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The co-evolution and co-movement of financial time series is of utmost importance in contemporary finance, especially when considering the joint behaviour of asset price realizations. The ability to model interdependencies and volatility spill-over effects introduces interesting dimensions in finance. This paper explores co-integrating relationships between crude oil and distillate fuel prices. Existence of multivariate co-integrating relations and bidirectional Granger-Causality is established among the series. It is also established that even after fitting a full VECM, the residuals are not necessarily multivariate normal suggesting the noise could as well be multivariate GARCH.

In empirical finance, most realistic applications are actually multivariate in nature. The co-evolution and covariation between the prices of financial assets is essentially important in Finance. Often, the interest is not merely in the behaviour of a single stock but rather in the joint behaviour of several stocks, and such joint behaviour is described using multivariate distributions. Multivariate analysis provides a framework for describing the properties of individual series as well as any possible correlations among series that are interrelated both contemporaneously and across time lags [

Co-integrated processes are characterized by short-term dynamics and long-run equilibria. Economic theory supposes a long term pricing relationship between prices of crude oil and revenues from distillate fuels. This implies that profit spreads tend to converge to a long term average. Co-integration theory, is useful in estimating and testing long-term equilibrium relationships among non-stationary asset prices and making meaningful statistical inference. Co-integrated series have stationary co-integrating residuals, and various spreads will be stationary [

The positive correlation of price variations or volatility clustering, evidenced in speculative markets motivated the introduction of the autoregressive conditional heteroscedastic (ARCH) process by Engle [

For dynamic volatilities, multivariate models provide the natural framework to account for cross sectional information. Haigh and Holt [

This study builds on the work of Aduda et al. [

In this section, we test for cointegration using the trace test, fit a full VECM and test for Grangercausality. The Durbin-Watson (D-W) is also used to check for spurious regression.

The error processes from two non-stationary series can be represented as a combination of two cumulated error processes. These cumulated error processes form stochastic trends which produce another non-stationary process if combined. If two series, say x_{t} and y_{t}, are related, they would be expected to move together and their two stochastic trends would be similar. If when combined, it would be possible to find a combination of them which eliminates the non-stationarity, then the two series would be co-integrated.

Co-integrated series evolve together, staying close to each other, even if individually, their systems drift about. If the series are not co-integrated, then, their spreads can deviate without bounds and spread trading for risk management would not be optimal [

According to Engle and Granger [_{t}} and {y_{t}} are co-integrated if _{t}} are called co-integrated of order (d,b), written briefly as, _{t}} are I(d) and there exists a linear combination

Co-integration analysis seeks to detect any common stochastic trends in price data and use these trends for dynamic analysis of correlations in returns, through, the error correction model (ECM). If two variables are I(1) and co-integrated, they can be modelled as having been generated by an ECM which corrects the deviations from the long-run equilibrium [_{t}}, which has both serial dependence within each component of the series {p_{ti}} and interdependence between different components {p_{ti}} and {p_{tj}}, for

For a k−dimensional time series

This which can be written in compact form as

The diagonal elements of the matrix in equation (1) are the auto-covariance functions of the univariate series {p_{ti}}, whereas the off-diagonal elements are the cross-covariances between p_{t}_{+h,i} and p_{t,j},

and the autocorrelation matrix is then defined as

the case of univariate stationary time series for which the auto-covariances of lag h and lag −h are identical, one must take the transpose of a positive-lag cross-covariance matrix to obtain the negative-lag cross-covariance matrix.

