Financial Econometrics Summer 2015, Rutgers Bu siness School Problem #1 (15 points) y x 2.8 21 3.4 24 3 26 3.5 27 3.6 29 3 25 2.7 25 3.7 30 i) Use the data above to fit a simple linear regress ion, write down the estimated regression equation. (

Financial Econometrics Summer 2015,

Rutgers Bu

siness School

Problem #1

(15 points)

2.8

3.4

3.5

3.6

2.7

3.7

i) Use the data above to fit a simple linear regress

ion, write down the estimated

regression equation.

(5 points)

ii) Interpret the slope coefficient. Interpret the intercept coefficient. (2 points)

iii) Calculate SSR, SST, and SSE. Calculate the coeffici

ent of determination and

interpret.

points)

iv)

Draw a plot consisting of the data and the fitted regression line.

(2 points)

Problem #2

(8 points)

Consider the following function

頀



鐀

Financial Econometrics Summer 2015,

Rutgers Bu

siness School

iii)

Calculate

谀㠀簀

頀

Ⰰ

. [hint: use the definition of variance and part i) and ii)]

points)

iv) Calculate

谀㠀簀

(

頀

)



頀

. (4

points)

Calculate

谀㠀簀

(



頀

Ⰰ

[hint: if it is not obvious, use the following fac

t from

probabilit

y theory,

谀㠀簀

頀



谀㠀簀

頀

鐀

谀㠀簀

(



頀

鐀

)

]

points)

vi)

Assuming again all Gauss‐Markov assumptions hold, in p

articular the

distributional assumption

堀

(

㠀

)

, how is

頀

distributed?

(4 points)

vii)

Write down the Likelihood function of the

regression model.

[hint: use vi) and if

needed the Appendix]

(6 points)

Problem #4

(10 points)

Consider the following regression models

log

頀



log

鐀



log

鐀

Financial Econometrics Summer 2015, Rutgers Business School

Write

in terms of the population coefficients of the true model. Calculate

what is the bias?

Problem #8 (30 points)

The following equation was estimated by OLS (values in parentheses are SE’s):

= 1.32 - 0.0135

+ 0.00148

+ 0.09

- 0.87

+ 0.108

(0.03) (0.00043) (0.002) (0.12) (0.022) (0.23) (0.099)

!"##

=0.78, n=101,

= 165.98

!"##

i) Individually, are all coefficients significantly different from 0? Perform the

appropriate hypotheses tests. Clearly state the hypotheses and test statistics, as well

as critical statistic at the 5% significance level. (5 points)

ii) Test for overall significance of the regression model. Clearly state the hypotheses

and the appropriate test and critical statistics. (5 points)

iii) Suppose a researcher is interested if

+ 0.22

is different from 1. Clearly

state the hypotheses and present an appropriate test statistic along with critical

statistic. (5 points)

iv) Interpret the OLS estimates in the above regression. Also interpret the coefficient

of determination. (5 points)

v) Suppose it is of interest to test, if jointly

= 0,

= 0. Write down the

restricted model and the hypotheses. The residual sum of squares for the restricted

model is given to be 198.43. Perform the test and clearly state the test and critical

statistics at the 1% significance level. (5 points)

vi) Finally, perform the following hypothesis test:

= 1,

= 0 vs.

= 1,

: . Suppose that the coefficient of

determination for this restricted model is given to be 0.45. Write down the models

you have to use for testing and the appropriate statistics at the 10% significance

level. (5 points)

Problem #9 (20 points)

Let be the sample average from a random sample where

has mean and

variance

. Consider two estimators for :

!!!

and

i) Check which estimator is unbiased or biased. State the bias. What happens to the

ii) Financial Econometrics Summer 2015, Rutgers Business School

iii) ii) Find the probability limits for the two estimators. [hint: the probability limit of a

iv) deterministic sequence is just the regular limit, rewrite the first estimator so that

v) you an use the Law of Large Numbers]. Which estimator is consistent? (5 points)

vi)

vii) iii) Find the variances of the estimators. (5 points)

viii)

ix) iv) Which estimator do you prefer? (5 points)

xi)

xii) Problem #10

xiii)

xiv) Consider the multiple regression model with three independent variables, under

xv) CLM assumptions MLR.1-MLR.6:

xvi) =

xvii) !

xviii) +

xix) You would like to test the null hypothesis

xx)

xxi) i) Let

xxii) !

xxiii) ,

xxiv) !

xxv) denote the OLS estimators of

xxvi) !

xxvii)

xxviii) !

xxix) +

xxx) !

xxxi) !

xxxii) !

xxxiii)

xxxiv) !

xxxv) :

xxxvi) ,

xxxvii) !

xxxviii) !

xxxix) +

xl) !

xli) - 3

xlii)

xliii) !

xliv) !

xlv) +

xlvi) = 1.

xlvii) . Find

xlviii) !

xlix) - 3

l) in terms of the

li) variances of

lii) !

liii) ,

liv) !

lv) !

lvi) and the covariance between them. What is the standard error of

lvii)

lviii) !

lix) - 3

lx) .

lxi)

lxii) ii) Write the t-statistic for the hypothesis test.

lxiii)

lxiv) !

lxv) iii) Define =

lxvi) !

lxvii) - 3

lxviii) !

lxix) and

lxx) =

lxxi) !

lxxii) - 3

lxxiii) . Write a regression equation involving

lxxiv)

lxxv) !

lxxvi) , ,

lxxvii) !

lxxviii) ,

lxxix) !

lxxx) !

lxxxi) that allows you to directly obtain

lxxxii) and its standard error.

lxxxiii)

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Financial Econometrics Summer 2015, Rutgers Bu siness School Problem #1 (15 points) y x 2.8 21 3.4 24 3 26 3.5 27 3.6 29 3 25 2.7 25 3.7 30 i) Use the data above to fit a simple linear regress ion, write down the estimated regression equation. (

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