
Yes, the correlation coefficient between two variables and the slope estimate from a regression of one of the variables on the other variable will always have the same sign.
This is so because positive values of correlation coefficient indicate direct proportionality while negative values indicate inverse proportionality between two variables. The same is true for slope of a line.
E.g. equation of line `y=2x` just says that variable `y` is twice as big (in terms of absolute value) as variable `x` while equation `y=-2x` says that variable `y` is twice as big but with opposite sign. Similarly, correlation coefficient of 0.95 means that the two variables are highly correlated and that as one variable increases, the other variable increases as well, while correlation coefficient of -0.95 means that the two variables are highly correlated but as one variable increases, the other decreases and vice versa.
The two images below show example of regression line for both positive and negative correlation coefficient.