Short-Run Forecasting of Commodity Prices: An Application of Autoregressive Moving Average Models

Ke-young Chu

doi:10.5089/9781451956498.024.A005

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Short-Run Forecasting of Commodity Prices: An Application of Autoregressive Moving Average Models

Author: Mr. Ke-young Chu

Publication Date:: 01 Jan 1978

eISBN:: 9781451956498

Language:: English

Keywords:: SP; cover book; commodity price; final equation; price series; form equation; series show; Agricultural commodities; Commodity markets; Arima equation

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KE-YOUNG CHU *

Abstract

KE-YOUNG CHU *

KE-YOUNG CHU ^*

Many time series—whether they are weekly prices, monthly interest rates, or the annual transaction volumes of commodities—have certain regularities, partially obscured by noises. Decomposing a time series into its systematic part and the random part is useful for understanding the nature of the series as well as for predicting its future values. The time-series technique extensively discussed in Box and Jenkins (1970) deals with the identification and estimation of autoregressive integrated moving average (ARIMA) models, which can be used for predicting the systematic parts of time series.

ARIMA models are now used not only for predicting time series but also for analytical purposes. ¹ In comparison with forecasting based on reduced forms derived from simultaneous equations models, ARIMA equations are very simple to use and require much less information for forecasting. Furthermore, ARIMA models are not entirely without an economic justification. It was shown by Tinbergen (1939), Zellner and Palm (1974), Palm (1977), and others that the “final equation” in an ARIMA form can be derived for each of the endogenous variables in a simultaneous equations model on a simple set of assumptions.

This paper reviews the relative merits of using ARIMA models for forecasting, compared with reduced forms derived from a simultaneous equations model. The paper also attempts to apply the method of ARIMA models to a short-run simulation of commodity prices. The results suggest that ARIMA models can yield quite satisfactory short-run forecasts for the monthly prices of some commodities and that they can be useful even for forecasting the yearly averages of monthly prices. ²

The paper is divided into three sections. Section I illustrates how the final equation for price in an ARIMA form can be derived from a structural system; it also reviews the relative merits of the two methods. Section II outlines the results obtained from applying the ARIMA scheme to short-run forecasting of the international prices of ten selected commodities (foods and raw materials). Section III summarizes the findings.

I. Two Representations of an Economic System—Reduced Forms and Final Equations

In this section, a simple system of equations containing the most rudimentary elements of a full-fledged commodity model is introduced to illustrate the conceptual relationship between reduced forms and final equations. It is shown that the final equation for price has an ARIMA form. The discussion begins with a deterministic system, which is later made stochastic. Since the model is introduced for an illustration, no attempt is made to discuss appropriateness of the explanatory variables or the lag structure of the model. Suppose that a commodity market is represented by the following system of equations:

\begin{array}{l} Q_{d t} & = α_{1} P_{t} + α_{2} Y_{t} \\ Q_{s t} & = β_{1} P_{t} \\ S_{t} & = γ_{1} S_{t - 1} + γ_{2} P_{t} \\ Q_{s t} - Q_{d t} & = S_{t} - S_{t - 1} (1) \end{array}

where

reduced forms and final equations

The four nonstochastic equations in (1) constitute a structural system. Each equation may include current and lagged endogenous and exogenous variables as regressors. From the structural system, a reduced form equation is derived for each of the endogenous variables. In a reduced form equation, only lagged endogenous variables and the current and lagged exogenous variables are used as regressors. The reduced form equation for price derived from the system in (1) is as follows:

\begin{matrix} P_{t} = \frac{1}{α_{1} - β_{1} + γ_{2}} [(1 - γ_{1}) S_{t - 1} - α_{2} Y_{t}] & (2) \end{matrix}

The reduced form equations are determined if the structural system is fully specified. If the coefficient values of the structural system are known, the values of the reduced form coefficients are also determined. Otherwise, the reduced form parameters can be either estimated directly or computed on the basis of estimated structural coefficients by deriving restricted reduced form equations. ³ In any case, forecasts can be obtained if the coefficient values, the future realizations of exogenous variables, and initial conditions are given.

The concept “final equation” was used in econometric analysis as early as 1939 by Tinbergen. For example, Tinbergen derives final equations in the form of autoregressive (AR) processes by eliminating endogenous variables from the right-hand sides of the equations in a structural system. ⁴ The final equation for price is derived from the structural system in (1) as follows: ⁵ Suppose that the income series Y_t(t = 1, 2, …,) in the system (1) is viewed as generated by a linear stochastic process. Specifically, let the process be specified as the following autoregressive integrated moving average (ARIMA) process:

\begin{matrix} Δ Y_{t} = θ_{1} Δ Y_{t - 1} + ϵ_{t} & (3) \end{matrix}

A combination of the four structural equations in (1) and the equation in (3) gives a final equation for each of the endogenous variables in the system. The final equation for price is as follows:

\begin{matrix} (1 - π_{1} L) (1 - π_{2} L) Δ P_{t} = (1 - γ_{1} L) u_{t} & (4) \end{matrix}

Where

L = lag operator, such that LP_t = P_t–1 and (1–L)P_t = ΔP_t

\begin{array}{l} π_{1} = \frac{γ_{1} (β_{1} - α_{1}) - γ_{2}}{β_{1} - α_{1} - γ_{2}} \\ π_{2} = θ_{1} \\ u_{t} = \frac{α_{2}}{β_{1} - α_{1} - γ_{2}} ϵ_{t} \end{array}

The final equation in (4) is an ARIMA (2, 1, 1) process. In a final equation, a current endogenous variable is expressed as a function of its own lagged values and the errors. The coefficients of the final equation in the form of (4) can be either estimated directly or identified from the estimates of structural parameters.

