Are commodity futures a hedge against inflation? A
Markov-switching approach
June 8, 2022
Abstract
This study examines the inflation-hedging ability of commodity futures. Applying a Markovswitching vector error correction model to a sample of commodity futures that cover the period
between January 1983 and December 2017, we find that total commodity futures fail to provide
a hedge against inflation. However, commodity futures in the markets of industrial metals exhibit
significant inflation-hedging properties. Other sub-indices, including energy, precious metals, agriculture, and livestock, do not have significant inflation hedging ability. The hedging capacity of
industrial metal futures exhibits substantial variation over time, with most of the inflation hedging
power occurring during the relatively longer and more common regimes covering the Great Moderation and post-subprime crisis. The results are robust to the inclusion of stocks and bonds in the
model.
Keywords: Inflation hedge; Commodity futures; Markov-switching; Vector error correction
JEL Codes: C50; E31; G11; G13
1
1
Introduction
The inflation protection ability of various asset classes (e.g., Treasury inflation-protected securities,
commodities, real estate) has attracted much attention from both researchers and the finance industry,
as unexpected inflation can have significant impact on the market and investors’ portfolio. Among these
assets, commodity futures have always been the topic of debate. Historically, commodity prices were
closely related to inflation and sometimes business cycles (Bernanke, 2009). In particular, commodities
and CPI tend to have a positive relationship, making them a natural candidate as an inflation hedge.
Many asset managers cite the inflation-hedging ability of commodities as they allocate such assets
into their portfolios on a regular basis.1 Recently money managers are over-weighting assets that are
traditionally believed to provide inflation hedge (e.g., commodities) under the pressure of inflation spike
as the economy rebounds from the pandemic-driven recession.2 The ever-increasing financialization of
commodity markets has significantly redefined the role of commodity indexes in portfolio allocation
of asset managers (e.g., index funds), and has significantly unleashed the potential of using this class
of assets as an inflation hedge (Singleton, 2012; Hamilton and Wu, 2014). However, research on
the hedging ability of commodity futures is far from conclusive, cautioning against using them as a
strategic inflation hedge. Our study seeks to investigate the inflation protection ability of this particular
asset class (Lawrence, 2003, Beckmann and Czudaj, 2013, Apergis, Cooray, Khraief, and Apergis,
2019, Shin, Yu, and Greenwood-Nimmo, 2014, Van Hoang, Lahiani, and Heller, 2016).
The existing literature has paid increased attention to the investigation of the inflation-hedging
effectiveness of commodity futures. Herbst (1985); Jensen, Johnson, and Mercer (2000); Gorton
and Rouwenhorst (2006); Erb and Harvey (2006); Hoevenaars, Molenaar, Schotman, and Steenkamp
(2008); Gorton, Hayashi, and Rouwenhorst (2013); and Spierdijk and Umar (2014) are among recent
1
Vanguard is one of the proponents of the idea that commodity futures could offer inflation protection if mixed with a
traditional stock/bond portfolio. On June 25 of 2019, Vanguard launched a fund that seeks to provide diversification benefits
and inflation protection to traditional stock/bond portfolios.
2
For instance, Ontario Teachers’ Pension Plan increased its net investment in commodities from 8% at the end of 2020
to 12% in the first half of 2021, as part of its effort to combat rising inflation that is mostly due to governments’ fiscal
responses as well as the loose monetary policy that aim to lift the economy. ATP of Denmark also overweights its exposure
to commodities recently to prevent the erosion of their assets by inflation risks.
2
papers that have focused on an aggregate commodity or commodity futures index.3 A variety of empirical techniques have been employed to investigate the inflation protection ability of commodities and
other asset classes. Conventional tools include: the Johansen cointegration vector (Mahdavi and Zhou,
1997); the vector error correction model (Levin, Montagnoli, and Wright, 2006); the vector autoregressive model (Hoevenaars, Molenaar, Schotman, and Steenkamp, 2008; Spierdijk and Umar, 2014);
the structural vector autoregressive model (Cologni and Manera, 2008; Lippi and Nobili, 2012; and
Blanchard and Riggi, 2013). However, none of the aforementioned methods are able to incorporate the
distinct nature of regimes present in the economy.4 Several recent papers are notable exceptions. For
instance, Bampinas and Panagiotidis (2015), Lucey, Sharma, and Vigne (2017), and Bilgin, Gogolin,
Lau, and Vigne (2018) look at the link between gold prices and inflation using a time-varying cointegration methodology. The evidence for instabilities in the relationship between commodity futures
prices and inflation verifies the application of the time-varying methodology.
This study extends the existing literature by adopting a Markov-switching vector error correction
model (MS-VECM) approach, which is a comprehensive framework to capture the relationship that
changes from state to state. The data used in this study are monthly data of returns on the S&P Goldman
Sachs Commodity Index (GSCI) Total Return Index, ranging from January 1983 to December 2017.
In additional to the total commodity futures index, we also examine the hedging ability of commodity
futures across different markets, including energy, industrial metals, precious metals, agriculture and
livestock. Finally, we evaluate the inflation hedging ability of commodity futures by including stocks
and bonds into the model.
The main findings of this study are as follows. Over the full sample period, the aggregate commodity futures return index does not show significant ability to hedge against inflation. We then dive
into exploring the inflation protection ability of several subindexes of commodity futures. According
to the coefficients related to the error-correction mechanism, industrial metals provide the best hedge.
3
There are also research that focuses on a specific commodity market, such as Hoevenaars, Molenaar, Schotman, and
Steenkamp (2008); Bekaert and Wang (2010); Baur and McDermott (2010); Wang, Lee, and Thi (2011); Beckmann and
Czudaj (2013); Lucey, Sharma, and Vigne (2017); and Bilgin, Gogolin, Lau, and Vigne (2018).
4
As demonstrated by Chinn and Coibion (2014), there exists significant heterogeneity within the sample period in terms
of inflationary regimes and commodity futures prices.
