FINANCE , STATISTICS, FORECASTING. ACCURACY TESTING AND ERROR MATRICES

International Journal of Forecasting 34 (2018) 1–16

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International Journal of Forecasting

journal homepage: www.elsevier.com/locate/ijforecast

Targeted growth rates for long-horizon crude oil price forecasts Stephen Snudden Queen’s University, Department of Economics, 94 University Ave, Kingston, ON, Canada K7L 3N6

a r t i c l e i n f o

Keywords: Forecasting and prediction methods Oil prices Filters Spectral analysis

a b s t r a c t

This paper proposes growth rate transformations with targeted lag selection in order to improve the long-horizon forecast accuracy. The method targets lower frequencies of the data that correspond to particular forecast horizons, and is applied to models of the real price of crude oil. Targeted growth rates can improve the forecast precision significantly at horizons of up to five years. For the real price of crude oil, the method can achieve a degree of accuracy up to five years ahead that previously has been achieved only at shorter horizons.

© 2017 The Author(s). Published by Elsevier B.V. on behalf of International Institute of Forecasters.

This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The existing oil price forecasting literature provides evidence that models of the real price of oil outperform the no-change forecast at short horizons (Alquist, Kilian, and Vigfusson, 2013 and Baumeister and Kilian, 2012, 2014, 2015, among others). However, these studies all focussed on horizons of less than two years and were not intended for forecasting at longer horizons. Much less is known about forecasting the real price of crude oil at longer hori- zons. This paper focuses on extending the forecasting suc- cess of individual models of the real price of oil to horizons of up to five years by proposing the method of targeted growth rate transformations.

Longer-term forecasts are of central interest for invest- ment decisions and public policy institutions. For example, if an oil producer decides to invest in drilling or an air- line company decides to purchase a new fleet of aircraft, they will care about the payoff over the lifetime of the investment. Many policy institutions produce forecasts for longer horizons in order to inform policy decisions. While some studies have focused on horizons beyond five years (Bernard, Khalaf, Kichian, & Yelou, 2017), less is known

E-mail address: [email protected].

about model performances for forecasting the real oil price at horizons of between two and five years. This paper fills this gap.

The paper proposes the method of targeted growth rate filtering, which is a modification of the standard fore- casting method. Lags in growth rate transformations are chosen in order to target lower frequencies. The method removes high frequencies and emphasizes the use of cer- tain low frequencies which correspond to particular fore- cast horizons. When applied to forecasting models of the real price of crude oil, the method improves the recursive forecast mean squared prediction error (MSPE) ratios and directional accuracy significantly at horizons of up to five years. The method of targeted growth rate transformations exhibits robust improvements in forecast performance, re- gardless of whether the real price of oil is in log levels or differences, as well as across sub-samples and for alterna- tive oil price series. Employing this method can achieve a degree of accuracy at longer horizons that previously has been achieved only at shorter horizons.

The analysis begins by considering simple univariate benchmark models for forecasting the real price of crude oil at horizons of up to five years. An attempt is made to answer the open question of benchmarks at longer hori- zons and to determine whether simple models can provide

http://dx.doi.org/10.1016/j.ijforecast.2017.07.002 0169-2070/© 2017 The Author(s). Published by Elsevier B.V. on behalf of International Institute of Forecasters. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

2 S. Snudden / International Journal of Forecasting 34 (2018) 1–16

better benchmarks than no-change forecasts. The evidence suggests that exponential smoothing and backward av- erages of the real price of oil can outperform no-change forecasts robustly for horizons beyond one year. The no- change forecast using the average real oil price over the last year provides a simple alternative rule for forecasting the real price of oil at horizons beyond one year, and works particularly well at the two- and three-year horizons.

Targeted growth rate transformations are then applied to univariate models for forecasting the real price of crude oil. This includes univariate autoregressive and fractional autoregressive models, as well as the low-frequency fore- casting technique proposed by Müller and Watson (2016). Univariate autoregressive models have been documented consistently to outperform no-change forecasts up to six months ahead (Alquist et al., 2013; Baumeister & Kilian, 2012), and serve as an intuitive way to illustrate the modi- fication of targeted growth rates to the standard univari- ate Box-Jenkins method (Box, Jenkins, & Reinsel, 2008). Applying targeted growth rate transformations to these methods can extend the out-of-sample forecast preference to longer horizons than is the case for models that rely on period-over-period growth rates. Univariate autoregres- sive models with targeted growth rate transformations can outperform no-change forecasts consistently for horizons of up to three years.

The method of targeted growth rates is also applied to multivariate forecast methods using vector autoregressive (VAR) models. Alquist et al. (2013) and Baumeister and Kil- ian (2014, 2015), among others, showed that monthly VAR models of the real price of crude oil produce forecasts that can beat the no-change forecast of the real price of crude oil for up to one year ahead. Applying targeted growth rate transformations to VARs can consistently outperform VAR models that rely on period-over-period growth rates in out-of-sample forecast performances at longer horizons.

Global real activity and global crude oil inventories are not observed directly or measured well. Various alternative series have been examined for forecasts at short hori- zons, by Baumeister and Kilian (2012, 2014) among others. The present paper is the first to extend this analysis to longer horizons by conducting a systematic investigation of alternative global real activity and crude oil inventory variables in VAR model forecasts for horizons of up to five years. Targeted growth rates applied to world industrial production and Kilian’s global real activity index (Kilian, 2009) can produce comparable forecasts at longer hori- zons. Moreover, US crude oil and petroleum inventories are found to produce superior forecasts at longer horizons. This extra predictive power of US crude oil inventory series is consistent with the improved forecast performance at short-run forecasts that was first established by Baumeis- ter, Guerin, and Kilian (2015) and used at shorter horizons by Baumeister, Kilian, and Lee (2014) and Baumeister and Kilian (2015).

This paper introduces targeted growth rate transforma- tions and is intentionally focused. For example, it ignores real-time data constraints that have been shown to be cru- cial in forecasting the real price of oil (see e.g. Alquist et al., 2013; Baumeister and Kilian, 2012). Moreover, none of the models allow for stochastic variances (see e.g. Baumeister,

Kilian, and Zhou, forthcoming). Similarly, no attempt is made to study forecast combinations (see e.g. Baumeister and Kilian, 2014, 2015). Finally, the analysis focuses on monthly data and forecast horizons. No quarterly models or forecasts are discussed (see e.g. Baumeister and Kilian, 2014, 2015). These extensions are exciting avenues for further research on targeted growth rate transformations.

The remainder of the paper is structured as follows. Sec- tion 2 introduces the method of targeted growth rate trans- formations for forecasting using spectral analysis. Section 3 analyzes the application of these methods to univariate real oil price forecasts. Section 4 extends the analysis to vector autoregression forecasts of the market for crude oil and examines their robustness to the use of alternative oil price series. Section 5 concludes.

2. Growth rate filter

Growth rate data transformations are applied com- monly in time series econometrics. For example, simple period-over-period growth rates are used to achieve sta- tionarity. Higher lags in growth rates are often used to produce more intuitive scales. For example, due to their intuitive format, macroeconomic data are often announced as year-over-year or quarter-on-quarter growth rates. This section provides the intuition behind targeted growth rate transformations and shows why it may be desirable to depart from the standard first lag when forecasting.

