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5.6 Suppose that, from a sample of 63 observations, the least squares estimates and the corresponding estimated covariance matrix are given by b₁ b₂ a. B₂=0 b. B₁ +28₂ = 5 c. B₁-B₂+ B₂ = 4 2 3 b₂ wollol ads nobizan QUAR 1X3101 tilbag Using a 5% significance level, and an alternative hypothesis that the equality does not hold, test each of the following null hypotheses: 3-2 1 cov (b₁,b₂, b3) = -2 4 0 1 0 3

5.6 Suppose that, from a sample of 63 observations, the least squares estimates and the corresponding estimated covariance matrix are given by b₁ b₂ a. B₂=0 b. B₁ +28₂ = 5 c. B₁-B₂+ B₂ = 4 2 3 b₂ wollol ads nobizan QUAR 1X3101 tilbag Using a 5% significance level, and an alternative hypothesis that the equality does not hold, test each of the following null hypotheses: 3-2 1 cov (b₁,b₂, b3) = -2 4 0 1 0 3

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please answer 5.6 a-c, thanks!
**5.5** For each of the following two time-series regression models, and assuming MR1-MR6 hold, find \( \text{var}(b_1 | x) \) and examine whether the least squares estimator is consistent by checking whether \( \lim_{T \to \infty} \text{var}(b_2 | x) = 0 \). **a.** \( y_t = \beta_1 + \beta_2 t + e_t, \, t = 1, 2, \ldots, T \). Note that \( x = (1, 2, \ldots, T) \), \( \sum_{t=1}^T (t - \bar{t})^2 = \sum_{t=1}^T t^2 - (\sum_{t=1}^T t)^2/T = T(T + 1)(2T + 1)/6 - (T(T + 1)/2)^2/T \). **b.** \( y_t = \beta_1 + \beta_2 (0.5)^t + e_t, \, t = 1, 2, \ldots, T \). Here, \( x = (0.5, 0.5^2, \ldots, 0.5^T) \). Note that the sum of a geometric progression with first term \( r \) and common ratio \( r \) is \[ S = r + r^2 + r^3 + \cdots + r^n = \frac{r(1 - r^n)}{1 - r}. \] **c.** Provide an intuitive explanation for these results. **5.6** Suppose that, from a sample of 63 observations, the least squares estimates and the corresponding estimated covariance matrix are given by \[ \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \\ -1 \end{bmatrix} \qquad \text{cov}(b_1, b_2, b_3) = \begin{bmatrix} -2 & 4 & 0 \\ 4 & 0 & 0 \\ 0 & 3 & 0 \end{bmatrix}. \] Using a
Transcribed Image Text:**5.5** For each of the following two time-series regression models, and assuming MR1-MR6 hold, find \( \text{var}(b_1 | x) \) and examine whether the least squares estimator is consistent by checking whether \( \lim_{T \to \infty} \text{var}(b_2 | x) = 0 \). **a.** \( y_t = \beta_1 + \beta_2 t + e_t, \, t = 1, 2, \ldots, T \). Note that \( x = (1, 2, \ldots, T) \), \( \sum_{t=1}^T (t - \bar{t})^2 = \sum_{t=1}^T t^2 - (\sum_{t=1}^T t)^2/T = T(T + 1)(2T + 1)/6 - (T(T + 1)/2)^2/T \). **b.** \( y_t = \beta_1 + \beta_2 (0.5)^t + e_t, \, t = 1, 2, \ldots, T \). Here, \( x = (0.5, 0.5^2, \ldots, 0.5^T) \). Note that the sum of a geometric progression with first term \( r \) and common ratio \( r \) is \[ S = r + r^2 + r^3 + \cdots + r^n = \frac{r(1 - r^n)}{1 - r}. \] **c.** Provide an intuitive explanation for these results. **5.6** Suppose that, from a sample of 63 observations, the least squares estimates and the corresponding estimated covariance matrix are given by \[ \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \\ -1 \end{bmatrix} \qquad \text{cov}(b_1, b_2, b_3) = \begin{bmatrix} -2 & 4 & 0 \\ 4 & 0 & 0 \\ 0 & 3 & 0 \end{bmatrix}. \] Using a
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