3
However, the hedging capacity exhibits substantial variation over time. In the model of inflation and industrial metals, inflation hedging is relatively fast in pace and is of relatively high statistical significance
in a stable regime, which prevails most of the time and covers the Great Moderation and post-subprime
crisis. The results of the model comprising inflation and precious metals reveal that the precious metals
index shows some ability to hedge inflation but with low statistical significance. In addition, the energy, agriculture, and livestock indices do not have good inflation hedging ability. In line with Bilgin,
Gogolin, Lau, and Vigne (2018), this relationship could be explained by the predominantly industrial
nature of industrial metals.
The results are also robust to the introduction of common stocks and government bonds into the
model. This exercise is meaningful as the bulk of portfolios are usually allocated to common stocks
and bonds, with commodity futures used as a tool to hedge against inflation. Besides, it enables us
to benchmark our results using findings in prior studies that examine the hedge and safe haven properties of commodities, such as previous metals (Baur and Lucey, 2010; Baur and McDermott, 2010).
After including these two assets in our system, we find that aggregate commodity futures are unable
to provide inflation hedging ability. However, industrial metals futures continue to show strong inflation hedging ability, with the error correction coefficient positive in sign and statistically significant.
The fact that industrial metals are a hedge against inflation in this analysis implies that investors that
hold industrial metals receive compensation through positive industrial metal returns as bond returns
turn negative in response to higher inflation. Moreover, the results also suggest that industrial metals
subindex is a hedge against stocks in the normal regime, but not a safe haven in the crisis regime. Despite being yield-free commodities, industrial metals should be considered as meaningful components
of a well-diversified investment portfolio.
Our paper contributes to the literature in the following ways. First, when we extend the previous
model to include latent regime shifts governed by one ergodic and irreducible Markov chain, we test
whether there is evidence of a need to incorporate additional parametric flexibility when investigating
the long-run relationship between commodity futures and inflation. Our paper sits in the growing line of
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literature that examine the potentially time-varying nature of relationships between commodity prices
and inflation (Bampinas and Panagiotidis, 2015; Lucey, Sharma, and Vigne, 2017; Bilgin, Gogolin,
Lau, and Vigne, 2018). Second, we also contribute to the literature that applies Markov-switching
models to the asset class of commodities (Alizadeh, Nomikos, and Pouliasis, 2008; Bae, Kim, and
Mulvey, 2014; and Giampietro, Guidolin, and Pedio, 2017). Different to existing studies that focuses
on the time-varying moments in commodity returns, we investigate the time-varying inflation hedging
ability of commodity futures. Our third contribution is that we test the inflation hedging ability of
commodity futures in a framework including stocks and bonds. This is particularly useful in light
of the recent developments in the commodity markets. Numerous studies examine the performance
of portfolios composed of stocks and bonds, and demonstrate the advantages of adding new asset
classes to portfolios (Jensen, Johnson, and Mercer, 2000; Gorton and Rouwenhorst, 2006; and Erb
and Harvey, 2006). We extend prior work by demonstrating that one of the economic rationales for
including commodity futures (i.e., industrial metals) as a portfolio component is their ability to hedge
against inflation.
The remainder of this paper is organized as follows. Section 2 describes the econometric methodology as well as the data used in this paper. In section 3, we estimate the autoregressive orders of the
considered vector autoregressions and implement the Johansen’s cointegration test. Section 4 presents
the main results concerning the Markov-switching multistate vector error-correction analysis. The final
section concludes.
2
Literature review
The literature has paid a particular attention to the relationship between commodity price and inflation
(Herbst, 1985; Erb and Harvey, 2006; Kat and Oomen, 2007; Spierdijk and Umar, 2014), but the empirical evidence is not always conclusive. For instance, utilizing the data of the GSCI, Erb and Harvey
(2006) perform a linear regression analysis with the annual data from 1969 to 2003, and document
5
that the sensitivity to inflation varies widely across individual commodities. Their results reveal that
the roll returns of energy futures index are positively correlated with unexpected inflation component.
Kat and Oomen (2007) then compare the inflation-hedging properties of roll and spot returns. Their
findings suggest that the correlation between commodity futures and inflation primarily stems from
the spot market, while the correlation of roll returns with unexpected inflation is largely insignificant, contradicting the results of Erb and Harvey (2006). In a recent paper, Spierdijk and Umar (2014)
implement a rolling-window analysis and document that the inflation hedging capacity of commodity
futures exhibits substantial variation over time.
The literature also deals with the inflation-performance of portfolios including commodity futures
(Jensen, Johnson, and Mercer, 2000; Becker and Finnerty, 2000; Gorton and Rouwenhorst, 2006;
Demidova-Menzel and Heidorn, 2007). Becker and Finnerty (2000) document that the inclusion of
commodity futures contracts in equity/bond portfolios improves the performance. As inflation has
positive impacts on commodity prices, the improvement is more pronounced during the high inflation
periods. Moreover, Gorton and Rouwenhorst (2006) examine the inflation-hedging properties of stocks,
bonds, and commodities at horizons covering one month to five years. They report that the stocks and
bonds are negatively correlated with inflation at all horizons, while the commodities are positively
associated with inflation. Demidova-Menzel and Heidorn (2007) further investigate the risk return
relationship in equity/bond portfolios incorporating commodities and document that commodities are
sensible addition assets during high inflationary periods.
The foundation of inflation-hedging properties investigation methodology comes from the the hypothesis of Fisher (1930). As stated in the hypothesis, the expected nominal return of any assets should
be equal to the expected inflation plus the expected real return of the asset. Thus, commodity futures
returns have a direct link to inflation. As we mentioned in the introductory section, previous studies implement a variety of empirical techniques (Fama and Schwert, 1977; Bodie, 1982; Mahdavi and Zhou,
1997; Levin, Montagnoli, and Wright, 2006; Hoevenaars, Molenaar, Schotman, and Steenkamp, 2008;
Spierdijk and Umar, 2014; and Blanchard and Riggi, 2013). However, these studies use single-state
6
models, failing to take into account the nonlinearity. More recently, Bampinas and Panagiotidis (2015),
Lucey, Sharma, and Vigne (2017), and Bilgin, Gogolin, Lau, and Vigne (2018) adopt a time-varying
cointegration method to investigate the relationship between gold prices and inflation. We add to this
line of research by applying MS-VECM to a broad class of commodity indexes, including the total
commodities index and several subindexes. While the literature has mainly utilized Markov-switching
models in the context of stocks and bonds (Guidolin and Timmermann, 2006), the number of studies
that apply the time-varying coefficient models to commodities is relatively limited (Alizadeh, Nomikos,
and Pouliasis, 2008; Beckmann and Czudaj, 2013; Bae, Kim, and Mulvey, 2014; Giampietro, Guidolin,
and Pedio, 2017).
