MODELLING NONLINEAR RELATIONSHIP AMONG VEGETABLE OIL PRICE TIME SERIES

Mohd Tahir Ismail

School of Mathematical Sciences, Universiti Sains Malaysia 11800 USM, Penang Malaysia

mtahir@cs.usm.my

Abstract The study of commodity price behaviour has

attracts the attention of many economists and finance specialists. This is due to the fact that many less developed countries rely on the revenues generated by the commodity exports. In this paper, the nonlinear relationship because of regime shifts in four vegetable oil price series was investigated. The multivariate Markov switching vector autoregressive (MS-VAR) model with regime shifts in both the mean and the variance was employed to capture common regime shifts behaviour among the four price series. Results revealed that all the series demonstrate common regime shifts trend of declining and increasing. In addition, the MS-VAR model fitted the data better than the linear vector autoregressive model (VAR).

1. Introduction

The international market for vegetable oils is growing at a significant pace because of the rapid growth in world population and income. Plus, the rise of petroleum price results in increasing demand for biodiesel, which consequently expanded the use of vegetable oils. Biodiesel is a mono alky ester that is made from a vegetable oil, such as soybean or rapeseed oil, or sometimes from animal fats. In Malaysia, scientists are focusing on the using the palm oil as a biodiesel fuel as alternative to crude oil. Moreover the growing development of food industry has increase the demand for vegetable oil.

There are number of studies that focus on interaction among vegetable oil and vegetable oil with other commodity. Reference [1] investigated the comovements among 8 vegetable oil prices and analyzes the long-run relationships that may exist between the prices of these oils. A multivariate cointegration approach is used to examine the price behaviour of vegetable oils. It was found that there were 4 cointegrating vectors in an eight-variable model with indicate that price of vegetable oil can be grouped according to their different end-uses. Then, [2] analyzed the price interdependence for vegetable and tropical oils in the world market using monthly observations from January 1971 to December 1993. They concluded that the cointegrating relationship does not exist among the five oils used in their studied. After that, [3] investigated the relationship between vegetable oil and crude oil prices using weekly data extending from the January 1999 to March 2006. Four world vegetable oils

prices, including soybean, rapeseed, sunflower and palm oil, along with a world crude oil price were analyzed. Their study suggests that shocks in crude oil prices have insignificant influence on the variation of edible oil prices.

Later, [4] also investigated the long-term relationship between the prices of petroleum and vegetable oils prices represented by palm, soybean, sunflower and rapeseed oils prices. The results from Granger test shows that there exists a long run unidirectional causality from petroleum price to each of palm, rapeseed, soybean and sunflower oil prices. Recently, [5] estimated the effect of El Nino Southern Oscillation (ENSO) over time on market for eight major vegetable oil prices: soybean, sunflower seed, groundnut, coconut, palm kernel, palm, cottonseed, and rapeseed using smooth transition autoregressive (STAR) and smooth transition vector error correction (STVEC) models. The results show that the presence of ENSO shock will increase the vegetable oil prices, and the opposite is true for the absence of ENSO shock.

The market prices of vegetable oils are assumed to interact with each other since they have to compete in the industry in which they are being used. Interaction between these vegetable oils is what we going to investigate throughout the analysis. We intend to develop a fundamental understanding of the price behaviour. However the approach being used in this paper is difference from previous studied mention above. In this paper we focus on finding whether nonlinear interaction because of common regime switching behaviour exists among the vegetable oil series by assuming that all the series are regime dependent. We use a two regimes multivariate Markov Switching Vector Autoregressive (MS-VAR) model with regime shifts that happened in both the mean and the variance to extract common regime switching behaviour from all the series.

This paper is organized as follows. The specification and estimation of the Markov Switching Vector Autoregressive model are given in Section II. Section III presents the empirical results and discussion on the results. Section IV contains the summary and the conclusion. 2. Markov Switching Vector Autoregressive

(MS-VAR) Model According to [6] and [7], Hamilton in 1989

developed the Markov Switching Autoregressive model

978-1-4577-0005-7/11/$26.00 ©2011 IEEE

(MS-AR) to identify changes between fast and slow growth regimes in the US economy. The model assume that a time series, ty is normally distributed with iμ in each of k possible regime where 1,2,...,i k= . A MS-AR model of two states with an AR process of order p ,

( )MS AR p− is given as follows:

( ) ( )( )

( )1

2~ 0, 1,2

p

t t i t i t i ti

t t t

y s y s u

u s NID and s

μ α μ

σ

− −=

⎡ ⎤= + − +⎢ ⎥⎣ ⎦

=

∑ (1)

where iα are the autoregressive parameters with 1,2,...,i p= .

