A boosting approach to forecasting the volatility of gold-price fluctuations under flexible loss

Resources Policy 47 (2016) 95–107

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A boosting approach to forecasting the volatility of gold-price fluctuations under flexible 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 21 July 2015 Received in revised form 10 November 2015 Accepted 4 January 2016

We use a boosting approach to study the time-varying out-of-sample informational content of various financial and macroeconomic variables for forecasting the volatility of gold-price fluctuations. We use an out-of-sample R2 statistic to evaluate forecasts as a function of the shape of a forecaster's loss function. We show that, when compared to an autoregressive benchmark forecast, those forecasters tend to benefit from using predictions implied by the boosting approach who encounter a larger loss when underestimating rather than overestimating the future volatility of gold-price fluctuations. We use a simulation experiment to study the significance of this benefit. & 2016 Elsevier Ltd. All rights reserved.

JEL classification: C53 E44 Keywords: Volatility of gold-price fluctuations Forecasting Boosting approach

1. Introduction Recent financial market jitters, geopolitical uncertainty, and a volatile financial and macroeconomic environment have spurred the interest of researchers and investors in the determinants of gold-price fluctuations. Some researchers have studied whether gold is a “safe haven” investment (Baur and McDermott, 2010; Baur and Lucey, 2010), while other researchers have studied the link between gold-price fluctuations and exchange-rate movements (Pukthuanthong and Roll, 2011; Reboredo, 2013). Still other researchers have studied whether investing in gold acts as an inflation hedge (Blose, 2010; Beckmann and Czudaj, 2013) and whether gold-price fluctuations are linked to interest rates (Diba and Grossmann, 1984; Fortune, 1987). Less is known, however, about the determinants of the volatility of gold-price fluctuations. Tulley and Lucey (2007) use a power GARCH model to study the volatility of gold-price fluctuations. While their results suggest that movements in the dollar influence the returns of the gold price, macroeconomic variables do not help to explain the dynamics of the volatility of gold-price fluctuations. Cai et al. (2001) also use GARCH techniques to study the properties of the volatility of gold-price fluctuations. Their study focuses on the intraday volatility of gold-price fluctuations. They find that some macroeconomic announcements have a significant but small effect on n

Corresponding author. E-mail address: [email protected] (C. Pierdzioch).

http://dx.doi.org/10.1016/j.resourpol.2016.01.003 0301-4207/& 2016 Elsevier Ltd. All rights reserved.

the intraday volatility. Hammoudeh and Yuan (2008) use GARCH models to study the effects of oil price shocks and interest-rate shocks on the volatility of gold-price fluctuations. Bentes (2015) studies the out-of-sample performance of various GARCH models and finds that a FIGARCH model performs better in terms of common measures of forecast performance than a GARCH and an IGARCH model. Bentes (2015) does not study whether financial and macroeconomic variables help to forecast the volatiltiy of gold-price fluctuations. Kristjanpoller and Minutolo (2015) show that combining the GARCH model with an artificial-neural-network approach improves forecast performance relative to a GARCH model, where the input variables of the neural network comprise movements of major exchange rates, stock-market returns, and oil price fluctuations. Batten et al. (2010) report that the volatility of gold-price fluctuations is mainly affected by the volatility of monetary macroeconomic variables, but this effect is unstable across the subsample periods they study. Such instability of effects is consistent with the results reported by Vivian and Wohar (2012), who find evidence of idiosyncratic breaks in the volatility of various commodity prices (including the gold price). In sum, our brief review of earlier literature yields two results. First, no consensus has emerged regarding the question which financial and macroeconomic variables should be taken into account as primary drivers of the volatility of gold-price fluctuations. Second, the effects of the various financial and macroeconomic variables may be unstable over time. We account for these two results by using a boosting approach to reconsider whether financial and macroeconomic variables help to forecast the volatility

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of gold-price fluctuations. Applications of the boosting approach in economics include the research by Audrino and Bühlmann (2003), who use the approach to estimate volatility, and Bai and Ng (2009), Carriero et al. (2011), Berge (2014), and Wohlrabe and Buchen (2014). Mittnik et al. (2012) use boosted regression trees to model the volatility of four different asset classes, including the GSCI commodity index. As compared to other index components (agriculture, energy, industrial metals, and livestock) precious metals make only a tiny contribution to the GSCI index. The boosting approach starts from a simple forecasting model and then increases the dimension of the forecasting model by endogenously identifying the usefulness of a large set of financial and macroeconomic variables as predictors of the volatility of gold-price fluctuations. The identification of forecasting models is based on a model-selection algorithm that balances the complexity and fit of a forecasting model. The boosting approach, thus, accounts for a forecaster's model uncertainty. Accounting for model uncertainty is particularly important given that earlier researchers have considered several alternative financial and macroeconomic variables as potential determinants of the volatility of gold-price fluctuations. Moreover, rather than using the full sample of data to implement the boosting approach, we implement the approach such that it only uses information to build a forecasting model available to a forecaster when a forecast is formed. To this end, we implement the boosting approach recursively by restarting the search for an optimal forecasting model as new information on the volatility of gold-price fluctuations and its potential financial and macroeconomic determinants become available to a forecaster. Hence, as new information become available, a forecaster can update the optimal forecasting model. Such a search-and-updating process accounts for the possibility that the optimal forecasting model may change over time because of unstable links between the volatility of gold-price fluctuations and its financial and macroeconomic determinants. In recent research, Pierdzioch et al. (2015a,b) use the boosting approach to study the determinants of returns of the price of gold. We go beyond their research in three respects. First, we use a boosting approach to study the determinants of the volatility of gold-price fluctuations rather than the returns of the gold price. Building on the results reported by Batten et al. (2010), we mainly focus on the link between the volatility of gold-price fluctuations and various monetary and financial variables (interest rates, stock market returns, inflation rate, etc.). Second, we use four widely studied (for example, Elliott et al., 2008; Patton, 2011) asymmetric loss functions to evaluate our volatility forecasts. The asymmetric loss functions that we consider in our research nest various loss functions widely studied in the literature on volatility forecasting, including a quadratic loss function. Evaluating volatility forecasts by means of a loss-function-based approach is advantageous given that earlier research has shown that the statistical performance of forecasts may not coincide with their economic value-added (Pierdzioch et al., 2015b). Moreover, an asymmetric loss function may arise due to the mechanisms described in the behavioral economics and finance literature (for surveys, see Shiller, 1999; Barberis and Thaler, 2003) or in a risk-management context when forecasts are used, for example, to setup option-trading strategies (Pierdzioch et al., 2015c). More generally, an asymmetric loss function arises whenever a forecaster implements a portfolio strategy that yields asymmetric payoffs depending on whether a forecaster overestimates or underestimates the future volatility of gold-price fluctuations. Third, we condense the rich information on forecast performance that we extract from the asymmetric loss functions by means of an easy-to-compute out-of-sample R2 statistic recently studied by Pierdzioch et al. (2014b). We apply the out-of-sample R2 statistic to assess the performance of forecasts relative to an autoregressive benchmark forecast. The out-of-

