1. **Problem C2.1 (Data file: 401K.dta):** (i) Find the average participation rate ($\overline{prate}$) and average match rate ($\overline{mrate}$). (ii) Estimate the simple linear regression: $$prate = \hat{\beta}_0 + \hat{\beta}_1 mrate$$ Report regression coefficients, sample size ($n$), and $R^2$. (iii) Interpret the intercept $\hat{\beta}_0$ and slope $\hat{\beta}_1$. (iv) Predict $prate$ when $mrate = 3.5$ using regression equation and assess reasonableness. (v) Evaluate how much variation in $prate$ is explained by $mrate$ using $R^2$. --- 2. **Problem C2.2 (Data file: CEOSAL2.dta):** (i) Find average salary ($\overline{salary}$) and average tenure ($\overline{ceoten}$). (ii) Count CEOs with tenure = 0 and find max tenure. (iii) Estimate regression: $$ \log(salary) = \beta_0 + \beta_1 ceoten + u $$ Report coefficients, sample size, $R^2$. Find approximate predicted percentage change in salary per 1 year increase in tenure. --- ### Solutions (conceptual, data needed to compute actual numbers): 1. C2.1 1. Compute sample means: $$ \overline{prate} = \frac{1}{n} \sum_{i=1}^n prate_i, \quad \overline{mrate} = \frac{1}{n} \sum_{i=1}^n mrate_i $$ 2. Perform OLS regression: $$ prate_i = \hat{\beta}_0 + \hat{\beta}_1 mrate_i + \hat{u}_i $$ Output includes $\hat{\beta}_0$, $\hat{\beta}_1$, $n$, and $R^2$. 3. Interpretation: - Intercept $\hat{\beta}_0$ represents average $prate$ when $mrate=0$ (plan with zero matching). - Slope $\hat{\beta}_1$ shows expected increase in $prate$ per unit increase in $mrate$. 4. Prediction: $$ \hat{prate} = \hat{\beta}_0 + \hat{\beta}_1 \times 3.5 $$ Assess if prediction is within plausible range (0-100%). If out of range, note model extrapolation limitations. 5. The $R^2$ shows fraction of variance in $prate$ explained by $mrate$; values closer to 1 indicate strong explanatory power. --- 2. C2.2 1. Compute averages: $$ \overline{salary} = \frac{1}{n} \sum_{i=1}^n salary_i, \quad \overline{ceoten} = \frac{1}{n} \sum_{i=1}^n ceoten_i $$ 2. Count CEOs with zero tenure: $$ \text{count} = \sum_{i=1}^n I(ceoten_i=0) $$ Find max tenure: $$ \max_{i=1}^n ceoten_i $$ 3. Estimate regression: $$ \log(salary_i) = \hat{\beta}_0 + \hat{\beta}_1 ceoten_i + \hat{u}_i $$ Interpretation: - $\hat{\beta}_1$ approximately equals the percentage increase in salary for one more year as CEO, since: $$ \%\Delta salary \approx 100 \times \hat{\beta}_1 $$ --- **Note:** Actual numerical computations require the data files and statistical software.