Forecasting energy prices : an analysis of two different approaches using time-varying parameter models to predict energy prices

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University of Amsterdam

Faculty of economics and Business, Amsterdam School of Economics

Forecasting energy prices

an analysis of two different approaches using time-varying parameter models to predict energy prices

Name: Youp van Steensel Student Number: 10786732

Subject: Time-varying parameter models for financial time series Course: Bachelor Thesis Econometrics

Group: 1

Thesis Supervisor: Prof. dr. C. Diks Date: December 22nd, 2017

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Statement of originality

This document is written by Student Youp van Steensel who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the super-vision of completion of the work, not for the contents.

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1 Introduction

The ever-changing energy prices have a crucial impact on numerous coun-tries worldwide. Now more than ever, predicting these future energy prices has become increasingly valuable for macroeconomic applications. Energy price fluctuations have important implications for economic growth, policy decisions, and future inflation.

Therefore, it is not surprising that much research has been done to obtain accurate forecasting models for energy prices. Still, predicting these prices remains challenging. In financial time series modelling, such as for energy prices, one often would like to accommodate for possible time varia-tion in parameters, especially for time series in which structural breaks are indispuTable. Time-varying parameter (TVP) models can capture unpre-dicTable shifts, when the persistence of a time series is changing over time. For instance, coefficients of the parameters in the model may be related to economic conditions, which vary throughout the years.

In this research, time series of natural gas prices, as well as crude oil prices, are being used since they play a vital role in the determination of future energy prices. The data of crude oil and natural gas may be subject to structural breaks, which becomes clear when looking at Figs 1 and 2.

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Figure 1

Figure 2

The TVP-models of the two time series mentioned above are esti-mated using two different approaches, the Rolling Window regression and the Rolling Window regression with exponentially decaying weights. The reasons behind the choice of these approaches are discussed in detail in the

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next Section.

The aim of this thesis is to assess the performance of the two dif-ferent methods in the estimation of TVP-models and the performance of their predictive ability in out-of-sample forecasts. The performance of both approaches can be compared by computing the unconditional mean and the unconditional variance of the forecasts. The root mean squared forecast error (RMSE) serves well to rank the unconditional mean and the score based approach of Giacomini and White (2006) evaluates the variance of the forecasts. Another crucial assessment in this thesis is to check whether the TVP-model forecasts indeed perform better than when no time-varying parameters are incorporated in the model.

The remaining part of this thesis is organized as follows. Section 2 contains a thorough theoretical background of the approaches used in this study. Section 3 provides information about the dataset and the formulation of the models. The estimation results of the TVP-models and statistical tests results are presented in Section 4, in which they are compared and analyzed as well. A summary and a discussion are stated in Section 5.

2 Literature Review

This Chapter further outlines the methodological choices for estimating TVP-models by discussing the literature behind it. Several studies are con-cerned with estimating time-varying coefficients, but only a few are available in the field of energy prices. Hence, it is crucial that competing estimation methods are compared. Due to the probable presence of structural breaks in the data concerning energy prices, time-varying coefficients are evident. Thus, a framework with time-varying parameters has to be considered, when forecasting energy spot prices, such as for crude oil and for natural gas (Naser, 2016).

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2.1 Rolling Window regression

The recent literature offers several approaches to implementing time-varying parameters in the models. Rolling Window regression is an obvious method to accommodate for possible time variation in the parameters. Quite com-monly, only the most recent observations are used to estimate the parameters of the model at hand. This is the case with the Rolling Window regression method. Parameter instability, which is proven to be a major concern in financial forecasting (Goyal Welsch, 2003), can be properly addressed by this method. It incorporates the Ordinary Least Squares (OLS) method at each separate window. These windows are subsamples of the total sample and make use of more recent data (Punales, 2011). Thus, it takes recent changes into account. Moreover, the Rolling Window approach is simple to interpret and to compute, even when time-varying coefficients are present in the model.

However, this approach has its limitations. In addition, the usage of shorter windows means that the data sample may exclude crucial data (Punales, 2011). As a result, the choice of the optimal window size is there-fore essential. Section 2.3 discusses this critical implication in further detail. Another limitation is the usage of the OLS estimation method. Strong assumptions are made when using OLS. For instance, the variance is con-sidered to be normally distributed, as in practice, this may not necessarily be true.

