Statistics 2021 Paper II 50 marks Compulsory Explain

Q5

(a) Explain Zellner's seemingly unrelated regression model and the feasible generalized least squares method of estimating the model. (10 marks) (b) Explain the functions of N.S.S.O. (10 marks) (c) Obtain the generalized least squares estimators in the two-variable model Yᵢ = β₁ + β₂Xᵢ + uᵢ assuming the heteroscedastic variances are known and obtain their variances. (10 marks) (d) Why is it considered desirable to convert gross scores to some standard scores? Define 'standard scores' and 'normalised scores' and describe how they are derived. (10 marks) (e) Fill in blanks which are marked with a query in the above skeleton life table and explain the meaning of the symbols at the heads of the columns. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) जेलनर के प्रतीयमानतः असंबंधित समाश्रयण निदर्श की व्याख्या कीजिए और निदर्श का आकलन करने के लिए सुसंगत व्यापकीकृत न्यूनतम वर्ग विधि को समझाइए। (10 अंक) (b) राष्ट्रीय प्रतिदर्श सर्वेक्षण संगठन (एन.एस.एस.ओ.) के प्रकार्यों को समझाइए। (10 अंक) (c) यह मानते हुए कि विषम विचलिता प्रसरण ज्ञात है, द्विचर निदर्श Yᵢ = β₁ + β₂Xᵢ + uᵢ में व्यापकीकृत न्यूनतम वर्ग आकलकों को प्राप्त कीजिए और उनके प्रसरणों को प्राप्त कीजिए। (10 अंक) (d) सकल समकों को किन्हीं मानक समकों में रूपांतरित करने को क्यों वांछनीय माना जाता है? 'मानक समकों' और 'प्रसामान्यीकृत समकों' को परिभाषित कीजिए और इन्हें कैसे व्युत्पन्न किया जाता है, इसका वर्णन कीजिए। (10 अंक) (e) रिक्त स्थानों को भरें, जो निम्नलिखित कंकाल बय-सारणी में एक क्वेरी के साथ चिह्नित हैं और कॉलम के प्रमुखों पर प्रतीकों का अर्थ समझाइए। (10 अंक)

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Directive word: Explain

This question asks you to explain. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

The directive 'explain' demands clear exposition with theoretical foundations and derivations. Allocate approximately 20% (10 marks) to each sub-part equally. For (a), present SURE model structure and FGLS estimation steps; for (b), enumerate NSSO functions with Indian statistical system context; for (c), derive GLS estimators with matrix algebra; for (d), clarify standardization rationale with formulae; for (e), complete life table calculations and interpret actuarial symbols. Structure: brief introduction, systematic part-wise treatment with equations, and concluding synthesis on statistical estimation methods.

Key points expected

Part (a): SURE model specification with contemporaneous correlation across equations, disturbance covariance matrix structure, and FGLS two-step estimation (estimated Ω used for feasible estimator)
Part (b): NSSO functions—conducting large-scale sample surveys (NSS, ASI), data dissemination, methodological research, coordination with state DES, and international reporting (SDGs)
Part (c): GLS transformation with known heteroscedastic variances σᵢ², weighted least squares derivation, variance-covariance matrix of estimators, and efficiency comparison with OLS
Part (d): Desirability of standard scores (comparability, norm-referenced interpretation), z-score formula, normalized scores (T-scores, stanines), and derivation steps
Part (e): Life table completion—calculating nqx, npx, nLx, Tx, ex from given lx and ndx columns; explanation of q (mortality), p (survival), L (person-years), T (total), e (expectation)

Evaluation rubric

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly specifies SURE system of equations with stacked notation for (a); accurately lists all 6-7 NSSO functions with constitutional/legal basis for (b); properly states heteroscedasticity assumption and weight matrix for (c); precisely defines both score types with formulae for (d); correctly identifies all life table column symbols for (e)	Partial specification of SURE without matrix form; lists 4-5 NSSO functions generically; states heteroscedasticity but confuses weighting; defines only one score type; identifies most symbols with minor errors	Confuses SURE with SUR or pooled OLS; vague NSSO description without specific surveys; ignores heteroscedasticity assumption; fails to define standard scores; fundamentally misunderstands life table notation
Method choice	20%	10	Selects appropriate FGLS two-step procedure for (a); structures NSSO answer by survey type and coordination role; chooses correct GLS transformation/weighted least squares for (c); applies appropriate standardization methods for different distributions for (d); uses correct life table relationships (nqx = ndx/lx, nLx formulae) for (e)	Describes FGLS but omits first-step estimation; functional but unstructured NSSO listing; uses WLS without showing transformation; presents only z-scores; applies some correct life table relationships	Proposes OLS for SURE or ignores feasibility issue; random NSSO facts without organization; applies OLS despite heteroscedasticity; confuses standardization with normalization conceptually; wrong relationships (e.g., ex = lx/ndx)
Computation accuracy	20%	10	Accurate matrix derivation of FGLS estimator β̂_FGLS = [X'(Σ̂⁻¹⊗I)X]⁻¹X'(Σ̂⁻¹⊗I)Y for (a); correct GLS estimator formulae β̂₂_GLS = Σwᵢxᵢyᵢ/Σwᵢxᵢ² with wᵢ=1/σᵢ² and variance derivation for (c); precise arithmetic in life table completion with consistent radix for (e)	Correct FGLS intuition without full matrix algebra; correct WLS formula but algebraic errors in variance; mostly correct life table calculations with minor arithmetic slips	Major errors in matrix dimensions or inversion; incorrect weighting or ignores σᵢ²; computational errors in life table (e.g., nLx = n × lx+n); missing derivations entirely
Interpretation	20%	10	Explains efficiency gains of SURE over equation-by-equation OLS when Σ is diagonal vs. non-diagonal; interprets NSSO's role in Indian evidence-based policy; clarifies when GLS reduces to WLS and efficiency implications; distinguishes norm-referenced vs. criterion-referenced score use; interprets life expectancy trends and mortality patterns from completed table	Mentions efficiency without explaining cross-equation information; lists NSSO surveys without policy linkage; notes GLS is BLUE but without conditions; basic score comparison; interprets some life table values	No interpretation of why SURE matters; no Indian statistical system context; no comparison with OLS; no practical application of scores; no demographic interpretation from life table
Final answer & units	20%	10	Clear final expressions for all estimators with proper notation; complete NSSO function summary; explicit variance-covariance matrix for GLS estimators; correct standard score formulae with mean/SD specifications; fully completed life table with all columns (lx, ndx, nqx, npx, nLx, Tx, ex) and correct units (years for eₓ)	Most final answers present with minor notation issues; incomplete NSSO summary; variance expressions without full simplification; score formulae with variable confusion; mostly complete table with some missing values	Missing final estimators or wrong expressions; incomplete or wrong NSSO functions; no variance derivation; no standard score formulae; largely blank or wrong life table completion

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