The role of comovement and time-varying dynamics in forecasting commodity prices.
Date: 2024-02-13
Institutions: European Central Bank
Subjects: econometrics, economic cycle, economic forecasting, primary product

Anastasia Allayioti, Fabrizio Venditti 

Working Paper Series 

The role of comovement and time-
varying dynamics in forecasting 
commodity prices 

No 2901 

Disclaimer: This paper should not be reported as representing the views of the European Central Bank 
(ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. 

 
 
 
Abstract

Commodity prices co-move, but the strength of this co-movement changes over time due 

to structural factors, like changing energy intensity in production and consumption as well 

as changing composition of underlying shocks.  This paper explores whether econometric 

models that exploit this co-movement and account for parameter instability provide more 

accurate  point  and  density  forecasts  of  ten  major  commodity  indices  viz-a-viz  constant-
coefficient mo dels. Improvements in point forecast accuracy are small, with predictability 

varying substantially across forecast horizons and commodity indices, but they are large and 

significant in terms of density f orecasting. An economic evaluation reveals that allowing for 

parameter time variation and commonalities leads to higher portfolios returns, and to higher 

utility values for investors.

JEL classification:  C32 , C52, C53, C55, E37

Keywords:  Commodities;  Commonalities;  Instabilities;  Density  forecasting;  Economic 

evaluation

ECB Working Paper Series No 29011Non-technical summary

Commodity prices have been a recurrent source of concern for policymakers, given their
effects on terms of trade, inflation, inflation expectations and disposable income. Forecasting

commodity prices, in particular, plays a key role in inflation projections in central banks, which

typically condition their inflation forecasts on a given path of oil and other commodity prices.

The predictability of commodity prices therefore remains an active and policy-relevant area of

applied research.

A striking feature of commodity prices, and one that might prove useful in forecasting, is the

extent of their co-movement. Significant co-movement is found not only for the prices of closely

related commodities but also for those that exhibit near-zero cross-price elasticities of demand

and supply. The strength of this co-movement, however, varies significantly over time, owing

to both structural as well as cyclical factors. Structural factors include the changing intensity

of commodity use (both in production and consumption) as well as the rising importance

of commodities in financial markets. Cyclical factors relate to the nature of the structural
shocks affecting prices. Whereas demand shocks are a common source of fluctuations across

prices, supply shocks are commodity specific, and they weaken co-movement across commodity

prices. Motivated by these two stylized features (co-movement and instability), this paper

investigates the merits of constructing forecasts for key commodity prices from models that

exploit co-movement in large data sets and that can deal with structural breaks. Among others,

we consider large dynamic factor models with time varying parameters (TVP) - including

specifications that impose the presence of specific blocks on the factor structure of commodity
prices - as well as their constant-coefficient counterparts. Evaluation is based on statistical

criteria but also on economic grounds, in particular on the gains accrued to an investor who

uses the model-implied forecasts for portfolio allocation.

In terms of forecast accuracy, we find that the largest improvements come from modeling

time variation (especially with respect to the second-order moments) rather than from the way

in which the factor structure of commodity prices is modeled. This is particularly evident when

we evaluate the ability of competing specifications in generating accurate and well-calibrated

predictive forecast densities. The economic evaluation of the competing forecasting models, on

the other hand, reveals an economic value of allowing for both parameter time variation and

commonalities.

ECB Working Paper Series No 290121

Introduction

Commodity prices have been a recurrent source of concern for policymakers, given their effects on

terms of trade, inflation , inflation expectations and disposable income. Forecasting commodity

prices is also an important block of inflation projections in central banks. Also, commodities are

now widely traded in derivative markets and form an important part of investors portfolios.

In spite of numerous empirical studies exploring whether commodity prices can be predicted,

and, if so, by which variables, forecasting commodity prices with reasonable accuracy remains a

challenging endeavor. Future prices, the most natural candidate, have proven to be inadequate

predictors of future spot prices (Hong and Yogo, 2012).

A striking feature of commodity prices, hence one that might prove useful in a forecasting

context, is the extent of their co-movement. Significant co-movement is found not only for the

prices of closely related commodities (think, for instance, of cocoa and sugar) but also for those

of unrelated commodities i.e. those for which the cross-price elasticities of demand and supply

are close to zero, according to the definition by Pindyck and Rotemberg (1990). Recent work

aiming at explaining the mechanism behind the high-degree of co-movement since the turn of

the twenty-first century includes, among others, Krugman (2008) who argues that the increase in

oil prices was responsible for the subsequent increase in food prices. More recently, Alquist et al.

(2014) show that the vast majority of historical commodity price movements can be attributed to

a general-equilibrium response to aggregate non-commodity shocks rather than specific shocks

to commodity markets. Building on this literature, Delle Chiaie et al. (2022), henceforth DFG

(2022), extract a common factor across a large panel of commodity prices and validate its use as

an indicator of the global business cycle.

Yet, the strength of the co-movement across commodity prices varies significantly over time,

as we document in Section 2. This is due to a host of factors, both structural as well as cyclical.

Structural factors include the changing intensity of commodity use (both in production and

consumption) as well as the rising importance of commodities in financial markets. Cyclical
factors relate to the nature of the structural shocks affecting prices. Indeed, whereas demand

shocks are a common source of fluctuations across prices, supply shocks are commodity specific,

and their occurrence weakens co-movement across commodity prices. In a recent contribution,

for instance, Peersman et al. (2021) document sizeable spillovers across commodity markets, but

find that the size of such spillovers (and therefore the intensity of price co-movement) changes

ECB Working Paper Series No 29013over time.

Motivated by these two stylized features (co-movement and instability), this paper studies the

performance of a specific class of econometric models, namely factor models with time-varying

parameters, in forecasting commodity prices and indices using 68 spot commodity prices.

Among the competing forecast specifications, we evaluate a dynamic factor model (DFM) with

10 factors, a hierarchical-DFM with the same number of factors that imposes a blocks structure,

in the spirit of DFG (2022), and their respective time-varying counterparts. Our work assesses

whether the accuracy of point and density forecasts for the price of a given commodity could

benefit from (i) imposing a block structure in cross-correlations and (ii) dealing with structural

breaks. The analysis is based on monthly data and evaluates the forecasting performance for the

period 2001:M1-2021M3.

The first result of our analysis is that the largest forecast improvements come from modeling

time variation, while the precise way in which co-movement is modeled is of lesser relevance.

This is particularly evident when we evaluate the ability of competing specifications in gen-

erating accurate and well-calibrated predictive forecast densities. Second, we document that

allowing for time-varying second-order moments substantially enhances the predictability of

commodity prices. Third, we find that predictability is heterogeneous across forecast horizons
and for different types of commodity prices. Finally, using a simple mean-variance portfolio

allocation framework we perform an economic evaluation of the competing forecasting models.

This analysis reveals an economic value of allowing for both parameter time variation and

comovement.

Our paper is connected to three strands of the literature. The first is the stream of papers
that explore the power of different predictors to forecast commodity prices. Chen et al. (2010)

and Pesenti and Groen (2011) look at the relationship between exchange rates and commodity

prices. Chen et al. (2010) find evidence that exchange rates predict commodity prices both

in-sample and out-of-sample, while Pesenti and Groen (2011) document much weaker support
for commodity exchange rates. Gargano and Timmermann (2014) focus on a set of different

predictors taken from the stock return predictability literature, in addition to macroeconomic

predictors such as inflation, money supply growth, growth in industrial production, and the

unemployment rate, along with exchange rates for commodity currencies, and indicators of

global economic activity. In an out-of-sample point forecasting exercise, DFG (2022) find that a

DFM with constant parameters has adequate point forecasting properties at short horizons. We

ECB Working Paper Series No 29014include their model in our empirical analysis and explore how it compares to other constant

and time-varying specifications in terms of point and density forecast accuracy, as well as in

terms of accrued economic gains. Our results illustrate how introducing time-varying dynamics

enhances both statistical and economic gains.

We also contribute to the literature exploiting empirical factor models to summarize a wide

range of information contained in commodity prices. For example, Byrne et al. (2013) investigate

the relationship between commodity prices and macroeconomic determinants using a Factor

Augmented Vector Auto Regression (FAVAR) approach. Moreover, Chen et al. (2014) show that

the bulk of movements of 51 tradeable commodities is mostly due to the first common component.

The main drawback with extracting a single common component from commodities is that
the estimated factor can be difficult to interpret and may not fully account for heterogeneity

(Moench et al., 2013; Byrne et al., 2019). Recent research has attempted to address these concerns.

For example, Yin and Han (2015) use a multilevel factor model to decompose commodity returns

into global, sectoral and idiosyncratic components.

Our paper is also related to the research on the role of structural instability in commodity

markets. The majority of existing work on the determinants of commodity prices relies upon a

time-invariant methodology (e.g. Poncela et al., 2014). There are, however, compelling reasons

to believe that changes occur in the composition of underlying structural shocks over time (e.g.

Kilian, 2009) and in their transmission mechanism (e.g. Riggi and Venditti, 2015; Blanchard and

Gali, 2007).

Our study differs from past empirical work in many aspects. First, we evaluate the forecasting

performance of various models instead of restricting our attention to a single model. The model

space includes specifications that account for the factor structure of commodities, potential

heterogeneity and the presence of structural breaks. This extensive comparison allows us to

disentangle the importance of each one of these characteristics in improving forecast accuracy.

To our knowledge, commodity heterogeneity and time-variation have not been simultaneously

considered in the forecasting literature within a unified framework.1 A second contribution of

this paper is that we evaluate not only point, but also density forecasts. Third, we look beyond

traditional statistical performance and perform an economic evaluation of the various forecast

models by analysing whether an investor conditioning on the models’ forecasts would be able

1For instance, DFG (2022) restrict their attention to a constant-parameter factor model that accounts for commodity
heterogeneity but does not allow for the presence of instabilities. Conversely, Chen et al. (2010) account for parameter
instability but only explore the predictive content of exchange rates for commodity prices.

ECB Working Paper Series No 29015to obtain tangible economic gains. To this purpose, we test whether an investor might rationally

use the predicted return (and its estimated variance) for portfolio optimization. Our economic

evaluation exercise is similar in spirit to existing studies forecasting stock returns (e.g. Rapach

et al., 2010) and exchange rate fluctuations (e.g. Abbate and Marcellino, 2014).

The rest of the paper is organized as follows. Section 2 introduces the data and presents

some stylized facts that motivate our analysis. Section 3 describes the theoretical framework of

the constant and time-varying specifications under examination. Section 4 discusses estimation

algorithms. Section 5 provides details about the nature of the forecasting setting. Section 6

presents the results of the analysis. Section 7 concludes.

2 Data and motivating evidence

The dataset consists of 68 monthly spot commodity prices in nominal U.S. dollars (see Table

A2 in Appendix A).2 The source of our data is the International Monetary Fund (IMF) primary

commodity price database and covers the period January 1992 to March 2021. The IMF also

constructs various price indices and sub-indices as the weighted average of the 68 individual

commodity prices. The weights of the indices correspond to the global import share over a

3-year period (2014-2016).3

We forecast ten core indices that reflect the different categories of individual prices described

in Table 1: one overall index for the entire set of commodities All Commodity that is constructed

as the weighted average of the 68 individual commodity prices and 9 sub-indices that further

decompose the individual prices into categories.4 Two core blocks are represented by, first, the

Non-Fuel index, which consists of 59 non-fuel prices and, second, the Fuel index which consists of

9 fuel prices. Subsequently, these two main blocks are further decomposed. The Non-Fuel index

consists of the Agriculture index with 42 prices, that is further decomposed into the Agriculture

Raw Materials index (9 prices), the Food index (29 prices) and the Beverages index (4 prices), the

Fertilizers index with 3 prices and the Metals index (14 prices). 5 Similarly, the Fuel index, which

consists of 9 fuel prices, is decomposed into the Oil index (3 prices) and the Coal, Natural Gas &

Propane (CNP) index (4 prices).