The cross-covariance matrices Γ(h) and cross-correlation matrices ρ(h) are positive definite since

The time paths of co-integrated variables are influenced by the extent of any deviation from long-run equilibrium. In any case, for the system to return to the long-run equilibrium, the movements of at least some of the variables must respond to the magnitude of the disequilibrium. These movements are captured using an ECM. When two random walk I(1) variables are co-integrated, an ECM can be formulated to study their short-run dynamics as influenced by the deviations from equilibrium [

Consider a simple long-term equilibrium model

A simple dynamic model of short-run adjustment is given by

where_{0} denotes the short-term reaction of y_{t} to changes in x_{t}. This implies that in the long run elasticity between y and x, assuming

potential problems including the likelihood of a high level of correlation between current and lagged values of a variable, which will result in problems such as multicollinearity, non-standard distributed parameter estimates and spurious correlation [_{t} − 1 from both sides of the short-run model in Equation (3) and further subtracting γ_{0}x_{t} − 1 from both sides of the resulting equation and then re-parametrizing to give

if we take

both short-run and long-run effects so that should the equilibrium hold, then the term _{t} − 1 must be negative. If the values of_{t} here is the disequilibrium error or the co-integrating residual. The expected value of z_{t} defines a long term equilibrium relationship between x_{t} and y_{t} and the periods of disequilibrium occur as the observed value of z_{t} varies around its expected value.

All terms in the ECM are stationary, so standard regression techniques are valid, assuming co-integration and that we have estimates for β_{0} and β_{1} [

The ECM can also be specified in multivariate form. In order to do this, we consider vector autoregressive and moving average (VARMA) or multivariate autoregressive and moving average (MARMA) models. A general V ARMA (pq) is written as

with p and q ≥ 0, ϕ_{0} = k-dimensional constant vector, ϕ_{i} and θ_{j} = k × k constant matrices and

Co-integrated variables are generally unstable in their levels, but exhibit mean-reverting “spreads” (generalized by the co-integrating relation) that force the variables to move around common stochastic trends. Modification of the VAR model to include co-integrated variables balances the short-term dynamics of the system with long-term tendencies. For the general VAR(p) model,

that the equation

where α and β are k × r matrices, though this representation is of course not unique. For interpretations, it is often convenient to normalize or identify the co-integrating vectors by choosing a specific coordinate system in which to express the variables. An arbitrary normalization, suggested by Johansen [_{t} − 1 may be interpreted as equilibrium relations and the elements of α, as adjustment coefficients which multiply the co-integrating relationship β'p_{t} − 1 to help counterbalance the deviations from the equilibrium. α can also be considered a loading matrix since it determines into which equation the co-integrating vectors enter and with what magnitudes. Normally, these coefficients would be expected to be negative. Suppose p_{t} is I(1) with E[p_{t}] = 0, then Equation (6) represents a VECM and the rank r of the matrix Π determines the number of co-integrating relationships.

The ECM also satisfies the assumptions of classical normal linear regression model (CNLRM) which include a linear regression model, normally distributed residuals, no serial autocorrelation of residuals, and no multicollinearity. As such, diagnostic tests must be carried out to ensure these assumptions are not violated. Among the tests employed are Jarque-Bera (JB) [

Testing for co-integration is necessary in checking if the models being built are empirically meaningful. With no evidence of co-integration, time series data is considered in differences. Spurious regression can also occur when completely unrelated time series appear to be related [^{2} > d where d is the Durbin-Watson (D-W) [

Co-integration allows for the regression of one integrated series over other integrated series [

The residual based E-G test [_{t} and y_{t} are co-integrated.

The Johansen [

This test is based on the examination of the long-run coefficient matrix Π = αβ’, described in Equation (6), so that testing for co-integration between variables is achieved by examining the rank of Π through the eigenvalues. These Johansen tests are the likelihood ratio test based on maximal eigenvalue of the stochastic matrix and the test based on the trace on the stochastic matrix. Before estimating the parameters of aVECM, you must choose the number of lags in the underlying VAR, the trend specification, and the number of co integrating equations.

The basic steps in Johansen’s methodology include 1) testing the order of integration of all variables, 2) setting the appropriate lag length of the model, 3) choose an appropriate model with regard the deterministic components in the multivariate system, 4) Construct likelihood ratio tests for the rank of Π to determine the number of co-integrating vectors, 5) impose normalization and identifying restrictions on the co-integrating vectors, and 6) given the normalized co-integrating vectors estimate the resulting co-integrated VECM by maximum likelihood (ML). The test procedure produces two statistics useful in determining the number of co-integrating vectors.

