1. Suppose the True (=Population regression) model is given by Y = ?1 + ?2 X + u. a) Is this model random? Why or why not? b) Provide the interpretation for slope coefficients. c) Provide OLS estimators for population regression coefficients by minimizing the residual sum of squares. d) What is the role of the random error term? e) Suppose E(u|X) = 0. Show that E(Xu) = 0. f) Show that OLS estimator of slope coefficient ??2,OLS = ?_{i=1}^n (X_i?X?)(Y_i??) / ?_{i=1}^n (X_i?X?)^2 can also be written as ??2,OLS = ?_{i=1}^n X_i(Y_i??) / ?_{i=1}^n X_i(X_i?X?). 2. Based on a sample of 10 observations, the following results were obtained: ? Y_i = 1110; ? X_i = 1700; ? X_iY_i = 205500; ? X_i^2 = 322000; ? Y_i^2 = 132100. Obtain OLS estimators for the simple linear regression model given in question 1 above. Provide the interpretation for slope coefficients. 3. In the model in question 1 above suppose E(u) = 0 and E(u|X) = 0. Find E(Y) and E(Y|X). What is the relation between conditional expectation of Y and unconditional expectation of Y? 4. Again the true model is Y = ?1 + ?2 X + u. Find the formula for the regression coefficients using the restrictions: E(u) = 0 and E(Xu) = 0. How does your finding relate to the answers in question 2 above? Explain.
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- The model \( Y = \beta_0 + \beta_1 X + u \) is not random because \(\beta_0\) and \(\beta_1\) are fixed parameters. The randomness comes from the error term \(u\), which captures the deviation of \(Y\) from the deterministic part \(\beta_0 + \beta_1 X\). Show more…
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Consider the standard simple regression model $y=\beta_{0}+\beta_{1} x+u$ under the Gauss-Markov Assumptions SLR.1, SLR.2, SLR.3, SLR.4 and SLR.5. The usual OLS estimators $\hat{\beta}_{0}$ and $\widehat{\beta}_{1}$ are unbiased for their respective population parameters. Let $\tilde{\beta}_{1}$ be the estimator of $\beta_{1}$ obtained by assuming the intercept is zero (see Section $2-6$ ). i. Find $E\left(\tilde{\beta}_{1}\right)$ in terms of the $x_{i}, \beta_{0},$ and $\beta_{1}$. Verify that $\tilde{\beta}_{1}$ is unbiased for $\beta_{1}$ when the population intercept $\left(\beta_{0}\right)$ is zero. Are there other cases where $\tilde{\beta}_{1}$ is unbiased? ii. Find the variance of $\tilde{\beta}_{1}$. (Hint: The variance does not depend on $\beta_{0} .$ ) iii. Show that $\operatorname{Var}\left(\tilde{\beta}_{1}\right) \leq \operatorname{Var}\left(\widehat{\beta}_{1}\right)$. [Hint: For any sample of data, $\sum_{i=1}^{n} x_{i}^{2} \geq \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2},$ with strict inequality unless $\bar{x}=0 .]$ iv. Comment on the tradeoff between bias and variance when choosing between $\widehat{\beta}_{1}$ and $\tilde{\beta}_{1}$.
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10. (a) Generate a sample of size n = 100 for (X, Y) as follows, where x contains n observations for X and y contains n observations for Y: set.seed(1) x = rnorm(100, 2, 1) es = rnorm(100, 0, 1) y = 1 + 2.5 * x + es (b) Use lm to fit a simple linear regression model to predict Y using X and obtain the estimated standard error σ̂1 of the estimated slope β̂1 via summary. Comment carefully on how σ̂1 is obtained by lm. (c) Apply bootstrap with B = 1000, i.e., obtain B bootstrap samples to obtain the estimated standard error σ̃1 of β̂1. Compare σ̃1 with σ̂1, and provide your findings. (d) Do (a), (b), and (c) with n = 3. (e) Can you do (b) with n = 2? Why or why not?
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