Comparison between Weibull distribution fitting and limit probability of a Markov chain of import commodities market: Oil price series

Introduction

Crude Oil is a non-renewable fossil fuel which is distributed randomly all over in the ground, Tuo and Yanbing (2011) [30] . Essentially, the industrialized world depends on crude oil as a chief source of energy supply, both as a major component in many manufacturing processes and for transportation purposes. Without doubt, crude oil represents the highest percentage of the world's energy consumption compared to other energy resources. The price of crude oil is one of the world's most influential global economic indicators and it is precisely observed by policy-makers, producers, consumers and financial market partakers.

It is also open to fluctuations and constant changes mostly due to the interaction between supply and demand forces on the international commodity markets. This instability is up to approximately 25% per annum according to He et al. (2012) [15] ; and it exerts considerable influence on economic situation of different countries. Modeling of crude oil prices with exactitude and high performance has been one of the most valuable academic topics explored to solve the problem of fluctuating oil prices. A number of approaches have been utilized to this effect, leading to diverse understandings of market with strong significance to global economic growth. This work aims to advance a method to deduce the trends of fluctuation of oil prices so as to assist investigators to access optimal model to solve this problem. The probability distribution and Markov Chain time dependence, while assuming the presence of time homogeneity without testing is the main tool for this task. This research focused on modeling of Brent crude oil prices with Weibull distribution and limit probability of a Markov Chain to deduce the changing trends of the oil prices.

Methods

Markov Chains Model

A Markov chain is a sequence of random variables 𝑋 , 𝑋 , 𝑋 , …, with the Markov property namely that, the conditional probability of any future event, given any past event and the present state, is independent of the past event and depends only on the present state. Formally,

𝑃[𝑋 = 𝑥|𝑋 = 𝑥 , … , 𝑋 = 𝑥 ]𝑃[𝑋 = 𝑥|𝑋 = 𝑥 ]

The set of possible values of 𝑥 is called the state space of the chain.

Transition probabilities

In a Markov chain, the conditional probabilities 𝑃{𝑋 = 𝑗|𝑋 = 𝑖} are called transition probabilities. In a time homogeneous Markov Chain, the transition probabilities are independent of the time index t: 𝑃{𝑋 = 𝑗|𝑋 = 𝑖} = 𝑃{𝑋 = 𝑗|𝑋 = 𝑖} for all 𝑡 = 1,2, ….

Stationary transition probabilities show that transition probabilities do not change over time. Markov Chains can also have n-step transition probabilities, which is the conditional probability that the process will be in state 𝑗 after n-steps provided that it starts in state 𝑖 at time 𝑡. n-step transition probabilities are defined as the conditional probability 𝑃{𝑋 = 𝑗|𝑋 = 𝑖} = 𝑃{𝑋 = 𝑗|𝑋 = 𝑖} for all 𝑡 = 0,1, ….

Evaluation of Markov Chain Tests

In this section, to ascertain the Markovian assumption that current state depends on that of the previous state, we validate the use of a Markov Chain. To achieve this, the conventional test for independence via chi-square statistic is used. is greater than 𝜒 (𝑖 -1)(𝑗 -1), 𝛼, we reject the null hypothesis that states are independent. Otherwise, we fail to reject the null hypothesis.

Limit of the transition probability matrix

A transition probability matrix is a square matrix describing the probabilities of moving from one state to another in a dynamic system. A stochastic matrix 𝑃 is ergodic if lim → 𝑃 exists, that is if each 𝑝 ( ) has a limit as 𝑛 → ∞. If we denote the limit matrix, necessarily a stochastic matrix, by 𝐿. The components of 𝑋 ( ) , defined by the equation 𝑋 ( ) = 𝑋 ( ) 𝐿 are the limiting state distributions and signify the approximate proportions of objects in the various states of a Markov chain after a large number of time periods.

Probability distributions of changing trends of oil prices

This section presents different probabilistic models as well as maximum likelihood method of estimating the unknown parameters of the distributions.

Gamma Distribution

The Gamma distribution is a two-parameter distribution and owing to its moderate skewness, is a useful model in many areas of statistics when the normal distribution is not appropriate.

