Analysis Of Oil Exchange Traded Funds

Published Date: 02 Nov 2017

1. Introduction

Motivation

Crude oil is one of the most important physical commodities in economy. The demands for oil are closely accompanied by the global economic growth. Crude oil also can be an important investment instrument for institutional and retail investors to diversify their portfolios or hedge against the economics. However, direct investment in physical oil is quite costly and not that practical. Oil ETFs offers investors an efficient way to gain exposure to various oil products without actually owning the oil itself. Most of these Oil ETFs have been designed to track crude oil price by investing in future contracts on crude oil.

This study attempts to understand the dynamic relationship between prices of Oil Exchange Traded Funds (ETF) and the underlying crude oil prices. It is interesting because oil ETFs usually consist of oil future contracts, and the futures prices are considered to provide the information for the future spot prices. Meanwhile, oil ETFs are tracking the price and movement of the current oil price. This makes the relationship between oil ETF and the underlying crude oil complex and interesting.

There are amounts of academic and empirical works on exchange-traded fund, most of them are focusing on Index ETFs, their efficiency and performance and the comparison to other investment products, such as hedge funds, mutual funds. Also, there are plenty of researches on the spot and future prices of crude oil. Those researches usually put emphasis on the determinant of the spot price of crude oil, such as supply and demand, storage cost, and other economic factors, or if the futures prices could be used as a predictor for spot prices, and the relationship between oil futures prices and oil spot prices. Research connecting the oil ETF and crude oil price, and concerning the dynamic relationship between the price of ETF and the underlying index is limited.

The main objective of this paper is to investigate the equilibrium relationship between Oil ETFs and crude oil prices and further examines the ability of Oil ETFs to track the underlying Crude Oil Benchmark.

Crude Oil Benchmark

Crude Oil which is also known by the name Petroleum, is the most actively traded Commodity in energy Markets. The Largest Market for Oil is located in London and New York. There are a couple of popular Benchmarks around the world the investors use for tracking the Market Price of Oil. The most recognized and widely quoted Oil Index in North America is known as The West Texas Intermediate or WTI. This Benchmark reflects the Market Price for a single Barrel of Light Sweet Crude Oil, most of which is pumped and refined in Texas and other locations located along the Gulf Coast.

1.3 ETFs and Oil ETFs Overview

Exchange-traded Fund (ETF) is a basket of securities, holding assets like stocks, commodities (such as precious metals and futures) and bonds. It is an investment fund to be traded at stock market like individual stocks. The most popular strategy of an ETF is to track a particular index, including Broad Market index, such as FTSE All-World index and MSCI US Broad Market index, major-index, such as S&P 500, Dow Jones Industrial Average and some are tracking the country index and cap-size index.

ETFs have several advantages compared to the traditional mutual funds. They have lower operating expenses, trade more flexible, as ETFs can trade throughout the trading day and the mutual funds only can trade at the end of trading day at the net asset value (NAV), and enjoy the tax-efficiency. Therefore, ETFs became popular immediately since have been available in the US in 1993. And it becomes one of the fastest growing sectors in financial market due to its advantages and features.

By the end of April 2012, the number of exchange-traded funds reached 1,175 in the United States with an estimated $1,075 billion in assets under management.

Figure_1.1 shows the explosive growth of the ETF sector in the United States regarding the number of ETFs and value of ETF assets since 2000.

Figure_1.1

Assets (US$bn)

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

12-Apr

ETF Assets

65.6

84.6

102.3

150.7

227.7

299.4

406.8

580.7

497.1

705.5

891

940.4

1,074.50

# of ETFs

81

101

113

117

152

201

343

601

698

772

896

1,098

1,175

* Source: ETF Landscape April 2012, BlackRock

ETFs can be divided into several types based on its structure, such as Index ETFs, Commodity ETFs, Bond ETFs, Currency ETFs and Leveraged ETFs.

Oil ETFs are Commodity ETFs. Commodity ETFs invest in commodities, including energy, precious metals, softs and agriculture.

Oil ETFs are tracking the underlying index same as other exchanged funds do. But the main difference between Oil ETFs and traditional ETFs is that traditional ETFs usually hold a basket of securities which comprise the underlying index, while Oil ETFs invest in the near term futures contracts of specialized oil prices. The reason is it’s quite costly to hold the oil products physically.

The very first exchange-traded fund for Oil is The United States Oil Fund ETF, which was launched on April 10, 2006.

Following are the four most popular ETFs which track the performance of Crude Oil, The West Texas Intermediate ("WTI") Benchmark:

ETF name

Trading Symbol

Fund Launched

Market Cap

Aveg. Daily Trading Volume

United States Oil Fund

USO

April 10th, 2006

S1,210 M

6,176,730

PowerShares DB Oil Fund

DBO

January 5th, 2007

$720.06 M

392,592

United States 12 Month Oil Fund

USL

December 5th, 2007

$101.14M

16,241

Teucrium WTI Crude Oil Fund

CRUD

February 23rd, 2011

$2.04 M

536

* Source: seekingalpha.com on 1/22/2013

In this paper, we will focus our study on the ETF of "USO" and "DBO", both of which are tracking the movement of crude oil "WTI", and are actively traded with high market caps.

Oil Futures

When study the Oil ETF, it is inescapable to mention the oil futures contracts, as the Oil ETFs are mainly composed of oil futures in different maturity periods. Oil futures contracts are the agreement between buyers and sellers on the price of the oil upon delivery at a specified date in the future. The prices of futures reflect the market expectation for the spot oil price in the future. The unit of the oil futures contract is 1,000 barrels of oil. New York Mercantile Exchange (NYMEX) and the Intercontinental Exchange (ICE) are the major markets for the trading of oil futures contracts.

The oil futures prices are widely used to be predictors of spot oil prices, and thought to be better than forecasts using economic models. Although Alquist and Kilian (2010) concluded by using a two-country, multi-period general equilibrium estimated model that "the price of crude oil futures is not the most accurate predictor of the spot price of crude oil in practice." The topic of the relationship between oil futures prices and spot oil price draws lots of interest in academics and empirical research.

1.5 Structure of the paper

The structure of the paper is arranged as follows: Chapter 2, we review the literatures on the researches of ETFs and energy futures. Chapter 3 describes the data. The empirical analysis begins in Chapter 4 with the testing of structural stability in data. A structural break point has been detected over the whole period of data. Then, in Chapter 5&6, we estimate the models both for the whole period and sub-periods. The co-integration relationship has been developed in Chapter 5. Result in Chapter 5 shows that the co-integration relationship between ETFs and underlying crude oil price is nonlinear. And Error Correction Models are developed for estimation. In Chapter 6, the Impulse Response Function is used to further describe the relationship between the ETFs and underlying crude oil price. Results in this chapter suggest that And Chapter 7 offers the conclusion.

2. Literature review

Prior to year 2000, there was few research on ETFs. Laurent Deville (2006) noticed that despite the increasing importance of ETFs markets, literature on these topics is still scarce, although research perspectives are promising. Research on ETFs is mainly of empirical studies and mostly focuses on ETFs’ characteristics and performs (return and tracking error); comparison to other investment instruments, such as mutual funds, Index funds and etc.

