Setup

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Suppose we observe an i.i.d. sample \((X_i, Y_i)_{i=1}^n\) and want to estimate the conditional mean function.

The estimator

The estimator solves the moment condition

\[\hat{\beta} = \arg\min_{\beta} \sum_{i=1}^n \left( Y_i - X_i^\top \beta \right)^2.\]

Under standard regularity conditions, \(\hat{\beta}\) is consistent and asymptotically normal:

\[\sqrt{n} \left( \hat{\beta} - \beta_0 \right) \xrightarrow{d} \mathcal{N}(0, \Sigma).\]

Discussion

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