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A real-time quantile-regression approach to forecasting gold returns under asymmetric loss
Resources Policy 45 (2015) 299–306
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A real-time quantile-regression approach to forecasting gold returns under asymmetric loss Christian Pierdzioch n, Marian Risse, Sebastian Rohloff Department of Economics, Helmut Schmidt University, Holstenhofweg 85, P.O.B. 700822, 22008 Hamburg, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 17 March 2015 Received in revised form 30 May 2015 Accepted 13 July 2015
We propose a real-time quantile-regression approach to analyze whether widely studied macroeconomic and financial variables help to forecast out-of-sample gold returns. The real-time quantile-regression approach accounts for model uncertainty, model instability, and the possibility that a forecaster has an asymmetric loss function. Forecasts are computed and evaluated using the same asymmetric loss function. When the loss function implies that an underestimation is somewhat more costly than an overestimation of the same size, the forecasts computed using the real-time quantile-regression approach outperform forecasts implied by an autoregressive benchmark model. & 2015 Elsevier Ltd. All rights reserved.
JEL classification: C53 E44 Keywords: Quantile regression Forecasting Asymmetric loss Gold returns
1. Introduction Against the background of recent financial market turbulences, research on the determinants of gold returns has mushroomed. Among the determinants that researchers have studied are the inflation rate (Beckmann and Czudaj, 2013a; Batten et al., 2014), the oil price (Zhang and Wei, 2010; Reboredo, 2013b), the exchange rate (Pukthuanthong and Roll, 2011; Reboredo, 2013b), and business-cycle fluctuations (Pierdzioch et al., 2014b). Studying whether the various determinants studied in earlier literature help to forecast gold returns is important because the properties of gold as a safe-haven investment, a low-correlation portfolio diversifier, and a hedge against fluctuating currency values have received much attention in recent research (Hillier et al., 2006; Joy, 2011; Ciner et al., 2013, to name just a few). Despite many research efforts, no consensus has emerged regarding the core determinants of gold returns. As a result, researchers have studied gold returns by applying flexible forecasting approaches that account for model uncertainty and model instability (see also Vrugt et al., 2007; Aye et al., 2015; Baur et al., 2014; Pierdzioch et al., 2014a, 2014b, 2015). Model uncertainty arises because gold returns may be linked to a potentially large number of determinants, none of which can be excluded a priori n
Corresponding author. E-mail address: [email protected] (C. Pierdzioch).
http://dx.doi.org/10.1016/j.resourpol.2015.07.002 0301-4207/& 2015 Elsevier Ltd. All rights reserved.
on economic grounds. Model instability arises because the relative importance of these determinants most likely has changed over time (Baur, 2011; Batten et al., 2014) and may be state dependent (Wang and Lee, 2011; Wang et al., 2011). We contribute to earlier literature in that we propose a realtime quantile-regression approach to forecast gold returns. A quantile-regression approach renders it possible to compute forecasts that target the conditional quantiles rather than the conditional mean of the distribution of gold returns. Quantile regressions have received growing attention in the recent finance literature (Basset and Chen, 2001; Engle and Manganelli, 2004; Chuang et al., 2009; Baur et al., 2012) and have been studied recently in a forecasting context by Meligkotsidou et al. (2014), Manzan (2015), and Pedersen (2015). Quantile regressions also have been applied to study gold returns. Ma and Patterson (2013) apply quantile regressions to study the links between the gold price and its macroeconomic and financial determinants. Mensi et al. (2014) use quantile regressions to study how emergingmarket stock-market returns depend on gold returns and other macroeconomic and financial factors. Dee et al. (2013) use quantile regressions to explore the link between gold returns, stock-market movements and inflation, and Zagaglia and Marzo (2013) use quantile regressions to study the link between gold returns and exchange-rate movements. Jeong et al. (2012) develop a test for Granger causality in conditional quantiles and apply their test to study the causal links between gold returns, oil-price returns, and
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exchange-rate movements. Baur (2013) uses quantile regressions to study the link between gold excess returns and the excess returns on a commodity index. Ciner (2015) shows that the quantile regressions render it possible to recover links between, on one hand, CAPM betas and returns of stocks of precious metal mining firms and, on the other hand, trading volume. Forecasting the conditional quantiles of the distribution of gold returns is a natural forecasting strategy if a forecaster has an asymmetric loss function (for an illustration, see Koenker and Hallock, 2001, p. 146). An asymmetric loss function easily arises in a risk-management context, or simply because of behavioral biases or strategic behavior of forecasters (Laster et al., 1999; Pierdzioch et al., 2013). In recent research on the determinants of gold returns, Pierdzioch et al. (2014b) use asymmetric loss functions to measure the accuracy of out-of-sample forecasts of gold returns. Building on research by Campbell and Thompson (2008), they evaluate forecasts using an out-of-sample R2 statistic that can be computed under a symmetric and an asymmetric loss function. The computation of forecasts, however, uses the real-time forecasting approach developed by Pesaran and Timmermann (1995, 2000). This approach accounts for model uncertainty and model instability but rests on the assumption that a forecaster has a symmetric loss function because forecasting regressions are estimated by the ordinary-least-squares technique. Hence, in case a forecaster has an asymmetric loss function, the problem arises that the loss function used to compute forecasts differs from the loss function used to evaluate forecasts. The real-time quantile-regression approach that we study in this research overcomes this problem because, as Koenker and Machado (1999) have shown, the potentially asymmetric loss function used for forecast computation can also be used for forecast evaluation. The real-time quantile-regression approach, thus, is an integrated approach to forecasting and evaluating gold returns under asymmetric loss. We organize the remainder of this research as follows. In Section 2, we outline the real-time quantile-regression approach and we describe how we evaluate forecasts under an asymmetric loss function. In Section 3, we describe our data and we lay out our empirical results. In Section 4, we conclude.
