LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

B.A. DEGREE EXAMINATION – ECONOMICS

FourTH SEMESTER – APRIL 2003

sT 4204  –  ECONOMETRICS

 

26.04.2003

9.00 – 12.00                                                                                                     Max : 100 Marks

 

                                                                PART – A                                       (10´ 2=20 marks)

      Answer ALL the questions.

 

1. Define Expectation of a random variable in the discrete and continuous cases.
2. Distinguish between parameter and statistic.
3. Define Best Linear Unbiased Estimator.
4. State any two conditions underlying the ordinary least square (O.L.S.) technique.
5. Define ‘residuals’ in the case of two variable linear model and state its properties.
6. Define coefficient of Determination.
7. Explain the term ‘Linear Hypothesis’ with an example.
8. State any two ways in which ‘specification Error’ occurs.
9. Write down the AR (1) model for the disturbance term stating the assumptions.
10. If X ~ N (3, 4), find P(3 < x < 5).

 

 

 

                                                                PART – B                                         (5´ 8=40 marks)

      Answer any FIVE questions.

 

11. X and Y are random variables whose joint distribution is as follows:

 

X Y	-1        0            1         2
2 3	1/10   3/10         0      2/10 2/10   1/10       1/10      0

Find means and variances of X and Y.

12. Construct a 95% confidence internal for the mean m of a normal population (variance

unknown) given the following observations:

3.25,         4.10,    4.72,    3.64,    3.50,    3.90,    4.85,    4.20,    4.30,    3.75 .

13. For the two variable linear model, obtain the decomposition of the total variation in the data. Present the ANOVA for testing H₀: b₂ = 0. Also, give a heuristic motivation for the ANOVA procedure.
14. Fit a regression line through the origin for the following data on the annual rate of return on a fund (Y) and market portfolio (X) and test the significance of the regression coefficient

Y	40.3	3.6	63.7	-35.2	67.5	37.5	20.0	19.3	-42.0	19.2
X	35.3	9.5	61.9	-29.3	19.5	31.0	14.0	45.5	-26.5	8.5

 

 

 

15. Briefly discuss the test for significance of a subset of regression coefficients in a
    k-variable linear model. In a five-variable model Y= b₁+b₂ X ₂ +b₃ X₃ + b₄ X₄ + b₅ X₅ , suppose that one wants to test H₀ : b₄ = b₅ = 0 with 15 observations and computes the residual sums of squares under the full and restricted regression as 12.25 and 21.37 respectively. Can the hypothesis be  rejected at 5% level of significance?
16. Explain the use of Dummy variables in regression analysis with an illustration.
17. Give the motivation for Generalized least squares (GLS). For two variable linear model, state the GLS estimate of the slope parameter.
18. “Econometrics is an amalgam of economic theory, mathematical economics and Mathematical Statistics; Yet, it is a subject on its own right” –

 

 

                                                                    PART – C                                    (2´20=40 marks)

Answer any TWO questions.

 

19. Let (X,Y)have joint p.d.f. f (x, y) = 2-x-y , , , Find correlation coefficient between X and Y.
20. To study the labour force participation of urban poor families, the following data were obtained from 12 regions:

Region	% Labour force (Y)	Mean family Income (in’ 100 Rupees) (X2)	Mean family size (X3)
1 2	64.3 45.4	19.98 11.14	2.95 3.40
3 4	26.6 87.5	19.42 19.98	3.72 4.43
5 6	71.3 82.4	20.26 18.53	3.82 3.90
7 8	26.3 61.6	16.66 14.34	3.32 3.80
9 10	52.9 64.7	15.13 20.08	3.49 3.85
11 12	64.9 70.5	17.04 15.25	4.69 3.89

Carry out the regression  of Y on (X₂ , X₃). Test the significance of the overall
regression at 5%level of significance.

21. (a) Explain the term structural change. Discuss the test procedure for the hypothesis of no
    structural change against the alternative hypothesis of structural change.