The MS-AR framework of Equation (1) can be readily extended to MS-VAR model with two regimes that allows the mean and the variance to shifts simultaneously across the regime. The model is given below:

( ) ( ) ( )( )( ) ( )( )

1 1 1 ...t t t t t

p t t p t p t

Y s A s Y s

A s Y s

ψ ψ

ψ ε− −

− −

− = − +

+ − + (2)

where ( )1 ,...,t t ntY Y Y= is the n dimensional time series vector, ψ is the vector of means, 1,..., pA A are the matrices containing the autoregressive parameters, and

tε is the white noise vector process such that ( )( )~ 0, .t t ts NID sε Σ

Other specifications of MS-

VAR model are being discussed by [8]. From Equation (1) and (2), ts is a random variable

that triggers the behaviour of tY to change from one regime to another. Therefore the simplest time series model that can describe a discrete value random variable such as the unobserved regime variable ts is the Markov chain. Generally, ts follow a first order Markov process where it implies that the current regime

ts depends on the regime one period ago, 1ts − and denoted as:

1 2

1

, ,...t t t

t t ij

P s j s i s k

P s j s i p− −

−

⎡ = = = ⎤⎣ ⎦= ⎡ = = ⎤ =⎣ ⎦

(3)

where ijp is the transition probability from one regime to another. From m regimes, these transition probabilities can be collected in a ( )m m× transition matrix denoted as P

11 12 1

21 22 2

1 2

...

...... ... ... ...

...

m

m

m m mm

p p pp p p

p p p

⎡ ⎤⎢ ⎥⎢ ⎥Ρ =⎢ ⎥⎢ ⎥⎣ ⎦

(4)

with 1

1, 1,2,..., 0 1m

ij ijj

p i m and p=

= = ≤ ≤∑ .

The transition probabilities also provide the expected duration that is the expected length the system is going to be stay in a certain regime. Let D define the duration of regime j . Then, the expected duration of the regime j is given by

( ) 1 1,2,...1 jj

E D jp

= =−

(5)

The conventional procedure for estimating the model parameters is to maximize the log-likelihood function and then use these parameters to obtain the filtered and smoothed inferences for the unobserved regime variable, ts . However, this method becomes disadvantageous as the number of parameters to be estimated increases. Generally, in such cases, the Expectation Maximization (EM) algorithm is used. This technique starts with the initial estimates of the unobserved regime variable, ts and iteratively produces a new joint distribution that increases the probability of observed data. Details about these two steps can be found in [9] and [10].

3. Modelling Dynamic Relationship

In this section, we will present the results of the econometric specifications used for modelling the four vegetable oil prices. This section starts by giving a description of the data. Next, we test the data for stationary. If the data is stationary, we use Johansen test to examine the existent of cointegration. Then, we justify the used of two regime Markov switching vector autoregressive model using likelihood ratio (LR) test and estimate the MS-VAR (1) model. Finally, we collect a series of the filtering and smoothing probabilities to identify the regime shift in the vegetable oil price returns series.

3.1 Data

Data for analysis are monthly vegetable oil prices (olive oil, coconut oil, palm Oil and soybean oil) as reported by International Monetary Fund (IMF). The estimation period is from January 1980 to June 2010. Each dataset consists of 366 observations. Prior to analysis, all the series are analysed in return, which is the first difference of natural algorithms multiplied by 100. This is done to express things in percentage terms. The monthly prices return series are used with an assumption that the regime shifts can be observed more clearly across time if in case there is low frequency data. Figure 1 and 2 below shows the behaviour of the four vegetable oil series over the study period (January 1980 - June 2010). 3.2 Stationarity and Cointegration Tests

Many of the econometric models require the knowledge of stationarity and order of integration for the variables. The unit root test is usually used to determine whether the order of integration of a variable is at level or first differences. Two of the common unit root tests are used in this paper namely the ADF test and the PP test. The ADF test was developed by [11] and the PP tests was suggested by [12].

From Table 1, most of the statistics for series at level are not significant. This suggests that the null hypothesis of unit root test cannot be rejected and the indices are not stationary at level. After first differencing has been employed for the series, the null

hypothesis of unit root test can be rejected at 5% level of significance for series with or without trend, Thus, the series are stationary at first difference and integrated of order 1, I(1). Thus, the cointegration test can be carried out after all the series are integrated at the same order. In addition this result justified the used of return series in modelling the relationship among the four vegetable oils.

Fig. 1: Original price series

Fig. 2: Return series

The results for testing cointegration relationships among the four series by using the Johansen and Juselius cointegration test (JJ test) [13] are on Table 2. As note on Table 2, the JJ test statistics fail to reject the null hypothesis of no cointegration vectors among all the vegetable oil series in both cases; the trace and maximal eigen-value forms of the test. This indicates that no long run relationship among the four series and stationary at the first differences.