2. The real-time quantile-regression approach We assume that a forecaster considers n macroeconomic and financial variables, x j, t , j = 1, … , n, as potential predictors for gold returns, rt + 1, in period of time tþ1. The forecasting model is of the general format where rt + 1 = β0 + β1x1, t + ⋯βn x n, t + ut + 1, ut + 1 = disturbance term and β j , j = 0, 1, 2, … , n are regression coefficients to be estimated. A key problem is that a forecasting model that features all predictor variables is not necessarily the best forecasting model. In principle, a forecaster can choose, in every period of time, t, among the competing forecasting models that feature alternative combinations of the predictor variables (Pesaran and Timmermann, 1995, 2000). Accordingly, we account for model uncertainty by estimating in every period of time, t, all possible combinations of forecasting models, given the n predictor variables. We use a quantile-regression approach (Koenker and Basset, 1978, for a textbook exposition, see Koenker, 2005) to estimate the forecasting models and, thereby, take into account that a forecaster may have an asymmetric loss function. The following period-loss function forms the foundation of the quantile-regression approach
3 (α, u^ jt + 1, m, α ) = u^t + 1, m, α (α − 1 (u^t + 1, m, α < 0)),
(1) ^ where 1 (·) = indicator function, and ut , m, α = forecast error for
model m in period of time t, given the quantile parameter, α ∈ (0, 1). The forecast error is defined as actual returns minus the forecast. If α ¼0.5, the loss function is symmetric in the absolute forecast error, while for α < 0.5 ( α > 0.5) the loss of a negative (positive) forecast error exceeds the loss of a positive (negative) forecast error. In the symmetric case with α ¼0.5, a forecaster should target the median of the distribution of gold returns. If the quantile parameter assumes a value α < 0.5 ( α > 0.5), a forecaster should target the α-quantile of the distribution of gold returns, requiring a downward (upward) adjustment of forecasts to make positive (negative) forecast errors more likely than in the case of α ¼0.5. Given a quantile parameter, α, we sum up over the period-loss functions to compute the total loss and choose, for every model, m, the parameters, βα , to minimize t
3 (α, m, t ) = min βα
∑ j = 01
3 (α, u^ j + 1, m, α ), (2)
where t denotes the latest period of time for which data that can be used to forecast gold returns are available, and u^ j + 1, m, α is interpreted as the in-sample forecast error. The notation βα emphasizes that the parameters of the forecasting models can differ across quantiles. In every period of time, t, we select an optimal forecasting model by comparing the m estimated models with a benchmark model, b. To this end, we compute min cα, m, t , with
cα, m, t = γm, α 3 (α , m , t ) /3 b (α , t ), where γm, α = (t − 1) /(t − l β, m, α ) penalizes model complexity, l β, m, α = length of the vector of regression parameters (neglecting the constant) for model m given the quantile parameter, α, and 3 b = loss under a benchmark model (autoregressive model of order one). In addition, we study various forecast-averaging schemes. Specifically, we compute the mean and the median of the out-of-sample forecasts implied by the m estimated models, and we compute a weighted out-of-sample forecast using, for simplicity, 1/cα, m, t as weights (weights are scaled to sum up to unity across models; for other averaging schemes, see Meligkotsidou et al., 2014). A quantile regression captures potential shifts in the links between gold returns and its macroeconomic and financial determinants across the distribution of gold returns. However, the links between gold returns and its determinants may also change over time due to, for example, financial crises and structural breaks. We account for the resulting model instability by recursively reestimating all possible combinations of forecasting models in every period of time, t, as new data become available. Nicolau and Palomba (2015) argue that using a recursive rather than a rolling-window estimation approach has the advantages that there is no need to specify the length of the rolling window and the information used for estimation is maximized since no observations are dropped to fix the length of the rolling window. For forecast evaluation, we use the out-of-sample R2 statistic studied in the context of forecasting gold returns by Pierdzioch et al. (2014b). Their out-of-sample R2 statistic is similar to the goodness-of-fit criterion for quantile regressions proposed by Koenker and Machado (1999), and it extends the out-of-sample R2 statistic analyzed by Campbell and Thompson (2008) to the case of an asymmetric loss function. For our quantile-regression approach, the out-of-sample R2 statistic is given by R2 (α , b) = 1 − 3 (α ) /3 b (α ), where 3 (α ) = sum of the out-of-sample losses, and 3 b (α ) = sum of the out-of-sample losses for a benchmark model. Hence, the loss function used for forecast evaluation is identical to the loss function used for forecast computation. Given an α-quantile, the cumulated loss, 3 (α ), is computed as the sum of losses implied by the one-period-ahead forecast errors obtained either from a
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sequence of selected optimal forecasting models or from one of the forecast-averaging schemes. If R2 (α , b) > 0 ( R2 (α , b) < 0), the cumulated loss is smaller (larger) relative to a benchmark model.