Estimation of the Parameters of Gamma Distribution

The probability density function of the Gamma distribution is given as 𝑓(𝑥: 𝛼, 𝜃) = ( ) 𝑒 , > 0, and 𝜃, 𝛼 > 0,

where 𝜃 > 0 is the shape parameter and 𝛼 > 0 is the scale parameter. We seek the conditions for the maximum likelihood estimates (MLE) of (𝜃, 𝛼).

The likelihood function for the Gamma distribution is obtained as follows:

Lognormal Distribution

A random variable X has the Lognormal distribution with parameters 𝜇 ∈ ℝ and 𝜎 ∈ (0, ∞) if ln(X) has the normal distribution with mean 𝜇 and standard deviation 𝜎. The parameter 𝜎 is the shape parameter of X while 𝑒 is the scale parameter of X. The probability density function of the Lognormal distribution with parameters 𝜇 and 𝜎 is given by

The Lognormal distribution is used to model continuous random quantities when the distribution is believed to be skewed, such as income and lifetime variables.

Estimation of the parameters of Lognormal Distribution

To compute the maximum likelihood estimators, we start with the likelihood function.

Where 𝑧 = The pdf can be written as

Estimation of the parameters of Gumbel Max Distribution

The probability density function of the Gumbel max distribution is given as:

We obtain the MLE of 𝜇 and 𝜎 as follows

The log-likelihood function is then derived by taking the natural log of the likelihood function

We take the partial derivative of ℓ with respect to 𝜇 and 𝜎 and set it equal to 0:

Solving the system ℓ 𝜇, 𝜎 𝑥 = ℓ 𝜇, 𝜎 𝑥 = 0 numerically yields the maximum likelihood estimates of 𝜇 and 𝜎 as:

Where 𝑥̅ denotes the sample mean

Weibull Distribution

The Weibull distribution is a very flexible life distribution model with two parameters. It has the probability density function

where 𝑘 > 0 is the shape parameter and 𝛾 > 0 is the scale parameter

Estimation of the parameters of Weibull Distribution

The probability density function of the Weibull distribution is given as

The likelihood function of the Weibull distribution is obtained as follows

Taking the natural logarithm gives the log likelihood function

Differentiating partially with respect to 𝑘 and 𝛾 in turn and equating to zero, we obtain the estimation as follows

We obtain an estimation of 𝛾 as

And on substitution, we obtain

Different Discrimination Procedure

This section is to succinctly describe methods for choosing the best fitted model to a given data set.

Theoretical Basis

In order to determine a suitable distribution to a data set theoretically, we can consider the Likelihood Ratio Test (LRT). Generally, given the observed data 𝑋 , … , 𝑋 , we can measure the relative plausibility of 𝐻 to 𝐻 by the log -likelihood ratio:

The log-likelihood ratio could help us choose which model (𝐻 or 𝐻 ) is a more likely explanation for the data.

Furthermore, the LRT is a statistical test relying on a test statistics obtained by taking the ratio of the maximum value of the likelihood function.

Let 𝑋 , 𝑋 , … , 𝑋 be i.i.d random variables from a known distribution (with p.d.f), if given the likelihood function, then the LRT is defined as:

Where 𝐿 (𝜃 , 𝜃 ) and 𝐿 (𝜆 , 𝜆 )are the likelihood function of a known different distributions and 𝜃 , 𝜃 , 𝜆 and 𝜆 are the MLE of 𝜃 , 𝜃 , 𝜆 and 𝜆 respectively. Now, we obtain the log-likelihood ratio statistic, denoted as 𝑇 and is defined as the natural logarithm of the ratio of the two maximized likelihood functions:

Given more than two models, the log likelihood ratio test applied to each pair of models implies that the model with the largest log likelihood is preferred.

Application

We applied the methods to monthly prices of Brent crude oil from January 2009 to June 2017. Results obtained are presented and discussed below. We also measured performance of methods according to their aptitude to capture the statistical behaviour of the observed data set. Finally, we carried out a comparative analysis of the methods.