2Note that the prices are period averages and are not seasonally adjusted.
3For some indices, data was not available starting from January 1992 so we re-constructed these using the IMF

methodology.

4For the aggregate commodity index, All Commodity and All will be used interchangeably.
5To keep our hierarchical factor model parsimonious, we combine the Agriculture and Fertilizers indices into one.

ECB Working Paper Series No 29016Table 1: Description of the adopted block structure. Values in parenthesis reflect trade weights
(%)

Global

Blocks

Subblock

Groups

All Commodities
(100)

Non-Energy
(59.1)

Agriculture
(34.5)

Agricultural Raw Materials
(4.3)

Beverages
(2.3)

Food
(27.8)

Energy
(40.9)

Fertilizers
(1.9)

Metals
(22.7)

Coal
(3.0)

Crude Oil
(28.6)

Natural Gas
(7.8)

Propane
(1.5)

N

(68)

(59)

(42)

(9)

(4)

(29)

(3)

(14)

(9)

(2)

(3)

(3)

(1)

ECB Working Paper Series No 29017To motivate our analysis, we start by looking at the cross-correlations and volatility of

selected core commodity indices over the period January 1992 to March 2021. Figure 1 presents
a diagram of the (unconditional) cross-correlation coefficients between five core indices and
across different sub-samples. Off-diagonal entries illustrate a scatterplot of the relevant variables
with a least squares reference line whose slope equals the displayed correlation coefficient. The
coefficients in red indicate that the null hypothesis of zero correlation cannot be rejected. Each

diagonal subplot depicts the distribution of each commodity index (returns) as a histogram.

The left-hand side graph in the first row (Full sample) depicts correlations over the full sample,
while the right-hand side graph runs from 1992 up until the beginning of the 2007/8 financial

turmoil. Moving to the second row, we present cross-correlations over December 2007 to June

2009 (Figure 1 - LHS ), and December 2019 to the end of sample (Figure 1 - RHS). The former

sample corresponds to the eighteenth-month recession that characterized the Global Financial

Crisis as identified by the NBER, and the latter sample signals the Covid-19 pandemic.

It is evident that the years prior to the 2007/08 turmoil exhibit weaker evidence of co-

movement. For example, while between 1992 and mid-2000s the lowest correlation (in absolute

value) is as low as (approximately) 1%, during the Global Financial Crisis years the lowest
correlation was approximately 25%. Also, the sign of certain correlation coefficients changes going

from the pre-GFC to the post-GFC years. During the first subsample we osberve (statistically

insignificant) negative correlation for specific commodity-pairs, e.g. Food & Beverages vs. Fuel.

In contrast, the Great Recession years are characterised by positive and statistically significant
correlations. A closer look at the correlation coefficients between December 2007 and June 2009

emphasizes the increased degree of co-movement during high-uncertainty periods. Similar

conclusions can be reached when examining the first half of the pandemic crisis.6 Overall,

these findings are consistent with previously documented results in the literature showing that,

for macroeconomic and financial variables, downturn periods are characterized by increased

co-movement (D’Agostino and Giannone, 2012).7

6The plots in the second row illustrate relationships based on a relatively small sample size. Employing the
Kendall correlation which offers a non-parametric alternative robust to smaller sample sizes results in qualitatively
similar results.

7The question of what drives commodities co-movement and whether these commonalities are originating from
short-run or long-run forces is beyond the scope of our paper. Notable (co-movement) drivers include the interest
rate (Byrne et al., 2013), US dollar exchange rate (Vansteenkiste, 2009), fluctuations in global economic activity that
are more relevant on the medium and long run (Delle Chiaie et al., 2022; Alquist et al., 2020; Baumeister et al., 2020)
and uncertainty that has been shown to exhibit an important role in determining short-run commonalities (Poncela
et al., 2014). In a recent application, Casoli and Lucchetti (2022) estimate factor models by taking cointegration into
account and find that commodity prices move together mostly due to long-term common forces.

ECB Working Paper Series No 29018Figure 1: Cross-correlations between main commodity indices over the full sample and three
subsamples

Notes: The chart presents correlation coefficients over the full sample, the first half between
1992 - 2006, the highly volatile period between 2007-2009 and the Covid-19 years for 5
main indices: Energy (Fuel), Oil (Oil), Agricultural (Agri), Food & Beverages (FoBe), and
Natural Gas (Gas).

ECB Working Paper Series No 29019Figure 2: Commodity prices volatility measured as the square monthly percentage changes in
the commodity reported in the title

ECB Working Paper Series No 290110Figure 2 provides some preliminary evidence on the presence of time-varying volatilities

(as measured by squared monthly percentage changes). Volatility was mostly low in the early
1990s but it spiked dramatically during the 2007/08 to fall again thereafter. More recently, the
Covid-19 shock affected prominently energy prices as containment measures led to a collapse of

traveling and transportation.8

In conclusion, summary statistics show evidence of time-variation in the degree of com-

monality and in the volatility of commodity prices. In the remaining of this paper, we explore

whether accounting for these two features can improve point and density forecasting accuracy

and yield meaningful economic gains.

3 Dynamic Factor Models (DFM): a suite of specifications

Our empirical investigation makes use of a suite of models that can be nested in the more general

specification of a dynamic factor model with time-varying parameters (TVP-DFM). Table 2

illustrates the full set of competing models and their respective estimation methodologies.9

Let xit = (x1t, ..., xnt) ′ be an n− dimensional vector of commodity returns expressed as the

log change in month t of a given commodity price i = 1, ..., n, that follows a dynamic factor

model of the form

xit = λitft + ϵit

ft = Btft−1 + ηt

(1)

(2)

where ft is the k × 1 vector of factors, λit is the n × k factor loadings vector, Bt is a k × k
matrix of VAR(1) coefficients and ϵit and ηt are disturbance terms. It is further assumed that
ϵt ∼ N(0, Vt) and ηt ∼ N(0, Qt) where Qt and Vt are the n × n and k × k diagonal covariance

matrices respectively. Note that the ϵit are uncorrelated with both ft or ηt at all leads and lags.

We let λt and βt evolve as driftless random walks:

λt = λt−1 + ut ut ∼ N(0, Rt)

βt = βt−1 + vt

vt ∼ N(0, Wt)

(3)

(4)

8Excess supply turned out to be so extreme and storing capacity so tight, that a given WTI crude oil futures

contract fell to a negative value on April 20, 2020, to -$38.

9We have also evaluated the statistical and economic gains of a large-dimensional non-parametric VAR proposed
by Kapetanios et al. (2019). This specification does not impose any type of factor structure but, instead, makes use of
the information content of the complete set of 68 commodities. Results are available upon request.

ECB Working Paper Series No 290111Table 2: Prediction models

Specification

Estimation

Constant Parameters
models

DFM
dfm

QML

DFM with block structure
dfmBlocks

Time-Varying Parameters
models

TVP-DFM
tvp

Forgetting Factors

TVP-DFM with block structure
tvpBlocks

The law of motion of the covariances Rt and Wt is described in Appendix A.2. We follow a

standard assumption in assuming that ut and vt are iid errors, uncorrelated with each other

as well as with ϵt and ηt at all leads and lags.10 The specification hereby adopted is largely

inspired by Del Negro and Otrok (2008), who explore the empirical performance of a TVP-DFM

in explaining international business cycles. Variants of this model have been used to study

the properties of global inflation (Mumtaz and Surico, 2012) and the interaction between the

macroeconomy and financial markets (Bianchi et al., 2009).

10See, for instance, Koop and Korobilis (2010).

ECB Working Paper Series No 290112We further impose a block structure that reflects the composition of the IMF commodity

price database and follows DFG (2022) in spirit.11 In particular, we extract one global factor

(All Commodities), two block factors (Non-Fuel and Fuel), four sub-block factors (Agriculture &

Fertilizers, Metals, Crude Oil and Coal, Natural Gas and Propane) and three group factors (Agricultural

Raw Materials, Food, and Beverage).12 Table 1 provides an overview of the imposed block structure.
Appendix A.2 offers details on the adopted factor specification and the corresponding block

structure.

The setup of DFM with structural instabilities given in Equations 1 and 2 nests the typical

time-invariant parameter DFM, when we restrict variation in all parameters. The time-invariant

parameter DFM is derived by replacing the time varying-specifications in equations 3 and 4

with

Rt = Wt = 0

and setting the time-varying covariances Vt and Qt to their constant values

Vt = V and Qt = Q

(5)

(6)

One additional nested model that imposes the assumption of time-invariant parameters

and similarly allows for the existence of group-specific comovement can be derived in a
straightforward manner. This constant-coefficient specification has been recently adopted by

DFG (2022) in order to study comovement in international commodity prices.

4 Estimation algorithms

In this section we introduce the estimation methodologies for both the time-varying and

constant-parameter specifications.

4.1 Time-varying models

Time variation is modelled through a forgetting factor framework, introduced in the macroe-

conomic forecasting literature by Koop and Korobilis (2013). This model can be considered as

an approximation of the Bayesian VAR with time-varying parameters and stochastic volatility

developed by Cogley and Sargent (2005) and Primiceri (2005). Instead of using a sampler to

11Note that our block structure differs from DFG (2022) in the number and composition of the distinct factors.
12Using the trade weights published in the IMF website, and with the purpose of maintaining a parsimonious
factor structure, we merge specific factors into one. In particular, we construct one common factor for Coal, Natural
Gas and Propane, and another one for Agriculture and Fertilizers.

ECB Working Paper Series No 290113draw the covariance matrices of the time-varying coefficients and volatility parameters, the

proposed framework estimates them as a weighted average of previous estimates and of the

current forecast error variance. It is a more parsimonious set up that considerably speeds up

computational time, and enables the analysis with a larger number of variables. See Appendix

A.3 for estimation details.

4.2 Time-invariant models

For the estimation of the constant-coefficient specifications we adopt an extension of the two-step

estimation algorithm for constant-parameter dynamic factor models of Doz et al. (2011). In

the first step, the algorithm is initialized by computing principal components, and the model

parameters are estimated by an OLS regression. The next step updates the factors estimates by

using the Kalman smoother as in Doz et al. (2011). The additional, and final, step we take is to

re-estimate the factor loadings based on the updated Kalman filter estimates of the previous

step.

5 The design of the forecasting exercise

We estimate the models over the period 1992M1-2000:M12 and evaluate their forecasting

performance for the period 2001:M1 through 2021M3-h for h = 1, 3, 6, 12 months ahead. In

addition to the assessment of the competing models over the full sample, we present an

evaluation based on distinct subsamples. The first subsample forecasts are for the period
2001M1 + h to 2007M1, while the second subsample begins just before the 2007/08 financial

turmoil and extends to the end of the sample.