The trace statistic

The trace test is a test whether the rank of the matrix Π = r. The null hypothesis of the trace statistic is that there are no more than r co-integrating relations. Restricting the number of co-integrating equations to be r or less implies that the remaining k − r eigenvalues are zero, where k is the maximum number of possible co-integrating vectors. Johansen [

where T is the number of observations and the

The maximum eigenvalue statistic

To test the null hypothesis of r co-integrating vectors versus the alternative of (r + 1) co-integrating vectors the test statistic is

where T is is the number of observations and ^{th} largest canonical correlation.

If two or more time-series are co-integrated, then there must be Granger causality between them―either one-way or in both directions. However, the converse is not true. Co-integration usually indicates the existence of a long-run relationship between variables. Even when the variables are not co-integrated in the long-run, they could still be related in the short-run. If the prediction of one time series is improved by incorporating the knowledge of a second time series, then the latter is said to have a causal influence on the first. Granger [

This study explores the co-integration relationships and interdependencies between the Cushing OK WTI and RBOB and the number 1 heating oil traded in the NYMEX, for the period running from 2^{nd} January 2006 to 22^{nd} May 2015. The data was obtained from the U.S. Energy Information Administration (EIA), the principal agency of the U.S. Federal Statistical System responsible for collecting, analysing, and disseminating energy information [

Spot and futures prices of crude oil and distillate fuels are non-stationary and integrated of order one [

Though the R^{2} = 0.9994, the D-W statistic is 0.7545 signalling a spurious regression. The null hypothesis of no co-integration is rejected with a p-value < 0.001 implying that indeed these series are co-integrated with the single co-integrating vector given by

and the long-run relationship given by

The co-integrating residuals which give the co-integrating relation are given by the representation

and the plot of the co-integrating relations (the error correction term), which depicts stationarity, is shown in

Parameter | Value | t-statistic | p-value | SE |
---|---|---|---|---|

α_{0} | 0.511765 | 10.69652 | 0.0000 | 0.047844 |

α_{1} | 0.986809 | 625.7656 | 0.0000 | 0.001577 |

α_{2} | −0.002571 | −1.251454 | 0.2109 | 0.002055 |

α_{3} | 0.001709 | 1.004433 | 0.3153 | 0.001701 |

α_{4} | −0.027940 | −5.154723 | 0.0000 | 0.005420 |

α_{5} | 0.035578 | 6.805134 | 0.0000 | 0.005228 |

single co-integrating relation, among what might be many such relations. When we consider more than two series, we will have a different co-integrating relationship for every dependent variable specified as shown in

For the Johansen co-integration trace test, we examine whether the rank of the matrix Π = r. As such, testing proceeds sequentially for

Rank | Trace statistic | Critical value | p-value | Eigenvalue |
---|---|---|---|---|

0* 1* 2* 3* 4 5* | 444.1103 155.9466 65.5717 34.3072 14.3214 3.9785 | 95.7541 69.8187 47.8564 29.7976 15.4948 3.8415C | 0.0010 0.0010 0.0010 0.0143 0.0746 0.0461 | 0.1150 0.0376 0.0132 0.0084 0.0044 0.0017 |

rejection of the null hypothesis at the 0.05 level.

The co-integrating relations are as shown in

and the vector of co-integrating relations

so that for most cases, the 6 ´ 6 matrix Π represented in Equation (6) would be

where α and β are given as above, _{t} are the shocks or innovations associated with each of the relations,

and

After fitting this model, an analysis of the residuals show that they are not white noise.