3
Econometric methodology and Data
3.1
Econometric methodology
The review of the most relevant literature in this area shows that single-state models used in the previous research may not be the best description of the dynamic relationships in the data. This paper
allows for regime shifts in autoregressive parameters and error-correction (EC) speed coefficients. The
model specification in this paper is an M-regime, p-th order autoregressive with r cointegrating vectors,
Markov-switching vector error-correction model.
∆Xt = µ(st ) +
p
X
i=1
Ai (st )∆Xt−i +
r
X
αj (st )Zt−1 + t , t |st ∼ N ID(0, Σst ),
(1)
j=1
where Xt denotes the time t column vector of observations, st = 1, 2, ..., M represents the regime
in time t, µ(st ) collects the regime-dependent intercepts, Ai (st ) is a row vector of i-th order autoregressive parameters in regime st , αj (st ) measures the speed of error correction in regime st , and Zt is
b t − CX
b t,
the column vector containing the residuals from the error-correction equation, i.e., Zt = CX
b is the estimated cointegration matrix. In order to provide a regime-specific equilibrium correcwhere C
7
tion, αj should be negative and significantly different from zero. These unobservable states are generated by a discrete-state, irreducible and, ergodic first-order Markov chain: P r(st = j | st−1 = i) = pij ,
where pij is the generic [i, j] element of the M × M transition matrix P .
The long-run error correction and cointegration is explained as a mixture of M different regimespecific equilibrium adjustments.






∆Xt |Ft−1 ∼ 





Pp

Pr
N µ(s1,t ) + i=1 Ai (s1,t )∆Xt−i + j=1 αj (s1,t )Zt−1 , Σ1 , with prob. qs1,t
P
P
N µ(s2,t ) + pi=1 Ai (s2,t )∆Xt−i + rj=1 αj (s2,t )Zt−1 , Σ2 , with prob. qs2,t
..
.
P
P
N µ(sM,t ) + pi=1 Ai (sM,t )∆Xt−i + rj=1 αj (sM,t )Zt−1 , ΣM , with prob. qsM,t
(2)
where Ft−1 is information content at time t − 1 and qsj,t is the predicted probability of being in regime
0
j at time t and qsj,t = ej P ξt−1|Ft−1 . Here, ej is a unit column vector with a unit value at the j-th
position, ξt|Ft is a column vector of the filtered regime probabilities at time t, and ξ is defined as
ergodic probabilities of the long-run equilibrium




 Pr(st = 1|Ft ) 
 Pr(st = 1) 








 Pr(s = 2|F ) 
 Pr(s = 2) 




t
t
t
 with ξ t = 
.
ξt|Ft = 




.
.




..
..












Pr(st = M |Ft )
Pr(st = M )
(3)
As a result, it can be concluded that unconditionally the MS(M)-VECM(p, r) has a long-run equilibrium
and provides an unconditional cointegration if the long-run error-correction coefficients are negative
and statistically different from zero.
Krolzig (1997) provides an exhaustive overview of Markov-switching vector autoregressions. In
the first stage, the Johansen (1991) cointegration test is applied in order to test if cointegration exists
and to determine the number of cointegrating vectors. The trace test and the L-Max test of Johansen’s
8





,





procedure are provided in this study.5 The second stage of the MS model estimation is performed
in Ox with the MS-VAR package and the MS-VECM code reported in Krolzig (1998). In particular,
estimation and inferences are based on the expectation-maximization algorithm described by Dempster,
Schatzoff, and Wermuth (1977) and Hamilton (1989). This algorithm is a filter that enables the iterative
calculation of the one-step ahead forecast of the state vector ξt|t−1 given the information set and the
construction of the log-likelihood function of the data. The cointegration matrix estimated in the first
stage is used in order to calculate the error-correction residuals, which enter equation (1) as Zt .
3.2
Data
We use monthly data of returns on the S&P GSCI Total Return Index, an index of commodity sector
returns that represents a broadly diversified and long-term investment in commodity futures. This index
is widely used in prior literature to assess the hedging potential of commodities, such as Hoevenaars,
Molenaar, Schotman, and Steenkamp (2008) and Spierdijk and Umar (2014). The aggregate S&P GSCI
Total Return Index includes 24 commodity futures contracts, categorized into five groups, namely energy, industrial metals, precious metals, agriculture, and livestock. We adopt the US seasonally adjusted
CPI for all urban consumers as our inflation index. The monthly returns on the S&P GSCI Total Return
Index were obtained from Thomson Reuters Datastream, and the inflation index was provided by the
Bureau of Labor Statistics. We also use the CRSP U.S. Treasury 10-Year Bond Index and the CRSP
value-weighted U.S. market index, a comprehensive proxy for U.S. stocks that are publicly traded on
NYSE, NASDAQ, and AMEX. Our sample covers the time period from January 1983 to December
2017. We choose January 1983 as the starting month of our sample period as it is when we start to have
data on energy futures index.
Table 1 provides summary statistics of the main variables. Commodity futures returns exhibit
substantially more volatility than the slowly moving process of inflation. The average monthly return
on the S&P GSCI Total Return Index is 0.486%, with a standard deviation of 5.677%. Among the five
5
The trace test tests the null hypothesis that the number of cointegrating vector is equal to r and the L-Max test has a null
hypothesis that the number of cointegrating vectors is no greater than r.
9
subsector indexes, industrial metals have the highest average monthly return of 0.820% and agriculture
has the lowest average monthly return of 0.082%. In terms of volatility, the return of energy futures
is the most volatile, with a standard deviation of 8.913%, and livestock futures return has the lowest
volatility of 4.176%. The average monthly inflation rate is 0.222%, and volatility equals 0.254%.