sample R2 statistic that we study is similar to the R2 statistic for quantile regressions proposed by Koenker and Machado (1999), and it nests in the special case of a symmetric quadratic loss function the out-of-sample forecast-evaluation criterion proposed by Campbell and Thompson (2008). Upon combining the various loss functions with the out-ofsample R2 statistic, we find that, relative to a simple autoregressive benchmark model, those forecasters tend to benefit from using financial and macroeconomic variables to set up the boosting approach who incur a larger loss when underestimating rather than overestimating the future volatility of gold-price fluctuations. We use a simulation experiment to assess the significance of this benefit under alternative functional forms of the loss function. We organize the remainder of this research as follows. In Section 2, we outline the boosting approach and we describe how we evaluate forecasts under an asymmetric loss function. In Section 3, we describe our data. In Section 4, we lay out our empirical results. In Section 5, we offer some concluding remarks.

2. Computation and evaluation of forecasts 2.1. Forecast computation We use a boosting approach to compute forecasts of the volatility of gold-price fluctuations. We define “volatility” in terms of the annualized monthly realized variance of gold-price fluctuations, σt2.1 As recommended by Bühlmann and Hothorn (2007), we estimate our forecasting model on demeaned data, where we compute the mean values of the volatility of gold-price fluctuations and the predictor variables using historical data available to a forecaster when a forecast is formed. For computing forecasts, we simply add the period-t mean to the forecast computed from the period-t forecasting model. The forecasting model, thus, features the volatility of gold-price fluctuations on the left-hand side and one or more of the predictor variables on the right-hand side. Moreover, similar to the modeling approach taken by, for example, Marquering and Verbeek (2004), Mittnik et al. (2012), and Christiansen et al. (2012), we do not model σt2 directly but rather prefer to forecast the natural logarithm of the realized volatility of goldprice fluctuations, ln (σt2 ).2 Forecasting ln (σt2 ) rather than σt2 is a convenient modeling strategy because it allows a standard linear forecasting model to be used. In analogy to return-prediction models, we set up the forecasting model as follows:

ln (σt2+ 1) = F (β, xt ) + ut + 1,

(1)

where ut denotes a stochastic disturbance term, and the function F (β , xt ) is a so-called strong learner that encapsulates information on the parameters, β, to be estimated and the vector, xt, of the financial and macroeconomic determinants of the (log of) realized volatility of gold-price fluctuations. We assume that the strong k learner takes the form F (β , xt ) = ∑ j = 1 βj xj, t , where xj, t denote the individual financial and macroeconomic j = 1, … , k determinants.3 Like Pierdzioch et al. (2015a,b), we choose the 1 In financial economics, many researchers use the term “volatility” to refer to the standard deviation of asset-price fluctuations. We prefer to measure the volatility of gold-price fluctuations in terms of the variance to make our use of terminology consistent with the one used by Marquering and Verbeek (2004), Patton (2011), Andreou et al. (2012), and others. 2 Christiansen et al. (2012) study the predictive power of several financial and macroeconomic predictors of commodity-price volatility. They use the S&P GSCI commodity index to measure commodity-price fluctuations. The contribution of precious metals to this index is rather small. 3 Our assumption regarding the form of the strong learner is consistent with modeling approaches adopted in earlier research. For example, Christiansen et al.

C. Pierdzioch et al. / Resources Policy 47 (2016) 95–107

strong learner as the solution of the following quadratic minimization problem:

^ F (β^ , xt ) = arg

⎡1 ⎤ min E ⎢ (ln (σt2+ 1) − F (β, xt ))2 ⎥, ⎣2 ⎦

F (β, xt )

(2)

where E denotes the expectation operator. Estimation is done by the ordinary-least-squares (OLS) technique, which is why this version of the boosting approach is known as L2-boosting (Bühlmann and Yu, 2003; Bühlmann and Hothorn, 2007). In contrast to the standard OLS technique, however, the minimization problem given in Eq. (2) is not simply solved by choosing optimal estimates of the parameter vector, β^ . Rather, the boosting ap-

^ proach also requires a decision on the optimal strong learner, F , and thus on the financial and macroeconomic variables to be included in the vector xt, implying that the boosting approach accounts for model uncertainty. We implement a functional-gradient-descent method (Friedman, 2001) to identify the optimal composition of xt. The functional-gradient-descent method rests on the insight that the OLS residuals, ut + 1, implied by Eq. (1) are equal to the negative of the derivative of the quadratic function given in Eq. (2) with respect to F (β , xt ). We, thus, estimate univariate regression models of the negative of the gradient vector on the j = 1, … , k financial and macroeconomic determinants of the volatility of gold-price fluctuations. The uni^ variate regression models produce k weak learners, fj = γ^j xj, t , where

γ^j denote the estimated coefficients of the univariate regression models. From the k weak learners, we choose the best weak learner, κ, that solves (Friedman, 2001; see also Bühlmann, 2006)

⎡ κ = arg min ⎢ j ⎢⎣

⎤

∑ (ut + 1 − γ^j xj, t )2⎥. t

⎥⎦

(3)

We then use the weak learner that solves Eq. (3) to update the strong learner, resulting in the following recursive formula:

^ ^ ^ Fm + 1 = Fm + sfm, κ ,

(4)

where s denotes a reasonably small step size (we set s¼0.1, which is a numerical value that is quite common in the boosting literature), and m denotes the number of the updating iteration. With the updated strong learner in hand, we compute a new gradient vector, new weak learners, and again an updated strong learner. In order to terminate the recursive updating of the strong learner, we use information criteria. First, like Bühlmann (2006), we compute a corrected Akaike information criterion (AIC) as follows (see Hurvich et al., 1998): 2 AICt, m = ln (σ^res, m ) +