The final equation for price can still be derived in a similar way even if a random error term is added to the supply equation. Let the structural system be as follows:

\begin{array}{l} Q_{d t} & = α_{1} P_{t} + α_{2} Y_{t} \\ Q_{s t} & = β_{1} P_{t} + η_{t} \\ S_{t} & = γ_{1} S_{t - 1} + γ_{2} P_{t} \\ Q_{s t} - Q_{d t} & = S_{t} - S_{t - 1} (5) \end{array}

where η_t has a property such that

l_t = η_t – η_t-1 = (1 – L)η_t

and ι_t is an identically and independently distributed (iid) random error independent of ϵ_t. The final equation in this case is as follows:

\begin{matrix} [1 - \frac{γ_{1} (β_{1} - α_{1}) - γ_{2}}{β_{1} - α_{1} - γ_{2}} L] (1 - θ_{1} L) (1 - L) P_{t} \\ = \frac{α_{2}}{β_{1} - α_{1} - γ_{2}} (1 - γ_{1} L) ϵ_{t} - \frac{1}{β_{1} - α_{1} - γ_{2}} (1 - γ_{1} L) (1 - θ_{1} L) ι_{t} & (6) \end{matrix}

\begin{matrix} (1 - π_{1} L) (1 - π_{2} L) Δ P_{t} \\ = (1 - γ_{1} L) u_{t} + (1 - γ_{1} L) (1 - θ_{1} L) w_{t} & \begin{matrix} (6^{'}) \end{matrix} \end{matrix}

where

w_{t} = - \frac{1}{β_{1} - α_{1} - γ_{2}} ι_{t}

The final equation in (6’) has an autoregressive (AR) part identical with that in (4) and a moving average (MA) part that is the sum of two independent MA processes. The two MA processes with u_t and w_t are independent of each other because ϵ_t and ι_t are mutually independent. The second term in

(1 – γ₁L)u_t + (1 – γ₁L)(1 – θ₁L)w_t

comes from the disturbance term added to the supply function. As Anderson (1975) and Rose (1977) show, the sum of two or more independent and invertible MA processes can be represented by a single invertible MA process that has a different random error. The MA part in (6’) can therefore be written as

\begin{matrix} (1 - π_{1} L) (1 - π_{2} L) Δ P_{t} = (1 - Φ_{1} L) (1 - Φ_{2} L) v_{t} & (7) \end{matrix}

where the MA process (1 – Φ₁L)(1 – Φ₂L)v_t is derived from the multiple MA process in (6′).

relative merits in forecasting

A successful prediction based on a model depends on a number of conditions: The model should be correctly specified; the coefficient values should be either known or consistently estimated with reasonably high asymptotic efficiency; and the system should be stable throughout the sample and prediction periods. Even if all these conditions are met, there can still be prediction errors if the equations have random error terms.

Although both can be derived from a given economic model, a reduced form and a final (ARIMA) equation have different capabilities in forecasting, since they are derived from different sets of assumptions and require different information bases for forecasting.

Reduced forms have an advantage in forecasting because a correct structural analysis can prevent the misspecification of a model. This is particularly true if the reduced form equations are derived from estimated structural equations. An economic and statistical evaluation can be made on whether specification and estimation are appropriate before forecasting is attempted. Furthermore, reduced forms facilitate an evaluation of alternative economic policies. This cannot be done with final equations, since final equations do not include exogenous variables as explanatory variables. Another merit of reduced forms in forecasting results from the fact that they permit an effective use of information on future realizations of the exogenous variables. As is explained later, the information is not so effectively utilized if final equations are used for forecasting.

To use reduced form equations for forecasting even one endogenous variable, however, generally the whole system must be estimated. Therefore, if only yearly data are available for some variables while quarterly series are available for the others, a quarterly model is difficult to use. Furthermore, separate forecasting should be made for the future values of each of the exogenous variables. Some scheme has to be found for predicting the future values of the exogenous variables in the system.

On the other hand, a foremost merit of a final (ARIMA) equation is simplicity. To forecast an endogenous variable, it is not necessary to deal with the entire model if a decision is made to “estimate” the final equation for the variable. Because no variables other than the endogenous variable that is explained enter the equation, the necessary predictions can be made on the basis of the one equation alone without separate predictions for the exogenous variables in the system. Because the whole system does not necessarily have to be estimated, final equations can be used for forecasting a number of endogenous variables even if observations on some variables in a model are not available.

It is obvious from the preceding discussion that final equations cannot be used for policy evaluations. Furthermore, the information on future exogenous variables cannot be effectively utilized if final equations are used for forecasting. Another weakness of final equations results from the insufficient state of present knowledge on their error terms: The error term in a final equation is a function of a number of the error terms in its associated structural system. In (4), for example, it is clear that the final equation error u_t is nothing more than $\frac{α_{2}}{β_{1} - α_{1} - γ_{2}} ϵ_{t},$ that is, the income equation error multiplied by a constant that is determined by structural coefficients. In many cases, however, this clear-cut relationship between structural and final equation errors is difficult to establish. In the second example in (7), for instance, the functional form that explicitly transforms the structural errors ϵ_t and η_t into the final equation error v_t cannot be established. Unlike forecasting based on reduced form equations, it is therefore difficult to effectively utilize extraneous information on future production, consumption, and speculation if final equations are used for forecasting.

In spite of the limitations just discussed, final (ARIMA) equations are still useful especially for a short-run prediction of commodity prices. For example, a simultaneous equations model is difficult to estimate for a short-run prediction of monthly prices of many commodities because monthly data are generally hard to obtain for other relevant variables. Furthermore, forecasts obtained through final equations can be modified by incorporating judgments into the forecasts. Several aspects of this problem are discussed in the following subsection.

error terms in final equations

A useful illustration can be made of the relationship between the error terms in a structural equation and a final (ARIMA) equation. The illustration will further clarify some of the merits and limitations of final equations and will suggest possible improvements in their use.

Let the problem of estimation be disregarded. That is, let the coefficient values be known. This assumption will simplify the discussion. The reduced form equation for price is as follows:

\begin{matrix} P_{t} = \frac{1 - γ_{1}}{α_{1} - β_{1} + γ_{2}} S_{t - 1} - \frac{α_{2}}{α_{1} - β_{1} + γ_{2}} Y_{t} + \frac{1}{α_{1} - β_{1} + γ_{2}} η_{t} & (8) \end{matrix}

On the other hand, the associated final equation with a multiple MA part has been shown to be

\begin{matrix} [1 - \frac{γ_{1} (β_{1} - α_{1}) - γ_{2}}{β_{1} - α_{1} - γ_{2}} L] (1 - θ_{1} L) (1 - L) P_{t} \\ = \frac{α_{2}}{β_{1} - α_{1} - γ_{2}} (1 - γ_{1} L) ϵ_{t} - \frac{1}{β_{1} - α_{1} - γ_{2}} (1 - γ_{1} L) (1 - θ_{1} L) ι_{t} & (6) \end{matrix}

which can be written as

\begin{matrix} (1 - π_{1} L) (1 - π_{2} L) Δ P_{t} \\ = (1 - γ_{1} L) u_{t} + (1 - γ_{1} L) (1 - θ_{1} L) w_{t} & \begin{matrix} (6^{'}) \end{matrix} \end{matrix}

\begin{matrix} (1 - π_{1} L) (1 - π_{2} L) Δ P_{t} = (1 - Φ_{1} L) (1 - Φ_{2} L) v_{t} & (7) \end{matrix}

if the MA part in (6’) is made single. ⁶ The final equation is derived by substituting the third structural equation for S_{t– 1} and the ARIMA process for Y_t in the reduced form equation. If the realizations of future η_t and Y_t are not known but reliable information is available for their prediction, using the reduced form equation in (2) or the final equation in the form of equation (6’) will give better results of prediction. On the other hand, the information cannot be effectively utilized for predictions if the final equation in the form of equation (7) is used, since the functional form that transforms u_t and w_t in (6′) into v_t in (7) is not known.