2.2 Exponential smoothing

A second method performed in this research to forecast time-varying param-eters is the Rolling Window regression with exponentially decaying weights. Intuitively, this approach has to be considered, while working with rolling windows. In essence, the smallest weight is assigned to the price the farthest

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away of of time t. Likewise, the biggest weight is assigned to the observation at time t−1. Thus, the observations of the price close to time t become more influential than observations further away from time t. For a more exten-sive analysis, four different λs are considered for the weighting scheme, 0.6, 0.9, 0.95, and 0.99 respectively. These λs behave differently, when several distinct window sizes are used.

2.3 Window size

Implementing a rolling regression approach involves a well-thought-out choice for the window size used. Based on their results, Giacomini and White (2006) state that a fixed window size is preferred over an expanding window. When misspecification of a forecasting model is the case, an expanding win-dow can give less reliable forecasts than a fixed winwin-dow size. Consequently, only a fixed window size is applied in this thesis. Choosing a suiTable win-dow size depends heavily on the structural breaks in the data. The dataset in this study consists of twenty years and thirty years of daily natural gas and crude oil spot prices, respectively. In this case, a window size of 10, 100, 250 or a 1000 may be favourable.

2.4 Forecasting model

Along with optimizing the window size for forecasting, the choice for the forecasting model is essential. The autoregressive model of order 1, the AR(1) model, serves as a benchmark model in this research. However, deal-ing with crude oil prices and natural gas prices implies the inclusion of a Random Walk model according to Hamilton (2008) and Felder (1995). In nature, the prices of these natural resources are not easy to forecast. Es-pecially predicting the price one quarter, one year, or even a decade ahead may be naive. Due to the fact that a Random Walk model assumes that the

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price moves randomly, it is an appropriate choice in this context. This can be seen from both time series graphs of the oil and gas spot prices in Figs 1 and 2.

The Random Walk model is a special case of the AR(1) model. The thesis by Hamilton (2008) shows with statistical tests that a Random Walk without drift (i.e. no intercept term) has to be considered for modelling crude oil prices. For natural gas, this is proved through research by Felder (1995).

As mentioned before, the TVP-AR(1) model is used as a benchmark model in this study. The TVP-AR(2) model is another model to enter the list of models being estimated in this thesis. This is due to a lower AIC-value of the AR(2)-model compared to the AR(1)-model, when estimating with-out time-varying parameters. On top of that, a Random Walk model with and without drift, a naive forecasting model and an ARMA(1,1)-model are compared with the two TVP-models. The inclusion of the Random Walk model follows from the literature, whereas the inclusion of the ARMA(1,1)-model follows from the computed AIC-value. This AIC-value tends to be lower than the AIC-values of the AR(1)-model as well as the AR(2)-model. A naive forecasting model is included to check whether the best forecast is that the price returns are zero at t + 1.

The TVP-models of the Rolling Window regression with and with-out exponentially decaying weights are slightly different. When regressing with exponentially decaying weights a λ is included in the model. This is explained in more detail in Section 3.2.

Due to the expected presence of structural breaks in the data concern-ing crude oil and natural gas spot prices, the model is assumed to perform better when time-varying parameters are present in the forecasting model.

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2.5 Comparison method

The main contribution of this thesis is the comparison of the two different methods implying time-varying parameters, which are estimated with sev-eral models described above. Moreover, a comparison is made between the TVP-models and the models, in which no time-varying parameters are in-cluded.

The different TVP-models can be compared over the total subsample (5714 step ahead predictions when predicting oil prices and 3222 one-step ahead predictions in the case of gas prices). However, when comparing the TVP-models with the non TVP-models, 100 steps ahead from point t are predicted for the non TVP-models. This is due to the fact that, the non TVP-models use all the observations up to time t. Larger forecast horizons than 100 steps ahead become more and more unreliable. For TVP-models, a hundred one-step ahead forecasts are compared with forecast horizon of the non TVP-models. The choice of the out-of-sample forecast region is based on the fact that one wants to forecast the most recent prices. Taking different periods in time as the out-of-sample forecast horizon, may result in different outcomes.

