(a) Explain Zellner's seemingly unrelated regression model and the feasible generalized least squares method of estimating the…

(a) SURE and FGLS. Zellner’s seemingly unrelated regression model is a system of M linear equations, yⱼ = Xⱼβⱼ + uⱼ, j = 1, …, M. The equations are “seemingly” unrelated because each has its own dependent variable and possibly different regressors, but they are related through the disturbance terms. The key assumption is no serial correlation, but contemporaneous correlation across equations: E(uⱼ uₖ′) = σ_jk Iₙ, with σ_jk possibly non-zero for j ≠ k. Stacking gives y = Xβ + u, where X is block diagonal, and Var(u) = Σ ⊗ Iₙ. Equation-by-equation OLS remains unbiased, but it is inefficient when Σ is not diagonal because it ignores the information in cross-equation covariance. FGLS makes the estimator feasible by estimating Σ. First estimate each equation by OLS, obtain residuals eⱼ, and set s_jk = n⁻¹∑ e_ij e_kj, or use a degrees-of-freedom correction. Then Ω̂ = Ŝ ⊗ Iₙ and β̂_FGLS = (X′Ω̂⁻¹X)⁻¹X′Ω̂⁻¹y. Equivalently, premultiply the stacked system by Ω̂⁻¹ᐟ² and apply OLS. If all Xⱼ are identical, FGLS reduces to OLS in each equation; when regressors differ, FGLS borrows strength across equations and has smaller variance.

(b) NSSO functions. The National Sample Survey Office, under the Ministry of Statistics and Programme Implementation, is the principal agency for large-scale sample surveys in India. Its first function is to design, field, process and analyse surveys such as the National Sample Survey rounds, the Annual Social Industry Survey, and special surveys on employment, household social consumption, agriculture, non-farm activity, education, health and housing. It prepares sampling frames, questionnaires, fieldwork manuals and quality-control procedures, and it disseminates results through reports, data tables, microdata and the NSO website. A second function is methodological research: improving sampling designs, weighting, imputation, data editing and estimation methods so that small-area and national estimates are reliable. A third function is coordination with state and district statistical offices, which support field operations and local data collection. Finally, NSSO supplies data for policy formulation, poverty and inequality measurement, and international reporting to bodies such as the UN, ILO and for SDG indicators. Thus it converts primary household data into official statistical evidence.

(c) GLS with known heteroscedastic variances. For Yᵢ = β₁ + β₂Xᵢ + uᵢ with known Var(uᵢ) = σᵢ², write Ω = diag(σ₁², …, σₙ²) and W = Ω⁻¹ = diag(wᵢ), where wᵢ = 1/σᵢ². GLS is weighted least squares: minimise ∑ wᵢ(Yᵢ − β₁ − β₂Xᵢ)². Equivalently, divide each observation by σᵢ and apply OLS to the transformed model. The normal equations are ∑ wᵢ(Yᵢ − β₁ − β₂Xᵢ) = 0 and ∑ w_iXᵢ(Yᵢ − β₁ − β₂Xᵢ) = 0. Let S₀ = ∑wᵢ, S₁ = ∑w_iXᵢ, S₂ = ∑w_iXᵢ², T₁ = ∑w_iYᵢ, T₂ = ∑w_iX_iYᵢ. Solving gives β̂₂ = (S₀T₂ − S₁T₁)/(S₀S₂ − S₁²) and β̂₁ = T₁/S₀ − β̂₂S₁/S₀. In matrix form β̂_GLS = (X′WX)⁻¹X′WY. The variance-covariance matrix is (X′WX)⁻¹ = (1/D) [[S₂, −S₁], [−S₁, S₀]], where D = S₀S₂ − S₁². Hence Var(β̂₁) = S₂/D, Var(β̂₂) = S₀/D and Cov(β̂₁, β̂₂) = −S₁/D. Because the weights are the true inverse variances, GLS is the BLUE; OLS is still unbiased but generally has larger variance and is inefficient under heteroscedasticity.

(d) Standard and normalised scores. Gross or raw scores are measured on the scale of the particular test, so two tests may have different means, ranges and units. Converting them to standard scores is desirable because it places different scores on a common metric, makes them comparable across tests, groups, years and selection contexts, and allows norm-referenced interpretation. A standard score is a linear transformation of a raw score to a fixed mean and standard deviation. The basic form is the z-score, z = (X − M)/SD; common normalised standard forms are T-scores, T = 50 + 10z, and stanines, stanine = 5 + 2z rounded to the nearest integer. When “normalised scores” is used in the stricter sense, it refers to scores obtained by forcing the transformed distribution to be approximately normal. They are derived by ranking the raw scores, assigning each score a percentile rank, converting that percentile to a z-value from the standard normal table, and then applying the chosen linear scale. This is useful when the raw-score distribution is skewed, because the final scores have a normal reference distribution while preserving the relative standing of the examinee.

(e) Life table blanks and symbols. Given l₃₀ = 762227, l₃₁ = 758580 and T₃₀ = 27296632, the deaths between ages 30 and 31 are d₃₀ = l₃₀ − l₃₁ = 3647. Therefore q₃₀ = d₃₀/l₃₀ = 3647/762227 = 0.004785 and p₃₀ = 1 − q₃₀ = 0.995215. The person-years lived between 30 and 31, assuming deaths are uniformly distributed, are L₃₀ = l₃₀ − 0.5d₃₀ = 762227 − 1823.5 = 760403.5. Using the usual convention Tₓ = ∑ₖ₌ₓ^∞ Lₖ, T₃₁ = T₃₀ − L₃₀ = 27296632 − 760403.5 = 26536228.5. The life expectancies are e₃₀ = T₃₀/l₃₀ = 35.81 and e₃₁ = T₃₁/l₃₁ = 34.98. The central death rate is m₃₀ = d₃₀/L₃₀ = 3647/760403.5 = 0.004796. The column symbols mean: x is age; lₓ is the number of survivors to exact age x; dₓ is the number of deaths between x and x+1; pₓ is the probability of surviving from x to x+1; qₓ is the probability of dying in that interval; Lₓ is the person-years lived between x and x+1; Tₓ is the total person-years lived from age x onward; eₓ, shown as e0x, is the expectation of life at age x; and mₓ is the central death rate, dₓ/Lₓ. If the skeleton table defines Tₓ as ∑lₓ rather than ∑Lₓ, T₃₁ would instead be T₃₀ − l₃₀ = 26534405, with e₃₁ still about 34.98.

Thus, these methods show how statistical inference and official statistics adjust for structure: SURE/FGLS and GLS use known or estimated covariance to obtain efficient estimators, NSSO provides the survey infrastructure for reliable Indian data, standard scores make measurements comparable, and life-table arithmetic converts survival counts into interpretable mortality and expectancy measures.