Determinants and underlying processes of the oil price

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Master Thesis

Determinants and underlying processes

of the oil price

Author: Christov Kleijn1 Supervisor: Prof. dr. C.G.H. Diks Second Reader: Prof. dr. F.R. Kleibergen September 17, 2015 Abstract

Oil price movements can be caused by shocks on the financial markets, natural events or political tensions. Whatever the cause is, impacts can be large, making it hard to build a model that has predictive power for future oil prices. This thesis is concerned with detecting causal relationships between the oil price, Eurodollar exchange rate, interest rate and the S&P 500 index. Regime switching models are used to distinguish periods with different types of dynamics, and subsamples are regarded for the same reason. The results are related to the events described above.

1_{Student number: 10383476}

Study program: Econometrics

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Acknowledgements

I would like to express my gratitude to two persons. My supervisor, Cees Diks, has been extremely helpful and provided me with many insights throughout the entire process. Some of the remarks he made in the beginning only reached me after a few months, leaving me with the suspicion that he knew where this thesis was going from the start. Also, I am especially grateful for being able to run the most computationally intensive program on his computer. This program, and that is where the second person comes in, was sent to me by Roland Langrock, who developed (together with a few co-authors) the new MS-GAM model. He warned me that it would be a ’bit of a pain’ to get it running, but that as soon as I had figured that out it would be easy for me to adapt the code according to my needs. In the end I managed to get everything working, although I had to make more changes to the program than I imagined when reading those words for the first time. I also thank Roland Langrock for allowing me to ask questions about both the code and the model, a privilege that I gratefully used a few times.

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1 Introduction

The price of oil is known to be of major importance for economic agents, which was brought to attention again with the recent price fall, starting in June 2014. The conse-quences of this price fall were widespread. There were immediate and very noticeable consequences such as a large price drop of fuel, which has an immediate impact on con-sumers’ budgets. At the same time airline companies announced expected price drops in 2016, due to delays caused by long-term contracts. Because crude oil and petroleum products that are made of crude oil are used in many industries, one can imagine that the consequences of an oil price shock go deeper than consumers’ wallets. A lot of re-search has been done, for example, on the relationships between oil price shocks and unemployment rates (e.g. Hamilton (1983)). Following Hamilton’s lead, Carruth et al. (1998) found a cointegrating relationship between unemployment, real interest rate and real oil prices. Even political situations influence the oil price and vice versa. When a revolution took place in Iran in 1979 this led to production difficulties in Iran. As a result oil companies started to increase their levels of stock, thereby driving the price up, which led to the second oil crisis.

Because of the many consequences, it is of importance to study the relationship between the oil price and its determinants. Prices of commodities normally depend on supply and demand. The supply is a powerful tool to influence prices that has been extensively used by the Organization of the Petroleum Exporting Countries (OPEC), es-pecially because oil demand is known to be very inelastic to price changes (e.g. (Cooper, 2003)). This inelasticity is, however, based on ceteris paribus conditions. When chang-ing certain macroeconomic conditions, such as an expensive dollar or high interest rates, the demand might become less inelastic. For Euro countries an expensive dollar is bad news in terms of American oil imports, as the prices are quoted in dollars. Turhan et al. (2014) found that since the Iraq war in 2003 oil and exchange rates have become more negatively correlated. Also, a high interest rate is bad news for all parties on the demand side, as many parties borrow money in order to buy oil (Arora and Tanner, 2013). Therefore, both of these factors can cause indirect increases in oil price, even if the direct price of oil remains unchanged. In this thesis West Texas Intermediate (WTI) crude oil prices are used, which is a benchmark for oil prices in the US, the world’s top consumer of oil. Thus, a strong dollar not only influences demand coming from Europe, but also the US demand, as importing oil becomes cheaper.

This thesis will focus on identifying the relationships between the oil price, the Eu-rodollar exchange rate, the interest rate and the S&P 500 index, and their dependence on macroeconomic conditions on one hand, and on forecasting directions of change on the other hand. The S&P 500 index was added later, when it was found to be a confounding variable. The dynamic relationships between oil and each of these variables have been extensively studied, but to the best of my knowledge these three explaining variables have not yet been used simultaneously. The relationships are studied by looking at the log-returns of the variables, which has the advantage that the remaining time series are stationary.

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We have used 20-years daily data, which are split up into four equally sized subsam-ples. The differences between the subsamples allow us to draw conclusions based on the differences in performance of models among the subsamples. Also, we look at potential differences between nominal and real data. A variety of models has been used in order to capture the dynamics between the time series. Those models will be compared based on in-sample and out-of-sample performance.

In periods covering a few years, as is the case with our five-year subsamples, many things can happen that may cause a structural change in the behavior of the oil price movements. This can range from political decisions to technological advances, each of which will influence the oil price in its own way. From a more behavioral point of view, it is known that traders act differently in times of busts than in times of booms. Therefore, regardless of the cause, an oil price getting below a certain lower threshold will lead to different dynamics than when the oil price exceeds a certain upper threshold. This phenomenon is known as the volatility smile, which states that bad news has more impact than good news, and is a common feature of financial time series. Instead of using the actual price of oil, oil price returns will be used because those series turn out to be stationary, a very convenient (if not necessary) feature for performing econometric analyses. A drawback of this approach is that it is hard to draw conclusions on absolute oil price levels. However, as we are more interested in the (direction and magnitude of) price movements than it’s actual levels this drawback is not of major importance.

To be able to handle structural shocks, we will consider regime-switching models. In general there are two types of regime-switching models, models that consider endogenous regime switching and models that consider exogenous regime switching. Endogenous regime switching occurs when one of the variables in the model attains a value above or below a certain threshold. Exogenous regime switching, on the other hand, is an

unobservable process that transitions from one state into another. The comparison

of endogenous and exogenous regime-switching models can lead to conclusions on the type of shocks that have most influence. Next to a variety of well established models a new model is used, the Markov-Switching Generalized Additive Model (Langrock et al., 2014), which is an extension of the Generalized Additive Model (GAM) framework (Wood, 2006) that is more flexible as it allows for multiple Markov regimes.

