#### 3.1. Data Description

To measure the degree of dependence across commodities, we compute the daily log-returns for the futures prices of the commodities listed in

Table 1. The 14 commodities include major metals, energy and agricultural products. The first panel of

Table 1 reports the commodities for which detailed results are discussed in

Section 3.2 and

Section 3.3. Commodity log-returns are obtained as the log differences of futures prices and are based on first generic futures contracts series extracted from Bloomberg. Specifically, we consider, at each date, the price of the contract closest to maturity. When a given contract approaches the expiration date, Bloomberg calculates a weighted average of the prices of two consecutive contracts, hence making a smooth transition between the contracts. Data cover the period from 11 August 2006 to 4 March 2020 for a total of 3500 observations for each commodity.

The descriptive statistics for the considered log-returns of the 14 commodities are reported in

Table 2. For the full period, the mean daily returns are all positive but extremely small. Oil and natural gas are the most volatile return series, with a daily standard deviation of 2.0% and 2.6%, respectively. In addition, commodity returns are negatively skewed and leptokurtic. Hence, the null of normality is strongly rejected by the Jarque–Bera tests. Furthermore, we assess if returns are heteroskedastic by applying the ARCH test ([

24]) and the Ljung–Box test to squared returns. For all commodities, the null of homoskedasticity is strongly rejected by both tests.

To calculate the Gerber statistic for the pair of returns $({r}_{i},{r}_{j})$, we consider two alternative pairs of thresholds: (1) ${Q}_{i}={\sigma}_{i}/2$ and ${Q}_{j}={\sigma}_{j}/2$ where ${\sigma}_{i}$ and ${\sigma}_{j}$ are the (unconditional) return volatilities; and (2) ${Q}_{i}={q}_{i}(90\%)$ and ${Q}_{j}={q}_{j}(90\%)$ where ${q}_{i}(90\%)$ and ${q}_{j}(90\%)$ are the (unconditional) $90\%$ quantiles of the returns.

#### 3.2. In-Sample Analysis

In this section, we estimate the CARML model in its reduced form, Equation (

5), and the DCC model for six selected pairs of commodity returns using the whole sample of 3500 observations. Based on the estimated parameters, it is then possible to obtain a time-varying version of probabilities appearing in Equation (

2) or Equation (

3). For instance, in the case of the CARML model, we use Equation (

8) with the estimated parameters and then Equation (

6) to recover the time-varying probabilities. In turn, these are used to obtain the time-varying Gerber statistic. We plot the resulting time-varying Gerber correlations of Equation (

3) in

Figure 1 and

Figure 2.

The shaded areas in the graphs—i.e., the periods from December 2007 to June 2009 and from February 2020 to the end of the period of investigation—describe the recession phases identified by the National Bureau of Economic Research. We find evidence of a significant degree of joint movements in our commodities and document that co-movements vary over time.

Figure 1 visually shows the changing patterns of co-movements on a daily basis and the significant amount of correlation pervasiveness in commodity returns, with some degree of heterogeneity. Sharp co-movements are detected for the pairs silver–gold (SI1,GC1), wheat–corn (W1,C1) and crude oil–gold (CL1,GC1).

The high co-movements between silver and gold is due to their characteristics of hedging commodities [

25,

26]. The study by Jaffe [

27] is one of the first analyses to empirically investigate the linkages between these two precious metals from 1971 to 1987. The pair was found to have a high correlation of 0.744. Erb and Harvey [

28], using data between December 1982 and May 2004, showed that silver has very low correlation with other major commodities, except gold (0.66). Our findings, based on the robust measure we consider, corroborate previous studies and suggest that, since the two precious metals display high correlation, the inclusion of both commodities in the same portfolio would be redundant from an investment perspective. An interesting result is for the pair gold and copper (not reported graphically). There is a very low positive correlation between the two commodities during the global downturn of 2007–2008, while their correlation significantly rises during normal times. This finding reflects the fact that the yellow precious metal is produced, in the great majority, for investment purposes, while the red metal is destined almost entirely for industrial usage. Hence, copper prices and returns tend to increase when economies are strong and growing, simply because there is a greater global demand [

29]. Conversely, gold prices and returns would rise when economies are weak, due to a risk aversion channel.

CARML and DCC models deliver similar results for the pairs crude oil–natural gas (CL1,NG1) and for crude oil–corn (CL1,C1). The joint return dynamics between crude oil and corn increased during the worldwide financial crisis. The rise in co-movements can be explained by the fact that an increase in oil prices generates contemporaneous upsurges in the price of other commodities, such as foodstuffs, via both the cost effect on the energy intensive agriculture sector and the substitution effect due to the increasing biofuel production which utilizes corn and soybeans [

30]. As an energy-intensive sector, agriculture is traditionally linked to the energy industry through its input channels. While fuel and electricity are used directly in agricultural production, fertilizers and pesticides represent the two most prominent indirect energy inputs. Through these energy input channels, higher energy prices increase the cost of producing and transporting agricultural commodities.