Let Y be a covariance-stationary series with absolutely summable autocovariances. Let sY (ω) be the population spectrum and gY (κ) the autocovariance generating func- tion of Y , where

sY (ω) = (2π) −1gY (e

−iω), (1)

and ω is the frequency with ω ∈ (0, π), and κ is a complex scalar. Let X be a transformation of Y given by X = h(L)Y , with

∑ ∞

−∞ |hj| < ∞. The autocovariance generating func-

tion of X is known to be calculated from Y by:

gX (κ) = h(κ)h(κ −1)gY (κ). (2)

The population spectrum of X is:

sX (ω) = (2π) −1h(e−iω)h(eiω)gY (ω), (3)

and hence, the population spectrum of X is related to that of Y by:

sX (ω) = h(e −iω)h(eiω)sY (ω). (4)

Thus, applying the h(L) filter to Y is the same as multiplying the spectrum of Y by h(e−iω)h(eiω). The original series must be covariance stationary in order for sY (ω) to exist. Oth- erwise, the population spectrum is not zero at frequency zero.

The growth rate of Y can be approximated by applying the first difference filter to the log of Y . Hence, h(e−ω) = 1 − L, and h(e−iω)h(eiω) is given by (1 − eiω)(1 − e−iω) = 2 − 2 cos(ω). Define the operator LZ such that LZ xt = xt−Z for Z ∈ R. More generally, transforming the logged variable Y using the difference on the Zth lag, (1 − LZ ), to produce

S. Snudden / International Journal of Forecasting 34 (2018) 1–16 3

Fig. 1. Spectral density of the growth rate filter at Z: time domain.

the spectrum of X, is equivalent to filtering the spectrum of Y by

h(e−izω)h(eizω) = 2 − 2 cos(Zω). (5)

Fig. 1 graphs the growth rate filter applied to the spectral density of Y using Z = 1, 2, 3 and 6, respectively. The frequency ω is converted into the period of a cyclical function T using T = 2π /ω. The growth rate filter using the first lag, Z = 1, preserves high frequencies, maximized at ω = π, but removes the lowest frequencies, minimized at ω = 0. Hence, the first difference filter is a form of a high-pass filter, as it retains higher frequencies. Applying growth rates on the third lag (q-o-q for monthly data) places zero weights at frequencies ω ∈ {0, 2/3π } and maximum weights at frequencies ω ∈ {1/3π , π}. Thus, for a monthly time series, q-o-q growth rates preserve cycles of two and six periods in length.

The derivation of the growth rate filter shows how lower frequencies beyond six months are filtered away for Z = 1. In addition, the filter not only preserves the highest frequencies but exacerbates them, with the max- imum weight of the filter being equal to four. Since the high frequencies are emphasized and preserved, they are a higher order of magnitude. Hence, a model built using data transformed with Z = 1 should not be expected to be able to either capture frequencies beyond six months in length or forecast well at longer horizons in small samples.

Conversely, growth rates using Z > 1 emphasize lower frequencies and may be able to generate better forecasts at their corresponding horizons. For example, year-on-year and quarter-on-quarter growth rate transformations on monthly data preserve selected lower frequencies. While such growth rates are special examples, this paper pro- poses a spectral representation that is maximized. The lag in the growth rate filter is selected so as to maximize the information at frequencies that correspond to the respec- tive forecast horizon.

The method of targeted growth rate transformations can be defined as follows. Target the lag in the growth rate, Z∗, to maximize the variance at frequencies related to the targeted horizon, H.

In practice, this implies that a forecaster estimates a model for each forecast horizon H, using data transformed

by Z∗ in order to maximize the cyclical information avail- able for that horizon. Despite the non-linearity of the spec- tral filter, the method can be summarized by a simple rule. Let H, C ∈ N+ and Z ∈ R+ so that the frequency corresponding to horizon H/C is targeted. Then, the opti- mal lag in the growth rate filtration, Z∗, is given by Z∗ = H/(2C). For example, to maximize the cyclical information available at forecast horizon H with C = 1, the rule is to select Z∗C=1 = H/2.

Targeted growth rate transformations have desirable properties for forecasting. Growth rates are backward- looking filters with no end-point bias. The method is easy to implement, and the level of a series can always be recovered. Moreover, the filter not only maximizes the se- lection of lower frequencies but also excludes select higher frequencies. For integer values of Z > 1, there are Z + 1 extrema in the filter. Hence, longer lags imply that zero weights are applied more often. The inclusion of some higher frequencies is advantageous, as model estimates are more stable in small samples. In contrast, applying a low- pass or band-pass filter to a series prior to estimation can also target low frequencies, but is known to result in over- fitting and introduces an end point bias (see Azevedo and Pereira, 2013).

The filtered series depends on both the filter and the spectral density of the original series. The choice of the cycle to target with Z∗ depends on the available sample size and the forecast horizon. Specifically, modeling lower fre- quencies is a problem in small samples, and the elimination of some higher frequencies allows the model to isolate the lower frequencies in estimation.

Let us now explore the use of C. One issue with low- frequency econometrics in small samples is that filtering the data to preserve only lower frequencies may exac- erbate the small sample problem, since there are fewer lower frequencies for a given fixed sample size. Hence, if the forecast horizon is too long or the sample too short in a small sample, the forecaster may choose to target a frequency that is a fraction of the full cycle. In this case, using C > 1 may be more appropriate, as it allows for a better preservation of lower frequencies but attempts to forecast the horizon using the Cth cycle of the forecast. For example, the rule suggests transforming the data using Z∗C=2 = H/4 when targeting cycles that are half of the length of the forecast horizon (C = 2). In this case, the second cycle of the forecast will correspond to the desired forecast horizon H. We explore the use of C when applied to the real price of crude oil, but in general C will depend on the sample size, and the information that is available at frequencies that correspond to the respective forecast horizons. While there are no formal tests for choosing C, the ultimate criterion is the forecast performance.

Up to this point, the filter derived was that for the log-difference growth rate transformation. However, the growth rates gt can be either calculated using the percent- age change, gt = (yt − yt−1)/yt−1, or approximated using log differences, g̃t = (1 − L)ln(yt). Since (1 − L)lnyt = gt −

1 2 g

2 t +

1 3 g

3 t . . . , it follows that gt = g̃t − et , where

et = ∑

∞

n=2(−1) n+1gnt /n. As et is a lower order of magnitude

than gt , it is a close approximation for low values of gt . The percentage change is a non-linear filter, so it is not possible

4 S. Snudden / International Journal of Forecasting 34 (2018) 1–16

Fig. 2. US refiners’ import real oil price at Z.

to derive exact simple rules for targeted transformations as can be done for the log-difference filter. However, the rules for targeted growth rate transformations that are derived above for the log-difference growth rate are a first- order approximation of the growth rate filter. This point is important, since there are two factors that will make the approximation less precise.

