(a) For the linear model Y = Xβ + u, obtain the expressions for Ordinary Least Squares (OLS) and Generalised Least Squares (GLS)…

What "Derive" is asking you to do

Reach the stated expression from a starting relation, justifying every step. The destination is printed in the question, so only the route earns marks, and the assumptions you work under are part of that route.

Structure that answers it

Assumptions and notation defined → starting relation or governing equation → each step with its justification → the required expression → limiting case or boundary check

Where marks are lost

Writing the standard result first and fitting three lines to it, which an examiner reads at a glance. Marks also go on assumptions left unstated — lossless medium, small amplitude, errors independent with zero mean — and on symbols used before they are defined, even when the question says usual notations.

All UPSC directive words, compared →

How this answer will be evaluated

Approach

Framework: UPSC Statistics Paper 2. (a) compare: paired headings or table > key differences > significance > conclusion | (b) explain: definition/context > points in order > small example > short close | (c) compare: paired headings or table > key differences > significance > conclusion Full marks: Rigorous derivations, correct calculations, clear distinctions, and proper interpretation of results.

Key points expected

• Derive OLS estimator β̂ = (X'X)⁻¹X'Y
• Derive GLS estimator β̂ = (X'Ω⁻¹X)⁻¹X'Ω⁻¹Y
• State Gauss-Markov theorem for OLS (BLUE under homoscedasticity)
• Compare efficiency of OLS vs GLS under heteroscedasticity
• Define Time Reversal Test (P₀₁ × P₁₀ = 1)
• Define Factor Reversal Test (P₀₁ × Q₀₁ = V₀₁)
• Calculate Laspeyres' index for 1979-1980 and 1980-1979
• Verify if product of indices equals 1

Evaluation rubric

Each sub-part is marked on its own, against the marks and word limit printed on the paper.

1. (a) Derive OLS and GLS estimators for Y=Xβ+u and compare their properties. 20 marks

   compare— paired headings or table → key differences → significance → conclusion

   Must cover

   • Derive OLS estimator β̂ = (X'X)⁻¹X'Y
   • Derive GLS estimator β̂ = (X'Ω⁻¹X)⁻¹X'Ω⁻¹Y
   • State Gauss-Markov theorem for OLS (BLUE under homoscedasticity)
   • Compare efficiency of OLS vs GLS under heteroscedasticity

   Loses marks

   • Deriving only one estimator without comparison
   • Failing to state assumptions for OLS (homoscedasticity)
   • Confusing OLS and GLS properties

   Earns more

   • Mention Aitken's theorem for GLS
   • Show OLS is consistent but inefficient under heteroscedasticity
   • Mention BLUE property of GLS under general covariance

   Extra mark

   • Mention Feasible GLS (FGLS) for unknown Ω
2. (b) Explain Time, Factor, and Circular tests; verify Laspeyres' Time Reversal Test. 15 marks

   explain— definition/context → points in order → small example → short close

   Must cover

   • Define Time Reversal Test (P₀₁ × P₁₀ = 1)
   • Define Factor Reversal Test (P₀₁ × Q₀₁ = V₀₁)
   • Calculate Laspeyres' index for 1979-1980 and 1980-1979
   • Verify if product of indices equals 1

   Loses marks

   • Defining tests without performing the calculation
   • Incorrect calculation of index values
   • Failing to conclude whether the test is satisfied

   Earns more

   • Define Circular Test (P₀₁ × P₁₂ × P₂₀ = 1)
   • Show step-by-step calculation of Σp₀q₀ and Σp₁q₀
   • State that Laspeyres' fails Time Reversal Test

   Extra mark

   • Mention Fisher's Ideal Index satisfies all three tests
3. (c) Distinguish population estimates from projections; describe component method. 15 marks

   compare— paired headings or table → key differences → significance → conclusion

   Must cover

   • Define population estimate (interpolating past data)
   • Define population projection (extrapolating to future)
   • Describe component method (births, deaths, migration)
   • Distinguish between the two concepts clearly

   Loses marks

   • Confusing estimates with projections
   • Failing to describe the component method
   • Vague definitions without distinction

   Earns more

   • Mention arithmetic/geometric methods for estimation
   • Mention cohort-component method for projection
   • Note that estimates are for past/present, projections for future

   Extra mark

   • Mention UN population projections as an example

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