The benchmark model against which we evaluate the predictive ability of the competing

specifications, unless otherwise stated, is the random-walk (no-drift) model. For both constant-

and time-varying specifications, we obtain the forecasts of the 68 individual commodity prices

iteratively using the Kalman filter formulation.

ˆxt+h|t ≡ E[xt+h|It] = Λ ˆE[ft+h|It] = Λ ˆft+h|t = ΛBhft|t

ˆxt+h|t ≡ E[xt+h|It] = ˆE[λt|tft+h|It] = λt|t ˆft+h|t = λt|tβt+hft|t

(7)

(8)

The respective point and density forecasts of the ten major commodity indices are, subse-

quently, calculated as the weighted averages of the individual commodity price forecasts using

ECB Working Paper Series No 290114the import weights published in the IMF primary commodity prices database.

For generating samples from the predictive density of the time-varying factor specifications

(eq. 8) we follow (Koop and Korobilis, 2013, henceforth KK, 2013). We derive the predictive

densities from the constant-parameter specifications (eq. 7) using bootstrap methods. The focus

of the forecasting literature has been steadily moving from point forecasts to density forecasts

that incorporate the uncertainty about the future evolution of the variables of interest; see,

among others, Jore et al. (2010) and Clark (2011). A multivariate forecast density for a given

horizon can be obtained assuming Gaussian forecast errors and known lag order and model

parameters. However, documented departures from these assumptions pose a serious concern

when forecasting with multivariate models, calling into question widely-used techniques for

constructing predictive densities. Bootstrap procedures provide a flexible methodology for

constructing forecast densities that incorporate parameter uncertainty and can, furthermore,

relax assumptions on the error distributions for a more general treatment. 13

The steps for obtaining a bootstrap forecast density and associated statistics for the 68

commodities in vector xit, i = 1, , , .68 are as follows:

1. For t = 1, ..., T , estimate the factor ( ˜ft) and associated factor loadings ( ˜Λ) and obtain the

residuals:

2. Generate x∗

˜ϵit = xit − ˜Λ ˜ft;
it = ˜Λ ˜ft + ϵ∗

it where ϵ∗

it is a resampled bootstrap version of the residuals ˜ϵit;

3. Using the new bootstrap set x∗

it, estimate the bootstrap factors f∗
t;

4. Estimate the corresponding bootstrap parameters, e.g. Λ∗, B∗

1 etc.;

5. The (bootstrap) forecast is given by Eq., (7), replacing the respective quantities with their

bootstrap counterparts. The forecast density F(xit+h) is given by the density function of

the bootstrap forecasts xit+h,b, b = 1, ..., B where B is the desired number of replications.

5.1 Forecast Evaluation Metrics

We perform a recursive out-of-sample forecasting exercise to evaluate the performance of

the competing forecasting methods in terms of both point and density h-step-ahead iterated

forecasts.

13Focusing on the constant-coefficient factor models (eq. 7), adopting a bootstrapping algorithm allows us to capture
uncertainty resulting from the estimation of the factors.14 The methodology is inspired by recent work on prediction
intervals for factor models. For example, Gonc¸alves and Perron (2014) propose a general residual-based bootstrap
method for inference in a factor-augmented regression, while Gonc¸alves et al. (2017) extend this bootstrapping
methodology to prediction intervals. Our algorithm, instead, can be applied to dynamic factor models of the form
described above and is based on a residual-based bootstrap scheme.

ECB Working Paper Series No 290115An extensive literature on statistical evaluation methods proposes various loss functions

and test statistics in order to assess statistical significance. We employ the Root Mean Squared

Forecast Error (RMSFE) criterion to explore whether it is possible to beat the random walk

forecast for a given forecast horizon. For each competing model, we compute the ratio of the

RMSFE of the individual specification relative to the RMSFE of the driftless random walk.

To evaluate the statistical significance of differences in forecast accuracy across the competing

specifications, we consider tests of equal predictive accuracy (Diebold and Mariano, 1995;

West, 1996). The tests compare the forecasts from the factor models against the benchmark.

Additionally, we incorporate the finite sample correction proposed by Harvey et al. (1997).15

We evaluate the entire predictive density and compare the cumulative differences in log-

predictive likelihoods of the competing models. Predictive density accuracy is evaluated on the

basis of the logarithmic score, see Mitchell and Hall (2005) and Amisano and Giacomini (2007).

See Appendix A.4 for additional details.

We investigate the correct specification of predictive densities using different tests. The

tests we consider include tests of uniformity, accounting for the possibility of serial correlation.

Focusing on tests based on the Probability Integral Transform (PIT), we consider the histogram-

based evaluation technique employed by Diebold et al. (1997) and Diebold et al. (1999). As

was highlighted in these papers, a correctly conditionally calibrated density produces PITs that

are uniformly distributed. For the one-step ahead forecasts, we evaluate the calibration of the

densities by looking at the results of Berkowitz (2001)’s LR test. For horizons beyond one step

ahead, we report the test of Kn ¨uppel (2015), which is robust to the presence of serial correlation

of the PITs.

6 Empirical results

6.1 Point forecasts

Tables 3 and 4 present RMSFE ratios for the monthly forecast horizons 1, 3, 6 and 12 months ahead

of the competing models relative to the benchmark random walk model over the subsamples

1992 - 2007 and 2008 - 2021. Entries with values less than one indicate that the forecast model

15It is important to note that the literature examining testing procedures shows that a number of issues arise when
testing for significant differences in forecasting performance, including the size of the in-sample relative to the size of
the out-of-sample period, and the type of estimation window used; see Corradi and Swanson (2006) for a useful
survey of some of these issues. In light of such evidence, the adopted test procedure is only used to provide a rough
guide for assessing statistical significance.

ECB Working Paper Series No 290116under examination delivers accuracy gains relative to the benchmark. 16

Looking at the overall forecasting performance of each model across the 10 indices, our
findings highlight the difficulties in finding a model that performs uniformly better than the

others across forecast periods, horizons and commodity indices. Focusing on the subsample

coinciding with the pre-GFC years (Table 3) , the competing factor models consistently outperform

the random walk for specific indices belonging to the NonFuel block, namely Agriculture and

Fertilisers, Metals and Agriculture Raw Materials. At the shortest horizon (h = 1) evidence of

predictability is weak, but as the forecast horizon increases, forecasting performance becomes

stronger. At h = 3, the time-invariant factor specification (dfm) improves upon the random walk

benchmark by 10% for the Non-Fuel index, while accounting for time-variation and commodity-

specific comovement appears to be particularly beneficial for the Metals index. Focusing on

the six-month ahead point forecasts, we observe that, for all but two indices (Oil and CoalGas),

introducing a factor structure is successful at outperforming the random walk benchmark.

For example, looking at the All Commodities index, the tvpBlocks specification outperforms the

competing factor models and the random walk benchmark, enhancing accuracy by more than

10%. At the longest horizon of one-year ahead, significant gains in point forecast accuracy

are documented across all ten commodity indices. On average, the forecasting model that

captures time-evolving dynamics and commodity-specific heterogeneity improves upon both its
constant-coefficient variant and the model that does not impose a specific block structure.

Results for the subsample from 2008 onward (Table 4) appear to reinstate the empirical
difficulty of finding a single modelling framework that outperforms its competitors at all

horizons. At the short and medium-term horizons (1-, 3-, and 6- months ahead), most models

deliver improvements over their pre-GFC performance.17 Nevertheless, we do note that at the

longest horizon of 12-months ahead, evidence of a superior post-Crisis performance is weaker.

For example, focusing on the Fuel index, we find statistically significant improvements over

the random walk model for the one-month and three-months forecast horizons. In contrast,

all competing specifications fail to outperform the benchmark model at the one-year ahead

16One concern related to macroeconomic and financial forecasting studies is the effect of data mining on the size of
tests of predictability. Data mining occurs when a researcher searches over alternative forecast specifications, but
only reports results for the model with the highest predictive content, causing the size of the test of predictability to
be inflated, and thereby resulting in spurious rejections of the no-predictability null (Inoue and Kilian, 2005). As a
first attempt in tackling issues related to collective data mining (Denton, 1985), we report results for high levels of
significance which should provide some confidence in the results.

17This finding is in line with prior literature documenting an enhanced point forecast performance of factor models

during the Great Recession, e.g. DFG (2022).

ECB Working Paper Series No 290117horizon. A similar pattern is observed for the majority of commodity indices. Results for the
Metals and Agriculture and Fertilisers indices paint a different picture, with all factor specifications

outperforming the benchmark random walk model at short-, medium-, and long-term horizons.

For instance, the evaluation exercise for the Metals index points to a particularly enhanced

performance of all models relative to the random walk, with the time-varying specifications
outperforming their constant-coefficient counterparts. At the longer horizons, no model exhibits

a consistently improved performance over the benchmark random walk. Nonetheless, a

comparison between the time-invariant forecasting models and their time-varying counterparts
suggests that introducing time-evolving dynamics to constant-coefficient factor models (with
or without a hierarchical structure) lowers forecast errors for h = 12. Turning to the effect of

imposing the block structure described in Table 1, we focus on the relationship between dfm and

dfmBlocks. Looking at the one-month ahead point forecasts, introducing a hierarchical structure

is beneficial for eight out of ten commodity indices. A similar performance is documented

at three- and six-months ahead. Improvements appear to diminish at the longest horizon of

one-year ahead, with the block structure model delivering marginal gains.

ECB Working Paper Series No 290118Table 3: RMSFE ratios of the competing models relative to a random walk for various forecast
horizons over the first sub-sample period 1992M1 to 2007M12. Underlined values denote
significantly different forecast errors according to a Diebold − Mariano test, modified using the
small sample correction of Harvey et al. (1998).

Model↓|Horizon→

h=1

h=3

h=6

h=12

||

h=1

h=3

h=6

h=12

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

All

1.03
0.95
1.04
0.95

NonFuel

0.90
0.93
0.96
0.94

Metals

0.95
0.95
0.96
0.94

CGP

1.09
1.06
1.10
1.11

0.93
0.88
0.98
0.86

0.84
0.83
0.87
0.81

0.89
0.92
0.93
0.91

0.82
0.83
0.85
0.81

0.94
0.94
0.95
0.93

0.93
0.92
0.93
0.91

1.07
1.04
1.07
1.05

0.98
0.98
1.04
0.96

Beverages

1.07
1.05
1.04
1.04

1.05
1.06
0.97
1.01

0.96
0.97
0.93
0.94

1.09
1.01
1.06
0.99

0.99
1.02
1.04
1.03

0.98
0.97
0.97
0.97

1.09
1.07
1.11
1.09

1.09
1.03
1.06
1.03

Fuel

1.08
0.99
1.10
1.05

Agr/Fuel

0.95
0.95
0.95
0.95

Oil

1.17
1.08
1.19
1.13

Food

1.01
1.00
1.12
1.01

1.00
0.94
1.03
0.96

0.90
0.89
0.94
0.88

0.94
0.94
0.95
0.93

0.94
0.94
0.93
0.93

1.11
1.05
1.16
1.08

0.94
0.92
0.99
0.90

0.95
0.96
1.04
0.96

0.90
0.92
1.01
0.91

AgriRaw

0.97
0.95
0.99
0.97

0.98
0.96
0.96
0.96

0.91
0.91
0.92
0.91

1.12
1.02
1.11
1.04

0.97
0.98
0.97
0.97

1.16
1.08
1.19
1.11

1.07
1.02
1.09
1.03

0.96
0.96
0.98
0.95

ECB Working Paper Series No 290119Table 4: RMSFE ratios of the competing models relative to a random walk for various forecast
horizons over the first sub-sample period 2008M1 to 2021M3. Underlined values denote
significantly different forecast errors according to a Diebold − Mariano test, modified using the
small sample correction of Harvey et al. (1998).