3. Empirical application We study monthly data for the sample period 01/1987–09/ 2014. Fig. 1 plots the gold price (end-of-month data). The gold price started to rise substantially and persistently at around 2000. This period of stable growth ended when Lehman Brothers collapsed in autumn 2008. When the ensuing financial crisis hit the U.S. economy and asset prices started to tumble, investment in gold started growing, which led to a rapid increase in the gold price and increasing volatility. By contrast, the volatility of goldprice fluctuations remained relatively low during the recent European debt crisis that started around 2009. The gold price reached a peak in autumn 2011. Thereafter, the market showed signs of “overheating”, and a substantial decline in the gold price started in September 2012 and continued until June 2013. Some commentators pointed out that this decline was triggered by the fact that inflation dynamics did not gather steam in the U.S. and the Eurozone despite the massive expansion of money supply, implying that there was no need to invest in gold as an inflation hedge and investors resolved gold positions. Our list of predictor variables is more or less standard. Researchers have studied similar predictor variables in earlier research in various configurations and contexts (Shafiee and Topal, 2010; Beckmann and Czudaj, 2013a, 2013b; Reboredo, 2013a, 2013b; Pierdzioch et al., 2014a, 2014b, among others). Our list of predictors comprises the following variables1:
Inflation rate (year-on-year change of the U.S. consumer price
index; all urban consumers, all items; publication lag of 1 month). Industrial production (year-on-year change; publication lag of 1 month). Continuously compounded returns of the exchange rate (yearon-year change; trade-weighted effective nominal U.S. exchange rate; publication lag of 1 month). Continuously compounded month-on-month returns of the nominal price of oil (West Texas Intermediate). Term spread (10-Year constant maturity T-Bond rate minus 3-months T-Bill rate). Corporate bond spread (Moody's seasoned corporate Baa minus Aaa bond yields). Continuously compounded month-on-month returns of the S&P 500 Composite index.
Our list of predictor variables is by no means exhaustive. Depending on the specific research purpose that a researcher focuses
Fig. 1. Gold price.
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on, additional predictor variables can be added to this list. In this research, our purpose is to illustrate the usefulness of the realtime quantile-regression approach for forecasting gold returns under the asymmetric loss, and so our list of predictor variables, on one hand, comprises several predictor variables that are fairly standard in the literature and, on the other hand, limits the amount of CPU time needed to implement the real-time quantileregression approach. In addition, we always include lagged gold returns in the forecasting model because an autoregressive model of order one forms the benchmark model. As all other regression parameters, the coefficient of lagged gold returns can vary across quantiles. Continuously compounded gold returns are defined as the differenced natural log of the end-of-month gold price. Fig. 2 plots the gold returns and the other predictor variables. As for the benchmark model, three remarks are in order. First, we do not consider higher-order autoregressive terms for our benchmark model. Inspection of the autocorrelation function (not reported) showed that the significant coefficient of first-order autocorrelation assumes a value of 0.12. Higher-order coefficients assume small values and become significant again only at lag 11. Second, we estimate the benchmark model recursively, implying that the coefficients of the model can change over time (and that the model nests the random-walk model as a special case). Third, the benchmark model is quantile specific, that is, we estimate a separate benchmark model for every quantile that we study. In order to analyze the incremental value of the forecasting models relative to the benchmark model, the forecasting models always include a constant plus the lagged gold returns and at least one of the other regressors. We use data up to and including 12/1999 to initialize the real-time quantile-regression approach. The choice of the initialization period is always to some extent arbitrary. Using 12/ 1999 as the last period of the initialization period implies that the training period and the forecasting period contain approximately the same number of observations. We obtain qualitatively similar results when we vary the initialization period (see Panel D of Fig. 3). We recursively repeat the process of model estimation and forecast computation in every new month and, at the end of the sample period, we compute the out-of-sample relative loss criterion.2 Table 1 shows how often (in percent) the various predictor variables are included in the selected optimal forecasting models (that is, when we do not average forecasts across models). The results show how the choice of the quantile parameter, α, governs the selection of the predictor variables. For example, industrial production is selected as a regressor in approximately 44% of all optimal forecasting models if the quantile parameter assumes the value α ¼ 0.25, but the importance of this predictor drops to only roughly 3% if the quantile parameter assumes the value α ¼0.75. Stock-market returns, in turn, are relatively unimportant if we fix the quantile parameter at α ¼ 0.25, but their importance increases from 12.5% to more than 96% if we switch the quantile parameter to α ¼0.75. In other words, forecasters who differ in the shape of their loss functions will come to different conclusions regarding the relative importance of the predictor variables. Fig. 3 illustrates further empirical results. Panel A shows that the out-of-sample R2 statistic is increasing in the quantile parameter, where forecast averaging yields more stable results than forecast selection. As expected, the results based on the mean forecast are very close to the results for the median forecast. Also, 1 The data were downloaded from the homepage of the Federal Reserve of St. Louis (see http://research.stlouisfed.org/fred2/). 2 All empirical results reported in this research were computed using the free R programming environment (R Development Core Team, 2014). The quantile regressions were estimated using the quantreg package developed by Koenker (2013).
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Fig. 2. Predictor variables.
because cα, m, t hovers around unity, the mean and the weighted forecasts are very similar. A forecaster who has a loss function characterized by α > 0.5 (that is, a forecaster for whom underestimating gold returns is more costly than overestimating) benefits from forecasts to a larger extent than a forecaster with a loss function featuring α < 0.5. When we average forecasts, the out-ofsample R2 statistic shows that a forecaster with a loss function characterized by α > 0.5 realizes benefits of up to roughly 5% relative to the benchmark model.3 Panel B plots the loss functions for three different quantile parameters, where a positive (negative) forecast error arises in 3 While in principle it may be possible that an investor can exploit a small outof-sample R2 statistic to generate economically meaningful profits (Campbell and Thompson, 2008), it is important to note that our results are silent with regard to the efficiency of the gold market.
case of an underprediction (overprediction). Two of the plotted loss functions represent a forecaster for whom an underprediction of gold returns is more costly than an overprediction of the same size ( α > 0.5). For such a forecaster, the loss functions mimic the payoff function of a so-called strap option strategy.4 A short strap option strategy requires that a forecaster sells more call options than put options, where the options should have the same strike price and the same time to expiry. If the strike price is equal to the expected gold price, and both the actual and the predicted change in the gold price are positive, a forecaster incurs a larger loss in case that the actual increase in the gold price turns out to be larger than the expected increase than in case of an overestimation of the price increase of the same absolute magnitude. Similarly, if the 4
For an overview of option trading strategies, see Hull (2012, p. 246).