As explained further in the following discussion, the information can be effectively utilized if reduced forms are used for forecasting. A justification for adding an error term to a structural equation can be found in those minor random incidents that affect the dependent variable. Without any information on the likelihood and magnitude of their occurrence, such incidents are expected to cancel each other out, resulting in a zero combined impact on the dependent variable. In other words, it is assumed that the unconditional mean of the error term vanishes. Once those incidents have occurred, however, it is possible to explain the realizations of the error term on the basis of the incidents that are observed in many cases. For example, it is possible to attribute a large positive residual of demand for fuel in a certain year to an unusually low temperature, or to explain a large positive residual of a hoarding equation for gold on the basis of an anticipation of a social or political turmoil. Similarly, it cannot but be maintained that the effects of those incidents on a dependent variable will cancel each other out during any prediction period if no information is available on the likelihood of their occurrence. In many cases, however, especially in short-run predictions, useful information can be obtained on possible magnitudes of those incidents that may occur during a prediction period. For example, a revised weather forecast may be obtained for a prediction period after the estimation of a model has been finished but immediately before a prediction is made. A rational forecaster in such a case will use the anticipated values of the error terms conditional on the new weather forecast. The information can be effectively utilized if either reduced forms derived from structural equations or “restricted” final equations are used for forecasting. Although the use of the information may not be equally effective, it can still improve forecasts obtained through (unrestricted) final equations if the information is utilized for a correct evaluation of those incidents’ net impact on the final equation error.

II. The Results of Estimation and Simulation for Ten Commodities

This section outlines the estimation and simulation results of the ARIMA equations for the international prices of ten commodities. Of these ten commodities, six are foods (bananas, beef, copra, maize, cocoa, and tea), and the others are raw materials (hides, jute, rubber, and copper). For each commodity, two ARIMA equations—one based on monthly price series and the other on annual data—are estimated. The price series used in this study are the spot prices in leading commodity markets.

estimation

In estimating the equations, a decision has to be made on proper prices to be used for the estimation. If a prediction of nominal prices is the main purpose of the estimation, using the nominal price has an obvious advantage of simplicity, because the deflator itself has to be predicted if relative prices are used for the estimation. During a period of inflation, nominal prices will show nonstationarity, whereas relative prices could be stationary. The nonstationarity problem can be dealt with, however, by successive differencing of the series in some cases.

In this study ARIMA equations are fitted to nominal prices. ⁷ Because most commodity prices rose sharply during the period of commodity price inflation (1973–75), the wisdom of using stationary models may be questioned. Solid evidence does not seem to exist, however, for maintaining that the commodity price inflation will be sustained in the future. Furthermore, graphical analyses of the monthly prices of the sample commodities do not reveal any strong indication of nonstationarity—at least until 1973, the initial year of the high inflationary period. The estimated autocorrelation functions for the ten monthly commodity prices also suggest that differencing operations may not be necessary. Stationary models are therefore used for the estimation, except for the annual models for the prices of bananas, copra, and maize. If the patterns of price movements are obviously different during a certain period, the months or the years (usually beginning or ending part of the period from January 1957 to September 1976) are excluded from the sample for estimation of the monthly models. ⁸ The sample period for the annual models is uniformly from 1965 to 1974.

Table 1 shows the orders of the lag polynomials used in the estimated equations, and Tables 2 and 3 present the results of estimation. A feature of the monthly models reported in Table 2 is that the models do not explicitly describe the cyclical price movements for durations longer than one year. Some of the models are designed to explain seasonal price fluctuations (3-month, 6-month, or 12-month). The models, although including prices lagged by, at most, 24 months as explanatory variables, could explain longer-run swings of the prices. The equations are selected, however, not for their capability to predict longer-run fluctuations but for their power to predict month-to-month changes in prices. The sample autocorrelation functions of the monthly prices of beef, cocoa, hides, rubber, and copper suggest longer cycles of varying duration. ⁹ The existence of such cycles is not explicitly reflected in the chosen functional forms, however, either because the evidence is not believed to be sufficiently strong or because the sample period is not sufficiently long. ¹⁰

Table 1.

Ten Commodities: Selected Functional Forms¹

^¹figures indicate the orders of the lag polynomials included in the equations.

^²Annual models are based on the first differences of prices.

Table 1.

Ten Commodities: Selected Functional Forms¹

		Monthly Model									Annual Model
		Autoregressive (AR) part				Moving average (MA) part					AR part	MA part
			Seasonal				Seasonal			Goodness of fit	AR part	MA part	Goodness of fit
Commodity		Regular	3-month	6-month	12-month	Regular	3-month	6-month	12-month	Goodness of fit	Regular	Regular	Goodness of fit
Foods
	Bananas²	3	2	0	2	0	0	0	0	0.644	0	0	0.000
	Beef	3	0	1	1	0	0	0	0	0.970	1	0	0.589
	Copra²	2	0	0	0	0	0	0	0	0.892	0	0	0.704
	Maize²	2	0	0	0	0	0	0	0	0.881	0	0	0.853
	Cocoa	2	0	1	1	0	0	0	0	0.968	2	0	0.735
	Tea	2	0	1	1	1	0	0	1	0.838	1	0	0.159
Raw Materials
	Hides	1	0	2	1	1	0	1	0	0.938	2	0	0.562
	Jute	2	0	0	0	0	0	0	0	0.895	0	1	0.208
	Rubber	2	0	0	0	0	0	0	0	0.941	1	0	0.214
	Copper	4	0	0	0	0	0	0	0	0.900	2	0	0.389

^¹figures indicate the orders of the lag polynomials included in the equations.

^²Annual models are based on the first differences of prices.

Table 1.