Consider when the two growth rate filters differ. First, if ∂g

∂Z > 0 then | ∂e ∂Z | > 0. This implies that taking larger

lags when log-differencing will create a larger discrepancy between these methods. Therefore, at low values of Z, the rules for targeted filtering may only be a good approxima- tion when applied to the exact percentage change. Second, suppose that we want to calculate dynamic forecasts for lnŷT +1|T with an estimated value of ˆ̃gT +1|T . Then, in general, lnŷT +h|T = lnyT +

∑h i=1(ĝt+i|T + êt+i|T ). Thus, the approx-

imation error between the two methods can compound when forecasting at long horizons. For these reasons, this paper focuses on forecast horizons of less than five years using monthly data, as the forecast performance is less consistent beyond five years ahead.

One downside of using log-differences instead of growth rates for forecasts at longer horizons is that if a constant is estimated in-sample where it does not exist in the population, it is applied linearly. Thus, the forecast can exhibit large trends at long horizons and the level of the forecast may become less than zero. This suggests that there are some advantages of calculating growth rates using the percentage change rather than log differences at long-forecast horizons for small samples. This paper therefore proceeds using exact growth rate filtering.

3. Univariate forecasts

This section applies targeted growth rate transforma- tions to univariate forecast models of the real price of crude oil. The real US dollar price of oil is measured as the monthly nominal U.S refiners’ acquisition cost of crude oil imports from the US Energy Information Administration (EIA), deflated by the US consumer price level from the Federal Reserve (henceforth real oil price). Alternative oil price series are examined in the next section. Fig. 2 graphs

both the log-level real price of crude oil and the filtered real oil price series with Z = 1 and Z = 6.

Dynamic, recursive, out-of-sample forecasts are eval- uated between 1992.1 and 2015.10. All models are esti- mated beginning in 1974.1. The forecast criteria reported include the recursive MSPE expressed as a ratio to that of the no-change forecast. Success ratios are calculated to quantify the accuracy of the forecast direction, and repre- sent the fraction of times the forecast predicts the direction of change in the real price of oil correctly. All forecast criteria are evaluated in the real level of the price of oil. The p-values of the success ratios are calculated following Pesaran and Timmermann (2009). The p-values of the re- cursive MSPE ratios are calculated following Diebold and Mariano (1995). One divisive caveat is that the Diebold and Mariano (1995) test is not valid with parameter un- certainty and there is no alternative test that can be used. Nevertheless, it is still reported with this caveat in mind.

3.1. Benchmarks

The analysis begins by considering a number of simple benchmark models for forecasting the real price of crude oil at longer horizons. An attempt is made to answer the open question of whether other types of simple models can provide better benchmarks than the standard no-change benchmark that is used at short horizons. At longer fore- cast horizons, mean reversion may be a more plausible assumption than a no-change forecast. Thus, two simple mean reverting models are examined and compared to the no-change forecast: exponential smoothing and backward- moving means.

The no-change forecast is motivated by the random walk model. Forecasts are generated using R̂oilT +H|T = R

oil T ,

where R̂oilT +H|T is the forecast of the level of the real price of crude oil at the H-step-ahead forecast horizon.

Recursive exponential smoothing has been employed with relative success for the real price of gasoline by Baumeister, Kilian, and Lee (2017), and is suitable for series without trends. This model converts the observed log-real level of the series, roilt , into a smoothed series r̃

oil, then uses the smoothed series as the forecast for all future horizons.

S. Snudden / International Journal of Forecasting 34 (2018) 1–16 5

Table 1 Univariate real oil price forecasts.

Monthly Exp. Smo. Backward AR(12) AR(12) ARFIMA(1) AR(12) AR(12) AR(12) Horizon α = 0.3 Mean (12) Log-levels Z = 1 Log-levels C = 1 C = 2 C = 3

MSPE ratios

1 4.39 7.60 0.76 0.77 0.73 – – – 3 1.40 1.78 0.90 0.91 0.83 0.89 – –

12 0.81 0.76 1.06 1.14 0.87 0.86** 0.97 0.99 24 0.89 0.87 1.13 1.15 1.15 0.81 0.87** 0.95** 36 0.90 0.88 1.23 1.12 1.42 0.97 0.98** 0.95* 48 0.94 0.94 1.26 1.18 1.46 0.98 0.92** 0.91* 60 0.92 0.92 1.22 1.27 1.41 1.21 1.35 1.11

Success ratios

1 0.43 0.44 0.59* 0.59* 0.58* – – – 3 0.41 0.41 0.55 0.56 0.54 0.58* – –

12 0.53 0.53 0.48 0.53 0.51 0.61* 0.57 0.55 24 0.54 0.54 0.50 0.49 0.50 0.55 0.64* 0.62 36 0.59* 0.62* 0.51 0.55 0.51 0.52 0.58 0.72* 48 0.47 0.48 0.53 0.59* 0.49 0.45 0.55* 0.61 60 0.43 0.43 0.53 0.61* 0.48 0.53 0.53 0.65*

Notes: Recursive, dynamic, out-of-sample-forecasts from 1992.1 to 2010.1, coverted into real levels. Bold values indicate improvements relative to the nos-change forecast. Based on the Pesaran and Timmermann (2009) test for the null hypothesis of no directional accuracy and the Diebold and Mariano (1995) test of equal MSPEs relative to random walk: * Denotes significance at the 5% level. ** Denotes significance at the 10% level.

That is, r̂oilT +H|T = r̃ oil T , ∀H, since r̂

oil T +H|T is converted to levels

for all forecasts, RoilT +H|T , by exponentiating when calculat- ing the forecast criteria. The smoothed series is constructed recursively from:

r̃oilT = αr oil t + (1 − α)r̃

oil t−1, t = 2, . . . , T , (6)

where α ∈ [0, 1] is the smoothing parameter. The smaller α is, the smoother the oil price series. The degree of backward-looking smoothing, α, is searched from 0.05 to 0.95 in increments of 0.05. An α of 0.3 is found to produce the lowest recursive MSPE ratios and the highest success ratios at horizons between one and five years. This is sim- ilar to the values that are commonly used in macro time series and exactly the value employed by Baumeister et al. (2017). However, the relative performance of this model is reasonably robust to changes in this parameter.

This paper introduces the model of backward-moving means for forecasting the real price of crude oil. The method is similar to the recursive exponential smoothing method, but places equal weights on all past observations. In particular, the forecast is generated using r̂oilT +H|T = r̄

oil T =

(P)−1 ∑P

p=0r oil T −p. For example, if P is equal to 24, the average

of the real oil price over the last two years is used as the forecast for all future horizons. The value of P is searched from three months up to the end of the available data in three-month increments. The results suggest that P = 12 produces the lowest recursive MSPE ratios and the highest success ratios for forecasts of between one and five years ahead.

The forecast performances of the simple benchmark models are presented in Table 1. The forecasts of the real price of oil from the exponential smoothing and backward mean models can outperform the no-change forecasts ac- cording to both the recursive MSPE ratios and the direc- tional accuracy for horizons of between one and three years. For the recursive MSPE ratios, the models outper- form the no-change forecast for up to five years ahead.

Interestingly, the model of backward-looking means with P = 12 outperforms the exponential smoothing model with α = 0.3 at all horizons. This suggests that the no- change forecast of the mean real oil price over the last year outperforms other simple methods, such as the no- change forecast at forecast horizons of between one and three years.