The remaning part of this thesis is organized as follows. In Section 2 the methods and models used are described in detail. In Section 3 the used data is described together with important statistical properties. Section 4 presents the results of the models, and Section 5 concludes.

2 Methods and preliminary results

2.1 Model specification

2.1.1 Augmented Dickey-Fuller test

The Augmented Dickey-Fuller (ADF) test as proposed in Dickey and Said (1984) is an extension to the classical Dickey-Fuller test that tests for the presence of a unit root in

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a time series rather than in an autoregressive model. When a unit root is present in a time series the time series are not stationary. For econometric purposes it is usually

convenient to work with stationary time series. This, amongst other things, avoids

spurious regression results. To test for the presence of a unit root the ADF test uses the regression yt= ct+ βyt−1+ p−1 X i=1 φi∆yt−i+ t. (1)

With H0 : β = 1 and Ha : β < 1, which is tested by performing a regular t-test on the

estimated value of β. Table 1 clearly shows that the null hypothesis of the presence of a unit root can not be rejected for all of the log-series, whereas it can be rejected for the first differences of the log-series. Therefore from now on the first-differences (FD’s) of the log-series will be used. Note that the value of the ADF statistic has an upper bound equal to zero, as the upper bound of β is equal to one. The test rejects for values that are too far below zero.

Table 1: ADF test statistics with p-values

Data type Oil FX IR SP

Log-series -2.05(0.5575) -1.94 (0.6038) -1.19(0.68) -2.61(0.3195)

FD’s -74.25 (0.0001) -72.01 (0.0001) -71.29 (0.0001) -77.22 (0.001)

2.1.2 Confounding variables

When investigating relationships between variables it is of great importance to take all possible confounders into account, in order to avoid spurious results. That is, to test for variables that influence both the dependent variable and the covariates. Originally my intention was to use the exchange rate and the interest rate as covariates and, of course, the oil price as dependent variable. There are two factors considered that might influence all three of these variables, namely the US GDP and the gold price. However, as there are no daily data available for the US GDP the S&P 500 index is used instead, as stock markets are generally considered to be closely related to GDP (e.g. Levine and Zervos, 2006).

An accompanying advantage of using the S&P 500 index is the fact that the crude oil prices are nowadays largely influenced by the financial markets and in particular the futures market Carollo (2012). Although WTI futures are traded on the NYMEX it is not unlikely that the S&P 500 index is influenced more by this than the US GDP.

Unlike the case with instrumental variables there is no straightforward test procedure for confounding variables. A widely used rule of thumb states that a variable can be seen as a confounder if the regression results change by more than 10% when the confounder is added as covariate. Table 2 gives the results of the first equation of a VAR model where the dependent variable is the log oil return. Including S&P 500 (SP) changes the

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coefficients of the other variables and the value of R-squared by more than 10% and is thus considered to be a confounding variable. Additionally including gold barely changes the estimates of the coefficients of the other variables and the value of R-squared. We thus conclude that gold is not a confounding variable and will not include it in the rest of the analysis.

Table 2: VAR results

Included variables Oil(-1) FX(-1) IR(-1) SP(-1) Gold(-1) R2

Oil,FX,IR -0.004 0.007 0.002 - - 0.0005

Oil,FX,IR,SP -0.034 0.068 0.014 0.114 - 0.004

Oil,FX,IR,Gold -0.032 0.054 0.013 0.113 -0.024 0.004

2.1.3 Johansen cointegration test

Testing for cointegration serves two goals. First, it is a misspecification test of the VAR model. If cointegration is detected a vector error correction model (VECM) should be used instead of a VAR model in log differences, as the latter does not capture the long-term cointegration relations . Second, cointegration tells something about the relation-ship between variables. Johansen (1991) developed a procedure to test for cointegration in a VAR model. The Johansen test for a VAR(1) model starts by rewriting the VAR(1) model:

∆rt= Πrt−1+ at, (2)

where Π = Φ1− I and at is the k × 1 error vector. The trace version of the test tests

each of the hypotheses for r cointegrating relationships Hr : rankΠ ≤ r which is done

by calculating the LR statistic

LRr = −T

k X

i=r+1

log(1 − ˆλi), (3)

where ˆλi denotes the i-th of the ordered eigenvalues of Π. The LR statistic has an

asymptotic Dickey-Fuller distribution under the null hypothesis. The results are given in Table 3 and show that there are no cointegrating relationships.

Because the test indicates there are no cointegrating relationships at the 0.05 level we proceed by using a VAR model.

2.1.4 Ramsey’s RESET test for nonlinearity

The RESET test (Ramsey, 1969) can be used to test for misspecification of the classical linear model. Therefore it can serve as a test that indicates whether it is of use to include nonlinear terms in the model. Consider the normal linear regression:

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Table 3: Johansen test results

r LR statistic critical value p-value

0 20.52 40.17 0.88

1 9.74 24.28 0.87

2 2.66 12.32 0.89

3 0.19 4.13 0.91

where y is the dependent variable and X a T x k matrix of explaining variables. The

RESET test obtains fitted values Xtβ of regression (1) and then runs the regression:ˆ

yt= Xtβ + γ1(Xtβ)ˆ 2+ · · · + γm(Xtβ)ˆ m+ t, (5)

where m denotes the maximum power value used in the regression. An F-test is used

to test the null hypothesis that the γi’s (i = 1, . . . , m) are equal to zero. Although this

test helps to identify the presence of nonlinear effects it has no power in identifying the type of nonlinear effects and will therefore not be used for that purpose. The RESET test clearly rejects the null hypothesis of nonlinearity (F=34.32, p=0.000) and indicates some possibly significant nonlinear terms for powers of 3, 5 and 7.

2.1.5 Ljung-Box test for serial correlation

The Ljung-Box test (Box and Ljung, 1978) for serial correlation uses the sample lag l

autocorrelation ρl to compute the test statistic

Qm= T (T + 2) m X l=1 ρ2_l T − l, (6)

where m denotes the number of lags for which is tested. Under the null hypothesis of

no serial correlation: ρ1 = ρ2= · · · = ρm= 0 the statistic has a chi-squared distribution

with m degrees of freedom.