Elton, Gruber, Comer and Li (2002) examine the performance of SPDR or Spiders, which is the most actively traded exchange-traded Index fund to replicate the S&P Index. They find out that the Spiders underperform the S&P Index by 28 basis points and low-cost index funds by 18 points over the 1993 – 1998 period. Gallagher and Segara (2006) investigate the ability of classical ETFs to track underlying equity benchmarks on the Australian Stock Exchange. They examine the tracking errors of ETFs on the Australian stock exchange and compare the tracking error volatility between ETFs and equity index funds operated off-market. They argue that ETFs are better to track their benchmarks than off-market index funds, and conclude that classical ETFs in Australia provide investors with returns commensurate with the underlying benchmark before costs. Patrick Chu (2011) studies the magnitude of tracking error and the determinants of tracking errors using the daily figures of the ETFs traded in Hong Kong stock Market. He finds out that the tracking errors for ETFs traded in HK stock exchange are comparatively higher than those documented in US and Australia. Shin and Soydemir (2011) estimate tracking errors and relative performance of 26 ETFs over their benchmark indexes. They find that tracking errors are significantly different from zero and display persistence. They also examine the factors affecting tracking errors, such as expense ratio, dividends, exchange rate and spreads of trading prices and find out that the main factor driving tracking errors is the change in the exchange rate.

Gastineau (2004) compares the benchmark index ETFs to indexed mutual funds, investigates the difference in returns between iShares Russell 2000 ETFs and Vanguard Small Cap Index Fund over 1994 to 2002. The results show that ETFs underperform the corresponding mutual fund. Kostovetsky (2003) develop a simple one-period model to examine the major differences between ETFs and index funds. The key areas of difference are management fees, share-holder transaction fees, taxation efficiency, as well as other qualitative differences. And the qualitative advantages of ETFs are attractive to more active larger investors.

3. Data

The data for this study includes the daily closing prices of two ETFs - the United States Oil Fund ("USO"), the PowerShares DB Oil Fund("DBO")which track the price of benchmark for the crude oil – the light, sweet crude oil ("West Texas Intermediate") ("WTI").

The United States Oil Fund is the first Crude Oil Based Fund launched on April 10, 2006. It is a domestic exchange traded security designed to track the movement of WTI. The USO portfolio will consist of listed crude oil futures and other oil related futures, forwards, and swap contracts. USO is also invest in obligations of the United States government with remaining maturities of two years or less and hold cash and cash equivalents to be used to meet its current or potential margin or collateral requirements with respect to its investments in crude oil futures contracts and other oil interests. The principle of USO tracking WTI is … percentage change in NAV tracking the

Another popular Oil ETF is the PowerShares DB Oil Fund. The Fund is designed to track the Market Performance of Crude Oil, which it achieves by following the performance of a Benchmark known as The Deutsche Bank Liquid Commodity Oil Index. This Index is comprised of Light Sweet Crude Oil Futures Contracts as well as investments made in highly liquid Short-Terms financial instruments such as 3 month United States Treasury Bills. The ETF Shares first began trading on the New York Stock Exchange on January 5th, 2007.

WTI light, sweet crude oil is the primary US benchmark for the crude oil. The price of WTI is often referenced in news reports on oil prices, alongside the price of Brent crude from the North Sea. Other important oil markers include the Dubai Crude and the OPEC Reference Basket. Historically, it has traded closely to Brent and the OPEC basket.

The sample period for the data set for USO and WTI covers from April 12, 2006 to March 31, 2012, includes 1,501 observations, and DBO from January 5, 2007 to March 31, 2012 with 1,319 observations. A plot of the raw data show that USO, DBO and WTI tend to move together.

Also, there is a feature of this data that makes it particularly interesting, which is the financial crisis and recession in year 2008.The financial crisis in 2008 had a big impact on the prices for USO, DBO and WTI, as all of them suffered a huge drop.

Graph_1 produced by using raw data series for price of USO, DBO & WTI

Graph_1

Graph_2 Shows the Monthly Return for ETFs and Benchmark

Graph_2

4. Methodology

4.1 Structural Break

4.1.1Technique for Testing Structural Stability

We begin our analysis to test for the structural stability of the data for ETFs "USO" and "DBO", and the benchmark crude oil "WTI".

Tests for structural stability are much discussed in the literature. Nyblom (1989) proposed the sup-F test to detect possible changes in parameters. Brown, Durbin, and Evans (1975) made an important contribution to assessing the constancy of regression coefficients, calculating updated coefficient vectors as additional observations are added to the regression. Stokes (1997) discussed thoroughly the Recursive Residuals (RR) procedure to detect the locations of potential structural breaks in a series. For an OLS procedure, the OLS residuals can be heteroscedastic and auto correlated even when the true errors are white noise. The Recursive Residuals procedure transforms the OLS residuals so that they do satisfy the OLS assumptions. The technique begins with estimating Ordinary Least Squares (OLS) and then calculating updated coefficient vectors as additional observations are added to the regression. And the recursive residuals satisfy the OLS properties, i.i.d. ~ N(0, σ ).

Stokes(1997) mentioned that the cumulated sum of recursive residuals test (CUSUM), cumulated sum of squared standardized recursive residuals test (CUSUMSQ) and the Harvery-Cller(1977) test are the three summary tests for parameter stability. The CUSUM test is particularly good at detecting systematic departure of the coefficients that results in a systematic sign on the first step ahead forecast error. The CUSUMSQ test is useful when the departure of the coefficients from constancy is haphazard rather than systematic but that there involves a systematic change in the accuracy of the estimated equation as observations are added. And the plotting of Quandt’s log-likelihood ratio statistic is used to detect the single time-point, at which there is a discontinuous change from one constant set of regression parameters to another.

The advantage of these test statistics is that they can be graphed, and we can identify not only their significance but also at what time point a possible break occurred.

4.1.2 Break point For Data

The recursive residuals plots in Figure 4.1, Figure 4.2 below show the results of CUSUM, CUSUMSQ and QLR test for price of USO & WTI and price of DBO & WTI, respectively, to examine the breakpoint in the data set for the complete sample.

Plots in Figure 3.1 are based on the model:

USOt = 1.007*USOt-1 + 0.1853*USOt-5 – 0.1949*USOt-6 – 0.1625*WTIt-5 + 0.1625*WTIt-6

(t=74.53) (3.33) (-3.55) (-4.76) (4.77)

The CUSUM test statistics plotted in Figure 3.1 are almost inside the CUSUM confidence bounds. But we noticed that during Apr. 1st 2008 to Aug. 31st 2008 (around n=450 to n=600), it shows instability, although still within 99% confidence level, but out of the 95% bound. And the CUSUMSQ plot clearly shows the instability as it breaks the upper confidence bound, and suggests that there should be model instability at some time-points during time period we study. And the Quandt-Likelihood Ratio shows that there is a dramatic drop from around n=575 to n=730, corresponding to the time period July 8th, 2008 to March 9th, 2009, indicates the coefficient shifts during this period. Further check the USO and WTI prices in this period. Prices of USO experienced an enormous decrease from $117.48/share on July 14, 2008 to $22.86/share on February 18, 2009, which represents a drop of 414% over 7 month and price of WTI dropped 380% from $145.31/barrel on July 3, 2008 to $30.28/barrel on December 23, 2008. This is mainly due to the financial crisis in late 2008. I suggest breaking the data series into two periods: April 12th, 2006 to December 31, 2008 and January 2nd, 2009 to March 31, 2012.

