Let's go step by step for each part: ### (a) Implication if \( \beta_0 = 0 \) and \( X = 0 \) is within scope **Given model:** \[ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i \] If \( X = 0 \) and \( \beta_0 = 0 \): - The regression function becomes: \( Y = \beta_1 X \). - **Implication:** When \( X = 0 \), \( Y = 0 \) (apart from error \( \epsilon \)). Thus, the regression line passes through the origin \((0, 0)\). --- ### (b) Least squares estimate of \( \beta_1 \) when \( \beta_0 = 0 \) If \( \beta_0 = 0 \), model: \( Y_i = \beta_1 X_i + \epsilon_i \). **Objective:** Minimize \(\sum_{i=1}^n (Y_i - \beta_1 X_i)^2\) Take derivative w.r.t. \( \beta_1 \) and set to 0: \[ \frac{\partial}{\partial \beta_1} \sum_{i=1}^n (Y_i - \beta_1 X_i)^2 = 0 \] \[ \sum_{i=1}^n -2 X_i (Y_i - \beta_1 X_i) = 0 \] \[ \sum_{i=1}^n X_i Y_i = \beta_1 \sum_{i=1}^n X_i^2 \] \[ \boxed{ \hat{\beta}_1 = \frac{ \sum_{i=1}^n X_i Y_i }{ \sum_{i=1}^n X_i^2 } } \] --- ### (c) Mean, variance, and unbiasedness of \( \hat{\beta}_1 \) Recall: \( Y_i = \beta_1 X_i + \epsilon_i \), \( E[\epsilon_i] = 0 \), \( Var(\epsilon_i) = \sigma^2 \) \[ \hat{\beta}_1 = \frac{ \sum_{i=1}^n X_i Y_i }{ \sum_{i=1}^n X_i^2 } = \frac{ \sum_{i=1}^n X_i (\beta_1 X_i + \epsilon_i) }{ \sum_{i=1}^n X_i^2 } = \beta_1 \frac{ \sum_{i=1}^n X_i^2 }{ \sum_{i=1}^n X_i^2 } + \frac{ \sum_{i=1}^n X_i \epsilon_i }{ \sum_{i=1}^n X_i^2 } = \beta_1 + \frac{ \sum_{i=1}^n X_i \epsilon_i }{ \sum_{i=1}^n X_i^2 } \] **Expectation:** \[ E[\hat{\beta}_1] = \beta_1 + \frac{ \sum_{i=1}^n X_i E[\epsilon_i] }{ \sum_{i=1}^n X_i^2 } = \beta_1 \] So, **it is unbiased**. **Variance:** \[ Var(\hat{\beta}_1) = Var\left( \frac{\sum X_i \epsilon_i}{\sum X_i^2} \right) = \frac{Var(\sum X_i \epsilon_i)}{(\sum X_i^2)^2} \] Since \( \epsilon_i \) are independent: \[ Var(\sum X_i \epsilon_i) = \sum X_i^2 Var(\epsilon_i) = \sigma^2 \sum X_i^2 \] So, \[ Var(\hat{\beta}_1) = \frac{ \sigma^2 \sum X_i^2 }{ ( \sum X_i^2 )^2 } = \frac{ \sigma^2 }{ \sum X_i^2 } \] --- ### (d) Fitting such a model using `lm` in R You fit the model **without an intercept**: ```r lm(Y ~ X + 0) # or equivalently lm(Y ~ 0 + X) ``` --- ### (e) Implication if \( \beta_1 = 0 \); regression plot If \( \beta_1 = 0 \): - Model: \( Y_i = \beta_0 + \epsilon_i \) - The regression function is flat (horizontal line): \( Y = \beta_0 \). - **Plot:** A horizontal line at \( Y = \beta_0 \), regardless of \( X \). --- ### (f) Least squares estimate of \( \beta_0 \) when \( \beta_1 = 0 \) Model: \( Y_i = \beta_0 + \epsilon_i \) Minimize \( \sum_{i=1}^n (Y_i - \beta_0)^2 \) Take derivative w.r.t \( \beta_0 \): \[ \frac{\partial}{\partial \beta_0} \sum_{i=1}^n (Y_i - \beta_0)^2 = -2 \sum_{i=1}^n (Y_i - \beta_0) = 0 \] \[ \sum_{i=1}^n Y_i = n \beta_0 \implies \boxed{ \hat{\beta}_0 = \frac{1}{n} \sum_{i=1}^n Y_i } \] --- ### (g) Expectation and variance of \( \hat{\beta}_0 \) in (f). Is it unbiased? \[ \hat{\beta}_0 = \bar{Y} \] \[ E[\bar{Y}] = E\left[\frac{1}{n} \sum Y_i\right] = \frac{1}{n} \sum E[Y_i] = \frac{1}{n} \sum \beta_0 = \beta_0 \] So, **unbiased**. \[ Var(\bar{Y}) = \frac{1}{n^2} \sum Var(Y_i) = \frac{1}{n^2} \cdot n \sigma^2 = \frac{\sigma^2}{n} \] --- ## **Summary Table** | Part | Solution/Answer | |------|----------------| | (a) | Regression passes through origin; at \( X=0, Y=0 \) | | (b) | \( \hat{\beta}_1 = \frac{\sum X_i Y_i}{\sum X_i^2} \) | | (c) | Mean: \( \beta_1 \), Variance: \( \frac{\sigma^2}{\sum X_i^2} \), Unbiased: Yes | | (d) | Use `lm(Y ~ X + 0)` in R | | (e) | Regression is a horizontal line at \( Y = \beta_0 \) | | (f) | \( \hat{\beta}_0 = \bar{Y} = \frac{1}{n} \sum Y_i \) | | (g) | Mean: \( \beta_0 \), Variance: \( \frac{\sigma^2}{n} \), Unbiased: Yes |

Let's analyze each part systematically: **(a) Implication if \( \beta_0 = 0 \) and \( X=0 \) is within scope** - Model: \( Y_i = \beta_0 + \beta_1 X_i + \epsilon_i \). - Given \( \beta_0=0 \), the model reduces to \( Y_i = \beta_1 X_i + \epsilon_i \). - If \( X=0 \) is within the data range (within scope), then at \( X=0 \), the expected value of \( Y \) is \( E[Y] = 0 \), so the regression line passes through the origin. - **Implication:** The regression function passes through the point \((0,0)\), indicating no intercept. --- **(b) Least squares estimate of \( \beta_1 \) when \( \beta_0=0 \)** - Model: \( Y_i = \beta_1 X_i + \epsilon_i \). - To estimate \( \beta_1 \), minimize the sum of squared residuals: \[ \min_{\beta_1} \sum_{i=1}^n (Y_i - \beta_1 X_i)^2 \] - Derivative w.r.t. \( \beta_1 \): \[ -2 \sum_{i=1}^n X_i (Y_i - \beta_1 X_i) = 0 \] - Solving for \( \beta_1 \): \[ \sum X_i Y_i = \beta_1 \sum X_i^2 \] \[ \boxed{ \hat{\beta}_1 = \frac{\sum_{i=1}^n X_i Y_i}{\sum_{i=1}^n X_i^2} } \] *This is the least squares estimator when intercept is zero.* --- **(c) Mean, variance, and unbiasedness of \( \hat{\beta}_1 \)** - Recall: \( Y_i = \beta_1 X_i + \epsilon_i \), with \( E[\epsilon_i]=0 \). - The estimator: \[ \hat{\beta}_1 = \frac{\sum X_i Y_i}{\sum X_i^2} = \frac{\sum X_i (\beta_1 X_i + \epsilon_i)}{\sum X_i^2} = \beta_1 + \frac{\sum X_i \epsilon_i}{\sum X_i^2} \] - **Expected value:** \[ E[\hat{\beta}_1] = \beta_1 + \frac{\sum X_i E[\epsilon_i]}{\sum X_i^2} = \beta_1 \] - **Variance:** \[ Var(\hat{\beta}_1) = Var\left( \frac{\sum X_i \epsilon_i}{\sum X_i^2} \right) = \frac{1}{(\sum X_i^2)^2} \cdot \sum X_i^2 \sigma^2 = \frac{\sigma^2}{\sum X_i^2} \] - **Unbiased?