• Discuss two methods of detecting heteroscedasticity and the remedies.      (10+10)

22. What is ‘Multi collinearity’ problem. Discuss a method of detecting multi collinearity in a given data. Also, describe in detail, the remedial measures to overcome the undesirable effects of multi collinearity.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.A., DEGREE EXAMINATION – ECONOMICS

FOURTH SEMESTER – APRIL 2004

ST 4204/STA 204 – ECONOMETRICS

07.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

 

SECTION -A

 

Answer ALL questions.                                                                               (10 ´ 2 = 20 marks)

 

1. Define the term ‘econometrics’.
2. Give any two properties of expectation.
3. List down the axioms of probability.
4. Mention the difference between statistic and parameter.
5. Give any two properties of least square estimates.
6. Consider the regression of y on x given by y = . How will you interpret the regression coefficients?
7. What is analysis of variance?
8. Give an example for structural change.
9. Mention the consequences of multicollinearity.
10. What is meant by heteroscedasticity?

 

SECTION – B

 

Answer any FIVE questions.                                                                                     (5 ´ 8 = 40 marks)

 

11. a) A, B and C are mutually exclusive and exhaustive events associated with a random experiment. find P(A) given that P (B) =  P(A);  P(C) =  P(B).
12. b) If 2 dice are thrown, what is the probability of getting a sum greater than 8?
13. c) If find P(A) and P(B). Hence show that A and B are independent.                                                                                                  (2+2+4)

 

12. a) Explain the term ‘linear’ with reference to a regression model.
13. b) Write down the assumptions in a simple linear regression model. (4+4)

 

13. Suppose that a researcher is studying the relationship between gallons of milk consumed by a family per month (y) and the price of milk each month (x in dollars / gallon). The sample consists of observations in 12 consecutive months.  Analysis of the data assuming a linear model of y on x reveals the following:

.  For this sample, find (i)  (ii)  (iii) Least squares slope (iv) Least squares intercept (v) Standard error of regression  (vi) standard error of slope (vii) Test the hypothesis that the slope coefficient is zero at 5% level.

 

14. Consider the following data set:

Sample No:       1          2          3        4          5

y :      15        10        14        8          3

x:         1          2          3        4          5

1. i) Calculate the least square estimates for and assuming the model

y =  for the above data.

1. ii) Calculate for each sample.

iii) Calculate TSS and RSS.

1. iv) Calculate the coefficient of determination.

 

15. Justify / interpret the following statements:
16. P { – 0.725 < b₁ < 2.35} = 0.90
17. R² = 0.6735

• the estimates obtained by the least squares method are the best when compared with the estimates obtained by some other method.

1. .

 

16. Explain the concept of interval estimation.

 

17. What are dummy variables? Explain how the data matrix is specified in the presence of dummy variables.

 

18. Explain the method of Generalized least squares.

 

SECTION – C

 

Answer any TWO questions.                                                                       (2 ´ 20 = 40 marks)

 

19. a) The joint probability distribution of two random variables X and Y in given by

 

X

-1      0      1

 

-1    0      0      1/3

Y     0     0      0      0

1     0     1/3   1/3

 

find i) the marginal distribution of X and Y.

1. ii) Var(X),  Var(Y)

iii) Cov(X,Y)

1.        iv)  Conditional probability of X given Y = 1.
2. v) Var(X + Y)

 

1. b) Define: Independent events, mutually exclusive events and sample space. (15+5)

 

 

 

 

 

 

 

20. a) Explain the test of overall significance of a multiple regression.

 

1. b) Consider the following ANOVA table:

 

SOURCE	df	Sum of Squares
All Variables (X1, X2, X3, X4)	?	800
First two Variables (X1, X2)	?	300
Difference	?	?
Residual	?	?
Total	25	1500

1. Complete the table by filling the missing values.
2. Test H₀: b₁ = b₂ = b₃ = b₄ = 0

Vs

H₁ :  atleast one b_k 0,  k = 1,2,3,4  at 5% level

iii)       Test H_0­:    b₁ = b₂ = 0

Vs

H₁:  b₁ 0 (or) b₂ 0 at 5% level.

