UPSC Previous Year Question Paper 2026 

Indian Economic Service - Indian Statistical Service Examination

STATISTICS III

1. (a) From a simple random sample of size n taken without replacement from N units, a simple random sample of size n₁ is duplicated without replacement and added to the original sample. Show that the mean based on (n + n₁) units is an unbiased estimate of population mean. Obtain the variance of this estimate. 15

1. (b) Obtain Generalised Least Square Estimator (GLSE) of β in,

Y_(n×1) = X_(n×k) β_(k×1), with E(U U′) = σ² Ω_(n×n),

Ω is a positive definite matrix. Also derive variance covariance matrix of this GLSE and write down its unbiased estimator. 15

1. (c) Explain price elasticities of supply and demand. For the demand function

D = C p^(−α), C > 0, α > 0

Obtain price elasticity of demand and comment on its shape. What happens when α = 1 ? 10

2. (a) In systematic sampling when N = nK, prove that the variance of the rth systematic sample is given by

V(ȳ_r) = ((N − 1)S²)/N − ((n − 1)/n) S²_wsy

where

S² = (1/(N − 1)) ∑_{r=1}^{K} ∑_{i=1}^{h} (y_ri − Ȳ)² and S²_wsy = (1/(n − 1)) ∑_{i=1}^{h} (y_ri − ȳ_r)²

Hence, show that systematic sampling is more efficient as compared to simple random sampling without replacement if within systematic sample variation is more as compared to the overall variation of population, that is, S². 10+5

2. (b) Consider the model

Y = Xβ + U, with E(UU′) = σ²I.

obtain the ordinary least square estimator (OLSE) of β and obtain the variances and covariances between its components. 10

2. (c) State the methods for estimating seasonal variation in time series. Compare "Ratio to trend" and "Ratio to moving average" methods.

Assuming that the trend is absent, determine seasonal indices for the following data and interpret your result.

Year 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
2007 30 40 36 34
2008 34 52 50 44
2009 40 58 54 48
2010 54 76 68 62
2011 80 92 86 82

UPSC Previous year question paper Indian Economic Service - Indian Statistical Service Examination, 2026 exam

SECTION 'B'

3. (a) Differentiate between ordered and unordered estimators in pps sampling without replacement. Write down the Des Raj Estimator, its mean and variance for estimating population total when a sample of size 2 is drawn from population of size N by ppswor. 15

3. (b) Writing down the assumptions usually made in general linear regression model

Y = Xβ + U, where U ~ N_p(0, σ²I).

Obtain the maximum likelihood estimator (mle) of σ². Also examine its unbiasedness. 15

3. (c) The Ministry of Labour retail price index was 104-6 given for a certain time period. The price indices in July 2004 were as follows : Rent and Rates 65, Clothing 220, Fuel and Light 110, Miscellaneous 125. Given that the weights of different items in the group were as follows : Food 60, Rent and rates 16, Clothing 12, Fuel and Light 8 and Miscellaneous 4. What was the price index of the food group ? 10

4. (a) From a list of 3042 names and addresses, a simple random sample of 200 names, taken without replacement, showed on investigation 18 wrong addresses. Estimate the total number of addresses needing corrections in the list and find the standard error of this estimate. 10

4. (b) State the rank and order conditions of identification for structural form of the system. Discuss the identification of the following Keynesian model of income determination :

C_t = α₀ + α₁Y_t + α₂T_t + U
I_t = β₀ + β₁Y_(t−1) + V
T_t = V₀ + V₁Y_t + W
Y_t = C_t + I_t + G_t

where C_t, I_t, T_t and Y_t are endogenous variables and Y_(t−1) and G_t are predetermined variables. 15

4. (c) Explain the concept of unit root test. How unit root test is used while applying Dickey Fuller (DF) test for testing stationarity of a given time series. Interpret the following models of time series y_t for testing stationarity of the time series.