1 + df (m)/Tt , 1 − (df (m) + 2)/Tt

(5)

2

where σ^res, m denotes the residual variance of the forecasting model in iteration m, the degrees of freedom are defined as df (m) = tr (Bm ), and the matrix Bm maps the vector of the log of the realized volatility of gold-price fluctuations onto the strong learner, F^m . This matrix is updated according to the recursive equation Bm = Bm − 1 + Hκ (I − Bm − 1), where Hκ = x κ , t (x κ , t )⊤ /||x κ , t ||2, I denotes a suitable identity matrix, and x κ , t denotes the vector of observations on the κth determinant of the volatility of gold-price fluctuations available in period of time t (see Bühlmann, 2006; Bühlmann and Hothorn, 2007). Second, we use the minimum description length (MDL) information criterion, which is defined as (see Bühlmann and (footnote continued) (2012) model realized volatility as a linear function of lagged volatility and its potential financial and macroeconomic determinants.

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Table 1 Inclusion of predictor variables in the boosted forecasting models. Predictor variable

AIC

MDL

AICa

MDLa

Realized volatility Stock returns Returns of the effective exchange rate Short-term interest rate Term spread Risk spread Inflation rate Growth rate of output Growth rate of M2

100.00 79.46 99.55 59.82 47.77 99.55 87.95 24.11 100.00

100.00 38.84 47.32 49.11 1.34 99.55 47.77 0.45 97.32

100.00 08.48 32.14 26.79 0.00 87.95 16.52 0.00 91.52

100.00 0.00 6.25 19.20 0.00 63.39 3.57 0.00 66.07

Iterations

AIC

MDL

AICa

MDLa

Min Max Mean Median

35.00 244.00 104.30 55.00

25.00 78.00 40.08 34.00

9.00 74.00 32.47 32.00

9.00 38.00 21.21 24.00

Note: The training period is 120 months. See Section 3 for a description of the predictor variables. Iteration statistics show the optimal number of iterations. The actual number is slightly larger because the algorithm only stops when an interior minimum is identified (unless m = mmax ).

Hothorn, 2007)

MDL t, m = ln (S ) + (df (m)/Tt ) ln (Z ), 2

where S =

Tt σ^res, m Tt − df (m)

, and Z =

2 ∑i (ln (σi2 ))2 − Tt σ^res, m

df (m) S

(6) , where the summation

runs over the observations being used for estimation. The AIC and MDL information criteria have been shown in earlier literature to take a relatively large number of iterations to converge and, thus, to select a relatively complex forecasting model (see also Table 1 in Section 4.1). A complex forecasting model has the advantage that it should have a good in-sample fit and the forecast bias due to skipping a predictor variable should be small. At the same time, however, a complex forecasting model is likely to be trained too strongly to the in-sample data such that a generalization to out-of-sample data works poorly and the variance of the out-of-sample error is large (see Hastie et al., 2009). Hence, Hastie (2007) suggests to replace the degrees of freedom, df(m), by the number of selected determinants (that is, the socalled “active set”) of the predictand. Accordingly, we also report results for two alternative information criteria, AICa and MDLa, that use the active set to compute the degrees of freedom.4 We use an algorithm similar to the one proposed by Mayr et al. (2012) to determine the final updating iteration, mn, that minimizes the information criterion, IC, being studied.5 Accordingly, we run the boosting algorithm mbreak times. If m⁎ = arg minm ICt , m satisfies m⁎ ≤ 0.9 × mbreak , we stop. Otherwise, we set mbreak = mbreak + 10, run 10 further updating iterations, and then check again if m⁎ ≤ 0.9 × mbreak . We continue until we reach some maximum iteration, mmax . In our empirical analysis, we set mmax = 250. We recursively restart the boosting approach in every period of time to account for potential temporal instability of the optimal forecasting model. Given this recursive forecasting approach, the length of the estimation window and, thus, the number of observations, Tt, that we use to implement the boosting algorithm increases over time. Moreover, as a robustness check, we use three different training periods (120 months, 150 months, 180 months of 4 Some researchers (see, for example, Hastie, 2007) recommend using cross validation as yet an alternative model-selection mechanism. Applying cross validation to our data yielded similar results (not reported, but available upon request) to those we shall report in Section 4. 5 Mayr et al. (2012) propose an extended stopping algorithm that features a resampling element.

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data) to initialize the boosting approach. Our decision to use a recursive forecasting approach and our choice of the three different training periods follows the way Pesaran and Timmermann (1995) implement their popular recursive modeling approach to forecast stock-market returns. The recursive modeling approach developed by Pesaran and Timmermann (1995) has been adopted in recent research by Pierdzioch et al. (2014a) to model the links between the returns of the price of gold and their financial and macroeconomic determinants.6 2.2. Forecast evaluation It remains to evaluate the informational content of the forecasts implied by the boosting approach. The boosting approach laid out in Section 2.1 operates under the assumed symmetric quadratic loss function given in Eq. (2).7 In practice, however, the need to evaluate forecasts under an asymmetric loss function easily arises if a forecaster uses options, futures, or other financial derivatives to set up a risk-management strategy. For example, a forecaster who is long in a straddle (that is, a forecaster who simultaneously buys a call option and a put option with the same underlying, strike price, and maturity) benefits when a forecast of the volatility of gold-price fluctuations underestimates the actual volatility. In contrast, an investment in a long straddle is likely to generate a negative payoff if a forecast of the volatility of goldprice fluctuations substantially overestimates the subsequently realized volatility. Different forecast errors, thus, yield different payoffs, and the loss function should reflect this asymmetry. For this reason, we opt for a forecast-evaluation approach that resembles the modeling strategy used by Patton (2011), who takes volatility forecasts as given and then proceeds, given volatility forecasts, studying implications for the optimal modeling of a forecaster's loss function. Similarly, we take the volatility forecasts implied by the boosting approach as given and then use a flexible forecast-evaluation approach to study how the shape of the loss function affects the benefits from the boosted volatility forecasts. An implication of this modeling strategy is that we can use alternative loss functions to evaluate the same set of forecasts, that is, we can isolate the effect on forecast performance of varying the loss function. Hence, rather than assuming that a forecaster's preferences can be expressed in terms of a single loss function, we consider three alternative loss functions.8 All three loss functions have the advantage that they easily accommodate the payoff structures of alternative risk-management strategies because differences in the loss from an underestimation of volatility relative to the loss from an overestimation of volatility can be expressed in terms of a single parameter.9 The first loss function that we consider is a simple quadratic loss function extended such that it can describe asymmetric preferences (for a similar approach in a different context, see e.g. Cukierman and Gerlach, 2003). The loss function 6 In contrast to a rolling-estimation approach, a recursive-estimation approach does not require to fix the length of the rolling window. Moreover, the information used for estimation is maximized because no data are deleted to fix the length of the rolling window. See Nicolau and Palomba (2015, p. 133). 7 The boosting approach could be implemented using other functional forms for the loss function. See Friedman et al. (2000) (for classification problems) and Bühlmann and Hothorn (2007) for a discussion of various boosting approaches from a statistical point of view. 8 For an evaluation of volatility forecasts under alternative loss functions, see Hansen and Lunde (2006), among others. 9 The utility-based approach developed by West et al. (1993) is an alternative to modeling an asymmetric loss function. Their approach gives rise to an asymmetric loss function insofar as an underestimation of volatility leads to a lower expected utility than an overestimation of the same magnitude. Yet another approach to model an asymmetric loss function is to simulate, for example, an artificial options market (see Engle et al., 1996; Pierdzioch et al., 2008).

