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

(a) OLS and GLS Estimators: Derivation, Properties and Comparison

Model: Y = Xβ + u, where Y is n × 1, X is n × k of full column rank k and non-stochastic, E(u) = 0.

OLS. Minimise S(β) = (Y − Xβ)′(Y − Xβ) = Y′Y − 2β′X′Y + β′X′Xβ. Setting ∂S/∂β = −2X′Y + 2X′Xβ = 0 gives the normal equations X′Xβ̂ = X′Y, so

β̂_OLS = (X′X)⁻¹X′Y.

GLS. Suppose Var(u) = σ²Ω, with Ω known and positive definite (heteroscedastic and/or autocorrelated errors). Choose a non-singular P with P′P = Ω⁻¹, so PΩP′ = I. The transformed model PY = PXβ + Pu has Var(Pu) = σ²I, so OLS on it is valid. Equivalently, minimise (Y − Xβ)′Ω⁻¹(Y − Xβ), which gives

β̃_GLS = (X′Ω⁻¹X)⁻¹X′Ω⁻¹Y.

Properties:

• Unbiasedness: E(β̂_OLS) = β and E(β̃_GLS) = β, since both are linear in Y and E(u) = 0; OLS stays unbiased even when Ω ≠ I.

• Variances: Var(β̃_GLS) = σ²(X′Ω⁻¹X)⁻¹. For OLS, Var(β̂_OLS) = σ²(X′X)⁻¹ when Ω = I, but σ²(X′X)⁻¹X′ΩX(X′X)⁻¹ in general.

• Optimality: by the Gauss–Markov theorem OLS is BLUE when Ω = I; by Aitken's theorem GLS is BLUE when Var(u) = σ²Ω. Hence Var(β̂_OLS) − Var(β̃_GLS) is positive semi-definite, and OLS is inefficient when Ω ≠ I.

• Inference: when Ω ≠ I the usual OLS estimate s²(X′X)⁻¹ of the variance is biased, so OLS t and F tests are invalid; GLS gives valid tests with σ̂² = (Y − Xβ̃)′Ω⁻¹(Y − Xβ̃)/(n − k).

• Normal errors: GLS is then also the maximum likelihood estimator of β, and both estimators are consistent under mild conditions.

Comparison: GLS reduces to OLS when Ω = I, and also whenever the columns of X span an invariant subspace of Ω (for example, equicorrelated errors with an intercept). When Ω is unknown, it is estimated from the OLS residuals and feasible GLS (FGLS) is used, which is asymptotically equivalent to GLS.

(b) Time Reversal, Factor Reversal and Circular Tests; Laspeyres and TRT

Theoretical Tests:

1. Time Reversal Test (TRT): A price index P₀₁ and its reciprocal P₁₀ should satisfy P₀₁ × P₁₀ = 1.

2. Factor Reversal Test (FRT): The product of the price index and quantity index should equal the value index (P₀₁ × Q₀₁ = V₀₁).

3. Circular Test: For three periods, P₀₁ × P₁₂ × P₂₀ = 1.

Verification of Laspeyres' Formula:

The Laspeyres Price Index is P₀₁^L = (Σ p₁ q₀)/(Σ p₀ q₀).

Step 1: Calculate P₀₁^L (1979 to 1980)

• Σ p₀ q₀ = (32 × 50) + (30 × 35) + (16 × 55) = 1600 + 1050 + 880 = 3530

• Σ p₁ q₀ = (30 × 50) + (25 × 35) + (18 × 55) = 1500 + 875 + 990 = 3365

• P₀₁^L = 3365/3530 ≈ 0.9533

Step 2: Calculate P₁₀^L (1980 to 1979)

• Σ p₁ q₁ = (30 × 50) + (25 × 40) + (18 × 50) = 1500 + 1000 + 900 = 3400

• Σ p₀ q₁ = (32 × 50) + (30 × 40) + (16 × 50) = 1600 + 1200 + 800 = 3600

• P₁₀^L = 3600/3400 ≈ 1.0588

Conclusion:

Product = (3365/3530) × (3600/3400) = 1.0093 ≠ 1.

Therefore, the Laspeyres' formula fails the Time Reversal Test.

(c) Population Estimates versus Projections; Component Method

Distinction:

• Population Estimates: These are statistical approximations of the population size for a specific point in time (usually the present or a recent past) using data from censuses and vital registration systems. They are essentially interpolations between census dates.

• Population Projections: These are estimates of the future population based on assumptions about future trends in fertility, mortality, and migration. They are extrapolations of current trends into the future.

Component Method of Projection:

The component method is the most rigorous approach to population projection. It involves the following steps:

1. Base Population: Start with the current population distribution by age and sex.

2. Survival: Apply life-table survival ratios (ₙLₓ₊ₙ/ₙLₓ) to each age–sex group to find how many survive into the next age group after n years.

3. Births: Apply age-specific fertility rates to the average number of women aged 15–49 over the period, split births by the sex ratio at birth, and survive them to the youngest age group.

4. Migration: Add or subtract the estimated net migration (in-migration minus out-migration) for each age group.

5. Iteration: Repeat the process year by year to project the population forward. This method is preferred over simple arithmetic or geometric methods because it accounts for demographic structure.