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Fig. 3. Empirical results.
strike price is equal to the expected gold price, and both the actual and the predicted change in the gold price are positive, a forecaster who invests in a short strip option strategy, which requires selling more put options than call options, would incur a larger loss in case of an overestimation than in case of an underestimation of the same absolute magnitude. The result is a payoff
function that mimics a loss functions that arise in the case of α < 0.5. Hence, depending on the specific option strategy being considered, the results given in Table 1 show that differences in option strategies and, thus, differences in loss functions require that forecasters pay attention to different macroeconomic and financial variables when forming their forecasts.
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Table 1 Selection of predictor variables. α
0.25
Δcpit − 1
32.386
84.659
80.114
53.409
81.250
Δipt − 1
44.318
46.023
43.750
51.705
2.841
Δext − 1
Δoilt
39.773 5.114
27.841 31.818
69.318 49.432
42.614 50.000
57.955 90.341
0.45
0.5
0.65
0.75
Δsp500t
12.500
0.000
44.886
92.045
96.023
term spreadt
0.000
58.523
47.727
92.045
96.023
bond spreadt
0.000
1.136
53.409
10.795
2.841
Note: We use data up to and including 12/1999 to initialize the real-time quantileregression approach. The predictors used to predict 1-month-ahead returns of the price of gold are defined as follows: Δcpit − 1 = the year-on-year change in the (ln) consumer price index (all urban consumers, all items) lagged one period, Δipt − 1 = the year-on-year change in the (ln) industrial production lagged one period, Δext − 1 = the year-on-year change in the (ln) trade-weighted nominal exchange rate lagged one period, Δdjiat = the continuously compounded month-onmonth rate of change in the S&P 500 index, Δoilt = the continuously compounded month-on-month rate of change in the nominal price of oil (West Texas Intermediate), term spreadt = the difference between the interest rate on a 10-year T-Bond and a 3-month T-Bill, bond spreadt = the difference between Moody's corporate Baa and Aaa bond yields. The numbers given in this table are in percent.
Panel C shows that a quantile parameter α < 0.5 requires a downward adjustment of forecasts relative to the case of a symmetric loss function because the loss from overestimating exceeds
the loss from underestimating gold returns. Conversely, a quantile parameter α > 0.5 leads to an upward adjustment of forecasts. The upward/downward adjustment of forecasts gets successively stronger when the quantile parameter reaches the boundaries of its domain (α reaches 0 or 1). As a result, for extreme values of the quantile parameter both the quantile regression model and the benchmark model persistently over-/underestimate gold returns. The losses implied by both models then get similar and the out-ofsample R2 approaches zero. In our analysis, we do not consider extreme values of the quantile parameter but focus on the more interesting question how intermediate values of the quantile parameter affect the out-of-sample R2. Panel D illustrates that the out-of-sample R2 statistic is insensitive to the choice of a somewhat shorter (ending 12/1996) or longer initialization period. When we increase the initialization period, the out-of-sample relative loss criterion remains positive for α > 0.5 and negative for α < 0.5. For a symmetric loss function, the out-of-sample relative loss criterion approaches zero as the initialization period gets longer. Panel E shows the results for a rolling rather than a recursive estimation window. For the rolling-estimation window, we drop one observation at the beginning of the estimation window when we add a new observation at the end of the estimation window. In other words, a rolling-estimation window implies that the number of observations being used for estimating the forecasting models is constant over time. Starting in 12/1999, results are similar to the
Fig. 4. Empirical results for longer-term returns.
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results plotted in Panel A with the exception that the out-ofsample R2 statistic is small and unstable for the forecast-selection scheme. Panel F plots kernel density estimates of the sampling distribution of the out-of-sample R2 statistic constructed using a bootstrap simulation. In order to retain the dynamics of the data, the bootstrap simulation is based on the block boostrap described by Politis and Romano (1994), where the mean block length is one quarter. We present results for 500 simulation runs and focus on the cases α ¼ 0.5 and α ¼ 0.65 (see also Panel B). Focusing on the case of model averaging (weighted forecasts), the one-sided null hypothesis is H0: R2 ≤ 0 and can be rejected for α ¼0.65 with a pvalue of approximately 5%, confirming a finding by Pierdzioch et al. (2014b) that even relatively small realizations of the out-ofsample R2 statistic can be statistically significant.5 The bootstrap simulation yields an insignificant result when we use a symmetric loss function. Finally, we plot in Fig. 4 the out-of-sample R2 statistic for longer-term returns. Results for one-quarter-ahead returns are similar to the results documented in Panel A of Fig. 3 except that the out-of-sample R2 statistic now also is positive for α < 0.4 . For 1-year-ahead returns, the out-of-sample R2 is smaller for α > 0.5 than in the case of monthly and quarterly returns, but we still get the result that a forecaster who has a loss function characterized by α > 0.5 benefits from the forecasts. For one-quarter-ahead rolling-window forecasts, the out-of-sample R2 statistic again increases in the quantile parameter for the forecast-averaging schemes. The results for the forecast-selection scheme are rather volatile and show no discernible pattern (see also Panel E of Fig. 3). For 1-year-ahead rolling-window forecasts, the out-of-sample R2 statistic is small and for several quantile parameters hardly different from zero.
4. Concluding remarks The real-time quantile-regression approach provides a unified platform to forecast and evaluate gold returns under asymmetric loss. The approach is easy to implement given that several popular statistical software programs feature commands or packages that make it straightforward to estimate quantile regressions. Moreover, the approach can be easily adapted to forecast other commodity prices under asymmetric loss whenever it is important to account for model uncertainty and model instability.