Data

The data used were obtained from the official website of The World Bank, "http:www.worldbank.org/en/research/commoditymarkets" and it consists of 102 observations. Table 1: gives the monthly prices of the Brent crude oil prices from January 2009 to June 2018 (US$/Barrel). Table 2 presents some sample distributional statistics for the monthly prices of the Brent crude oil prices. This consists of the mean, variance, skewness, kurtosis etc. Table 3 shows the transformed value of the monthly prices of Brent Crude oil. Histogram and CDF plots of an empirical distribution for a continuous variable (transformed value of the monthly prices of Brent Crude oil) as provided in R software by the plotdist function. Four goodness-of-fit plots for various distributions fitted to continuous data (Weibull, Gamma and Lognormal distributions fitted to transformed value of the monthly prices of Brent Crude oil) as provided in R software by functions denscomp, qqcomp, cdfcomp and ppcomp.

The Oil Price Transition States

In this section, we categorized the oil prices as five states: low state, middle -low state, middle state, middle -high state and high state. These states constitute the following full space for stochastic events of crude oil prices: With m = 5, the degrees of freedom (𝑚 -1) = (5 -1) = 16. Using a 5% significance level and referring to the 𝜒 tables with 16 degrees of freedom, we find that 𝜒 ( , . ) = 26.296. The observed value of the sample statistics 𝜒 is 224.8321, which is higher than 𝜒 ( , . ) = 26.296. Thus, we reject the null hypothesis that states are independent. As a result, it confirms that a state transition chain of Brent crude oil prices from January 2009 to June 2017 follows a Markov Chain.

Limit Probability of a Markov Chain for Changing Trends of Brent Oil Prices

Let P represent the first -stage transition matrix. Having each element 𝑛 of respective row from Table 3 divided by the sum of its row (𝑛 . ), then we obtain the first -stage transition matrix P: The row -vector, which has been converged to the same value, is the Markov chain's limit of oil price series denoted by 𝐿.

With the initial state probability vector, 𝑋 ( ) = [0 1 0 0 0 ], we have The limit probability of oil -price state transition chain as a Markov Chain is the ultimate state of oil price series. It approximates the changing trends of oil price in the medium-and long-terms, but not in the short-term.

The Probability Distribution of the Changing Trends of Oil Prices

In this research work, four distributions have been considered, namely, Gamma, Lognormal, Gumbel Max and Weibull distribution. We carry out some statistical tests among the competing models to determine which distribution best fits the oil price data.

Parameter Estimates of the Distributions

The maximum likelihood estimates of each of the parameters of the distributions are obtained. Table 6 shows parameter estimates for the distributions.

Log -Likelihood Ratio Values of the Distributions

The log-likelihoods form the basis of the selection of the distribution that fits the data. The distribution with the maximum loglikelihood value is considered the most suitable. The log -likelihood values are provided in Tables 7. The log -likelihood ratio value is gotten by finding the difference between likelihood values of competing distributions

The Weibull distribution arises as the most appropriate distribution for the data as it has the highest log -likelihood value.

Goodness-of-Fit Test

We employed the Kolmogorov -Smirnov (see Kolmogorov (1941) and Smirnov (1948)) and Anderson-Darling (1952) goodnessof-fit test to measure the "distance" between the data and the distribution we are testing, and compare that distance to some threshold value. Since the goodness-of-fit test statistics indicate the distance between the data and the fitted distributions, it is obvious that the distribution with the lowest statistic value is the best fitting model. The Weibull distribution indicated to have the best fit of the four distributions. The Kolmogorov -Smirnov and Anderson -Darling statistics have the least figure under the Weibull distribution. These results show that in comparison to Gamma, Lognormal and Gumbel Max distribution, the Weibull distribution is a more acceptable fit for the Brent crude oil price.

Comparison of Model

We compare the suitability of the distribution models. Specification measures such as Akaike Information Criteria (AIC) and Bayes Information Criteria (BIC) were employed. We noticed that the AIC and BIC for the Weibull distribution is smaller than those of the Gamma, Lognormal and Gumbel Max distribution. Thus, the Weibull distribution is a better fit for the Brent oil Price.