1. iv) Test H₀ : b₃ = b₄ = 0 Vs H₁: b_3­ 0 (or) b₄ 0 at 1% level                             (8+12)

 

21. a) Consider the following data

y          X₁        X₂

1          1          2

3          2          1

8          3          -3

Based on this data, estimate the following regressions:

y_i­ = a₀+ a₁ X_{1 i} + u_1i

                                y_i = b₀ + b₁X_1i+ b₂X_2i +u_2i

Is a₁ = b₁?  Why or why not?

 

1. b) i) Explain the concept of structural change.
2. ii) Give the steps involved in ‘Chow test’ to test for structural change. (10+10)

 

22. a) Explain the remedial measures for multicollinearity.

 

1. b) Write short notes on:
2. Coefficient of determination
3. Statistical inference

Random variables.                                                                                   (10+10)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.A. DEGREE EXAMINATION – ECONOMICS

AC 13

FOURTH SEMESTER – APRIL 2006

                                                             ST 4204 – ECONOMETRICS

 

 

Date & Time : 22-04-2006/9.00-12.00         Dept. No.                                                        Max. : 100 Marks

 

 

                                                            Section A

Answer all questions                                                                          ( 10 x 2 = 20 )

1. What is the difference between a ‘Mathematical model’ and an ‘Econometric model’?
2. Give any two properties of variance.
3. Mention any two properties of OLS estimates.
4. Let the sample space for a random experiment be S = {1, 2,…, 50}.Let A and B be two events defined on S with the event space A={all odd numbers between 1 to 50} and B={prime numbers between 1 to 50}.Are A and B independent?
5. Give one example each for discrete and continuous random variable.
6. What is meant by ‘Time series’ data? Give an example for the same.
7. Interpret the following regression equation.

Y = 1.7 + 0.251X₁ + 1.32X₂

where  Y denotes the weekly sales( in ‘000’s)

X₁ denotes the weekly advertisement expenditure

X₂ denotes the number of sales persons.

1. In a multiple regression model, the value of R² is found to be 0.894.How would you interpret it?
2. For a two variable regression model, the observed and estimated (under OLS) values of Y are given below:

Observed Y:    10         14        13        12        17

Estimated Y:    10         13        11        14        15

Calculate the standard error of the estimate.

1. What is meant by ‘dummy variable’?

 

Section B

Answer any five questions                                                                 ( 5 x 8 = 40 )

1. Define the following:

1. Sample space
2. Independent events
3. Conditional probability
4. Random variable

1. If X is a random variable distributed as normal with mean 10 and variance 3  Find  a.) P(X<8)   b.) P(-1<X<5) c.) P(X>15)
2. The following data relates to the family size(X) and family food spending(Y)

of 8 persons randomly selected from a small urban population.

Y: 40   50   50   70   80   100  110  105

X: 1      1     2     1     4      2      4      6

Assuming there is a linear relationship between Y and X, perform a regression of Y on X and estimate the regression coefficients. Also find the standard error of the estimate.

 

1. Consider the following information from a 4 variable regression equation:

Residual sum of squares = 94;

Y = 10,12,14,9,7,8,2,22,4,12.

a.) Find TSS and ESS.

b.) Test the hypothesis that R² = 0 at 5% level.

1. Explain the concept of point and interval estimation with an example.
2. Give the procedure for Chow’s test.
3. Explain the ANOVA and ANCOVA models in regression analysis using

example.

1. Consider the following data set:

Sample no.: 1  2           3          4          5

Y: 15 10        14        8          3

X: 1  2           3          4          5

Calculate the standard errors of the intercept and slope coefficients.

 

Section C

 

Answer any two questions                                                                 ( 2 x 20 = 40 )

 

1. a.) Explain the different steps involved in an Econometric study.

b.) Mention the various assumptions in a Classical Linear Regression model.

(10 + 10 )

1.  A bag contains three balls numbered 1, 2, 3 . Two balls are drawn at random , with replacement , from the bag. Let X denote the number of the first ball drawn and Y the number of the second ball drawn.

