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08统计期中试题


Midterm Exam Applied Regression Analysis
November 2010 Name____________ ID_____________ Score___________

You have up to 1.5 hours to complete the exam. Use your time wisely. Good Luck! Instructions: Calculators are permitted.

一、Multiple Choice
1. Which of the following statements is correct concerning the conditions required for OLS to be a usable estimation technique? ( ) (a) The model must be linear in the parameters (b) The model must be linear in the variables (c) The model must be linear in the variables and the parameters (d) The model must be linear in the residuals. 2. Which one of the following is NOT an assumption of the classical linear regression model? ( ) (a) The explanatory variables are uncorrelated with the error terms. (b) The disturbance terms have zero mean (c) The dependent variable is not correlated with the disturbance terms (d) The disturbance terms are independent of one another.

3. Consider a simple regression model with coefficient standard errors calculated using the usual formulae. Which of the following statements is correct regarding the standard error estimator for the slope coefficient? ( ) ? (a) It varies positively with the square root of the residual variance ( σ ) (b) It varies positively with the spread of X about its mean value (c) It varies positively with the spread of X about zero (d) It varies positively with the sample size T 4. What result is proved by the Gauss-Markov theorem? ( ) (a) That OLS gives unbiased coefficient estimates (b) That OLS gives minimum variance coefficient estimates (c) That OLS gives minimum variance coefficient estimates only among the class of linear unbiased estimators (d) That OLS ensures that the errors are distributed normally

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5. you are given the following data

(

)

? 1.3 2.1 ?1.4 ? ? ?1.6? ? ( X ' X ) ?1 = ? 2.1 0.8 1.9 ? , ( X ' y ) = ? 2.9 ? , σ 2 = 0.86, T = 103 ? ? ? ? ? ?1.4 1.9 3.4 ? ? 0.8 ? ? ? ? ?
The regression equation is yt = β1 + β2X2t + β3X3t + ut
? Which of the following is the correct value for β 1 ?

(a) 2.89 (b) 1.30 (c) 0.84
? (d) We cannot determine the value of β 1 from the information given in the question

二、 following table shows the regression output ,with some numbers erased, when The a simple regression model relating a response variable y to a predictor variable x. Compute the 12 missing numbers
Model Summary Adjusted R Model 1 R (10)__a R Square (9)_____ Square (11)______ Std. Error of the Estimate (12)_____

a. Predictors: (Constant), x

ANOVAb Model 1 Regression Residual Total a. Predictors: (Constant), x b. Dependent Variable: Sum of Squares df Mean Square (4)______ (5)______ F (6)____ Sig. .000a

13060.419 (1)____ (3)______ 574611.720 (2)____ 1387

Coefficientsa Standardized Unstandardized Coefficients Model 1 (Constant) x a. Dependent Variable: B 119.772 -.514 (8)_____ Std. Error .572 Coefficients Beta t 209.267 -.151 (7)____ Sig. .000 .000

2

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

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三、Estimate a model with unknown slope and intercept of a line, assuming the Gauss-Markov Assumption hold. Suppose
n = 12, X = 70, Y = 100,

∑ X Y = 80450, ∑ X
i i

2 i

= 61050, ∑ Yi 2 = 4992040,

.please compute coefficient estimates and their standard errors:

? ? (1) β 0 和 β 1

? (2) σ
? ? (3) se( β 0 ), se( β1 )

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四、Suppose that Yi = β X i + ui , where (ui , X i ) satisfy the Gauss ? Markov conditions. a. Derive the least squares estimators of β and show that it is a linear function of Y1 , Yn . b. Show that the estimator is conditionally unbiased. c. Derive the conditional variance of the estimator. d . Prove that the estimator is BLUE.

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五、Suppose that the simple regression model yi =β 0 +β1x i +u i , where (u i ,x i ) satisfy the Gauss-Markov conditions. ? ? Let β and β be the OLS intercept and slope estimators,respectively.
0 1

What happens to the least squares regression coefficient estimates. (i)yi is regressed on x i +5 rather than x i . In other words,we add a constant 5 to each observation of explanatory x i and return the regression. (ii)yi +2 is regressed on x i . In other words, a constant 2 is added to yi . (iii)yi is regressed on 2x i . (A constant 2 is multiplied to x i ).

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六、Consider the simple regression model yi = β 0 + β1 xi + ui , please prove the following results: ? β1 Lxx n ? 2r (1)t= = ? σ 1? r2 ?2 β 1 Lxx 2 SSE /1 (2) F = = =t ? SSR / (n ? 2) σ2

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七、A multiple regression of y on a constant x1 and x2 produces the following results:
? y = 4 + 0.4 x1 + 0.9 x2 , R 2 = ? 29 0 0 ? ? ? X ' X = ? 0 50 10 ? ? 0 10 80 ? ? ? 8 , SSR = e ' e = 520, n = 29 60

Test the hypothesis that the two slopes sum to 1.

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