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

(a) Zellner's SURE model and feasible GLS

The seemingly unrelated regression (SURE) model is a system of M equations, y_j = X_jβ_j + u_j (j = 1, …, M). Each equation has its own dependent variable and possibly different regressors, so they look unrelated; they are linked only through the disturbances. The assumptions are no serial correlation but contemporaneous correlation across equations: E(u_j u_k′) = σ_jk I_n, with σ_jk possibly non-zero for j ≠ k. Stacking gives y = Xβ + u, with X block diagonal and Var(u) = Σ ⊗ I_n.

Equation-by-equation OLS stays unbiased but ignores the cross-equation covariance and is inefficient. GLS would use Σ, which is unknown, so FGLS estimates it in two steps. First apply OLS to each equation and take the residuals e_j to form s_jk = e_j′e_k/n (or with a degrees-of-freedom correction). Then use Ω̂ = Ŝ ⊗ I_n in β̂ = (X′Ω̂⁻¹X)⁻¹X′Ω̂⁻¹y. The efficiency gain disappears if Σ is diagonal or if all equations have identical regressors, and it is larger when the disturbance correlation is high and the regressors differ across equations.

(b) Functions of the NSSO

The National Sample Survey Office, under the Ministry of Statistics and Programme Implementation (now part of the National Statistics Office), is India's main agency for large-scale sample surveys. Its functions are:

• to design, field, process and analyse surveys, such as the National Sample Survey rounds on household consumption, employment, agriculture, health, education and housing, and the Annual Survey of Industries fieldwork;
• to prepare sampling frames and designs, questionnaires and instruction manuals, and to apply quality checks through supervision and scrutiny;
• to carry out methodological work on sampling, weighting, estimation and data editing;
• to collect price data and support other surveys through its field operations division, working with state statistical bureaus;
• to release reports, tables and unit-level data, which feed poverty and inequality measurement, national accounts and policy.

(c) GLS with known heteroscedastic variances

For Yᵢ = β₁ + β₂Xᵢ + uᵢ with known Var(uᵢ) = σᵢ², let wᵢ = 1/σᵢ². GLS is weighted least squares: minimise Σwᵢ(Yᵢ − β₁ − β₂Xᵢ)², which is the same as dividing every term by σᵢ and applying OLS. The normal equations are Σwᵢ(Yᵢ − β₁ − β₂Xᵢ) = 0 and ΣwᵢXᵢ(Yᵢ − β₁ − β₂Xᵢ) = 0.

Write S₀ = Σwᵢ, S₁ = ΣwᵢXᵢ, S₂ = ΣwᵢXᵢ², T₁ = ΣwᵢYᵢ, T₂ = ΣwᵢXᵢYᵢ and D = S₀S₂ − S₁². Then

β̂₂ = (S₀T₂ − S₁T₁)/D and β̂₁ = (T₁ − β̂₂S₁)/S₀.

The covariance matrix is (X′WX)⁻¹, giving Var(β̂₁) = S₂/D, Var(β̂₂) = S₀/D and Cov(β̂₁, β̂₂) = −S₁/D. Since the weights are the true inverse variances, these are BLUE; OLS stays unbiased but has larger variance.

(d) Standard scores and normalised scores

Raw (gross) scores depend on the scale, mean and spread of each test, so scores from different tests, groups or years cannot be compared directly. Converting them to standard scores puts everything on a common scale and gives a norm-referenced meaning to each score.

A standard score is a linear transformation of the raw score X to a fixed mean and standard deviation. The basic form is the z-score, z = (X − mean)/SD, which has mean 0 and SD 1. Others are rescaled forms such as the T-score, T = 50 + 10z.

Normalised scores are obtained by forcing the distribution to be normal. Rank the raw scores, convert each to a percentile rank, read the corresponding z-value from the standard normal table, and rescale (for example T = 50 + 10z, or the stanine 5 + 2z rounded, limited to 1 to 9). This is used when raw scores are skewed, since the relative order is kept while the shape becomes normal.

(e) Life table blanks and symbols

Given l₃₀ = 762227, l₃₁ = 758580 and T₃₀ = 27296632:

• d₃₀ = l₃₀ − l₃₁ = 3647
• q₃₀ = d₃₀/l₃₀ = 0.004785 and p₃₀ = 1 − q₃₀ = 0.995215
• L₃₀ = (l₃₀ + l₃₁)/2 = 760403.5
• T₃₁ = T₃₀ − L₃₀ = 26536228.5
• e₃₀ = T₃₀/l₃₀ = 35.81 years and e₃₁ = T₃₁/l₃₁ = 34.98 years
• m₃₀ = d₃₀/L₃₀ = 0.004796

Meaning of the symbols: x is age; lₓ is the number surviving to exact age x; dₓ is the deaths between x and x + 1; pₓ and qₓ are the probabilities of surviving and of dying in that year; Lₓ is the person-years lived between x and x + 1; Tₓ is the total person-years lived from age x onward (the sum of L from x); eₓ⁰ is the expectation of life at age x, Tₓ/lₓ; and mₓ is the central death rate, dₓ/Lₓ.