is of the format: 2 3 A (b, σt2, σ^IC , t ) =

1⎡ ⎢2 2⎣

2 2 2 ⎤ − bI (σt2 − σ^IC , t < 0) + bI (σt2 − σ^IC , t ≥ 0) ⎥⎦ (σt2 − σ^IC , t )2 , (7)

2

where I (·) denotes the indicator function, and σ^IC , t denotes the forecast implied by the boosting approach under the model-selection criterion IC. The asymmetry parameter can assume values in the interval b ∈ [ − 2, 2]. Setting b ¼0 yields a standard quad2 2 ratic symmetric loss function: 3A (0, σt2, σ^IC , t ) = (σt2 − σ^IC , t )2. Setting b = − 2, in turn, implies that the loss of underestimation is zero while the loss of overestimation is quadratic in the forecast error. Similarly, the loss of overestimation is zero for b¼ 2, while the loss of underestimation is quadratic. The second loss function that we consider is the loss function proposed by Elliott et al. (2005, 2008) and studied by Pierdzioch et al. (2014b) to evaluate forecasts of gold-price returns. This loss function can be interpreted as a generalization of Eq. (7). The loss function is of the format p

2 2 2 3E (b, σt2, σ^IC , t ) = [b + (1 − 2b) I (σt2 − σ^IC , t < 0)] σt2 − σ^IC , t .

(8)

A so-called lin–lin loss function obtains for p ¼1, and a quad–quad loss function obtains for p¼ 2. The asymmetry parameter can assume values in the interval b ∈ (0, 1), where setting b ¼0.5 yields a symmetric loss function. If the shape parameter assumes a value of b > 0.5 ( b < 0.5), the loss from an underestimation (overestimation) exceeds the loss from an overestimation (underestimation) of the same size. The third loss function that we consider is a loss function advocated by Patton (2011). He shows that, in addition to encompassing a wide range of loss functions, the loss function given in Proposition 4 of his study implies that the loss function is robust to both the presence of noise in the volatility proxy being studied and a rescaling of the data.10 The loss function is given by ⎧ 1 2b + 4 σt2b + 4 − σ^IC , t ⎪ ⎪ (b + 1)(b + 2) ⎪ ⎛ 2 ⎞ ⎪ ⎪ σ^ 2 − σt2 + σt2 ln ⎜ σt ⎟ 2 ^2 P ⎜ ^2 ⎟ 3 (b, σt , σ IC , t ) = ⎨ IC , t ⎝ σ IC , t ⎠ ⎪ ⎪ ⎛ 2 ⎞ 2 ⎪ σt ⎜ σt ⎟ ⎪ 2 − ln ⎜ 2 ⎟ − 1 ^ ⎪ ⎝ σ^IC , t ⎠ ⎩ σ IC , t

(

) for b ≠ − 2, b ≠ − 1 for b = − 1

.

for b = − 2

(9)

For b ¼0, a symmetric loss function obtains. For b > 0, an overestimation of the volatility of gold-price fluctuations is more costly than an underestimation. Conversely, for b < 0, an underestimation of the volatility of gold-price fluctuations is more costly than an overestimation. Fig. 1 illustrates the three loss functions for alternative values of the shape parameter, b. In order to quantify the benefit from using forecasts implied by the boosting approach, we combine the loss functions given in Eqs. (7)–(9) with the forecast-evaluation approach recently studied by Pierdzioch et al. (2014b, 2015c). Their approach extends the outof-sample R2 statistic studied by Campbell and Thompson (2008) to the case of an asymmetric loss function. The resulting out-ofsample R2 resembles the goodness-of-fit criterion that Koenker and Machado (1999) have developed to study quantile regressions. The out-of-sample R2 statistic is defined as 2 R2 (b, σt2, σ^IC , t , σ¯t2 ) = 1 −

2 T ∑t = τ 3l (b, σt2, σ^IC , t ) T

∑t = τ 3l (b, σt2, σ¯t2 )

,

(10)

10 For a recent application of Patton's (2011) loss function to forecasting volatility, see Andreou et al. (2012). For results regarding the precision of estimators of realized variance (or realized standard deviation), see Barndorff-Nielsen and Shephard (2002).

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Fig. 1. Examples of the loss functions. Note: The figure shows the loss function given in Eqs. (7)–(9) for alternative values of the shape parameter, b, as a function of the forecast, σ^t , of the realized volatility of goldprice fluctuations. The actual value of volatility assumes the value σ 2 = 5 (dashed vertical line).

where l ∈ {A , L , P}, σ¯t denotes a benchmark forecast, and τ (T) denotes the first (final) period for which a forecast is available. We compute a benchmark forecast by a version of Eq. (1) that only features various lags of ln (σt2 ) but no other determinants as explanatory variables (again using demeaned data). In other words, a simple autoregressive model is our benchmark model (see also Christiansen et al., 2012). We use the OLS technique to estimate the benchmark model. If R2 > 0, then our approach implies that the predictive power of the boosting approach that potentially features financial and macroeconomic determinants exceeds the predictive power of a benchmark model. Similarly, if R2 < 0, a forecaster would benefit from using the forecasts implied by the benchmark model rather than those implied by the boosting approach.