Ten Commodities: Selected Functional Forms¹

		Monthly Model									Annual Model
		Autoregressive (AR) part				Moving average (MA) part					AR part	MA part
			Seasonal				Seasonal			Goodness of fit	AR part	MA part	Goodness of fit
Commodity		Regular	3-month	6-month	12-month	Regular	3-month	6-month	12-month	Goodness of fit	Regular	Regular	Goodness of fit
Foods
	Bananas²	3	2	0	2	0	0	0	0	0.644	0	0	0.000
	Beef	3	0	1	1	0	0	0	0	0.970	1	0	0.589
	Copra²	2	0	0	0	0	0	0	0	0.892	0	0	0.704
	Maize²	2	0	0	0	0	0	0	0	0.881	0	0	0.853
	Cocoa	2	0	1	1	0	0	0	0	0.968	2	0	0.735
	Tea	2	0	1	1	1	0	0	1	0.838	1	0	0.159
Raw Materials
	Hides	1	0	2	1	1	0	1	0	0.938	2	0	0.562
	Jute	2	0	0	0	0	0	0	0	0.895	0	1	0.208
	Rubber	2	0	0	0	0	0	0	0	0.941	1	0	0.214
	Copper	4	0	0	0	0	0	0	0	0.900	2	0	0.389

^¹figures indicate the orders of the lag polynomials included in the equations.

^²Annual models are based on the first differences of prices.

Table 2.

Ten Commodities: Estimated ARIMA Equations for Monthly Prices¹

^¹Multiplicative seasonal models are used for some commodities. The lag operator is indicated by L. The figures accompanying the estimated coefficients are the ratios between the estimated coefficients and the associated standard errors. The ratios would behave asymptotically like normal random variables. (See footnote 8 in the text for the sample periods.) The coefficient of determination (R²) is computed by using the formula $1 - \hat{V} (a_{t}) / \hat{V} (P_{t}) .$

Table 2.

Ten Commodities: Estimated ARIMA Equations for Monthly Prices¹

Commodity		Equation
Foods
	Bananas	$\begin{matrix} (1 - \underset{9.98}{0.666 L} - \underset{0.77}{0.060} L^{2} - \underset{3.02}{0.223} L^{3}) (1 - \underset{2.87}{0.218} L^{3} + \underset{1.61}{0.113} L^{6}) \\ (1 - \underset{2.47}{0.164 L^{12}} - \underset{1.33}{0.090 L^{24}}) P_{t} = \underset{1.67}{0.179} + {\hat{a}}_{t} \end{matrix}$
			T = 237	R² = 0.644
	Beef	$\begin{matrix} (1 - \underset{17.31}{1.146 L} - \underset{2.84}{0.284} L^{2} - \underset{1.79}{0.120} L^{3}) (1 - \underset{1.82}{0.124} L^{6}) \\ (1 - \underset{1.75}{0.119 L^{12}}) P_{t} = \underset{1.46}{0.695} + {\hat{a}}_{t} \end{matrix}$
			T = 239	R² = 0.970
	Copra	$\begin{matrix} (1 - \underset{16.46}{1.164} L + \underset{3.24}{0.230} L^{2}) P_{t} = \underset{2.78}{12.817} + {\hat{a}}_{t} \end{matrix}$
			T = 192	R² = 0.892
	Maize	$\begin{matrix} (1 - \underset{18.84}{1.338} L + \underset{5.56}{0.395} L^{2}) P_{t} = \underset{2.63}{0.078} + {\hat{a}}_{t} \end{matrix}$
			T = 192	R² = 0.881
	Cocoa	$\begin{matrix} (1 - \underset{23.48}{1.492} L + \underset{7.78}{0.516} L^{2}) (1 + \underset{1.08}{0.087} L^{6}) (1 - \underset{1.57}{0.123} L^{12}) P_{t} \\ = \underset{1.94}{0.786} + {\hat{α}}_{t} \end{matrix}$
			T = 204	R² = 0.968
	Tea	$\begin{matrix} (1 - \underset{5.03}{1.271} L + \underset{1.93}{0.422} L^{2}) (1 - \underset{0.79}{0.058 L^{6}}) (1 - \underset{38.27}{0.934 L^{12}}) P_{t} \\ = \underset{190}{0.577} + (1 - \underset{6.51}{0.422} L^{12}) (1 - \underset{0.87}{0.238} L) {\hat{α}}_{t} \end{matrix}$
			T = 237	R² = 0.838
Raw Mater Hides
	Hides	$\begin{matrix} (1 - \underset{44.94}{0.967} L) (1 + \underset{9.20}{0.0662 L^{6}} - \underset{4.50}{0.330} L^{12}) (1 - \underset{0.03}{0.002 L^{12}}) P_{t} \\ = \underset{1.62}{0.820} + (1 + \underset{47.31}{0.944} L^{12}) (1 - \underset{1.00}{0.070} L) {\hat{α}}_{t} \end{matrix}$
			T = 237	R² = 0.938
	Jute	$(1 - \underset{17.31}{1.111} L + \underset{2.47}{0.158} L^{2}) P_{t} = \underset{2.68}{12.778} + {\hat{α}}_{t}$
			T = 237	R² = 0.895
	Rubber	$(1 - \underset{19.55}{1.213} L + \underset{3.94}{0.246} L^{2}) P_{t} = \underset{2.21}{0.875} + {\hat{α}}_{t}$
			T =237	R² = 0.941
	Copper	$\begin{matrix} (1 - \underset{18.16}{0.491} L + \underset{6.11}{0.880} L^{2} - \underset{3.80}{0.550} L^{3} + \underset{2.99}{0.247} L^{4}) P_{t} \\ = \underset{3.26}{5.398} + {\hat{α}}_{t} \end{matrix}$
			T = 239	R² = 0.900

Table 2.

