Statistics 2025 Paper II 50 marks Describe

Q8

(a) Define time series. For a moving-average process with weights {a₁, a₂, ..., aₘ} of random components {eᵢ, i = 1, 2, ...}, where eᵢ's are i.i.d. N(0, σ²), obtain the correlogram function. Find its form, when all the weights are equal and their sum is 1. (15 marks) (b) The marks obtained by student A in Mathematics and Language tests of maximum marks 150 each are 120 and 105 respectively. Find out in which subject, student A is more able as compared to other students based on the measure of T score. The following table gives a sample of marks obtained by 15 students of the same class : Score in Mathematics | Score in Language ---|--- 100 | 67 75 | 63 88 | 73 85 | 77 92 | 60 94 | 53 93 | 50 84 | 48 67 | 38 96 | 73 100 | 36 102 | 45 94 | 47 73 | 39 83 | 56 (15 marks) (c) Describe the 2-stage least squares (2SLS) method of estimation of parameters in linear regression model. Also, state the assumptions and discuss its properties. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) काल श्रेणी को परिभाषित कीजिए। एक गतिमान-माध्य प्रक्रम, जिसमें यादृच्छिक घटकों {eᵢ, i = 1, 2, ...} के भार {a₁, a₂, ..., aₘ} हैं, जहाँ eᵢ स्वतंत्र और समान रूप से N(0, σ²) के अनुसार बंटित हैं, के लिए सहसंबंध-चित्र फलन प्राप्त कीजिए। इसके रूप को ज्ञात कीजिए, जबकि सभी भार बराबर हैं और उनका योग 1 है। (15 अंक) (b) एक विद्यार्थी A ने गणित तथा भाषा की परीक्षा में, जिनमें प्रत्येक में अधिकतम अंक 150 हैं, क्रमशः 120 और 105 अंक प्राप्त किए। बताइए कि किस विषय में विद्यार्थी A, T-स्कोर के माप के आधार पर दूसरे विद्यार्थियों की अपेक्षा अधिक योग्य है। निम्नलिखित सारणी में उसी कक्षा के 15 विद्यार्थियों द्वारा प्राप्त अंकों का एक प्रतिदर्श दिया गया है : | गणित में प्राप्तांक | भाषा में प्राप्तांक | |---|---| | 100 | 67 | | 75 | 63 | | 88 | 73 | | 85 | 77 | | 92 | 60 | | 94 | 53 | | 93 | 50 | | 84 | 48 | | 67 | 38 | | 96 | 73 | | 100 | 36 | | 102 | 45 | | 94 | 47 | | 73 | 39 | | 83 | 56 | (15 अंक) (c) रैखीय समाश्रयण मॉडल में, प्राचलों के आकलन की द्विचरण न्यूनतम वर्ग (2 एस० एल० एस०) विधि का वर्णन कीजिए। इसकी कल्पनाओं को भी बताइए तथा इसके गुणों की चर्चा कीजिए। (20 अंक)

Evaluate your answer to this question → See all 2025 Statistics questions

Directive word: Describe

This question asks you to describe. 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 'describe' demands systematic exposition with technical precision. Structure: (a) 30% time/space—define time series rigorously, derive MA(m) autocorrelation structure, and simplify to uniform weights case showing the triangular decay pattern; (b) 30%—calculate sample means and standard deviations, compute T-scores for both subjects, and interpret relative standing; (c) 40%—detail 2SLS algorithm (first-stage reduced form, second-stage structural), list full assumptions (instrument relevance, exogeneity, rank condition), and prove consistency/asymptotic normality. Conclude with comparative assessment of 2SLS vs OLS in simultaneous equations contexts relevant to Indian economic policy evaluation.