(i) ŷ*_t = 0.00576 y_(t−1)
t = (5.7980)

(ii) ŷ*_t = 28.2054 − 0.00136 y_(t−1)
t = (1.1576) (−0.2191)

(iii) ŷ*_t = 190.3857 + 1.4776t − 0.0603 y_(t−1)
t = (1.8389) (1.6109) (−1.6252)

where y*_t = (y_t − y_(t−1)) and t = (τ) value are the critical values for the models. (The tabulated values at 5% for (i) there is no constant is −1.9434 (ii) there is a constant is −2.8947 (iii) there is a constant and trend is −3.4614) 15

5. (a) If variates y_i, x_i are measured on each unit of a simple random sample of size n (assumed to be large) then show that variance of R̂ = Ȳ/X̄ is approximately

V(R̂) = ((1 − f)/(n X̄²)) × (∑_{i=1}^{N} (y_i − R x_i)² / (N − 1))

where R = Ȳ/X̄ is the ratio of population means and f = n/N, N is the size of population. 15

5. (b) Considering the model

Y_t = α + β X_t + U_t,

use the Durbin-Watson test for autocorrelation, with the following observations on Y and X and give your conclusions :

X: 1 2 3 4 5 6 7
Y: 2 2 2 1 3 5 6

(Given : the values of d_L and d_u on 5% level of significance with n = 7 and one explanatory variable are d_L = 0.700 and d_u = 1.356) 10

5. (c) Describe autocorrelation function and correlogram. If a given time series of profit consists of 90 observations then while doing correlogram what is appropriate lag length to be taken and why ? Explain the role of correlogram in testing stationarity of given time series. 15

6. (a) If there are two strata and φ be the ratio of actual (n₁/n₂) and (n₁/n₂) obtained through Neyman Allocation in stratified random sampling, then ignoring f.p.c, show that whatever be the values of N₁, N₂, S₁ and S₂ the ratio

V(ȳ_st)_N / V(ȳ_st)

is never less than 4φ/(1 + φ)² where V(ȳ_st)_N and V(ȳ_st) represent the variances of sample mean under Neyman Allocation and actual allocation respectively. 15

6. (b) Obtain the Least Square Estimator of β in General Linear Model (GLM)

Y_(n×1) = X_(n×k) β + U_(n×1)

with usual assumptions, when linear restrictions are imposed in the form

r_(g×1) = R_(g×k) β_(k×1), r and R are known constants.

Also obtain variance-covariance matrix of this estimator.

6. (c) Describe the scope and problems which generally occur in the construction of consumer price index numbers. 10

7. (a) What is the main drawback of Cumulative Total Method for drawing a pps sample? Explain Lahiri's Method of drawing a pps sample.

Select a pps sample of size 4 by Lahiri's Method from the population given below if random numbers selected are (8, 16), (10, 13), (4, 26), (5, 35), (7, 26) 6+4

Serial Number of unit Size of the unit
1 50
2 30
3 45
4 25
5 40
6 26
7 44
8 35
9 28
10 27

7. (b) Explain the concept of heteroscedastic disturbances in General Linear Regression Model (GLRM). Also explain their consequences. 15

7. (c) Differentiate between fixed base and chain base indices. In which type of situations chain base indices are derived? Write merits and demerits of chain base indices.

For the following series of index numbers of price of four commodities, calculate the chain index number and fixed base index number. Write your conclusion. 15

Commodity Price in rupees 2007 2008 2009 2010 2011
A 2 3 4 3 7
B 3 6 9 1 3
C 4 12 20 8 19
D 5 7 18 11 22

8. (a) In case of cluster sampling with N clusters, each of size M, n(< N) clusters are selected using simple random sampling without replacement (SRSWOR). Write down an unbiased estimator (ȳ_c) for estimating population mean (Ȳ) in this case. Obtain its variance and show that

V(ȳ_c) = (1/n − 1/N) S_b²

S_b² = ((NM − 1)S² − N(M − 1)S_v²)/(M(N − 1)) (15)

8. (b) What is the nature of autocorrelation disturbances in General Linear Regression Model (GLRM)? Also state the consequences of autocorrelated disturbances on GLRM. 10

8. (c) Explain the following terms using proper examples :

(i) Random walk with Drift
(ii) Trend stationary time series
(iii) Difference stationary time series
(iv) Deterministic trend
(v) Time series integrated of order d

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Indian Economic Service - Indian Statistical Service Examination STATISTICS III

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