Population Parameter Estimation 09C704

1. **State the problem:** We have a population regression function (PRF) given by $$Y = \beta_1 + \beta_2 X + u$$ where $Y$ is the dependent variable (consumption), $X$ is the independent variable, $\beta_1$ and $\beta_2$ are the true population parameters, and $u$ is the error term. 2. **Sample Regression Function (SRF):** The SRF estimates the PRF using sample data: $$Y = \hat{\beta}_1 + \hat{\beta}_2 X + \hat{u}$$ where $\hat{\beta}_1$ and $\hat{\beta}_2$ are estimators of the true parameters, and $\hat{u}$ is the residual. 3. **Residual definition:** The residual $\hat{u}$ is the difference between observed $Y$ and predicted $Y$ from the SRF: $$\hat{u} = Y - \hat{\beta}_1 - \hat{\beta}_2 X$$ 4. **Goal:** Estimate the true population parameters $\beta_1$ and $\beta_2$ from the SRF. 5. **Method:** Use Ordinary Least Squares (OLS) to minimize the sum of squared residuals: $$\min_{\hat{\beta}_1, \hat{\beta}_2} \sum (Y_i - \hat{\beta}_1 - \hat{\beta}_2 X_i)^2$$ 6. **OLS estimators formulas:** $$\hat{\beta}_2 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$$ $$\hat{\beta}_1 = \bar{Y} - \hat{\beta}_2 \bar{X}$$ where $\bar{X}$ and $\bar{Y}$ are sample means. 7. **Interpretation:** - $\hat{\beta}_2$ measures the estimated change in consumption $Y$ for a one-unit change in $X$. - $\hat{\beta}_1$ is the estimated consumption when $X=0$. 8. **Summary:** The SRF provides estimates $\hat{\beta}_1$ and $\hat{\beta}_2$ that approximate the true parameters $\beta_1$ and $\beta_2$ in the PRF. 9. **Residuals:** The residuals $\hat{u}$ capture the deviations of observed values from the estimated regression line. This completes the estimation of the true population parameters from the hypothetical SRF.