Figure 4.1 CUSUM, CUSUMSQ and QLR plot for USO & WTI

CUSUMSQ plot for USO price and crude oil price during 4/12/2006 to 3/31/2012

CUSUM plot for USO price and crude oil price during 4/12/2006 to 3/31/2012

QLR plot for USO price and crude oil price during 4/12/2006 to 3/31/2012

Plots in Figure 4.2 are based on the model:

DBOt = 0.8628*DBOt-1 + 0.1304*DBOt-4 + 0.036*WTIt-1 – 0.035*WTIt-5

(t=32.40) (4.95) (5.13) (-5.03)

The recursive residuals plots for DBO & WTI are very similar to those of USO & WTI. CUSUM are almost within the confidence bound, CUSUMSQ goes outside of the upper bound to indicate the coefficient shift. And Quandt-Likelihood Ratio experienced tremendous drop during July 2008 to March 2009. Price of DBO dropped from $55.01/share on July 14th, 2008 to $15.83/share on February 18th, 2009. The breakpoint would be set on December 31, 2008. The data series could be truncated into two subsets for January 5th, 2007 to December 31st, 2008 and January 5th, 2009 to March 31st, 2012.

Figure 4.2 CUSUM, CUSUMSQ and QLR plot for DBO & WTI

CUSUMSQ plot for DBO price and crude oil price during 1/5/2007 to 3/31/2012

CUSUM plot for DBO price and crude oil price during 1/5/2007 to 3/31/2012

QLR plot for DBO price and crude oil price during 1/5/2007 to 3/31/2012

Figure 4.3 shows the whole data separated into two sub-period data on December 31, 2008

Figure 4.3

4.2 Testing for Coefficient and Variance Changes

Stock and Watson (2002) proposed a test to distinguish between changes in the coefficients or the changes in the variance of a VAR model. The basic idea is that within a sample there may be changes in the coefficients of the model and or there may be changes in the innovation variance, or both may occur.

The Stock-Watson procedure can be used in a two period model where it is possible to test whether at one date there was a change in variance or a change in structure or in a moving period mode where inspection of graphs indicates what is happening in terms of structural change or shock change.

In our case, a VAR model of order 12 was estimated for USO & WTI and DBO & WTI, and the point to break the data into two periods was set on December 31, 2008.The Stock-Watson test can be used to validate whether in fact there was a shift and if that was the case the nature of the shift. While and represent factual data, and represent counterfactuals. The appropriate values to test are to test for coefficient shifts, to test for variance shifts and to test when both the variance and the coefficients have shifted. Karras-Lee-Stokes (2003) contains an application that in addition developed critical values.

Table 4.1 Factual& counterfactual data for Stock-Watson Test

Stock-Watson Value

σ11

σ12

σ21

σ22

|σ11-σ12|

|σ21- σ22|

|σ11 – σ21|

|σ12 – σ22|

|σ12 - σ21|

USO & WTI

85.33

35.21

75.03

35.2

50.12

39.83

10.3

0.01

39.82

DBO & WTI

93.43

46.41

82.79

41.34

47.02

41.45

10.64

5.07

36.38

The result of the Stock-Watson Test indicates that it was an innovation change rather than the coefficient change for both USO & WTI and DBO & WTI.

For series USO & WTI, the shock change for the first period |σ11-σ12|= 50.12, which is much greater than the structural change |σ11 – σ21| = 10.3. And for the second period, the shock change |σ21- σ22| = 39.83 while the structure change |σ12 – σ22| = 0.01. It is very similar for data series DBO & WTI, which shows that the variance change is much greater than coefficient change.

Figure 4.3 and 4.4 show the plots of factual and counterfactual data for Stock-Watson Test.

We compare the graphs in pairs vertically for coefficient shift and horizontally for shock change.

Figure 4.3 Stock-Watson values for USO & WTI (raw data)

Figure 4.4 Stock-Watson values for DBO & WTI (raw data)

The above graphs confirm our prior statements that it was mainly change in the variance, as shapes changed more if compared horizontally than vertically.

In the following sections, we will discuss based on the whole sample and sub-samples.

Sample (I) for USO: 4/12/2006 – 3/31/2012

for DBO: 1/5/2007 – 3/31/2012

Sample (II) for USO: 4/12/2006 - 12/31/2008

for DBO: 1/5/2007 – 12/31/2008

Sample (III) for both USO & DBO: 1/2/2009 – 3/31/2012

5. Co-integration and Long-term Equilibrium Relationship

5.1 Co-integration Theory

Ender (2004) stated that Equilibrium theories involving non-stationary variables require the existence of a combination of the variables that is stationary. And co-integration often means that a linear combination of individually unit-root non-stationary time series becomes a stationary and invertible series.

The concept of integration and co-integration were introduced by Engle and Granger (1987).

A time series xt is said to be integrated of order d, xt~I(d), if (1-B)dxt is stationary and invertible, where d > 0.

In a multivariate case, vector xt are said to be co-integrated of order d, b, xt ~ CI(d,b), if (i) all components of xt are I(d); (ii) there exists a vector α(≠0) so that zt = α’xt is integrated of order I(d – b), b>0. The vector αis called the co-integrating vector.

If interpreting α’xt = 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary.

Since zt = α’xt is stationary, the l-step ahead forecast of zT+l at the forecast origin T satisfies

zT(l)→p E(zt) = uz. l → ∞

This also implies that a’zT(l) →uz as l increases. Then point forecast of xt satisfy a long-term stable constraint.

In the case of Oil ETF and Crude Oil, we noticed that USO, DBO and WTI move dependently with each other, so next, we will determine whether there exists an equilibrium relationship between USO & WTI and DBO & WTI.

5.2 Linear Co-integration

5.2.1 Engle and Granger Two-Step Methodology

Ender (2004) illustrated two methodologies to test for co-integration. One is the Engle-Granger testing procedure, which was initially proposed by Engle and Granger (1987).This methodology seeks to determine whether the residuals of the equilibrium relationship are stationary. The other is the Johansen (1988) and Stock-Watson (1988) methodologies, which determine the rank of π by using maximum likelihood estimator.

Engle and Granger (1987) suggested the following two-step estimator.

The first step is to determine the order of integration for each variable and generate the error series{êt}. The Dickey-Fuller (DF) or augmented Dickey-Fuller (ADF) can be used to detect the number of unit root in each variable. It is important because if the variables are integrated of different orders, it’s possible to conclude that they are not co-integrated. If the results indicate that two series {yt} and {zt} are I(1), the long-run equilibrium relationship can be estimated using OLS:

yt= β0 + β1zt + et

Then define êt = yt–βˆ0 – βˆ1zt

{êt} is the series of the estimated residuals of the long-run relationship. If the deviations from long-run equilibrium are stationary, then {yt} and {zt} are co-integrated of order (1,1). So we could perform the Dickey-Fuller test and Augmented Dickey-Fuller test on the residuals by using the following two equations to test if α1 = 0

Δêt = α1 êt-1 + εt

Δ êt = α1 êt-1 + Σαi+1 Δêt-i +εt

The only difference from the traditional ADF to (this version of) the Engle-Granger test are the critical values. The critical values to be used here are no longer the same provided by Dickey-Fuller, but instead provided by Engle and Yoo (1987). This happens because the residuals above are not the actual error terms, but estimated values from the long run equilibrium equation of USO against WTI.