1. Find the joint distribution of X and Y
2. Find the marginal distributions of X and Y
3. Find variance of X and variance of Y
4. Are X and Y independent ?
5. Calculate the correlation between X and Y

 

1. a.) Explain the procedure for testing structural change using dummy

variables.

b.) Consider the following data on annual income (in 000’s $) categorized by

gender and age.

Income: 12         10       14       15        6       11       17

Gender:  0         1          1          0        0         1         1

Age:  1         1          0         1         0         0         1

where Gender = 1 if male; 0 if female

Age = 1 if less than or equal to 35; 0 if greater than 35.

Perform a regression of Income on Gender and age. Interpret the results.

What is the benchmark category for the above model?                        ( 8 + 12 )

22. a) Explain the construction of 95% confidence interval for the slope

coefficient in a two variable regression model.

b.) Define the following:

1.) Standard error of the estimate

2.) Correlation coefficient

3.) Differential intercept

4.) Mutually exclusive events.

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 13

B.A. DEGREE EXAMINATION – ECONOMICS

FOURTH SEMESTER – APRIL 2007

ST 4204 – ECONOMETRICS

 

 

 

Date & Time: 19/04/2007 / 9:00 – 12:00            Dept. No.                                                      Max. : 100 Marks

 

 

SECTION A

Answer all questions.                                                             (2*10=20)

 

1. Define the term ‘Econometrics’.
2. What are mutually exclusive and exhaustive events?
3. Let P(A) = 0.3; P(B^C) = 0.4; P(A|B) = 0.54. Find P(A∩B).
4. Consider the following distribution function of X:

X:         -2         -1         1          2

P[X=x]:         0.25     0.5       0.13     0.12

Find E(2X+3).

1. Give any two properties of normal distribution.
2. Define: null and alternative hypothesis.
3. Show that the observed mean and estimated mean of Y for a simple linear model of Y on X are equal.
4. Obtain ESS from the following data given that RSS = 133.

Y:         10        14        17        20        25        30        19        27

1. What is Variance Inflating Factor? When will its value be equal to one?
2. Mention the limitations of OLS estimates in the presence of heteroscedasticity?

 

SECTION B

Answer any FIVE questions.                                                          (5*8=40)

 

11. Explain the concept of population regression function and sample regression

function with an example.

12. A family consists of 4 boys and 4 girls. If 3 among them are selected at

random, what is the probability that a.) all are girls b.) exactly 2 are boys

c.) first and last are girls and middle one is a boy.

13. Fit a binomial distribution to the following data:

X:       0          1          2          3          4

Frequency:       7          10        8          4          1

14. a.) What is a standard normal distribution?

b.) Let X be a normally distributed random variable with mean 24 and

variance 9. Find the percentage of observations  i.) above 10 ii.) between

22 and 25.

15. What is meant by structural change? Explain the procedure of Chow’s test to

test for the presence of structural change.

16. What are dummy variables? Explain its usefulness in regression analysis with

example.

17. Consider the following regression result:

Estimate           Standard Error             T-Ratio

Constant           10.325             2.147                           ?

Intercept           5.12                 ?                                  13.56

Slope               -7.16                1.45                             ?

 

ANOVA TABLE

 

 

Source             df         Sum of Squares

Regressors       2                      ?

Error                ?                     122

Total                22                    348

• Find the missing values.
• Find R² and test the overall significance of the model at 5% level.

18. Explain the method of Generalized Least Squares to estimate the regression

parameters in the presence of heteroscedasticity.

 

SECTION C

Answer any TWO questions.                                                             (2*20=40)

 

19. Five families live in an apartment. The number of cats each family keeps as

pets are indicated in the following table:

Family:            1          2          3          4          5

Number of cats:            0          4          0          0          6

• What is the “mean number of cats” for the population of families?
• Suppose that a researcher took a random sample of two families. What is the sample mean number of cats for each of the 10 possible samples?

 

• Calculate the frequency and relative frequency of each possible value of the mean number of cats for all samples of size two and give the sampling distribution.

 

• What are the mean and variance of this sampling distribution?

