Đề thi giữa kì môn kinh tế lượng

Question 1 (1) Model 6: OLS, using observations 1-177 Dependent variable: Y coefficient std. error t-ratio p-value ------------------------------------------------------------------------------------------------------------ const 619.800 65.4907 9.464 2.18e-017 X1 13.2968 [A] 2.361 0.0194 X2 0.0196671 0.0109819 [B] 0.0751 X3 [C] [D] 2.067 0.0402 Mean dependent var 865.8644 S.D. dependent var 587.5893 Sum squared resid [E] S.E. of regression 532.3780 R-squared 0.193089 Adjusted R-squared 0.179096 F([F], 173) 13.79925 P-value(F) 4.13e-08 We have the following 95% Confidence interval for X3 coefficient: VARIABLE COEFFICIENT 95% CONFIDENCE INTERVAL X3 [C] 0.0153950 0.666662 a. Calculate A, B, C, D, E, F b. Is X2 statistically significant at 5% significance level for (i) 2-sided test ? (H0: βX2 = 0 , H1 : βX2 ≠ 0 ) (ii) 1-sided test ? (H0: βX2 = 0 , H1 : βX2 > 0 ) c. Suppose now we want to add a new variable (X4) to the model (2) And get the R2 for the new model (2) equal to 0.198404. Using F-test, decide whether we should add X4 to the model (1) or not, given Fcritical = 3.896 ( ). Remember to write down the null and alternative hypothesis. Question 2 While evaluating the effect of various firm-specific factors on the returns of a sample of 200 firms, a financial analyst estimates a model with the following result (with standard errors in parenthesis): = 0.080 + 0.801Si + 0.321MBi + 0.164PEi - 0.084BETAi (0.064) (0.147) (0.136) (0.420) (0.120) Where: ri is the percentage annual return for the stock of the firm i Si is the size of the firm i measured in terms of sales revenue MBi is the market to book ratio of the firm PEi is the price/earnings (P/E) ratio of the firm BETAi is the stock CAPM beta coefficient. a. Interpret the result of each coefficient in this regression. b. Calculate t-ratio for each coefficient. c. Given tcritical=1.972 for 2-sided test at 5% of significant level, which factors have significant effect on stock returns? d. If a stock’s beta increased from 1 to 1.3, what would be the expected effect on the stock’s returns? e. Supposed we have two firms with the same MB and same P/E, but firm A has sales revenue (S) of 1 (units) higher than firm B; the Beta of firm A is 0.7, and of firm B is 1.1. What is the difference in annual returns (r) between firm A and firm B? Which firm has higher returns?

Question 1

Y =β0 +β1X1+β2X2 +β3X3 +u (1) Model 6: OLS, using observations 1-177

Dependent variable: Y

coefficient std error t-ratio p-value

const 619.800 65.4907 9.464 2.18e-017

X1 13.2968 [A] 2.361 0.0194

X2 0.0196671 0.0109819 [B] 0.0751

X3 [C] [D] 2.067 0.0402

Mean dependent var 865.8644 S.D dependent var 587.5893

Sum squared resid [E] S.E of regression 532.3780

R-squared 0.193089 Adjusted R-squared 0.179096

F([F], 173) 13.79925 P-value(F) 4.13e-08

We have the following 95% Confidence interval for X3 coefficient:

VARIABLE COEFFICIENT 95% CONFIDENCE INTERVAL

X3 [C] 0.0153950 0.666662

a Calculate A, B, C, D, E, F

b Is X2 statistically significant at 5% significance level for

(i) 2-sided test ? (H0: βX2 = 0 , H1 : βX2 ≠ 0 )

(ii) 1-sided test ? (H0: βX2 = 0 , H1 : βX2 > 0 )

c Suppose now we want to add a new variable (X4) to the model

Y = β0 +β1X1+β2X2 +β3X3 +β4X4 +u (2) And get the R2 for the new model (2) equal to 0.198404

Using F-test, decide whether we should add X4 to the model (1) or not, given Fcritical = 3.896 (F0.05(1,172) 3.896= ) Remember to write down the null and alternative hypothesis

Question 2

While evaluating the effect of various firm-specific factors on the returns of a sample of 200 firms,

a financial analyst estimates a model with the following result (with standard errors in parenthesis):

ˆr = 0.080 + 0.801S i i + 0.321MB i + 0.164PE i - 0.084BETA i

(0.064) (0.147) (0.136) (0.420) (0.120)

Where: r i is the percentage annual return for the stock of the firm i

S i is the size of the firm i measured in terms of sales revenue

MB i is the market to book ratio of the firm

PE i is the price/earnings (P/E) ratio of the firm

BETA i is the stock CAPM beta coefficient

a Interpret the result of each coefficient in this regression

b Calculate t-ratio for each coefficient

c Given tcritical=1.972 for 2-sided test at 5% of significant level, which factors have significant effect on stock returns?

d If a stock’s beta increased from 1 to 1.3, what would be the expected effect on the stock’s returns?

e Supposed we have two firms with the same MB and same P/E, but firm A has sales

revenue (S) of 1 (units) higher than firm B; the Beta of firm A is 0.7, and of firm B is 1.1 What is the difference in annual returns (r) between firm A and firm B? Which firm

has higher returns?

Question 3

X is a population with mean μ and variance σ2

Draw a sample of size 100 from X we have X1, X2,… X100 Therefore, X1, X100 are i.i.d (µ, σ2) Calculate var( )X

X is a population with mean μ and variance σ2 Draw a sample of size n from X we have X1, X2,…

Xn That means X1, Xn are i.i.d (µ, σ2) Prove that

2

var( )X

n

σ

=

Question 4

Regression without regressor (without independent variable).

Suppose you have the model: Yi = + β ui

Use OLS to find the estimator forβ What is the variance of this estimator

Question 5

In class, you know how to retrieve the value of covariance among coefficients using Coefficient covariance matrix menu in Gretl

Supposed now you have a simple regression model (Y =β β1+ 2X u+ ) and you want to compute

cov( , )b b using mathematical formula.

Prove that cov( , )b b1 2 = −X.var( )b2

[Hint: using definitions, basic properties and theorems can be a good way to solve the problem Keep in mind: b1= −Y b X2 ; E b( )1 =β1; E b( )2 =β2 ]