Is the slope coefficient on its own an indication of correlation coefficent? More simply, how are the two related?
I know the slope will take on the sign of the correlation, but what beyond that? Or is the coefficient of determination the primary indication of correlation from a regression ? Thanks in advance!
No the slope coefficiant has nothing to do with the correlation coefficiant. They are bery different. The slope coefficiant measures the change in or the value of the dependant variable given a change in or the value of the independant variable. This measure can take on any value and is actually the beta of the dependant variable. Consider the CAPM and the beta of a stock. Its the change in the value of the stock for a 1% change in the market. Beta in this case is the slope coefficiant of a regression line that links the return on the stock (dependant variable) to the return of the independant variable (independant variable). Beta can of course be greater than one and will tell you by how much will Y move for a change in x. Of course this does not tell you on its own how will why move if x stays the same. This is why in the regression equation you have the intercept bo, and finally the error term to account for imprecisions in forecasts based on this regression line. Now the correlation coefficiant. In the regression equation, the correlation coefficiamt does not appear. The correlation between x and y will be give you an idea about the fit of the regression line you have. Consider a scatter plot. If all the points observed can fit on one perfect straight line, then the correlation coefficiant between the 2 variables is 1 or -1 if the line is downoard sloping. Now as you can imagine, if you plot your observations of the change in citigroup stock and the S&P500 index, there is no way that you will obtain a relationship that is absolutely the same for all periods. When you construct a straigth line to represent those points, the degree to which this straight line fits those points is your correlation coefficiant. So if your line perfectly passes by all yours points, then you have positive or negative perfect correlation and r is. If your line does not really represent your points, meaning that many of your points are very far from the line, that your line does not fit well your data and correlation is low. Now join both concepts. A regression line can have a slope of 4.7 which is b1 in your equation and the 2 variables might have a correlation of between -1 and +1 which does not appear in the equation but gives you an idea about how well your straight line fits the data used to draw this line. As for the coefficiant of determination it serves one purpose. It will tell you how much of the change in y is explained by the change in x. So if this coefficant, which is the square of correlatiom for simple regressions, is 89%, then this means that 89% of the change in y is explained by the change in x. Which also meams that 11% of the change in y is due to the change in another factor, i.e. It is not explained by the change in x. Hope that helps.
Of course if you want to see the whole picture, there is a relationship between the slope coefficiant and the correlation coefficiant. The slope is the cov of x and ydivided by the variance of x. Which reduces to the corr of x and y times the ratio of standard deviation y to standard deviation x. So yes the slope is a function of the corr coefficant but the slope on its own cannot give you an idea about the correlation coefficiamt unless you know standard deviations of x and y.
Thank you, this is all very helpful. I guess I overlooked this obvious point, but one of the core assumptions of linear regression is the assumption of linearity btw the dependent and independent variables. So while it is not explicit, it is assumed. I think your points to a good job explaining the technicality of this assumption. Thanks!