Table 1. Unit root test Variable ADF PP

Level First difference

Level First difference

Olive -2.018 -15.76** -1.811 -15.76**

Coconut -2.937 -21.42** -2.937 -21.38**

Soybean -3.139 -13.77** -2.746 -13.52** Palm -3.316 -7.558** -2.881 -13.38**

**indicate significant at 5% levels

Table 2. JJ Cointegration Test

0H 1H Maximum eigenvalues statistcs

Trace statistics

maxλ 1% critical value

traceλ 1% critical value

0=r 1≥r 51.85 54.46 23.13 32.24 1≤r 2≥r 28.72 35.65 18.99 25.52

2≤r 3≥r 9.72 20.04 6.20 18.63 3≤r 4≤r 3.52 6.65 3.52 6.65

3.3 Estimating MS-VAR Model

Table 3. ML estimation results for the MS-VAR (1) tOlive tCoconut tSoybean tPalm Regime-dependent means

1μ -0.012 -0.224 -0.222 -0.387

2μ 0.364 2.42 1.88 2.00

1−tOlive 0.157 -0.030 -0.039 0.102

1−tCoconut 0.038 -0.113 -0.028 -0.160

1−tSoybean -0.025 0.307 0.234 0.007

1−tPalm -0.018 0.0889 0.075 0.360

1Σ 4.62 4.59 5.01 6.49

2Σ 2.04 14.86 8.55 10.88 Fitting

( ) ( )12 VARMSMH −

( )1VARlinear

LikLog -4462.26 -4565.23 AIC 24.77 25.24 HQ 24.96 25.37 SC 25.26 25.56

VARMSLR − 205.93 [.000]

ijp

11 =−ts

21 =−ts

Duration

1=ts 0.9135 0.0865 11.57

2=ts 0.3742 0.6258 2.67

Refer to Equation (2) for full specification of the equation Figures in the bracket [] is the Davies p-value

Adhere to the principle of parsimony, we found that two regimes Markov Switching Vector Autoregressive model of order one with switching in the mean and the variance or MS-VAR(1) manage to capture the interaction among the four series very well. As shown on Table 3 the likelihood ratio test for testing the null hypothesis of linear model against an alternative

1000

2000

3000

4000

5000

6000

7000

1980 1985 1990 1995 2000 2005 2010

OLIVE

0

400

800

1200

1600

2000

1980 1985 1990 1995 2000 2005 2010

COCONUT

0

200

400

600

800

1000

1200

1980 1985 1990 1995 2000 2005 2010

PALM

200

400

600

800

1000

1200

1400

1600

1980 1985 1990 1995 2000 2005 2010

SOYBEAN

-30

-20

-10

0

10

20

30

1980 1985 1990 1995 2000 2005 2010

OLIVE

-60

-40

-20

0

20

40

60

1980 1985 1990 1995 2000 2005 2010

COCONUT

-40

-30

-20

-10

0

10

20

30

1980 1985 1990 1995 2000 2005 2010

PALM

-30

-20

-10

0

10

20

30

40

1980 1985 1990 1995 2000 2005 2010

SOYBEAN

of regime switching model found that the null hypothesis can be rejected because the [14] p-value show significance results. Therefore, a nonlinear MS-VAR(1) model is better than linear VAR(1) model in describing the data. Moreover, the minimum value of AIC (Akaike), HQC (Hannan-Quinn) and SBC (Schwartz Bayesian) criteria indicate that the performance of the MS-VAR(1) models are better than the nested linear VAR(1) model.

As note on Table 3, each of the two regimes identified for the vegetable oil prices returns has a clear interpretation. The first regime for each of the prices returns captures the behaviour of the decreases in prices and the second regime describes the increases in prices. The negative signs of the expected returns in Regime 1,

1μ indicate that an average all the prices returns fall around 0.01% to 0.38% monthly. While, the positive signs of the expected returns of regime 2,

2μ specify that an average the prices returns tend to increase about 0.4% to 2.4% monthly.

In addition, the percentage monthly variance for regime 2, 2Σ is higher for coconut oil, soybean oil and palm oil than regime 1, 1Σ . While, the percentage monthly variance for regime 2, 2Σ is lower for olive oil than regime 1, 1Σ . This result reveal that coconut oil, soybean oil and palm oil are more volatile during increasing period while olive oil is more volatile during decreasing period. Table 3 also provide the probabilities of staying in regime 1 and regime 2 which are 0.9135 and 0.6258 respectively. It means on average the duration of staying regime 1 is 11.57 months and regime 2 is 2.67 months.

