Problem 3 (20 points) Answer the following questions: a. Suppose that each value of the predictor variable in a simple least squares problem (with intercept) is duplicated. Therefore we have two independent observations y, and y/. The data will look like below: y 12 1/2 112 Yn Consider the regression of y, on r, using 2n observations and the regression of ston, using n observations. Are the least squares estimates of the intercept and slope the sake for both models? b. Refer to question (a). Compare the regression sum of squares of the two models. Please explain your answer mathematically, c. Refer to question (a). Will the two models have the same t and F statisties? How about R? Please explain your answer d. Consider the simple regression model y = Bir + The Gauss-Markov conditions hold and also N(0,a). Suppose n = 30, 2007 = 169031.7.2017 - 206135.6. Test the hypothesis Ho:8, = 0 against the alternative H, Bio. Use a 0.05. e. Consider the model y = Bo + B14, + The Gauss-Markov conditions hold and also ~ N(0,c). Develop the likelihood ratio test for testing the hypothesis Ho : Bo - 231 = 1 against the alternative that it is not true and show that the test statistic follows the F distribution. (This test statistic should be the same as the one that uses the ratio of two chi-square random variables. These chi-square random variables come from the distribution of Bo 23, -1 and the distribution of (2) to construct the Fstatistic.) Problem 3 (20 points) Answer the following questions: a. Suppose that each value of the predictor variable in a simple least squares problem (with intercept) is duplicated. Therefore we have two independent observations y, and y/. The data will look like below: y 12 1/2 112 Yn Consider the regression of y, on r, using 2n observations and the regression of ston, using n observations. Are the least squares estimates of the intercept and slope the sake for both models? b. Refer to question (a). Compare the regression sum of squares of the two models. Please explain your answer mathematically, c. Refer to question (a). Will the two models have the same t and F statisties? How about R? Please explain your answer d. Consider the simple regression model y = Bir + The Gauss-Markov conditions hold and also N(0,a). Suppose n = 30, 2007 = 169031.7.2017 - 206135.6. Test the hypothesis Ho:8, = 0 against the alternative H, Bio. Use a 0.05. e. Consider the model y = Bo + B14, + The Gauss-Markov conditions hold and also ~ N(0,c). Develop the likelihood ratio test for testing the hypothesis Ho : Bo - 231 = 1 against the alternative that it is not true and show that the test statistic follows the F distribution. (This test statistic should be the same as the one that uses the ratio of two chi-square random variables. These chi-square random variables come from the distribution of Bo 23, -1 and the distribution of (2) to construct the Fstatistic.)