2.1.6 Jarque-Bera test for normality

The Jarque-Bera test (Bera and Jarque, 1981) statistic uses two properties of normally distributed random variables, namely that the skewness is zero and the kurtosis is three. Letting S be the sample skewness and K the sample kurtosis of a series the Jarque-Bera statistic is computed as J B = S 2 6/T + (K − 3)2 24/T , (7)

which is, under the null hypothesis of normality, distributed as a chi-squared random variable with 2 degrees of freedom.

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2.2 Vector autoregressive model

Vector autoregressive (VAR) models are a multivariate generalization of autoregressive (AR) models and are widely used for modeling financial time series. The main advantage

of VAR models is that they allow for bi-directional causal relationships. A general

VAR(p) model as in Tsay (2009) can be written as

rt= φ0+ p X j=1 Φjrt−j+ at (8) or, equivalently rt= φ0+ Φ(L)rt+ at, (9)

where rt = (roilt , rF Xt , rIRt , rSPt ) is a 4-dimensional vector of return series, φ0 is a

4-dimensional vector of intercepts, Φ is a 4 x p matrix of coefficients, L is the lag operator

and at is a sequence of serially uncorrelated random vectors with mean zero and

covari-ance matrix Σ. A crucial aspect of fitting VAR models is the lag selection, i.e. selecting the value of p. This is done by using the Akaike Information Criterion (AIC), which measures the tradeoff between goodness of fit and the number of estimated parameters. The AIC is calculated as

AIC = 2q − 2L, (10)

where L is the value of the maximized log-likelihood and q is the number of estimated parameters. The appropriate number of lags is then determined by choosing the value p that minimizes the AIC.

2.3 Nonlinear models

2.3.1 Threshold vector autoregressive model

The threshold vector autoregressive (TVAR) model is motivated by nonlinear charac-teristics of data, and was proposed by Tsay (1998) as an extension to the univariate TAR model. Based on the preliminary VAR results we estimate the following TVAR(1) model:

rt=

Φ1rt−1+ a1t if st≤ γ

Φ2rt−1+ a2t if st> γ,

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which simply specifies one VAR model for each regime. Here stis the threshold variable,

γ is the threshold value and the subscripts 1 and 2 denote the regimes. Note that constants have not been included because these turned out to be insignificant after

estimation. Throughout this thesis the threshold variable st that is chosen is poilt−1,

i.e. the lagged oil price, where st is required to be stationary (Tsay, 2009). Since this

threshold variable is used some prefer to speak of a so called self-exciting TVAR model. The threshold γ is estimated such that the total sum of squares of the two regimes is minimized. The most important advantage of threshold models is that these models can be robust against structural breaks in the data. A disadvantage is that the combined result does not lead to a continuous regression function. As the threshold variable comes from within the model, it assumes endogenous regime switching.

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2.3.2 Exponential smooth transition autoregressive model

To avoid discontinuity problems caused by the threshold used in the TVAR model, vector smooth transition autoregressive (VSTAR) models replace the threshold by a smooth transition function. There are two main types, the exponential smooth transition function (leading to the EVSTAR model) and the logistic smooth transition function (leading to the LVSTAR model). The EVSTAR model seems more appropriate because it’s general structure is such that it distinguishes between a high and a low regime, where the terms high and low are based on the amount of volatility. That is, a EVSTAR model generally estimates a model in which the high regime consists of observations that are

large in absolute value. The model as in Ter¨asvirta and Yang (2014) with one lag has

been used and is as follows:

rt= Φ1rt−1G(sit, γi, κi) + Φ

0

1rt−1(1 − G(sit, γi, κi)) + at, (12)

where Φ0₁ denotes the matrix under the alternative regime, sit is the threshold variable

used for variable i, γi is the estimated threshold for equation i and κi is the transition

speed. The transition function G for the univariate case is defined as

G(st, γ, κ) = 1 − exp[−κ(st− γ)2], (13)

which is a function on [0,1]. Thus, the function value can be seen as the probability

that the system is in the first regime and rtis forecasted as a weighted average of those

regime probabilities.

2.3.3 Generalized additive model

The generalized additive model (GAM) framework has been introduced by Hastie and Tibshirani (1990) as a generalized linear model with an additive structure. We use the following GAMs as in Wood (2006), modified to our purposes:

yt= β0+ f1(x1,t) + f2(x2,t) + f3(x3,t) + t (14)

yt= β0+ f0(yt−1) + f1(x1,t−1) + f2(x2,t−1) + f3(x3,t−1) + t, (15)

where (10) is estimated to investigate the dynamics between the variables and (11) is estimated to evaluate the model’s forecasting performance. The functions f are smooth functions that are generally based on cubic B-splines (see Wood (2006)). One of its attractive features is that the smooth functions f show the nonlinear conditional rela-tionships between the dependent variable and the covariates. When such a conditional relationship shows large differences between certain intervals this can be used as input for regime switching models.

2.3.4 Markov-Switching generalized additive model

Langrock et al. (2014) propose the Markov-switching GAM (MS-GAM) as a very flexible new model that combines the advantages of the GAM framework with Markov

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regime-switching. The model basically specifies one GAM for each Markov state, i.e. y(st) t ∼ N (µ (st) t , σ2st) (16) µ(st) t = β (st) 0 + f (st) 1 (yt−1) + f2(st)(x1,t−1) + f3(st)(x2,t−1) + f4(st)(x3,t−1). (17) For a detailed description of the estimation procedure see Langrock et al. (2014). Here instead we will focus on the computation of the state probabilities, which is used for the computation of the fitted and predicted values. It is necessary to assume a distribution

of the dependent variable to be able to compute the state probabilities ξ_tj that give the

probability of being in state j at time t. Next to state probabilities the Markov chain has transition probabilities that give the probability of going from state i ∈ 1, 2 to state

j ∈ 1, 2. We will denote the transition probabilities γij. Note that the sum γi1+ γi2

equals one, hence the transition matrix Γ looks like: Γ = γ11 1 − γ11 1 − γ22 γ22 . (18)

Following Hamilton (1989) we compute the state probabilities in the following way. Let

η_tj be the value of the density of state j evaluated in yt and let ζt be the combined

conditional density. That is

ζt= ηt1(ξt1γ11+ ξt2(1 − γ22)) + ηt2(ξ2tγ22+ ξ1t(1 − γ11)), (19)

which follows from basic probability theory. Finally, the probability of being in state j at time t + 1 is computed as