Acknowledgments We thank an anonymous reviewer for helpful comments. The usual disclaimer applies.
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A real-time quantile-regression approach to forecasting gold returns under asymmetric loss Christian Pierdzioch n, Marian Risse, Sebastian Rohloff Department of Economics, Helmut Schmidt University, Holstenhofweg 85, P.O.B. 700822, 22008 Hamburg, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 17 March 2015 Received in revised form 30 May 2015 Accepted 13 July 2015
We propose a real-time quantile-regression approach to analyze whether widely studied macroeconomic and financial variables help to forecast out-of-sample gold returns. The real-time quantile-regression approach accounts for model uncertainty, model instability, and the possibility that a forecaster has an asymmetric loss function. Forecasts are computed and evaluated using the same asymmetric loss function. When the loss function implies that an underestimation is somewhat more costly than an overestimation of the same size, the forecasts computed using the real-time quantile-regression approach outperform forecasts implied by an autoregressive benchmark model. & 2015 Elsevier Ltd. All rights reserved.
JEL classification: C53 E44 Keywords: Quantile regression Forecasting Asymmetric loss Gold returns
1. Introduction Against the background of recent financial market turbulences, research on the determinants of gold returns has mushroomed. Among the determinants that researchers have studied are the inflation rate (Beckmann and Czudaj, 2013a; Batten et al., 2014), the oil price (Zhang and Wei, 2010; Reboredo, 2013b), the exchange rate (Pukthuanthong and Roll, 2011; Reboredo, 2013b), and business-cycle fluctuations (Pierdzioch et al., 2014b). Studying whether the various determinants studied in earlier literature help to forecast gold returns is important because the properties of gold as a safe-haven investment, a low-correlation portfolio diversifier, and a hedge against fluctuating currency values have received much attention in recent research (Hillier et al., 2006; Joy, 2011; Ciner et al., 2013, to name just a few). Despite many research efforts, no consensus has emerged regarding the core determinants of gold returns. As a result, researchers have studied gold returns by applying flexible forecasting approaches that account for model uncertainty and model instability (see also Vrugt et al., 2007; Aye et al., 2015; Baur et al., 2014; Pierdzioch et al., 2014a, 2014b, 2015). Model uncertainty arises because gold returns may be linked to a potentially large number of determinants, none of which can be excluded a priori n
Corresponding author. E-mail address: [email protected] (C. Pierdzioch).
http://dx.doi.org/10.1016/j.resourpol.2015.07.002 0301-4207/& 2015 Elsevier Ltd. All rights reserved.
on economic grounds. Model instability arises because the relative importance of these determinants most likely has changed over time (Baur, 2011; Batten et al., 2014) and may be state dependent (Wang and Lee, 2011; Wang et al., 2011). We contribute to earlier literature in that we propose a realtime quantile-regression approach to forecast gold returns. A quantile-regression approach renders it possible to compute forecasts that target the conditional quantiles rather than the conditional mean of the distribution of gold returns. Quantile regressions have received growing attention in the recent finance literature (Basset and Chen, 2001; Engle and Manganelli, 2004; Chuang et al., 2009; Baur et al., 2012) and have been studied recently in a forecasting context by Meligkotsidou et al. (2014), Manzan (2015), and Pedersen (2015). Quantile regressions also have been applied to study gold returns. Ma and Patterson (2013) apply quantile regressions to study the links between the gold price and its macroeconomic and financial determinants. Mensi et al. (2014) use quantile regressions to study how emergingmarket stock-market returns depend on gold returns and other macroeconomic and financial factors. Dee et al. (2013) use quantile regressions to explore the link between gold returns, stock-market movements and inflation, and Zagaglia and Marzo (2013) use quantile regressions to study the link between gold returns and exchange-rate movements. Jeong et al. (2012) develop a test for Granger causality in conditional quantiles and apply their test to study the causal links between gold returns, oil-price returns, and
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exchange-rate movements. Baur (2013) uses quantile regressions to study the link between gold excess returns and the excess returns on a commodity index. Ciner (2015) shows that the quantile regressions render it possible to recover links between, on one hand, CAPM betas and returns of stocks of precious metal mining firms and, on the other hand, trading volume. Forecasting the conditional quantiles of the distribution of gold returns is a natural forecasting strategy if a forecaster has an asymmetric loss function (for an illustration, see Koenker and Hallock, 2001, p. 146). An asymmetric loss function easily arises in a risk-management context, or simply because of behavioral biases or strategic behavior of forecasters (Laster et al., 1999; Pierdzioch et al., 2013). In recent research on the determinants of gold returns, Pierdzioch et al. (2014b) use asymmetric loss functions to measure the accuracy of out-of-sample forecasts of gold returns. Building on research by Campbell and Thompson (2008), they evaluate forecasts using an out-of-sample R2 statistic that can be computed under a symmetric and an asymmetric loss function. The computation of forecasts, however, uses the real-time forecasting approach developed by Pesaran and Timmermann (1995, 2000). This approach accounts for model uncertainty and model instability but rests on the assumption that a forecaster has a symmetric loss function because forecasting regressions are estimated by the ordinary-least-squares technique. Hence, in case a forecaster has an asymmetric loss function, the problem arises that the loss function used to compute forecasts differs from the loss function used to evaluate forecasts. The real-time quantile-regression approach that we study in this research overcomes this problem because, as Koenker and Machado (1999) have shown, the potentially asymmetric loss function used for forecast computation can also be used for forecast evaluation. The real-time quantile-regression approach, thus, is an integrated approach to forecasting and evaluating gold returns under asymmetric loss. We organize the remainder of this research as follows. In Section 2, we outline the real-time quantile-regression approach and we describe how we evaluate forecasts under an asymmetric loss function. In Section 3, we describe our data and we lay out our empirical results. In Section 4, we conclude.