The kurtosis of the returns on the S&P GSCI Total Return Subsector Indexes covers from 3.560% to
7.465%. By contrast, the inflation index has some excess kurtosis. We adopt Jarque-Bera statistics
to test the hypothesis of normality and reject the null of normality in all series. Figure 1 depicts
the evolution of these commodity futures prices during 1983-2017. Unsurprisingly, the subindexes
of commodity futures are highly correlated with each other. Agriculture and livestock subindexes are
exceptions. Different to other subindexes, they exhibit a downward trend throughout the sample period.
In order to perform the augmented Dickey-Fuller (ADF) unit root test, an assumption on the most
appropriate maximum lag of an autoregressive (AR) representation of each of the time series needs to be
made. Table A1 presents the Akaike and Schwarz information criterion (AIC and SIC, respectively) of
each series. Due to the fact that the AIC tends to choose models that include relatively more parameters
and the SIC information chooses more parsimonious models, whenever there is conflicting evidence
from the two information criteria, the preference is given to the SIC with which we can keep the
saturation ratio high. The results show that the most appropriate AR order for the CPI is equal to 3,
indicating that the inflation index has some persistence. The AR orders for total commodity, energy and
precious metals are 2, while the AR orders for industrial metals, agriculture, and livestock are equal to
1. Table A2 shows the results of the ADF for all the seven considered series. It includes three different
specifications: an unrestricted specification with a trend and with an intercept, as well as two restricted
specifications, one without trend and the other without trend and without intercept. With reference
to three different types of the ADF test on a level series, the null hypothesis of a unit root cannot be
rejected for all the nine indexes at a 1% significant level. When all series are first-differenced, the null
hypothesis of a unit root is significantly rejected.
10
4
Single-state cointegration analysis
In this section, we investigate whether commodity futures index and CPI is cointegrated in our sample period. We first determine the autoregressive orders of the VAR models and then implement the
Johansen’s cointegration test.
4.1
Selection of the autoregressive order of the VAR multivariate models
Panel A of Table 2 presents the values of AIC and SIC corresponding to different autoregressive orders.
In the model of inflation and commodity, AIC indicates an autoregressive order equal to 3, while SIC
suggests an order equal to 2. In order to keep the saturation ratio high, we rely on SIC statistics and
estimate a VAR model with an autoregressive order of 2. Moreover, the autoregressive orders selected
based on SIC are equal to 2 for the rest of the models except the one of industrial metals. With reference
to the VAR models with two series, AIC suggests autoregressive orders of 4 for the models of industrial
metals and livestock, while showing autoregressive orders of 3 for the rest of the models.
4.2
Johansen’s cointegration test
An application of Johansen’s cointegration test requires that the VAR models are changed into their
corresponding vector error-correction form. This means that lag p of a vector autoregressive model
corresponds to lag p−1 of a vector error-correction model. Panel B of Table 2 summarizes the results of
Johansen’s cointegration test applied to the VAR models and shows both L-Max and trace test statistics.
For the first model that examine the cointegration between CPI and the aggregate commodity futures return index, Johansen’s trace test suggests the existence of one cointegrating vector. The Mackinnon, Haug, and Michelis (1999) p-values of the trace test as well as the L-Max test p-value suggest that
the null hypothesis of no cointegration is rejected at a 1% significance level. Neither of these two tests
can reject the null hypothesis of one cointegrating vector at conventional level. We therefore demonstrate that there is one cointegrating vector in the model and hence a long-run relationship between CPI
11
and commodity futures index.
The estimation of the cointegrating vector, normalized with respect to the inflation index, is also
presented in Panel C. To save space, we only present the normalized coefficients of the corresponding
commodity index. The coefficient attached to the commodity index has a negative sign, indicating that
the larger difference between the two series has a larger impact on the error-correction mechanism. An
intercept in the cointegrating vector allows for a long-run relationship in which the series do not need
to reach the same level for the error-correction mechanism to revert its direction of influence.
In the second model of CPI and energy subsector index, Mackinnon, Haug, and Michelis (1999)
p-values of both the tract test and the L-Max test indicate one cointegrating equation. Column (2) of
Panel C presents the cointegrating equation of the CPI and energy index. The statistical significance
of the coefficient suggests a long-run relationship between these two series. The coefficient of energy
equals -0.114, implying that the larger the difference between the CPI index and the energy subsector
index, the stronger is the error-correction strength. In the third model of CPI and industrial metals
index, we identify one cointegrating vector. The coefficient of industrial metals equals -0.195 and its
t-statistic is -6.650, implying a very strong error-correction mechanism embedded in the system.
According to the fourth column of Panel B, we are able to identify one cointegration vector for the
VAR model of CPI and the precious metals index. However, we fail to find a significant cointegration
vector, although the sign of the coefficient is still negative, as shown in Column (4) of Panel C. Similar
conclusions can be drawn according to the coefficient estimates of the rest two models that examine the
relationship between CPI and agriculture as well as livestock futures indexes. In sum, our findings here
support a cointegrating relationship between CPI and the commodity futures index. There exists some
heterogeneity within subindexes of commodity futures with regard to their cointegration relationship
with inflation. Over the long-run, commodity futures (i.e., energy and industrial metals futures) are
useful as inflation hedges since a cointegrating relationship prevails. However, during some periods
when no price adjustment is observed, they are not able to protect a portfolio against inflation risks.6
6
For instance, fund managers may invest in commodities at the beginning of a month with no price adjustment and sells
at the end of the corresponding month. Hence, they can not use commodities to hedge against inflation risks.
12
5
Markov-switching multistate vector error-correction analysis
In this section, we estimate an Markov-switching vector error-correction model using various commodity futures indexes as well as CPI. Based on information criteria statistics, we first select the right form
of model specification and then present the corresponding model estimation results.