In addition to evaluating the forecast performance to date, Fig. 4 also presents its evolution. Ideally, the models will outperform the no-change forecast robustly not just by the end of the sample, but consistently over the entire sam- ple. The first 30 periods are dropped to allow for the law of large numbers to begin working. Both the exponential smoothing and backward-looking mean models are unable to outperform the no-change forecast consistently, using both criteria, over the full sample at any forecast horizon. In particular, the run-up in the real price of oil in the middle of the 2000s causes the no-change forecast to outperform both models at horizons of up to three years. Moreover, despite being lower by the end of the sample, the forecast from the model of backward-looking means with P = 12 is more volatile than that from the exponential smoothing model with α = 0.3 for the time horizon considered. Over- all, these simple models still outperform the no-change forecast most of time for the two- and three-year horizons, especially in directional accuracy, in spite of their flaws. This suggests that a better rule for forecasting the real price of crude oil at the two- and three-year horizons would be to use the mean of the real price over the last year, rather than this month’s price.

3.2. ARI and ARFI

Univariate autoregressive (AR) models have been well documented to outperform no-change forecasts up to six months ahead (Alquist et al., 2013; Baumeister & Kilian,

6 S. Snudden / International Journal of Forecasting 34 (2018) 1–16

Fig. 3. Univariate AR parametric spectral density.

2012), and serve as an intuitive way of illustrating the mod- ification of targeted growth rates to the standard univariate Box-Jenkins method (Box et al., 2008). Let y be a stationary process, c a constant and εt the error, and let φp denote the p th autoregressive parameter. Then, the AR representation takes the following form:

yt = c + P∑

p=1

φpyt−p + εt . (7)

At shorter horizons, AR models estimated in log-real levels are preferred (Alquist et al., 2013). At longer hori- zons, it is an open question whether the AR models esti- mated in log-real levels or in growth rates generate better forecasts. For comparison, the analysis includes AR models estimated in both log-real levels and period-over-period growth rates.

Fig. 3 illustrates the sample parametric spectral density for the AR model estimated in growth rates with Z = 1 and targeting frequencies that correspond to the two- year horizon, with C ∈ {1, 2, 3}. The model is estimated using 24 lags. The parametric sample density is similar for models estimated with 12 or 18 lags, or when selected using the AIC (Akaike, 1974). The model is estimated using maximum likelihoods, although the model estimated using least squares is similar. The parametric spectral density can be calculated as:

f (ω; φ, σ 2ε , γ0)

= σ 2ε

(2π γ0)(1 − φ1e−iω − φ2e−i2ω − · · · − φPe−iPω) , (8)

where ω ∈ [0, π], γ0 is the variance of the variable, and σ 2ε is the variance of the error, see Box et al. (2008).

The area under the sample periodogram represents the portion of the variance of the series that can be attributed to cycles of different frequencies, ω. The sample peri- odograms show that targeted growth rates increase the portion of the variance that can be attributed to frequencies of 24 periods. This suggests that the model estimated using data transformed by targeted growth rates is able to isolate

frequencies corresponding to two years in length better. The forecast performance may be superior at horizons that correspond to frequencies for which the maximum portion of the variance is explained.

Simple autoregressive fractionally integrated models (ARFI) are included for comparison with the model esti- mated in log-real levels, since they may increase the effi- ciency of estimation for series with long memory. The ARFI model takes the form

ρ̂(L)(1 − L)dyt = ϵt , (9)

where d is the order of integration estimated using maxi- mum likelihoods. The model is estimated in log-real levels, and one autoregressive lag is found to be sufficient.

The forecasts obtained from AR models estimated with 12 lags are generally found to be superior at shorter hori- zons to those obtained using either other fixed lag-lengths or lag selection via information criteria (Alquist et al., 2013; Baumeister & Kilian, 2012). However, the issue of which lag order is most suitable for longer horizon forecasting is an open question for two reasons. First, previous papers focused on short horizons, and the choice of lag may affect the forecast performance at longer horizons. The second novelty is using Z > 1. For example, the use of period-over- period growth rates implies that one lag is lost relative to a model estimated in log-real levels. In contrast, the use of targeted growth rates with Z > 1 emphasizes the lower frequencies, which results in more partial autocorrelation relative to Z = 1. Given the presence of model uncertainly and the commonly-employed benchmark of 12 lags, mod- els estimated using Z > 1 are also estimated with 12 lags in the benchmark results. The models estimated with 18 or 24 lags or with lag selection using the AIC (Akaike, 1974) are qualitatively similar.

The method of targeted growth rate filters is applied to AR models. The method is directly comparable to Z = 1 us- ing targeted filtration with C ∈ {1, 2, 3} and with Z∗ given by Z∗ = H/2C. The parameters of all models considered are estimated recursively in each time period, and the out-of- sample dynamic forecasts are evaluated against observed data for 1992.1–2015.10. The model is estimated using data from 1974.1. Forecasts in log-real levels are converted back to real levels by exponentiating. Forecasts using growth rates, Xoilt,z = 100(R

oil t /R

oil t−z − 1), are mapped to real levels

using the data available at the point when the forecast was produced. That is,

R̂oilt = (1 + X̂ oil t,z/100)R̂

oil t−z, (10)

where R̂oilt−z uses historical data up to T and recursively estimated forecasts thereafter.

Table 1 reports the forecasts from these models. Con- sistent with previous findings, the model estimated in log- real levels has lower MSPE ratios than the model in growth rates, with Z = 1 at horizons of less than a year. The ARFI model is able to outperform both the AR model in log-real levels and that with Z = 1, as well as the no- change forecast for up to one year. However, as Fig. 4 shows, the improvement in the MSPE ratios from the ARFI at the one-year horizon at the end of the sample comes at the cost of poor performances between 2004 and 2008. The

S. Snudden / International Journal of Forecasting 34 (2018) 1–16 7

Fig. 4. Evolution of benchmark forecasts, 1992.1–2015.10.

8 S. Snudden / International Journal of Forecasting 34 (2018) 1–16

ARFI forecasts also provide directional accuracies that are comparable to those of the AR models, although slightly less than those of the model estimated with Z = 1. None of these methods are able to outperform the no-change forecast in both directional accuracy and MSPE ratios for horizons of between two and five years.

In contrast, the models for which the data were trans- formed using targeted growth rates produce improve- ments in both the success and MSPE ratios for horizons of up to four years. Targeted filtration with C = 1 produces forecasts that are superior to those of the AR and ARFI models at horizons of between one and three years. At horizons of two years and longer, values of C greater than 1 are desirable, as the small sample makes it more difficult to isolate the lower frequencies. In particular, C = 1 has lower MSPE ratios in the first two years, while C = 3 performs best for horizons beyond two years. The direc- tional accuracy of the forecasts using targeted filtration is quite high for all horizons. Success ratios of over 0.6 are found for horizons of one year and beyond for the model estimated with C = 3. These values are large relative to those found in the empirical finance literature (Pesaran & Timmermann, 1995), and are even slightly larger than the short-horizon success ratios of the multivariate modeling method of Baumeister and Kilian (2014).

Univariate autoregressive models with targeted growth rate transformations outperform no-change forecasts in both MSPE and success ratios for most of the sample up to the three-year-ahead horizon. Fig. 5 shows the model with C = 1 for the first two years, and the model with C = 3 for horizons of three years and longer. At longer horizons, targeted transformations can improve the out-of-sample forecast performance consistently relative to models that rely on period-over-period growth rates. Up to the three- year-ahead horizon, the success ratios are consistently over 0.5 and the recursive MSPE ratios are more often below one.