Ten Commodities: Estimated ARIMA Equations for Monthly Prices¹

Commodity		Equation
Foods
	Bananas	$\begin{matrix} (1 - \underset{9.98}{0.666 L} - \underset{0.77}{0.060} L^{2} - \underset{3.02}{0.223} L^{3}) (1 - \underset{2.87}{0.218} L^{3} + \underset{1.61}{0.113} L^{6}) \\ (1 - \underset{2.47}{0.164 L^{12}} - \underset{1.33}{0.090 L^{24}}) P_{t} = \underset{1.67}{0.179} + {\hat{a}}_{t} \end{matrix}$
			T = 237	R² = 0.644
	Beef	$\begin{matrix} (1 - \underset{17.31}{1.146 L} - \underset{2.84}{0.284} L^{2} - \underset{1.79}{0.120} L^{3}) (1 - \underset{1.82}{0.124} L^{6}) \\ (1 - \underset{1.75}{0.119 L^{12}}) P_{t} = \underset{1.46}{0.695} + {\hat{a}}_{t} \end{matrix}$
			T = 239	R² = 0.970
	Copra	$\begin{matrix} (1 - \underset{16.46}{1.164} L + \underset{3.24}{0.230} L^{2}) P_{t} = \underset{2.78}{12.817} + {\hat{a}}_{t} \end{matrix}$
			T = 192	R² = 0.892
	Maize	$\begin{matrix} (1 - \underset{18.84}{1.338} L + \underset{5.56}{0.395} L^{2}) P_{t} = \underset{2.63}{0.078} + {\hat{a}}_{t} \end{matrix}$
			T = 192	R² = 0.881
	Cocoa	$\begin{matrix} (1 - \underset{23.48}{1.492} L + \underset{7.78}{0.516} L^{2}) (1 + \underset{1.08}{0.087} L^{6}) (1 - \underset{1.57}{0.123} L^{12}) P_{t} \\ = \underset{1.94}{0.786} + {\hat{α}}_{t} \end{matrix}$
			T = 204	R² = 0.968
	Tea	$\begin{matrix} (1 - \underset{5.03}{1.271} L + \underset{1.93}{0.422} L^{2}) (1 - \underset{0.79}{0.058 L^{6}}) (1 - \underset{38.27}{0.934 L^{12}}) P_{t} \\ = \underset{190}{0.577} + (1 - \underset{6.51}{0.422} L^{12}) (1 - \underset{0.87}{0.238} L) {\hat{α}}_{t} \end{matrix}$
			T = 237	R² = 0.838
Raw Mater Hides
	Hides	$\begin{matrix} (1 - \underset{44.94}{0.967} L) (1 + \underset{9.20}{0.0662 L^{6}} - \underset{4.50}{0.330} L^{12}) (1 - \underset{0.03}{0.002 L^{12}}) P_{t} \\ = \underset{1.62}{0.820} + (1 + \underset{47.31}{0.944} L^{12}) (1 - \underset{1.00}{0.070} L) {\hat{α}}_{t} \end{matrix}$
			T = 237	R² = 0.938
	Jute	$(1 - \underset{17.31}{1.111} L + \underset{2.47}{0.158} L^{2}) P_{t} = \underset{2.68}{12.778} + {\hat{α}}_{t}$
			T = 237	R² = 0.895
	Rubber	$(1 - \underset{19.55}{1.213} L + \underset{3.94}{0.246} L^{2}) P_{t} = \underset{2.21}{0.875} + {\hat{α}}_{t}$
			T =237	R² = 0.941
	Copper	$\begin{matrix} (1 - \underset{18.16}{0.491} L + \underset{6.11}{0.880} L^{2} - \underset{3.80}{0.550} L^{3} + \underset{2.99}{0.247} L^{4}) P_{t} \\ = \underset{3.26}{5.398} + {\hat{α}}_{t} \end{matrix}$
			T = 239	R² = 0.900

Table 3.

Ten Commodities: Estimated ARIMA Equations for Annual Prices¹

^¹No satisfactory equations are obtained for bananas, copra, and maize. The sample period is uniformly from 1965 to 1974. The lag operator is denoted by L. The figures under the coefficients are the ratios between the estimated coefficients and the associated standard errors.

Table 3.

Ten Commodities: Estimated ARIMA Equations for Annual Prices¹

Commodity		Equation
Foods
Bananas	$(1 - L) P_{t} = α_{t} R^{2} = 0.000$
Beef	$\begin{matrix} (1 - \underset{3.39}{0.741} L) P_{t} = \underset{1.41}{17.752} + {\hat{a}}_{t} R^{2} = 0.589 \end{matrix}$
Copra	$(1 - L) P_{t} = α_{t} R^{2} = 0.704$
Maize	$(1 - L) P_{t} = α_{t} R^{2} = 0.853$
Cocoa	$(1 - \underset{4.37}{1.679} L + \underset{1.41}{0926} L^{2}) P_{t} = \underset{0.62}{11.503} + {\hat{a}}_{t} R^{2} = 0.735$
Tea	$(1 - \underset{1.23}{0.405} L^{}) P_{t} = \underset{1.52}{28.720} + {\hat{a}}_{t} R^{2} = 0.159$
Raw Materials
Hides	$\begin{matrix} (1 - \underset{2.88}{0.044} L + \underset{1.57}{0.825} L^{2}) P_{t} = \underset{2.28}{12.801} + {\hat{a}}_{t} R^{2} = 0.562 \end{matrix}$
Jute	$\begin{matrix} P_{t} = \underset{20.71}{288.388} + (\underset{7.12}{1 + 0.837 L}) {\hat{a}}_{t} R^{2} = 0.208 \end{matrix}$
Rubber	$\begin{matrix} (1 - \underset{1.47}{0.628} L) P_{t} = \underset{0.98}{8.751} + {\hat{a}}_{t} R^{2} = 0.214 \end{matrix}$
Copper	$\begin{matrix} (1 - \underset{1.52}{0.585} L + \underset{1.71}{0.624} L^{2}) P_{t} = \underset{2.30}{62.773} + {\hat{a}}_{t} R^{2} = 0.389 \end{matrix}$

Table 3.

Ten Commodities: Estimated ARIMA Equations for Annual Prices¹

Commodity		Equation
Foods
Bananas	$(1 - L) P_{t} = α_{t} R^{2} = 0.000$
Beef	$\begin{matrix} (1 - \underset{3.39}{0.741} L) P_{t} = \underset{1.41}{17.752} + {\hat{a}}_{t} R^{2} = 0.589 \end{matrix}$
Copra	$(1 - L) P_{t} = α_{t} R^{2} = 0.704$
Maize	$(1 - L) P_{t} = α_{t} R^{2} = 0.853$
Cocoa	$(1 - \underset{4.37}{1.679} L + \underset{1.41}{0926} L^{2}) P_{t} = \underset{0.62}{11.503} + {\hat{a}}_{t} R^{2} = 0.735$
Tea	$(1 - \underset{1.23}{0.405} L^{}) P_{t} = \underset{1.52}{28.720} + {\hat{a}}_{t} R^{2} = 0.159$
Raw Materials
Hides	$\begin{matrix} (1 - \underset{2.88}{0.044} L + \underset{1.57}{0.825} L^{2}) P_{t} = \underset{2.28}{12.801} + {\hat{a}}_{t} R^{2} = 0.562 \end{matrix}$
Jute	$\begin{matrix} P_{t} = \underset{20.71}{288.388} + (\underset{7.12}{1 + 0.837 L}) {\hat{a}}_{t} R^{2} = 0.208 \end{matrix}$
Rubber	$\begin{matrix} (1 - \underset{1.47}{0.628} L) P_{t} = \underset{0.98}{8.751} + {\hat{a}}_{t} R^{2} = 0.214 \end{matrix}$
Copper	$\begin{matrix} (1 - \underset{1.52}{0.585} L + \underset{1.71}{0.624} L^{2}) P_{t} = \underset{2.30}{62.773} + {\hat{a}}_{t} R^{2} = 0.389 \end{matrix}$

As for seasonal fluctuations, the autocorrelation functions of the levels of the monthly prices reveal that there are conspicuous 12-month and 6-month cycles in the tea price and a strong 12-month cycle in the banana price (Chart 1), while the autocorrelation functions of the first differences of the prices indicate possible 6-month cycles in the prices of beef, cocoa, and hides and 3-month cycles in the prices of bananas and tea. Multiplicative seasonal models are therefore used for bananas, beef, cocoa, tea, and hides. The results show that most of the parameters of the seasonal lag polynomials are statistically significant. For example, the estimated equation for the monthly price of bananas includes two significant seasonal autoregressive coefficients (for P_t–3 and P_t–12), while the equation for the monthly price of tea has two highly significant seasonal (12-month) coefficients (one autoregressive and the other moving average). The equation for the monthly price of hides also includes highly significant seasonal (6-month) autoregressive and moving average parameters.