** Yes, because \( E[\hat{\beta}_1] = \beta_1 \). --- **(d) How to fit such a model using `lm` in R** - To fit a model **without intercept**: ```r lm(Y ~ X + 0) ``` or equivalently: ```r lm(Y ~ 0 + X) ``` This instructs R to estimate the slope **only**, passing through the origin. --- **(e) Implication if \( \beta_1=0 \)** - Model reduces to: \( Y_i = \beta_0 + \epsilon_i \). - The regression is **horizontal** at \( Y = \beta_0 \). - **Plot:** A flat horizontal line at the mean of Y (if estimated from data). **Graphically:** The regression line is a straight horizontal line at \( Y = \hat{\beta}_0 \), indicating no relationship between \( X \) and \( Y \). --- **(f) Least squares estimate of \( \beta_0 \) when \( \beta_1=0 \)** - Model: \( Y_i = \beta_0 + \epsilon_i \). - Minimize: \[ \sum_{i=1}^n (Y_i - \beta_0)^2 \] - Derivative: \[ -2 \sum (Y_i - \beta_0) = 0 \Rightarrow \sum Y_i = n \beta_0 \] - So, \[ \boxed{ \hat{\beta}_0 = \frac{1}{n} \sum_{i=1}^n Y_i = \bar{Y} } \] --- **(g) Expectation, variance, and unbiasedness of \( \hat{\beta}_0 \)** - Since \( \hat{\beta}_0 = \bar{Y} \): \[ E[\hat{\beta}_0] = E[\bar{Y}] = \frac{1}{n} \sum E[Y_i] = \frac{1}{n} \sum (\beta_0 + 0) = \beta_0 \] - Variance: \[ Var(\hat{\beta}_0) = Var(\bar{Y}) = \frac{\sigma^2}{n} \] - **Unbiased?** Yes, because \( E[\hat{\beta}_0] = \beta_0 \). --- **Summary:** | Part | Key Result | |---------|------------------------------| | (a) | Regression line passes through origin at \( (0,0) \). | | (b) | \( \hat{\beta}_1 = \frac{\sum X_i Y_i}{\sum X_i^2} \). | | (c) | Mean: \( \beta_1 \); Variance: \( \sigma^2 / \sum X_i^2 \); Unbiased: Yes. | | (d) | Fit with `lm(Y ~ X + 0)` in R. | | (e) | Regression is a horizontal line at \( Y=\beta_0 \). | | (f) | \( \hat{\beta}_0 = \bar{Y} \). | | (g) | Mean: \( \beta_0 \); Variance: \( \sigma^2 / n \); Unbiased: Yes. |

Let's analyze each part systematically: **(a) Implication if \( \beta_0 = 0 \) and \( X=0 \) is within scope** - Given the model: \( Y_i = \beta_0 + \beta_1 X_i + \epsilon_i \) - With \( \beta_0=0 \), the model reduces to \( Y_i = \beta_1 X_i + \epsilon_i \). - Since \( X=0 \) is within the data range, at \( X=0 \), the expected \( Y \) is \( E[Y] = 0 \). - **Implication:** The regression line passes through the origin \((0,0)\), meaning the intercept is zero, and the predicted \( Y \) is zero when \( X=0 \). --- **(b) Least squares estimate of \( \beta_1 \) when \( \beta_0=0 \)** - Model: \( Y_i = \beta_1 X_i + \epsilon_i \) - To estimate \( \beta_1 \), minimize the sum of squared residuals: \[ \min_{\beta_1} \sum_{i=1}^n (Y_i - \beta_1 X_i)^2 \] - Setting derivative to zero: \[ -2 \sum_{i=1}^n X_i (Y_i - \beta_1 X_i) = 0 \] \[ \Rightarrow \sum_{i=1}^n X_i Y_i = \beta_1 \sum_{i=1}^n X_i^2 \] - Solving for \( \beta_1 \): \[ \boxed{ \hat{\beta}_1 = \frac{\sum_{i=1}^n X_i Y_i}{\sum_{i=1}^n X_i^2} } \] *This is the least squares estimator assuming no intercept.