UPSC IES ISS 2017 Question Paper for Statistics Paper-III

lES/ISS EXAM,2017
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STATISTICS
Paper - III

Time Allow ed: Three Hours Maximum M a rk s: 200

Question Paper Specific Instructions

Please read each o f the following instructions carefully before attempting
questions:
There are E IG H T questions divided under TW O sections.
Candidate has to attempt F IV E questions in all.
Both the T W O questions in Section A are com pulsory.

Out o f the S IX questi4)n8 in Section B, any T H R E E questiotis are to he attempted.
Attempts o f questions shall be counted in sequential order. Unless struck off, attempt o f a
question shall be counted even i f attempted partly.

The number o f marks carried by a question /part is indicated against it.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.

Assume suitable data, i f necessary and indicate the same clearly.
Any page or portion o f the page left blank in the Question-cum-Answer Booklet must be
clearly struck off.

Answers must be written in E N G L IS H only.

SECTION A

Both the questions are compulsory.

Q l. (a) For SRSWOR (N, n), show that the sample proportion p is unbiased for
the population proportion P. Also derive the sampling variance o f this
estimator. 10

(b) What is the problem in estimating a linear regression model in presence
o f multicollinearity ? How is multicollinearity detected ? Explain how
ridge estimation tackles this issue. 15

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(c) Consider the MA(1> process =* e„ + Pe„ _ j, where e„ N(0,1>.

For a data set it is noted that autocovarianccs are Yq = 1 and Yj = - 0-25.

(i) Estimate p. Which value o f the estimate do you think we should
choose and why ?

(ii) What problem do we have i f = - 0-5 ? How would the variance o f
the error have affected the change ? 15

Q2. (a) Given below are the figures on production (in thousand metric tons) o f a
cooperative sugar factory:

Y ear: 2010 2011 2012 2013 2014 2015 2016
Production : 77 88 84 85 91 98 90

(i) Fit a linear trend by least squares method. Tabulate the trend
values.

(ii) Compute the monthly estimated increase in production during the
period. 10

(b) If, in every stratum, the simple estimator is unbiased, then show
that

L
yst
hsl

is unbiased for population mean y , where Wj^ is the proportion o f
population units in the strata and L denotes the total number o f strata
in the population.

Derive the sampling variance o f ygt and state how you would
unbiasedly estimate the same. 15

(c) In the context o f a finitely distributed lag model, discuss the problem o f
OLS estimation and suggest how to obtain good (consistent) estimates o f
the parameters in such a model by bringing in some restrictions on lag
w'eights. 15

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S E C T IO N B

A n sw er any th re e qu estions o f th e six qu estion s ip v en b elow .

Q3. (a) Explain and illustrate the following ;

(i) Two-stage sampling

(ii) Two-phase sampling

Pinpoint the difference bet\vcen the two types o f sampling schemes. 10

(b) Write briefly on

(i) Sample size determination in surveys;

(ii) Cumulative total method for PPSWR sampling;

(iii) Rao-Hartley-Cochran Scheme. 15

(c) Discuss the following allocations o f the sample size in stratified random
sam pling:

(i) Proportional allocation

{ii) Ncyman allocation

{iii) Optimum allocation with a linear cost function

Explain the practical implications o f these methods. 15

Q4. (a) Discuss Koyck approach to an infinitely distributed lag model and obtain
the mean lag for Koyck’s model. What are the basic features o f Koyck’s
transformed model ? 10

(b) For the following linear regression model

Yj = (3i + (igXi + Uj i = l,...,n

E(ui)s^O, Cov(Uj,Uj) = 0 i95j, E ( u ? ) = a j ,

obtain the OLS estimates o f Pj and P2 when a ? , i = 1, n are known.

Discuss what steps could be taken when a?, i = 1, n are unknown.

Revise the least squares estimates o f and 02 when o? = a^E^(Yj)

where is unknown. 15

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(c) Discuss the problem o f estimating parameters by OLS in the presence o f
serial correlation in the following m odel;

U j « p U j _ j + £ t , - l < p < l , p is known.

E (e^) = 0, V (€j^) = . Cov (Cj, ^ = 0 s ?£0.

Propose suitable estimates o f Pj and 02- calculate the variance of
the estimate o f p2 - estimate be modified when p is
unknown ? J5

Q5. (a) Demand and supply functions o f a certain commodity are respectively

Xj = 240 + 10 ^ - 4p kg per month;
dt

Xg = 100 ” + 6p - 60 Iq; per month,

where p is price o f the commodity at time t.

Find the time path o f p for dynamic equilibrium i f the initial price is to
be ? 72 per kg. 10

(b) Explain briefly the methods o f computing price index numbers

(i) by simple average o f price relatives;

(ii) by simple aggregate o f prices; and

(iii) by weighted a ^ re g a te o f prices. 15

(c) Discuss the different forms o f the Engel curve that are usually employed
for fitting to family-budget data. In such fitting, how would you tackle
the following complications ? 15

(i) Household expenditure on a particular item depends, besides
depending on income, on the number o f persons per family.

(ii) Consumption o f families o f the same size differs because o f varying
age and sex consumption.

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Q6. (a) Give an illustration for linear systematic sampling. Show that, under
this method, a positive correlation between units in the same sample
inflates the sampling variance o f the estimator o f population total. K

(b) Consider a populution o f N 6 units with values 1, 2, 3,4, 5 and 6.

(i) Write down all possible samples o f size 2 drawn by SRSWOR
scheme. Verify that the sample mean is unbiased for the population
moan.

(ii> Also compute the sampling variance o f the sample mean. 15

(c) Explain the ratio method o f estimation for estimating a population total.
Show that it is generally biased. Evaluate the mean squared error o f the
estimator to the first order o f approximation. Assume SRSWOR o f
n units from the population. 15

Q7. (a) Using standard notations, briefly explain the instrumental variable
technique in the context o f estimating the coefBcients in a linear
regression model. State the situations when this technique is applicable. 10

(b) State the rank and order conditions for identiflability o f parameters in a
system o f structural equations. Which one o f these two conditions is
sufficient for identiflability ? Establish this condition mathematically. IS

(c) Discuss the estimation o f parameters o f an equation appearing in a
simultaneous equation system by Limited Information Maximum
Likelihood method. State whether the estimator (if it exists) is unique.
(An outline o f the approach is adequate) 15

Q8. (a) Show that the relationship * 0-7X^_j + 0 3X^_2 ■*"
(where denotes white noise) defines A R IM A d , 1, 1) model. 10

(b) What do you understand by the seasonal variations in a time series ?
Give example.
Explain the method o f link relatives o f computing the seasonal indices. 15

(c) Define correlc^am.
For an infinite series generated by the average o f a random component
with equal weights, show that the correlogram is

1- for k £ m
m 15
0 for k > m

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