The moduli of all the roots of the estimated lag polynomials lie outside the unit circle, indicating that all the prices are stationary during the sample periods and that the moving average parts are invertible. ¹¹ For the monthly models, the statistic R², defined as

1 - \frac{\hat{V} (a_{t})}{\hat{V} (P_{t})}

where $\hat{V} (a_{t})$ and $\hat{V} (P_{t})$ denote estimated variances of a_t and P_t, ranges from 0.644 (bananas) to 0.970 (beef). ¹² At least 88 per cent of the variations in the monthly price is explained by a second-order AR lag polynomial for copra, maize, jute, and rubber, while a fourth-order AR lag polynomial is required to explain 90 per cent of the variations in the price of copper. The monthly prices of two commodities (tea and hides) are better explained by equations with MA parts. Both of the equations have MA parts with 6-month seasonal lag polynomials, and the estimated parameters of the polynomials are highly significant.

The estimated monthly models may be used for a short-run prediction of the commodity prices: they can be used for a prediction of the monthly prices; they can also be used for a prediction of the quarterly, half-yearly, or yearly averages of the monthly prices. If a prediction of the yearly averages of the monthly prices is all that is needed, however, annual—rather than monthly—models may be used. For a comparison of the predictive powers of monthly and annual models, simple autoregressive or moving average models are estimated on the basis of the yearly price series of the ten commodities. Since the yearly price series are not sufficiently long, the identification process utilized for the monthly models is bypassed. First-order and second-order autoregressive and moving average models are tried, and a model that is considered to be most reasonable is selected for each commodity. The results are presented in Table 3. ¹³ It is interesting to note that no satisfactory annual models are found for three commodities (bananas, maize, and copra). Therefore, the annual prices of these commodities are simply assumed to be generated by random walk processes. The coefficients of determination for these prices reported in Table 3 are the coefficients of determination between the current prices and the prices lagged by one year. The coefficients are 0.704 and 0.853, respectively, for copra and maize. For bananas, the coefficient is zero; this implies that using the yearly price series for forecasting the price of bananas is not feasible for the sample period. The yearly prices of the remaining seven commodities yield models with coefficients of determination ranging from 0.159 (tea) to 0.735 (cocoa). The coefficients of determination associated with the monthly models are generally higher than those associated with the annual models, since the former are based on monthly predictions while the latter are based on yearly predictions. The difference between the coefficients of determination associated with the monthly model and the annual model is more pronounced in the seasonal commodities than in the others.

simulation

The procedure described in Box and Jenkins (1970) is used for two sets of dynamic simulations for the monthly models—a series of 12-month and a series of 6-month simulations. ¹⁴ For example, a simulated path of the monthly maize price is computed as a result of a series of 12-month dynamic simulations beginning in January each year. The simulations are dynamic in the sense that within any one year (or within any 6-month period for 6-month dynamic simulations) some or all of P_t–j+l^s (j> 0), the lagged prices that are used as explanatory variables, are replaced by their simulated values. These dynamic simulations are carried out 12 months (or 6 months) at a time. For maize, the equation that is used for simulation is

\begin{matrix} {\hat{P}}_{t + e} = 0.078 + 1.338 {\hat{P}}_{t - 1 + e} - 0.395 {\hat{P}}_{t - 2 + e} & (9) \end{matrix}

with ℓ = 1, 2, …, 12, for 12-month simulations. By nature of the forecasting scheme and because the models are stationary, the simulated prices tend to converge to the series’ mean values. For example, the time path of the simulated maize price will converge to its mean value if ℓ goes to infinity

\begin{matrix} lim_{ℓ \to \infty} {\hat{P}}_{t + ℓ} = \frac{0.078}{1 - 1.338 + 0.395} & (10) \end{matrix}

Therefore, if the price of a commodity is higher (lower) than the mean during the months immediately preceding a simulation period, the simulated path of the price tends to fall (rise) ultimately.

An evaluation of the scheme’s predictive power of monthly prices is made by comparing the actual path with the simulated path of the price. First, the correlation coefficient (R₆) is calculated between the actual monthly price and the simulated monthly price resulting from 6-month dynamic simulations for each commodity for the entire simulation period (1965–74). Second, the correlation coefficient (R₁₂) is calculated between the actual monthly price and the simulated monthly price resulting from 12-month dynamic simulations. The correlation coefficient R₁₂ indicates the average degree of accuracy of the monthly model’s short-run (one-year) predictions of monthly prices, whereas the correlation coefficient R₆ indicates that of 6-month forecasts of monthly prices. In other words, R₁₂ indicates the degree of accuracy of predictions made at the end of each year for the following 12 monthly prices, while R₆ indicates that of predictions made at the end of June and December of each year for the following 6 months. If predictions are made at the end of each year for the following 12 months and if the forecasts for the second half of each year are revised on the basis of the predictions made at the end of June, the degree of accuracy of the revised monthly forecasts will be indicated by R₆, which should generally be higher than R₁₂. Also, the simulation period is a part of the sample period for each commodity. For an evaluation of the scheme’s predictive power of the yearly averages of monthly prices, the simulated monthly prices obtained from the 12-month simulations and the 6-month simulations are converted into yearly averages. The correlation coefficients (R_6a and R_12a) are computed from the actual yearly price series and the yearly averages of the simulated monthly prices based on the 12-month and 6-month dynamic simulations.

