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

Identification. 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 M=3 and K=3. Equation 1 includes y1 and y2, so m=2; it includes x1 and x2, so k=2; K−k=1=m−1. Excluded y3 and x3 have coefficients (−1,0) and (1,2) in equations 2 and 3, rank 2; hence just-identified. Equation 2 includes y2 and y3, m=2; includes x2, k=1; K−k=2>1. Excluded y1, x1 and x3 have coefficients (1,2,0) and (−1,0,2), rank 2; hence over-identified. Equation 3 includes y1, y2 and y3, m=3; includes x3, k=1; K−k=2=m−1. Excluded x1 and x2 have coefficients (2,−1) and (0,−1), rank 2; hence just-identified.

Mortality comparison. 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^s/ΣP_a; 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.

Reliability. 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.

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]. Thus identification, standardisation and reliability all require removing confounding—simultaneity, age composition and measurement error—to recover the underlying parameter.