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LE.S/.S.5.
Examination - 2025 BGSP-T-STS
STATISTICS
PAPER—III
Time Allowed : Three Hours Maximum Marks : 200
QUESTION PAPER SPECIFIC INSTRUCTIONS
Please read each of the following instructions carefully before
j attempting questions
There are EIGHT questions divided under TWO Sections.
Candidate has to attempt FIVE questions in all.
Both the questions in Section—A are compulsory.
Out of the SIX questions in Section—B, any THREE questions are to be attempted.
The number of marks carried by a question/part is indicated against it.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary, and indicate the same clearly.
Attempts of questions shall be counted in sequential order. Unless struck off, attempt of
a question shall be counted even if attempted partly.
Any page or portion of the page left blank in the Question-cum-Answer (QCA) Booklet must be
clearly struck off.
Answers must be written in ENGLISH only.
BGSP-T-STS/69 1 [P:£:0,
\s. SECTION—A:
1. (a) If the cost function is of the form C=C) + t,./n;, where Cy and t; are known
numbers, then in order to minimize V(Y,;) for fixed total cost, show that n;
must be proportional to
t:
2/3
15
(b) Define the general form of the simultaneous equations model. Based on the
likelihood function, explain the problem of identification. When is an equation
called just identified, under-identified or over-identified? Also, give the order
condition for identification. 15
(c) Show that the demand curve with constant price elasticity of demand is a
hyperbola whose shape depends on the value of the parameter. Hence, show
that when the elasticity of demand is 1, then the demand curve is an
equilateral hyperbola. 10
2. (a) Consider the following observations of a population of size 5 :
110; 60; :-80;- 20,430
If a simple random sample of size 2 is taken from this population, then using
the given observations, show that simple random sampling without
replacement gives a more efficient estimator of population mean than simple
random sampling with replacement. 15
(b) In general linear model y= XB+u, how does the presence of multicollinearity
affect the squared length of the OLS estimator of B ? Describe multicollinearity
index and variance inflation factor for the detection of multicollinearity. 10
(c) Define the following terms :
() Autocovariance function
(ii) Autocorrelation function
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(iii) Spectral density function
For the stochastic processes
Z, =10+a,; +0-7a;_)
Z, =10+a, —0-7a,_,
where {a,} is a sequence of uncorrelated N(0,1) variables, obtain the covariance
function and spectral density function. 1s
SECTION—B
3. (a) Define product estimator for estimating population total. Obtain its bias and
variance to the first order of approximation. 15
(b) Derive the mixed regression estimator for the regression coefficients vector of a
general linear model involving stochastic linear restrictions for the regression
parameters. Show that the resulting estimator is unbiased. Obtain the
variance-covariance matrix of the mixed regression estimator and prove that
the difference between the variance-covariance matrix of OLS estimator and
that of mixed regression estimator is positive definite. 15
(c) Define time series. When is a time series called stationary time series? Give an
example of stationary time series. Check the stationarity of the following time
series y; :
Yt = YUr-1 + Ut 10
4. (a) Avvillage has 10 orchards containing 150, 80, 50, 200, 100, 40, 160, 60, 140
and 220 trees respectively. Select a sample of 3 orchards with probability
proportional to the number of trees in the orchards by cumulative total
procedure if the random numbers drawn from the random number table are
850, 300 and 650. 10
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(b) Consider the general structural model (the symbols have their usual meanings)
y¥ Bot Koos
(nx M)(MxM) (nxK)(KxM) (nx M)
For estimating the coefficients of the first equation
u= Y% Bek Ay ty
(nxl) (nxm-1)(m-1x1) (nxk)(kx1) (nx1)
which is assumed to be (just or over) identified, derive the two-stage least
squares (2SLS) estimator. Prove that for the just identified case, the 2SLS
estimator reduces to the indirect least squares (ILS) estimator. 15
(c) Explain briefly the major steps involved in the applications of the Box-Jenkins
methodology for forecasting. Interpret the following model of time series Y; for
forecasting :
G =23-08+0-34y;_, -0-29y;_g -0-26y;_15
where y; =(Y; -Y%-1) 15
5. (a) Describe non-sampling errors and their sources. 15
(b) Consider the linear regression model y; =8,x,; +BoX9; tuj; i=1, ---, n with
2 2
Di = DXi =O, Ye = x = Vix =p
i i i i i
Obtain the OLS estimator of B = (8; B.)’. Derive the variance-covariance matrix
and explain how the presence of multicollinearity affects the performance of
the OLS estimators of B; and B. 10
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(c) Describe briefly the various methods of weighted indices for constructing an
index number of prices.
A worker in the city of Ahmedabad earns € 45,000 per month. The cost of living
index for a particular month is given as 130. Using the following information,
find the amount of money he spent on clothing and house rent :
Group Expenditure Group index
(in )
Food 14,000 180
Clothing ? 150
House rent ? 100
Fuel and lighting 6,600 110
Miscellaneous 7,500 80 is
6. (a) Define Horvitz-Thompson estimator for estimating population mean. Show that
it is unbiased and give its variance. What is the main drawback of this
estimator and how can it be tackled? Explain. 15
(b) Consider the following structural form of the simultaneous equations model :
Yat +BroYar +¥i1%e + ¥i2Xae = Ua
Yor +Baiit +Y23%at = Vat
Here y's are the endogenous and x’s are the predetermined variables.
(i) Obtain the reduced form of the model.
(ii) Using rank and order conditions, check the identifiability of both the
equations of the model. 15
(c) Describe briefly the various mathematical tests of consistency for the
construction of index numbers. Show that Fisher’s index number does not
satisfy the circular test using proper example. 10
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7. fal
A sampler proposes to draw a stratified random sample. He expects that his
field cost will be of the form YC;n;. His advance estimates of relevant quantities
for the two strata are as follows :
Strata Size of | C,; (in®) S;
strata
1 400 4 10
2 600 9 20
(j) Find the values of n, and ny that minimize the variance of the estimator of
population mean if the sample size is 264.
(ii) What will be the total field cost?
(b) Consider the multiple linear regression model
y=XBt+u; E(u=0, E(uu)= o71,
() Derive an ordinary least squares (OLS) estimator of B and its
variance-covariance matrix.
(i) For a forecast period, say f, with X variables uncertain, give unbiased
point prediction and obtain its variance.
(c) 1fU=CX°%Y8 is an individual’s utility function of two commodities, show that
his demand for the goods is
jee Boe ae
at+B py a+B p,
where p,, Py are the fixed prices of the two commodities, and m is the
individual’s fixed income. Deduce that the elasticity of demand for either good
with respect to income or price is equal to unity in absolute value.
8. (a) Explain double sampling. Define double sampling ratio estimator for
estimating the population mean and obtain an expression for its variance to
the first order of approximation.
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10
15
15
15
(b) For the linear model with AR(1) disturbances y, =Bx;+uU,z; U; =PUy_) +E;,
obtain the OLS estimator of B and derive its variance. Compare it with the
variance of OLS estimator when no autocorrelation is present. Comment on the
effect of ignoring autocorrelation. 10
(c) Define demand and supply functions and price elasticities of demand and
supply functions. Explain how the change in price on the total expenditure
made by the population affects the purchase of commodity using price
elasticity of demand. 15
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