Two comparison methods are being used in this thesis. The first one is the RMSE, a well known measure to rank forecast accuracy. The RMSE is in most cases preferred over other measurement errors. Since this method includes squaring, large errors are penalized by assigning bigger weights to them (Chai & Draxler, 2014). However, one important assumption has to be made when using the RMSE. The variances have to be unbiased and normally distributed. What is more, Chai and Draxler (2014) state that the RMSE may be sensitive to outliers. In line with their findings, it might even be justified to ommit outliers that are exorbitantly large compared to other values in the sample. These outliers have more impact on the RMSE, when

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the data sample is limited.

As a second method the score-based approach of Giacomini and White (2006) is performed in this thesis. Their approach produces a framework for out-of-sample predictive ability and is based on research by Diebold and Mariano (2002) and West (1996). They serve as a starting point by eval-uating the accuracy of a forecasting method, which Giacomini and White (2006) improve in their study. That is, they perform forecasts based on limited memory estimators. Their method evaluates point, interval, proba-bility, and density forecasts for a general loss function and can be applied to compare forecast performances of the Rolling Window regression with and without exponentially decaying weights. The definition of this score-based approach of Giacomini and White (2006) is presented in the next Section.

All in all, Chapter 2 gave a thorough explanation about the methods used in this thesis to estimate models with time-varying parameters and two ways to assess their predictive ability. The explicit definitions of these methods are given in the next Chapter.

3 Methodology

This Chapter will discuss the models and data used for estimating the TVP-coefficients of the Rolling Window regression with and without exponentially decaying weights. On top of that, this Chapter outlines a statistical test, which can be used to assess if the data is stationary. Moreover, a detailed methodology is outlined to compare the estimation performance and the performance of the predictive ability of the out-of-sample forecasts.

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3.1 Test for stationarity

Before estimating and comparing forecasting methods, the data have to be examined, as the data should be stationary. If this is not the case, the data become unpredicTable and unable to forecast or model. For both oil prices and gas prices this can be evaluated with the Augmented Dickey-Fuller (ADF) Test. When the prices appear to follow a random walk it means that they are non stationary, which can be expected from the time series at hand. Taking first differences of the data solves this issue. Outcomes of the ADF test are discussed in Chapter 4.

3.2 Models

The forecasting model used as a benchmark in this study is the AR(1) model and is defined as

Pt = α + βPt−1+ t, t∼ N (0, σ2), (1)

in which Pt is the logarithmic spot price for crude oil or natural gas,

de-pending on which natural resource is being estimated. The α and t are the

intercept and the white noise error term, respectively. As mentioned before, a Random Walk is a special case of an AR(1) model with β = 1 and is a non stationary process. The Random Walk without drift has α = 0 and β = 1 in the model above.

As the outcomes of the ADF-test (denoted in Chapter 4) suggest that the model defined in (1) is non stationary, the variables in the model should be transformed into first differences. The model becomes

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with ∆Pt= Pt− Pt−1 and likewise ∆Pt−1= Pt−1− Pt−2.

The models of equation (1) and (2) can be estimated with the Rolling Window regression using OLS. The same holds true for the method involving Rolling Windows with exponentially decaying weights. However, there is a slight difference between the two methods. When using exponentially decaying weights a λ is introduced in the model. The value of the first observation used of the window gets multiplied by λ1999_{, whereas the value}

at t − 1 gets multiplied by a way bigger value namely λ0 = 1. To compare distinct weighting schemes, four different values for λ are taken: 0.6, 0.9, 0.95 and 0.99.

After estimating the rolling coefficients for both methods, the one-step ahead forecasts for the price differences can be computed. For example, this is done by

ˆ

∆Pt+1= ˆα + ˆβ∆Pt (3)

when estimating an AR(1)-model, in which ˆα and ˆβ are the rolling coef-ficients. Notice that the error term does not influence the one-step ahead forecast for the price difference. This is due to the fact that the error term is assumed to be white noise, in which the expectation of is equal to zero. In general, for testing the accuracy of the competing forecasting meth-ods with regards to the unconditional mean, the RMSE serves as a great measure and is given by

RM SE = v u u t 1 T T X i=1 ( ˆyi− yi)2. (4)

Giacomini and White (2006) proposed a score-based approach to compare the accuracy of different forecasting methods. The loss function is defined

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by

∆St+1= log(g(yt+1)) − log(f (yt+1)) = S1,t+1− S2,t+1, t = m + 1, ..., T + m.