3. The data 3.1. Volatility of gold-price fluctuations We measure the gold price, Gt, using daily gold fixing prices (3:00 P.M., London time) denominated in terms of U.S. dollars.11 We let Rn = 100 × ln (Gn/Gn − 1) denote the returns of the gold price at day n, and we compute the monthly realized volatility as 1 RVt = N ∑n Rt2, n , where Rt , n denotes returns at day n of month t, and t

Nt denotes the total number of daily data available for month t.12 Finally, we annualize the realized monthly volatility of gold-price fluctuations by computing σt2 = 12 × RVt . Fig. 2 shows the volatility of gold-price fluctuations, the log of volatility (that is, ln (σt2 )), and the distribution of the log of volatility. Our sample period starts in February 1987 and ends in September 2015.13 The volatility of gold-price fluctuations shows several peaks. Volatility was relatively high in the aftermath of the U.S. recession of 1990/1991, while it remained relatively low and stable during the mid 1990s. Since around the later years of the 1990s it again experienced discernible upward jumps. Several

11 For information on the London market, see the web page http://www.lbma. org.uk/pages/index.cfm. For the gold-price data and, unless otherwise noted, the other data, see the web page of the Federal Reserve Bank of St. Louis at http:// research.stlouisfed.org/fred2/. 12 See Andersen et al. (2003) for an early in-depth analysis of realized volatility. 13 The choice of the sample period is governed by the availability of end-ofmonth data on Moody's Baa-rated bonds, which are only available from 1986 onwards. In order to be consistent with Pierdzioch et al. (2014a), we let our sample period start in February 1987.

Fig. 2. Realized volatility of gold-price fluctuations. Note: See Section 3 for details on how we calculated the realized volatility of goldprice fluctuations. This graph (and all other empirical results reported in this research) has been computed using the free R programming environment (R Core Team, 2014).

peaks of the volatility at around 2000 coincide with the bursting of the DotCom Bubble. The volatility of gold-price fluctuations reached an all-time high in October 2008 following the collapse of Lehman Brothers. Before the collapse of Lehman Brothers and the ensuing market turmoil, when the U.S. as well as most European countries faced a period of stable economic growth, the volatility was low and relatively stable. However, when the financial crisis hits the U.S. and asset prices started to tumble, investment in gold started growing, which lead to rapidly increasing gold prices and increasing volatility. By contrast, the volatility of gold-price fluctuations remained relatively low during the recent European debt crisis that started around 2009, which can be interpreted to support the “safe haven” hypothesis discussed in earlier literature (Baur and Lucey, 2010; Baur and McDermott, 2010). When the European debt crisis started, the gold price had already reached a high plateau. While the European debt crisis fostered the demand for gold as a hedge against tumbling asset prices, the ensuing rise

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in the gold price was relatively steady when compared with the situation before the crisis. The steady growth resulted in a comparatively low volatility of gold-price fluctuations during the European debt crisis. The volatility of gold-price fluctuations peaked again around October 2011, perhaps because the market showed signs of “overheating” after the gold price had reached an all-time high in September 2011. The last peak in volatility in April 2013 reflects the rapid decline in the price of gold that started in September 2012 and continued until June 2013. Some commentators speculated that this decline in the price of gold was triggered by the observation that inflation dynamics did not gather steam in the U.S. and the Eurozone despite the massive expansion of money supply. Because inflation dynamics only developed moderately there was no need for investors to buy gold as an inflation hedge and gold positions were resolved. The distribution of the logarithm of the volatility of gold-price fluctuations resembles a normal distribution (normality cannot be rejected) but is not fully symmetric. 3.2. Predictor variables Based on the findings reported by Batten et al. (2010) that monetary and financial variables help to explain the volatility of gold-price fluctuations, and earlier research (see for example Vrugt et al., 2007; Bialkowski et al., 2012; Pierdzioch et al., 2014a, b, among others) we use (in addition to the log of the lagged realized volatility) the following U.S. financial and macroeconomic variables to forecast the volatility of gold-price fluctuations: the returns on the Dow Jones Industrial Index (DJIA), to take into account the potential quality of gold investments to act as a hedge or a “safe haven” against stock market volatility (see, for example, Baur and Lucey, 2010; Baur and McDermott, 2010, among others). In order to analyze the link between gold-price movements and exchange-rate movements (see Tulley and Lucey, 2007; Sjaastad, 2008; Pukthuanthong and Roll, 2011; Reboredo, 2013), we use returns on the trade-weighted nominal effective exchange rate.14 As a measure of the stance of monetary policy and because it has been studied extensively as a leading indicator of the business cycle (see, for example, Estrella and Mishkin, 1998), we use the term spread defined as the difference between the 10-year treasury constant maturity rate and the 3-month treasury constant maturity rate (see also Pierdzioch et al., 2014a). We also include the corporate bond spread, defined as the difference between Moodys' Baa- and Aaa-rated bonds, as a proxy for financial risk and financial crises (Hartmann et al., 2008). Because the properties of gold as an inflation hedge have been extensively studied in earlier research (see, among others, Fortune, 1987; Gosh et al., 2004; Levin and Wright, 2006; Blose, 2010), we use the inflation rate (defined as the year-on-year change of the U.S. Consumer Price Index (all goods)) as another potential predictor of gold-price volatility. We use the year-on-year growth rate of output calculated using data on industrial production to study the link between gold-price volatility and business cycle movements (see Pierdzioch et al., 2014b). Given that Batten et al. (2010) emphasize the importance of monetary variables as determinants of goldprice volatility, we further use the year-on-year change of the monetary aggregate M2 as a predictor variable. If data are available at a daily frequency, we use end-of-month data. In case of the inflation rate, the growth rate of output, and the growth rate of the monetary aggregate M2, we use lagged values in order to take into 14 We use data on the nominal effective exchange rate (narrow definition) as published by the Bank of International Settlements. See the web page of the Bank of International Settlements (http://www.bis.org/) for further information. Our choice of the narrow index is governed by data availability, since data on the broad index is only available from 1994 onwards.

account that a forecaster faces a publication lag for these macroeconomic variables.15 Fig. 3 plots our predictor variables. A few points warrant a closer discussion. At the beginning of our sample period, we observe highly negative stock market returns (the 1987 crash) which coincide with the first peak in the volatility of gold-price fluctuations shown in Fig. 2. Similar to the volatility of gold-price fluctuations, most predictor variables show a noticeable reaction to the recent financial market jitters. For example, the corporate bond (that is, risk) spread reached an (even by historical standards) high level in autumn/winter 2008. The same applies to the returns of the nominal effective exchange rate, while the short term interest rate reached the zero-lower bound at the end of the sample period. The change in the monetary aggregate M2 and the inflation rate reacted with a time lag to the financial crisis, reaching a trough in 2009 and 2010, respectively. Fig. 3 also illustrates the impact of the financial crisis on the growth rate of industrial production, which reached a historical trough following the crisis, but bounced back to more standard growth rates at the end of the sample period.

