The OLS estimator with one regressor is defined as $\widehat{\beta}_1 = COV(X,Y)/VAR(X)$. It can be interpreted as the marginal effect of $X$ on $Y$. My question: I don't understand intuitively, why I get the marginal effect if I divide $COV(X,Y)$ by $VAR(X)$. Can you provide some intuition?
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$Cov(X, Y) = SD(X) SD(Y) Cor(X, Y)$ so this formula is equivalent to $\hat \beta_1 = Cor(X,Y) \times \frac{SD(Y)}{SD(X)}$. The correlation is what links $X$ and $Y$, and then the respective standard deviations scale it. Note that if we standardized $X$ and $Y$ to have unit variance then the slope actually just is the correlation.
So we know that $Cor(X,Y)$ measures how linear the relationship between $X$ and $Y$ is. It doesn't matter how the point cloud is orientated: it could be almost vertical, it could be almost horizontal, or anything in between. It's just how close to linear it is. Note that $Cor(X,Y)$ is unitless.
It is $\frac{SD(Y)}{SD(X)}$ that gives us our scale, and this is where $\hat \beta_1$ gets its units from. Imagine that $SD(X) >> SD(Y)$. This means that the point cloud varies way more in $X$ than in $Y$, so it will look flat. This in turn means that no matter what the correlation is we'll still have a small value for $\hat \beta_1$ because a 1 unit step in $X$ won't change $Y$ by much, since $X$ spreads over such a large range. And vice versa for $SD(X) << SD(Y)$
