(a) What is autocorrelation ? What are its consequences ? Explain the Goldfeld-Quandt test and Glesjer test for…

Autocorrelation and heteroscedasticity tests. Autocorrelation is serial correlation between disturbance terms in a time-series regression, eₜ correlated with eₜ₋₁; it is different from heteroscedasticity, where error variance changes across observations. If regressors are strictly exogenous and no lagged dependent variable is included, OLS remains unbiased and consistent, but it is no longer efficient. The usual standard errors are biased, often too small, so t- and F-tests overstate significance, R² can look deceptively high, and forecasts are inefficient.

For heteroscedasticity, Goldfeld-Quandt is used when variance is suspected to rise with an ordered variable Z. Order observations by Z, omit the middle c observations, split the remaining data into lower and upper halves, and run the same regression on each half. Let RSS_U and RSS_L be residual sums of squares for the upper and lower ordered groups, the upper group being the suspected higher-variance group. The test statistic is F=(RSS_U/(n_U-k))/(RSS_L/(n_L-k)), compared with the F distribution under H0 of equal variances; a large value rejects H0 in favour of increasing variance. It is simple but needs an ordering variable and loses observations.

The Glesjer test is more general. Estimate the model, take absolute or squared residuals, and regress them on the explanatory variables, e.g. |eᵢ|=α₀+α_1x₁ᵢ+...+α_pxₚᵢ+vᵢ. The null is all αⱼ=0; the auxiliary F-statistic (R²/p)/((1-R²)/(n-p-1)) tests it. Using all observations and testing several variables simultaneously, it is usually more applicable and often more powerful than Goldfeld-Quandt, though it depends on auxiliary specification and can be sensitive to outliers.

Identifiability of the two-equation system. There are two endogenous variables (M=2) and two exogenous variables (K=2), so the order condition for an equation is K-k≥M-1=1, where k is the number of exogenous variables included in that equation. The rank condition requires that the excluded exogenous variables appear in the other equation with coefficients whose rank is at least 1.

Under (i) γ12=0 and γ21=0, equation 1 includes x1 only and excludes x2; equation 2 includes x2 only and excludes x1. For each equation K-k=1, satisfying the order condition. The excluded x2 appears in equation 2 with coefficient γ22, and the excluded x1 appears in equation 1 with coefficient γ11; if these coefficients are non-zero, the rank condition holds. Both equations are therefore exactly identified.

Under (ii) γ11=0 and γ12=0, equation 1 includes no exogenous variable, so K-k=2≥1. The excluded x1 and x2 appear in equation 2 with coefficients γ21 and γ22; if at least one is non-zero, the rank condition holds, so equation 1 is identified, and overidentified when both are non-zero. Equation 2, however, includes both x1 and x2, so K-k=0<1. The order condition fails, and equation 2 is not identified.

Leslie matrix projection. A Leslie matrix is an age-class matrix used to project a population. If there are n age classes, it is n×n: the first row contains age-specific fertility rates Fᵢ, the sub-diagonal contains survival probabilities Pᵢ from class i to i+1, and all other entries are zero. Multiplying the matrix by the current population vector n(t) gives n(t+1)=L n(t); the first row counts births by age group, while the sub-diagonal moves survivors into the next age class. Repeated multiplication, n(t+T)=L^T n(t), gives projections. The dominant eigenvalue λ is the finite rate of increase, and the intrinsic rate of natural increase is r=ln λ; the associated eigenvector gives the stable age distribution. In India, the Registrar General and Census Commissioner of India use age-specific fertility and survival rates from SRS and life tables to project population by age and sex, helping plan health, education and welfare programmes. Thus, valid inference and projection depend on correct assumptions, restrictions, and demographic parameters.