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by

H0 : E(∆St+1) > 0 and H1 : E(∆St+1) < 0. (6)

Equation (6) can be viewed as the test that g has a better unconditional predictive ability than f against the alternative that it is the opposite. The worst performing forecast is g based on the RMSE and f is the better one. Thus, when the null hypothesis is rejected, f has a better score than its coun-terpart, which would be in line with the RMSE criterion. This hypothesis can be tested with the test statistic stated as

tm,T = ∆ ¯Sm,T ˆ σT/ √ T, ∆ ¯Sm,T = 1 T T +m X t=m ∆Sm,t, (7)

where ˆσT in this test statistic is the standard deviation of the ∆St+1. It is

assumed that tm,T is standard normal distributed. Thus, when |tm,T| > zα/2

holds true then the level α test rejects the null hypothesis. Zα/2with α = 5%

is equal to 1.96.

3.3 Data

Observations for the daily spot prices of Henry Hub Natural Gas as well as the Crude Oil Europe Brent FOB are retrieved from the U.S. Energy Information Administration (EIA), the nation’s premier source of energy in-formation. Only weekdays are included in the data of both energy resources. Data for the spot prices of crude oil are available for May 20, 1987 through

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October 10, 2017 and include 7716 observations. The spot prices are in US dollars per barrel. For the natural gas spot prices, the data include 5224 observations and covers the period from January 7, 1997 through October 10, 2017. The prices are given in US dollars per million Btu, a standard measure for natural gas.

4 Results & Analysis

In this Chapter the results of the statistical test as well as the estimation results of the TVP-models of Chapter 3 are reported and analyzed. On top of that, the different forecasting methods are compared.

4.1 Stationarity

Before discussing the estimation results, the data should be checked for sta-tionarity. The outcomes of the ADF-test show for both oil and gas spot prices that the null hypothesis of non stationary data cannot be rejected at a 5% significance level. For the logarithmic oil spot prices the Dickey Fuller (DF) test statistic is -2.6061 with a p-value of 0.3216. The DF test statistic for the logarithmic gas spot prices is -2.7348 and a p-value of 0.2672. As a result, the model of equation (2) is used, which implies that the variables in the model become first differences. The outcomes of the ADF-test of ∆Pt

show that the transformed data are stationary. For oil prices the DF test statistic is -18.773 with corresponding p-value 0.01 and for the gas prices the values are -18.993 and 0.01, respectively. When inspecting the Figs 4-7, one can draw the same conclusions.

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Figure 4 Figure 5

Figure 6 Figure 7

4.2 Window size

Another crucial part of the rolling estimation is the window size. On the one hand, a too small window can exclude crucial observations in time. On the other hand, when the estimation window becomes too large it may include observations, which are not relevant when estimating at time t. For comparing purposes, forecasts with window sizes of 10, 20, 50, 100, 250, 1000, and 2000 observations are included for both oil and gas prices. The results

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show that a small window is preferred over a large window of observations. A reason for this is that prices in the past seem to be irrelevant quite soon for predicting future prices. This is in line with the reasoning that both oil and gas prices have unpredicTable behaviour. The best performing window for both TVP-methods are discussed in more detail in sections 4.3 and 4.4. The window size has important implications for the data. As a window of, for example, 2000 observations is used, the first rolling beta estimate is at time t = 2001. In that way, the number of estimated ∆Pt becomes 2000

observations less. Still, enough observations remain in the sample to give a valid assessment about forecast accuracy.

4.3 Forecast results of TVP-models

4.3.1 Oil spot prices

For different window sizes as well as a stationary dependent variable, the forecast results can now be presented. First of all, the RMSEs are shown of the Rolling Window regressions with and without decaying weights of the two TVP-models in Tables 1-8.

