Slope of LS line = Cov(X,Y)/Var(X). Intuitive explanation?

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SUMMARY

The slope of a fitted line in linear regression is defined as the ratio of Cov(X,Y) to Var(X). This relationship highlights that while Cov(X,Y) indicates the degree to which X and Y vary together, it is essential to normalize this by Var(X) to account for the variability of X itself. The discussion emphasizes the importance of understanding that fluctuations in X and Y may not always indicate a true relationship, as some covariance can be attributed to noise. This insight is crucial for accurately interpreting the significance of the slope in regression analysis.

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pluviosilla
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The slope of a fitted line = Cov(X,Y)/Var(X). I've seen the derivation of this, and it is pretty straightforward, but I am still having trouble getting an intuitive grasp. The formula is extremely suggestive and it is bothering me that I can't quite see its significance.

Perhaps, my mental block comes from thinking of X values as points on a line and I should instead be thinking of two parallel series, one of x values and another of y values that are parameterized using a third variable, t for time, for example. Thus values for X would not be sequential, like x values on the x-axis. They will fluctuate around a mean. Sometimes y values will "co-vary", i.e. fluctuate in *tandem* with x values but some of that apparent covariance is deceptive. It is really just noise, which is why the slope of the fitted line cannot be simply Cov(X,Y). We must divide by Var(X) in order to subtract out that *accidental* coincidence (or covariance) of X & Y.

Is it something like that?
 
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I like to think of Cov(X,Y) as the mean risk of a portfolio with two shares in X and Y. As Var(X) = Cov(X,X), the quotient is somehow the mean risk of a 2 shares portfolio compared to the risk of an all X shares portfolio. I know this view of a font manager is not really a mathematical point of view but it may help to grasp it.
 
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