3.3. Müller-Watson

The application of targeted growth rate transformations can be useful for any method when growth rate trans- formations are present. This section examines the perfor- mance of the method when it is applied to a variant of the modeling techniques of Müller and Watson (2016). In particular, the real price of crude oil is forecast using a weighted average of trigonometric functions. Note that the model used to forecast the real price level is based on the specification of Müller and Watson (forthcoming), not on an average growth rate as per Müller and Watson (2016). In particular, the r̂oilT +h|T forecast is constructed using:

roilt = r̄ + Ψ((t − 1/2)T) ′ β̂t + εt , (11)

where Ψj(s) = √ 2cos(jsπ), and Ψj(s) has period 2T /j.

Ψj(s) is a R+ valued function and ΨT is a T × q matrix by evaluating Ψ at s = (t − 12 T), t = 1, . . . , T . As per Müller and Watson (2016), q, j = 12, although variations are not found to affect the results qualitatively.

Table 2 reports the forecasts from this model when prices are estimated in log-real levels, Z = 1, and C ∈

Table 2 Müller-Watson real oil price forecasts.

Horizon Log-levels Z = 1 C = 1 C = 2 C = 3

MSPE ratios

12 0.72 2.40 2.05 2.15 2.24 24 0.96 4.14 2.24 3.00 3.37 36 1.52 6.09 1.99 3.43 4.37 48 1.90 8.47 2.76 4.06 5.50 60 1.89 11.84 4.41 5.41 6.38

Success ratios

12 0.53 0.49 0.52 0.54 0.54 24 0.57 0.53 0.53 0.52 0.48 36 0.41 0.66 0.72* 0.63 0.62 48 0.32 0.68 0.65 0.68 0.70 60 0.36 0.62 0.53 0.58 0.59

Notes: Recursive, dynamic, out-of-sample-forecasts from 1992.1 to 2010.1, coverted into real levels. Bold values indicate improvements rela- tive to the no-change forecast. * denotes significance at the 5% level and ** at the 10% level based on the Pesaran and Timmermann (2009) test for the null hypothesis of no directional accuracy and the Diebold and Mariano (1995) test of equal MSPEs relative to random walk. All models estimated with q = 12.

{1, 3}. The model is able to achieve directional accuracies and MSPE ratios of less than one at the one- and two- year horizons. At horizons greater than two years, though, the method does not fare well. Moreover, when the model is estimated with Z = 1, the MSPE ratios does not out- perform the no-change forecast at any horizon, though it still produces success ratios greater than 0.5. With the tar- geted growth rate transformation, the MSPE ratios improve relative to the model with Z = 1, although it still does not outperform the no-change forecast. Interestingly, the success ratios are quite large for the model with C = 1, approaching 0.72 at the three-year-ahead horizon.

The results suggest that the method of targeted growth rate transformation is able to improve the accuracy relative to the model with period-over-period growth rates. The failure of the Müller-Watson framework to produce low MSPE ratios by the end of the sample is due mainly to its inconsistent forecast performance over time; its forecast accuracy is very poor during selected historical episodes, even though its mean-squared errors are very low for other parts of the sample. Thus, its forecast performance is quite variable, albeit often with the correct sign for directional accuracy. As the parameters are estimated recursively, the results may suggest overfitting in small samples. An explo- ration of alternative forms of information updating may be worthwhile, but is left to future research.

4. VAR forecasts

This section evaluates the forecast performance of tar- geted filtration when it is applied to vector autoregression (VAR) models. The VAR takes the form

yt = c + P∑

p=1

Blyt−p + αDt + ϵt , (12)

where ϵt is a vector of innovations, c is a vector of con- stants, Dt is a matrix of seasonal dummy variables, α is the corresponding coefficient matrix, Bp (p = 1, . . . , P) is the

S. Snudden / International Journal of Forecasting 34 (2018) 1–16 9

Fig. 5. Evolution of univariate forecasts, 1992.1–2015.10.

10 S. Snudden / International Journal of Forecasting 34 (2018) 1–16

matrix of autoregressive coefficients, and yt is a vector of endogenous variables.

The multivariate analysis favors the use of variables that are related to economic fundamentals for the global crude oil market. In particular, the VAR models use series that are related to the latent global crude oil supply, demand, and inventories. This would allow for structural analysis if additional restrictions were imposed following Kilian and Lee (2014) and Kilian and Murphy (2014), for example. However, the models used for forecasting in this paper are unrestricted, since the focus of this paper is on forecast performances. Some liberties are taken by allowing the use of a US petroleum inventory series that may not be appropriate for structural analysis but that improves the long-run forecast performance. The author makes notes when it is used, and also presents results for the next best inventory measure. Such series are included in the forecast exercise to allow researchers to evaluate for themselves the potential trade-off involved in deviating from a po- tential structural interpretation in order to increase the forecast precision.

Observable measures of the global crude oil production and the real price of crude oil are measured reliably at the monthly frequency. In contrast, there also exist alternative measures of global real activity and inventory variables. The multivariate analysis begins by conducting a system- atic investigation of the forecast performance of alterna- tive measures of global real activity and oil inventories. The purpose is to examine whether the measures that are used currently for forecasting up to the two-year horizon continue to be the preferred variables when forecasting at longer horizons.

4.1. Alternative series

As with the univariate forecasts, the real price of oil is measured using the monthly nominal U.S refiners’ acquisi- tion cost of crude oil imports, deflated by the US consumer price level. Forecasts of the Brent real price of crude oil are also considered in Section 4.3. Global crude oil production is measured as the international production of crude oil available from the EIA, which includes lease condensate but excludes natural gas plant liquids.

The forecasting performances of five crude oil related inventory series are evaluated systematically: crude oil inventories from Kilian and Murphy (2014), US crude oil inventories with and without the strategic petroleum re- serve (SPR), US petroleum inventories excluding SPR, and an OECD crude oil series. The OECD crude inventory series uses the level of OECD crude inventories when available, and extrapolates the series prior to 1987.11 using the period-over-period growth rate of US crude oil inventories excluding SPR. The use of petroleum series is not consistent with a structural interpretation of the global market for crude oil, but is included nonetheless in order to evaluate the forecast accuracy. Moreover, two measures of global real economic activity are evaluated as well: world in- dustrial production (WIP) from Global Data Services and Kilian’s global real economic activity index (REA) (Kilian, 2009). A complete list of the variables, along with their sources, date range, and summary statistics, is given in Table A.1 in the online appendix.

The WIP series includes both advanced economies and emerging market economies. The series incorporates the measured industrial production (IP), making adjustments for the inclusion of countries’ data series once they be- come available. For example, China’s IP was included when it began to be measured starting in 1992.1, Brazil starting 1985.1, and India starting in 1971.1. While not reported here, the forecast performances of IP indexes for advanced economies, developing economies, and MER weighted were also compared, but the global series was consistently found to have a better forecast performance.

