# Showing an equality graphically (estimation)

Suppose $$Y_i = \beta_1 + \beta_2 X_i + u_i$$ is our population regression function with $\beta_1$ the population regression line intercept, $\beta_2$ the population regression line slope, $u_i$ the disturbance term, $X_i$ the regressor and $Y_i$ the regressand.

How can I explain this relationship $\operatorname{E}(\hat \beta_2 u_i) = \operatorname{Cov}(\hat \beta_2,u_i)$ graphically? (where $\hat \beta_2$ is the OLS estimator of $\beta_2$).

• Are you asking how to illustrate graphically covariance? Commented Oct 12, 2016 at 15:37
• @A.E, yes. Wouldn't that also be showing the expectation part of the equality, since both are equal? Commented Oct 12, 2016 at 15:53
• Yes. Do you assume that $u_i \sim N(0, \sigma^2)$? Commented Oct 12, 2016 at 15:55
• @A.E, now that I think about it, yes we did say that about $u_{i}$, could you point out how that can help us? Commented Oct 12, 2016 at 15:59

I would assume that $(u, \hat{\beta_1})$ follow the normal bivariate distribution and would draw $n$ pairs $(u_i, \beta_{1i})$ from this function, plot it and draw $u=\frac{S_u}{S_{\beta_1}}r(b_1-\beta_1)$ to illustrate $\mathbb{E}[u|\beta_1=b_1]$. Where the parameters for the bivariate distribution you can take as the MLE estimators of $\sigma^2, \beta_1, \rho...$