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Weekly UpdatesUsing a simple unobserved components model, we show that explicitly modelling asymmetric cycles on crude oil prices improves the forecast ability of univariate time series models of the oil price.
1. Introduction
The existence of commodity cycles has long been documented in the economic literature (see, e.g., Kondratieff, 1925; Dewey and Dakin, 1947). Commodity prices may be influenced by (potentially cyclical) monetary factors, a typical example being the dollar exchange rate, given that most commodity prices are denominated in dollars. (1) Carson (2006) argues that commodity prices generally tend to be a very good barometer of the business cycle.
On the other hand, Deaton and Laroque (1992) report a variety of statistical features of commodity prices such as non-normal skewness and leptokurtosis in returns, coupled with serial correlation, as evidence of non-linear dynamic price behaviour and non-normality in the price distribution. From a theoretical point of view, the non-negativity constraint on inventories has been put forward as a cause of non-linear dynamic behaviour in the prices of storable commodities. (2)
In this paper we investigate empirically whether time series models with asymmetric cycles are able to better forecast oil prices than their linear symmetric alternatives. We use a model recently proposed by Crespo Cuaresma (2003) that generalizes univariate unobserved component models a la Harvey (1989) by assuming asymmetric cyclical behaviour in crude oil prices around a reasonably general trend, thus improving on other studies on the cyclicality of commodity prices (e.g., Reinhart and Wickham, 1994), which tend to assume symmetric behaviour when modelling crude oil price time series. Our results present evidence of better forecasting abilities for the nonlinear model when compared to a benchmark autoregressive model and to its symmetric counterpart.
This note is organized as follows. Section two presents the unobserved components model with an asymmetric cycle which will be used in the forecasting exercise. Section three presents an empirical modelling and forecasting exercise aimed at assessing the potential improvements in out-of-sample forecasting. Section four concludes and briefly sketches future paths of research.
2. OIL PRICES AND ASYMMETRIC CYCLES : THE UNOBSERVED COMPONENTS MODEL
Time series of oil prices present very strong persistence, local trends over relatively long periods of time and apparent cyclical behaviour at higher frequencies. Figure 1 presents monthly data on nominal and real crude oil prices for the period January 1983-August 2007 (source: Energy Information Administration, U.S. Department of Energy), (3) which will be the series of interest for our analysis. The strong persistence of the series mirrors itself in the fact that the null hypothesis of a unit root cannot be rejected at any sensible significance level when performing an Augmented Dickey-Fuller (ADF).
The change in oil price, nominal and real, presents strong excess kurtosis and slightly negative skewness. The null hypothesis of normally distributed data can be rejected at all reasonable significance levels when using a Jarque-Bera test for Gaussianity of the distribution. In principle the deviation of normality can be taken as some preliminary evidence that nonlinear dynamic structures will be necessary in the specification of the model for oil prices.
We use a simple univariate unobserved components models in order to jointly capture the stochastic trend and the cyclical behaviour of the oil price series. In particular, we propose a model including a stochastic trend (a general I(2) trend nesting the cases of a random walk and a linear deterministic trend) and a general asymmetric cyclical component which nests the case of a symmetric cycle as a special testable case. This class of models was proposed in Crespo Cuaresma (2003) to evaluate the existence of nonlinear cycles in macroeconomic time series.
[FIGURE 1 OMITTED]
We assume that the data generating process for oil prices can be decomposed multiplicatively into a trend, cyclical and irregular (white noise) component which are uncorrelated among each other, so that the log of oil prices ([p.sub.t]) can be written as
[p.sub.t] = [[tau].sub.t] + [[phi].sub.t] + [[epsilon].sub.t], [[epsilon].sub.t] [approximately equal to] NID (0, [[sigma.sup.2.sub.e]). (1)
The trend component ([[tau].sub.t] ) is specified as
[[tau].sub.t] = [[tau].sub.t]-1 + [[beta].sub.t+1] + [v.sub.t], [v.sub.t] [approximately equal to] NID (0, [[sigma.sup.2.sub.v]). (2)
[[beta].sub.t] = [[beta].sub.t+1] + [xi].sub.t], [xi].sub.t] [approximately equal to] NID (0, [[sigma.sup.2.sub.[xi]]). (3)
where [v.sub.t] and [xi].sub.t] are shocks which are assumed to be uncorrelated mutually and with the irregular component, [[epsilon].sub.t]. The parameter restrictions [[sigma.sup.2.sub.[xi]] = 0 and [[sigma.sup.2.sub.v] > 0 define the model with a random walk trend and the smooth trend case appears for the case [[sigma.sup.2.sub.[xi]] > 0 and [[sigma.sup.2.sub.v] = 0. If [[sigma.sup.2.sub.[xi]] = 0 and [[sigma.sup.2.sub.v] = 0 are imposed, the trend component is a linear deterministic trend.
