** LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034**

B.A. DEGREE EXAMINATION – ECONOMICS

FOURTH SEMESTER – April 2009

# ST 4207/ ST 4204 – ECONOMETRICS

Date & Time: 27/04/2009 / 9:00 – 12:00 Dept. No. Max. : 100 Marks

**SECTION A **

** Answer all the questions 10 x 2 = 20 marks**

1 If A and B are two events such that P(AU B) = 0.57 , P( A^{c} ) = 0.50 and

P( B^{c }) = 0.60 , find P( A ∩ B ).

- Mention any two properties of variance.
- If X is a continuous random variable having the probability density function

f (x) = (1/9) x^{2} , 0 ≤ x ≤ 3 ; 0 , elsewhere

find P(0 < X < 1).

- Define maximum likelihood estimation.
- Write a note on interval estimation.
- Define sample regression function.
- Distinguish between R
^{2 }and adjusted R^{2}.
- Write the importance of dummy variables in regression models.
- Define variance inflating factor.
- Write any two consequences of multicollinearity.

** SECTION B**

** Answer any five questions 5 x 8 = 40 marks **

- If 10 fair coins are tossed simultaneously , find the probability of getting

(i) exactly 4 heads (ii) at least 8 heads (iii) at most 9 heads (iv) between 7 and 9 heads inclusive.

- Given the following probability distribution:

X=x : -3 -2 -1 0 1 2 3

p(x) : 0.05 0.10 0.30 0 0.30 0.15 0.10

Compute E(X) and V(X).

- Let X be normally distributed with mean 8 and standard deviation 4.

Find (i) P(5≤X≤10) (ii) P(10≤X≤15) (iii) P(X≥15) (iv) P(X≤5).

- A random sample of 10 boys had the following I.Q.’s: 70 120 110 101

88 83 95 98 107 100 .Construct 95% confidence limits for the population mean.

- Two random samples drawn from two normal populations are:

Sample I : 20 16 26 27 23 22 18 24 25 19

Sample II : 27 33 42 35 32 34 38 28 41 43 30 37

Test whether the populations have the same variances. Use 1% significance level. 16. Fit a regression model of the form

Y = β_{1 }+ β_{2}X + u for the following data:

Y : 55 88 90 80 118 120 145 135 145 175

X : 80 100 120 140 160 180 200 220 240 260

Also find residual sum of squares.

- Explain the log-linear and semi log models.
- Explain the ANOVA for testing the equality of parameters for a k-

variable linear regression model.

** SECTION C**

** Answer any two questions 2×20 = 40 marks **

- Let X
_{1} and X_{2} have the joint p.d.f.

f(x_{1},x_{2}) = 2 , 0<x_{1}<x_{2}<1 : 0 , elsewhere.

- Find the marginal distributions of X
_{1} and X_{2}
- Find the conditional mean and variance of (i) X1 given X
_{2} = x_{2} and

(ii) X_{2} given X_{1}= x_{1}.

- Find the correlation between X
_{1} and X_{2}.

- (a) Obtain the maximum likelihood estimators of μ and s
^{2}, if X_{1},X_{2},…X_{n} is

a random sample from normal distribution with mean μ and variance s^{2}. .

(b) Fit a linear regression model of the form

Y_{i }= β_{2} X_{i }+ u_{i } for the following data:

Y : 10 20 25 22 27 15 12

X : 8 12 15 13 16 10 9

Also find

- Standard error of slope parameter
- Residual sum of squares
- 95% confidence interval for β
_{2}.

- (a) Mention the assumptions underlying the method of least squares in the

classical regression model.

(b) Write a note on:

(i) t-distribution (ii) F-distribution (iii) Chi-square distribution

(iv) Normal distribution

- Fit a regression model of the form

Y_{i } = β_{1} + β_{2} X_{2i} + β_{3} X_{3i} + u_{i} for the following data:

Y : 1 3 8 11 15 14

X_{2} : 1 2 3 5 7 6

X_{3} : 2 1 4 3 5 4

Also find :

(i) Standard errors of estimators of β_{2} and β_{3}.

(ii) Covariance between the estimators of β_{2} and β_{3}.

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