Statistics 2022 Paper II 50 marks Explain

Q6

(a) Explain Akaike information criterion for order selection in an ARMA (p, q) process. 15 marks (b) Define autocorrelation coefficient. What are its consequences for ordinary least squares? Discuss the maximum likelihood estimation of the model, in the usual notations, Y = Xβ + u with AR (autoregressive)(1) disturbance. 20 marks (c) Explain the method of collection of industrial data. Describe the (i) official publications for data collection and (ii) statistics collected by the various official agencies pertaining to industrial production. 15 marks

हिंदी में प्रश्न पढ़ें

(a) एक ए० आर० एम० ए० (p, q) प्रक्रम में क्रम चयन के लिए अकैके सूचना मानदंड की व्याख्या कीजिए। 15 (b) स्वसहसंबंध गुणांक को परिभाषित कीजिए। साधारण न्यूनतम वर्गों के लिए इसके परिणाम क्या हैं? प्रचलित संकेतों में, ए० आर० (स्वसमाश्रयी)(1) विघोष के साथ, निदर्श Y = Xβ + u के अधिकतम संभाविता आकलन का वर्णन कीजिए। 20 (c) औद्योगिक आँकड़ों के संग्रह की विधि की व्याख्या कीजिए। (i) आँकड़ों के संग्रह के लिए राजकीय प्रकाशनों का और (ii) औद्योगिक उत्पादन से संबंधित विभिन्न राजकीय एजेंसियों द्वारा एकत्र किये गये आँकड़ों का वर्णन कीजिए। 15

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Directive word: Explain

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How this answer will be evaluated

Approach

Explain the theoretical foundations across all three parts with appropriate mathematical derivations where required. Allocate approximately 30% effort to part (a) on AIC and ARMA order selection, 40% to part (b) on autocorrelation and MLE estimation given its higher marks, and 30% to part (c) on industrial data collection methods. Structure with clear sectional headings, begin each part with precise definitions, develop through step-by-step reasoning, and conclude with practical implications or limitations.

Key points expected

  • Part (a): Definition of AIC as -2log(L) + 2k where L is likelihood and k is number of parameters; trade-off between goodness-of-fit and model complexity; comparison with BIC/AICc; application to ARMA(p,q) via minimization over candidate orders
  • Part (b): Autocorrelation coefficient ρ_k = Cov(u_t, u_{t-k})/Var(u_t); consequences for OLS: biased standard errors, inefficient estimates, invalid t/F tests; MLE derivation for AR(1) errors with transformation matrix Ω, concentrated likelihood, and iterative estimation
  • Part (c): Methods: census vs sample surveys, ASI (Annual Survey of Industries) schedule, establishment surveys; Official publications: ASI Summary Results, Index of Industrial Production (IIP), Economic Census; Agencies: CSO (now NSO), DIPP, Labour Bureau, RBI industrial data
  • Mathematical rigor: Proper likelihood functions, matrix notation for GLS transformation, stationarity conditions for AR(1) parameter |ρ| < 1
  • Applied context: Indian industrial statistics system, ASI coverage of registered manufacturing, limitations of informal sector data

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly defines AIC formula with proper likelihood specification for (a); accurately states autocorrelation definition and AR(1) error structure u_t = ρu_{t-1} + ε_t for (b); precisely identifies ASI, NSS, and relevant Indian statistical machinery for (c)Basic definitions present but with minor notational errors; AR(1) structure vaguely described; lists some agencies but confuses their functionsIncorrect AIC formula, wrong autocorrelation definition, or fundamental misunderstanding of Indian industrial data sources; omits key institutional names
Method choice20%10For (a) explains penalization rationale and order selection algorithm; for (b) derives GLS/ML approach with proper Ω matrix and Prais-Winsten transformation; for (c) distinguishes between census, survey, and administrative data methods with ASI methodologyMentions MLE and GLS without full derivation; describes data collection superficially without methodological distinctionsConfuses AIC with information criteria properties; applies OLS to autocorrelated errors without correction; conflates different data collection mechanisms
Computation accuracy20%10Accurate log-likelihood expression for ARMA: log L = -T/2 log(2π) - T/2 log(σ²) - 1/2 log|Ω| - 1/2σ² u'Ω^{-1}u; correct concentrated likelihood and first-order conditions; proper variance formulas under AR(1)Correct general likelihood form but errors in matrix specifics or missing |Ω| term; incomplete derivation stepsMajor computational errors in likelihood construction; incorrect variance formulas; no derivation attempted for MLE
Interpretation20%10Interprets AIC's bias-variance tradeoff and overfitting prevention; explains why OLS becomes BLUE only when ρ=0 and efficiency loss magnitude; evaluates reliability of Indian industrial statistics, informal sector gaps, and ASI limitationsGeneric statements about model selection and autocorrelation problems; descriptive account of data sources without critical evaluationNo interpretation of why methods matter; fails to discuss consequences or limitations; purely descriptive without analytical insight
Final answer & units20%10Synthesizes across parts: connects model selection (a) with estimation under misspecification (b) and data quality implications (c); notes consistency of MLE vs AIC's asymptotic properties; concludes with integrated view on statistical reliability for policyConcludes each part separately without cross-references; missing synthesis of time series and industrial statistics themesNo conclusion or abrupt ending; parts treated as disconnected questions; missing any integrative insight

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