A Deduction Model of Integrating the Limit Probability of a Markov Chain With a Probability Distribution

In the previous sections, we discuss the Markov chain model and the probability function model, separately. Here, we integrate the two models, in order to infer the changing trends of Brent oil prices from the short -term to the middle -and long -terms.

Theoretical Frequency of a Weibull Distribution and the Limit Frequency of a Markov Chain

Suppose that the length of a deduction period is the same as 102 months of the sample statistic. Multiplying the probability of a Weibull distribution and the limit probability of a Markov Chain in Table 11 by the sample observations of 102, we have the theoretical frequency of a Weibull distribution and the limit frequency of a Markov chain of oil prices, as shown in Table 12.

Time Homogeneity

One important aspect of Markov probabilities is the assumption of time homogeneity. That is, the probability of moving from a given state to the next state is assumed to remain constant over time. In the case of Brent crude oil price movements we speculate that this is a requirement that is not strictly adhere to. This study tested time dependence, while assuming the presence of time homogeneity without testing. The Brent oil prices exhibits periods of greater and lesser volatility. It does not demonstrate time homogeneity. This will be subject to the next paper.

Summary

Crude oil prices have been unstable over time and with huge disparities. It is this instability in oil price series that makes deduction of the changing trends of oil prices in the middle-and long-terms problematic and hence renders the prediction of their price levels in the short-term challenging.We certified and corroborated the use of a Markov chain to ascertain the Markovian assumption that current state depends on that of the previous state by using the conventional test for independence via chi -square statistic. The observed value of the sample statistics 𝜒 , 224.8321 is much higher than 𝜒 ( , . ) = 26.296. Consequently, it verified that there is an existence of a Markov chain in the observed Brent crude oil price series from January 2009 to June 2017. We then modelled the oil prices series using the limit probability of a Markov chain and observed at the middle and long -term, that the probability of low level state is 0.0396, middle -low level state is 0.2772, the middle level state is 0.1881, middle -high level state is 0.1386 and high level state is 0.3564, having the proportion is 3.96%, 27.72%, 18.81%, 13.86% and 35.64%, respectively in the series. We also confirmed that the Weibull distribution models of the oil price series performed better among the selected probability distributions using some test criterion. From our investigation, coupled with the results in Tables 6 -9, we discovered that the oil price series can best be modelled using the Weibull distribution. From the table, it is apparent that the Weibull distribution has the highest log -likelihood function of -473.236 and also has the lowest AIC and BIC values of 950.4721 and 955.7220 respectively. This indicates that among the chosen statistical distributions, the Weibull distribution stands a better chance in providing a good fit for the oil price data. The maximum likelihood estimates of the parameters of the distributions were also obtained. It was observed at the short -term that the probability of low level state is 0.0578, middle -low level state is 0.1673, middle level state is 0.2861, middle-high level state is 0.2856, and high level state is 0.1565. Our work has, correspondingly, established a trend deduction model of fluctuating oil prices. This model integrates the probability distribution of oil price series with the limit probability distribution of a Markov Chain. The difference between them indicates specific changes in a variety of oil price states from the short -term to the middle -and long -terms. The probability of oil prices below 40 US$/barrel has been reduced by 31.49% while that of oil prices over 40 US$/barrel has increased markedly.

Conclusion

Our analysis and the above summary undoubtedly show that there is an existence of a markov chain in the oil price data i.e. oil price state are clearly not independent and the Weibull distribution would provide a good fit to the oil price data when compared with the Gamma distribution, Lognormal distribution and Gumbel max distribution.

We then integrated the Markov chain model and Weibull distribution to model the oil price series and obtained results which shows that the oil prices will be over 40 US$/barrel. However, this deduction is only meaningful for the recent changing trends of oil prices. A lot of factors determine the changing trends of oil prices such as current supply and output, future supply and reserves, demand from major countries, political events and crises.

A change in the oil-price level as well as its variability poses an arduous challenge to a diverse range of actors in the global market. The overall effect of changes and uncertainties in oil prices on the economy as a whole depends on many parameters, including a country's position in the oil market and her degree of competitiveness.

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