(a) Explain Box-Jenkins methodology to build ARIMA models. (15 marks) (b) Prepare the cost of living index for 2006 on the basis…

(a) Box–Jenkins methodology for ARIMA models

The Box–Jenkins approach builds an ARIMA(p, d, q) model, φ(B)∇ᵈyₜ = θ(B)εₜ, through an iterative cycle of identification, estimation and diagnostic checking, followed by forecasting.

• Stationarity. Plot the series and test for a trend, seasonality and a unit root (ADF, KPSS). If it is non-stationary, difference it d times (and use a log transform for a changing variance, or seasonal differencing) until the mean and autocovariances are stable.
• Identification. Examine the ACF and PACF of the stationary series. A PACF that cuts off after lag p with a decaying ACF suggests AR(p). An ACF that cuts off after lag q with a decaying PACF suggests MA(q). If both decay, a mixed ARMA(p, q) is tried, and information criteria (AIC, BIC) help choose among rival orders.
• Estimation. The parameters are estimated by maximum likelihood or non-linear least squares, checking that the AR roots lie outside the unit circle (stationarity) and the MA roots outside it (invertibility), and that the coefficients are significant.
• Diagnostic checking. The residuals should be white noise: the residual ACF shows no significant spikes and the Ljung–Box test is not rejected. If they are autocorrelated, the model is re-specified and the cycle repeats.
• Forecasting. The accepted model gives recursive forecasts with confidence intervals that widen with the horizon. Parsimony, meaning the fewest parameters that give white-noise residuals, is the guiding principle.

(b) Cost of living index, 2006 on 2005

Base-year outlays p₀q₀ are A 34.50, B 30, C 6, D 48, E 8 and F 20, so Σp₀q₀ = 146.5. Current prices at base quantities p₁q₀ are 36, 48, 9, 60, 6 and 15, so Σp₁q₀ = 174.

(i) Aggregative method: index = (Σp₁q₀/Σp₀q₀) × 100 = (174/146.5) × 100 = 118.77.

(ii) Weighted relatives: the price relatives R = (p₁/p₀) × 100 are A 104.35, B 160, C 150, D 125, E 75 and F 75, with weights w = p₀q₀. Then ΣRw/Σw = 17400/146.5 = 118.77.

Comment: the two methods agree because, with base-year value weights, the weighted relatives index is algebraically identical to the Laspeyres aggregative index. The cost of living rose by about 18.8% over 2005. The rise comes from B, C and D, while E and F became cheaper. As a fixed-basket index it ignores substitution, quality change and new goods, so it tends to overstate the rise.

(c) Price quotations and foreign-trade publications

Price statistics rest on price quotations: observed or reported prices of specified items in specified markets at specified dates. Their quality depends on:

• Item selection: commodities are chosen by their expenditure share and availability, with the grade, quality and pack size precisely specified so that like is compared with like.
• Market selection: centres are chosen by population, trade volume and price variation, with wholesale or retail quotations kept distinct.
• Timing and source: prices are collected at fixed intervals from shopkeepers, traders or official records, then edited for outliers, missing quotes and quality changes.

In India, retail price quotations feed the consumer price indices. The CPI (rural, urban and combined) is compiled by the National Statistics Office under MoSPI, while the Labour Bureau compiles CPI-IW for industrial workers and CPI-AL and CPI-RL for agricultural and rural labourers. Wholesale quotations feed the WPI, compiled by the Office of the Economic Adviser in the Department for Promotion of Industry and Internal Trade (Ministry of Commerce and Industry). Expenditure weights come from household consumption surveys.

For foreign trade, the Directorate General of Commercial Intelligence and Statistics (DGCI&S), Kolkata, under the Ministry of Commerce and Industry, compiles data from customs documents, classified under the ITC-HS. Its publications include:

• monthly foreign trade statistics by principal commodities and countries, and 8-digit commodity-level data for exports and imports;
• the weekly Indian Trade Journal;
• the annual Selected Statistics of the Foreign Trade of India, and annual statistics on foreign and coastal cargo movement and on customs and excise revenue.

The Ministry of Commerce also releases monthly quick estimates of exports and imports, and the RBI reports trade within the balance of payments.

Limitations are the representativeness of quotations, quality change, new goods and delays between collection and release; trade data are also revised as late returns arrive.