Is there a difference between estimating the slope of a line using OLS vs calculating the slope using the formula Cov(x,y)/var(x) ? . If these methods are different, what are the advantages of using either?
Estimating slope of regression using OLS vs Cov(x,y)/var(x)
least squaresregression
Related Solutions
The correlation coefficient $r$ gives you a measurement between $-1$ to $+1$. This gives you information about the strength of the linear relationship that can be interpreted independently of the scale of the two variables. Again, when $sd_y=sd_x$, then $m=r$.
So, $r$ is the slope of the regression line when both $X$ and $Y$ are expressed as z-scores (i.e. standardized). Remember that $r$ is the average of cross products, that is,
$r=\frac{\sum Z_xZ_y}{N}$
So, it turns out that $r$ is the slope of $Y$ on $X$ in z-score form. This correlation coefficient tells us how many standard deviations that $Y$ changes when $X$ changes $1$ standard deviation. When there is no correlation ($r = 0$), $Y$ changes zero standard deviations when $X$ changes $1$ standard deviation. When $r$ is $1$, then $Y$ changes $1$ standard deviation when $X$ changes $1$ standard deviation.
The regression $m$ weight is expressed in raw score units rather than in z-score units. To move from the correlation coefficient to the regression coefficient, we can simply transform the units:
$m=r(sd_y/sd_x)$
This says that the regression weight is equal to the correlation times the standard deviation of $Y$ divided by the standard deviation of $X$. Note that $r$ shows the slope in z-score form, that is, when both standard deviations are $1.0$, so their ratio is $1.0$. But we want to know the number of raw score units that $Y$ changes and the number that $X$ changes. So to get new ratio, we multiply by the standard deviation of $Y$ and divide by the standard deviation of $X$, that is, multiply $r$ by the raw score ratio of standard deviations.
Look for example William Greene's book Econometric Analysis.
Systems of equations need to be estimated jointly and only in special case, like in recursive system, can be OLS used without biases. Usually Full Information Maximum Likelihood and Limited Information Maximum Likelihood methods are used. They are quite similar as 2SLS and 3SLS.
SUR is a method of estimation for the system of equations when there is no endogenoity bias but disturbances across system can be related. This might be possible in examples like estimation of consumption system.
SUR is similar as generalized least squares estimation in single equation linear models where you believe homoskedasticity assumption cannot be met.
Best Answer
If we are talking about simple linear regression, the formula you mention is the least squares solution. You can use it when the model contains intercept and only one feature, otherwise, you need the solution for the matrices.