ξ_t+1j = ηj_t(γjjξtj+ (1 − γii)ξti)/ζt, (20)

where i 6= j. The probabilities are initialized as

ξ1₀ = 1 − γ22 2 − γ11− γ22 (21) ξ₀2 = 1 − γ11 2 − γ11− γ22 . (22)

Ideally values of γ11 and γ22 are higher than 0.8. If lower estimates are obtained there

might be identifiability problems, making it hard to distinguish between the two Markov

states. The starting values of γ11and γ22are not of major importance, although it seems

logical to set γ11 = γ22 to avoid initial biasedness of the system in favor of one of the

two states. Estimation is done by finding the parameter vector that maximizes the log-likelihood, where the parameter vector consists of all the smoothing terms, the con-ditional mean and variance and the transition probabilities. The state probabilities are not a part of the parameter vector as they can be computed using only inputs from

the parameter vector. To avoid singularity problems the oil returns yt are multiplied

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known and is used to obtain unbiased forecasts in the following way. The forecast ˆyt is

computed as a weighted average of µ1_t and µ2_t, with the weights being used equal to the

state probabilities.

2.4 Model performance measures

2.4.1 Magnitude measures

Common measures to evaluate a model’s forecasting performance are the root mean squared error (RMSE) and the mean absolute error (MAE). Defining the error vector e

as the difference between the observed value y and the forecasted value ˆy these measures

are computed as:

RM SE = v u u t 1 T T X t=1 e2_t (23) M AE = 1 T T X t=1 |e_t|. (24)

For most error vectors the RMSE and MAE will point to the same conclusion. However, using both can lead to insights when relating the error measures to data properties such as the mean and standard deviation.

2.4.2 Direction measures

We follow the approach described in Tsay (2009) to test whether a model’s direction forecasts are significantly better than the direction forecasts of a random walk model with equal upward and downward probabilities. This approach uses the following contingency table, containing ”hits” and ”misses”.

Table 4: Direction forecast contingency table

Actual / Predicted Up Down Total

Up m11 m12 m10

Down m21 m22 m20

Total m01 m02 m

Where m is the total number of forecasts made, m11 is the number of correctly

predicted upward movements, m22 the number of correctly predicted downward

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of wrongly predicted downward movements. The test statistic χ2= 2 X i=1 2 X j=1 (mij− mi0m0j/m)2 mi0m0j/m d − → χ2(1) (25)

is used to evaluate the null hypothesis that the random walk model provides forecasts that are at least just as good as those of the model used. Throughout the entire thesis a significance level of 0.05 is used. Thus, if the p-value is below 0.05 the null hypothesis is rejected. Returns in the data that are equal to zero are adjusted such that they become negative or positive with equal probability, as the returns are discrete (measured in tick sizes) and therefore it is likely that they have become zero because of rounding procedures. The number of degrees of freedom is equal to 1, which is common procedure in a 2 x 2 contingency table.

Within the data the number of positive returns is not necessarily equal to the num-ber negative returns. Since the test described above only tests whether the used model outperforms the random walk with equal probabilities for upward and downward move-ments, it is useful to construct an additional test that uses variables

Pu =

number of positive returns

total number of returns (26)

and

Pd=

number of negative returns

total number of returns . (27)

Defining the success rate SR as

SR = m11 + m22

m , (28)

we can now conduct a t-test to see if SR is significantly larger than the fraction of correctly predicted movements from the random walk that predicts an upward movement

with probability Pu. Using that the fraction of correct predictions P0 of this model will

be equal to the maximum of Pu and Pd we get the following test statistic

t = SR − P0

pP0(1 − P0)/N

d −

→ N (0, 1). (29)

Note that this test only needs to be conducted in the case that SR is greater than the

maximum of Pu and Pd.

2.4.3 Information criteria

Many models are estimated by maximizing their corresponding log-likelihood. The AIC and Bayesion information criterium (BIC) are measures that give a penalty to the log-likelihood value based on the number of parameters. The definition of the AIC has

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already been discussed (see 2.2). The BIC is computed in a similar manner, but uses an extra input.

BIC = q log(T ) − 2L, (30)

where as before L is the value of the maximized log-likelihood and q is the number of estimated parameters. Instead of a factor 2, as the AIC uses, the BIC multiplies this by the log of the number of observations. Typically this number is greater than 2 so the BIC punishes models harder for including extra parameters. Because of the structure of the AIC and BIC we prefer models with the lowest AIC and BIC values.

3 Data

3.1 Background

We have used daily data from 1995-2014. The total number of trading days in that time period is equal to 5218, hence the total number of observations N = 5218. When performing out-of-sample forecasts the last two years are left out. That is, the models are estimated on all data from 1995-2012, and the daily values of 2013 and 2014 are forecasted. This yields a total of 521 forecasts. As said, we used the West Texas Inter-mediate (WTI) crude oil price, which is commonly used in the literature as a benchmark for oil prices in the United States. Despite the fact that the Eurodollar exchange rate is traded continuously there is a daily close value available, which is measured at 22:00 UTC. Interest rates are not available in daily form, so we used (as is also common in the literature) the ten year US government bond yields as an approximation for the interest rates. Finally, the S&P 500 daily close values have been used. As has been pointed out, we have distinguished between nominal and real data. The real data are corrected for the appropriate (consumer) price indices in the following way. The oil price and S&P index have been corrected by dividing by the consumer price index. The real exchange

rate is obtained as follows. Let F Xtbe the nominal value at time t, then the real value

F X_treal= F Xt∗

CP I_teu

CP I_tus. (31)

Where CP I_teu and CP I_tus are the consumer price indices at time t in the Eurozone and

the US respectively. Finally, the real interest rate is calculated by using the Fisher equation, which states that the real interest rate is the nominal interest rate minus the inflation rate (Fisher, 1977). It is worth noting that there are only monthly consumer price index and inflation rate data available. The interpolated values have been used as a reasonable estimate of the daily values. Furthermore, to avoid problems with taking logs of negative real interest rates the interest rates have been transformed such that the minimum real interest rate is equal to 1.