Table 4 shows the four correlation coefficients (R_6, R₁₂, R_6a, and R_12a) and the correlation coefficient (R_a) between the actual yearly price and the price statically simulated by the annual model for each commodity. A few relevant facts should be mentioned: First, the figures in the table are correlation coefficients—not coefficients of determination; second, they are based on short-run (at most 12-month) simulations. It is not surprising therefore that the coefficients are relatively high in many cases and as high as 0.985 (R_12a, maize) in one case. Longer-run simulations would inevitably produce lower coefficients. Also, the simulations conducted in this study are based on no information other than the price of the commodity in consideration. The results suggest that, even if a prediction of the yearly average of the monthly prices of a commodity is the only purpose of the model, working with monthly—rather than yearly—data can give better results. For example, the monthly models for bananas, copra, and maize yield, respectively, 0.536, 0.903, and 0.985 as coefficients, although no satisfactory annual models can be found for the prices. ¹⁵ The correlation coefficients improve when the monthly forecasts are converted into yearly forecasts, probably because the monthly prediction errors are canceled when yearly averages are derived.

Table 4.

Ten Commodities: Goodness of Fit of Simulated Prices

Table 4.

Ten Commodities: Goodness of Fit of Simulated Prices

			Monthly Model
			Simulation of monthly price		Simulation of annual price
Commodity		Annual Model (Ra)	6-month dynamic simulation (R₆)	12-month dynamic simulation (R₁₂)	6-month dynamic simulation (R_6a)	12-month dynamic simulation (R_12a)
Foods
	Bananas	0.000	0.509	0.417	0.738	0.536
	Beef	0.767	0.877	0.723	0.924	0.759
	Copra	0.839	0.912	0.868	0.966	0.903
	Maize	0.923	0.944	0.835	0.984	0.985
	Cocoa	0.857	0.901	0.838	0.974	0.900
	Tea	0.399	0.386	0.410	0.662	0.378
Raw Materials
	Hides	0.749	0.869	0.798	0.958	0.844
	Jute	0.456	0.607	0.132	0.706	0.123
	Rubber	0.462	0.901	0.681	0.919	0.748
	Copper	0.624	0.763	0.650	0.916	0.679

Table 4.

Ten Commodities: Goodness of Fit of Simulated Prices

			Monthly Model
			Simulation of monthly price		Simulation of annual price
Commodity		Annual Model (Ra)	6-month dynamic simulation (R₆)	12-month dynamic simulation (R₁₂)	6-month dynamic simulation (R_6a)	12-month dynamic simulation (R_12a)
Foods
	Bananas	0.000	0.509	0.417	0.738	0.536
	Beef	0.767	0.877	0.723	0.924	0.759
	Copra	0.839	0.912	0.868	0.966	0.903
	Maize	0.923	0.944	0.835	0.984	0.985
	Cocoa	0.857	0.901	0.838	0.974	0.900
	Tea	0.399	0.386	0.410	0.662	0.378
Raw Materials
	Hides	0.749	0.869	0.798	0.958	0.844
	Jute	0.456	0.607	0.132	0.706	0.123
	Rubber	0.462	0.901	0.681	0.919	0.748
	Copper	0.624	0.763	0.650	0.916	0.679

Simulated paths of two monthly prices (for bananas and tea) are compared with the actual monthly prices in Charts 2 and 3 for an illustration. The two price series show pronounced seasonality, which seems to become less regular for tea during the second half of the simulation period. Also, the monthly model for tea persistently underpredicts the price during the second half of the simulation period. This may be due to the changes in the seasonal pattern of the price that began to emerge during the period.

possible improvements

The limitations of the ARIMA model in forecasting commodity prices and other economic time series have already been discussed. The most important of these is probably the fact that the model in the form of (7) cannot effectively utilize the information that a forecaster may have on the exogenous variables. Even if accurate forecasts of Y_t are available, the forecaster who uses the equation in (7) would have to make predictions as if he did not have the information. This is an important waste of information. This is also true for the information that the forecaster may have on the future realizations of the structural errors.

Although no attempt to do so is made in this paper, judgmental forecasts based on economic analysis of various factors in commodity markets can be used to revise forecasts made through ARIMA models. An autoregressive final form (or a transfer function) may also be used for prediction if one has accurate information on future realizations of the exogenous variables. For example, one may derive the autoregressive final form for price from the structural system in (1) as

\begin{matrix} (1 - ψ_{1} L) Δ P_{t} = ψ_{2} (1 - γ_{1} L) Δ Y_{t} + (1 - γ_{1} L) n_{t} & (11) \end{matrix}

where

\begin{array}{l} ψ_{1} = \frac{γ_{1} (α_{1} - β_{1} + γ_{2}) + (1 - γ_{1})}{α_{1} - β_{1} + γ_{2}} \\ ψ_{2} = \frac{α_{2}}{α_{1} - β_{1} + γ_{2}} \\ n_{t} = \frac{1}{α_{1} - β_{1} + γ_{2}} ι_{2} \end{array}

by combining the stock equation in (5) with the reduced form in (8). The equation can be estimated and used for prediction. An estimation of the whole system is not necessary for an effective utilization of an equation like (11) for forecasting.

III. Summary and Conclusions

The relative merits of final (ARIMA) equations in forecasting economic time series are reviewed in this paper. The foremost advantage is simplicity. No series other than the one that is predicted is required in forecasting based on final equations. This simplicity facilitates effective use of existing data, which may have to be underutilized if forecasting is based on reduced forms. Thus, a monthly series can be fully utilized through a final equation, while it may have to be converted into quarterly, half-yearly, or yearly series if reduced forms are used for forecasting and if other necessary series are not collected on a monthly basis. On the other hand, reduced forms are essential in policy analysis and more effective in utilizing the relevant information on the future realizations of exogenous variables and structural errors.

The relationship between the reduced form and the final equation suggests that there is some economic justification for using ARIMA models for forecasting: ARIMA models are not mere statistical constructs. An ARIMA equation compatible with a correctly specified structural model could be estimated if the estimation of the equation is constrained by the structural relationships among relevant variables. Even if the ARIMA equation is estimated without any constraints, the estimated equation should be expected to be consistent with the correct structural system if identification and estimation are appropriate.

The results of an application of the scheme to short-run simulations of the international prices of ten commodities are also described in the paper. Annual and monthly models were estimated and used for a simulation of the yearly averages of the monthly prices for the period 1965–74. The annual model gave slightly better results of simulation for beef, tea, and jute, while the monthly model performed better for the other commodities. For bananas, copra, and maize, no satisfactory annual model could be found for the sample period, while the monthly models gave quite accurate short-run predictions for copra and maize.

The time-series technique discussed in this study is a simple and economical device that can be used for short-run forecasting of commodity prices. The scheme has, however, only limited capability to predict unusual movements in prices. For example, the scheme can neither predict the sudden increase in the price of tea nor forecast the wide fluctuations in the price of bananas during the period 1972–74. The predictive power can possibly be improved by incorporating into the scheme judgments based on an economic analysis of the market or by introducing relevant exogenous variables into the equation.