5.1
Inflation and commodity futures
To determine the model specification, we compare the information criteria statistics between two major types of models, i.e., linear and Markov-switching vector error-correction models. Within each
category, we consider models with different specifications. Specifically, for Markov-switching vector error-correction models, we allow for multiple regimes (e.g., two or three) and different forms of
VECM. Table 3 presents relevant statistics of models that use the total commodity futures index as
well as the five subindexes. As shown in Panel A, both AIC and SIC imply that Markov-switching
models are better than the linear models. In particular, the AIC shows that the best model specification is an MSIAH(3)-VECM(1,1), while the SIC suggests MSIAH(2)-VECM(1,1).7 Because SIC is
known to suggest more parsimonious models than AIC does, the latter was chosen as the decisive one
in our analysis. Therefore, we estiamte an MSIAH(2)-VECM(1,1) model that features two regimes,
heteroscedastic errors and an autoregressive order of 1. The Krolzig (1997) algorithm converged after 26 iterations. The number of observations in the system is 836, and the number of parameters in
Markov switching system equals 24, while the corresponding linear model has 11 parameters. With
two nuisance parameters specified in the multi-state model, the likelihood ratio linearity test should be
implemented under a chi-square with 11 degrees of freedom.8
Table 4 presents the results of the estimation of an MSIAH(2)-VECM(1,1) using CPI and the total
7
MSIAH stands for Markov-switching regime dependent intercept, matrix of autoregressive coefficients, and variance
model, while MSIA stands for Markov-switching regime dependent intercept and matrix of autoregressive coefficients model.
8
The p-values of both the LRT and the Davies (1977) test, which avoids the estimation of nuisance parameters, are less
than 0.01%, indicating the rejection of the null hypothesis at a 1% level. As a result, the tests confirm that the Markovswitching model is better than the linear model.
13
commodity futures index. The long-run relationship is explained by the coefficients corresponding to
the error-correction component. In regime 1, there is no significant adjustment in the inflation index
since the corresponding coefficient is not statistically significant (t-statistic = -1.005). In regime 2,
the coefficient equals -0.002 and its t-statistic is -3.924, implying a significant error-correction in the
inflation index. The positive sign of the coefficient on the error-correction term for the commodity
index in regime 2 is the correct one for the error-correction mechanism to work. It implies that if the
previous month CPI increases, then the total commodity price in this month would increase. But this
coefficient is not statistically significant (t-statistic = 0.456). As a result, we do not find evidence that
total commodity futures have significant inflation hedging ability.
Figure 2 depicts the two regimes identified by the MS model, showing the smoothed probabilities
of being in regime 1 and regime 2. Regime 2 prevailed from 1983 through 1987 (with a short regime
1 in early 1986); from 1991 through 1998; from 2000 through 2005 (with regime 1 appearing at late
2001 and early 2003 for a short time period); and from mid-2009 through the end of 2017. Regime
1 prevailed from 1988 through 1991, and from 2007 through 2009. Between January 2005 and July
2007, both regimes seem to have been present. The estimated transition matrix and ergodic probabilities
suggest that regime 2 predominates, characterizing 73.1 % of months with spells lasting 11.28 months
on average; regime 1 occurs during only 26.9 % of months, with spells lasting just 4.15 months on
average.
5.2
Inflation and subindexes of commodity futures
In this section, we present estimation results of models that use CPI and each of the five subindexes of
commodity futures. Based on the estimated coefficients, we evaluate the ability of these subindexes in
providing inflation protection.
14
5.2.1
The energy futures subindex
According to AIC and SIC statistics in Panel B of Table 3, we estimate an model in the form of
MSIA(2)-VECM(1,1) using CPI and the energy futures index. In other words, we estimate a MSVECM model with two distinct regimes and a homoscedastic error covariance matrix. The number
of observations in the system is again 836, which corresponds to approximately 40 observations per
estimated parameter since there are 21 parameters in total.9 Panel A of Table 5 presents the estimation
results. The coefficients related to the error-correction mechanism implies that the long-run relationship between the CPI and the energy futures index changes from regime to regime. In regime 1, the
negative sign and the related t-statistic of the long-run error-correction coefficient in the energy index
indicate that energy prices diverge from the inflation index, suggesting the opposite to adjusting toward
equilibrium. However, regime 1 lasts a relatively short time because its expected duration is less than
4 months. During regime 2, the sign of the error-correction coefficient on the energy index is negative,
suggesting that the divergence from equilibrium won’t be closed. In regime 2, there is strong long-run
equilibrium adjustment in inflation, which is confirmed by the negative error-correction coefficient and
the corresponding t-statistic of -5.455. Figure 3 (a) shows the smoothed probabilities, with regime 1
prevailing during 1983, 1986, 2001, 2005, 2006, 2008 and 2014. Regime 2 prevailed during 1984,
1985, 1987-2000, 2002-2004, 2007, and covered the Great Moderation and 2009-2017. Both regimes
are strongly persistent, with regime 2 lasting 42.64 months on average and prevailing 91.6 % of the
time, while regime 1 lasts 3.93 months on average and prevails during the remaining 8.5 % of months.
5.2.2
The industrial metals futures subindex
A model in the form of MSIAH(2)-VECM(1,1) is selected based on the information criteria shown in
Panel C of Table 3. Here, the model features two regimes, heteroscedastic errors and an autoregressive
order of 1. Panel B of Table 5 summarizes the results of estimation of an MSIAH(2)-VECM(1,1) for
9
Both the LRT and the Davies (1977) test confirm that the Markov-switching model is better than the linear model. For
brevity, we do not report these statistics for later models.
15
the CPI index and the industrial metals futures index. In regime 1, the EC coefficient on inflation
indicates an error-correction in inflation because it has a negative sign, but the equilibrium adjustment
is not statistically significant. At the same time, the negative sign on industrial metals provides an
indication that industrial metals diverge from the inflation index. In regime 2, there is a significant
error-correction in inflation and a significant long-run adjustment in industrial metals, confirmed by the
negative error correction coefficient on the inflation index (coefficient = -0.003, t-statistic = -3.976) and
by the positive speed coefficient on industrial metals (coefficient = 0.057, t-statistic = 2.353). Regime
1 lasts a much shorter time period, namely 2.83 months on average, and prevails 20.1% of the time.
Regime 2 lasts 11.23 months on average and prevails during 79.9% of months. These results suggest
that if the inflation increases, the industrial metals price would increase. The ability of industrial
metals to hedge inflation risks could be due to the predominantly industrial nature of such assets. This
finding is in line with the previous paper by Bilgin, Gogolin, Lau, and Vigne (2018) who examine the
inflation-hedging effectiveness of white precious metals and indicate that platinum and palladium are
more reliable inflation hedges because of their industrial nature.
