The OLS estimator

Summary

Key idea 1 from lecture.

• The linear regression model is an example of a data generating process (DGP) as it allows us to simulate data on \(y_i\) .

Key idea 2 from lecture.

• In the linear regression model

\[y_i=β_1+β_2x_i+ε_i , i=1,…,n\]

• the OLS estimator

\[b_2= \frac{\sum_{i=1}^{n}{ \left( x_i-\bar{x} \right)\left( y_i-\bar{y} \right) }}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]

• is an estimator of \(β_2\) and the OLS estimator

\[b_1=\bar{y}-b_2\bar{x}\]

• is an estimator of \(β_1\) .
• Many other estimators for \(β_1,β_2\) exist in the LRM.
• The OLS fitted values

\[{\hat{y}}_i=b_1+b_2x_i\]

• are estimates of the conditional expectations

\[β_1+β_2x_i=E\left( y_i | x_i \right)\]

• Similarly

\[\hat{y}=b_1+b_2x\]

• are estimates of the conditional expectations

\[β_1+β_2x=E\left(y | x \right)\]

• for arbitrary values of \(x\) not necessarily in the sample.
• Since the residuals are given by

\[e_i=y_i-b_1-b_2x_i\]

• and the error terms are given by

\[ε_i=y_i-β_1-β_2x_i\]

• if \(b_1\) is close to \(β_1\) and \(b_2\) is close to \(β_2\) then \(e_i\) is close to \(ε_i\) .

Key idea 3 from lecture.

• The OLS estimators \(b_1,b_2\) are random variables (so are \({\hat{y}}_i\) and \(e_i\) )