The real price of crude oil is estimated in either log-real levels or percentage changes. The REA is always estimated in levels. Measures of oil production and inventories are always estimated in growth rates. The use of growth rates instead of differences for inventories differs from the work of Kilian (2009) but does not affect the relative forecast performance. Importantly, the transformation is consistent with the other variables, and does not change the results; however, it is not the approach recommended by Kilian (2009). Forecasts are always converted into real levels be- fore evaluating the forecast performance.

At shorter horizons, VAR models estimated with 12 lags are generally found to produce forecasts that are supe- rior to those from other fixed lag lengths or when using information criteria (Alquist et al., 2013; Baumeister & Kilian, 2012). Again, the appropriate lag selection at longer horizons and with Z > 1 is an open question, so models estimated using 24 lags are explored. Both of these lag orders ensure that the residuals pass the portmanteau (Q ) test for white noise (Box & Pierce, 1970; Ljung & Box, 1978), regardless of whether the models are estimated in log-real levels or percentage changes.

Ten models are compared in order to evaluate the fore- cast performances of the alternative series: the combina- tions of the five inventory series and the two measures of global real activity. Recursive, dynamic, out-of-sample forecasts are computed over the 1995.1–2015.10 evalua- tion period.1 All models are estimated beginning in 1974.1.

The online appendix provides a complete summary of forecast performances across various lags, series, and oil price transformations. Let us begin by considering the case where the real price of oil is estimated in log-real levels and all other variables, excluding the REA, are estimated in percentage changes with Z = 1. Of the various inventory series, using either WIP or REA, US petroleum inventories are found to produce superior forecasts for horizons of between one and two years for the MSPE ratios, followed by US crude oil inventories. At the one- and two-year horizons, the success ratios of the forecasts slightly favor the use of US crude oil inventories including SPR, followed closely by US petroleum oil inventories. At the three- to five-year horizons, the US petroleum inventories have suc- cess similar to the OECD crude oil inventories. The OECD crude oil inventory series outperforms the crude oil inven- tory series of Kilian and Murphy (2014) in both the MSPE

1 The later start of the sample evaluation period is due to a few cases where Z is very large. In this case, too much of the sample is lost to the growth rate transformation, and the forecasts in the early part of the sample are clearly non-sensible.

S. Snudden / International Journal of Forecasting 34 (2018) 1–16 11

ratios and directional accuracy, especially at the four- to five-year horizons. The extra predictive power of US crude oil inventories at longer horizons, relative to the Kilian and Murphy (2014) crude oil inventory series, is consistent with the improvement in predictive power for short-run forecasts that was established first by Baumeister et al. (2015), and used by Baumeister and Kilian (2014, 2015).

The largest MSPE improvements at horizons between one and five years can be made by using WIP, rather than the level of the REA. This holds for all oil inventory series. However, the improvement in the MSPE using the WIP sometimes comes at the cost of smaller success ratios when compared to REA, especially at the four- and five-year horizons.

The forecast performances when all variables, excluding the REA, are estimated in growth rates with Z = 1 differ from those when the real price of crude oil is estimated in log-real levels. The MSPE ratios are substantially lower for the models estimated with WIP than for those with the REA, especially beyond the one-year-ahead horizon. However, the success ratios are slightly higher in some cases at the four- and five-year horizons using the REA. The performances of the OECD oil inventories are generally consistent when the real price of oil is estimated in log-real levels. The US crude oil and petroleum inventory series, with and without SPR, are generally the best predictors, followed by the OECD inventory series.

When using WIP, the results are quite similar for VAR models where oil prices are estimated in log-real levels or period-over-period growth rates. The model estimated with the oil price in log-real levels performs better for MSPE ratios at horizons of up to a year, and four and five years, but performs worse than the model with real prices transformed with Z = 1 at the two- to three-year horizons. In contrast, when the forecasts use the level of the REA, the log-real price is clearly preferred. The superior performance of WIP at longer horizons may be due to not using real-time data. As is often the case, the forecast per- formance in real time may differ from that using historical data. Nevertheless, the evidence does suggest that alterna- tive series provide unique information at select horizons and specifications. This is explored further when they are expressed with targeted growth rates.

The relative performances of the series persist when the VAR model is estimated with 24 lags. Generally, the models estimated with 12 lags outperform those estimated with 24 lags at horizons of up to one year. For the MSPE ratios, the models estimated with 24 lags generally perform better at all horizons between one and five years. This is complicated by a general loss of directional accuracy at three years and beyond for the models estimated with 24 lags. This leads to a trade-off between MSPE ratios and success ratios at three years ahead and beyond. Note that this trade- off has not been found in previous studies due to their focus on shorter horizons. However, it is confirmed by the univariate analysis of the previous section. Since the MSPE ratios and the directional accuracy should ideally confirm each other, this suggests that the model estimated with 24 lags may be preferable at the one- and two-year horizons and the VAR model with 12 lags at the three- to five-year horizons.

Table 3 VAR forecasts with the WIP, US crude oil inventories.

Horizon Oil prices in log-levels Oil prices in growth rates

Z = 1 C = 3 C = 1 Z = 1 C = 3 C = 1

MSPE ratio

12 0.98 0.93 0.97 1.13 1.05 1.14 24 0.89 0.92 0.86 1.16 1.20 1.23 36 0.87 0.86 0.82** 1.26 1.36 1.15 48 0.82 0.81 0.87** 1.26 1.21 1.09 60 0.90* 0.88* 1.14 1.38 1.20 1.17

Success ratio

12 0.53 0.54 0.51 0.45 0.49 0.48 24 0.54 0.52 0.58* 0.50 0.54 0.54 36 0.56 0.55 0.51* 0.49 0.59 0.62** 48 0.53 0.50 0.41 0.56 0.57 0.40 60 0.57 0.52 0.45 0.53 0.66 0.32

Notes: Recursive, dynamic, out-of-sample-forecasts from 1995.1 to 2015.10. converted into real levels. Estimated with 12 lags beginning in 1974.1. Bold values indicate improvements relative to the no-change forecast. Based on the Pesaran and Timmermann (2009) test for the null hypothesis of no directional accuracy and the Diebold and Mariano (1995) test of the null hypothesis of equal MSPEs relative to random walk. Model estimated using WIP and US crude oil inventories, with all non-oil price data in growth rates using Z or C: * Denotes significance at the 5% level. ** Denotes significance at the 10% level.

4.2. Targeted growth rates

The method of targeted growth rates is applied to VAR models in order to improve the forecast performance at horizons of between one and five years. The forecasting exercise is identical to that described in the previous sec- tion, except that now, targeted growth rates are employed on all transformed series. We begin by analyzing the effect on the forecast performance when targeted growth rates are applied to the VAR model with WIP and US crude oil inventories, using 12 lags. The filtered series depends on the filter as well as on the spectral density of the original se- ries. Hence, the performances of the alternative series may not continue to hold when the targeted transformation is employed.