Model↓|Horizon→

h=1

h=3

h=6

h=12

||

h=1

h=3

h=6

h=12

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

dfm
dfmBlocks
tvp
tvpBlocks

All

0.98
0.97
0.99
0.98

NonFuel

0.98
0.96
0.98
0.95

Metals

0.93
0.93
0.95
0.93

CGP

0.94
0.97
1.00
0.98

1.04
1.03
1.02
1.01

1.14
1.13
1.08
1.05

1.02
1.02
0.99
0.99

1.11
1.10
1.07
1.07

0.93
0.91
0.90
0.90

0.92
0.91
0.89
0.90

0.99
1.01
1.00
0.99

1.12
1.11
1.07
1.05

Beverages

1.05
1.03
1.05
1.03

1.02
1.02
1.02
1.03

1.11
1.09
1.09
1.07

0.95
0.91
0.94
0.93

0.95
0.92
0.94
0.92

0.98
0.97
0.98
0.97

0.97
0.94
0.97
0.96

0.99
0.98
0.99
0.98

Fuel

1.00
0.98
1.00
1.01

Agr/Fer

0.95
0.95
0.97
0.96

Oil

1.07
1.04
1.05
1.05

Food

0.98
0.97
0.98
0.98

0.99
1.04
1.03
1.03

1.13
1.13
1.09
1.08

0.95
0.93
0.95
0.94

0.95
0.94
0.95
0.94

1.11
1.09
1.08
1.06

1.19
1.18
1.15
1.13

1.04
1.03
1.01
1.01

1.11
1.10
1.10
1.08

AgriRaw

0.95
0.97
0.94
0.96

1.01
1.01
1.00
1.00

1.08
1.06
1.05
1.05

0.96
0.92
0.95
0.96

0.98
0.99
0.99
0.99

1.05
1.03
1.02
1.00

0.97
0.95
0.96
0.94

0.94
0.97
0.92
0.97

ECB Working Paper Series No 2901206.2 Density forecasts

In this section, we present an assessment of commodity price density forecasts. First, we

investigate the importance of imposing a factor block structure. Figure 3 presents the results of
the analysis. It shows the cumulative differences in log-predictive likelihoods between the model
that adopts a block structure (dfmBlocks) and the baseline constant-coefficient factor model (dfm).
Increasing and positive differentials indicate a better performance of the dfmBlocks specification.

For the overall commodity index, a strong superiority of the dfmBlocks model emerges at

almost all horizons. This result, at the aggregate index level, is driven by the energy block,

that is oil (panel b) and coal, gas and propane (panel g), as well as by metals (panel e). For the

non-energy block as a whole (panel c), results are instead in favor of the dfm. This is mostly

due to the better performance the dfm specification for the food block (panel h), which accounts

for almost half of the non-energy block. This first comparison shows that, for some specific

commodities, exploiting within block information may improve density forecast accuracy.

Among the total 40 examined cases (10 indices, 4 horizons), a superior performance of the block

structure model is indeed documented for 27 cases (67%).

ECB Working Paper Series No 290121We test whether the differences observed in Figure 3 are statistically significant, using

the Amisano and Giacomini (2007) test, see Table 5. Positive (negative) values with asterisk

denote forecast horizons at which the benchmark constant-factor model performs significantly

better (worse) than the model indicated in the row header.18 Focusing on the comparison with

the dfmBlocks specification, the test confirms that imposing a hierarchical structure yields a

significantly lower density forecast error for 7 out of the 10 indices under investigation.

Next, we compare the density forecasting performance of the factor model that allows
for time-varying slope and volatility parameters (tvp) and its respective constant-coefficient

variant (dfm). This exercise tells us whether time variation improves forecast accuracy. 19

Figure 4 plots the cumulative log-likelihood score at one-month to one-year forecast horizons,
where the benchmark is again the constant-coefficient factor model dfm.20 As evident from

increasing and positive log-scores, the model that allows for time-variation in both slope and

volatility emerges uniformly as the best model for all commodities at all horizons. At the

shortest horizon of one-month ahead, the time-varying parameters specification consistently

outperforms its time-invariant competitor for nine out of ten currency pairs. The largest gains

are documented for the Non-fuel block of commodities, with the bulk of forecast improvements

driven by the Agriculture & Fertilisers index. The test statistics of the equal predictive ability test

corresponding to this comparison can be found in the middle panel of Table 5.21 The statistics
are all negative and significantly different from zero, confirming the benefits of introducing

time-varying dynamics in the context of density forecasting. The results obtained with the

model allowing for time-varying parameters and a block structure (tvpBlocks) are very similar to

the ones obtained on the basis of the tvp model. We do not report the log-scores for brevity, but

comparing the middle to the bottom panels of Table 5 provides a clear indication that adding a

block structure to the tvp model does not lead to any further material improvements in density

forecast accuracy.

18For the purpose of evaluating the importance of adopting a block structure for density forecasting, we set the

constant factor model dfm as the benchmark model.

19Density forecast comparisons of selected models and the random walk benchmark can be found in Appendix

A.5.1 (Figures A.1).

20Figure A.2 in Appendix A.5.1 presents a comparison between the tvpBlocks and the dfmBlocks specifications,

also aiming at illustrating the benefits of introducing time-variation in terms of density forecast accuracy.

21Additional comparisons for this test can be found in Appendix A.5.1 - Table A3.

ECB Working Paper Series No 290122Figure 3: Cumulative differences in log-predictive likelihood of the constant-parameters dynamic
factor model with block structure (dfmBlocks) relative to the constant-parameters dynamic
factor model (dfm). Increases in the statistic denote dates in which dfmBlocks outperforms the
alternative.

(a) All Commodity

(b) Fuel

(c) Non-Fuel

(d) Agriculture & Fertilisers

(e) Metals

(f) Oil

2005201020152020051015h=120052010201520200510h=320052010201520200246h=62005201020152020-0.500.511.52h=1220052010201520200246810h=120052010201520200510h=32005201020152020246810h=62005201020152020246h=122005201020152020-6-4-20h=12005201020152020-20-15-10-5h=32005201020152020-20-15-10-5h=62005201020152020-8-6-4-202h=1220052010201520200246810h=12005201020152020-8-6-4-202h=32005201020152020-10-50h=62005201020152020-4-20h=1220052010201520201020304050h=1200520102015202020406080h=3200520102015202020406080h=62005201020152020102030405060h=122005201020152020051015h=12005201020152020-20246h=320052010201520200246h=62005201020152020024h=12ECB Working Paper Series No 290123(g) Coal, Gas & Propane

(h) Food

(i) Beverages

(j) Agriculture Raw Materials

20052010201520200510h=120052010201520200510h=320052010201520200246h=620052010201520200246h=122005201020152020-10-8-6-4-20h=12005201020152020-25-20-15-10-5h=32005201020152020-20-15-10-50h=62005201020152020-4-3-2-101h=1220052010201520200510h=1200520102015202005101520h=3200520102015202005101520h=62005201020152020051015h=1220052010201520202468h=12005201020152020-6-4-202h=32005201020152020-10-50h=62005201020152020-10-8-6-4-20h=12ECB Working Paper Series No 290124Figure 4: Cumulative differences in log-predictive likelihood of the time-varying dynamic factor
model (tvp) relative to the constant-coefficient dynamic factor model (dfm). Increases in the
statistic denote dates in which tvp outperforms the alternative.

(a) All Commodity

(b) Fuel

(c) Non-Fuel

(d) Agriculture & Fertilizers

(e) Metals

(f) Oil

200520102015202050100150h=1200520102015202020406080100120140h=3200520102015202020406080100120140h=6200520102015202020406080100h=12200520102015202020406080100h=1200520102015202010203040506070h=3200520102015202020406080h=620052010201520200102030405060h=12200520102015202050100150200h=12005201020152020050100150h=32005201020152020050100150h=62005201020152020050100h=122005201020152020100200300400500h=12005201020152020100200300400h=32005201020152020100200300400h=62005201020152020100200300400500h=1220052010201520200102030405060h=1200520102015202020406080h=3200520102015202020406080100h=6200520102015202020406080h=122005201020152020020406080h=120052010201520200102030405060h=320052010201520200102030405060h=6200520102015202001020304050h=12ECB Working Paper Series No 290125(g) Coal, Gas & Propane

(h) Food

(i) Beverages

(j) Agriculture Raw Materials

200520102015202020406080100120h=1200520102015202020406080h=32005201020152020102030405060h=62005201020152020102030405060h=12200520102015202050100150200h=12005201020152020020406080100120h=32005201020152020020406080100120h=62005201020152020020406080100h=122005201020152020050100150200h=12005201020152020050100150h=32005201020152020050100150h=62005201020152020020406080100120h=12200520102015202050100150200h=1200520102015202050100150h=3200520102015202050100150h=6200520102015202050100150h=12ECB Working Paper Series No 290126Table 5: Equal predictive ability test.

All

Fuel

NF

AF Metals

Oil

CGP

Food

Beve

AR

-2.17*
-2.39*
-1.80
-0.79

-9.94*
-8.66*
-6.28*
-5.84*

-10.45*
-9.63*
-9.88*
-9.87*

-1.91*
-3.03*
-3.60*
-3.46*

-6.34*
-4.75*
-5.28*
-5.80*

-6.86*
-5.29*
-5.83*
-5.58*

0.77
3.70*
4.32*
2.25*

-11.16*
-11.10*
-6.83*
-4.26*

-11.38*
-10.20*
-6.34*
-5.09*

-1.54*
1.41
3.26*
1.98*

-8.47*
-8.19*
-9.87*
-7.67*

-8.24*
-7.47*
-8.80*
-7.00*

-5.23*
-7.57*
-5.60*
-5.03**

-6.25*
-7.78*
-4.54*
-4.45*

-5.58*
-6.91*
-4.61*
-4.38*

-3.58*
-1.80*
-1.70*
-2.08*

-7.67*
-4.68*
-4.83*
-3.63*

-7.83*
-5.03*
-6.05*
-4.09*

-2.84*
-3.43*
-3.98*
-4.92*

-8.44*
-7.35*
-4.65*
-4.74*

-7.81*
-7.48*
-4.97*
-5.06*

1.00
5.25*
4.85*
2.26*

-10.15*
-7.78*
-7.42*
-2.97*

-10.28*
-8.37*
-8.36*
-4.54*

-2.73*
-5.04*
-6.96*
-6.73*

-12.17*
-8.21*
-12.50*
-7.74*

-12.50*
-9.89*
-13.54*
-7.22*

-0.56
1.34
2.01*
2.16*

-10.93*
-7.27*
-8.60*
-9.00*

-11.16*
-7.80*
-8.65*
-9.18*

dfmBlocks
h=1
h=3
h=6
h=12
tvp
h = 1
h = 3
h = 6
h = 12
tvpBlocks
h = 1
h = 3
h = 6
h = 12

1 The table reports the t-statistics for the null hypothesis that the model under investigation has the
same predictive ability as the dynamic factor model with constant coefficients. Positive values with
one asterisk (*) denote higher log-predictive likelihood at 5% significance level.