2. The real-time quantile-regression approach We assume that a forecaster considers n macroeconomic and financial variables, x j, t , j = 1, … , n, as potential predictors for gold returns, rt + 1, in period of time tþ1. The forecasting model is of the general format where rt + 1 = β0 + β1x1, t + ⋯βn x n, t + ut + 1, ut + 1 = disturbance term and β j , j = 0, 1, 2, … , n are regression coefficients to be estimated. A key problem is that a forecasting model that features all predictor variables is not necessarily the best forecasting model. In principle, a forecaster can choose, in every period of time, t, among the competing forecasting models that feature alternative combinations of the predictor variables (Pesaran and Timmermann, 1995, 2000). Accordingly, we account for model uncertainty by estimating in every period of time, t, all possible combinations of forecasting models, given the n predictor variables. We use a quantile-regression approach (Koenker and Basset, 1978, for a textbook exposition, see Koenker, 2005) to estimate the forecasting models and, thereby, take into account that a forecaster may have an asymmetric loss function. The following period-loss function forms the foundation of the quantile-regression approach
3 (α, u^ jt + 1, m, α ) = u^t + 1, m, α (α − 1 (u^t + 1, m, α < 0)),
(1) ^ where 1 (·) = indicator function, and ut , m, α = forecast error for
model m in period of time t, given the quantile parameter, α ∈ (0, 1). The forecast error is defined as actual returns minus the forecast. If α ¼0.5, the loss function is symmetric in the absolute forecast error, while for α < 0.5 ( α > 0.5) the loss of a negative (positive) forecast error exceeds the loss of a positive (negative) forecast error. In the symmetric case with α ¼0.5, a forecaster should target the median of the distribution of gold returns. If the quantile parameter assumes a value α < 0.5 ( α > 0.5), a forecaster should target the α-quantile of the distribution of gold returns, requiring a downward (upward) adjustment of forecasts to make positive (negative) forecast errors more likely than in the case of α ¼0.5. Given a quantile parameter, α, we sum up over the period-loss functions to compute the total loss and choose, for every model, m, the parameters, βα , to minimize t
3 (α, m, t ) = min βα
∑ j = 01
3 (α, u^ j + 1, m, α ), (2)
where t denotes the latest period of time for which data that can be used to forecast gold returns are available, and u^ j + 1, m, α is interpreted as the in-sample forecast error. The notation βα emphasizes that the parameters of the forecasting models can differ across quantiles. In every period of time, t, we select an optimal forecasting model by comparing the m estimated models with a benchmark model, b. To this end, we compute min cα, m, t , with
cα, m, t = γm, α 3 (α , m , t ) /3 b (α , t ), where γm, α = (t − 1) /(t − l β, m, α ) penalizes model complexity, l β, m, α = length of the vector of regression parameters (neglecting the constant) for model m given the quantile parameter, α, and 3 b = loss under a benchmark model (autoregressive model of order one). In addition, we study various forecast-averaging schemes. Specifically, we compute the mean and the median of the out-of-sample forecasts implied by the m estimated models, and we compute a weighted out-of-sample forecast using, for simplicity, 1/cα, m, t as weights (weights are scaled to sum up to unity across models; for other averaging schemes, see Meligkotsidou et al., 2014). A quantile regression captures potential shifts in the links between gold returns and its macroeconomic and financial determinants across the distribution of gold returns. However, the links between gold returns and its determinants may also change over time due to, for example, financial crises and structural breaks. We account for the resulting model instability by recursively reestimating all possible combinations of forecasting models in every period of time, t, as new data become available. Nicolau and Palomba (2015) argue that using a recursive rather than a rolling-window estimation approach has the advantages that there is no need to specify the length of the rolling window and the information used for estimation is maximized since no observations are dropped to fix the length of the rolling window. For forecast evaluation, we use the out-of-sample R2 statistic studied in the context of forecasting gold returns by Pierdzioch et al. (2014b). Their out-of-sample R2 statistic is similar to the goodness-of-fit criterion for quantile regressions proposed by Koenker and Machado (1999), and it extends the out-of-sample R2 statistic analyzed by Campbell and Thompson (2008) to the case of an asymmetric loss function. For our quantile-regression approach, the out-of-sample R2 statistic is given by R2 (α , b) = 1 − 3 (α ) /3 b (α ), where 3 (α ) = sum of the out-of-sample losses, and 3 b (α ) = sum of the out-of-sample losses for a benchmark model. Hence, the loss function used for forecast evaluation is identical to the loss function used for forecast computation. Given an α-quantile, the cumulated loss, 3 (α ), is computed as the sum of losses implied by the one-period-ahead forecast errors obtained either from a
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sequence of selected optimal forecasting models or from one of the forecast-averaging schemes. If R2 (α , b) > 0 ( R2 (α , b) < 0), the cumulated loss is smaller (larger) relative to a benchmark model.
