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 $e_i$ )