The Markov switching vector autoregressive model also provides the conditional regime probabilities which is the probabilities of being in regime 1 and regime 2 at time t . These probabilities are very useful in helping to understand more about the interpretation that was made earlier by using the estimated parameters. Figure 3 shows the smoothed probability plots for the two regimes MS-VAR (1) model. It shows the probability of staying in either regime 1 or regime 2 at time t and will helped to further assist with the interpretation of those two regimes. It appears that some form of correlation relationship exists between the smoothed probabilities of regime 1 and regime 2. This is because, when the probability of regime 1 is close to one, the probability of regime 2 is close to zero. And it also happens vice versa. The finding indicate that the MS-VAR (1) perform well in getting the direction of change in a series either the series is in regime 1 or in regime 2. Finally Figure 4 illustrates the fit of the MS-VAR(1) model.

4. Conclusion

This paper investigates the nonlinear behaviour because of regime shifts of four vegetable oil prices using data from January 1980 until June 2010. For preliminary tests, the results are as follows. Firstly, for stationarity test, the results indicate that all the price series are I(1) processes. Secondly, for cointegration test, the result indicates that the price series are not cointegrated. Therefore, based on the results, two

regimes multivariate Markov switching vector autoregressive (MS-VAR) model with regime shifts in both mean and the variance is employed instead of modelling the vegetable oil price series as a linear VAR model. This is done to extract the common regime switching behaviour. Overall the MS-VAR (1) model manages to reveal common regime shifts behaviour among the four series. In addition the MS-VAR(1) model outperform linear VAR(1) in modelling the interaction.

Fig. 3: Smoothed probability plots of the MS-VAR(1)

model

Fig. 4: Fit of the MS-VAR(1) model

Acknowledgments

This research is financed by Short Term grant (304/PMATHS/6310055) from Universiti Sains Malaysia.

References

[1] In, F. and Inder, B., Long-Run Relationships Between World Vegetable Oil Prices, The Australian Journal of Agricultural and Resource Economics, Vol. 41, (1997), pp. 455-470. [2] Owen, A. D., Chowdhury, K. and Garrido, J. R. R., Price Interrelationships in the Vegetable and Tropical Oils Market, Applied Economics, Vol. 29, (1997), pp. 119-124.

1980 1985 1990 1995 2000 2005 2010

0.25

0.50

0.75

1.00 Probabilities of Regime 1

1980 1985 1990 1995 2000 2005 2010

0.25

0.50

0.75

1.00 Probabilities of Regime 2

1990 2000 2010

-20

-10

0

10

20

30 Olive in the MSMH(2)-VAR(1)Olive fitted

1990 2000 2010

-25

0

25

50

Coconut in the MSMH(2)-VAR(1)Coconut fitted

1990 2000 2010

-20

0

20

Soybean in the MSMH(2)-VAR(1)Soybean fitted

1990 2000 2010

-20

0

20

Palm in the MSMH(2)-VAR(1)Palm fitted

[3] Yu, T.H., Bessler, D. A. and Fuller, S., Cointegration and Causality Analysis of World Vegetable Oil and Crude Oil Prices: Proceedings of American Agricultural Economics Association Annual Meeting, (2006). [4] Amna, A. A. H and Fatimah, M. A., The Impact of Petroleum Prices on Vegetable Oils Prices. Proceedings of International Borneo Business Conference On Global Changes: Corporate Responsibility, (2008). [5] Ubilava, L and Holt, F., Nonlinearities in the World Vegetable Oil Price System: El-Nino Effects: Proceedings of Agricultural & Applied Economics Association 2009 AAEA & ACCI Joint Annual Meeting, (2009). [6] Ismail, M. T. and Isa, I., Modelling the Interactions of Stock Price and Exchange Rate in Malaysia, Singapore Economic Review, Vol. 54, No. 4, (2009), pp. 605-619. [7] Ismail, M. T. and Rahman, R. A., Modelling the Relationships between US and Selected Asian Stock Markets, World Applied Sciences Journal, Vol. 7, No. 11, (2009), pp. 1412-1418. [8] Krolzig, H,-M., Markov Switching Vector Autoregressions: Modelling, Statistical Inference and Application to Business Cycle Analysis: Lecture Notes in Economics and Mathematical Systems, Vol. 454, Berlin, Springer-Verlag, 1997. [9] Hamilton, J.D., Time Series Analysis, Princeton, Princeton University Press, 1994. [10] Kim, C.J. and Nelson, C.R., State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications, MIT, MIT Press, 1999. [11] Dickey, D.A. and Fuller, W.A., Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association, Vol. 74, (1979), pp. 427-431. [12] Phillips, P.C.B. and Perron, P., Testing for unit root in time series regression, Biometrika, Vol. 75, (1988), pp. 335- 346. [13] Johansen, S. and Juselius, K., Maximum likelihood estimation and inference on cointegration with applications to the demand for money, Oxford Bulletin of Economics and Statistics, Vol. 52, (1990), pp. 169–210. [14] Davies, R.B., Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, Vol. 74, (1987), pp. 33-43.