1. We are given the theoretical model: $$\text{Inflation}_t = \beta_0 + \beta_1 \text{Interest\_Rate}_t + u_t$$ where both variables are in percentage points. We need to find formulas and compute the OLS slope estimate and its standard error. 2. (a) The formula for the OLS slope coefficient $$\hat{\beta}_1$$ is: $$\hat{\beta}_1 = \frac{\text{Cov}(Y,X)}{\text{Var}(X)}$$ where $Y = \text{Inflation}$ and $X = \text{Interest\_Rate}$. 3. (b) The formula for the standard error of the slope coefficient $$SE(\hat{\beta}_1)$$ is: $$SE(\hat{\beta}_1) = \sqrt{\frac{\hat{\sigma}^2}{\sum (X_i - \bar{X})^2}}$$ This can also be written as: $$SE(\hat{\beta}_1) = \sqrt{\frac{\text{RSS}/(N-2)}{(N-1)\text{Var}(X)}}$$ where RSS is residual sum of squares and $N$ is sample size. 4. (c) Compute the OLS slope estimate using given values: $$\hat{\beta}_1 = \frac{\text{Cov}(\text{Inflation}, \text{Interest\_Rate})}{\text{Var}(\text{Interest\_Rate})} = \frac{-25}{50} = -0.5$$ 5. Compute $$SE(\hat{\beta}_1)$$: First, compute residual variance estimate: $$\hat{\sigma}^2 = \frac{\text{RSS}}{N-2} = \frac{49}{100-2} = \frac{49}{98} = 0.5$$ Sum of squares of $X$: $$\sum (X_i - \bar{X})^2 = (N-1) \times \text{Var}(X) = 99 \times 50 = 4950$$ Thus, $$SE(\hat{\beta}_1) = \sqrt{\frac{0.5}{4950}} = \sqrt{0.00010101} \approx 0.01005$$ 6. Final answers: - OLS Slope Estimate: $$\hat{\beta}_1 = -0.5$$ - Standard Error: $$SE(\hat{\beta}_1) \approx 0.01005$$