This test is applicable since individual variables from a set of variables that are jointly multinomial distributed are also normally distributed, although if a number of variables are normally distributed individually, they are not necessarily also multivariate normal [

The null hypothesis for this test is that the residuals are multivariate normal and this is rejected as can be seen from the p-values reported in

A visual inspection of

For this study, the results for Granger-causality in levels are reported in

From

Series | Skewness | χ^{2} statistic | df | p-value |
---|---|---|---|---|

CF | 0.056937 | 1.274020 | 1 | 0.2590 |

CS | −4.240484 | 7066.810 | 1 | 0.0000 |

GF | −0.589093 | 136.3830 | 1 | 0.0000 |

GS | 0.592101 | 137.7792 | 1 | 0.0000 |

HF | −1.138971 | 509.8211 | 1 | 0.0000 |

HS | −0.125048 | 6.145373 | 1 | 0.0132 |

7858.213 | 6 | 0.0000 | ||

Series | Kurtosis | χ^{2} statistic | df | p-value |

CF | 10.21375 | 5112.752 | 1 | 0.0000 |

CS | 101.6174 | 955,519.8 | 1 | 0.0000 |

GF | 17.20323 | 19,820.15 | 1 | 0.0000 |

GS | 15.34322 | 14,968.89 | 1 | 0.0000 |

HF | 24.22178 | 44,248.25 | 1 | 0.0000 |

HS | 31.65732 | 80,687.03 | 1 | 0.0000 |

1,120,357 | 6 | 0.0000 | ||

Series | JB statistic | df | p-value | |

CF | 5114.026 | 2 | 0.0000 | |

CS | 962,586.6 | 2 | 0.0000 | |

GF | 19,956.53 | 2 | 0.0000 | |

GS | 15,106.67 | 2 | 0.0000 | |

HF | 44,758.07 | 2 | 0.0000 | |

HS | 80,693.18 | 2 | 0.0000 | |

Joint | 1,128,215 | 12 | 0.0000 |

Joint test: | |||||
---|---|---|---|---|---|

χ^{2} statistic | df | p-value | |||

6278.398 | 1176 | 0.0000 | |||

Individual | components: | ||||

Dependent | R^{2} | F (56, 2301) | p-value | χ^{2} (56) | p-value |

res1 * res1 | 0.233479 | 12.51559 | 0.0000 | 550.5426 | 0.0000 |

res2 * res2 | 0.211971 | 11.05254 | 0.0000 | 499.8271 | 0.0000 |

res3 * res3 | 0.085742 | 3.853501 | 0.0000 | 202.1805 | 0.0000 |

res4 * res4 | 0.260093 | 14.44374 | 0.0000 | 613.2987 | 0.0000 |

res5 * res5 | 0.142756 | 6.842568 | 0.0000 | 336.6190 | 0.0000 |

res6 * res6 | 0.100405 | 4.586018 | 0.0000 | 236.7544 | 0.0000 |

res2 * res1 | 0.222538 | 11.76124 | 0.0000 | 524.7442 | 0.0000 |

res3 * res1 | 0.145531 | 6.998214 | 0.0000 | 343.1617 | 0.0000 |

res3 * res2 | 0.146352 | 7.044489 | 0.0000 | 345.0987 | 0.0000 |

res4 * res1 | 0.134115 | 6.364236 | 0.0000 | 316.2435 | 0.0000 |

res4 * res2 | 0.131406 | 6.216239 | 0.0000 | 309.8558 | 0.0000 |

res4 * res3 | 0.097981 | 4.463300 | 0.0000 | 231.0398 | 0.0000 |

res5 * res1 | 0.127599 | 6.009785 | 0.0000 | 300.8780 | 0.0000 |

res5 * res2 | 0.126302 | 5.939869 | 0.0000 | 297.8197 | 0.0000 |

res5 * res3 | 0.133311 | 6.320207 | 0.0000 | 314.3473 | 0.0000 |

res5 * res4 | 0.119110 | 5.555916 | 0.0000 | 280.8617 | 0.0000 |

res6 * res1 | 0.100783 | 4.605250 | 0.0000 | 237.6472 | 0.0000 |

res6 * res2 | 0.100042 | 4.567614 | 0.0000 | 235.8994 | 0.0000 |

res6 * res3 | 0.101842 | 4.659118 | 0.0000 | 240.1439 | 0.0000 |

res6 * res4 | 0.107252 | 4.936366 | 0.0000 | 252.9014 | 0.0000 |

res6 * res5 | 0.100744 | 4.603270 | 0.0000 | 237.5553 | 0.0000 |

test also. This actually shows the presence of spillover effects across the series. From