4. Empirical results 4.1. Results for a baseline model We start our empirical analysis with a baseline model. The baseline model features the lagged (log) volatility and our financial and macroeconomic determinants as potential predictors, but we do not include higher-order lags. Table 1 summarizes (for a training period of 120 months) how often (in percent) the boosting approach includes the various financial and macroeconomic predictor variables in the forecasting model. The results demonstrate that the baseline AIC criterion often leads to a relatively complex forecasting model that features many predictors. Accordingly, the number of iterations is on average quite large for the AIC information criterion. The baseline MDL information criterion implies a more restrictive selection of predictor variables than the baseline AIC criterion, and the number of iterations is substantially smaller than for the AIC criterion (40 versus 104 optimal iterations on average). In terms of model parsimony, however, the AICa and MDLa criteria dominate the other two information criteria. Moreover, it only takes on average approximately 32 and 21 optimal iterations to minimize the AICa and MDLa criteria. Irrespective of the information criterion being studied, the optimal forecasting model always features the lagged realized volatility of gold-price fluctuations. When evaluated across all four IC, the risk spread and the growth rate of M2 also are important predictor variables. In order to illustrate how the composition of the forecasting models varies over time, we plot in Fig. 4, as an example, when the inflation rate is included in the boosted forecasting models. In earlier literature, there has been an intense debate on whether investors can use investments in gold as a hedge against inflation. Results reported in earlier literature are mixed. Results of recent research suggest that one reason for the mixed results documented in earlier literature is that the gold-inflation link has not been stable over time (Batten et al., 2014). The results depicted in Fig. 4 suggest that, depending on the information 15 We use the latest revised macroeconomic data available at the time of writing of this paper. Pierdzioch et al. (2008) show that the choice between realtime and revised macroeconomic data does not affect volatility forecasts in an outof-sample forecasting experiment that uses as data input entire vintages of macroeconomic data in a systematic way. In other contexts, the choice between realtime and revised macroeconomic data may have interesting effects, see Kizys and Pierdzioch (2010a,b).

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Fig. 3. Predictor variables. Note: The level of the log of the realized volatility of gold-price fluctuations is also being used as a predictor variable, but is not shown in this graph. See Fig. 2 for a graph of realized volatility.

Fig. 4. Inclusion of the inflation rate in the boosted forecasting models. Note: The dummy variable assumes the value one when the inflation rate is included in the boosted forecasting model, and zero else. The training period is 120 months.

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criterion being studied, a similar instability holds for the link between the volatility of gold-price fluctuations and the inflation rate. The inflation rate is often included when we study the AIC information criterion. In contrast, we observe much more variation in the volatility–inflation link under the MDL information criterion. The inflation rate is only occasionally selected under the AICa information criterion, and it is hardly selected under the MDLa information criteria. Fig. 5 shows the realized volatility, σt2, of gold-price fluctuations 2 and the forecasts, σ^IC , t , implied by the boosting approach under the four information criteria being studied. The forecasts often track the realized volatility quite closely. We also observe, however, that

the forecasts tend to fall short of the realized volatility in times when the volatility of gold-price fluctuations suddenly rises, where the extent of underestimation varies across the information 2 criteria. The correlation between σt2+ 1 and σ^IC , t + 1 assumes values between approximately 0.52 and 0.57 (and is significant for all four IC). 4.2. Out-of-sample R2 statistic Fig. 6 plots the out-of-sample R2 statistic as a function of the asymmetry parameter, b, for the three different loss functions given in Eqs. (7)–(9). To compute the out-of-sample R2 statistic, we

Fig. 5. Forecasts versus realized volatility. Note: The time since start of sample starts after the training period when the first forecast is being formed. Realized volatility is measured in annualized terms. The training period is 120 months.

Fig. 6. Out-of-sample R2 statistic. Note: The figure shows the out-of-sample R2 statistic as a function of the shape, b, of a forecaster's loss function. For a description of the alternative loss function, see Equations (7)–(9). For the definition of the out-of-sample R2 statistic, see Equation (10). The training period is 120 months. It should be noted that the scaling of the horizontal axis is not the same across the three graphs shown in this figure.

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Fig. 7. Comparison of forecast errors. Note: Vertical axis: σt denotes the actual value of the realized volatility. σ^t denotes the forecast of the realized volatility of gold-price fluctuations implied by the boosting approach. Horizontal axis: σ¯t denotes the forecast of the realized volatility of gold-price fluctuations implied by the AR(1) benchmark model. The training period is 120 months.

Fig. 8. Out-of-sample R2 statistic for adjusted volatility. Note: The figure shows the out-of-sample R2 statistic as a function of the shape, b, of a forecaster's loss function. Volatility is measured as defined in Eq. (11). For a description of the alternative loss function, see Eqs. (7)–(9). For the definition of the out-of-sample R2 statistic, see Eq. (10). The training period is 120 months. It should be noted that the scaling of the horizontal axis is not the same across the three graphs shown in this figure.

use a simple autoregressive model of order one as our benchmark model. The training period is 120 months. Fig. 6 shows that, as one would have expected, the specific form of the dependence of the out-of-sample R2 statistic upon the shape parameter, b, depends

on the IC being studied. The general picture that emerges, however, shows that the out-of-sample R2 statistic assumes positive values if the loss of underestimating volatility exceeds the loss of overestimating volatility by the same size. Hence, if

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Fig. 9. Out-of-sample R2 Statistic for 3E (lin–lin functional form). Note: The figure shows the out-of-sample R2 statistic as a function of the shape, b, of a forecaster's loss function. The loss function is of the type 3E . TP, training period in months. Dashed lines (dashed and dotted) denote the boundaries of the 95% interval of the sampling distribution based on the percentiles (BCa) method. The sampling distribution was computed using a block bootstrap (smoothing parameter 1/12, 5000 simulation runs) as described by Politis and Romano (1994).