2 NF correspond to the Non-Fuel index, AF to the Acriculture & Fertilizers, and AR to the Agriculture

Raw Materials.

Finally, we test uniformity for both short (one-month) and long (one-year) horizon forecasts.

Figure 5 reports results based on the Diebold et al. (1999) test for one-month ahead forecasts

of the All Commodities index.22 We focus on the distribution functions of the PITs for four

core models, namely dfm, dfmBlocks, tvp and tvpBlocks. It is evident that models with constant
coefficients (dfm & dfmBlocks) tend to produce densities that are somewhat U-shaped and in

which some realisations fall outside the bounds implied by i.i.d uniform PITs. The two models

with time-varying slope and volatility, instead, produce well-calibrated one-month ahead density
forecasts. Figure 6 shows the relevant results for one-year ahead forecasts. Here the difference

is all the more striking, and the role of time-varying parameters in delivering well-calibrated

forecasts emerges more clearly. As a last exercise, we report results based on the inverse normal

of the PIT. Table 6 shows the results for Berkowitz (2001) test at the one-month ahead horizon.

According to Berkowitz (2001), this is a test of joint uniformity and (lack of) serial correlation. It

is more powerful than some of the alternatives tests. For the longer-term horizons we employ the

Kn ¨uppel (2015) test which is robust to the presence of serial correlation of the PITs. The results

22Relevant figures for the remaining commodity indices can be found in Appendix A.5.2 and A.5.3 for the

one-month and one-year horizon, respectively.

ECB Working Paper Series No 290127confirm that the density forecasts of constant-parameters models are poorly calibrated, both

at short and long-horizons, with the exception of Fuel and Agricultural Raw Materials.23 For

time-varying models, density forecasts are better calibrated and rejections of the null hypothesis

of uniformity are sporadic.

23Relevant figures for the remaining commodity indices can be found in Appendix A.5.2 and A.5.3 for the

one-month and one-year horizon, respectively.

ECB Working Paper Series No 290128Figure 5: Probability density functions of the PITs for four core models for one-month ahead
forecasts of the All Commodity index. The dashed lines are 95% confidence intervals, constructed
using a normal approximation to a binomial distribution, as per Diebold et al. (1998).

(a) dfm

(b) dfmBlocks

(c) tvp

(d) tvpBlocks

00.10.20.30.40.50.60.70.80.90102030405060708090100h=1All Commodities00.20.40.60.810102030405060708090100h=1All Commodities00.20.40.60.810102030405060708090100h=1All Commodities00.10.20.30.40.50.60.70.80.910102030405060708090100h=1All CommoditiesECB Working Paper Series No 290129Figure 6: Probability density functions of the PITs for four core models for one-year ahead
forecasts of the All Commodity index. The dashed lines are 95% confidence intervals, constructed
using a normal approximation to a binomial distribution, as per Diebold et al. (1998).

(a) dfm

(b) dfmBlocks

(c) tvp

(d) tvpBlocks

00.20.40.60.810102030405060708090100h=12All Commodities00.20.40.60.810102030405060708090100h=12All Commodities00.20.40.60.810102030405060708090100h=12All Commodities00.10.20.30.40.50.60.70.80.910102030405060708090100h=12All CommoditiesECB Working Paper Series No 290130Table 6: Berkowitz (2001) Likelihood Ratio and Kn ¨uppel (2015) tests.

All

Fuel NonFuel AgrFer Metals

Oil

CGP

Food Beve AgrRaw

0.05
0.07

0.79
0.08

0.00*
0.20

0.03*
0.15

0.07
0.05

0.00*
0.00*

0.09
0.03*

0.00*
0.09

0.00*
0.08

0.00*
0.16

0.00*
0.04*

0.06
0.09

0.00*
0.12

0.00*
0.34

0.01*
0.08

0.00*
0.09

0.00*
0.01*

0.02*
0.80

0.37
0.07

0.00*
0.03*

0.25
0.28

0.10
0.21

0.08
0.16

0.05
0.24

0.05
0.24

0.01*
0.24

0.09
0.08

0.04*
0.59

0.23
0.04*

0.11
0.23

0.27
0.52

0.86
0.52

0.57
0.36

0.05
0.29

0.48
0.21

0.50
0.49

0.52
0.13

0.12
0.65

0.04*
0.40

0.56
0.14

dfm
h = 1
h = 12

dfmBlocks
h = 1
h = 12

tvp
h = 1
h = 12

tvpBlocks
h = 1
h = 12

1 Notes: Results for h = 1 are based on the p.value of the test proposed by Berkowitz (2001).Results for
h = 12 are based on the p.value of Knuppel’s (2015) test. * marks rejection at the 5% significance level.

Discussion: Decomposing parameter time variation

Our empirical analysis shows that introducing time-variation in slope and volatility improves

forecast accuracy for most commodity indies. We explore here which source of parameter time
variation matters the most: whether it is in the parameters of the slope coefficients, or in the

volatility of the innovations.

We start by comparing the performance, in terms of density forecast accuracy, of the

specification that accounts for time-variation in both the slopes and volatility with that of a

dynamic factor model with time-varying volatiliy but constant slopes. The results are shown in
Figure 7. For most commodities, the cumulative differences in log-predictive likelihoods are

negative, and are either constant or (especially at longer horizons) falling. This implies that
switching-off time-variation in the slope coefficients actually yields an improvement in density

ECB Working Paper Series No 290131forecast accuracy.

The second exercise consists of comparing the model with time-variation in both the slope and

volatility with that of a model that has constant volatility but time-varying slope (Figure 8). In

this case, for almost all the 10 commodities, we observe a steady increase in cumulative predictive
likelihood differences. These positive, rising lines indicate that switching-off time-variation
in volatility parameters, while retaining time-evolving dynamics for slope coefficients, reduces

forecast accuracy.

ECB Working Paper Series No 290132Figure 7: Cumulative differences in log-predictive likelihood of the model allowing for both
time-varying slope and volatility (tvp) relative to the model that only allows for time-varying
volatility. Increases in the statistic denote dates in which tvp outperforms the alternative.

(a) All Commodity

(b) Fuel

(c) Non-Fuel

(d) Agriculture & Fertilisers

(e) Metals

(f) Oil

2005201020152020-4-3-2-101h=12005201020152020-6-4-202h=32005201020152020-4-202h=62005201020152020-8-6-4-20h=122005201020152020-2-101h=12005201020152020-8-6-4-2h=32005201020152020-4-3-2-101h=62005201020152020-2-10h=1220052010201520200246h=1200520102015202002468h=32005201020152020-8-6-4-20h=62005201020152020-8-6-4-20h=122005201020152020-4-202468h=12005201020152020-4-202h=32005201020152020-10-8-6-4-2h=62005201020152020-6-4-202h=122005201020152020-20246h=12005201020152020-6-4-20h=32005201020152020-3-2-101h=620052010201520200246h=12200520102015202002468h=12005201020152020-1-0.500.51h=32005201020152020-4-3-2-10h=62005201020152020-3-2-10h=12ECB Working Paper Series No 290133(g) Coal, Gas & Propane

(h) Food

(i) Beverages

(j) Agriculture Raw Materials

2005201020152020-4-202h=12005201020152020-6-4-202h=32005201020152020-6-4-20h=32005201020152020-1012h=122005201020152020-6-4-202h=12005201020152020-101234h=32005201020152020-15-10-50h=62005201020152020-15-10-50h=1220052010201520200246810h=12005201020152020-20246h=32005201020152020-4-3-2-101h=62005201020152020-6-4-202h=12200520102015202002468h=12005201020152020-8-6-4-20h=32005201020152020-10-50h=62005201020152020-10-8-6-4-20h=12ECB Working Paper Series No 290134Figure 8: Cumulative differences in log-predictive likelihood of the model allowing for both
time-varying slope and volatility (tvp) relative to the model that only allows for time-varying
slope. Increases in the statistic denote dates in which tvp outperforms the alternative.

(a) All Commodity

(b) Fuel

(c) Non-Fuel

(d) Agriculture & Fertilisers

(e) Metals

(f) Oil

2005201020152020051015h=12005201020152020102030405060h=3200520102015202020406080100120h=620105101520253035h=12200520102015202005101520h=12005201020152020010203040h=3200520102015202010203040506070h=620101020304050h=122005201020152020510152025h=1200520102015202020406080h=3200520102015202050100150h=610203040h=122005201020152020-50-40-30-20-100h=120052010201520200102030h=3200520102015202020406080100120h=620406080h=122005201020152020-25-20-15-10-5h=12005201020152020-60-50-40-30-20-10h=32005201020152020-15-10-50510h=610203040h=1220052010201520200510h=1200520102015202010203040h=3200520102015202010203040506070h=61020304050h=12ECB Working Paper Series No 290135(g) Coal, Gas, & Propane

(h) Food

(i) Beverages

(j) Agriculture Raw Materials

200520102015202005101520h=12005201020152020102030h=32005201020152020102030405060h=6200520102015202020406080h=1220052010201520205101520253035h=1200520102015202020406080100h=3200520102015202050100150h=65101520253035h=12200520102015202010203040h=1200520102015202020406080100h=3200520102015202020406080100120140h=6201010203040h=122005201020152020051015h=120052010201520201020304050h=3200520102015202020406080100h=6201020406080100h=12ECB Working Paper Series No 290136In conclusion, this exercise shows that incorporating time-varying volatility is the key

ingredient to obtain a more accurate calibration of density forecasts. Our findings are broadly

in line with the macroeconomic literature that, similarly, emphasizes the importance of time-

evolving second-order moments for forecast improvements.24

6.3 Economic evaluation

So far, we have described various (purely) statistical criteria that can be used to evaluate the

competing forecast models. However, for practical purposes related to portfolio allocation, an

evaluation based on economic criteria might also be interesting. An economic evaluation of

the competing models is obtained following Marquering and Verbeek (2004), Campbell and

Thompson (2008), and Welch and Goyal (2008). In particular, we calculate realized utility gains

for a mean-variance investor. We first compute the average utility for a mean-variance investor

with relative risk aversion parameter γ who allocates her portfolio on a monthly basis between

commodities and risk-free bills using forecasts of commodity returns based on the competing

forecasting specifications under investigation. For the forecast of the variance of commodity

returns, and similar to Campbell and Thompson (2008), we assume that the investor estimates

the variance using a rolling window of monthly returns. We compute the average utility for

a mean-variance investor who forecasts commodity returns using an individual forecasting

specification as specified in Table 3. At the end of period t the investor will decide to allocate the

following share of her portfolio in period t + 1:

wj,t =

(cid:33)

(cid:19) (cid:32)

(cid:18) 1
γ

ˆrj,t+1|t
ˆσ2

t+1|t

and realizes an average utility level of

ˆvj = ˆµj −

(cid:19)

(cid:18) 1
2

γˆs2
j

(9)

(10)

where ˆσ2 is the rolling-window estimate of the variance of commodity returns, and ˆµj and ˆs2

j are

the sample mean and variance, respectively, over the out-of-sample period for the return on the

portfolio formed using forecasts based on an individual forecasting method indexed by j.

