6.11 Looking ahead

We have a target, a model and a criterion.

The target is the population regression function \(E(Y \mid X = x)\), which we have chosen to approximate with a straight line. The model \(y_i = \beta_0 + \beta_1 x_i + u_i\) adds a name for the part of \(y\) that attendance does not explain. The criterion says to pick the line minimising the sum of squared residuals.

What we do not yet have is the line.

The next unit finds it. Two derivatives set to zero give \(\hat{\beta}_0\) and \(\hat{\beta}_1\) in closed form, and the formulas turn out to say something interpretable: the slope is the covariance of \(x\) and \(y\) divided by the variance of \(x\).

With estimates in hand we can then ask what Units 2 to 4 taught us to ask of any estimator. Is it centred on the truth? How much does it vary from sample to sample? How do we test a claim about it? And how much of the variation in \(y\) has it actually accounted for?