3.2 Properties of the data

All the log-series were found to be integrated of order 1. In Figure 3 it can be seen that the structure of the interest rate series differs from the other three in terms of volatility.

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Table 5 shows some descriptive statistics of all the log return series. It is noticeable that the kurtosis of the IR series is approximately 23 times higher than that of the SP series. As is confirmed by the graph, the volatility of the IR series is, in general, much lower than that of the other three series which can be explained by the fact that the government bond yields are not traded on the financial markets. For some models it is

Table 5: Descriptive Statistics nominal log return series

Series mean standard deviation skewness kurtosis max min

Oil 0.0002 0.0237 -0.2020 8.3263 0.1641 -0.1709

FX 0.0000 0.0060 -0.1879 5.6491 0.0384 -0.0462

IR -0.0002 0.0118 -0.7126 256.0206 0.2929 -0.2911

S&P 0.0003 0.0120 -0.2480 11.5060 0.1096 -0.0947

necessary to assume a distribution of the dependent variable. It can be seen that it is highly unlikely that the oil returns are normally distributed as the excess kurtosis is too high. However, due to the lack of an appropriate alternative we will, when necessary, assume that the oil return series has a normal distribution with mean and variance taken from the sample.

Table 6: Correlation matrix real data

Oil Oil(-1) FX(-1) IR(-1) SP(-1)

Oil 1.000 -0.026 0.019 -0.013 0.054

Oil(-1) -0.026 1.000 -0.137 0.007 0.154

FX(-1) 0.019 -0.137 1.000 0.040 -0.032

IR(-1) -0.013 0.007 0.040 1.000 0.077

SP(-1) 0.054 0.154 -0.032 0.077 1.000

In Table 6 the correlation values are given for oil and all explaining variables. The common property of mean reversion is confirmed by the negative correlation between oil and it’s lagged values. The signs of the contemporaneous correlations are as expected. It is noticeable that the correlation between IR(-1) and oil is bigger in absolute value than that of IR(-1) and oil(-1). As stated earlier, rising interest rates make it more expensive to buy oil for parties that borrow money, therefore causing lower demand. This correlation matrix raises the hypothesis that there is a delayed effect of interest rates on oil demand.

When considering macroeconomic conditions, a useful part of the initial analysis is to see on which dates the largest values in the data are observed. It appears that the five largest values and five smallest values are observed in only three of the twenty years, namely 1998, 2001 and 2008, with the largest price drop taking place in 2001. This happened only a few days after the 9-11 terrorist attacks, that is known to have caused major turbulence on the financial markets. Also 1998 and 2008 have been turbulent

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years, where in 1998 a long period of stable growth in the oil industry came to an end and 2008 was the start of the financial crisis. It can be expected that models have difficulty in minimizing errors in such periods, where volatility is high and sometimes caused by political events.

4 Results

Before presenting the results, a few comments are appropriate. The differences between subsamples will be discussed, but the graphic and numerical output can be found in the Appendix. In this section we will focus on the results of using the whole sample instead. Also, there is very little difference in the results obtained by using nominal data and the results obtained by using real data, especially when considering forecasting performance. In such cases the results obtained by using nominal data can also be found in the Appendix. Therefore when discussing forecasting performances only the results obtained by using real data will be displayed.

4.1 Relationships between variables

The relationships between the variables are investigated by comparing the visual GAM results to the VAR coefficients. Figure 1 shows the conditional relationships that have been estimated by estimating (14).

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Figure 1: GAM whole sample (a) Real data

-0.04 -0.02 0.00 0.02 0.04 -0 .1 5 -0 .0 5 0.05 FX s(F X, 28 .9 4) -0.4 -0.2 0.0 0.2 0.4 0.6 -0 .1 5 -0 .0 5 0.05 IR s(I R ,7 .8 ) -0.10 -0.05 0.00 0.05 0.10 -0 .1 5 -0 .0 5 0.05 SP s(SP, 39 .7 6) (b) Nominal data -0.04 -0.02 0.00 0.02 0.04 -0 .1 0 -0 .0 5 0.00 0.05 FX s(F X, 22 .8 9) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0 .1 0 -0 .0 5 0.00 0.05 IR s(I R ,1 5. 59 ) -0.10 -0.05 0.00 0.05 0.10 -0 .1 0 -0 .0 5 0.00 0.05 s(SP, 3. 77 ) 16

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The first thing that is worth noting is that there barely is a difference between real data and nominal data. Only for IR it seems that in case of nominal data this variable is not significant, as the confidence bounds are wide and the line is a bit fickle. This is indeed confirmed by the numerical output of the GAM estimation. Apart from that we can see that there is in general a negative relationship between oil and FX, and a positive relationship between oil and SP, as we already expected. Interestingly, for large values of FX the oil returns nosedive towards very low values. It is however unclear whether that is significant, as there are only a handful FX returns observed that are greater than 0.03, where the oil returns start to drop. Table 7 give the estimated VAR coefficients of the first VAR equation.

Table 7: VAR results

Data Oil(-1) FX(-1) IR(-1) SP(-1)

Nominal -0.02903 0.07411 0.01641 0.11758**

Real -0.03276* 0.06818 -0.03381 0.12015**

*: significant at 0.05 level **: significant at 0.01 level

We can see a few things in this table. First, only the estimated SP coefficient is significant in both cases, the estimated oil coefficient is only significant for real data and the other two are not significant in either case. The significance of SP coincides with the correlation matrix in Table 6, as there SP showed the largest absolute correlation with oil. The property of mean reversion is hidden in the table as the estimated oil(-1) coefficients are negative. From the GAM estimation we have already seen that there is a positive relationship between oil and SP. This table shows that the lagged value of SP has a positive influence on oil, which also coincides with the correlation matrix.

4.2 Forecasting performance

The tables below give the in sample and out of sample performances for the models that have been used.

