(a) Explain the problem of identification with a suitable example. Also discuss the conditions of identification. Check the…

(a) Problem of Identification; Order and Rank Conditions

In simultaneous equations, identification asks whether the structural parameters of one equation can be recovered from the reduced form. In a simple market, demand Q=α+βP+γIncome+u and supply Q=δ+θP+v. If only P and Q are observed and no variable shifts only one curve, the reduced form gives only the equilibrium line; β and θ cannot be separated. If Income is excluded from supply, demand is identified; if two excluded variables are available, it is over-identified. An equation is under-identified if too few excluded exogenous variables are available, just-identified if exactly enough, and over-identified if more than enough.

The order condition is K−k≥m−1, where K is total exogenous variables, k included exogenous variables, and m endogenous variables in the equation. The rank condition requires the matrix of coefficients of variables excluded from the equation but present in other equations to have rank M−1, where M is total endogenous variables. If this rank is lower, the equation is not identified even if the order condition holds.

Here the endogenous variables are y₁, y₂, y₃ (M = 3) and the exogenous variables are x₁, x₂, x₃ (K = 3). Writing each equation with all terms on one side:

• Equation 1 (y₁ = 3y₂ − 2x₁ + x₂ + u₁): m = 2, k = 2, so K − k = 1 = m − 1 (order condition met exactly). The excluded variables y₃ and x₃ have coefficients (−1, 0) in equation 2 and (1, 2) in equation 3; this matrix has determinant −2 ≠ 0, so its rank is 2 = M − 1. Equation 1 is exactly identified.
• Equation 2 (y₂ = y₃ + x₂ + u₂): m = 2, k = 1, so K − k = 2 > m − 1 = 1. The excluded y₁, x₁, x₃ have coefficients (1, 2, 0) in equation 1 and (−1, 0, 2) in equation 3, rank 2. Equation 2 is over-identified.
• Equation 3 (y₃ = y₁ − y₂ − 2x₃ + u₃): m = 3, k = 1, so K − k = 2 = m − 1. The excluded x₁, x₂ have coefficients (2, −1) in equation 1 and (0, −1) in equation 2; the determinant is −2 ≠ 0, rank 2. Equation 3 is exactly identified.

(b) Crude versus Standardised Death Rates; Comparative Mortality Index

Crude death rates are total deaths divided by total population. They cannot be compared directly because age-sex composition differs: Kerala has a larger elderly share, so its crude rate may be higher than Bihar or Uttar Pradesh even if age-specific risks are lower; younger states appear artificially healthier. A high CDR may therefore reflect an old age structure rather than poor health. Standardisation removes this compositional effect. Direct standardisation applies the study population’s age-specific rates m_a to a standard population P_a^s: ASR=ΣP_a^s m_a/ΣP_a^s. It is useful when age-specific rates are stable. Indirect standardisation applies standard age-specific rates M_a^s to the study population P_a: expected deaths E=ΣP_a M_a^s; SMR=ΣP_a m_a/ΣP_a M_a^s (observed/expected deaths); if a rate is needed, multiply SMR by the standard crude rate. It is useful when age-specific rates are unstable.

A comparative mortality index compares a region’s mortality with a standard. For indirect standardisation, CMI=ΣP_a m_a/ΣP_a M_a^s×100, i.e. observed deaths divided by expected deaths using the region’s own age distribution. For direct-standardised comparison, CMI=[ΣP_a^s m_a/ΣP_a^s]/[ΣP_a^s M_a^s/ΣP_a^s]×100=ΣP_a^s m_a/ΣP_a^s M_a^s×100. A CMI above 100 indicates mortality above the standard, below 100 below it; it allows ranking states or districts relative to a common standard.

(c)(i) Reliability and the Effect of Test Length

Reliability is the proportion of observed score variance due to true score variance: r=σ_T^2/σ_X^2, where X=T+E and error is uncorrelated with true score. Test length affects reliability through the Spearman-Brown prophecy formula, r_L=n r/[1+(n−1)r], where n is new length divided by original length. Longer tests reduce random error and raise reliability, but with diminishing returns. A longer test with similar items is therefore more reliable than a shorter one.

(c)(ii) Methods of Estimating Reliability

Methods include test-retest, which correlates the same test administered twice but is affected by practice effects; parallel forms, which correlates equivalent forms; and split-half, which divides items into halves, correlates them, and corrects with Spearman-Brown. For dichotomous items, KR-20 is r_20=k/(k−1)[1−Σp_iq_i/σ_X^2]; KR-21 assumes equal item difficulty and is r_21=k/(k−1)[1−k p(1−p)/σ_X^2], where p=M/k. Cronbach’s alpha generalises this to continuous items: α=k/(k−1)[1−Σσ_i^2/σ_X^2].