Fig. 10. Out-of-sample R2 statistic for 3E (quad–quad functional form). Note: The figure shows the out-of-sample R2 statistic as a function of the shape, b, of a forecaster's loss function. The loss function is of the type 3E . TP, training period in months. Dashed lines (dashed and dotted) denote the boundaries of the 95% interval of the sampling distribution based on the percentiles (BCa) method. The sampling distribution was computed using a block bootstrap (smoothing parameter 1/12, 5,000 simulation runs) as described by Politis and Romano (1994).

underestimation is more costly than overestimation, then using our financial and macroeconomic predictor variables to forecast the realized volatility of gold-price fluctuations by means of the boosting approach tends to lead to a superior forecasting performance relative to the forecasting performance of a recursively estimated autoregressive benchmark model that features the (log) realized volatility of gold-price fluctuations as the only predictor variable. While the out-of-sample R2 statistic may also assume positive values in case of symmetry of the loss function (see, for example, 3P for AIC), a symmetric loss function does not lead to the best forecasting performance in terms of the out-of-sample R2 statistic. In order to build intuition, we compare in the scatterplots shown in Fig. 7 the forecast error implied by the boosting approach with the forecast error implied by the benchmark model. The comparison illustrates that many points can be found below the dashed bisecting line. The benchmark model, thus, often produces a larger underestimation of the actual volatility of goldprice fluctuations than the boosting approach. Moreover, the benchmark model produces a few relatively large positive forecast errors. As a result, the loss under the benchmark model is relatively large as compared to the loss under the boosting approach when an underestimation of the volatility of gold-price

fluctuations is more costly than a comparable overestimation. 4.3. An alternative measure of volatility Unlike returns, volatility cannot be directly measured and different estimators may yield different results in terms of the out-of-sample R2 statistic. As an alternative measure of volatility, we consider, like Marquering and Verbeek (2004), the possibility that autocorrelation in gold-price returns distorts our measure of the volatility of gold-price fluctuations (see also, for example, Akgiray, 1989). Accordingly, we compute an adjusted volatility as follows:

⎞ ⎛ Nt − 1 σt2, a = σt2 ⎜⎜ 1 + 2Nt−1 ∑ (Nt − j ) ϕt j ⎟⎟, ⎠ ⎝ j=1

(11)

where ϕt denotes the first-order coefficient of autocorrelation of gold-price returns in month t. Fig. 8 shows for our baseline model the out-of-sample R2 statistic for the three different loss functions, ( 3A, 3E , 3P ), and a training period of 120 months. Corroborating the results given in Fig. 6, a forecaster who penalizes underestimation to a larger extent than overestimation benefits from using the boosted

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Fig. 11. Out-of-sample R2 statistic for 3A . Note: The figure shows the out-of-sample R2 statistic as a function of the shape, b, of a forecaster's loss function. The loss function is of the type 3A . TP, training period in months. Dashed lines (dashed and dotted) denote the boundaries of the 95% interval of the sampling distribution based on the percentiles (BCa) method. The sampling distribution was computed using a block bootstrap (smoothing parameter 1/12, 5,000 simulation runs) as described by Politis and Romano (1994).

Fig. 12. Out-of-sample R2 statistic for 3P . Note: The figure shows the out-of-sample R2 statistic as a function of the shape, b, of a forecaster's loss function. The loss function is of the type 3P . TP, training period in months. Dashed lines (dashed and dotted) denote the boundaries of the 95% interval of the sampling distribution based on the percentiles (BCa) method. The sampling distribution was computed using a block bootstrap (smoothing parameter 1/12, 5,000 simulation runs) as described by Politis and Romano (1994).

forecasts rather than the forecasts implied by the benchmark model. Further, the shape of the functions shown in Fig. 8 in many cases resembles the shape of the functions shown in Fig. 6. At the same time, a more detailed comparison with the results summarized in Fig. 6 yields some further insights. First, the out-of-sample R2 statistic for the 3A and 3E loss functions are close to zero and occasionally assume small positive values for the AICa information criterion. Second, the out-of-sample R2 statistic for the 3P loss function exhibits a somewhat larger dispersion for a small negative parameter b when compared across the information criteria than in Fig. 6. Third, the out-ofsample R2 statistic converges to zero more quickly under the 3P loss function as b assumes large positive values than in Fig. 6. The reason for this convergence is that, as we move from left to right, overestimation of volatility becomes more costly relative to underestimation and the boosting approach successively loses its advantage over the benchmark model under the 3P loss function. Moreover, the loss function becomes steep around σt2 and any forecast error becomes very costly. As a result, R2 → 0 as b becomes large.16 16

By the same token, Fig. 1 illustrates that the loss function becomes

4.4. Further extensions As an extension, we replace the recursive estimation window with a rolling estimation window. A rolling estimation window may capture more rapidly than a recursive estimation window potential structural shifts in the link between volatility and its determinants. At the same time, the number of observations per estimation window is smaller than in the case of a recursive estimation window and this is likely to increase the influence of occasional outbursts of volatility on the forecasts. Results (not reported) show that a rolling estimation window implies qualitatively similar results as the baseline model studied in Section 4.2, that is, the general picture that emerges is that the out-of-sample R2 statistic becomes positive when underestimation is more costly than overestimation.17 (footnote continued) insensitive to small over- and underestimations of volatility as b gets negative and large in absolute value. Again, the boosting approach and the benchmark model become more similar in terms of their implied loss, and R2 → 0 as |b| becomes large. 17 As compared to the results for a recursive estimation window, the relative importance of the short-term interest rate and the inflation rate as predictors of the volatility of gold-price fluctuations increases when we study a rolling estimation