Figure 3 (b) depicts the smoothed probabilities of being in regime 1, which prevailed during 1983,
1986, 1988-1990, 2001, 2005-2006, and 2008-2009, 2015, and regime 2, which prevailed during in
the rest of the time in our sample periods and covers the Great Moderation and post-subprime crisis.
The results of the MS-VECM model show that industrial metals provide a good long-run hedge against
inflation for most of the time. The regime-specific EC coefficient implies that under regime 2, the most
persistent regime prevailing around 80% of months, inflation hedging is relatively fast in pace and is
of relatively high statistical significance. It is also worth noting that regime 2 covers a relatively stable
inflationary period, such as the Great Moderation and post-subprime crisis. In this regime, low inflation
tends to go hand in hand with low industrial metals futures returns.
16
5.2.3
The precious metals futures subindex
Again, the information criteria provided in Panel D of Table 3 reveals that MSIAH(2)-VECM(1,1) is the
appropriate model. The model estimation results are summarized in Panel C of Table 5. As reflected
by the transition matrix, both regimes are very persistent. In regime 1, there exists an equilibrium
adjustment in CPI, with the estimated error-correction term marginally insignificant. At the same time,
there is no significant adjustment in precious metals. Regime 1 lasts for a fairly short time period
(6.10 months) and occurs around once in every four observations. Regime 2 prevails during 73.1%
of our sample period and lasts 16.57 months on average. In regime 2, there is significant equilibrium
adjustment in the inflation index as the EC coefficient has a significantly negative sign. In addition, the
EC coefficient on precious metals reveal that the precious metals index provides some ability to hedge
against inflation but with low statistical significance (coefficient = 0.023, t-statistic = 1.592). According
to Figure 3 (c), regime 2 prevails during the period 1983-2004 (with a short period of regime 1 in 1986,
1990 and 2000-2001). In the period between 2007 and 2009, both regimes seem to have been present.
Overall, we do not find that precious metals have significant inflation hedging ability. This is perhaps
due to the multi-facet nature of precious metals, with industrial use being one of the many attributes of
this asset (Bilgin, Gogolin, Lau, and Vigne, 2018).
5.2.4
The agriculture futures subindex
The estimation results of an MSIAH(2)-VECM(1,1) model using CPI and the agriculture futures index
are presented in Panel D of Table 5. In regime 2, the EC coefficient on the CPI is statistically significant,
implying an error correction mechanism in inflation (coefficient = -0.001, t-stat = -3.990). The error
correction coefficient is only marginally significant in regime 1. Moreover, the EC coefficients on the
agriculture index in both regimes are insignificant. It can be concluded that the agriculture futures are
not able to provide a hedge against inflation. Interestingly, the inflation index significantly depends on
its own lagged returns in both regimes; it also depends on the past return of the agriculture index, but
17
only in regime 1. Regime 1 lasts a relatively short period (7.01 months), prevailing 35.1% of the time,
while regime 2 lasts 12.96 months on average and prevails during the remaining time of our sample
period. The smoothed probabilities of being in regime 1 and regime 2 are depicted in Figure 3 (d).
5.2.5
The livestock futures subindex
Among all the models that use CPI and the livestock futures index, the SIC chooses the specification of
MSIAH(2)-VECM(1,1). The estimation results of this model are presented in Panel E of Table 5. The
transition matrix indicates that both regimes, especially regime 2, are highly persistent. The expected
duration of regime 2 is 16.67 months. According to Figure 3 (e), regime 2 prevailed in years 1984-1985,
1987-1988, 1991-1998, 2002-2004, from mid-2009 to mid-2011 and 2015-2017. Regime 1 prevailed
during the rest of the sample period. In regime 2, the significantly negative EC coefficient on CPI
indicates an error correction mechanism embedded in inflation. However, we cannot find significant
error-correction coefficient in regime 1. Moreover, in both regimes, the estimated EC coefficients of
the livestock index are statistically insignificant, implying that the livestock futures lack the ability to
hedge inflation risk.
5.3
Including stocks and bonds
In this section, we further evaluate the inflation hedging ability of commodity futures by including
common stocks and government bonds into the model. This exercise is meaningful as the bulk of
portfolios are usually allocated to common stocks and bonds, with commodity futures used as a tool
to hedge against inflation risk. Several prior studies examine the hedging and safe haven properties of
precious metals against stocks and bonds (Baur and Lucey, 2010; Baur and McDermott, 2010; Lucey
and Li, 2015; Li and Lucey, 2017). In particular, Baur and Lucey (2010) document that gold is in
general a hedge against stocks and a safe haven in bear stock markets, while gold is never a safe haven
for bonds. In this section, we follow this line of literature and identify the inflation hedging potential
of commodity futures, taking into account the inter-relatedness between commodities and conventional
18
financial assets (i.e., stocks and bonds).
In unreported results, Johansen’s test indicates one cointegrating vector for the model consisting of
inflation, commodity, stocks, and bonds. When comparing different specifications of linear and MSVECM models, the SIC suggests MSIAH(2)-VECM(1,1) as the best model.10 The estimation results of
the new model is presented in Panel A of Table 6. The estimated transition matrix and ergodic probabilities suggest that regime 2 dominates, characterizing 76.0% of months with spells lasting 13.48 months
on average; regime 1 occurs during 24.0% of months with spells lasting just 4.25 months on average.
In both regimes, there is a significant short-term dependence of the inflation index on the lagged commodity returns, as reflected by the significantly positive coefficients of ∆COM M (t − 1). However,
we fail to find such a relationship between CPI and stocks or bonds. The error-correction coefficient
explains the direction and pace of the long-run adjustment mechanism. Based on the coefficient of the
error-correction term (coefficient = -0.003, t-statistic = -4.787), there is a significant long-run equilibrium adjustment within the inflation index in regime 2. The same coefficient is negative but statistically
insignificant in regime 1. Similar to the main analysis in Table 4, we find that inflation hedging is slow
in pace (coefficient = 0.009) and is of low statistical significance (t-statistic = 0.530) after we take common stocks and government bonds into consideration. The smoothed probabilities of being in regime
1 and regime 2 are depicted in Figure 3 (e).