24Among other contributions, Clark (2011) illustrates how introducing stochastic volatility to BVAR models
improves real-time accuracy of U.S. macroeconomic density forecasts, Jore et al. (2010) show that allowing for discrete
breaks in variances improved U.S. density forecasts made in the Great Moderation period, while Marcellino et al.
(2016) also document that introducing stochastic volatiilty to a mixed-frequency factor model contributes to higher
accuracy of euro-area GDP forecasts.

ECB Working Paper Series No 290137The utility gain (or certainty equivalent return) can be interpreted as the portfolio management

fee that an investor would be willing to pay to have access to the additional information available

in a forecast method j relative to the information coming from the benchmark model alone. We

express the utility gain in average annualized percentage return. 25 The constant dynamic factor

model (dfm) is considered to be the benchmark model and, as such, positive values indicate

that the alternative competing specifications perform better. We report results for γ = 3; the

results are qualitatively similar for γ = 6. In the context of mean-variance analysis, another

commonly used measure of economic value is the Sharpe ratio. This measure, which is defined

as the ratio between the mean return of a portfolio and its standard deviation, summarizes the
mean–variance trade-off of a given investment strategy. Despite evidence (e.g. Han (2006)) that

Sharpe ratios tend to overestimate the conditional risk an investor faces at each point in time,

thereby underestimating the performance of dynamic trading strategies, we report Sharpe ratios
as a complement to the reported utility differences. Following Della Corte et al. (2009) and Lo

(2002), we adjust for serial correlation in the monthly returns by multiplying the monthly Sharpe

ratios by the adjustment factor

(cid:113)

12 + 2

(cid:80)

12

11
k=1(12 − k)ρk

(11)

where ρk is the autocorrelation coefficient of returns at lag k.

Table 7 presents results for the economic significance of the competing forecasting models.

In particular, we report utility gains ¯U and Sharpe ratios for four main forecasting strategies
(dfm, dfmBlocks, tvp, tvpBlocks). We use the baseline constant-coefficient dfm as the benchmark
model. Three main results emerge. First, introducing a time-varying or/and a hierarchical

factor structure to the baseline (time-invariant) dynamic factor model enhances its economic

value. For all ten commodity indices, one of three alternative competing factor specifications

delivers higher Sharpe ratios and positive utility. Second, focusing on the relevance of capturing

25The impact of transaction costs is an essential consideration in assessing the profitability of trading strategies.
Depending on the cost of trading and the extent of weights fluctuation, different trading strategies could be more
costly to implement. A realistic evaluation of the profitability of competing strategies should, therefore, take into
account the effect of transaction costs. In general, making an accurate determination of the size of transaction costs is
challenging because it involves three (main) factors: (i) whether the investor is an individual or institutional, (ii) the
precise value of the transaction, and (iii) whether the broker belongs to a brokerage firm or whether she engages
in direct internet trading. We account for the effect of transaction costs by computing the relevant performance
measures for the investor’s realized returns net of transaction costs. A wide range of estimates has been used in
empirical studies, ranging from transaction costs between 0.1% and 2%. We set the proportional transaction costs to
1%, representing a high-cost regime.

ECB Working Paper Series No 290138time-evolving dynamics, strategies based on time varying parameter factor models yield (on

average) higher Sharpe ratios than the benchmark constant parameter model (dfm). In particular,

this strong performance is mostly driven by the time-varying hierarchical model (tvpBlocks)

which improves upon the benchmark for all ten commodity indices. With the exception of

the Beverages index, utility gains are mostly in agreement with the documented performance

based on Sharpe ratios. Lastly, accurately capturing commodity heterogeneity appears to be

particularly important for economic gains. Precisely, our economic evaluation criteria suggest

that, for 9 out of 10 commodity indices, the model that delivers the highest economic gains is
either the constant-coefficient or the time-varying block-structure specification. As previously,

the only exception is the Beverages index for which a standard time-varying factor model yields

the highest economic value. Nonetheless, it becomes easily apparent that time-variation helps to

‘reveal’ the economic value of hierarchical structure models.

Table 7: For each strategy in the row header, we report the Sharpe ratio SR and utility gains ¯U
(monthly utility changes are annualized) of an investor using any of the three predictive models
(dfmBlocks, tvp, tvpBlocks) relative to an investor using the benchmark constant-coefficient factor
model (dfm). Underlined values denote the best performing strategy for each statistic.

dfm
SR
¯U

dfmBlocks
SR
¯U

tvp
SR
¯U

tvpBlocks
SR
¯U

All

Fuel NonFuel AgrFer Metals Oil CGP Food Beve AgrRaw

0.06
-

0.04
-

0.12
-

0.13
-

0.09
-

0.04
-

0.01
-

0.08
-

0.05
-

0.10
-

0.09
0.09

0.09
0.31

0.10
-0.03

0.14
0.01

0.08
-0.03

0.10
0.33

0.02
0.11

0.07
-0.05

0.03
-0.08

0.15
0.05

0.10
0.08

0.06
0.12

0.11
0.00

0.15
0.03

0.05
-0.13

0.07
0.05

0.03
0.19

0.14
0.07

0.07
0.03

0.10
-0.01

0.14
0.15

0.08
0.15

0.17
0.04

0.22
0.09

0.11
0.07

0.07
0.13

0.04
0.28

0.17
0.13

0.06
-0.01

0.14
0.01

ECB Working Paper Series No 2901397 Conclusions

Recent research has shown that commodity prices exhibit substantial co-movement, which

can be captured by few “common” factors. These factors are broadly related to global demand

for commodity shocks, which are pervasive across all commodity prices, and idiosyncratic

(commodity-specific) supply shocks. A separate literature has stressed how the composition

of underlying structural shocks that drive commodity prices has changed over time, with

the importance of global demand shocks becoming higher since 2000 due to strong growth

in emerging economies, potentially resulting in unstable unconditional correlations across

commodity prices. These two findings suggest that (i) forecast accuracy for the price of a given

commodity could benefit from the information contained in other commodity prices and that (ii)

dealing with potential structural breaks could also improve forecast accuracy.

In this paper we investigate the merits of constructing forecasts for key commodity prices and

indices from models that make use of a large information set and that can deal with structural

breaks. Among others, we consider large TVP dynamic factor models and TVP hierarchical

dynamic factor models that impose the presence of specific blocks on the factor model structure
of commodity prices, as well as their constant-coefficient counterparts. Given that standard

estimation methods for small-dimensional models fail in a data-rich environment, we adopt

estimation techniques that allow us to tackle the issue of dimensionality.

Overall, we find that the out-of-sample predictability of commodity prices varies substantially
across economic states and different commodities. In terms of point forecast accuracy, competing

models exhibit similar performance, with significant gains being documented throughout
different periods. Focusing on the potential significance of adopting a specific block structure, we

document how this feature can, in some cases, improve density forecast accuracy at short-term

horizons. In contrast, we find that modeling parameter time variation exhibits greater predictive

content, with gains being consistent across commodities and forecast horizons. In particular, we

illustrate how appropriately modeling time-varying volatility drives the bulk of density forecast

improvements coming from time-varying specifications. Lastly, trading strategies based on the

various forecast models show that controlling for the high degree of commonalities leads to

higher Sharpe ratios, and to higher values for investors.

ECB Working Paper Series No 290140A Appendix

Our Appendix contains a description of the 10 commodity indices and 68 individual commodity

prices, and explores additional density forecast evaluation exercises for the specifications

presented in the main section.

ECB Working Paper Series No 290141A.1 Data

Table A1: Commodity Price Indices

Description (Mnemonic)

Weight

All Commodity (PALLFNF)
Fuel (PNRG)
Non-Fuel (PNFUEL)
Agriculture and Fertilizers (N/A)
Metals (PMETA)
Oil (POILAPSP)
Coal, Natural Gas and Propane (N/A)
Food (PFOOD)
Beverages (PBEVE)
Agricultural Raw Materials (PRAWM)

100.0
59.1
40.9
36.4
22.7
28.6
12.3
27.8
2.3
4.3

ECB Working Paper Series No 290142Table A2: Individual Commodity Prices

Data Type

Commodity

Commodity.Description

USD

PALUM

USD

PBANSOP

USD

PBARL

USD

PBEEF

Aluminum, 99.5% minimum purity,
LME spot price, CIF UK ports, US$ per metric ton

Bananas, Central American and
Ecuador, FOB U.S. Ports, US$ per metric ton

Barley, Canadian no.1 Western
Barley, spot price, US$ per metric ton

Beef, Australian and New Zealand
85% lean fores, CIF U.S. import price, US cents per pound

USD

PCOALAU

Coal, Australian thermal coal,
12,000- btu/pound, less than 1% sulfur, 14% ash, FOB Newcastle/Port Kembla,
US$ per metric ton

USD

PCOALSA

Coal
South African export price
US$ per metric ton

USD

PCOCO

Cocoa beans, International Cocoa
Organization cash price, CIF US and European ports, US$ per metric ton

USD

PCOFFOTM

Coffee, Other Mild Arabicas,
International Coffee Organization New York cash price, ex-dock New York, US
cents per pound

USD

PCOFFROB

Coffee, Robusta, International
Coffee Organization New York cash price, ex-dock New York, US cents per pound

USD

PROIL

USD

PCOPP

Rapeseed oil, crude, fob
Rotterdam, US$ per metric ton

Copper, grade A cathode, LME
spot price, CIF European ports, US$ per metric ton

USD

PCOTTIND

Cotton, Cotton Outlook ’A
Index’, Middling 1-3/32 inch staple, CIF Liverpool, US cents per pound

USD

PFSHMEAL

USD

PGNUTS

USD

PHIDE

USD

PIORECR

USD

PLAMB

USD

PLEAD

USD

PLOGORE

Fishmeal, Peru Fish meal/pellets
65% protein, CIF, US$ per metric ton

Groundnuts (peanuts), 40/50 (40
to 50 count per ounce), cif Argentina, US$ per metric ton

Hides, Heavy native steers, over
53 pounds, wholesale dealer’s price, US, Chicago, fob Shipping Point, US
cents per pound

China import Iron Ore Fines 62%
FE spot (CFR Tianjin port), US dollars per metric ton

Lamb, frozen carcass Smithfield
London, US cents per pound

Lead, 99.97% pure, LME spot
price, CIF European Ports, US$ per metric ton

Soft Logs, Average Export price
from the U.S. for Douglas Fir, US$ per cubic meter

USD

PLOGSK

Hard Logs, Best quality
Malaysian meranti, import price Japan, US$ per cubic meter

USD

PMAIZMT

Maize (corn), U.S. No.2 Yellow,
FOB Gulf of Mexico, U.S. price, US$ per metric ton

USD

PNGASEU

Natural Gas, Netherlands TTF
Natural Gas Forward Day Ahead, US$ per Million Metric British Thermal Unit

USD

PNGASJP

Natural Gas, Indonesian
Liquefied Natural Gas in Japan, US$ per Million Metric British Thermal Unit

USD

PNGASUS

Natural Gas, Natural Gas spot
price at the Henry Hub terminal in Louisiana, US$ per Million Metric British
Thermal Unit

USD

PNICK

Nickel, melting grade, LME spot
price, CIF European ports, US$ per metric ton

USD

POILAPSP

average of three spot prices; Dated Brent, West Texas Intermediate, and the

Crude Oil (petroleum), simple

USD

POILBRE

Crude Oil (petroleum), Dated Brent, light blend 38 API, fob U.K.,

USD

POILDUB

API, fob Dubai Crude Oil (petroleum), Dubai Fateh Fateh 32 API, US$ per

Oil; Dubai, medium, Fateh 32

Dubai Fateh

US$ per barrel

barrel

USD

POILWTI

Crude Oil (petroleum), West

Texas Intermediate 40 API, Midland Texas, US$ per barrel

ECB Working Paper Series No 290143Data Type

Commodity

Commodity.Description

USD

POLVOIL

Olive Oil, extra virgin less
than 1% free fatty acid, ex-tanker price U.K., US$ per metric ton

USD

PORANG

Oranges, miscellaneous oranges
CIF French import price, US$ per metric ton

USD

PPOIL

Palm oil, Malaysia Palm Oil
Futures (first contract forward) 4-5 percent FFA, US$ per metric ton

USD

PPORK

Swine (pork), 51-52% lean Hogs,
U.S. price, US cents per pound.

