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

Box–Jenkins methodology. Box–Jenkins builds ARIMA(p,d,q) models by treating a time series as a stochastic process that can be made stationary, identified, estimated and checked. The model is φ(B)∇^d y_t = θ(B)ε_t, where ∇^d is the d-th difference, φ(B) the autoregressive polynomial of order p and θ(B) the moving-average polynomial of order q. First, the series is examined for trend, seasonality and unit root; ADF and KPSS tests help decide whether levels or differences are stationary. If non-stationary, differencing is applied until shocks do not persist, because only a stationary process has stable mean and autocovariances. Next, ACF and PACF are used for identification: a PACF that cuts off after p suggests AR(p), an ACF that cuts off after q suggests MA(q), while mixed decay suggests ARMA. Parameters are then estimated, usually by maximum likelihood, with constraints that roots lie outside the unit circle. Diagnostic checking asks whether residuals are white noise; the Ljung–Box test and residual ACF/PACF are used. If residuals show autocorrelation, the model is re-specified. Finally, forecasts are produced recursively with confidence intervals that widen over the horizon. In India, RBI has used ARIMA-type models for short-run forecasting of inflation and monetary aggregates, supporting monetary-policy analysis.

Cost of living index. From the given table, base-year outlays v0=p0q0 are A 34.5, B 30, C 6, D 48, E 8 and F 20, so Σv0=146.5. Current-year outlays at base quantities p1q0 are 36, 48, 9, 60, 6 and 15, so Σp1q0=174. By the aggregative Laspeyres method, the cost of living index for 2006 on 2005 base is (174/146.5)×100=118.77. By the weighted relatives method, price relatives R=(p1/p0)×100 are 104.35, 160, 150, 125, 75 and 75. Using exact relatives, Rw=100p1q0, so ΣRw=17400; the index is 17400/146.5=118.77. If the rounded relative 104.35 is used, ΣRw=17400.075, still giving 118.77 to two decimals. Thus the cost of living rose by about 18.77 per cent. The rise is driven mainly by B, C and D, while E and F fell. The two methods agree because the weighted relatives method is algebraically equivalent to the Laspeyres aggregative method when base-year expenditure weights are used. The result should be interpreted cautiously: index numbers can suffer substitution bias, quality changes and omission of new goods, so the 18.77 per cent is a fixed-basket measure rather than a full welfare measure.

Price quotations and foreign trade publications. Price statistics depend on price quotations, i.e. observed or reported prices for specified commodities in specified markets. The key problem is representativeness: items must be selected according to expenditure share and availability; quality, grade and pack size must be specified; markets must be chosen by population, trade volume and price variation; and collection timing must be fixed to avoid seasonal or temporary distortions. In India, NSSO household consumption surveys provide expenditure weights and item lists, while CSO/NSO and Labour Bureau construct CPI series such as CPI-AL, CPI-RL, CPI-U and CPI-IW; Labour Bureau is specifically responsible for CPI-IW. WPI, however, is compiled by the Office of Economic Adviser, Ministry of Commerce and Industry/DPIIT, not by CSO. For foreign trade, DGCI&S under the Ministry of Commerce publishes Monthly Statistics of Foreign Trade and the Annual Report on Foreign Trade, with commodity-wise and country-wise data, using the ITC-HS classification. These publications help track exports, imports, trade partners and commodity trends. The main limitations are the representativeness of price quotations, quality changes, new goods, and the timeliness of trade data, since monthly and annual releases may lag real-time economic conditions. Overall, Box–Jenkins gives a disciplined forecasting framework, the index shows a substantial fixed-basket cost increase, and Indian price and trade statistics are useful but require attention to collection quality and limitations.