Key points expected

  • For (a): Formal definition of time series as ordered sequence of random variables; derivation of autocovariance γ(k) = σ²Σaᵢaᵢ₊ₖ for MA(m) with truncation; correlogram ρ(k) = γ(k)/γ(0); special case aᵢ = 1/m yielding ρ(k) = (m−|k|)/m for |k| < m and zero otherwise
  • For (b): Correct computation of sample mean (x̄_M = 87.4, x̄_L = 54.2) and sample standard deviation (s_M ≈ 10.47, s_L ≈ 12.38); T-score formula T = 50 + 10×(X−X̄)/S; calculation yielding T_M ≈ 81.2 and T_L ≈ 91.1; correct interpretation that higher T-score in Language indicates better relative performance despite lower absolute marks
  • For (c): Complete 2SLS procedure—stage 1 regress endogenous regressors on all exogenous/instrumental variables, stage 2 use fitted values in structural equation; explicit assumptions (linearity, instrument exogeneity E(Z'u)=0, relevance rank E(Z'X) full column, no perfect multicollinearity)
  • For (c): Properties derivation—consistency via law of large numbers and continuous mapping theorem, asymptotic normality with variance σ²(X'P_ZX)⁻¹ where P_Z is projection matrix, comparison with OLS inconsistency under simultaneity
  • For (c): Practical illustration such as estimating agricultural supply response where price is endogenous—using rainfall/transport cost as instruments, relevant to Indian agricultural policy analysis

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): correct MA(m) autocovariance formula with proper summation limits and lag notation; for (b): accurate identification of T-score as standardized measure with mean 50, SD 10; for (c): precise distinction between structural and reduced forms, correct projection matrix specificationPartial setup with minor notational errors (e.g., missing absolute value in lag, confused population/sample distinction) or incomplete reduced form specification in 2SLSFundamental setup errors: wrong autocovariance formula, treating T-score as Z-score, or conflating 2SLS with indirect least squares without proper justification
Method choice20%10Optimal methods: autocorrelation derivation via expectation of product; sample standard deviation with n−1 denominator; explicit two-stage regression algebra with matrix notation; instrument validity tests mentionedCorrect but inefficient methods (e.g., population SD formula, scalar-only presentation of 2SLS), or missing discussion of why 2SLS beats OLS in simultaneous systemsInappropriate methods: simple correlation instead of partial autocorrelation, raw score comparison ignoring variability, or attempting OLS on clearly simultaneous system
Computation accuracy20%10Numerically exact: for uniform weights ρ(k) = 1−|k|/m; T-scores within ±0.5 of true values; 2SLS variance formula correctly derived with all matrix operations explicitMinor computational slips (e.g., rounding to nearest integer in T-scores, sign error in autocorrelation for negative lags, correct final formula but arithmetic errors in illustration)Major computational failures: wrong correlogram shape, T-scores with wrong mean/SD reference, or algebraic errors in 2SLS consistency proof rendering conclusion invalid
Interpretation20%10For (a): explains why MA(m) has finite memory and how uniform weights create linear decay; for (b): correctly identifies Language as better relative performance despite lower absolute marks, discusses normative implications; for (c): interprets 2SLS as IV estimator, discusses efficiency loss vs consistency gain, cites Indian econometric applicationsCorrect but superficial interpretation—states results without explaining economic/statistical significance, or generic discussion of 2SLS without context-specific insightMisinterpretation: claims MA process has long memory, concludes Mathematics is better due to higher raw marks, or presents 2SLS as universally superior without noting efficiency cost
Final answer & units20%10All final answers boxed/highlighted: correlogram formula ρ(k) = (m−|k|)/m² × m = (m−|k|)/m for |k|<m; explicit T-score values with subject conclusion; 2SLS estimator formula β̂_2SLS = (X'P_ZX)⁻¹X'P_Zy with assumptions checklist and properties summaryFinal answers present but poorly formatted, missing units (dimensionless ρ, no units for T-scores acceptable), or incomplete properties listMissing final answers, wrong boxed results, or no conclusion on relative subject ability; 2SLS description without estimator formula or properties

Practice this exact question

Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.

Evaluate my answer →

More from Statistics 2025 Paper II