From the total prediction sample of the first differenced oil spot prices, we can draw the following conclusions from Tables 1-8 based on the com-puted RMSEs. Table 1 shows almost identical RMSEs of the AR(1)-model and the AR(2)-model when the same window size is considered for the mod-els. The AR(2)-model is slightly preferred, as it has a lower RMSE for all distinct window sizes. The best performing window size for the AR(1)-model as well as the AR(2)-model, is when only 10 previous observations are in-cluded. The RMSEs are 0.020717 and 0.019706, respectively.

When looking at Tables 2-8, for every λ the AR(2)-model performs better than its counterpart. Both models suggest that the lower the λ, the lower the RMSE becomes. The AR(2)-model with λ = 0.6 and window size

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of a 1000 observations is the best performing TVP-model for oil prices with a RMSE of 0.009302. However, this is due to the fact that the weights are zero for almost the entire window. Thus, only the last few observations influ-ence the price predictions. This underlines the fact that a smaller window is favourable over a large window. As most of the observations are redundant for considerable windows (250,1000, and 2000 observations) when λ = 0.6, a window of 10 observations is preferred. Therefore, the RMSE (0.009888) of the AR(2)-model with λ = 0.6 is considered to be the best performing TVP-model. All the computed RMSEs of the Rolling Window method including exponentially decaying weights in Table 2, are smaller than the ones in Table 1. Consequently, the optimal Rolling Window forecasts with exponentially decaying weights performs better than without, based on the RMSE.

The second method to assess the forecasts’ predictive ability is done with the score-based approach of Giacomini and White (2006). The abso-lute value of the test statistic for the Rolling Window forecast of competing forecasts AR(1)-model (g) and AR(2)-model (f ) is 31.52038. The null hy-pothesis is easily rejected, as the test statistic has to be bigger than 1.96. This supports the fact that the AR(2)-model performs better. The absolute value of the test statistic of the Rolling Window forecasts with decaying weights with the optimal window size of 10 observations is 134.3697, when comparing the AR(2)-model and AR(1)-model with λ = 0.6. As the AR(2)-models with a window of 10 observations performed best in both forecasting methods, based on the RMSEs, it is crucial to check if the null hypothesis is rejected between those two models. The absolute value of the test statistic is 53.14091, which underlines that the AR(2)-model with λ = 0.6 is indeed better than the AR(2)-model without decaying weights.

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4.3.2 Gas spot prices

The prediction sample of the gas spot prices is smaller compared to the one of oil spot prices. As before, Tables 9-16 represent the computed RMSEs of the one-step ahead forecasts of the Rolling window with and without expo-nentially decaying weights.

Tables 9-16 show similar findings as earlier in the case of the RMSEs of the oil spot prices in Tables 1-8. The AR(2)-model performs slightly better than the AR(1)-model for predicting gas spot prices with a Rolling Window. The RMSEs 0.039064 for the AR(1)-model and 0.036136 for the AR(2)-model are the lowest of this method. The difference between those two RMSEs is somewhat larger than in Table 1.

Both the AR(1)-model and the AR(2)-model perform better when fore-cast with a Rolling Window with exponentially decaying weights, when com-paring the same window size. The RMSEs with the distinct λs follow a similar pattern as in Table 2-8. The best performing forecast, based on the RMSE, is the AR(2)-model with λ = 0.6 and a window of 2000. It has a RMSE of 0.001732. This is a significantly higher value than the RMSE of the AR(2)-model of the oil prices in Table 8. Then again, the AR(2)-model with λ = 0.6 with a window of 10 observations (Table 10) is considered to perform better than forecasting with a window of 2000 observations. The RMSE is slightly higher, 0.018475. Only a couple of previous observations with actual weights are included when forecasting with a window size of 2000 and a λ = 0.6. For this reason a short window is similar and in this case favourable.

When making use of the method of Giacomini and White (2006), the following remarks can be made. Firstly, the best performing AR(1)-model and the AR(2)-model of the Rolling Window forecasts without decaying weights are assessed to check if the null hypothesis can be rejected. As the

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absolute value of the test statistic is equal to 9.588352, the null hypothesis is rejected. Therefore, the AR(2)-model is preferred over the AR(1)-model when forecasting without decaying weights. When forecasting with Rolling Windows with exponentially decaying weights, the AR(1)-model and the AR(2)-model with the best performing window are assessed. This is, as explained before, the window of 10 observations. The outcome of the test statistic is 45.13048, which is why the null hypothesis is rejected.