The case of crude oil–gold (CL1,GC1) is interesting. Specifically, during critical economic and financial phases, gold is confirmed to be a ‘safe haven’ given that the common movements between the two commodities fall significantly during the recession phases (both during the period of the global financial crash and the period of sovereign crisis).

The pair soybean–corn (S1,C1) exhibits relatively high co-movements throughout the sample period. A possible reason for the strong common movements between the two commodities could be the demand for biofuels, since corn and soybean are the main crops that are used in the production of biofuels (biodiesel and ethanol) and are good substitutes.

The result further suggests that spikes in co-movements were more marked during the global financial crisis than in other periods. It also indicates that the joint return dynamics among energy, agricultural and metal commodities is always positive (excluding very few negative relations between oil and natural gas). These positive co-movements point to a reduced possibility of diversification across commodities in the short-run.

Finally, DCC models provide more volatile Gerber estimates than CARML models owing to the fact that they do not directly produce the probabilities needed for the Gerber statistic. DCC models indeed focus on the dynamics of the conditional correlation matrix and the probabilities are subsequently obtained via the integrals reported in

Section 2.1.3. For CARML models, instead, the (logit transforms of) the required probabilities are directly modeled via the recursions described by Equation (

4) or (

5).

#### 3.3. Out-of-Sample Analysis

To assess the out-of-sample forecasting performances of the considered models, we estimate them recursively. We use a moving window of length 1000 and end up with

$N=2500$ forecasts for the probabilities appearing in the Gerber correlations. We consider again the reduced form of CARML models, Equation (

5). We compare the forecasting accuracy of different models using the Brier Scores defined as:

where

${\widehat{p}}_{hk,t|t-1}\left(\mathcal{M}\right)$ represents the forecast obtained using model

$\mathcal{M}$ (conditional on the information up to

$t-1$) and

${I}_{hk,t}$ is the time

t indicator associated to the region

$(h,k)$. For instance, when the investigated commodity returns are

${r}_{i}$ and

${r}_{j}$, the indicator variables for the regions

$UD$ and

$NU$ are

The Brier score is a tool commonly used to evaluate probability forecasts (see, e.g., [

19,

31]). Since the expected value of the score is lowest for the true probability, the Brier score represents a proper scoring rule [

32]. We compare the estimated models with the reference model (HS) using the Brier skill score defined as

Models that present a Brier skill score larger than zero have a superior forecasting accuracy compared to the reference model (historical simulation). Brier skill scores can also be used to compare the three alternative models considered in the present study as models with larger Brier skill scores are more accurate.

The Brier skill scores for the selected pairs of commodities are reported in

Table 3. The results indicate that CARML models perform better in term of forecasting power when the thresholds are half the volatility. DCC models are more suitable when forecasting extreme commodity returns for energy and metals. Moreover, CARML models have higher forecasting power for agricultural commodities in the case in which the thresholds are set equal to the 90% quantiles. In all cases, CARML and DCC are more accurate than filtered historical simulation models.

Table 4 instead refers to all 91 pairs of commodities we can form. In the table, we provide the number of pairs (and frequency, i.e., the number divided by 91) for which a model has a positive Brier skill score. In the table, we also give the number (and frequency) of pairs for which a given method presents a Brier skill score larger than an alternative model. The results confirm that both the CARML and DCC models outperform FHS. CARML models perform better than DCC models when the threshold is half the unconditional volatility, whereas the opposite is true when the threshold is set equal to the 90% quantile. From the analysis of all the possible pairs of commodities, we notice that CARML models tend to be more accurate when both commodities are of the agricultural type, while DCC models are more accurate when both commodities are energy products or metals.

As a robustness check, we consider two additional thresholds for the Gerber correlation.

Table 5 reports the Brier skill scores for the six selected pairs of commodities when the thresholds are chosen as

${Q}_{i}={q}_{i}(85\%)$,

${Q}_{j}={q}_{j}(85\%)$,

${Q}_{i}={q}_{i}(95\%)$ and

${Q}_{j}={q}_{j}(95\%)$.

Table 6 summarizes the results for all the pairs of commodities for these thresholds.

Table 5 and

Table 6 confirm the superiority of all three models with respect to HS. Comparing the two tables, we see that, when we increase the quantile from 85% to 95%, the number of pairs for which the CARML models outperforms DCC models seems to decrease.