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[5 points] In this exercise we investigate what happens to the distribution of $\beta_0$ in linear regression when the homoscedasticity assumption is not satisfied. Assume that, given $x_1, \dots, x_n$, $Y_i = \beta_0 + \beta_1 x_i + \epsilon_i$, where $\epsilon_i \sim N(0, \sigma^2)$. Assume that $\epsilon_i$ are independent. This is similar to the simple linear model we have used in class with the exception that the assumption of homoscedasticity is not satisfied. Consider the estimator for the slope $\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{x}$, where $\hat{\beta}_1 = \frac{\sum_{i=1}^n (x_i - \bar{x})Y_i}{\sum_{i=1}^n (x_i - \bar{x})^2}$. Find the distribution of $\hat{\beta}_0$. (Keep in mind that with heteroschedasticity you loose independence between $\bar{Y}$ and $\hat{\beta}_1$.)

Uploaded: 2022-11-18T11:13:52-08:00
Duration: 2 min 46 s
Channel: Sri K

          [5 points] In this exercise we investigate what happens to the distribution of $\beta_0$ in linear regression when the homoscedasticity assumption is not satisfied. Assume that, given $x_1, \dots, x_n$, $Y_i = \beta_0 + \beta_1 x_i + \epsilon_i$, where $\epsilon_i \sim N(0, \sigma^2)$. Assume that $\epsilon_i$ are independent. This is similar to the simple linear model we have used in class with the exception that the assumption of homoscedasticity is not satisfied. Consider the estimator for the slope $\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{x}$, where $\hat{\beta}_1 = \frac{\sum_{i=1}^n (x_i - \bar{x})Y_i}{\sum_{i=1}^n (x_i - \bar{x})^2}$. Find the distribution of $\hat{\beta}_0$. (Keep in mind that with heteroschedasticity you loose independence between $\bar{Y}$ and $\hat{\beta}_1$.)

[5 points] In this exercise we investigate what happens to the distribution of β0 in linear regression when the homoscedasticity assumption is not satisfied. Assume that, given x1, …, xn, Yi = β0 + β1 xi +, where ∼ N(0, σ^2). Assume that are independent. This is similar to the simple linear model we have used in class with the exception that the assumption of homoscedasticity is not satisfied. Consider the estimator for the slope β̂0 = Y̅ - β̂1 x̅, where β̂1 = (∑i=1^n (xi - x̅)Yi)/(∑i=1^n (xi - x̅)^2). Find the distribution of β̂0. (Keep in mind that with heteroschedasticity you loose independence between Y̅ and β̂1.)

Added by Henry B.

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