The cyclical component, [[phi].sub.t], is specified as a first-order stochastic cycle where the frequency of the cycle may depend on past realizations of the process, so that asymmetric cycles may arise. We hypothesize the existence of two regimes with different cyclical frequencies depending on some function of past values of [p.sub.t]. (4) The specification we propose for [[phi].sub.t] is therefore given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [[phi].sup.*.sub.t] appears by construction, [rho] [member of] [0,1) is a damping factor, [[lambda].sub.1] and [[lambda].sub.2] are the frequencies of the cycle in the two possible regimes ([[lambda].sub.1] [member of] [0,[pi]], [[lambda].sub.2] [member of] [0,[pi]]), [[omega].sub.t] and [[omega].sup.*.sub.t] are iid normally distributed disturbances, mutually uncorrelated and with equal, fixed variance [[sigma].sup.2.sub.[omega]], I ([{[p.sub.[tau]}.sup.t-1.sub.[tau]=1]) is an indicator function taking value one if a given function of the realized values, f ([{[p.sub.[tau]}.sup.t-1.sub.[tau]=1]) is positive and zero otherwise. If [[lambda].sub.1] = [[lambda].sub.2], the model is the symmetric trend plus cycle model developed by Harvey (1985, 1989). Once the function f (*) has been specified, symmetry is a testable hypothesis, so that the existence of asymmetric cycles can be verified statistically with the data at hand. Given the fact that in period t the realization of the f (*) function is known, the model is conditionally Gaussian and can be easily estimated using Kalman filtering methods. The details on the estimation of the model are found in Crespo Cuaresma (2003).
3. EMPIRICAL RESULTS : MODELLING AND FORECASTING OIL PRICES USING ASYMMETRIC CYCLES
The aim of this section is to evaluate if the explicit modelling of asymmetric cycles in oil prices can contribute to improvements in out-of-sample forecasts. We will use the sample ranging from January 1983 to December 1995 as the first in-sample period. Based on this sample, we estimate a simple autoregressive model in the first difference of the log of oil prices, an unobserved components model with symmetric cycles (that is, imposing [[lambda].sub.1] = [[lambda].sub.2] in the specification given by (4) and (5)) and an unobserved components model with the yearly growth rate of oil prices as threshold function f, that is, for our monthly data, f ([{[p.sub.s]}.sup.t-1.sub.s=1])] = [p.sub.t-1] - [p.sub.t-13]. We chose a smooth trend specification for the unobserved components models in all cases, since they tended to explain the data better in-sample. With the estimated model, forecasts up to three years (36 months) ahead are obtained, and the forecasting error is calculated using the actual data. (5) The one-step ahead forecast of an estimated model (1) with the trend and cyclical components above can be obtained in a straightforward manner. The one-step ahead forecast of the oil price can then be used to obtain the forecast of the f (*) function and the two-step ahead forecast. This is repeated until three-year ahead forecasts are obtained. (6) A new observation (the one corresponding to January 1996) is added to the in-sample subsample, and the procedure is repeated. This is done until no out-of-sample observations are available (that is, until the observation corresponding to August 2007 is reached). At this stage two statistics evaluating the forecast accuracy of the point forecasts of the models being studied (Root Mean Square Forecasting Error, RMSFE, and the Direction of Change statistic, DOC) are computed by comparing the forecasts with the actually realized values. As an extra evaluation instrument, we test whether the differences in forecasting ability are significant across models using the Diebold-Mariano test (Diebold and Mariano, 1995).
We also evaluate the direction of change statistics by comparing the computed statistics for each model with the "coin toss" (p = 0.5) benchmark. This benchmark roughly coincides with the frequencies of price increases/decreases in the fist in-sample period (January 1983- December 1995) for both the nominal oil price (50%) and the real oil price (56%). We use a normal approximation for the binomial distribution, and obtain a test statistic using the Z score, which allows us to test the null hypothesis p = 0.5 against p [not equal to] 0.5.
The estimation of the symmetric and asymmetric models corresponding to the first in-sample period is given in Table 2 and gives significant evidence of asymmetry in the estimated cycle. The model estimated is given by a simple linear trend and an asymmetric cycle. For the regime defined by [p.sub.t-1] - [p.sub.t-13] > 0 the estimate of the cyclical frequency is 0.27 (with a standard error of 0.05) for the nominal price and 0.25 (with a standard error of 0.06) for the real price, and for the complementary regime, [p.sub.t-1] - [p.sub.t-13] < 0, the cyclical frequency is estimated to be 0.07 (with standard error 0.02) for the nominal price and 0.10 (with standard error 0.02) for the real price. A Wald test strongly rejects the null of equality of cyclical frequencies across regimes in both cases.
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