3. Empirical application We study monthly data for the sample period 01/1987–09/ 2014. Fig. 1 plots the gold price (end-of-month data). The gold price started to rise substantially and persistently at around 2000. This period of stable growth ended when Lehman Brothers collapsed in autumn 2008. When the ensuing financial crisis hit the U.S. economy and asset prices started to tumble, investment in gold started growing, which led to a rapid increase in the gold price and increasing volatility. By contrast, the volatility of goldprice fluctuations remained relatively low during the recent European debt crisis that started around 2009. The gold price reached a peak in autumn 2011. Thereafter, the market showed signs of “overheating”, and a substantial decline in the gold price started in September 2012 and continued until June 2013. Some commentators pointed out that this decline was triggered by the fact that inflation dynamics did not gather steam in the U.S. and the Eurozone despite the massive expansion of money supply, implying that there was no need to invest in gold as an inflation hedge and investors resolved gold positions. Our list of predictor variables is more or less standard. Researchers have studied similar predictor variables in earlier research in various configurations and contexts (Shafiee and Topal, 2010; Beckmann and Czudaj, 2013a, 2013b; Reboredo, 2013a, 2013b; Pierdzioch et al., 2014a, 2014b, among others). Our list of predictors comprises the following variables1:
Inflation rate (year-on-year change of the U.S. consumer price
index; all urban consumers, all items; publication lag of 1 month). Industrial production (year-on-year change; publication lag of 1 month). Continuously compounded returns of the exchange rate (yearon-year change; trade-weighted effective nominal U.S. exchange rate; publication lag of 1 month). Continuously compounded month-on-month returns of the nominal price of oil (West Texas Intermediate). Term spread (10-Year constant maturity T-Bond rate minus 3-months T-Bill rate). Corporate bond spread (Moody's seasoned corporate Baa minus Aaa bond yields). Continuously compounded month-on-month returns of the S&P 500 Composite index.
Our list of predictor variables is by no means exhaustive. Depending on the specific research purpose that a researcher focuses
Fig. 1. Gold price.
301
on, additional predictor variables can be added to this list. In this research, our purpose is to illustrate the usefulness of the realtime quantile-regression approach for forecasting gold returns under the asymmetric loss, and so our list of predictor variables, on one hand, comprises several predictor variables that are fairly standard in the literature and, on the other hand, limits the amount of CPU time needed to implement the real-time quantileregression approach. In addition, we always include lagged gold returns in the forecasting model because an autoregressive model of order one forms the benchmark model. As all other regression parameters, the coefficient of lagged gold returns can vary across quantiles. Continuously compounded gold returns are defined as the differenced natural log of the end-of-month gold price. Fig. 2 plots the gold returns and the other predictor variables. As for the benchmark model, three remarks are in order. First, we do not consider higher-order autoregressive terms for our benchmark model. Inspection of the autocorrelation function (not reported) showed that the significant coefficient of first-order autocorrelation assumes a value of 0.12. Higher-order coefficients assume small values and become significant again only at lag 11. Second, we estimate the benchmark model recursively, implying that the coefficients of the model can change over time (and that the model nests the random-walk model as a special case). Third, the benchmark model is quantile specific, that is, we estimate a separate benchmark model for every quantile that we study. In order to analyze the incremental value of the forecasting models relative to the benchmark model, the forecasting models always include a constant plus the lagged gold returns and at least one of the other regressors. We use data up to and including 12/1999 to initialize the real-time quantile-regression approach. The choice of the initialization period is always to some extent arbitrary. Using 12/ 1999 as the last period of the initialization period implies that the training period and the forecasting period contain approximately the same number of observations. We obtain qualitatively similar results when we vary the initialization period (see Panel D of Fig. 3). We recursively repeat the process of model estimation and forecast computation in every new month and, at the end of the sample period, we compute the out-of-sample relative loss criterion.2 Table 1 shows how often (in percent) the various predictor variables are included in the selected optimal forecasting models (that is, when we do not average forecasts across models). The results show how the choice of the quantile parameter, α, governs the selection of the predictor variables. For example, industrial production is selected as a regressor in approximately 44% of all optimal forecasting models if the quantile parameter assumes the value α ¼ 0.25, but the importance of this predictor drops to only roughly 3% if the quantile parameter assumes the value α ¼0.75. Stock-market returns, in turn, are relatively unimportant if we fix the quantile parameter at α ¼ 0.25, but their importance increases from 12.5% to more than 96% if we switch the quantile parameter to α ¼0.75. In other words, forecasters who differ in the shape of their loss functions will come to different conclusions regarding the relative importance of the predictor variables. Fig. 3 illustrates further empirical results. Panel A shows that the out-of-sample R2 statistic is increasing in the quantile parameter, where forecast averaging yields more stable results than forecast selection. As expected, the results based on the mean forecast are very close to the results for the median forecast. Also, 1 The data were downloaded from the homepage of the Federal Reserve of St. Louis (see http://research.stlouisfed.org/fred2/). 2 All empirical results reported in this research were computed using the free R programming environment (R Development Core Team, 2014). The quantile regressions were estimated using the quantreg package developed by Koenker (2013).
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Fig. 2. Predictor variables.
because cα, m, t hovers around unity, the mean and the weighted forecasts are very similar. A forecaster who has a loss function characterized by α > 0.5 (that is, a forecaster for whom underestimating gold returns is more costly than overestimating) benefits from forecasts to a larger extent than a forecaster with a loss function featuring α < 0.5. When we average forecasts, the out-ofsample R2 statistic shows that a forecaster with a loss function characterized by α > 0.5 realizes benefits of up to roughly 5% relative to the benchmark model.3 Panel B plots the loss functions for three different quantile parameters, where a positive (negative) forecast error arises in 3 While in principle it may be possible that an investor can exploit a small outof-sample R2 statistic to generate economically meaningful profits (Campbell and Thompson, 2008), it is important to note that our results are silent with regard to the efficiency of the gold market.
case of an underprediction (overprediction). Two of the plotted loss functions represent a forecaster for whom an underprediction of gold returns is more costly than an overprediction of the same size ( α > 0.5). For such a forecaster, the loss functions mimic the payoff function of a so-called strap option strategy.4 A short strap option strategy requires that a forecaster sells more call options than put options, where the options should have the same strike price and the same time to expiry. If the strike price is equal to the expected gold price, and both the actual and the predicted change in the gold price are positive, a forecaster incurs a larger loss in case that the actual increase in the gold price turns out to be larger than the expected increase than in case of an overestimation of the price increase of the same absolute magnitude. Similarly, if the 4
For an overview of option trading strategies, see Hull (2012, p. 246).
