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

(b) Define autocorrelation coefficient. What are its consequences for ordinary least squares? Discuss the maximum likelihood estimation of the model, in

Question

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

(b) Define autocorrelation coefficient. What are its consequences for ordinary least squares? Discuss the maximum likelihood estimation of the model, in the usual notations, Y = Xβ + u with AR (autoregressive)(1) disturbance. 20 marks

(c) Explain the method of collection of industrial data. Describe the (i) official publications for data collection and (ii) statistics collected by the various official agencies pertaining to industrial production. 15 marks

UPSC Answer Check · Accepted Answer

(a) Akaike Information Criterion for ARMA Order Selection

The Akaike Information Criterion (AIC) is a penalised likelihood measure used to select the orders (p, q) of an ARMA process. For a candidate model with likelihood L and k free parameters, AIC = −2 log L + 2k. The first term falls as fit improves; the second penalises complexity, so AIC trades goodness-of-fit against parsimony. For ARMA(p, q), k = p + q + 1 (including the variance), and one estimates each candidate model by conditional or exact MLE, computes AIC, and selects the (p, q) minimising it. Because AIC is not consistent, the BIC (−2 log L + k log n) is preferred for large samples, while AICc (−2 log L + 2kn/(n−k−1)) corrects small-sample bias. The selected order balances underfitting (biased forecasts) against overfitting (inflated variance).

(b) Autocorrelation Coefficient and MLE with AR(1) Disturbances

The autocorrelation coefficient at lag k is ρₖ = Cov(uₜ, uₜ₋ₖ)/Var(uₜ), with ρ₀ = 1 and |ρₖ| ≤ 1. Under spherical errors OLS is BLUE; with autocorrelated errors, E(uu′) = σ²Ω ≠ σ²I, so OLS remains unbiased and consistent but is inefficient, its standard errors are biased (usually understated), and the t and F statistics are invalid, leading to spurious significance. GLS is BLUE.

For Y = Xβ + u, uₜ = ρuₜ₋₁ + εₜ, |ρ| < 1, εₜ ~ N(0, σ²), the covariance matrix is Ω = (1/(1−ρ²)) times the AR(1) correlation matrix. The likelihood is L(β, ρ, σ²) = (2πσ²)^−n/2|Ω|^−1/2 exp[−(Y−Xβ)′Ω⁻¹(Y−Xβ)/(2σ²)]. Since Ω⁻¹ = P′P with P the Prais–Winsten transformation matrix, the model becomes PY = PXβ + Pu, whose errors are spherical. Maximising the concentrated log-likelihood in ρ and β yields GLS estimates β̂ = (X′Ω⁻¹X)⁻¹X′Ω⁻¹Y, with σ̂² = û′Ω⁻¹û/n. Because ρ is unknown, estimation is iterative: estimate β by OLS, obtain residuals, estimate ρ, transform, re-estimate, and repeat until convergence (Cochrane–Orcutt). The estimator is consistent and asymptotically efficient, and testing ρ̂ = 0 validates the OLS specification.

(c) Collection of Industrial Data

Industrial data are collected by census (complete enumeration, e.g., Economic Census) and sample surveys (e.g., Annual Survey of Industries, ASI), using establishment schedules that record output, inputs, employment, wages and capital. The ASI, under the Collection of Statistics Act, covers registered manufacturing establishments; the IIP uses a weighted basket of 697 items from selected industries.

Official publications include the ASI Summary Results, Index of Industrial Production, Economic Census reports, and RBI Handbook of Statistics. Agencies: the National Statistical Office (erstwhile CSO) compiles IIP and ASI; DIPP (now DPIIT) publishes industrial licensing and production data; the Labour Bureau reports wages and employment; RBI compiles industrial statistics for monetary policy. Limitations include weak coverage of the informal sector and time lags in ASI release.

In sum, AIC provides a principled order-selection rule, GLS/MLE corrects autocorrelation-induced inefficiency, and India's industrial statistics rest on ASI and IIP, whose coverage gaps call for periodic revision.

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

Model answer

What "Explain" is asking you to do

Structure that answers it

Where marks are lost

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Must cover

Loses marks

Earns more

Extra mark

Must cover

Loses marks

Earns more

Extra mark

Must cover

Loses marks

Earns more

Extra mark

Practice this exact question

More from Statistics 2022 Paper II