Let Y = Z3+ € denote the standard form of the multivariate regression model where Y is a matrix of n observations on m dependent variables, Z is the nx (r + 1) design matrix, 3 is the matrix of regression coefficients and denotes the error matrix. Let Σ denote the m x m covariance matrix of any row of € and assume the rows are independent. The least square estimate for 3 is 3 (Z'Z)-¹Z'Y. The projection matrix is H = Z(Z'Z)-¹Z'. Let the ith column of 3 be denoted by 3, and the i, jth element of Σ by σij. Define the sample covariance matrix of the residuals by = Σ = Prove 1 n-p-1 (c) E() = 0 (d) cov(i, k) = Oik (I-H) -d'ê

Suppose you want to test whether X2 and X3 can jointly explain Y in the following regression model: Y = Bo + B1X1 + B,X2 + B3X3 + u You obtain data for 85 observations and conduct a joint test of significance at 1% level. Your restricted model is given by, OY = Bo + B3X3 +u OY = Bo + B1X1 + B3 X3 + u OY = Bo + B1X1 + u OY = Bo + u

Each of 25 teenage girls with one brother was asked to provide her own height (y), in inches, and the height (x), in inches, of her brother. The scatterplot below displays the results. Only 22 of the 25 pairs are distinguishable because some of the (x,y) pairs were the same. The equation of the least-squares regression line is ŷ = 35.1 + 0.427x.*Girls are on the y-axis and brothers are on the x-axis* a.) Draw the least-squares regression line on the scatterplot above. b.)  One brother’s height was x = 67 inches and his sister’s height was y = 61 inches. Circle the point on the scatterplot above that represents this pair and draw the segment on the scatterplot that corresponds to the residual for it. Give a numerical for the residual. c.)  Suppose the point x = 84 , y = 71 is added to the data set. Would the slope of the least squares regression line increase, decrease, or remain about the same?Explain. Would the correlation increase, decrease, or remain about the same? Explain.

(Yi, X1i, X2i) satisfy the assumptions. You are interested in β1, the causal effect of X1 on Y . Suppose that X1 and X2 are uncorrelated. You estimate β1 by regressing Y onto X1 (so that X2 is not included in the regression). Does this estimator suffer from omitted variable bias? Explain.

Heteroscedasticity   Stigler and Friedland (1983) conducted a study to determine whether the separation of company control from company ownership affects company profits. Using data from 69 companies in the United States, the authors estimate the following model:   profiti = α + β1asseti + β2management_controli + ei Where i is company and: profiti = annual profit (in million dollar) asseti = company asset (in million dollar) management_controli = dummy variable that is worth one if the control of the company is held by the manager   The regression results are presented in the table in the picture   a. Explain whether the statements below are TRUE, FALSE, or CANNOT BE DETERMINED. "If there is a heteroscedasticity problem, the confidence interval of the OLS estimator is not valid." b. Determine the 95% confidence interval for the parameter 2, what can you conclude?

The null hypothesis being tested in the least-squares regression output for B is B1 = B1,0=1. True False

The y-interept bo of a least-squares regression line has a useful interpretation only if the x-values are either all positive or all negative.  Determine if the statement is true or false. Why? If the statement is false, rewrite as a true statement.

Suppose we have a multiple regression model with 2 predictors and an intercept. (Without any interaction or higher order terms, we have only the 2 predictors in the model and the intercept.) We have only n= 6 observations (so it would be rather silly to fit this model to this data, but let's pretend it is reasonable). We find the values of the first 5 residuals are: 2.6, 2.3, 2.5, -1.5, -1.4 What is the value of MSRes for this multiple regression model?

Students who complete their exams early certainly can intimidate the other students, but do the early finishers perform significantly differently than the other students? A random sample of 37 students was chosen before the most recent exam in Prof. J class, and for each student, both the score on the exam and the time it took the student to complete the exam were recorded. a. Find the least-squares regression equation relating time to complete (explanatory variable, denoted by x, in minutes) and exam score (response variable, denoted by y) by considering Sx = 15, sy = 17,r = 39.706, x = 90, ỹ = 78 b. The standard error of the slope of this least-squares regression line was approximately (Sp) is 20.13. Test for a significant positive linear relationship between the two variables exam score and exam completion time for students in Prof. J's class by doing a hypothesis test regarding the population slope B1. Write the null and Alternate hypothesis and conclude the results. (Assume that…

A manufacturer of phone batteries determines that the average length of talk time for one of its batteries is 470 minutes.  Suppose that the standard deviation is known to be 32 minutes and that the data are approximately bell-shaped. Estimate the percentage of batteries whose talk time is between 406 minutes and 534 minutes. Estimate the percentage of batteries whose talk time is less than 438 minutes. Estimate the percentage of batteries whose talk time is more than 534 minutes. answer in excel