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As yet another extension, we vary the step size that the boosting approach uses to update the strong learner from s ¼0.1 to s¼ 0.05. A smaller step size increases the average number of iterations until the updating is terminated but leads to similar patterns regarding the inclusion of macroeconomic variables. Hence, results (not reported) using the smaller step size corroborate the results for the baseline model (see Section 4.2). 4.5. Simulation experiments for a dynamic model Our baseline model illustrates the intuition underlying the boosting approach and the out-of-sample R2 statistic, but the relatively simple model may not capture the potentially complex dynamic links between the volatility of gold-price fluctuations and its financial and macroeconomic determinants. Hence, we next study a dynamic model. To this end, we extend the list of potential determinants of gold-price fluctuations to include up to six lags of all predictor variables (that is, we use the most recent data plus five additional lags). Because our plan is to trace out the incremental informational content of the financial and macroeconomic determinants, we use an autoregressive model of order six as a benchmark model.18 A natural question is whether the out-of-sample R2 statistics are significant. Some of the out-of-sample R2 statistics plotted in Figs. 6 and 8 assume relatively small numerical (absolute) values and a natural question is whether such values are different from zero in a statistical sense. Hence, we use simulation experiments to study the sampling distribution of the out-of-sample R2 statistic. To implement the simulation experiments, we resample B time 2 series of pairs {σt , σ¯t , σ^t , IC } using the block boostrap described by Politis and Romano (1994). The block boostrap accounts for the temporal dependence of volatility (e.g., volatility clustering) often observed in empirical research. We set B = 5000. We then compute B realizations of the out-of-sample R2 statistic and determine the boundaries of the 95% interval of its sampling distribution. We use 95% percentiles and the BCa method to determine the relevant boundaries of the sampling distribution (see Efron and Tibshirani, 1998). We interpret a 95% interval of the sampling distribution that does not include zero as evidence of a significant out-ofsample R2 statistic (two-sided test). Figs. 9 (for p ¼1) and 10 (for p ¼2) summarize the simulation results for the loss function 3E . The simulation results show that, as far as a lin–lin functional form is assumed for the loss function, the out-of-sample R2 statistic is in general significant for the AIC information criterion if an underestimation is substantially more costly than an overestimation of the same size, irrespective of the choice of σt or σ^t . The out-of-sample R2 statistic is significant in case of the MDL information criterion if the shape parameter gets sufficiently large (it is significant at the 90% level). Moreover, the out-of-sample R2 statistic is significant if an overestimation is more costly than an underestimation for the AIC and MDL (adjusted volatility) information criterion. It is interesting to note that, in terms of significance of the results, the AIC information criterion, which has been criticized in earlier literature for leading to model overfitting, performs well in terms of the out-of-sample R2 statistic if the parameter b is sufficiently small. We further observe that the out-of-sample R2 statistic is also significant in case of a sufficiently asymmetric loss function under the AIC information criterion for the quad–quad loss function, especially for the (footnote continued) window. 18 In case a dynamic model features several lags of a potentially large number of predictor variables, the blockwise L2 boosting algorithm proposed by Bai and Ng (2009) is a useful alternative to the boosting algorithm that we study in this research.

adjusted volatility. Finally, the simulation results show that the out-of-sample R2 statistic is not significantly different from zero in case a forecaster has a symmetric loss function. This is a general result that also holds for the other two loss functions. Figs. 11 and 12 summarize the results for the loss functions 3A and 3P . For the loss function 3A , the out-of-sample R2 statistic scratches the boundary of significance in case of underestimation (b > 0) under the AIC information criterion if we use σ^t , and it gets significant if we measure the volatility of gold-price fluctuations in terms of σ^t , a . The other information criteria do not yield significant benefits in case of underestimation. For the loss function 3P , in contrast, we observe strong evidence of a significant out-of-sample R2 statistic for the case that the costs of underestimation volatility exceed the costs of a corresponding overestimation ( b < 0) across the four different IC and across the three different training periods. In those cases where Fig. 12 displays the convergence R2 → 0 as b becomes large, we also observe that we get, as expected, a degenerate sampling distribution under the 3P loss function as the shape parameter, b, becomes sufficiently large.

5. Concluding remarks We have applied a boosting approach to forecast the volatility of gold-price fluctuations by means of various financial and macroeconomic variables. Forecasts account for model uncertainty and model instability. We then have applied the modified version of the out-of-sample R2 statistic of Campbell and Thompson (2008) studied by Pierdzioch et al. (2014b, 2015c), to evaluate forecasts for different shapes of a forecaster's loss function. Summing up, our results show that (i) the lagged realized volatility, the risk spread, and the growth rate of M2 are key predictor variables often selected across all IC under the recursive estimation scheme being studied in this research, (ii) a forecaster would benefit from using financial and macroeconomic variables to forecast the volatility of gold-price fluctuations by means of our forecasting model if the loss from underestimating the volatility of gold-price fluctuations exceeds the loss of an overestimation of the same magnitude, and (iii) the significance of this benefit depends on the information criterion and loss function being studied. We observe a significant out-of-sample R2 statistic mainly for the case that the costs of underestimation exceed the cost of an overestimation of the same size for the AIC information criterion and the loss function 3P . In future research, it is interesting to extend our analysis along several dimensions. First, it is interesting to use the boosting approach to study the predictive power for the volatility of goldprice fluctuations of other predictor variables than we have studied in our empirical study. For example, while we have studied the predictive power of U.S. macroeconomic data to ensure comparability of our results with results documented in earlier research, it is interesting to explore in future research the predictive power of macroeconomic developments in emerging market economies and the informational content of the dynamics of global variables. Second, the combination of a boosting approach with the out-of-sample R2 statistic criterion can be applied to study the informational content of financial and macroeconomic variables for the returns and the volatility of other metal and commodity prices. Such an analysis would be particularly important given that it has been emphasized by earlier researchers that precious metals cannot be considered as a single asset class (Batten et al., 2010). Third, while we have considered a simple autoregressive model as our benchmark model, it is straightforward to adapt the out-ofsample R2 statistic to situations where a researcher wants to study other and potentially more complex benchmark models. Finally, in technical terms, it is interesting to use other boosting approaches (see, for example; Mittnik et al., 2012) than the boosting approach

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used in this research to model the volatility of fluctuations of gold and other precious metals and commodity prices. While this research has focused on the evaluation of forecasts computed by means of a relatively simple boosting approach under alternative asymmetric loss functions, it is also interesting to examine, given a loss function, the comparative performance of alternative boosting approaches for forecasting the volatility of gold-price fluctuations.

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