The second step is using the residuals {êt} to estimate the error-correction model, then estimate the long-run equilibrium relationship. If {yt} and {zt} ~ CI(1,1), the variables have the error-correction form:

Δyt = α01 + αyêt-1 + Σ α11(i)Δyt-i + Σ α12(i)Δzt-i + εyt

Δzt = α02 + αzêt-1 + Σ α21(i)Δyt-i + Σ α22(i)Δzt-i + εzt

The residual εyt, εzt will be checked whether they are serially correlated. The model should be re-estimated by using longer lag lengths if the residuals are serially correlated until yield serially uncorrelated errors. Then we may test the speed of adjustment parameters αy and αz, and if Δyt and Δzt converge to the long-run equilibrium relation.

Ender (2004) mentioned that although the Engle-Granger procedure is convenient, there are two important defects. First, the procedure requires to place one variable on the left-hand side and to use the others as regressors on the right-hand side. If in tests, three or more variables are used since any of the variables can be selected as the left-hand side variable, the result for the test will be different. Second, the coefficient α is obtained by estimating a regression using the residuals from another regression, so any error introduced in step 1 is carried into step 2.

5.2.2 Co-integration Testing Result

The variables USO, DBO and WTI for the whole period and sub-period were all tested using Dickey-Fuller test (DF) and the Augmented Dickey-Fuller test (ADF) with 4 lags. The results are reported in Table 4.1.

Table 4.1 The Associated t-statistic for ΔUSOt, ΔDBOt and ΔWTIt

 

 

Sample (I)

Sample (II)

Sample (III)

 

 

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

ΔUSOt

DF test

-1.034

-0.901

-0.1393

 

ADF (4 Lags)

-1.395

-0.4734

-2.849

ΔWTIt

DF test

-0.00409

-0.6202

0.7436

 

ADF (4 Lags)

-1.760

-0.7968

-2.081

 

 

1/5/2007 - 3/31/2012

1/5/2007 - 12/31/2008

1/2/2009 - 3/31/2012

ΔDBOt

DF test

-0.5891

-0.4858

0.3695

 

ADF (4 Lags)

-1.100

-0.63

-2.243

ΔWTIt

DF test

0.1353

-0.4312

0.7436

 

ADF (4 Lags)

-1.988

-0.9514

-2.081

The 95% critical value for DF is -1.940 and for ADF using 4 lags is -2.865. All the absolute values of t-statistics are below the critical value, so we cannot reject the null hypothesis that there is a unit root in any of the series. Therefore, the whole period and sub-periods series of USO and WTI are I(1). Then the long-run equilibrium regression can be estimated. The estimates of the long-run relationship for USO & WTI, DBO& WTI with whole data and sub-period data:

(full sample) USOt = 6.3163 + 0.5441WTIt + e1t (4.1)

(3.67) (26.34)

(full sample) DBOt = 5.0661 + 0.2872 WTIt + e2t (4.2)

(14.38) (69.60)

(subsample I) USOt = 5.8391 + 0.7527WTIt + e1t (4.3)

(9.82) (107.09)

(subsample II) USOt = 22.2154 + 0.1761WTIt + e1t (4.4)

(56.27) (36.74)

(subsample I) DBOt = 4.145 + 0.334 WTIt + e2t (4.5)

(19.68) (142.72)

(subsample II) DBOt = 11.4465 + 0.1844 WTIt + e2t (4.6)

(55.71) (73.93)

Where, e1t and e2t are the residuals from the equilibrium regressions.

The question of greatest interest is whether or not the residuals {ê1t} and {ê2t} are stationary. If the residuals are I(0), then the variables are said to be co-integrated of order (1,1).

The Associated t-statistic for Δê1t and Δê2t are reported in Table 4.2. The plots in Figure 4.2 are of scatter plot of residuals {ê1t} and {ê2t} for full samples and subsamples.

Table 5.2 The Associated t-statistic for Δê

 

 

Sample (I)

Sample (II)

Sample (III)

 

 

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

Δê1t

No Lags

-0.754

-2.676

-3.548

for USOt

4 Lags

-0.771

-2.334

-3.066

 

 

1/5/2007 - 3/31/2012

1/5/2007 - 12/31/2008

1/2/2009 - 3/31/2012

Δê2t

No Lags

-1.668

-3.907

-2.912

for DBOt

4 Lags

-0.963

-2.149

-2.793

Figure 5.2 Scatter Plot of residuals

Here, the critical value from an ordinary Dickey-Fuller table might not be appropriate to test whether the residual series is stationary. The reason is that the {êt} sequence are generated from the regression yt= β0 + β1zt + et, and we do not know the actual error et, only the estimate of the error êt. Only if β0and β1 were known in advance and used to construct the true {êt}, the Dickey-Fuller test can be used. Engle and Yoo (1987) provided a table of critical value for Engle-Granger co-integration test. To test for co-integration between two variables, the critical value of DF test at the 5 percent significance level is 3.35, at 1 percent level 4.00, and critical value for ADF at 5% is 3.25, 1% 3.78.

Obviously, based on Engle and Granger methodology of co-integration testing, variable USO and WTI for the whole period from 4/12/2006 to 3/31/2012 are not co-integrated at any significance level. DF and ADF statistics -0.754 and -0.771 are less than the critical values 3.35 and 3.25 at 5% level, respectively. And it is the same for the variable DBO and WTI from 1/5/2007 to 3/31/2012. DF statistics is -1.668, ADF statistics -0.963, which is far below the DF critical value at 5% level 3.35, and ADF 3.25. So we cannot reject the null of I(1) that residuals from the equilibrium regression is stationary.

For the sub-periods, ADF test shows all variables are not co-integrated for sub-period I and sub-period II. But if using DF statistics, USO and WTI are co-integrated at 5% level for period of 1/2/2009 to 3/31/2012. Also, we noticed that the DF statistics for DBO and WTI at period 1/7/2007 to 12/31/2008 is -3.907, which is greater than 5% level of 3.35 and close to 1% level of 4.00. That might mean the linear relationship of DBO & WTI for this period is quite strong.

The difference in results for full sample and the subsamples is mainly due to the structural break cross the series, and it is consistent with previous conclusion in section 3.

In general, by using Engle and Granger methodology, under the null hypothesis, {êt} is I(1), so that co-integration is not found for the variable USO & WTI and DBO & WTI in either whole period or sub-periods. But we still have doubt about the result, because the daily price of USO and DBO is tracking the performance of WTI, there must be some potential connections between those of ETFs and the underlying crude oil price of WTI.

In next section, we will investigate if there is non-linear relationship existed in prices of USO and DBO with WTI.