We also examine the inflation hedging ability of the five aforementioned subindexes of commodity
futures separately in the new model. Consistent with the model without stocks and bonds, we find
that industrial metals futures continue to provide significant hedging against inflation. The coefficient
estimates of an MSIAH(2)-VECM(1,1) model on industrial metals futures are shown in Panel B of
Table 6. The smoothed probabilities of being in regime 1 and regime 2 are depicted in Figure 4.
Compared with the model using total commodity futures index, the positive error correction coefficient
of industrial metals in regime 2 (coefficient = 0.028, t-statistic = 2.084) implies that the hedging ability
of industrial metals futures is much stronger than the total commodity futures. Moreover, combined
10
The estimation algorithm converged after 23 iterations. The number of observations in the system is 1,672. The number
of parameters in our Markov-switching model is 70.
19
with the significantly negative EC coefficient on bond returns (coefficient = -0.017, t-statistic = -3.357),
our results indicate that investors who hold industrial metals receive compensation through positive
industrial metal returns as bond returns turn negative in response to higher inflation. Moreover, the
coefficient on stock returns for industrial metals in regime 1 is not significant, while that in regime 2 is
significantly negative. These results suggest that industrial metals subindex is a hedge against stocks
in the normal regime covering the Great Moderation and post-subprime crisis. Taken together, these
results suggest that our main findings are robust to the inclusion of stocks and bonds into the model.
6
Conclusion
This study aims to answer the question whether commodity futures provide the ability to hedge against
inflation. Markov-switching framework, a comprehensive framework to describe the relationship that
changes from state to state, has been used in the past but not in the area of inflation hedging by commodity futures. Because of the heterogeneity of the sample period in terms of inflationary regimes
and commodity futures prices, we adopt an MS-VECM approach in order to characterize the changing
nature of the long-run inflation hedging properties of commodity futures.
We find that commodity futures fails to hedge US inflation over a sample period ranging from January 1983 to December 2017. By analyzing the coefficients related to the error-correction mechanism,
we show that commodities futures in the markets of industrial metals are the best inflation hedges.
However, the hedging capacity exhibits substantial variation over time. Most of the actual inflation
hedging is obtained under the relatively longer and more common regimes covering the Great Moderation and post-subprime crisis periods. We do not find evidence that energy, precious metals, agriculture
and livestock subindexes have significant inflation hedging ability. Our main findings are robust to the
inclusion of common stocks and government bonds into the model.
Results in this paper have implications to future academic research on this topic as well as industry
practitioners. Applying refined Markov-switching models to historical data, we demonstrate the time-
20
varying nature of the inflation-hedging ability of commodity futures. As demonstrated at the beginning
of this paper, various types of assets managers tend to over-weight commodities, motivated by the
wisdom that commodities provide protection from inflation spikes as the economy rapidly recovers
from the pandemic recession. Our findings provide two alerts to such practice. First, we find that not
all commodities can hedge inflation. In fact, historical data reveals that only industrial metals display
inflation-hedging ability. Second, from an investor’s point of view, the effectiveness of commodity
futures as an inflation hedge crucially depends on the time horizon. Over the long run, commodity
futures are useful as a partial hedge since a cointegrating relationship prevails. However, during some
periods where no price adjustment is observed, commodity futures are not able to shield a portfolio
from inflation risks. In terms of future research, we believe it to be interesting to calculate hedge ratios
as well as the degree of hedging effectiveness.
21
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26
Table 1: Summary statistics
The table displays sample statistics (in %) of the inflation rate as well as the nominal returns on bonds,
stocks, the S&P GSCI Total Return Index and subsector indexes during the period January 1983 December 2017. It shows mean, median, maximum, minimum, standard deviation, skewness, excess
kurtosis, and the p-value of Jarque-Bera (JB) test. The JB statistic is used to test the hypothesis of
normality of the series. COMM stands for the aggregate index, EN for energy, IM for industrial metals,
PM for precious metals, AGR for agriculture, LS for livestocks. All sample statistics are on a monthly
basis.
Variables
Mean
Median
Max.
Min.
Std. Dev.
Skewness
Kurtosis
JB p-values
CPI
Bonds
Stocks
COMM
EN
IM
PM
AGR
LS
0.222
0.615
0.969
0.486
0.778
0.820
0.332
0.082
0.320
0.225
0.630
1.342
0.478
0.554
0.453
-0.042
-0.334
0.288
1.377
8.538
12.850
22.940
37.713
38.435
15.578
17.815
17.175
-1.771
-6.682
-22.536
-28.199
-31.199
-26.662
-18.615
-18.968
-15.757
0.254
2.119
4.333
5.677
8.913
6.565
4.785
5.256
4.176
-1.375
0.096
-0.886
-0.207
0.383
0.686
0.026
0.236
-0.139
14.257
3.713
5.960
5.184
4.875
7.465
4.150
4.547
3.560
0.000
0.009
0.000
0.000
0.000
0.000
0.000
0.000
0.033
27
Table 2: Johansen cointegration test
The table presents results of the cointegration test, using monthly index series between January 1983
and December 2017. We test whether total commodity futures index as well as each of the five
subindexes have cointegration relationship with CPI. Among the five subindexes, COMM stands for
the aggregate index, EN for energy, IM for industrial metals, PM for precious metals, AGR for agriculture and LS for livestocks. Panel A shows the information criteria with different autoregressive orders
that determine the VAR order. The minimum is displayed in bold fonts. Panel B shows the L-Max and
Trace statistics, along with the corresponding p-value. Null hypotheses of cointegration and one cointegration vector are tested respectively. Panel C reports the coefficients of the estimated cointegration
vector, with the t-statistics provided in the brackets below.
