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

(a) Autocorrelation and tests for heteroscedasticity

Autocorrelation is serial correlation between the disturbances of a time-series regression, so that uₜ is correlated with uₜ₋₁ (for example uₜ = ρuₜ₋₁ + εₜ). It differs from heteroscedasticity, where the error variance changes across observations.

Consequences. With strictly exogenous regressors, OLS estimators remain unbiased and consistent but are no longer efficient (not BLUE). The usual standard errors are biased, typically too small under positive autocorrelation, so t and F tests overstate significance, R² looks inflated and forecasts are inefficient. If a lagged dependent variable is a regressor, OLS is also inconsistent.

Goldfeld–Quandt test. It is used when the error variance is suspected to rise with an ordering variable Z. Order the observations by Z, omit the c middle observations, and fit the same regression separately to the lower and upper groups. Under H₀ (equal variances), F = [RSS_upper/(n₁ − k)] / [RSS_lower/(n₂ − k)] follows an F distribution, and a large value indicates variance increasing with Z. It is simple but needs a known ordering variable and loses the omitted observations.

Glejser test. Fit the model by OLS and take the absolute residuals |eᵢ|. Regress them on the suspect variable in forms such as |eᵢ| = a + bXᵢ, |eᵢ| = a + b√Xᵢ or |eᵢ| = a + b/Xᵢ. If b is significantly different from zero, heteroscedasticity is indicated, and the best-fitting form shows its pattern. It uses all the observations and also suggests a transformation, but the auxiliary errors are not well-behaved, so it is mainly a large-sample test.

(b) Identifiability of the two-equation system

Here the number of endogenous variables is M = 2 and of exogenous variables is K = 2. The order condition for an equation is K − k ≥ M − 1 = 1, where k is the number of exogenous variables it contains. The rank condition needs the matrix of coefficients, in the other equation, of the variables excluded from this one to have rank M − 1 = 1.

Case (i): γ₁₂ = 0 and γ₂₁ = 0. Equation 1 excludes x₂ and equation 2 excludes x₁, so K − k = 1 for each and the order condition holds with equality. The excluded x₂ appears in equation 2 with coefficient γ₂₂, and the excluded x₁ appears in equation 1 with coefficient γ₁₁. If γ₂₂ ≠ 0 and γ₁₁ ≠ 0 the rank condition holds, so both equations, and the system, are exactly identified.

Case (ii): γ₁₁ = 0 and γ₁₂ = 0. Equation 1 excludes both x₁ and x₂, so K − k = 2 > 1. The excluded variables appear in equation 2 with coefficients γ₂₁ and γ₂₂, so the rank is 1 if either is non-zero. Equation 1 is then over-identified. Equation 2 contains both x₁ and x₂, so K − k = 0 < 1 and it is not identified. Hence the system as a whole is not identified.

(c) Leslie matrix and population projection

The Leslie matrix L is a square matrix for a female population divided into n age groups. Its first row holds the age-specific fertility terms Fᵢ (births per woman in group i, allowing for survival of the newborns), its sub-diagonal holds the survival probabilities Pᵢ from group i to i + 1, and every other entry is zero.

Projection technique: let n(t) be the vector of the numbers in each age group at time t. Then n(t + 1) = L n(t), where the first row creates the new births and the sub-diagonal moves survivors up one age group. Repeating this gives n(t + k) = Lᵏ n(t), so a population projection needs only the initial age distribution, fertility and survival rates.

If the rates stay constant, L has a dominant positive eigenvalue λ. In the long run the population grows by a factor λ per period (intrinsic rate r = ln λ) and settles into the stable age distribution given by the corresponding eigenvector, whatever the initial age distribution. The projection assumes constant rates and ignores migration unless it is added.