USD

PPOULT

Poultry (chicken), Whole bird
spot price, Ready-to-cook, whole, iced, Georgia docks, US cents per pound

USD

PRICENPQ

Rice, 5 percent broken milled
white rice, Thailand nominal price quote, US$ per metric ton

USD

PRUBB

Rubber, Singapore Commodity
Exchange, No. 3 Rubber Smoked Sheets, 1st contract, US cents per pound

USD

PSALM

Fish (salmon), Farm Bred
Norwegian Salmon, export price, US$ per kilogram

USD

PSAWMAL

Hard Sawnwood, Dark Red Meranti,
select and better quality, C&F U.K port, US$ per cubic meter

USD

PSAWORE

Soft Sawnwood, average export
price of Douglas Fir, U.S. Price, US$ per cubic meter

USD

PSHRI

Shrimp, No.1 shell-on headless,
26-30 count per pound, Mexican origin, New York port, US$ per kilogram

USD

PSMEA

Soybean Meal, Chicago Soybean
Meal Futures (first contract forward) Minimum 48 percent protein, US$ per
metric ton

USD

PSOIL

Soybean Oil, Chicago Soybean Oil
Futures (first contract forward) exchange approved grades, US$ per metric ton

USD

PSOYB

Soybeans, U.S. soybeans, Chicago
Soybean futures contract (first contract forward) No. 2 yellow and par, US$
per metric ton

USD

PSUGAISA

Sugar, Free Market, Coffee Sugar
and Cocoa Exchange (CSCE) contract no.11 nearest future position, US cents
per pound

USD

PSUGAUSA

Sugar, U.S. import price,
contract no.14 nearest futures position, US cents per pound (Footnote: No. 14
revised to No. 16)

USD

PSUNO

Sunflower oil, Sunflower Oil, US
export price from Gulf of Mexico, US$ per metric ton

USD

PTEA

USD

PTIN

USD

PURAN

Tea, Mombasa, Kenya, Auction
Price, US cents per kilogram, From July 1998,Kenya auctions, Best Pekoe
Fannings. Prior, London auctions, c.i.f. U.K. warehouses

Tin, standard grade, LME spot
price, US$ per metric ton

Uranium, NUEXCO, Restricted
Price, Nuexco exchange spot, US$ per pound

USD

PWHEAMT

Wheat, No.1 Hard Red Winter,
ordinary protein, Kansas City, US$ per metric ton

USD

POATS

USD

PSORG

Oats,
Generic 1st ‘O’ Future,
USD/bushel

Sorghum,
U.S., Number 2 yellow, fob Gulf of Mexico
USD cents per pound

USD

PWOOLC

Wool, coarse, 23 micron,
Australian Wool Exchange spot quote, US cents per kilogram

USD

PWOOLF

Wool, fine, 19 micron,
Australian Wool Exchange spot quote, US cents per kilogram

USD

PZINC

Zinc, high grade 98% pure, US$
per metric ton

ECB Working Paper Series No 290144Data Type

Commodity

Commodity.Description

USD

PLMMODY

Molybdenum,
57 to 63% purity contained in roasted molybdenum concentrate,
LME spot price,
USD/ton

USD

PCOBA

USD

PGOLD

USD

PSILVER

USD

PPALLA

USD

PPLAT

Cobalt,
U.S. cathodes,
spot

Gold,
Fixing Committee of the London Bullion Market Association,
London 3 PM fixed price,
USD/troy ounce

Silver,
London Bullion Market Association,
USD/troy ounce

Palladium,
LME spot price,
USD/troy ounce

Platinum,
LME spot price,
USD/troy ounce

USD

PPROPANE

North American Spot LPG,
Propane Price/Mont Belvieu LST

USD

PUREA

USD

PPOTASH

USD

PDAP

US Gulf NOLA,
Urea Granular Spot Price,
USD/ST

Potassium Chloride,
Standard Grade: FOB Vancouver Spot Price,
USD/metric tonne

Diammonium phosphate,
US Gulf NOLA DAP Epxort Spot Price per MT,
USD/metric tonne

USD

PTOMATO

Monthly average consumer prices in metropolitan France,
Tomatoes (1 Kg), EUR

USD

PMILK

USD

PCHANA

USDA Class 3 Milk Spot Price,
USD/cwt

MCX India, Chana Spot
INR/100 Kgs

USD

PAPPLE

Monthly average consumer prices in metropolitan France,
Apples (1 Kg), EUR

ECB Working Paper Series No 290145A.2 Dynamic factor models

We retain the factor model representation given in equations 1 and 2. For the idiosyncratic

components ϵit, we impose the following decomposition

K(cid:88)

ϵit =

lijgjt + vit

j=1






lij =

(cid:44) 0,

if i ∈ j

0,

otherwise

(A.1)

(A.2)

where gjt is the vector of block factors, lij the block factor loadings and vit the purely

idiosyncratic component. Equation A.2 describes how the block structure is imposed: whenever

the commodity i does not belong to the block j, the associated factor loadings lij are set equal to

zero. The block diagonal matrix is of the form:

l11

0

. . .

l22
...

. . .
. . .














0
...

0

0

0

0














0

. . .

lKK

where K denotes the different blocks specified in Table 1.

Moreover, the block factors gjt and the purely idiosyncratic component vit are assumed to

follow independent autoregressive processes

gjt = ϕjgjt−1 + wjt wjt ∼ i.i.dN(0, 1)

vit = ρivit−1 + eit

eit ∼ i.i.dN(0, σ2
i)

(A.3)

(A.4)

ECB Working Paper Series No 290146A.3 Estimation of time-varying specifications

Let xit = (x1t, ..., xnt) ′ be an n− dimensional vector of variables that follows a dynamic factor

model of the form:

xit = λitft + ϵit,

ft = Btft−1 + ηt,

(A.5)

(A.6)

where ft is the k × 1 vector of factors, λit is the n × k factor loadings, Bt is a k × k matrix of
VAR(1) coefficients and ϵit and ηt are disturbance terms. It is further assumed that ϵt ∼ N(0, Vt)
and ηt ∼ N(0, Qt) where Vt and Qt are the n × n and k × k diagonal covariance matrices

respectively. Note that the ϵit are uncorrelated with both ft and ηt at all leads and lags. In order

to complete the description of the TVP-DFM model we need to define how the time-varying

parameters evolve. We allow λt and βt to evolve as driftless random walks:

λt = λt−1 + ut ut ∼ N(0, Rt),

βt = βt−1 + vt

vt ∼ N(0, Wt).

(A.7)

(A.8)

The model has a standard state space representation where equations A.5 are the measurement

equations and A.6 to A.8 are the state equations. The state vector ft, λt, βt are estimated via

the Kalman smoother, provided that an estimate of the covariances, Vt, Qt, Rt, Wt is available.

We assume that errors across blocks of equations are uncorrelated, i.e. that ut and vt are i.i.d.

errors, uncorrelated with each other as well as with ϵt and ηt at all leads and lags.26 The model

covariances are estimated using a standard forgetting factor algorithm. First, Rt and Wt evolve

as follows:

Rt =

Wt =

(cid:18) 1 − θR
θR
(cid:18) 1 − θW
θW

(cid:19)

(cid:19)

Pλ
t−1/t−1,

Pβ
t−1/t−1,

where Pλ

t−1/t−1 and Pβ

t−1/t−1 are the estimated covariance matrices of the unobserved state
vectors λt and βt in the model. The smoothing parameters θR and θW are set at 0.96. The

matrices Vt and Qt are estimated by suitably discounting past squared one step ahead prediction

26See, for instance, Cooley (1971); Koop and Korobilis (2012)

ECB Working Paper Series No 290147errors:

(cid:98)Vt = κv (cid:98)Vt−1 + (1 − κv)ϵtϵ′
t

(cid:98)Qt = κQ (cid:98)Qt−1 + (1 − κQ)ηtη′
t

(A.9)

where ϵt is the vector that collects the measurement errors in equation A.5 and κv and κQ are

also set at 0.96.27

27This framework is flexible enough to enable the estimation of the degree of evolution of the model’s parameters
from the data, therefore allowing for modeling different degrees of time variation and, when necessary, no variation
at all. Results based on such an approach do not improve upon our baseline time-varying specifications. Moreover,
we have estimated specifications with different pre-specified values than the ones reported above. None of these
additional specifications enhance predictability and, therefore, results are not reported.

ECB Working Paper Series No 290148A.4 Density scores evaluation criteria

Suppose the forecasting exercise aims at comparing two density forecasts of models i = 1, 2

log(pi,h,t(yt+h|Fi,t−1)),

i = 1, 2

(A.10)

where pi,h,t(∗) denotes the predictive likelihood of model i at horizon h, y is the vector of target
variable(s), Fi,t−1 is the information set of model i available at time t. The KLIC differential
between them is the expected difference in their log-predictive likelihood. In particular, we will
focus on the cumulative differences between the two likelihoods

Sj,h =

T −h(cid:88)

t=1

[log(p1,h,t(yt+h|F1,t−1)) − log(p2,h,t(yt+h|F2,t−1))]

(A.11)

When comparing two different predictive densities, the average difference between their

logarithms is inherently related to their KLIC distance. Among alternative models, choosing

the one with the highest log-predictive likelihood is equivalent to selecting the model with the
minimal KLIC distance.28 To assess whether any detected differences in the log-likelihoods

of the models under investigation are statistically significant, we employ the equal predictive

ability test proposed by Amisano and Giacomini (2007).

A.5 Additional density forecast evaluations

A.5.1 Additional log-score differentials

Although the focus of our work is identifying the benefits of accounting for time-variation and

commodity-specific co-movement for the purpose of forecasting, Figure A.1 presents a density

forecasting comparison between the model that combines both of these features (tvpBlocks) and

the random walk benchmark. Overall, the strictly positive and increasing line provides strong

evidence in favour of the predictive power of the time-varying hierarchical specification.

In the main section presenting our main forecasting results we have already established

that introducing time-variation to dynamic factor models is particularly beneficial to density

forecasting accuracy. As a robustness check, here we illustrate whether this result holds for

hierarchical factor models as well. Figure A.2 presents an illustration of the cumulative log-score
differential between the time-varying hierarchical model (tvpBlocks) and its constant-coefficient

counterpart (dfmBlocks). An increasing and positive-values line signals a better performance of

28See the discussion in Hall and Mitchell (2007) and Geweke and Amisano (2010).