Table 8: In sample model performances for real data

model RMSE MAE BIC AIC SR

VAR 0.024 0.017 -27607 -27810 0.51(0.85,0.97)

TVAR 0.024 0.017 -28676 -28791 0.51(0.97,0.87)

EVSTAR 0.024 0.017 -24215 -24287 0.50(0.17,0.99)

GAM 0.024 0.017 -24049 -24313 0.53(0.00,0.18)

MS-GAM 0.023 0.016 -42144 -42325 0.53(0.00,0.10)

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Table 9: Out of sample model performances for real data

model RMSE MAE BIC SR

VAR 0.014 0.010 350.23 0.413(NaN)

TVAR 0.020 0.016 -149.86 0.512(0.43)

EVSTAR 0.014 0.010 13.58 0.514(0.49)

GAM 0.015 0.010 -65.77 0.474(0.94)

MS-GAM 0.014 0.010 -120.82 0.496(0.27)

p-values of independence test are in parentheses

For in sample forecasting all models perform very similar, with MS-GAM performing slightly better. It must be noted that the sample standard deviation of the whole sample is also 0.024, so the RMSE of all model forecasts is (approximately) equal to one time the standard deviation. The information criteria prefer the MS-GAM model, as both values are by far the smallest. As for the success rate, we see that both the GAM and MS-GAM succeed in predicting directions of change significantly better than the random walk. However, they do not outperform the naive forecast of always predicting the direction that was observed most often. Note that the p-value of the independence test of the VAR model forecasts cannot be calculated, which is due to the fact that the model predicts every downward change correctly and every upward change wrongly. Out of sample the differences in performance between the models are greater, where again the MS-GAM seems to perform best. However, none of the models significantly outperforms the random walk in forecasting directions of change, which is consisted with Hamilton (2009).

4.3 Regime switching

We have seen that the models that assume two regimes perform (slightly) better for both in sample and out of sample forecasts. In this section we will compare the results of the regime switching models, by looking at a turbulent period in the data: the last 7 months in 2008. In June 2008 the oil prices hit a record high. Six months later, they were down almost 100$. The volatility in this period was about twice as large as the volatility of the whole sample. The threshold that was estimated by the TVAR for the whole sample was 0.026, while for sample 3 (that included 2008) it was 0.027. The threshold estimated by the EVSTAR was 0.011 for the whole sample and 0.008 for sample 3, where for sample 3 it was not significantly different from zero. In general both models estimate very different thresholds, where normally the EVSTAR model estimates a threshold closer to the mean of the data. Given that this model uses a smooth transition function this does not seem surprising.

In Figure 2 the state probabilities are plotted that have been estimated by the EVS-TAR and the MS-GAM respectively. For both models the red line corresponds to the high regime and the blue line corresponds to the low regime. It is clear that the

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EVS-Figure 2: State probabilities (a) EVSTAR 0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0 function v alue Transition function (b) MS-GAM 0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0 time Probability

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TAR model has some more problems in identifying the appropriate regime in the last few months 2008, where the MS-GAM model sees some switches in the first few months (consisting of June, July and August). We see that the state probabilities are larger than 0.2 (or 0.8) relatively often, indicating that there is a lot of insecurity.

When looking at the whole sample the models all three estimate one dominant and one less dominant regime, where here dominant means that the system is more often in that regime. TVAR and EVSTAR estimate the frequency of the dominant regime at 90% while MS-GAM is a bit more careful and estimates it at 85%. It must be noted however that Markov-switching models estimate regimes in a completely different way. The estimation is done based on an assumed distribution of the data, and conditional on this distribution the estimation is done. TVAR and EVSTAR on the other hand estimate the threshold that minimizes the sum of squares. As has already been noted, the latter approach is endogenous whereas the MS-GAM assumes exogenous regime switching.

A natural hypothesis that comes with this difference is that the TVAR and EVSTAR models are better able to minimize errors in times of oil price shocks that are caused by endogenous factors, while MS-GAM should be better able to minimize errors in times of exogenous shocks, such as the hurricane Katrina. To see how the different models perform we compare the RMSE for three turbulent periods, of which one was known to be caused by an exogenous shock. We compare the RMSE for the last few months of 2005, 2008 and 2014. The last months of 2014 featured rapidly declining oil prices and interest rates, and rapidly inclining Eurodollar rates and S&P 500 index, and in 2005 hurricane Katrina reached the shore in the final days of August. The RMSE of the periods are given in Table 10.

Table 10: Regime switching model performances in turbulent periods

Period TVAR EVSTAR MS-GAM

2005 0.020 0.020 0.036

2008 0.047 0.046 0.057

2014 0.023 0.023 0.084

The MS-GAM model performs significantly worse than the other two models. There are two notes to be placed on this result. First, it is hard to distinguish between endoge-nous and exogeendoge-nous shocks with the bare eye. There could be identifiability problems and there for sure is imperfect information in the market. Second, the large difference could be caused by the way the MS-GAM forecasts are computed. Recall that they are established as a weighted average of the forecasts of the two regimes. Langrock et al. (2014) instead use the Viterbi algorithm to calculate the most likely state sequence and take the forecast of the most likely state for each time point. This approach might work better for periods where the state probabilities do not indicate clear regimes.

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4.4 Residual analysis

All model residuals have been tested on serial correlation and normality. With no ex-ception the tests indicate for each model in each sample for each type of data that there is no serial correlation in the residuals and that they are not normally distributed. Plots of the autocorrelation functions can be found in the Appendix.

5 Conclusions

We have seen that generally there is no significant difference in using nominal or real data, especially for forecasting. This indicates that correcting the data for price levels is of no benefit for forecasting purposes.

There is some difference between the subsamples. Especially sample 2 and 3 lead to larger forecasting errors. As these periods are the most turbulent ones this is unsur-prising. Sample 2, ranging from 2000 up until 2004 included the strike of the employers of the largest Venezuelan oil company in 2002, which caused a huge drop in oil prices as Venezuela is one of the OPEC members. After this strike, the production has never returned to its previous levels. Also, it suffered from the 9/11 terrorist attacks and the subsequent Iraq war in 2003. Sample 3 on the other hand included the effects of the hurricane Katrina, laying down almost the entire oil production in the Southern golf coast of the US. Next to that there was the turbulent year 2008, in which the latest financial crisis started. Wars in Iraq and Afghanistan influenced production levels and then a phenomenon that is commonly observed in the oil industry was set in motion, namely buying a lot to increase the stocks. This obviously drives the price up, leading it to be a self-fulfilling prophecy.

