1. **Stating the problem:** We want to understand the objectives of econometric estimation and explain how to estimate the regression model $$Y_i = \beta_1 + \beta_2 X_i + \mu_i$$ using the Ordinary Least Squares (OLS) method. 2. **Objectives of econometric estimation:** - To estimate the unknown parameters (\(\beta_1, \beta_2\)) of the model. - To test hypotheses about the relationships between variables. - To make predictions or forecasts of the dependent variable \(Y_i\). - To quantify the strength and form of the relationship between \(Y_i\) and \(X_i\). 3. **The regression model:** The model is: $$Y_i = \beta_1 + \beta_2 X_i + \mu_i$$ where: - \(Y_i\) is the dependent variable. - \(X_i\) is the independent variable. - \(\beta_1\) is the intercept. - \(\beta_2\) is the slope coefficient. - \(\mu_i\) is the error term capturing unobserved factors. 4. **OLS estimation method:** OLS estimates \(\beta_1\) and \(\beta_2\) by minimizing the sum of squared residuals: $$\text{Minimize } S(\beta_1, \beta_2) = \sum_{i=1}^n (Y_i - \beta_1 - \beta_2 X_i)^2$$ 5. **Deriving the OLS estimators:** Set partial derivatives of \(S\) with respect to \(\beta_1\) and \(\beta_2\) to zero: $$\frac{\partial S}{\partial \beta_1} = -2 \sum (Y_i - \beta_1 - \beta_2 X_i) = 0$$ $$\frac{\partial S}{\partial \beta_2} = -2 \sum X_i (Y_i - \beta_1 - \beta_2 X_i) = 0$$ 6. **Solving the normal equations:** From the first: $$\sum Y_i = n \beta_1 + \beta_2 \sum X_i$$ From the second: $$\sum X_i Y_i = \beta_1 \sum X_i + \beta_2 \sum X_i^2$$ 7. **Expressing estimators:** Define sample means: $$\bar{X} = \frac{1}{n} \sum X_i, \quad \bar{Y} = \frac{1}{n} \sum Y_i$$ Then, $$\hat{\beta}_2 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$$ $$\hat{\beta}_1 = \bar{Y} - \hat{\beta}_2 \bar{X}$$ 8. **Interpretation:** - \(\hat{\beta}_2\) measures the estimated change in \(Y\) for a one-unit change in \(X\). - \(\hat{\beta}_1\) is the estimated value of \(Y\) when \(X=0\). 9. **Summary:** OLS provides a simple, unbiased, and efficient way to estimate linear relationships by minimizing squared errors. Final answer: $$\hat{\beta}_2 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}, \quad \hat{\beta}_1 = \bar{Y} - \hat{\beta}_2 \bar{X}$$