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Fig. 3. Empirical results.
strike price is equal to the expected gold price, and both the actual and the predicted change in the gold price are positive, a forecaster who invests in a short strip option strategy, which requires selling more put options than call options, would incur a larger loss in case of an overestimation than in case of an underestimation of the same absolute magnitude. The result is a payoff
function that mimics a loss functions that arise in the case of α < 0.5. Hence, depending on the specific option strategy being considered, the results given in Table 1 show that differences in option strategies and, thus, differences in loss functions require that forecasters pay attention to different macroeconomic and financial variables when forming their forecasts.
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Table 1 Selection of predictor variables. α
0.25
Δcpit − 1
32.386
84.659
80.114
53.409
81.250
Δipt − 1
44.318
46.023
43.750
51.705
2.841
Δext − 1
Δoilt
39.773 5.114
27.841 31.818
69.318 49.432
42.614 50.000
57.955 90.341
0.45
0.5
0.65
0.75
Δsp500t
12.500
0.000
44.886
92.045
96.023
term spreadt
0.000
58.523
47.727
92.045
96.023
bond spreadt
0.000
1.136
53.409
10.795
2.841
Note: We use data up to and including 12/1999 to initialize the real-time quantileregression approach. The predictors used to predict 1-month-ahead returns of the price of gold are defined as follows: Δcpit − 1 = the year-on-year change in the (ln) consumer price index (all urban consumers, all items) lagged one period, Δipt − 1 = the year-on-year change in the (ln) industrial production lagged one period, Δext − 1 = the year-on-year change in the (ln) trade-weighted nominal exchange rate lagged one period, Δdjiat = the continuously compounded month-onmonth rate of change in the S&P 500 index, Δoilt = the continuously compounded month-on-month rate of change in the nominal price of oil (West Texas Intermediate), term spreadt = the difference between the interest rate on a 10-year T-Bond and a 3-month T-Bill, bond spreadt = the difference between Moody's corporate Baa and Aaa bond yields. The numbers given in this table are in percent.
Panel C shows that a quantile parameter α < 0.5 requires a downward adjustment of forecasts relative to the case of a symmetric loss function because the loss from overestimating exceeds
the loss from underestimating gold returns. Conversely, a quantile parameter α > 0.5 leads to an upward adjustment of forecasts. The upward/downward adjustment of forecasts gets successively stronger when the quantile parameter reaches the boundaries of its domain (α reaches 0 or 1). As a result, for extreme values of the quantile parameter both the quantile regression model and the benchmark model persistently over-/underestimate gold returns. The losses implied by both models then get similar and the out-ofsample R2 approaches zero. In our analysis, we do not consider extreme values of the quantile parameter but focus on the more interesting question how intermediate values of the quantile parameter affect the out-of-sample R2. Panel D illustrates that the out-of-sample R2 statistic is insensitive to the choice of a somewhat shorter (ending 12/1996) or longer initialization period. When we increase the initialization period, the out-of-sample relative loss criterion remains positive for α > 0.5 and negative for α < 0.5. For a symmetric loss function, the out-of-sample relative loss criterion approaches zero as the initialization period gets longer. Panel E shows the results for a rolling rather than a recursive estimation window. For the rolling-estimation window, we drop one observation at the beginning of the estimation window when we add a new observation at the end of the estimation window. In other words, a rolling-estimation window implies that the number of observations being used for estimating the forecasting models is constant over time. Starting in 12/1999, results are similar to the
Fig. 4. Empirical results for longer-term returns.
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results plotted in Panel A with the exception that the out-ofsample R2 statistic is small and unstable for the forecast-selection scheme. Panel F plots kernel density estimates of the sampling distribution of the out-of-sample R2 statistic constructed using a bootstrap simulation. In order to retain the dynamics of the data, the bootstrap simulation is based on the block boostrap described by Politis and Romano (1994), where the mean block length is one quarter. We present results for 500 simulation runs and focus on the cases α ¼ 0.5 and α ¼ 0.65 (see also Panel B). Focusing on the case of model averaging (weighted forecasts), the one-sided null hypothesis is H0: R2 ≤ 0 and can be rejected for α ¼0.65 with a pvalue of approximately 5%, confirming a finding by Pierdzioch et al. (2014b) that even relatively small realizations of the out-ofsample R2 statistic can be statistically significant.5 The bootstrap simulation yields an insignificant result when we use a symmetric loss function. Finally, we plot in Fig. 4 the out-of-sample R2 statistic for longer-term returns. Results for one-quarter-ahead returns are similar to the results documented in Panel A of Fig. 3 except that the out-of-sample R2 statistic now also is positive for α < 0.4 . For 1-year-ahead returns, the out-of-sample R2 is smaller for α > 0.5 than in the case of monthly and quarterly returns, but we still get the result that a forecaster who has a loss function characterized by α > 0.5 benefits from the forecasts. For one-quarter-ahead rolling-window forecasts, the out-of-sample R2 statistic again increases in the quantile parameter for the forecast-averaging schemes. The results for the forecast-selection scheme are rather volatile and show no discernible pattern (see also Panel E of Fig. 3). For 1-year-ahead rolling-window forecasts, the out-of-sample R2 statistic is small and for several quantile parameters hardly different from zero.
4. Concluding remarks The real-time quantile-regression approach provides a unified platform to forecast and evaluate gold returns under asymmetric loss. The approach is easy to implement given that several popular statistical software programs feature commands or packages that make it straightforward to estimate quantile regressions. Moreover, the approach can be easily adapted to forecast other commodity prices under asymmetric loss whenever it is important to account for model uncertainty and model instability.
Acknowledgments We thank an anonymous reviewer for helpful comments. The usual disclaimer applies.
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