ECB Working Paper Series No 290149the tvpBlocks specification. Looking at the overall picture, it becomes apparent that time-varying

dynamics help revealing the predictive content of hierarchical factor structures. At the shortest

forecast horizon of one-month ahead, Non-fuel commodity indices such as Food, Beverages and

Agricultural Raw Materials benefit the most. Gains continue to be strong for longer horizons,

albeit of smaller magnitude. At the longest horizon of one-year ahead the largest density forecast

improvements are documented for the Agricultural Raw Materials index, while the index that

seems to benefit the least is Oil. In summary, findings presented in both the main section and

appendix emphasize the important role of appropriately capturing structural instabilities in

delivering accurate density forecasts.

Table A3: Equal predictive ability test.

All

Fuel

NF

AF Metals

Oil

CGP

Food

Beve

AR

tvpBlocks
h = 1
h = 3
h = 6
h = 12

-10.45*
-9.63*
-9.88*
-9.87*

-6.86*
-5.29*
-5.83*
-5.58*

-11.38*
-10.20*
-6.34*
-5.09*

-8.24*
-7.47*
-8.80*
-7.00*

-5.58*
-6.91*
-4.61*
-4.38*

-7.83*
-5.03*
-6.05*
-4.09*

-7.81*
-7.48*
-4.97*
-5.06*

-10.28*
-8.37*
-8.36*
-4.54*

-12.50*
-9.89*
-13.54*
-7.22*

-11.16*
-7.80*
-8.65*
-9.18*

1 The table reports the t-statistics for the null hypothesis that the model under investigation has the
same predictive ability as the dynamic factor model with constant coefficients. Positive values with
one asterisk (*) denote higher log-predictive likelihood at 5% significance level.

2 NF correspond to the Non-Fuel index, AF to the Acriculture & Fertilizers, CGP to the Coal, Natural

Gas & Propane, and AR to the Agriculture Raw Materials.

ECB Working Paper Series No 290150Figure A.1: Cumulative differences in log-predictive likelihood of the time-varying hierarchical
dynamic factor model (tvpBlocks) relative to the random walk benchmark. Increases in the
statistic denote dates in which tvpBlocks outperforms the alternative.

(a) All Commodity

(b) Fuel

(c) Non-Fuel

(d) Agriculture & Fertilizers

(e) Metals

(f) Oil

2005201020152020204060h=1 2005201020152020102030h=3 200520102015202005101520h=6 20052010201520200246810h=122005201020152020102030h=1 200520102015202051015h=3 200520102015202002468h=6 20052010201520200123h=12200520102015202020406080100h=1 2005201020152020204060h=3 200520102015202010203040h=6 20052010201520205101520h=12200520102015202020406080100h=1 2005201020152020102030405060h=3 200520102015202010203040h=6 20052010201520205101520h=122005201020152020204060h=1 200520102015202010203040h=3 2005201020152020510152025h=6 20052010201520200510h=12200520102015202005101520h=1 200520102015202024681012h=3 20052010201520200246h=6 2005201020152020-1.5-1-0.500.51h=12ECB Working Paper Series No 290151(g) Coal, Natural Gas & Propane

(h) Food

(i) Beverages

(j) Agriculture Raw Materials

20052010201520200102030h=1 200520102015202051015h=3 200520102015202002468h=6 20052010201520200123h=12200520102015202020406080h=1 20052010201520201020304050h=3 2005201020152020102030h=6 20052010201520205101520h=12200520102015202020406080h=1 200520102015202010203040h=3 2005201020152020510152025h=6 2005201020152020051015h=12200520102015202020406080h=1 20052010201520201020304050h=3 200520102015202051015202530h=6 20052010201520200510h=12ECB Working Paper Series No 290152Figure A.2: Cumulative differences in log-predictive likelihood of the time-varying hierarchical
dynamic factor model (tvpBlocks) relative to its constant-coefficient counterpart (dfmBlocks).
Increases in the statistic denote dates in which tvpBlocks outperforms the alternative.

(a) All Commodity

(b) Fuel

(c) Non-Fuel

(d) Agriculture & Fertilizers

(e) Metals

(f) Oil

200520102015202050100150200h=1200520102015202020406080100120140h=3200520102015202050100150h=6200520102015202020406080100120h=12200520102015202020406080100120h=1200520102015202020406080h=3200520102015202020406080h=6200520102015202020406080h=122005201020152020050100150200250h=12005201020152020050100150200h=32005201020152020050100150h=62005201020152020050100150h=12200520102015202050100150h=1200520102015202050100150h=3200520102015202050100150h=6200520102015202050100150h=122005201020152020020406080h=1200520102015202020406080h=3200520102015202020406080h=6200520102015202020406080100h=122005201020152020020406080100120h=1200520102015202020406080h=3200520102015202020406080h=620052010201520201020304050h=12ECB Working Paper Series No 290153(g) Coal, Natural Gas & Propane

(h) Food

(i) Beverages

(j) Agriculture Raw Materials

200520102015202020406080100120140h=1200520102015202020406080h=3200520102015202010203040506070h=6200520102015202010203040506070h=12200520102015202050100150200250h=1200520102015202050100150200h=3200520102015202050100150h=62005201020152020050100150h=122005201020152020050100150200250h=12005201020152020050100150h=3200520102015202050100150h=62005201020152020050100150h=12200520102015202050100150200250h=1200520102015202050100150200h=3200520102015202050100150200h=6200520102015202050100150200h=12ECB Working Paper Series No 290154A.5.2 Additional PITs (h = 1)

Figure A.3: Probability density functions of the PITs for the dfm model for one-month ahead
forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998)

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100h=1Fuel00.20.40.60.810102030405060708090100h=1Non Fuel00.20.40.60.810102030405060708090100h=1Agriculture and Fertilizers00.20.40.60.810102030405060708090100h=1Metals00.20.40.60.810102030405060708090100h=1Oil00.20.40.60.810102030405060708090100h=1Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100h=1Food00.20.40.60.810102030405060708090100h=1Beverage00.20.40.60.810102030405060708090100h=1Agricultural Raw MaterialsECB Working Paper Series No 290155Figure A.4: Probability density functions of the PITs for the dfmBlocks model for one-month
ahead forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998)

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100h=1Fuel00.20.40.60.810102030405060708090100h=1Non Fuel00.20.40.60.810102030405060708090100h=1Agriculture and Fertilisers 00.20.40.60.810102030405060708090100h=1Metals00.20.40.60.810102030405060708090100h=1Oil00.20.40.60.810102030405060708090100h=1Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100h=1Food00.20.40.60.810102030405060708090100h=1Beverages00.20.40.60.810102030405060708090100h=1Agricultural Raw MaterialsECB Working Paper Series No 290156Figure A.5: Probability density functions of the PITs for the tvp model for one-month ahead
forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998)

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100h=1Fuel00.20.40.60.810102030405060708090100h=1Non Fuel00.20.40.60.810102030405060708090100h=1Agriculture and Fertilizers00.20.40.60.810102030405060708090100h=1Metals00.20.40.60.810102030405060708090100h=1Oil00.20.40.60.810102030405060708090100h=1Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100h=1Food00.20.40.60.810102030405060708090100h=1Beverages00.20.40.60.810102030405060708090100h=1Agricultural Raw MaterialsECB Working Paper Series No 290157Figure A.6: Probability density functions of the PITs for the tvpBlocks model for one-month
ahead forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998)

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100Fuel00.20.40.60.810102030405060708090100h=1Non Fuel00.20.40.60.810102030405060708090100h=1Agriculture and Fertilizers00.20.40.60.810102030405060708090100h=1Metals00.20.40.60.810102030405060708090100h=1Oil00.20.40.60.810102030405060708090100h=1Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100h=1Food00.20.40.60.810102030405060708090100h=1Beverages00.20.40.60.810102030405060708090100h=1Agricultural Raw MaterialsECB Working Paper Series No 290158A.5.3 Additional PITs (h = 12)

Figure A.7: Probability density functions of the PITs for the dfm model for one-year ahead
forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998)

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100h=12Fuel00.20.40.60.810102030405060708090100h=12Non Fuel00.20.40.60.810102030405060708090100h=12Agriculture and Fertilizers00.20.40.60.810102030405060708090100h=12Metals00.20.40.60.810102030405060708090100h=12Oil00.20.40.60.810102030405060708090100h=12Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100h=12Food00.20.40.60.810102030405060708090100h=12Beverage00.20.40.60.810102030405060708090100h=12Agricultural Raw MaterialsECB Working Paper Series No 290159Figure A.8: Probability density functions of the PITs for the dfmBlocks model for one-year
ahead forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998).

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100h=12Fuel00.20.40.60.810102030405060708090100h=12Non Fuel00.20.40.60.810102030405060708090100h=12Agriculture and Fertilisers00.20.40.60.810102030405060708090100h=12Metals00.20.40.60.810102030405060708090100h=12Oil00.20.40.60.810102030405060708090100h=12Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100h=12Food00.20.40.60.810102030405060708090100h=12Beverages00.20.40.60.810102030405060708090100h=12Agricultural Raw MaterialsECB Working Paper Series No 290160Figure A.9: Probability density functions of the PITs for the tvp model for one-year ahead
forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998)

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100h=12Fuel00.20.40.60.810102030405060708090100h=12Non Fuel00.20.40.60.810102030405060708090100h=12Agriculture and Fertilizers00.20.40.60.810102030405060708090100h=12Metals00.20.40.60.810102030405060708090100h=12Oil00.20.40.60.810102030405060708090100h=12Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100h=12Food00.20.40.60.810102030405060708090100h=12Beverages00.20.40.60.810102030405060708090100h=12Agricultural Raw MaterialsECB Working Paper Series No 290161Figure A.10: Probability density functions of the PITs for the tvpBlocks model for one-year
ahead forecasts of the nine remaining commodity indices. The dashed lines are 95% confidence
intervals, constructed using a normal approximation to a binomial distribution, as per Diebold
et al. (1998)

(a) Fuel

(b) Non-fuel

(c) Agriculture & Fertilizers

(d) Metals

(e) Oil

(f) Coal, Gas & Propane

(g) Food

(h) Beverages

(i) Agricultural Raw Materials

00.20.40.60.810102030405060708090100h=12Fuel00.20.40.60.810102030405060708090100h=12Non Fuel00.20.40.60.810102030405060708090100hh=12Agriculture and Fertilizers00.20.40.60.810102030405060708090100hh=12Metals00.20.40.60.810102030405060708090100h=1Oil00.20.40.60.810102030405060708090100hh=12Coal, Natural Gas and Propane00.20.40.60.810102030405060708090100hh=12Food00.20.40.60.810102030405060708090100h=12Beverages00.20.40.60.810102030405060708090100h=12Agricultural Raw MaterialsECB Working Paper Series No 290162References

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ECB Working Paper Series No 290167Acknowledgements 

We want to thank for their comments an anonymous referee, Andrea De Polis, participants at the Computational and Financial 
Econometrics 2019 conference (UK), the International Association for Applied Econometrics Annual Conference 2019 (Cyprus), the 
Symposium of the Society for Nonlinear Dynamics and Econometrics 2020 (Virtual) and the Vienna Workshop on Economic Forecasting 
2020 (Virtual). 

Anastasia Allayioti 
European Central Bank, Frankfurt am Main, Germany; email: Anastasia.Allayioti@ecb.europa.eu 

Fabrizio Venditti 
Bank of Italy, Rome, Italy; email: fabrizio.venditti@bancaditalia.it 

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