BIBLIOGRAPHY

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Mr. Chu, economist in the Commodities Division of the Research Department, is a graduate of Kyung Hee University (Seoul) and Columbia University. Before joining the Fund, he was an instructor at Columbia University.

There have been studies on applications of time-series models for forecasting macroeconomic variables (Naylor, Seaks, and Wichern, 1972, and Nelson, 1972), for predicting birth rates (Saboia, 1977), and for analyzing telephone data (Thompson and Tiao, 1971); time-series models have also been used for specifying an econometric model (Zellner and Palm, 1974) and for evaluating economic causality (Brillembourg, 1976, Feige and Pearce, 1976, and Frenkel, 1977).

All the equations identified and estimated in this study exhibit stationarity: all of them are in autoregressive integrated moving average (ARIMA; p, d, q) forms with d = 0, or autoregressive moving average (ARMA; p, q1) forms, where p, d, and q indicate the orders of the AR part, differencing, and the MA part, respectively.

See Dhrymes. (1973).

The nonstochastic second-order AR equations in Tinbergen (1939, p. 131) are essentially ARIMA (2, 0, 0) or ARMA (2, 0) processes.

For derivation of final equations from a structural system with a more general moving average part, see Zellner and Palm (1974) or Palm (1977).

The equation in (6) may be called a restricted final equation if the parameter estimates are derived from estimated structural coefficients.

The series used in preparing this paper are similar to but not identical with those published in International Monetary Fund, International Financial Statistics.

The sample periods for the estimation of the monthly models are not uniform. Bananas: January 1957–September 1976; beef: January 1957–November 1976; copra: January 1957–December 1972; maize: January 1957–December 1972; cocoa: January 1957–December 1973; tea: January 1957–September 1976; hides: January 1957–September 1976; jute: January 1957–September 1976; rubber: January 1957–September 1976; copper: January 1957–November 1976.

Beef: 2.5 years; cocoa: 3 years; hides: 3 years; rubber: 2.5 years; copper: longer than 3.5 years.

The sample autocorrelation function $({\hat{ρ}}_{k})$ attains a local maximum at k = 24 (months) for the price of hides and at k = 30 (months) for the price of rubber. The maximum values of the correlation functions are 0.38 and 0.26, respectively. For copper, where the cycle is believed to be longer, the sample period is not long enough to make a clear indication of the exact duration of any cycle.

The third-order regular autoregressive lag polynomials for bananas and beef have two imaginary roots and a real root each, and the autoregressive lag polynomials for bananas, tea (both second-order), and copper (fourth-order) have all imaginary roots, whereas all the other roots are real. The moduli of the imaginary roots range from 1.705 (tea, regular) to 2.975 (bananas, 3-month seasonal), and those of the real roots of the second-order, or higher-order, lag polynomials range from 1.006 (hides, regular) to 4.306 (bananas, 12-month seasonal). The real roots of the second-order or higher-order lag polynomials for bananas and hides are negative.

See Nelson (1976) for an interpretation of the statistic. Since the statistic is a measure of the degree of static simulation and is comparable with R² in standard multiple regressions, the results of dynamic simulation may be different.

No attempt was made to find an annual model consistent with the estimated monthly model for any commodity. The annual models therefore may not be consistent with the monthly models in the way in which equation (6) is consistent with equation (3) in den Butter (1976).

The simulations are carried out for the ten-year period 1965–74, during which most prices showed both regular and unusual movements. This period is a subperiod of the sample period used for estimating the equations for all commodity prices.

The random walk model gives 0.839 and 0.923 for copra and maize, but the coefficients are lower than those associated with the monthly models. (See the R_a column and the R_l2a column of Table 4.) The difference between R_l2a and R_a (0.536–0.000 for bananas, 0.903–0.839 for copra, and 0.985–0.923 for maize) indicates the increase in the goodness of fit made possible by the use of the monthly model. Random walk models based on yearly price series would generate the simulated prices no better than those generated by the estimated annual AR or MA models in Table 3, since the estimated models maximize the coefficients of determination between the actual and the predicted prices.

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IMF Staff papers: Volume 25 No. 1

Author: International Monetary Fund. Research Dept.

Volume/Issue:: Volume 1978: Issue 001

Publisher:: International Monetary Fund

ISBN:: 9781451956498

ISSN:: 1020-7635

Pages:: 234

DOI:: https://doi.org/10.5089/9781451956498.024

Keywords
I. Two Representations of an Economic System—Reduced Forms and Final Equations
II. The Results of Estimation and Simulation for Ten Commodities
III. Summary and Conclusions
BIBLIOGRAPHY

View raw image

Autocorrelation Functions of Monthly Prices of Bananas and Tea

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Monthly Price of Bananas: Series of 12-Month Dynamic Simulations, Beginning in January, 1965–74

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Monthly Price of Tea: Series of 12-Month Dynamic Simulations, Beginning in January, 1965–74

Table 4.

Ten Commodities: Goodness of Fit of Simulated Prices

			Monthly Model
			Simulation of monthly price		Simulation of annual price
Commodity		Annual Model (Ra)	6-month dynamic simulation (R₆)	12-month dynamic simulation (R₁₂)	6-month dynamic simulation (R_6a)	12-month dynamic simulation (R_12a)
Foods
	Bananas	0.000	0.509	0.417	0.738	0.536
	Beef	0.767	0.877	0.723	0.924	0.759
	Copra	0.839	0.912	0.868	0.966	0.903
	Maize	0.923	0.944	0.835	0.984	0.985
	Cocoa	0.857	0.901	0.838	0.974	0.900
	Tea	0.399	0.386	0.410	0.662	0.378
Raw Materials
	Hides	0.749	0.869	0.798	0.958	0.844
	Jute	0.456	0.607	0.132	0.706	0.123
	Rubber	0.462	0.901	0.681	0.919	0.748
	Copper	0.624	0.763	0.650	0.916	0.679

Q_dt	=	quantity demanded
Q_st	=	quantity supplied
S_t	=	quantity of inventory demanded
P_t	=	price of the commodity
Y_t	=	income
α₁, α₂, β₁, γ₁, γ₂	=	coefficients

Q_dt	=	quantity demanded
Q_st	=	quantity supplied
S_t	=	quantity of inventory demanded
P_t	=	price of the commodity
Y_t	=	income
α₁, α₂, β₁, γ₁, γ₂	=	coefficients

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Abstract

I. Two Representations of an Economic System—Reduced Forms and Final Equations

reduced forms and final equations

relative merits in forecasting

error terms in final equations

II. The Results of Estimation and Simulation for Ten Commodities

estimation

simulation

possible improvements

III. Summary and Conclusions

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