* --- **(c) Mean, variance, and unbiasedness of \( \hat{\beta}_1 \)** - Recall: \( Y_i = \beta_1 X_i + \epsilon_i \), with \( E[\epsilon_i]=0 \). - The estimator: \[ \hat{\beta}_1 = \frac{\sum X_i Y_i}{\sum X_i^2} = \frac{\sum X_i (\beta_1 X_i + \epsilon_i)}{\sum X_i^2} = \beta_1 + \frac{\sum X_i \epsilon_i}{\sum X_i^2} \] - **Expected value:** \[ E[\hat{\beta}_1] = \beta_1 + \frac{\sum X_i E[\epsilon_i]}{\sum X_i^2} = \beta_1 \] - **Variance:** \[ Var(\hat{\beta}_1) = Var\left( \frac{\sum X_i \epsilon_i}{\sum X_i^2} \right) = \frac{1}{(\sum X_i^2)^2} \cdot \sum X_i^2 \sigma^2 = \frac{\sigma^2}{\sum X_i^2} \] - **Unbiased?** Yes, because \( E[\hat{\beta}_1] = \beta_1 \). --- **(d) How to fit such a model using `lm` in R** - To fit the model **without intercept**: ```r lm(Y ~ X + 0) ``` or equivalently: ```r lm(Y ~ 0 + X) ``` This specifies a model passing through the origin. --- **(e) Implication if \( \beta_1=0 \)** - The model simplifies to: \( Y_i = \beta_0 + \epsilon_i \). - The regression line is **horizontal** at \( Y = \hat{\beta}_0 \). - **Graphically:** The fitted line is a flat horizontal line, indicating **no relationship** between \( X \) and \( Y \). --- **(f) Least squares estimate of \( \beta_0 \) when \( \beta_1=0 \)** - Model: \( Y_i = \beta_0 + \epsilon_i \) - Minimize: \[ \sum_{i=1}^n (Y_i - \beta_0)^2 \] - Derivative: \[ -2 \sum_{i=1}^n (Y_i - \beta_0) = 0 \Rightarrow \sum_{i=1}^n Y_i = n \beta_0 \] - Therefore, \[ \boxed{ \hat{\beta}_0 = \frac{1}{n} \sum_{i=1}^n Y_i = \bar{Y} } \] --- **(g) Expectation, variance, and unbiasedness of \( \hat{\beta}_0 \)** - Since \( \hat{\beta}_0 = \bar{Y} \): \[ E[\hat{\beta}_0] = E[\bar{Y}] = \frac{1}{n} \sum_{i=1}^n E[Y_i] = \frac{1}{n} \sum_{i=1}^n (\beta_0 + 0) = \beta_0 \] - Variance: \[ Var(\hat{\beta}_0) = Var(\bar{Y}) = \frac{\sigma^2}{n} \] - **Unbiased?** Yes, because \( E[\hat{\beta}_0] = \beta_0 \). --- **Summary:** | Part | Key Result | |---------|------------------------------| | (a) | Regression line passes through \((0,0)\). | | (b) | \( \hat{\beta}_1 = \frac{\sum X_i Y_i}{\sum X_i^2} \) | | (c) | Mean: \( \beta_1 \); Variance: \( \frac{\sigma^2}{\sum X_i^2} \); Unbiased: Yes. | | (d) | Fit with `lm(Y ~ X + 0)` in R. | | (e) | Regression is a flat horizontal line at \( Y=\hat{\beta}_0 \). | | (f) | \( \hat{\beta}_0 = \bar{Y} \). | | (g) | Mean: \( \beta_0 \); Variance: \( \frac{\sigma^2}{n} \); Unbiased: Yes. | **In brief:** - When \( \beta_0=0 \) and \( X=0 \) is in scope, the line passes through the origin. - The estimator for \( \beta_1 \) (if intercept is zero) is the ratio of the covariance of \( X \) and \( Y \) over the variance of \( X \). - Estimators are unbiased with known variance formulas.