Lastly, based on the RMSE, the best model of both competing fore-casting methods is the AR(2)-model. The absolute value of the test statistic is 32.21762, when comparing those two. Consequently, the best TVP-model for predicting the gas spot prices is the AR(2)-model forecast by the Rolling Window with exponentially decaying weights.

4.4 Forecast comparison between TVP-models and non

TVP-models

In this Section, the best performing TVP-model forecasts are compared with forecasting models, which do not take time-varying parameters into account. These models without time-varying parameters are the AR(1)-model, the AR(2)-model, the Random Walk model with and without drift, the naive forecast model and an ARMA(1,1)-model. As mentioned in Chapter 2, a forecast horizon of 100 observations is assessed with the RMSE. The last 100 observations of the dataset of oil spot prices and gas spot prices are predicted out-of-sample. The TVP-models use the 10 observations, the win-dow size, before time t and the non TVP-models use every observation up to time t to forecast the 100 steps ahead. Thus, the non TVP-models use 7704 observations for the oil spot prices and 5112 observations for gas spot prices are included to forecast. The RMSEs of the oil price predictions are given in Table 17 and the gas price predictions are given in Table 18.

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From Table 17, it can be concluded that the best non TVP-models are the AR(1)-model, the AR(2)-model and the ARMA(1,1)-model with RMSEs around 0.017558. The best performing TVP-models are the AR(1)-model as well as the AR(2)-AR(1)-model with λ = 0.95. This means that the TVP-models for these 100 observations predict a little less accurate than some models without the inclusion of time-varying parameters. Between the TVP-models, there are mixed results. For λ = 0.6 the Rolling Window forecast with exponentially decaying weights performs worse than without the decaying weights. The forecasts for other λs, however, perform better than the TVP-method without decaying weights.

The results of the RMSEs of the predicted gas prices in Table 18 show a similar pattern as in Table 17. Still, the best performing non TVP-models are the AR(1)-model, the AR(2)-model, and the ARMA(1,1)-model with a RMSE around 0.0222. The Rolling Window forecasts with and without exponentially decaying weight performs slightly worse than the non TVP-models for both the AR(1)-model and the AR(2)-model. When predicting gas prices for these 100 steps, the Rolling Window with exponentially de-caying weights is more accurate than without the dede-caying weights when λ = 0.9, 0.95, 0.99. The opposite is true when λ = 0.6.

5 Summary and Discussion

Predicting future energy prices, such as oil and gas prices, has become in-creasingly important, as it has a major impact on future inflation and eco-nomic growth. When fluctuations across the time in a financial time series are frequent, such as for energy prices, one could introduce time-varying parameter models to forecast these prices.

This thesis investigated two methods to incorporate these time-varying parameters. These were the Rolling Window regression with and without

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ex-ponentially decaying weights. On top of that, these two methods, including several forecasting models, was assessed for their out-of-sample predictive ability. The results of the computed RMSEs over the total prediction sam-ple of these distinct TVP-methods turned out to favour the Rolling Window forecast with exponentially decaying weights. A shorter window turned out to perform better than longer windows. In particular, the AR(2)-model with window size 10 and λ = 0.6 for both the oil as well as the gas predictions performed best. The rejected null hypothesis of the score-based approach of Giacomini and White (2006) supported this outcome.

Another comparison was made between the TVP-models and the non TVP-models over a forecast horizon of a 100 steps ahead. The prediction results showed for these 100 predictions that a couple non TVP-models per-formed slightly better than the TVP-models.

However, a few crucial remarks about these outcomes have to be made. First of all, in these research it is assumed that the variances of the esti-mated models were normally distributed for calculating purposes. This may in practice not be the case. Secondly, the out-of-sample region for the 100 steps ahead forecasts is chosen to be the last 100 observations of the dataset. Results may differ when choosing a different time period as an out-of-sample region. Lastly, there are more methods for introducing time-varying param-eters, which are not assessed in this thesis. These could improve energy price forecasts even more and is left for further research.

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Forecasting energy prices : an analysis of two different approaches using time-varying parameter models to predict energy prices

University of Amsterdam