f,g please (a)~(g) is question The following results were obtained from 10 sets
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f,g please
(a)~(g) is question
The following results were obtained from 10 sets of observations on Y, X: sigma Y_i = 20, sigma X_i = 30, sigma Y^2_i = 88.2, sigma X^2_i = 92, sigma Y_i X_i = 59. Now we regress Y on X including an intercept term: Y_i = alpha + beta X_i + elementof_i. Assume that usual conditions for the standard classical linear regression, A_0 - A_5, hold. (a) Get the OLS estimators, alpha and beta. (b) Get 95% confidence intervals for each of beta_0, beta_1 and sigma^2. (c) Calculate R^2. (d) Get the Wald statistic for testing H_0: beta = 0. What is the distribution of the test statistic under H_0? (e) You've already studied another approach to getting the test statistic in this situation, which is using restricted and unrestricted sums of squares of residuals. Using this approach, construct the test statistic for testing H_0: beta = 0. Is it the same as the test statistic calculated in (d)? (f) Test H_0: beta_0 + beta_1 = 2 vs. H_1: beta_0 + beta_1 notequalto 2. (g) Suppose that elementof_i is not normally distributed. Explain possible problems of the above tests in this case.Explanation / Answer
(a) OLS estimators:
beta= cov(x,y)/var x
varX=E[x^2] - E^2[x] = (92-30)/10 =6.2
varY=(88.2-20)/10=6.82
Cov(X,Y) = E[XY]-E[X]E[Y] = 50/10-20*30/100 =-1
SO, beta= -1/6.2 = -0.1613
alpha= y_mean- beta*x_mean = 20/10-(-.1613*30/10) = 2.484
(c) R-sq= correlation^2 = (-1/(sqrt(6.2*6.82))^2 = 0.024
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