In this study, both the E-G test [

CF | CS | GF | GS | HF | HS | Joint | |
---|---|---|---|---|---|---|---|

CF | 2356 | 254 | 239 | 195 | 1044 | 951 | 1819.2 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

CS | 321 | 382 | 293 | 159 | 833 | 732 | 1564 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

GF | 40 | 46 | 33369 | 6 | 54 | 40 | 146.6 |

(<0.001) | (<0.001) | (<0.001) | (0.2132) | (<0.001) | (<0.001) | (<0.001) | |

GS | 25 | 19 | 49 | 6077 | 36 | 34 | 103 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

HF | 61 | 125 | 110 | 109 | 13655 | 148 | 458.9 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

HS | 102 | 122 | 78 | 104 | 48 | 14178 | 455.3 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) |

p-values are indicated in the parenthesis.

∆CF | ∆CS | ∆GF | ∆GS | ∆HF | ∆HS | Joint | |
---|---|---|---|---|---|---|---|

∆CF | 51.5493 | 81.5114 | 158.8971 | 231.7504 | 871.1525 | 598.9422 | 1463.8 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

∆CS | 37.6260 | 28.3827 | 210.8375 | 176.4722 | 857.9510 | 579.0799 | 1212.4 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

∆GF | 23.9313 | 32.1628 | 22.5417 | 3.8876 | 51.2044 | 30.3673 | 175 |

(<0.001) | (<0.001) | (<0.001) | (0.4214) | (<0.001) | (<0.001) | (<0.001) | |

∆GS | 6.3281 | 2.8095 | 13.2817 | 6.7035 | 21.7478 | 14.5890 | 72.2 |

(0.176) | (0.5902) | (0.1) | (0.1524) | (<0.001) | (0.0056) | (<0.001) | |

∆HF | 61.3676 | 192.8939 | 45.3097 | 122.6379 | 173.5127 | 112.1761 | 498.4 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

∆HS | 34.5446 | 90.3696 | 61.5954 | 114.1517 | 30.1818 | 27.3170 | 451.6 |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) |

p-values are indicated in the parenthesis.

in a multivariate context.

Under the E-G test, the null hypothesis of no co-integration is rejected and this result is backed by the DW statistic. Similarly, Johansen’s test results reveal up to 4 co-integrating vectors with an optimal lag length of 4 if all six series are considered simultaneously. After fitting the full VECM, a residual analysis is carried out and it suggests some heteroscedasticity in the noise process, contravening the Gaussian assumption on the residuals. Tests reveal ARCH effects in the residuals, and the JB hypothesis of multivariate normality is rejected. This suggests a VECM with GARCH errors could have been better. For noise that is MGARCH, fitting a VEC-GARCH. This involves writing the VEC part as a seemingly unrelated regression (SUR) model and combining to obtain SUR-GARCH model. That is however not covered in this study.

Causal relationships are also explores and bidirectional Granger-causality is exhibited for all cases except for the gasoline futures which do not seem to Granger cause gasoline spot prices. This underscores the importance of the spill-over effects in the volatilities of these price series. It confirms the presence of cointegrating relations and hence spill-over effects.

Aduda, J., Weke, P. and Ngare, P. (2018) A Co-Integration Analysis of the Interdependencies between Crude Oil and Distillate Fuel Prices. Journal of Mathematical Finance, 8, 478-496. https://doi.org/10.4236/jmf.2018.82030