5.3Non-linear Co-integration Relationship

5.3.1 Non-linear Co-integration Generalization

In Engle-Granger methodology, co-integration refers to a linear combination of non-stationary variables

zt = xt – Ayt that is stationary. Actually, in many macroeconomic and financial cases, linear relationship is not found in non-stationary contexts, and it is possible that nonlinear long-run equilibrium exist among those integrated variables, even if the variables are not linearly co-integrated.

Escanciano and Escribano (2011) defined nonlinear co-integration as "if two or more series are of extended memory, but a nonlinear transformation of them is short memory, then the series are said to be nonlinearly co-integrated".

Granger (1991) proposes generalizations extended to nonlinear co-integration.

The first generalization is that nonlinear transformation of the time series that will be co-integrated in g(x) and h(y), and the linear combination of nonlinearly transformed variables zt = g(x) - Ah(y) is short memory in mean.

Here Granger (1991) also defined the variable that is short memory / long memory in mean. Given information It at time t, the conditional mean of a variable x at time t+h, E(xt+h | It ) converges to a constant, when h → ∞, then we say the variable x is short memory in mean (SMM). If E(xt+h | It) depends on It for all h, variable x is long memory in mean (LMM). In long memory series, the shocks have persistent effects.

A second generalization is using time-varying parameters, and the error-correction model equations are in form of:

Δxt = ρ1(t)zt-1 + lagsΔxt, Δyt + residual

ρ1(t) is the speed of adjustment parameter that is allowed to change over time.

Michael, Nobay and Peel(1997) used nonlinear error-correction in the residuals from linear co-integration to capture the deviations from purchasing power parity (PPP). The nonlinear adjustment process was characterized in terms of an exponential smooth transition autoregressive (ESTAR) model. And conclude "The failure to find co-integration on the basis of a linear model does not necessarily invalidate long-run PPP".

5.3.2 ACE Algorithm

Granger and Hallman (1991) suggested that two series are not co-integrated linearly, but if there exists a nonlinear attractor, it can be viewed as a nonlinear co-integration.

In linear case, if xt, yt are I(1) and there exists a linear combination zt = xt- Ayt which is I(0), the line x=Ay can be thought of an attractor. In nonlinear case, if xt, yt are not linearly co-integrated, but we haveqt = g(xt) – h(yt) ~ I(0), define A = (x,y:g(x)=h(y) or f(x,y)=0), then A is a nonlinear attractor for xtandyt.

Granger and Hallman (1991) showed that the ACE algorithm provides a practical estimation to obtain the nonlinear attractor if there is no prior information about the shape of a possible attractor.

The Alternating Conditional Expectations (ACE) algorithm was originally proposed by Breiman and Friedman (1985). The ACE model can be written as:

Θ(y) = α0 + j (xj)

The ACE algorithm maximize the correlation between Θ(y) and + j (xj), which is equivalent to minimizing the squared error E{Θ(y) - α0 - j (xj)}2 subject to var{Θ(y)} = 1.

The procedure to estimate ACE algorithm includes four steps:

(i) Initialize to set Θ(y) = {y - ȳ} / √var(y), and set αj(xj) the regression of y on xj

(ii) Fit an additive model to Θ(y) to obtain new function α1(x1)…αk(xk)

(iii) ComputeΘ ̂(y) = E{j (xj) | y} and standardize the new Θ(y),

Θ(y) = Θ ̂(y) / √var(Θ ̂(y))

(iv) Alternate by repeating (ii) and (iii) until E{Θ(y) - j (xj)}2 converges

5.4 Nonlinear Co-integration testing results

5.4.1 ACE Transformation for the series

In section 4, we applied data by using Engle and Granger methodology to estimate the linear co-integration relationship between series USO & WTI and DBO & WTI. Although whole period and sub-period dataset of USO, DBO and WTI are I(1), the result suggested there is no linear co-integration in those series for either full period or sub-period. In this section, we are interested in finding out if any nonlinear co-integration relationship in series.

According to Granger’s generalization of nonlinear co-integration, if the residual of transformed series x, y, qt = g(xt) - h(yt) ~ I(0), then we say x and y are non-linearly co-integrated.

Figure 5.1 shows the ACE transformations of the series g1(USOt) and h1(WTIt) in full data period from 4/12/2006 to 3/31/2012 and sub-data periods from 4/12/2006 to 12/31/2008 and 1/2/2009 to 3/31/2012, respectively.

Similarly, Figure 5.2 shows the ACE transformations of the series g2(DBOt) and h2(WTIt) from whole period 1/7/2007 to 3/31/2012 and sub-periods 1/7/2007 to 12/31/2008 and 1/2/2009 to 3/31/2012.

Figure 5.1 ACE transformation of series USO and WTI

Figure 5.2 ACE transformation of series DBO and WTI

Almost all the transformed series clearly show the evidence of nonlinearity, except for series DBO and WTI in sample period of 1/7/2007 to 12/31/2008. The lines are almost straight and suggest the linearity of the series.

We go back to section 4.2, where it is the test of linear co-integration between the series. The conclusion indicates that the linear relationship of DBO & WTI for period of 1/7/2007 to 12/31/2008 is quite strong, which is consistent to the result of ACE transformation.

Table 5.1 compares the R2 for linear model and ACE transformation. Obviously, the ACE transformation produces a better fit than linear model, as all the R2from ACE transformation is greater than that of from the estimated linear model of (4.1) – (4.6), which suggest that the nonlinear transformation is necessary.

Table 5.1 Comparison of R2 value for linear model and ACE transformation

 

 

Sample (I)

Sample (II)

Sample (III)

 

 

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

R2

Linear Model

0.3160

0.9438

0.6231

ACE Transformation

0.9841

0.9968

0.8580

 

 

1/5/2007 - 3/31/2012

1/5/2007 - 12/31/2008

1/2/2009 - 3/31/2012

R2

Linear Model

0.7861

0.9760

0.8701

ACE Transformation

0.9762

0.9936

0.9690

Table 5.2 shows the DF statistics in the three samples for

q1t = g(USOt) - h(WTIt)

q2t = g(DBOt) - h(WTIt)

Function g(.) and h(.) is the ACE transformation of the series.

Table 5.2 DF test statistics for q1t and q2t

 

 

Sample (I)

Sample (II)

Sample (III)

 

 

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

q1t

DF

-3.550***

-4.861***

-5.448***

 

 

1/5/2007 - 3/31/2012

1/5/2007 - 12/31/2008

1/2/2009 - 3/31/2012

q2t

DF

-4.797***

-9.061***

-4.638***

*p < 0.1, **p < 0.05, ***p < 0.01

Critical Values for DF test at 99% is -2.569. All the t- statistics significant at 99% level suggest that the null hypothesis of long memory (ie. I(1)) for q1t and q2t should be rejected. Therefore, we conclude that both q1t = g(USOt) - h(WTIt) and q2t = g(DBOt) - h(WTIt) ~ I(0), at full period and sub periods. If define A1 = (USO,WTI: g(USO)=h(WTI)) and A2 = (DBO, WTI: g(DBO)=h(WTI)), A1 and A2 are the nonlinear attractor for series USOt and WTIt, and DBOt and WTIt.

