(a) Discuss the Indian Statistical System. State some important organisations and explain the main working of the National…

Indian Statistical System

The Indian Statistical System (ISS) is a decentralised, multi-agency framework for collection, compilation and dissemination of official statistics, whose foundations were laid by P.C. Mahalanobis with the establishment of the Indian Statistical Institute (1931) and the National Sample Survey (1950). It is broadly divided into Central and State statistical machinery, coordinated through the Ministry of Statistics and Programme Implementation (MoSPI). Key central organisations include the Central Statistics Office (CSO), the National Sample Survey Office (NSSO), the Registrar General of India (RGI, responsible for census and vital statistics), NITI Aayog, and the Reserve Bank of India for monetary and external sector data. State Directorates of Economics and Statistics handle state-level data, while the National Statistical Commission (NSC), constituted on the Rangarajan Committee's recommendation, acts as the apex advisory body.

The National Statistical Organisation (NSO), formed by merging the CSO and NSSO in 2019, functions through three broad mandates: data collection through large-scale sample surveys (employment-unemployment, consumer expenditure, health, education); coordination and standardisation of concepts, definitions and classifications across agencies to ensure comparability; and dissemination of statistical products such as national accounts estimates, CPI, IIP and survey reports. It works under NSC oversight, which safeguards methodological integrity and independence. Thus the ISS, despite coordination challenges and calls for a unified National Statistical Office, remains the backbone of evidence-based policymaking in India.

AR(1) Forecasting

For the AR(1) model with non-zero mean, Xₜ = μ + φ(Xₜ₋₁ − μ) + εₜ, with μ = 74.3293, φ = 0.5705 and xₙ = 67.

One-step-ahead forecast: X̂ₙ₊₁ = μ + φ(xₙ − μ) = 74.3293 + 0.5705(67 − 74.3293) = 74.3293 + 0.5705(−7.3293) = 74.3293 − 4.1814 = 70.1479.

For h steps ahead, X̂ₙ₊ₕ = μ + φ^h(xₙ − μ). For h = 5, φ⁵ = (0.5705)⁵ ≈ 0.0604. Hence X̂ₙ₊₅ = 74.3293 + 0.0604(−7.3293) = 74.3293 − 0.4427 ≈ 73.8866. The forecast converges towards the mean μ as h increases, since φ^h → 0 for |φ| < 1.

SLSE Model: Structure and Inappropriateness of OLS

A Simultaneous Linear Statistical Equations (SLSE) model describes interdependence among endogenous variables, where each endogenous variable may appear as regressor in other equations. Its structure consists of a set of structural equations derived from economic theory, e.g. demand and supply functions, with endogenous variables on both sides and stochastic disturbances. The reduced form expresses each endogenous variable solely in terms of predetermined (exogenous and lagged) variables, obtained by solving the structural system.

Applying OLS to structural equations is inappropriate because the endogenous regressor is correlated with the disturbance term. In a demand-supply system, price and quantity are jointly determined, so the regressor (price) covaries with the error, violating the Gauss-Markov assumption of zero covariance between regressors and errors. This produces simultaneous equation bias: OLS estimators are biased and inconsistent, and the bias does not vanish even asymptotically. Identification issues compound this — a structural equation may be under-identified, making estimation impossible without restrictions. Hence methods such as Indirect Least Squares, Instrumental Variables, Two-Stage Least Squares or Limited Information Maximum Likelihood are used, which exploit predetermined variables as instruments to break the correlation between regressors and errors.

Average Age at Death and Force of Mortality

(i) The average age at death of those dying between ages x and x + n is the mean of the ages at death in that interval, weighted by the distribution of deaths. It is given by

nₐₓ = [∫₀ⁿ (x + t) μ(x + t) l(x + t) dt] / [l(x) − l(x + n)]

equivalently expressible using life table functions as nₐₓ = x + n·aₓ, where n·aₓ is the average number of years lived in the interval by those dying within it. This measure is useful in constructing life tables and in actuarial valuation.

(ii) The force of mortality is μ(x) = −d[ln l(x)]/dx. Given l(x) = 100(100 − x)¹/2, ln l(x) = ln 100 + ½ ln(100 − x), so d[ln l(x)]/dx = −1/[2(100 − x)]. Hence μ(x) = 1/[2(100 − x)]. At x = 84, μ(84) = 1/[2(16)] = 1/32.

Intelligence Tests, Mental Age and IQ

Intelligence tests are standardised instruments designed to measure cognitive abilities such as reasoning, comprehension, memory, numerical and verbal aptitude, and problem-solving. They originated with Binet and Simon (1905) and were refined by Terman (Stanford-Binet) and Wechsler (WAIS, WISC). Tests may be individual or group, verbal or performance-based. They are used in education for identifying gifted and slow learners and for streaming; in clinical settings for diagnosing intellectual disability and learning disorders; in occupational selection and placement; and in research on cognitive development. In India, adaptations such as the Binet-Kamat Test and Bhatia's Battery of Performance Tests are widely used.

Mental age (MA) is the chronological age at which the average child performs at the level demonstrated by the examinee — i.e., the age-equivalent of test performance. Intelligence Quotient (IQ) was originally defined by Stern as the ratio IQ: IQ = (MA / CA) × 100, where CA is chronological age. Wechsler later introduced the deviation IQ, based on standard scores relative to age-group norms, which avoids the statistical problems of ratio IQ in adults. Thus intelligence tests provide a norm-referenced, quantifiable measure of cognitive functioning, though their cultural fairness and predictive limits must be acknowledged.

Overall, the five parts together illustrate the breadth of applied statistics — from national data infrastructure and time-series forecasting to simultaneous-equation econometrics, actuarial life-table methods and psychometrics — each requiring both conceptual clarity and computational precision.