Table 3 presents the forecast performances of the VAR models estimated when the real price of crude oil is in log-real levels and in growth rates. All other variables are transformed into growth rates. The model uses US crude oil inventories and WIP. When the model is estimated with real oil prices in log-real levels, the forecast performances for both the MSPE ratios and the directional accuracy are improved by using C = 1 for the one- to three-year-ahead forecasts. Using C = 3 improves the MSPE ratios at almost all horizons relative to the model using growth rates with Z = 1. Using C = 2 results in MSPE ratio and success ratio performances of between those for C = 1 and C = 3, and so these results are not reported in the tables for the sake of brevity. Targeted growth rates are unable to improve the directional accuracy at the four- and five-year horizons, despite the improvements in the MSPE ratios. Consistency of the MSPE ratios and success ratios is achieved when modeling the growth rate in percentage changes. In par- ticular, the model with C = 3 is able to outperform the

12 S. Snudden / International Journal of Forecasting 34 (2018) 1–16

Table 4 VAR forecasts with WIP, log-real oil prices, 24 lags.

Horizon US petroleum inventories US crude oil inventories

Z = 1 C = 3 C = 1 Z = 1 C = 3 C = 1

MSPE ratio

12 0.99 0.93 0.94 1.01 0.96 0.91 24 0.92 0.75 0.83 0.99 0.86 0.92 36 0.87 0.77 0.96* 0.93** 0.88 1.11 48 0.77 0.76** 0.86** 0.81 0.85 1.06 60 0.90* 0.87* 1.01 0.94* 0.92* 1.14

Success ratio

12 0.50 0.55* 0.58* 0.45 0.48 0.51 24 0.50 0.59* 0.58* 0.47 0.56** 0.57** 36 0.47 0.52* 0.47 0.49 0.54** 0.43 48 0.44 0.33 0.34 0.49 0.45 0.40 60 0.37 0.30 0.27 0.48 0.41 0.35

Notes: Recursive, dynamic, out-of-sample-forecasts from 1995.1 to 2015.10. converted into real levels. Estimated with 24 lags beginning in 1974.1. Bold values indicate improvements relative to the no-change forecast. Based on the Pesaran and Timmermann (2009) test for the null hypothesis of no directional accuracy and the Diebold and Mariano (1995) test of the null hypothesis of equal MSPEs relative to random walk. Model estimated using WIP and real oil prices in log-real levels, with all non-oil price data in growth rates using Z or C: * Denotes significance at the 5% level. ** Denotes significance at the 10% level.

model with Z = 1 for almost all periods according to both the MSPE ratios and the directional accuracy at horizons of one year and beyond.

The performances of the targeted transformations are improved further at the one- and two-year horizons when applied to VAR models estimated with 24 lags and oil prices in log-real levels (see Table 4). Although the MSPE ratios are generally better than those from the model estimated with 12 lags, this comes at the cost of slightly lower success ratios at horizons of three years and beyond. The model with variables transformed using C = 3 improves both the directional accuracy and the MSPE ratios, relative to the VAR model with Z = 1. As has been noted, it may be preferable to use the VAR model with 24 lags at the one- and two-year horizons, but the model with 12 lags at the three-year horizon and beyond, for consistency of the forecast criteria.

For most specifications, US petroleum inventories pro- duce superior forecasts when targeted filtered. While this may negate the structural interpretation, the forecast per- formances of US crude oil and petroleum inventories are presented in Table 4. In almost all cases, the MSPE ratios are lower for the model using US petroleum inventories. This comes with the trade-off of slightly lower success ratios, especially at longer horizons. This major improvement in MSPE ratios using US petroleum inventories forms a trade- off for researchers who are looking for a structural inter- pretation.

Ideally, the VAR forecast of the models using targeted growth rates will outperform the traditional VAR models consistently over time, not just at the end of the sample. Moreover, it would be remarkable if an individual model could outperform the no-change forecast consistently over the entire sample. To this end, Fig. 6 presents the evolution of the MSPE ratios and success ratios of the baseline VAR

for horizons of one to five years. The model uses WIP and US petroleum inventories. The results from using US crude oil inventories have the same quantitative model ranking and maintain the same qualitative insights. At the one- and two-year horizons, the VAR model employs 24 lags, while at the three- to five-year horizons, the models employ 12 lags. The MSPE ratios and success ratios are reported by month, with the first 30 periods being dropped to allow the law of large numbers to take effect.

The VAR model with crude oil prices estimated in log- real levels or with Z = 1 is unable to outperform the no- change forecast consistently for MSPE precision. Moreover, the success ratios are consistently below 0.5 for most of the sample at the one- and two-year-ahead horizons. This evidence is consistent with the VAR forecasts from indi- vidual monthly VAR models having difficulty forecasting consistently well over all sample periods (Baumeister & Kil- ian, 2014, 2015). Interestingly, and unlike in the univariate case, the two models have similar performances at hori- zons of up to two years, with the two models’ performances improving and deteriorating at similar times.

When the real price of oil is estimated in log-real levels, transforming the other variables using C = 3 consistently outperforms the use of Z = 1 for both the MSPE ratios and directional accuracy. At the four- and five-year hori- zons, the MSPE ratio improvements as a result of using targeted growth rates come at the cost of lower success ratios. Similar improvements hold when all variables are modeled in percentage changes. When growth rates of C = 3 are used for all variables, the model outperforms the model with Z = 1 consistently up to the three-year horizon. At horizons of four years and beyond, employing C = 3 outperforms the model with Z = 1 for most of the sample. The model estimated in percentage changes and with targeted growth rates is able to outperform the no-change forecast consistently for horizons of up to three years. The performance of the model exploiting targeted growth rates is particularly notable at the two-year-ahead horizon, with the MSPE ratio being close to 0.8 and the success ratio close to 0.6 for most of the sample. This consistency of performance is better than those observed at shorter forecast horizons, or ever observed previously by any monthly individual VAR forecast.

The insight that data transformations are explicit filters is quite useful. Previous attempts to forecast the real price of oil have focused on log-real levels or log-differenced data with Z = 1, so it would be expected that these models would be designed to forecast well at shorter horizons. Moreover, other forms of filters, including backward-moving mean filters, could also be beneficial for targeting selected lower frequencies. For example, using data at lower observation frequencies, such as annual or quarterly data, is a form of moving average filter. This may explain why there is evidence that time series models which model trends explicitly and are estimated using annual data observations can outperform random walk forecasts in the long run (see for example Bernard et al., 2017). It could also explain why VAR models estimated with quarterly data generally perform better at horizons of up to two years (Baumeister & Kilian, 2014, 2015).

S. Snudden / International Journal of Forecasting 34 (2018) 1–16 13

Fig. 6. Evolution of VAR forecasts 1995.1–2015.10.

14 S. Snudden / International Journal of Forecasting 34 (2018) 1–16

Table 5 VAR forecasts with the REA, US crude oil inventories.