What distinguishes oil from other commodities is that there are (or were, although other forms of energy remain expensive) no satisfying alternatives. In other words, a political or natural event that causes a shock in production easily leads to panic. Needless to say that emotional (or irrational) behavior of economic agents is the main cause of bubbles. The models find it harder to make forecasts for these periods. It does seem however that the GAM performs better in terms of direction forecasts in those periods than in less volatile times. It would be interesting to investigate the relationship between the performance of the GAM and the volatility of a time series. When considering out of sample performance we have seen that it is indeed hard to beat the random walk in forecasting oil returns. The MS-GAM model provides very interesting directions for future research, as is shown by the results. It is however very computationally intensive. For comparison, estimating any of the other four models on the whole sample is done in a few seconds while estimating the MS-GAM takes about 9 hours. For this purpose the benefits are not large enough to justify such a computation time. However, the model is still being developed and therefore also the code is subject to a lot of changes in the foreseeable future.

A few relationships between the oil returns and covariates have been detected. It is shown that the S&P 500 index has a positive influence on the oil returns. By using the

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conditional relationship feature of the GAM we have also seen that if there are increases in the Eurodollar exchange rate, the oil return decreases. This was especially clear for very large increases in the exchange rate, to which the oil price very negatively responds. For such cases it can be of interest to employ a Value-at-Risk type of framework, in which extreme percentiles are treated different than observations that are near the mean. The models that employ two regimes outperform the models that assume one regime, indicating that the use of multiple regime models is beneficial.

At this point there is, to my knowledge, not a clear procedure for testing how many regimes (exogenous or endogenous) are dictated by a given time series. It would be very useful if it would be possible to start the model specification by performing such a test on the data, just like model specification often starts with conducting an ADF test. A final potentially interesting direction of future research for the MS-GAM is to explore the difference between computing unbiased weighted average forecasts and the use of the Viterbi algorithm for the most likely state sequence. Such research could indicate that depending on, for example, the volatility or extreme values in the data one approach might be preferred over the other.

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6 Appendix

Figure 3: Real data figures (a) Oil 1995 2000 2005 2010 2015 2.5 3.0 3.5 4.0 4.5 5.0 year pri ce

Real log oil prices

(b) FX 1995 2000 2005 2010 2015 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0.0 0.1 0.2 year exch an ge ra te

Real log Eurodollar rates

(c) IR 1995 2000 2005 2010 2015 0.0 0.5 1.0 1.5 2.0 year in te re st ra te

Real log interest rates

(d) SP 1995 2000 2005 2010 2015 6.0 6.5 7.0 7.5 year in de x va lu e

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Figure 4: Nominal data figures (a) Oil 1995 2000 2005 2010 2015 2.5 3.0 3.5 4.0 4.5 5.0 year pri ce

Nominal log oil prices

(b) FX 1995 2000 2005 2010 2015 -0 .4 -0 .3 -0 .2 -0 .1 0.0 0.1 0.2 year exch an ge ra te

Nominal log Eurodollar rates

(c) IR 1995 2000 2005 2010 2015 0.0 0.5 1.0 1.5 2.0 year in te re st ra te

Nominal log interest rates

(d) SP 1995 2000 2005 2010 2015 6.5 7.0 7.5 year in de x va lu e

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Table 11: VAR results real data

Sample Oil(-1) FX(-1) IR(-1) SP(-1)

1 -0.00745 0.04335 -0.03300 0.04375

2 -0.03240 0.10207 -0.23767* 0.06129

3 -0.04807 0.13365 -0.06698 0.22427**

4 -0.0789* 0.04361 0.02880 0.13944*

Whole -0.03276* 0.06818 -0.03381 0.12015**

Table 12: Model performances for all subsamples real data

Model RMSE MAE BIC AIC SR threshold

Sample 1 VAR 0.023 0.016 -5983 -6066 0.52(0.21,0.42)) TVAR 0.023 0.016 -5861 -6032 0.54(0.00,0.02) 0.015 EVSTAR 0.023 0.016 -6045 -6102 0.53(0.04,0.14) -0.009 GAM 0.023 0.016 -6056 -6087 0.53(0.12,0.23) Sample 2 VAR 0.025 0.019 -5771 -5854 0.53(0.05,0.52) TVAR 0.025 0.018 -5651 -5822 0.54(0.00,0.18) -0.013 EVSTAR 0.025 0.018 -5801 -5858 0.53(0.93,0.51) 0.017 GAM 0.025 0.018 -5561 -5846 0.58(0.00,0.00) Sample 3 VAR 0.027 0.019 -5593 -5676 0.51(0.81,0.62) TVAR 0.027 0.019 -5663 -5834 0.53(0.03,0.32) 0.027 EVSTAR 0.027 0.019 -5642 -5699 0.54(0.00,0.23) -0.011* GAM 0.025 0.018 -5290 -5743 0.58(0.00,0.00) Sample 4 VAR 0.017 0.012 -6799 -6882 0.49(0.51,0.89) TVAR 0.017 0.012 -7033 -7204 0.53(0.02,0.17) -0.019 EVSTAR 0.017 0.012 -6844 -6901 0.52(0.07,0.26) 0.013 GAM 0.017 0.012 -6819 -6903 0.50(0.40,0.80)

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Table 13: Model performances for all subsamples nominal data

Model RMSE MAE BIC AIC SR threshold

Sample 1 VAR 0.023 0.016 -5983 -6066 0.51(0.54,0.62) -TVAR 0.023 0.016 -5863 -6034 0.53(0.01,0.09) 0.014 EVSTAR 0.023 0.016 -6046 -6103 0.51(0.80,0.68) -0.01 GAM 0.023 0.016 -6056 -6087 0.52(0.11,0.17) -Sample 2 VAR 0.025 0.019 -5765 -5848 0.53(0.02,0.49) TVAR 0.025 0.019 -5645 -5816 0.52(0.96,0.68) -0.013 EVSTAR 0.025 0.019 -5799 -5856 0.52(0.97,0.53) 0.021 GAM 0.025 0.018 -5718 -5863 0.57(0.00,0.00) Sample 3 VAR 0.027 0.019 -5589 -5672 0.51(0.81,0.62) TVAR 0.027 0.019 -5667 -5838 0.53(0.03,0.32) 0.027 EVSTAR 0.027 0.019 -5642 -5699 0.54(0.00,0.23) -0.011* GAM 0.025 0.018 -5290 -5743 0.58(0.00,0.00) Sample 4 VAR 0.017 0.012 -6797 -6880 0.48(0.76,0.99) TVAR 0.017 0.012 -7031 -7202 0.53(0.02,0.13) -0.019 EVSTAR 0.017 0.012 -6845 -6901 0.52(0.10,0.30) 0.014 GAM 0.017 0.012 -6818 -6904 0.52(0.06,0.47)

The * indicates that the estimated value was not significant. Benchmarks: Sample 1: 51.1% positive. Sample 2: 52.6% positive. Sample 3: 52.4% positive. Sample 4: 51.7% positive.

