(a) Explain Akaike information criterion for order selection in an ARMA (p, q) process. 15 marks (b) Define autocorrelation…

(a) Akaike information criterion for ARMA(p, q) order selection

The likelihood never falls when parameters are added, so fit alone cannot choose p and q. Akaike's criterion, based on the Kullback–Leibler distance between the true and the fitted model, penalises complexity:

AIC = −2 ln L̂ + 2k,

where L̂ is the maximised likelihood and k the number of estimated parameters. For a Gaussian ARMA(p, q), ln L̂ = −(n/2) ln σ̂² + constant, so AIC(p, q) = n ln σ̂² + 2(p + q) + constant, with σ̂² the ML estimate of the innovation variance.

Procedure:

1. Fix upper bounds P and Q (for example from the ACF and PACF).
2. Fit every ARMA(p, q) with 0 ≤ p ≤ P and 0 ≤ q ≤ Q by maximum likelihood on the same n observations.
3. Compute AIC for each and choose the pair with the smallest value; only differences matter, and a gap below about 2 means the models are practically equivalent.
4. Check that the residuals of the chosen model are white noise.

The first term falls as parameters are added and the second rises, so the minimum balances the bias of under-fitting against the variance of over-fitting.

AIC is not consistent: even as n → ∞ it can select too large an order. Consistent alternatives are BIC = −2 ln L̂ + k ln n and Hannan–Quinn = −2 ln L̂ + 2k ln ln n. For small samples use AICc = AIC + 2k(k + 1)/(n − k − 1).

(b) Autocorrelation, its effect on OLS, and ML estimation with AR(1) errors

The autocorrelation coefficient is the correlation between disturbances k periods apart: ρₖ = Cov(uₜ, uₜ₋ₖ)/Var(uₜ), with −1 ≤ ρₖ ≤ 1. Autocorrelation means ρₖ ≠ 0 for some k ≥ 1, so E(uu′) ≠ σ²I.

Consequences for OLS (X non-stochastic):

• β̂ stays unbiased and consistent but is no longer BLUE; GLS has smaller variance.
• The usual s²(X′X)⁻¹ is a biased estimate of Var(β̂), typically too small when ρ > 0, so t and F tests and confidence intervals are unreliable and R² is overstated.
• If a lagged dependent variable is a regressor, OLS is also inconsistent.

ML estimation of Y = Xβ + u, uₜ = ρuₜ₋₁ + εₜ, εₜ ~ iid N(0, σ²), |ρ| < 1: here Var(uₜ) = σ²/(1 − ρ²) and Cov(uₜ, uₜ₋ₛ) = σ²ρˢ/(1 − ρ²), so u ~ N(0, σ²V) with |V| = 1/(1 − ρ²). With eₜ = yₜ − xₜ′β, S(β, ρ) = (y − Xβ)′V⁻¹(y − Xβ) = (1 − ρ²)e₁² + Σₜ₌₂ⁿ (eₜ − ρeₜ₋₁)², ln L = −(n/2) ln 2π − (n/2) ln σ² + ½ ln(1 − ρ²) − S(β, ρ)/(2σ²).

Likelihood equations:

• σ̂² = S/n.
• For given ρ: β̂ = (X′V⁻¹X)⁻¹X′V⁻¹y, the GLS (Prais–Winsten) estimator.
• For given β: ∂ln L/∂ρ = 0 is a cubic in ρ.

Because the equations are non-linear, iterate (Beach–MacKinnon): start from OLS residuals, solve for ρ, recompute β̂ by GLS, and repeat to convergence. Dropping the first observation gives the conditional ML (Cochrane–Orcutt) estimate. The ML estimators are consistent and asymptotically efficient.

(c) Collection of industrial data

Method: data are collected from establishments by complete enumeration (census) or sample survey, using returns and schedules filled in by occupiers, supplemented by monthly production returns from source agencies. The Collection of Statistics Act, 2008 gives statutory backing.

• Census: the Economic Census (MoSPI) counts all non-farm establishments; the Annual Survey of Industries (ASI) enumerates all large factories (generally 100 or more employees).
• Sample: the remaining ASI factories are sampled by a stratified design; the Annual Survey of Unincorporated Sector Enterprises (NSO) covers informal units.

(i) Official publications for data collection:

• ASI results and reports, and the Statistical Year Book India (MoSPI)
• IIP monthly press release (NSO) and the Index of Core Industries release (OEA, DPIIT)
• Economic Survey (Ministry of Finance); RBI Handbook of Statistics on the Indian Economy

(ii) Statistics collected by official agencies on industrial production:

• NSO: IIP, a monthly volume index (base 2022-23 from June 2026, with 1,042 products in 463 item groups; the 2011-12 series had 839 items in 407 groups); ASI data on output, inputs, employment, wages, fixed capital and value added by industry and state
• OEA, DPIIT: Index of Core Industries (nine industries, including iron ore, in the 2022-23 series) and the Wholesale Price Index
• Source ministries and bodies: Indian Bureau of Mines, Central Electricity Authority, Ministries of Coal, Petroleum and Steel
• Labour Bureau: factory employment, wages and disputes
• RBI: capacity utilisation survey