Previously, we concluded that the data series USO and WTI are not linearly co-integrated, as the residual of linear combination of these two series comes to be I(1). In this section, we extend the concept of linear co-integration to generalization of nonlinear co-integration, and transform two series by use of the ACE algorithm. Result shows that q1t = g(USOt) - h(WTIt) is stationary, and a nonlinear attractor A1was found as well, which indicate that the nonlinear co-integration relations do exist in series USO and WTI.

Similar situation is for series DBO and WTI, we also obtain the nonlinear attractor A2. Therefore, the data series USO & WTI and DBO & WTI are said to be co-integrated nonlinearly, in both whole date period and sub data periods.

5.5Error Correction Model (ECM) for the Prices of ETF and WTI

Error Correction Model (ECM) is used to model the co-integrated processes by estimating the short-term and long-term effects of Xon Y between two co-integrated series {xi} and {yj}, and the speed of Y returns to the equilibrium after a deviation occurred. Here, we will estimate the ECM with ∆USO and ∆DBO by introducing both linear and nonlinear model, and to see if the nonlinear model will better off under the nonlinear context.

To estimate the ECM model, we are going to use the MARS approach. The reason to choose MARS approach is because the MARS procedure is powerful to detect and fit models in situation where there are distinct breaks in the model, such as, if there is a change of the coefficients. As evidence shows in section 3, that there is a distinct breakpoint in the data series of USO, DBO and WTI. Both ECM models will be estimated for USO&WTI and DBO&WTI over three time periods, which have been identified in 3.1.2.

5.5.1 Linear Error Correction Model (ECM) with nonlinear cointegration

With the ACE transformation of series, we obtain the long-run equilibrium for USO & WTI and DBO & WTI. Therefore, in the linear ECM model, we will include the nonlinear co-integration residuals instead of the residuals of the linear combination of (yt – βˆzt).

The linear form of ECM for USO&WTI:

ΔUSOt = α01 + αusoq̂1t-1 + Σ α11(i)ΔUSOt-i + Σ α12(i)ΔWTIt-i + εyt (5.1)

ΔWTIt = α02 + αwtiq̂1t-1+ Σ α21(i)ΔUSOt-i + Σ α22(i)ΔWTIt-i + εzt (5.2)

Where q̂1t-1 is the residuals of the ACE algorithm q1t = g(USOt) - h(WTIt) at the time t-1, αuso and αwti are the speed of adjustment.

In equation (5.1) and (5.2), αuso, αwti are the parameters to adjust the change of USO and WTI in response to the previous period’s deviation from long-term equilibrium g(USOt) - h(WTIt).

Similarly, for DBO&WTI, the ECM in linear form:

ΔDBOt = α01 + αdboq̂2t-1+ Σ α11(i)ΔDBOt-i + Σ α12(i)ΔWTIt-i + εyt

ΔWTIt = α02 + αwtiq̂2t-1+ Σ α21(i)ΔDBOt-i + Σ α22(i)ΔWTIt-i + εzt

And q^2t-1 is the residuals of q2t = g(DBOt) - h(WTIt)

5.5.2 Nonlinear Error Correction– MARS

Multivariate adaptive regression splines (MARS) approach is a procedure to describe nonlinear relationship between the response variable and set of explanatory variables by defining the spline knots, which are breakpoints or changes in a model coefficient. MARS can be written in the form of

y = α + c1(x – τ*)+ - c2(τ* - x)+ + e

where Ï„* is the knot point, (. )+ is the right truncated spline function which takes the maximum value on max(0, (. )).

MARS model also can identify the complex nonlinear interactions between variables. An interaction model for y = f(x,z) can be written

y = α + c1(x – τ1*)+ - c2(τ1* - x)+ + c3(x – τ1*)+ (z – τ2*)+ + e

when x >τ1* and z >τ2*, y = α + c1(x – τ1*)+c3(x – τ1*)(z – τ2*)+ e,

when either x <τ1* or z <τ2*, or both x <τ1*, z <τ2*, the interaction term c3(x – τ1*)+ (z – τ2*)+ equals 0.

Friedman introduced the MARS approach in 1991. Stokes (2005) explained in detail how to perform MARS method by using B34S ProSeries Econometric System and SCA WorkBench.

MARS approach is such a powerful data mining methodology that it has extensive and increasing applications in different fields, such as macro economy, finance and social science.

In our case to study the dynamic relationship between the prices of ETFs and the oil price WTI, I will involve the same variables in MARS ECM model as those of in the linear ECM models. For example, when to estimate the ECM for USO & WTI, in equation (1), the left hand-side variable is ΔUSOt, and right-hand side variables include q̂1t-1, lags in ΔUSOt and lags in ΔWTIt, equation(2) the left hand-side variable is ΔWTIt, and right-hand side variables q̂1t-1, lags in ΔUSOt and lags in ΔWTIt. Then we can compare the results of Error Correction Model by using linear and nonlinear models.

5.5.3 Results of Error Correction Model

(I) Linear Model

To estimate the Error Correction Model, I use one lag in the model for both linear and nonlinear ECM over three sample periods, as the residuals appear to be clean with these lags. The models are as follows:

ΔUSOt = α01 + αuso(1)q̂1t-1 + α11(1)ΔUSOt-1 + α12(1)ΔWTIt-1 + εyt

ΔWTIt = α02 + αwti(1)q̂1t-1 + α21(1)ΔUSOt-1 + α22(1)ΔWTIt-1 + εzt

ΔDBOt = α10 + αdbo(2)q̂2t-1 + α11(2)ΔDBOt-1 + α12(2)ΔWTIt-1 + εyt

ΔWTIt = α20 + αwti(2)q̂2t-1 + α21(2)ΔDBOt-1 + α22(2)ΔWTIt-1 + εzt

Estimates for linear models are shown in table 5.2.

Table 5.3 _ Coefficients for Linear ECM models for sample (I) (II) (III).

Sample

(I)

(II)

(III)

 

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

 

ΔUSOt

ΔWTIt

ΔUSOt

ΔWTIt

ΔUSOt

ΔWTIt

α

-0.0224

0.0362

-0.0578

-0.0312

0.0039

0.0862

 

(-0.70)

(0.70)

(-0.93)

(-0.36)

(0.14)

(1.43)

qt-1

-0.0004

-0.0051

-0.0257

-0.0615***

0.0325**

0.0574**

 

(-0.16)

(-1.39)

(-1.54)

(-2.64)

(2.50)

(2.01)

ΔUSOt-1

-0.105*

0.3137***

-0.137

0.3445***

0.0008

0.7636***

 

(-1.91)

(3.55)

(-1.60)

(2.89)

(0.01)

(3.88)

ΔWTIt-1

0.039

-0.2097***

0.0541

-0.3016***

-0.0125

-0.3116***

 

(1.13)

(-3.82)

(0.88)

(-3.55)

(-0.31)

(-3.50)

 

1/5/2007 - 3/31/2012

1/5/2007 - 12/31/2008

1/2/2009 - 3/31/2012

 

ΔDBOt

ΔWTIt

ΔDBOt

ΔWTIt

ΔDBOt

ΔWTIt

α

0.0047

0.0586

-0.0125

-0.037

0.011

0.0706

 

(0.27)

(1.02)

(-0.35)

(-0.34)

(0.65)

(1.17)

qt-1

-0.0037

-0.0611**

-0.1728***

-1.2525***

0.0358**

0.0288

 

(-0.50)

(-2.50)

(-3.34)

(-8.00)

(2.28)

(0.51)

ΔDBOt-1

-0.1439***

0.4637***

-0.2626***

0.2875

0.0861

0.7559***

 

(-2.64)

(2.58)

(-3.18)

(1.15)

(1.10)

(2.71)

ΔWTIt-1

0.0362**

-0.162***

0.0583**

-0.2315***

-0.0265

-0.1781**

 

(2.20)

(-2.98)

(2.22)

(-2.91)

(-1.21)

(-2.28)

*p < 0.1, **p < 0.05, ***p < 0.01

The qt is the deviation from previous period’s equilibrium. The prices of USO and WTI changed in response to the previous period’s deviation by the speed of adjustment parameters αuso and αwti.

