Statistics 2024 Paper II 50 marks Derive

Q6

(a) For the linear model Y = Xβ + u, obtain the expressions for Ordinary Least Squares (OLS) and Generalised Least Squares (GLS) estimators of the parameters. Discuss their properties and compare them. 20 marks (b) Explain Time Reversal Test, Factor Reversal Test and Circular Test in the Index Number Theory. Using the following data, verify whether the Laspeyres' formula satisfies Time Reversal Test. 15 marks (c) Distinguish between population estimates and population projections. Briefly describe the component method of population projection. 15 marks

हिंदी में प्रश्न पढ़ें

(a) रैखिक मॉडल Y = Xβ + u के लिए, प्राचलों के साधारण न्यूनतम वर्ग (OLS) तथा व्यापकीकृत न्यूनतम वर्ग (GLS) आकलकों के व्यंजक प्राप्त कीजिए। उनके गुणों की विवेचना कीजिए और उनकी तुलना कीजिए। 20 (b) सूचकांक सिद्धांत में समय उत्क्रमता परीक्षण, उपादान (तत्व) उत्क्रमता परीक्षण तथा चक्रीय परीक्षण की व्याख्या कीजिए। निम्नलिखित आँकड़ों का उपयोग करके जाँच कीजिए कि क्या लासपेयर्स का सूत्र समय उत्क्रमता परीक्षण को संतुष्ट करता है। 15 (c) जनसंख्या आकलन और जनसंख्या पूर्वानुमान के बीच अंतर स्पष्ट कीजिए। जनसंख्या पूर्वानुमान के लिए घटक विधि का संक्षेप में वर्णन कीजिए। 15

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Directive word: Derive

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

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How this answer will be evaluated

Approach

Begin by deriving OLS and GLS estimators using matrix notation for part (a), allocating approximately 40% of time/space given its 20 marks. For part (b), explain the three index number tests conceptually before applying the Time Reversal Test to Laspeyres' formula with the given data (~30%). For part (c), clearly distinguish estimates from projections and outline the component method with fertility, mortality, and migration components (~30%). Conclude with a brief synthesis of how these statistical tools inform policy-making in India.

Key points expected

  • For (a): Derivation of β̂_OLS = (X'X)^(-1)X'Y and β̂_GLS = (X'Ω^(-1)X)^(-1)X'Ω^(-1)Y with proper assumptions
  • For (a): Comparison of BLUE properties under homoscedasticity vs. heteroscedasticity/autocorrelation; efficiency of GLS over OLS when Ω ≠ σ²I
  • For (b): Clear statement of Time Reversal (P01 × P10 = 1), Factor Reversal (P01 × Q01 = Σp1q1/Σp0q0), and Circular (P01 × P12 × P20 = 1) tests
  • For (b): Numerical verification showing Laspeyres' index fails Time Reversal Test with explicit calculation
  • For (c): Distinction between estimates (current/retrospective) and projections (future-oriented, assumption-dependent)
  • For (c): Component method: projection of births by age-specific fertility, deaths by life tables, migration by net migration rates

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly specifies model assumptions for (a): E(u)=0, E(uu')=σ²I for OLS and E(uu')=σ²Ω for GLS; for (b) correctly identifies Laspeyres formula as Σp1q0/Σp0q0; for (c) defines estimates vs projections with official Indian census contextStates most assumptions correctly but misses non-singularity conditions or confuses population estimates with projections partiallyIncorrect model specification, wrong index number formula, or fundamental confusion between estimates and projections
Method choice20%10Uses matrix differentiation for deriving estimators in (a); applies exact algebraic manipulation for Time Reversal verification in (b); selects appropriate cohort-component methodology for (c) citing Registrar General of India practicesUses correct general approach but with suboptimal notation or skips transformation steps in GLS derivationUses scalar algebra throughout for (a), arithmetic instead of algebraic proof for (b), or describes mathematical method instead of demographic component method for (c)
Computation accuracy20%10Flawless matrix algebra with correct inverse operations for (a); precise numerical demonstration that P01^L × P10^L ≠ 1 for (b); accurate age-group calculations for (c) showing how cohorts survive through projection intervalsMinor arithmetic errors in matrix multiplication or index number calculations that don't affect final conclusionMajor computational errors in derivation steps, incorrect index number values, or missing quantitative demonstration entirely
Interpretation20%10Explains why GLS reduces to OLS when Ω=I; interprets Laspeyres' failure of Time Reversal as upward bias in price measurement; connects component method to India's National Population Policy and Sustainable Development GoalsStates properties without economic/statistical intuition or provides generic interpretation without Indian policy contextNo interpretation of results, or misinterprets efficiency properties, or fails to explain why tests matter for index number construction
Final answer & units20%10Presents estimators in standard matrix form with all conditions; explicitly concludes Laspeyres fails Time Reversal with numerical evidence; specifies projection outputs in person-years with time horizon clearly statedCorrect final forms but missing explicit conclusion statements or units in demographic projectionsIncomplete final expressions, missing conclusion on index number test verification, or no specification of projection timeframe

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