Statistics 2021 Paper II 50 marks Explain

Q8

(a) What is autocorrelation ? What are its consequences ? Explain the Goldfeld-Quandt test and Glesjer test for heteroscedasticity. (20 marks) (b) Check the identifiability of the following two-equation system : β₁₁y₁ₜ + β₁₂y₂ₜ + γ₁₁x₁ₜ + γ₁₂x₂ₜ = u₁ₜ β₂₁y₁ₜ + β₂₂y₂ₜ + γ₂₁x₁ₜ + γ₂₂x₂ₜ = u₂ₜ Given the restrictions (i) γ₁₂ = 0, γ₂₁ = 0 and (ii) γ₁₁ = 0, γ₁₂ = 0 (15 marks) (c) Describe Leslie matrix and describe Leslie Matrix Technique for the population projection. (15 marks)

हिंदी में प्रश्न पढ़ें

(a) स्वसहसंबंध क्या है ? इसके परिणाम क्या हैं ? विषम विचलितता (हेटेरोस्किडास्टिसिटी) के लिए गोल्डफेल्ड-क्वांट (Goldfeld-Quandt) परीक्षण और ग्लेसजर (Glesjer) परीक्षण को समझाइए । (20 अंक) (b) निम्नलिखित द्वि-समीकरण प्रणाली की अभिज्ञेयता (आइडेंटिफायबिलिटी) की जाँच कीजिए : β₁₁y₁ₜ + β₁₂y₂ₜ + γ₁₁x₁ₜ + γ₁₂x₂ₜ = u₁ₜ β₂₁y₁ₜ + β₂₂y₂ₜ + γ₂₁x₁ₜ + γ₂₂x₂ₜ = u₂ₜ दिये गये प्रतिबंध हैं (i) γ₁₂ = 0, γ₂₁ = 0 और (ii) γ₁₁ = 0, γ₁₂ = 0 (15 अंक) (c) लेस्ली (Leslie) आव्यूह का वर्णन कीजिए और समष्टि प्रक्षेपण के लिए लेस्ली आव्यूह तकनीक का वर्णन कीजिए । (15 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 causal reasoning and illustrative examples. Allocate approximately 40% of time/words to part (a) given its 20 marks weight, covering autocorrelation definition, consequences, and both heteroscedasticity tests with step-wise procedures. Devote roughly 30% each to parts (b) and (c): for (b), construct the identification analysis using order and rank conditions for both restriction sets; for (c), explain Leslie matrix structure, fertility/survival parameters, and iterative projection mechanics with Indian demographic application. Structure as: definitional clarity → methodological exposition → worked application → concluding synthesis.

Key points expected

  • Part (a): Precise definition of autocorrelation (correlation between error terms across observations) and distinction from heteroscedasticity; enumeration of consequences including inflated t-statistics, inefficient OLS estimates, and misleading R²
  • Part (a): Goldfeld-Quandt test: correct procedure of ordering observations, splitting samples, computing F-ratio of residual variances, and interpretation against critical values
  • Part (a): Glesjer test: auxiliary regression of absolute/squared residuals on explanatory variables, test statistic derivation, and comparison with Goldfeld-Quandt in terms of power and applicability
  • Part (b): Application of order condition (K-k ≥ m-1) and rank condition for identification under restriction set (i) γ₁₂=0, γ₂₁=0, showing both equations are identified
  • Part (b): Analysis under restriction set (ii) γ₁₁=0, γ₁₂=0, demonstrating identification failure for equation 1 due to rank deficiency
  • Part (c): Leslie matrix structure: age-specific fertility rates (Fᵢ) in first row, survival probabilities (Pᵢ) on sub-diagonal, zeros elsewhere; matrix dimensions matching age classes
  • Part (c): Population projection technique: iterative multiplication n(t+1) = L × n(t), stable population properties, intrinsic growth rate extraction, and application to Indian census projections

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): correctly distinguishes autocorrelation (temporal/spatial correlation in errors) from heteroscedasticity (non-constant variance) with precise mathematical notation E(uᵢuⱼ)≠0; for (b): properly defines endogenous/exogenous variables and writes identification conditions; for (c): accurately specifies Leslie matrix dimensions and element positionsDefines autocorrelation loosely without mathematical precision; attempts identification conditions but confuses order/rank; describes Leslie matrix elements but misplaces survival probabilitiesConfuses autocorrelation with multicollinearity or heteroscedasticity; fails to set up identification framework; describes Leslie matrix as generic transition matrix without demographic specificity
Method choice20%10For (a): selects appropriate test statistics (F for G-Q, t or LM for Glesjer) with correct auxiliary regression specifications; for (b): applies order condition first, then rank condition via coefficient matrix analysis; for (c): chooses eigenvalue decomposition or iterative projection method appropriatelyStates test procedures mechanically without justifying why specific tests are chosen; applies order condition only, omitting rank verification; uses simple matrix multiplication without discussing convergence propertiesSelects inappropriate tests (e.g., Durbin-Watson for heteroscedasticity); applies identification rules incorrectly or reverses conditions; proposes non-matrix methods for population projection
Computation accuracy20%10For (b): constructs exact coefficient matrices, computes determinants or ranks correctly, shows |γ|≠0 for identification; derives degrees of freedom for G-Q test ((n-c)/2 - k); for (c): demonstrates characteristic equation det(L-λI)=0 and extracts dominant eigenvalueSets up matrices correctly but makes arithmetic errors in rank computation; states G-Q degrees of freedom approximately; mentions eigenvalues without derivationMajor errors in matrix construction (e.g., ignoring zero restrictions); incorrect F-statistic formula; no computational demonstration for Leslie matrix properties
Interpretation20%10For (a): interprets G-Q F-statistic relative to critical value for one-tailed test, explains Glesjer's advantage in detecting heteroscedasticity form; for (b): clearly states 'exactly identified' vs 'underidentified' with economic intuition; for (c): interprets dominant eigenvalue as intrinsic growth rate and eigenvector as stable age distribution, with Indian demographic contextStates test conclusions without economic significance; identifies equations as 'identified' without degree specification; mentions population growth without connecting to demographic transition theoryMisinterprets test outcomes (accepting null when rejection needed); confuses identification concepts; treats Leslie projection as deterministic forecast without discussing uncertainty
Final answer & units20%10Synthesizes across parts: contrasts autocorrelation vs heteroscedasticity remedies; summarizes identification outcomes in tabular form for both restriction sets; provides complete population projection formula with time subscripts and interprets Leslie matrix powers; concludes with policy relevance for Indian statistical practicePresents fragmented answers per part without cross-referencing; states final identification results clearly but without synthesis; gives projection formula without interpretationIncomplete answers for one or more parts; missing final conclusions; no integration of econometric and demographic techniques despite common matrix algebraic foundation

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