1. **Problem Statement:** Estimate the linear regression model $$Y = \beta_1 + \beta_2 X_2 + \beta_3 X_3 + U$$ using the given summary statistics for $N=9$ observations. 2. **Given Data:** $$\sum Y = 1052, \quad \sum X_2 = 1017, \quad \sum X_3 = 954,$$ $$\sum YX_2 = 119750, \quad \sum YX_3 = 111433, \quad \sum X_2 X_3 = 107690,$$ $$\sum Y^2 = 124288, \quad \sum X_2^2 = 115571, \quad \sum X_3^2 = 101772.$$ 3. **Step a: Estimate coefficients $\beta_1, \beta_2, \beta_3$** - Construct matrices for regression: Let $\mathbf{X}$ be the design matrix with a column of ones for intercept and columns for $X_2$ and $X_3$. - Compute means: $$\bar{Y} = \frac{1052}{9} = 116.89, \quad \bar{X}_2 = \frac{1017}{9} = 113, \quad \bar{X}_3 = \frac{954}{9} = 106.0.$$ - Compute elements of $\mathbf{X}^T \mathbf{X}$ and $\mathbf{X}^T \mathbf{Y}$: $$\mathbf{X}^T \mathbf{X} = \begin{bmatrix} 9 & 1017 & 954 \\ 1017 & 115571 & 107690 \\ 954 & 107690 & 101772 \end{bmatrix}, \quad \mathbf{X}^T \mathbf{Y} = \begin{bmatrix} 1052 \\ 119750 \\ 111433 \end{bmatrix}.$$ - Estimate coefficients by: $$\hat{\beta} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{Y}.$$ - Calculate inverse and multiply (using matrix algebra or software): 4. **Step b: Interpretation of coefficients** - $\hat{\beta}_1$ is the estimated intercept, the expected value of $Y$ when $X_2=0$ and $X_3=0$. - $\hat{\beta}_2$ measures the expected change in $Y$ for a one-unit increase in $X_2$, holding $X_3$ constant. - $\hat{\beta}_3$ measures the expected change in $Y$ for a one-unit increase in $X_3$, holding $X_2$ constant. 5. **Step c: Estimate variance-covariance matrix of $\hat{\beta}$** - Compute residual sum of squares (RSS): $$RSS = \sum Y^2 - \hat{\beta}^T \mathbf{X}^T \mathbf{Y}.$$ - Estimate variance of residuals: $$\hat{\sigma}^2 = \frac{RSS}{N - k}$$ where $k=3$ (number of parameters). - Variance-covariance matrix: $$Var(\hat{\beta}) = \hat{\sigma}^2 (\mathbf{X}^T \mathbf{X})^{-1}.$$ 6. **Step d: Hypothesis test $H_0: \beta_2 = 0$** - Compute test statistic: $$t = \frac{\hat{\beta}_2}{\sqrt{Var(\hat{\beta}_2)}}.$$ - Compare $t$ to critical $t$-value with $N-k=6$ degrees of freedom. - If $|t|$ is greater than critical value, reject $H_0$; otherwise, do not reject. **Final answers require numerical matrix inversion and multiplication, which can be done using software tools.**