 

• Suppose that the researcher took a random sample of four families. What is the sample mean number of cats for each of the 5 possible samples?

 

• Calculate the frequency and relative frequency of each possible value of the mean number of cats for all samples of size four and give the sampling distribution.

 

• What is the mean and variance of this sampling distribution?

 

• Is the mean of the samples an unbiased estimator of the population mean for samples of sizes two and four?

 

• What conclusion can be drawn about the variance of the sampling distribution as the sample size increases?

 

20. Suppose that a researcher is studying the relationship between gallons of milk

consumed by a family per month (Y) and the price of milk each month ( X in

dollars per gallon). The sample consists of observations in 12 consecutive

months. Analysis of the data reveals the following:

∑ Y = 480       ∑ X = 36         ∑ xy = -440

∑ x² = 20         RSS = 528

where x and y are the deviations of X and Y from their respective means.

• Find the least squares intercept and slope.
• Find the standard error of the slope.
• Test whether the slope is significantly different from zero at 5% level.
• Assume that the Total Sum of Squares (TSS) is 1100. Form an ANOVA table and test the significance of the overall model at 5% level.

 

 

 

21. Consider the following data on Y, X₁ and X₂.

Y:         10        20        40        30        50

X₁:       2          5          3          8          7

X₂:       1          0          1          2          1

a.) Fit a linear model of Y on X₁ and X₂. Interpret the regression coefficients.

b.) Calculate R² and interpret it.

c.) Test at 5% Level H₀: R² = 0 Vs H₁: R² ≠ 0.

 

22. ) Explain the various methods of detecting multicollineartiy.
    • Consider the following observed and expected Y values obtained from a linear regression model of Y on X.

Observed Y:     12.4     14.4     14.6     16.0     11.3     10.0     16.2                             10.4     13.1     11.3

Expected Y:     12.1     21.4     18.7     21.7     12.5     10.4     20.8                             10.2     16.0     12.0

Use spearmen’s rank correlation test to test for the presence of

heteroscedasticity (Assume the level of significance to be 0.05).

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.A. DEGREE EXAMINATION – ECONOMICS

NO 16

FOURTH SEMESTER – APRIL 2008

ST 4207 / 4204 – ECONOMETRICS

 

 

 

Date : 25/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION- A

Answer all the questions. Each carries TWO marks.                  (10 x 2 =  20 marks)

 

1. Define sample space and event of a random experiment.
2. If P(A) = ¼  , P(B) = ½  and P(AB) = 1/6 , find (i) P(AB) and (ii) P(A^cB).
3. Given:

X= x :    0      1        2        3        4

P(X=x): 1/6  1/8     ¼      1/12    3/8

Find E(2X + 11).

1. If  f ( x, y) is the joint p.d.f. of X and Y, write the marginals and conditional

distributions.

1. Write any two properties of expected values.
2. Define BLUE.
3. Define the population regression coefficient .
4. Write variance inflating factor of an estimator in the presence of multicollinearity.
5. Define autocorrelation.
6. Define point and interval estimation.

 

SECTION –B

Answer any FIVE questions. Each carries EIGHT marks.                (5 x 8 =  40 marks)

 

1. Consider 3 urns. Urn I contains 3 white and 4 red , Urn II contains 5 white and 4 red and  Urn III contains 4 white and 4 red balls. One ball was drawn from each urn. Find the probability that  the sample will contain 2 white and 1 red balls.
2. If a fair coin is tossed 10 times, find the chance of getting (i) exactly 4 heads

(ii) atleast 6 heads (iii) atmost  8 heads (iv) not more than 4 heads.

13. Derive the least square estimators of the linear model  Y = ₁  + ₂ X + u .
14. State any six assumptions of the linear regression model.
15. How to fit a non-linear regression model of the form Y = ₁ + ₂ X + ₃ X ² ?.
16. Consider the model Y = ₁  + ₂ X + u  where X and Y denote respectively

consumer income (hundreds of dollars per person) and consumption of purple

oongs (pounds per person) . The sample size is 20 , sum of X is 300, sum of Y

is 120 , sum of squares of deviations of X from its mean is 500 , sum of product

of deviations of X and Y from their respective means is 66. 5 and sum of squares

of is 3.6.