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Table 14: VAR results nominal data

Sample Oil(-1) FX(-1) IR(-1) SP(-1)

1 -0.00769 0.045686 -0.001903 0.044441

2 -0.03173 0.09941 -0.20616** 0.06591

3 -0.05246 0.12913 0.10514 0.19842**

4 -0.0789* 0.04444 0.01713 0.13949*

Whole -0.02903 0.07411 0.01641 0.11758**

Table 15: In sample model performances for nominal data

model RMSE MAE BIC AIC SR

VAR 0.024 0.017 -24145 -24250 0.51(0.98,0.84)

TVAR 0.024 0.017 -24226 -24442 0.51(0.91,0.89)

EVSTAR 0.024 0.017 -24214 -24268 0.50(0.39,0.98)

GAM 0.023 0.016 -23733 -24284 0.54(0.00,0.00)

MS-GAM 0.023 0.016 -22194 -22397 0.53(0.00,0.06)

p-values of independence and t-test are in parentheses

Table 16: Out of sample model performances for nominal data

model RMSE MAE BIC SR

VAR 0.014 0.010 401.56 0.24(0.30)

TVAR 0.020 0.016 -110.68 0.48 (0.77)

ESTAR 0.014 0.010 18.87 0.47(0.79)

GAM 0.015 0.010 -40.52 0.51(0.65)

MS-GAM 0.015 0.010 -93.51 0.491(0.27)

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Figure 5: ACF of residuals of the models (a) VAR 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_var (b) TVAR 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_var (c) EVSTAR 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_nls (d) GAM 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_gam (e) MS-GAM 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF Series 1

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Figure 6: ACF of residuals of the models (a) VAR 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_var (b) TVAR 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_tvar (c) EVSTAR 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_nls (d) GAM 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag AC F Series residuals_gam (e) MS-GAM 0 10 20 30 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF Series 1

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Figure 7: GAM sample 1 (a) Real data

-0.02 -0.01 0.00 0.01 0.02 -0 .0 2 -0 .0 1 0.00 0.01 FX s(F X, 1) -0.10 -0.05 0.00 0.05 0.10 -0 .0 2 -0 .0 1 0.00 0.01 IR s(I R ,1 ) -0.06 -0.02 0.02 -0 .0 2 -0 .0 1 0.00 0.01 SP s(SP, 1) (b) Nominal data -0.02 -0.01 0.00 0.01 0.02 -0 .0 2 0.00 0.01 0.02 FX s(F X, 1) -0.10 -0.05 0.00 0.05 0.10 -0 .0 2 0.00 0.01 0.02 IR s(I R ,1 ) -0.06 -0.02 0.02 -0 .0 2 0.00 0.01 0.02 SP s(SP, 1)

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-0.03 -0.01 0.00 0.01 0.02 -0 .0 6 -0 .0 2 0.02 FX s(F X, 3. 06 ) -0.1 0.0 0.1 0.2 -0 .0 6 -0 .0 2 0.02 IR s(I R ,2 .4 4) -0.06 -0.02 0.02 0.04 0.06 -0 .0 6 -0 .0 2 0.02 SP s(SP, 1. 8) (b) Nominal data -0.03 -0.01 0.00 0.01 0.02 -0 .6 -0 .2 0.2 0.6 FX s(F X, 2. 7) -0.10 -0.05 0.00 0.05 0.10 -0 .6 -0 .2 0.2 0.6 IR s(I R ,4 3. 9) -0.06 -0.02 0.02 0.04 0.06 -0 .6 -0 .2 0.2 0.6 SP s(SP, 2. 15 )

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-0.04 -0.02 0.00 0.02 0.04 -0 .1 5 -0 .0 5 0.05 0.15 FX s(F X, 35 .2 5) -0.4 -0.2 0.0 0.2 0.4 0.6 -0 .1 5 -0 .0 5 0.05 0.15 IR s(I R ,1 ) -0.10 -0.05 0.00 0.05 0.10 -0 .1 5 -0 .0 5 0.05 0.15 SP s(SP, 38 .1 3) (b) Nominal data -0.04 -0.02 0.00 0.02 0.04 -0 .1 5 -0 .0 5 0.05 0.15 FX s(F X, 33 .4 2) -0.20 -0.10 0.00 0.10 -0 .1 5 -0 .0 5 0.05 0.15 IR s(I R ,8 .7 1) -0.10 -0.05 0.00 0.05 0.10 -0 .1 5 -0 .0 5 0.05 0.15 s(SP, 38 .6 1) 34

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-0.02 -0.01 0.00 0.01 0.02 -0 .1 0 -0 .0 5 0.00 0.05 FX s(F X, 1. 01 ) -0.4 -0.2 0.0 0.2 -0 .1 0 -0 .0 5 0.00 0.05 IR s(I R ,9 .7 9) -0.06 -0.02 0.02 0.04 -0 .1 0 -0 .0 5 0.00 0.05 SP s(SP, 28 .6 8) (b) Nominal data -0.02 -0.01 0.00 0.01 0.02 -0 .1 0 -0 .0 5 0.00 FX s(F X, 1. 6) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0 .1 0 -0 .0 5 0.00 IR s(I R ,6 .7 6) -0.06 -0.02 0.02 0.04 -0 .1 0 -0 .0 5 0.00 s(SP, 25 .9 6) 35

Determinants and underlying processes of the oil price

Master Thesis