Horizon Oil prices in log-levels Oil prices in growth rates

Z = 1 C = 3 C = 1 Z = 1 C = 3 C = 1

MSPE ratio

12 1.10 1.06 1.06 1.33 1.23 1.25 24 1.02 1.02 0.85 1.62 1.66 1.40 36 0.83 0.79 0.72 1.69 1.60 1.56 48 0.79 0.77 0.80 1.83 1.93 1.32 60 0.97* 0.97* 1.07 2.40 2.02 1.64

Success ratio

12 0.57 0.58 0.56 0.54 0.52 0.54 24 0.49 0.50 0.62* 0.45 0.55 0.57 36 0.56 0.57 0.63* 0.45 0.57 0.64** 48 0.59 0.56 0.41 0.53 0.55* 0.45 60 0.63 0.55 0.44 0.56 0.64 0.38

Notes: Recursive, dynamic, out-of-sample-forecasts from 1995.1 to 2015.10. converted into real levels. Estimated with 12 lags beginning in 1974.1. Bold values indicate improvements relative to the no-change forecast. Based on the Pesaran and Timmermann (2009) test for the null hypothesis of no directional accuracy and the Diebold and Mariano (1995) test of the null hypothesis of equal MSPEs relative to random walk. Model estimated using Kilian (2009) global real activity index in levels and US crude oil inventories, with all other non-oil price data in growth rates using Z or C: * Denotes significance at the 5% level. ** Denotes significance at the 10% level.

4.3. VAR robustness

As has been shown, the forecast performance up to five years ahead is similar to those previously found at shorter horizons by Alquist et al. (2013) and Baumeister and Kilian (2012). However, the VAR models used in these previous studies were not intended for forecasting beyond short horizons, except for use in forecast combinations. The im- proved performance at longer horizons that is found in this paper can be attributed to both targeted transformations and the choice of variables.

We now employ the REA index of Kilian (2009) for VAR forecasting at longer horizons. The REA may be advanta- geous because it is better able to be employed in real- time studies. Table 5 replicates the baseline VAR results using the level of the REA. When the price is modeled in log-real levels, the performance of the level of the REA at the end of the sample is very similar to that of the model using WIP. Applying targeted growth rates to the oil production and inventory variables improves the forecast performance further, producing low MSPE ratios and high success ratios up to the three-year horizon with C = 1. In contrast, when the real price of oil is modeled in percentage changes and uses the REA, the forecast performance is poor at longer horizons. These results suggest that the REA of Kilian (2009) matches the performance of the WIP closely.

Fig. 7 reports the evolution of the MSPE and success ratios for the out-of-sample recursive VAR forecasts from 1995.1 to 2015.10, estimated using the real Brent crude oil price, global crude oil production in Z = 1, the WIP in Z = 1, and the US petroleum inventory in Z = 1. The forecast performances of the OECD inventory measures are found to be consistent with those of the US refiners’ import price. Methods of extrapolating the oil price series back to 1974.1

lead to a deterioration in the forecast performance, so the model is estimated from the beginning of the data series in 1987.1. Similar to the VAR of the US refiners’ import price, 24 lags are found to produce superior forecasts at the one- and two-year horizons. The consistency of the MSPE and success ratios again suggests the use of the VAR estimated with 12 lags at the three- to five-year horizons. When the model is estimated with prices in log-real levels, targeted growth rates with C = 3 improve the MSPE performance consistently up to the two-year horizon.

When the model is estimated with Brent crude prices in percentage changes (Fig. 7), targeted growth rates with C = 3 improve the MSPE performance consistently up to the two-year horizon, for both the MSPE ratios and direc- tional accuracy. At longer horizons, the targeted growth rates are still able to achieve improvements in the MSPE ratios, but sometimes at the cost of lower success ratios. The use of targeted growth rates at horizons beyond two years improves the forecasts relative to relying on Z = 1, but is not reported here for the sake of brevity. Similarly to the case of the US refiners’ import price, the model with prices in targeted growth rates achieves the lowest MSPE ratios consistently across horizons. The results suggest that targeted transformations are useful for forecasting the real price of Brent crude oil.

The Brent price forecasts are slightly less precise than those of the real US refiners’ import price of crude oil. This is due mainly to the fact that the estimation begins in 1987.1, leaving less of a training sample. If the evaluation period is pushed back, even to 2000.1, the forecast criteria are quite similar to those for the refiners’ import price.

5. Conclusion

This paper suggests a modification of traditional fore- casting methods when producing forecasts at longer hori- zons. The method of targeted filters to transform data for each forecast horizon is proposed. Lags in growth rate transformations are chosen so as to maximize the infor- mation at certain targeted lower frequencies. This is in contrast to standard approaches, which consist of esti- mating either in levels, which makes lower frequencies hard to model in small samples, or first differences, which explicitly filters lower frequencies of the data.

The analysis suggests that the simple rule of using the mean of the real price over the last year can generally outperform the no-change forecast at the two- and three- year horizons when forecasting the real price of crude oil. When targeted growth rates are applied to AR models, the success ratios are consistently over 0.5 and the MSPE ratios are more often below 1 up to the three-year-ahead horizon.

The method of targeted growth rate transformations exhibits robust improvements in forecast performances across sub-samples, regardless of the choice of modeling method, whether the real price of oil is in log levels or differences, and for alternative oil price series. VAR models with 24 lags are generally found to perform better at the one- and two-year horizons. For the three- and five-year horizons, VAR models with 12 lags consistently produce low MSPE ratios and high success ratios. The US inventories series are found to produce the best forecasts at longer

S. Snudden / International Journal of Forecasting 34 (2018) 1–16 15

Fig. 7. Evolution of VAR Brent forecasts.

horizons, followed by the OECD inventories. The WIP and REA measures of global real activity have similar perfor- mances at longer horizons.

Regardless of the specification of the model, targeted growth rates are generally found to improve the out-of sample forecasts of the real price of crude oil. The method is able to improve the forecast performance by the end of the sample for horizons of up to five years for a va- riety of models. Particular specifications of VAR models employing targeted growth rates are able to outperform the no-change forecast consistently over the full sample for horizons of up to three years. The method and its results are a step forward in oil price forecasting, achieving a level of success at longer horizons in individual monthly models that has been found previously only at shorter horizons.

Any forecasting exercise with small sample sizes that attempts to exploit information at a business cycle fre- quency may benefit from targeted transformations. The method could also be used for forecasting other commodi- ties such as metal prices, or asset prices such as housing prices, stock prices and interest rates.

The method could even be used to forecast the effects of climate change. Other forms of filters, including backward- moving mean filters, could also be beneficial for target- ing select lower frequencies. Likewise, the method could be used with quarterly data in an attempt to extend the success of the forecasts beyond five years. The ability of the model to provide unique information at longer forecast horizons makes this method potentially useful for fore- cast combinations. The success of the method for fore- casting the price of crude oil in real time should also be

evaluated. These and other applications are left for future research.

Acknowledgments

The author thanks Ron Alquist, Allan Gregory, Lutz Kilian, Dirk Muir, Morton Nielsen, Hashem Pesaran, and Gregor Smith for comments. He also thanks referees and conference participants at the Canadian Economics Asso- ciation Conference Bank of Canada Student Paper Award Competition (2016), the International Symposium on En- ergy and Finance Issues (2016), and the International As- sociation for Applied Econometrics Conference (2016).

Funding: This research was supported by the IAAE Con- ference Travel Grant (2016) and by the Social Sciences and Humanities Research Council of Canada, SSHRC Award Number: 767-2013-2637.

Appendix A. Supplementary data

Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ijforecast.2017. 07.002.

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Stephen Snudden has held positions as a Project Officer in the IMF’s Research Department Economic Modeling Unit, where he developed and applied macroeconomic policy models. He held a similar position in the International Department of the Bank of Canada (2008–10). He is pursuing a Ph.D. in economics at Queen’s University in Kingston, Ontario.