αuso and αwti shows the negative sign for the whole time period in sample (1) and sub-period I in sample (2) for ∆USOt. But for sample (3), these parameters are positive and significant. And in sample (1), αusois -0.0004 and αwti is -0.0051, which are quite small and not significant. That means the response of the price of USO and WTI to the previous period’s deviation from the long term equilibrium is not notable. But in sample (2), αwti is significant and sample (3), both αuso and αwti are significant. As explained at the beginning, that the ETF USO is constructed to track the performance of WTI, but results of data analysis show that price of crude oil is not the exogenous as expected in the long-term model.

The reason why the parameters for the previous period’s deviation show opposite signs and different behaviors for sample (1) sample (2) and sample (3) is mainly due to the structure break in the data.

Learn from previous discussion, there is a structure break in the whole data from 4/12/2006 to 3/31/2012, and the break point is exactly at the time of 2008 financial crisis. During that time, the crude oil price experienced dramatic drop, and it began to recover gradually since year of 2009. But for both crude oil market and stock market, it turned to be more volatile and unpredictable after the financial crisis due to the economy uncertainties. Since the ETF of USO consists of listed crude oil futures and other oil related futures, and tries to follow the performance of crude oil, it’s reasonable to find that the price of USO and WTI in sample (3) (which is from 1/2/2009 – 3/31/2012, after the financial crisis period) has the different movement toward / away the equilibrium when the market is greatly volatile.

From Table 5.3, the significant t-statistics in model ∆WTIt over three sample periods suggest that crude oil prices are respond to changes of previous period in USO and WTI, which confirms the analysis of response to deviation from the long term equilibrium, that crude oil price will be respond to lag of both USO price and WTI price itself. We will discuss further in next section by using Impulse Response Functions.

Similarly for ∆DBOt, in the full sample (1/5/2007 – 3/31/2012) and the sample period from 1/5/2007 to 12/31/2008 , the speed of adjustment parameters αdboand αwti gives the negative sign to suggest the prices of DBO converge to the equilibrium. But in the sample that represents the time period after 2008 financial crisis, α shows positive sign, indicating the deviation from the equilibrium. This confirms the explanation that different sign of speed of adjustment parameters in full sample and two sub samples is due to the structure change in the data and the unstable market condition during the time period for sample (3).

(II) MARS model

Table 5.3 shows the result of estimating the ECM by using MARS.

Table 5.3 _ Coefficients for ECM models using MARS for sample (I) (II) (III).

(I)

(II)

(III)

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

ΔUSOt

Slope

 ΔWTIt

Slope

 ΔUSOt

Slope

 ΔWTIt

Slope

 ΔUSOt

Slope

 ΔWTIt

Slope

ΔUSO{1} < -3.58

0.640

ΔUSO{1} > -3.58

0.374

ΔUSO{1} < -3.58

0.700

ΔWTI{1} > -1.31

-0.817

ΔWTI{1} > 3.34

-0.149

ΔUSO{1} > -1.73

1.029

 

(3.27)

 

(3.92)

 

(2.67)

 

(-5.92)

 

(-2.57)

 

(4.90)

ΔWTI{1} > -3.19

-7.931

ΔUSO{1} > 1.34

-0.440

ΔWTI{1} > -1.31

-0.324

q̂t-1< -8.84

2.766

q̂t-1< -2.95

0.294

ΔWTI{1} > 3.34

1.784

 

(-2.98)

 

(-2.14)

 

(-2.86)

 

(4.96)

 

(2.32)

 

(4.58)

ΔWTI{1} > -2.77

2.343

ΔWTI{1} > 4.53

-5.904

ΔWTI{1} > -4.57

0.230

 

 

ΔWTI{1} < 3.34

-0.595

 

(2.29)

 

(-3.88)

 

(2.47)

 

 

 

 

 

(-2.37)

ΔWTI{1} > -3.46

5.555

ΔWTI{1} > 4.13

5.284

 

 

 

 

ΔWTI{1} > -2.64

16.991

 

-3.28

 

(3.54)

 

 

 

 

 

 

 

(3.56)

 

ΔWTI{1} > -2.10

-0.247

 

 

 

 

ΔWTI{1} > -2.47

-9.595

 

 

 

(-3.64)

 

 

 

 

 

 

 

(-3.03)

 

 

 

 

 

 

ΔWTI{1} > -3.04

-8.483

 

 

 

 

 

 

 

 

 

 

 

(-4.26)

 

 

 

 

 

 

q̂t-1< -4.27

-0.716

 

 

 

 

 

 

 

 

 

 

 

(-3.28)

1/5/2007 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

ΔDBOt 

Slope

ΔWTIt 

Slope

 ΔDBOt

Slope

 ΔWTIt

Slope

 ΔDBOt

Slope

 ΔWTIt

Slope

ΔDBO{1} > 1.51

-1.281

ΔWTI{1} > 4.65

-4.262

ΔDBO{1} < -1.65

1.393

ΔDBO{1} < -1.65

1.986

ΔWTI{1} > 3.34

0.249

ΔDBO{1} > -3.03

1.037

 

(-3.67)

 

(-3.82)

 

(3.63)

 

(2.12)

 

(3.20)

 

(3.86)

ΔDBO{1} < 1.51

0.873

ΔWTI{1} > 4.15

3.365

ΔDBO{1} > -1.25

-0.165

ΔWTI{1} > 4.18

-0.944

q̂t-1> -2.02

6.175

ΔWTI{1} > 3.34

1.099

 

(3.29)

 

(3.24)

 

(-3.10)

 

(-6.97)

 

(3.81)

 

(3.57)

ΔDBO{1} > -1.65

0.823

q̂t-1> 4.26

-0.270

ΔWTI{1} < -0.24

-0.151

q̂t-1> -0.91

-0.476

q̂t-1> -2.32

-2.323

ΔWTI{1} > -2.64

13.252

 

(3.09)

 

(-2.62)

 

(-3.47)

 

(-2.49)

 

(-3.00)

 

(2.88)

ΔWTI{1} > -2.21

-0.106

 

q̂t-1< -0.78

0.531

q̂t-1< -0.91

5.863

q̂t-1> -1.80

-3.840

ΔWTI{1} > -2.47

-8.105

 

(-3.36)

 

 

 

(2.85)

 

(10.30)

 

(-4.24)

 

(-2.55)

 

 

q̂t-1> 0.26

-0.260

q̂t-1> 1.35

-2.270

 

ΔWTI{1} > -3.04

-5.478

 

 

 

 

 

(-2.91)

 

(-2.66)

 

 

 

(-3.36)

 

 

 

 

 

 

q̂t-1> -2.02

4.545

 

 

 

 

 

 

 

 

 

 

 

(2.89)

 

 

 

 

 

 

q̂t-1> -1.80

-4.604

 

 

 

 

 

 

 

 

 

 

 

(-2.88)

The results of MARS are similar to that of the linear models, but more in detail. In ΔUSOt and ΔWTIt model, the t-statistics are not significant for the error correction term αq̂t-1 in sample (1), but both are significant in sample (3) at knot -2.95 and -4.27, respectively. And also, in sample (2) the αwti is significant, which are the same as in linear models.

