1. **Problem Statement:** We have data for $C_t$ and $Y_t$ and want to estimate the parameters $\beta_1$ and $\beta_2$ in the linear regression model: $$ C_t = \beta_1 + \beta_2 Y_t + u_t $$ 2. **Normal Equations:** The normal equations for OLS estimation are: $$ X' y = X' X \beta $$ where $X$ is the design matrix with a column of ones and a column of $Y_t$, and $y$ is the vector of $C_t$ values. 3. **Constructing Matrices:** Let $n=15$ be the number of observations. $$ X = \begin{bmatrix} 1 & 13.0 \\ 1 & 13.2 \\ \vdots & \vdots \\ 1 & 15.0 \end{bmatrix}, \quad y = \begin{bmatrix} 20.2 \\ 20.4 \\ \vdots \\ 21.8 \end{bmatrix} $$ 4. **Calculate sums:** $$ \sum Y_t = 213.2, \quad \sum C_t = 318.7, \quad \sum Y_t^2 = 3033.66, \quad \sum Y_t C_t = 4535.06 $$ 5. **Form $X'X$ and $X'y$:** $$ X'X = \begin{bmatrix} n & \sum Y_t \\ \sum Y_t & \sum Y_t^2 \end{bmatrix} = \begin{bmatrix} 15 & 213.2 \\ 213.2 & 3033.66 \end{bmatrix} $$ $$ X'y = \begin{bmatrix} \sum C_t \\ \sum Y_t C_t \end{bmatrix} = \begin{bmatrix} 318.7 \\ 4535.06 \end{bmatrix} $$ 6. **Solve for $\beta$:** $$ \beta = (X'X)^{-1} X'y $$ Calculate determinant: $$ \det = 15 \times 3033.66 - 213.2^2 = 45504.9 - 45444.2 = 60.7 $$ Inverse: $$ (X'X)^{-1} = \frac{1}{60.7} \begin{bmatrix} 3033.66 & -213.2 \\ -213.2 & 15 \end{bmatrix} $$ Multiply: $$ \beta = \frac{1}{60.7} \begin{bmatrix} 3033.66 & -213.2 \\ -213.2 & 15 \end{bmatrix} \begin{bmatrix} 318.7 \\ 4535.06 \end{bmatrix} = \frac{1}{60.7} \begin{bmatrix} 3033.66 \times 318.7 - 213.2 \times 4535.06 \\ -213.2 \times 318.7 + 15 \times 4535.06 \end{bmatrix} $$ Calculate numerator: $$ \begin{cases} 966,497.5 - 966,497.5 = 0 \\ -67,956.8 + 68,025.9 = 69.1 \end{cases} $$ So: $$ \beta_1 = 0/60.7 = 0, \quad \beta_2 = 69.1/60.7 \approx 1.139 $$ 7. **Calculate residuals and estimate variance:** Predicted $C_t$: $$ \hat{C}_t = 0 + 1.139 Y_t $$ Calculate residuals $u_t = C_t - \hat{C}_t$ and residual sum of squares $RSS$. 8. **Estimate variance of residuals:** $$ \hat{\sigma}_u^2 = \frac{RSS}{n-2} $$ 9. **Calculate standard errors:** $$ \hat{\sigma}_{\beta_k} = \hat{\sigma}_u \sqrt{(X'X)^{-1}_{kk}} $$ where $(X'X)^{-1}_{11} = 3033.66/60.7 \approx 50.0$ and $(X'X)^{-1}_{22} = 15/60.7 \approx 0.247$. **Final answers:** $$ \hat{\beta}_1 = 0, \quad \hat{\beta}_2 \approx 1.139 $$ Standard errors depend on $\hat{\sigma}_u$ computed from residuals.