• Compute the slope and intercept.
• Compute the standard error of regression.
• Compute the standard error of slope.

 

 

17. In a book of 520 pages , 390 typo- graphical errors occured. Assuming Poisson

law  for the number of errors per page, find the probability that a random sample

of 5 pages contain (i) no error (ii) atleast 3 errors.

 

18. The mean yield for one-acre plot is 662 kg with a standard deviation of 32 kg.

Assuming normal distribution find how many one-acre plots in a batch of 1000

plots will have yield (i) over 700 kg  (ii) below 65 kg .

 

SECTION – C

 

Answer any TWO questions. Each carries TWENTY marks.          (2 x 20 =  40 marks)

 

19. Consider the following joint distribution of (X,Y):

 

X          0            1            2               3

0        1/27      3/27       3/27           1/27

Y      1        3/27      6/27       3/27              0

2        3/27      3/27        0                  0

3        1/27        0           0                  0

 

(a) Find the marginal distributions of X and Y.

(b) Find E( X ) and V ( X )

(c) Find the correlation between X and Y.

(d) Find E ( Y | X = 2 )

(e) Verify whether or not X and Y are independent.

 

20. (a) Explain the following methods of estimation used in the analysis of regression

models:

(i) Maximum likelihood      (ii) Moments

(b) The heights of 10 males of a given locality are found to be 70 , 67, 62 , 68 , 61

68 ,70 , 64 , 64 , 66 inches. Is it reasonable to believe that the average height is

greater than 64 inches ? Test at 5% significance level.

 

21. For the following data on consumption expenditure (Y ) , income ( X₂ ) and wealth

( X₃ ):

 

Y($)   : 70     65      90       95      110      115     120     140      155       150

X₂ ($) : 80    100    120     140    160      180      200     220      240       260

X₃ ($) : 810  1009  1273   1425  1633    1876    2052   2201    2435     2686

• Fit a regression model Y = ₂ X₂ + ₃ X₃ + u .
• Find the correlation coefficients between Y and X₂ , Y and X₃ , X₂ and X₃.
• Find unadjusted and adjusted R² .
• Test H₀ : ₂ = ₃ = 0 at 5% significance level .

 

22. (a) For the k-variate regression model Y = ₁ + ₂ X₂ +…+_k X_k + u

carry out the procedure for testing  H₀ : ₂ = ₃ = … =  _k = 0 against

H₁: atleast one _k 0.

(b) Write the properties of ordinary least square(OLS) estimators under the

normality assumption.

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.A. DEGREE EXAMINATION – ECONOMICS

YB 16

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) x²   , 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 R² and adjusted R².
• 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  

11. 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.

12. 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).

13. 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).

14. 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.

15. 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 = β₁ + β₂X + 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.

17. Explain the log-linear and semi log models.
18. 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  

19. Let X₁ and X₂ have the joint p.d.f.

f(x₁,x₂) = 2  ,  0<x₁<x₂<1   : 0   , elsewhere.

• Find the marginal distributions of X₁ and X₂
• Find the conditional mean and variance of (i) X1 given X₂ = x₂ and

(ii) X₂ given X₁= x₁.

• Find the correlation between X₁ and X₂.

20. (a) Obtain the maximum likelihood estimators of μ and s², if X₁,X₂,…X_n is

a random sample from normal distribution with mean μ and variance s². .

(b) Fit a linear regression model of the form

Y_i = β₂ 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 β₂.

21. (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

22. Fit a regression model of the form

Y_i  = β₁ + β₂ X_2i + β₃ X_3i + u_i   for the following data:

Y :  1     3    8     11   15   14

X₂ : 1     2    3      5     7     6

X₃ : 2     1    4      3     5     4

Also find :

(i) Standard errors of estimators of β₂ and β₃.

(ii) Covariance between the estimators of β₂ and β₃.

 

 

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