For the ETF and crude oil price response to previous period changes, we get more detail information by MARS model. As we know, the MARS function is to use the combination of the truncated spline functions to estimate the model, and the knots represent the potential thresholds in the independent variables. In linear model, the response of ΔUSOt is not significant to lag(1) price of USO and WTI for all three samples. By using MARS model, ΔUSOt is respond significantly to previous change of price in USO at knot of -3.58, in WTI at knot -3.19, -2.77 and -3.46 for sample (1), knot of -3.58 in ΔUSOt-1,-1.31 and -4.57 in ΔWTIt-1 for sample (2), and knot at 3.34 in ΔWTIt-1 for sample (3). The MARS model captures more information in the relationship of change of price in USO and WTI to the previous period price changes.

As for DBO, the t-statistics for deviation from the previous equilibrium q̂t-1 is not significant in sample (1) for ΔDBO, but significant for both ΔDBO and ΔWTI in sample (2) & (3), which are almost the same as the results from linear model, except for ΔWTI in sample (3). In linear model, it is insignificant, but significant by MARS model. It is because MARS estimate the model flexibly in different threshold levels, which is more advanced than the regular linear model. The similar situation is in sample (3) for ΔDBO. In linear model estimation, price change of DBO is not respond significant to the previous period change of DBO and WTI. But by using MARS model, the response of ΔDBO is statistically significant to ΔWTI at knot of 3.34.

Therefore, it’s not a surprise to see that the RSS for MARS will be better than in linear model.

Table 5.4 shows the RSS for linear and nonlinear (MARS) models.

Table 5.4 Residual Sum of Squared for OLS and MARS

 

 

RSS

 

Sample (I)

Sample (II)

Sample (III)

 

 

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

ΔUSOt

OLS

2287.2

1773.87

490.26

MARS

2260.42

1755.99

487.30

ΔWTIt

OLS

5903.64

3431.93

2368.96

MARS

5752.56

3259.45

2264.47

 

 

1/5/2007 - 3/31/2012

1/5/2007 - 12/31/2008

1/2/2009 - 3/31/2012

ΔDBOt

OLS

515.48

311.74

188.66

MARS

510.10

301.75

182.76

ΔWTIt

OLS

5635.68

2861.17

2402.00

PPREG

5504.67

2375.90

2313.07

6 Impulse Response Function (IRF)

In section 5, when estimating the ECM model, we find out that the feedbacks between the ETF and WTI is not one-way, but two-ways. The prices of ETF and WTI are both respond to the previous period price of ETF and WTI. To confirm this, we will use Granger (1969) to test for Granger Causality between change in price of the ETF and WTI. And further to use the estimates from the Impulse Response Function (IRF) to understand more about the interaction between the oil ETF and crude oil price.

Here yt represents the price of an ETF, xt is the price change in crude oil price. Granger (1969) said that only the stationary series will be involved in the test, as in non-stationary case, the existence of causality may alter over time.

xt is said to cause yt if in the model:

yt = α + Σγiyt-i+ Σδjxt-j + et(6.1)

xt-j contains information which is statistically significant to predict the value of Δyt.

To test for Granger causality, we use the VAR model of yt and xt with lag 6

Φ(B) = (6.2)

Equation (6.1) can be written as:

(1 - ΣγiBi) yt = α + ΣδjBixt + et (6.3)

B is the backshift operator defined as Bixt ≡ xt-i

Given Σ|γi| < ∞, which means (1 - ΣγiBi ) is invertible, (6.3) can be written as:

yt = xt + et (6.4)

(6.4) is called the impulse response function (IRF).

Box, Jenkins and Reinsel (2008) mentioned that for IRF, it restricts there is no feedback from yt to xt.

In our study, we found that feedbacks exist in the model from ETF to WTI in Error Correction Model estimations. Alternatively, we will transform the VAR to vector moving average form (VMA) to estimate the impulse response in the shocks, as VMA allows the measurement of shock going in two ways.

(6.2) can be transformed in form of VMA model:

=Θ(B) (6.5)

Θ(B) measures the dynamic responses of change in price of ETF or WTI to a shock in the model.

(6.5) can be expanded to

= (6.6)

In detail, θ12(B) measures the effect of shocks in ETFs on WTI and θ21(B) measures the effect of shocks in WTI on ETFs.

6.1 Results of Granger Causality Test

Table 6.1 reports the results of Granger Causality test for ETFs USO & DBO and the underlying crude oil price WTI.

The results suggest that the price of ETFs and the price of WTI are inter-related. In sample (1), the significance of WTI Granger-Causality of ETFs is 1.000 for USO and 0.998 for DBO, and the significance that USO and DBO Granger-Cause the WTI is 1.000 and 0.999, respectively. In sample (2), the situations are similar, only the effect of WTI to granger cause the DBO is 0.942, which is a little bit under 95% level. But in sample (3), there is an interesting point for both USO and DBO. The significance of F test for WTI Granger-Causality of USO is 0.558, DBO is 0.864, which are not significant. As we know, the price of USO and DBO are tracking the performance of WTI, so the price of WTI should have impact on the price of ETFs. As it happened in sample (3), it is just during the period after the 2008 financial crisis, we explain this phenomenon as because the market is highly volatile and unpredictable, which weaken the impact of crude oil on the price of ETFs.

The results indicate that the feedbacks between ETFs and WTI are not only from WTI to ETFs, but also from ETFs to WTI, therefore the transfer function model is not proper to be used. We will use VAR model and Impulse Response Function to obtain the information of the dynamic relationship between the oil ETF USO & DBO with the benchmark oil price WTI.

Table 6.1 Granger Causality F test for ETFs and WTI

Sample

(I)

(II)

(III)

 

4/12/2006 - 3/31/2012

4/12/2006 - 12/31/2008

1/2/2009 - 3/31/2012

 

USOt

WTIt

USOt

WTIt

USOt

WTIt

USOt

22059.5

3.932

376.4

3.803

537.8

1.005

 

(1.000)

(0.999)

(1.000)

(0.999)

(1.000)

(0.558)

WTIt

4.806

8808.3

4.269

339.3

2.970

2350.2

 

(1.000)

(1.000)

(0.999)

(1.000)

(0.999)

(1.000)

 

1/5/2007 - 3/31/2012

1/5/2007 - 12/31/2008

1/2/2009 - 3/31/2012

 

DBOt

WTIt

DBOt