(a) Explain the multicollinearity problem in a regression model. What are its consequences? State the different indicators of…

The five issues link statistical inference with demographic measurement: diagnostics protect regression coefficients, while fertility and stability assumptions protect population projections.

(a) Multicollinearity. Multicollinearity occurs when regressors in a linear model are nearly linearly dependent, so one explanatory variable can be predicted with high R² from the others. It is distinct from simple pairwise correlation: with several regressors, near dependence can exist even when pairwise correlations are moderate. In applied Indian data, household income, education and family size in NSSO regressions may move together. It does not violate Gauss-Markov; OLS remains unbiased and consistent, but X′X is near singular. The causal chain is: near dependence creates small eigenvalues of X′X, inflates diagonal elements of (X′X)⁻¹, and raises Var(β̂ⱼ)=σ²[(X′X)⁻¹]ⱼⱼ. Large variances make coefficients unstable to small data changes and deflate t-statistics, so individually insignificant terms may appear even when the overall F-test is significant. Consequences include wide confidence intervals, sign reversals, and unreliable policy coefficients, because estimated partial effects may not be interpretable. Indicators are a high pairwise correlation matrix; auxiliary regressions of each Xⱼ on the others with high R²ⱼ; VIFⱼ=1/(1−R²ⱼ), where VIF>10 signals serious collinearity; a condition index, the square root of the ratio of largest to smallest eigenvalue of X′X, above 30 (severe above 1000); and high overall R² with insignificant t-ratios.

(b) Fertility rates. Let n(a) be the number of women of age a, P_F=∫₁₅⁴⁹ n(a)da, P total population, and B births in a year. If f(a) is the age-specific fertility rate, B=∫ f(a)n(a)da. Crude birth rate CBR=B/P; general fertility rate GFR=B/P_F. Hence CBR=GFR(P_F/P). Total fertility rate TFR=∫ f(a)da. Therefore GFR=[∫ f(a)n(a)da]/[∫ n(a)da]=TFR×R, where R=∫ f(a)n(a)da/[∫ n(a)da ∫ f(a)da] is an age-distribution adjustment factor. Thus CBR=TFR×R×(P_F/P). If women are uniformly distributed over the reproductive span L=34, R=1/L, so CBR≈(TFR/34)(P_F/P); the conversion factor is the span or weighted ratio, not the mean age of childbearing. The continuous formulation shows why CBR can fall even when TFR is unchanged if the share of reproductive-age women falls, as in ageing populations. Conversely, a young population can keep CBR high despite lower TFR. TFR is synthetic because it combines age-specific rates from one period, not the experience of one cohort. CBR and GFR are conventionally per 1000, TFR per woman. CBR is crude and age-structure dependent; GFR refines the denominator to reproductive-age women but still depends on the age mix within 15–49; TFR is age-standardized, a synthetic period measure, comparable across populations and useful for replacement-level analysis.

(c) Stable and quasi-stable populations. A stable population assumes constant age-specific fertility and mortality, and no net migration or a constant migration pattern. Lotka’s equation ∫ e⁻ʳᵃl(a)m(a)da=1 gives a constant growth rate r and a fixed age distribution proportional to e⁻ʳᵃl(a); the population grows exponentially. This simplifies projections, intercensal estimation, and separation of demographic momentum. But India’s fertility and mortality have changed over recent decades, so a strictly stable model can misstate growth and age structure. A quasi-stable population allows vital rates to change slowly; the age distribution is nearly stable and the growth rate changes gradually. This is closer to Census of India and UN medium-variant projections, where fertility decline, mortality improvement and migration are modelled over time. Quasi-stable assumptions are more realistic but more sensitive to trend assumptions and data quality. They capture momentum: even after fertility reaches replacement, a young age structure can sustain growth. Stable assumptions are useful as theoretical benchmarks; quasi-stable assumptions are preferred for planning, but both must be checked against Census and sample registration data.

(d) Heteroscedasticity. Heteroscedasticity means Var(Uᵢ) is not constant; here Var(Uᵢ)=K²Xᵢ². Assuming E(Uᵢ)=0, OLS remains unbiased but inefficient, and usual standard errors are invalid. Divide by Xᵢ (Xᵢ≠0): Yᵢ/Xᵢ=β+α(1/Xᵢ)+Uᵢ/Xᵢ. Let zᵢ=1/Xᵢ, yᵢ*=Yᵢ/Xᵢ, vᵢ=Uᵢ/Xᵢ; then Var(vᵢ)=K². OLS on this transformed model is WLS for the original model with weights 1/Xᵢ² and is BLUE. Its variance for β, the intercept, is K²/Σ(Xᵢ−X̄_w)²/Xᵢ², where X̄_w=Σ(Xᵢ/Xᵢ²)/Σ(1/Xᵢ²); its variance for α, the slope on zᵢ, is K²/Σ(zᵢ−z̄)², z̄=n⁻¹Σzᵢ. The transformed model downweights observations with large Xᵢ because their errors are more variable. The original OLS slope variance can be written as K²ΣXᵢ²(Xᵢ−X̄)²/Sxx², which is larger because it gives too much weight to high-variance observations. To prove the inequality, express original OLS in transformed variables. β_OLS=Σaᵢyᵢ*, aᵢ=Xᵢ(Xᵢ−X̄)/Sxx, Sxx=Σ(Xᵢ−X̄)². It is linear unbiased for β because Σaᵢ=1 and Σaᵢzᵢ=0. The transformed OLS intercept uses dᵢ=1/n−z̄(zᵢ−z̄)/Szz, Szz=Σ(zᵢ−z̄)², the minimum-norm vector satisfying the same constraints; since aᵢ−dᵢ is orthogonal to dᵢ, Σaᵢ²≥Σdᵢ². Hence Var(β_OLS)=K²Σaᵢ²≥K²Σdᵢ²=Var(β_WLS), strictly when Xᵢ vary. For α, α_OLS=Σcᵢyᵢ*, cᵢ=Xᵢ/n−X̄Xᵢ(Xᵢ−X̄)/Sxx. It satisfies Σcᵢ=0 and Σcᵢzᵢ=1. Cauchy-Schwarz gives 1=Σcᵢ(zᵢ−z̄)≤(Σcᵢ²)¹ᐟ²(Szz)¹ᐟ², so Var(α_OLS)=K²Σcᵢ²≥K²/Szz=Var(α_WLS). Thus OLS has greater variance.

(e) Validity and reliability. Validity is the degree to which a test measures what it claims to measure; it concerns systematic error and construct accuracy. Content validity asks whether items cover the domain, as in NSSO consumption surveys covering food, fuel, education and health. Criterion validity asks whether scores predict an external criterion, such as prelims scores predicting Mains performance. Construct validity asks whether the instrument captures the intended theoretical dimension, such as a poverty index capturing multidimensional deprivation. Reliability is precision or repeatability: consistency of scores under repeated measurement, assessed by test-retest, split-half or Cronbach’s alpha. Reliability estimates random error; validity estimates bias. A test can be reliable but invalid, consistently measuring the wrong thing; a valid test is generally reliable, but reliability alone does not establish validity. In survey design, reliability is improved by clear wording and trained enumerators, while validity requires correct sampling frames, measurement invariance and linkage to administrative records.

Together, these diagnostics ensure that regression and demographic models used in Indian planning produce robust, policy-relevant estimates.