UNIVERSITY OF EXETER

BUSINESS SCHOOL

January 2009

Quantitative and Research Techniques 1

Module Convenor: Dr Andreea Halunga

Duration: TWO HOURS

Answer QUESTION 1 from Section A [34 points]

Answer ONE question from Section B [33 points]

Answer ONE question from Section C [33 points]

This is a closed note paper.

1

Section A

Question 1

Consider the following "true" regression model y = X1

1

+u

(1)

where y is (n 1) ; X1 is a (n k1 ) non-stochastic matrix with full column rank k1 ; 1 is a (k1 1) vector of parameters and u N 0; 2 In :

However, instead of (1), the following regression is estimated y = X1

1 +X2

2 +e

(2)

where X2 is a (n k2 ) non-stochastic matrix with full column rank k2 ,

1) vector of parameters and e is the disturbance term.

2 is a (k2

In regression (2), 2 = 0: Let the OLS estimators of 1 and 2 from the regression (2) be

^1 =

^2 =

X0 M2 X1

1

1

X0 M2 y

1

X0 M1 X2

2

1

X0 M1 y

2

where Mi = In Xi (X0 Xi ) 1 X0 , for i = 1; 2: Denote the OLS i i residual vector from regression (2) as ^ = y X1 ^ 1 X2 ^ 2 . e (a) Is the OLS estimator ^ 2 an unbiased estimator of 2 ? [6 points]

(b) Obtain the variance matrix of ^ 2 , i.e. var[ ^ 2 ]: [7 points]

^0^

ee be the estimated disturbance variance from the nk regression (2), where k = k1 + k2 : Show that s2 is an unbiased estimator of 2 : [7 points]

(c) Let s2 =

(d) Given that the residual vector from regression (2) can be written as ^ = M1 u M1 X2 ^ 2 ; show that cov [ ^ 2 ; ^] = 0: [7 points] e e

(e) Let 2j be the j th element of the parameter vector 2 : Stating clearly all your steps, construct a t-test for testing the null hypothesis H0 : 2j = 0: [7 points]

(BEEM102 January 2009)

2

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Section B

Question 2

Consider the following regression model y =X +u

(3)

where y is (n 1) ; X is a (n k ) non-stochastic matrix with full column rank k; is a (k 1) vector of parameters, E [u] = 0 and

0 ] = 2 I : Let the OLS estimator of

E [uu be ^ = (X0 X) 1 X0 y: n (a) Let e = ^ + Ay be an alternative estimator of ; where A is a (k n) non-stochastic matrix. Under what condition is e an unbiased estimator of ? [7 points]

(b) Given that e is an unbiased estimator of ; obtain the variance matrix of e and show that var[ e ] v ar[ ^ ] is a positive semide…nite matrix. Brie‡ discuss the importance of this result: [10 y points]

(c) Let Z = XH; where H is a (k k ) non-stochastic matrix of rank k: Show that the OLS estimator obtained from regressing y on Z is equal to H 1 ^ . [9 points]

(d) Show that the residual vector obtained from the OLS regression in (c) is equal to the residual vector obtained from the OLS regression of y on X? [7 points]

Question 3

Consider the following regression model y =X +u where X is a (n

k ) stochastic matrix with rank k; the elements of

X0 u

X0 X u are independently distributed, plim

6= 0 and plim

=

n n QXX is a positive de…nite non-random matrix.

(a) Is the OLS estimator ^ from the above regression a consistent estimator of ? [7 points]

(BEEM102 January 2009)

3

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(b) Outline a method by which a consistent estimator of can be obtained, stating any additional assumptions which are necessary.

[10 points]

(c) Derive the limiting distribution of the estimator