Panel A: VAR order selection
Model
Lag
1
2
3
4
Lag
1
2
3
4
COMM
EN
IM
PM
AGR
LS
-12.422
-12.675
-12.692
-12.684
-12.229
-12.438
-12.462
-12.451
-12.696
-12.865
-12.897
-12.899
-12.364
-12.579
-12.556
-12.509
-12.171
-12.342
-12.326
-12.277
-12.639
-12.769
-12.762
-12.724
AIC
-12.166
-12.799
-12.824
-12.818
-11.282
-11.867
-11.882
-11.880
-11.863
-12.051
-12.092
-12.095
SIC
-12.108
-12.703
-12.689
-12.644
-11.224
-11.770
-11.746
-11.706
-11.805
-11.954
-11.956
-11.921
Panel B: Johansen co-integration test
Model
COMM
EN
IM
PM
AGR
LS
Null: no co-integration
L-Max
156.315
p-value
0.000
Trace
160.448
p-value
0.000
138.084
0.000
142.389
0.000
88.043
0.000
92.477
0.000
68.356
0.000
74.693
0.000
94.496
0.000
99.970
0.000
95.131
0.000
103.560
0.000
Null: one co-integration vector
L-Max
4.133
p-value
0.393
Trace
4.133
p-value
0.393
4.305
0.369
4.305
0.369
4.434
0.351
4.434
0.351
6.337
0.166
6.337
0.166
5.474
0.235
5.474
0.235
8.429
0.069
8.429
0.069
Panel C: Co-integration vectors
Index
Commodity
Constant
(1)
COMM
(2)
EN
(3)
IM
(4)
PM
(5)
AGR
(6)
LS
-0.161
[-2.415]
-5.135
[-12.347]
-0.114
[-2.109]
-5.333
[-14.899]
-0.195
[-6.650]
-4.345
[-22.249]
-0.089
[-0.955]
-5.519
[-12.211]
-0.069
[-0.264]
-5.860
[-4.482]
-0.157
[-1.044]
-5.237
[-5.960]
28
Table 3: Model Selection
The table provides AIC and SIC for different specifications of linear and Markov-switching vector error-correction models of the inflation
index and the various commodity index. MSIA stands for Markov-switching regime dependent intercept and matrix of autoregressive coefficients model, while MSIAH stands for Markov-switching regime dependent intercept, matrix of autoregressive coefficients, and variance
model. Model 1 uses CPI index and total commodity index. Models 2-6 use CPI and each of the five subindex of commodities, including
energy, industrial metals, precious metals, agriculture and livestock. The sample spans the period between January 1983 and December 2017.
Max LogLikelihood
No.
Obs.
#Par.
Saturation
ratio
SIC
AIC
LR
Linearity
Davies
Itrarations
-12.75
-12.75
-12.93
-12.92
-13.00
-12.96
-12.92
-12.95
-13.01
-12.96
NA
NA
113.94
117.70
151.78
147.59
149.89
190.79
210.14
216.84
NA
NA
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
NA
NA
28
53
26
27
72
40
39
43
-11.81
-11.81
-12.06
-12.03
-12.07
-12.03
-12.01
-12.04
-12.02
-11.96
NA
NA
140.01
142.67
152.93
152.54
163.14
204.62
186.02
192.65
NA
NA
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
NA
NA
29
44
37
36
61
52
40
56
NA
NA
60.31
62.43
177.86
177.53
136.78
152.31
239.41
241.87
NA
NA
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
NA
NA
31
74
31
49
39
35
86
52
NA
NA
83.29
98.84
145.81
150.30
145.04
164.49
184.21
189.54
NA
NA
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
NA
NA
37
55
74
30
40
43
100
100
-12.38
-12.39
-12.46
-12.49
-12.67
-12.64
-12.61
-12.58
-12.61
-12.55
NA
NA
68.73
92.67
165.50
165.31
175.24
188.17
197.52
195.51
NA
NA
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
NA
NA
100
100
23
25
45
30
78
51
-12.81
-12.82
-12.96
-12.96
-13.04
-13.02
-12.93
-12.98
-13.03
-13.03
NA
NA
98.97
108.91
143.17
141.55
129.55
174.13
194.03
213.77
NA
NA
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
NA
NA
38
37
32
27
100
67
35
39
HQIC
Panel A: The selection of models using commodity futures index
Model 1
2 States
3 States
Linear-VECM (1,1)
Linear-VECM (2,1)
MSIA(2)-VECM(1,1)
MSIA(2)-VECM(2,1)
MSIAH(2)-VECM(1,1)
MSIAH(2)-VECM(2,1)
MSIA(3)-VECM(1,1)
MSIA(3)-VECM(2,1)
MSIAH(3)-VECM(1,1)
MSIAH(3)-VECM(2,1)
2683.69
2686.32
2740.66
2745.18
2759.58
2760.12
2758.63
2781.72
2788.76
2794.74
Linear-VECM (1,1)
Linear-VECM (2,1)
MSIA(2)-VECM(1,1)
MSIA(2)-VECM(2,1)
MSIAH(2)-VECM(1,1)
MSIAH(2)-VECM(2,1)
MSIA(3)-VECM(1,1)
MSIA(3)-VECM(2,1)
MSIAH(3)-VECM(1,1)
MSIAH(3)-VECM(2,1)
2488.32
2489.49
2558.32
2560.82
2564.78
2565.76
2569.89
2661.66
2581.33
2585.82
418
417
836
834
836
834
836
834
836
834
11
15
21
29
24
32
33
45
39
51
38.00
27.80
39.81
28.76
34.83
26.06
25.33
18.53
21.44
16.35
-12.68
-12.67
-12.81
-12.75
-12.86
-12.78
-12.72
-12.69
-12.78
-12.67
-12.79
-12.81
-13.01
-13.03
-13.09
-13.08
-13.04
-13.13
-13.16
-13.16
Panel B: The selection of models using energy futures index
Model 2
2 States
3 States
418
417
836
834
836
834
836
834
836
834
11
15
21
29
24
32
33
45
39
51
38.00
27.80
39.81
28.76
34.83
26.06
25.33
18.53
21.44
16.35
-11.75
-11.72
-11.94
-11.86
-11.93
-11.84
-11.82
-11.78
-11.79
-11.66
-11.85
-11.87
-12.14
-12.14
-12.16
-12.15
-12.14
-12.21
-12.16
-12.16
Panel C: The selection of models using industrial metals futures index
Model 3
2 States
3 States
Linear-VECM (1,1)
Linear-VECM (2,1)
MSIA(2)-VECM(1,1)
MSIA(2)-VECM(2,1)
MSIAH(2)-VECM(1,1)
MSIAH(2)-VECM(2,1)
MSIA(3)-VECM(1,1)
MSIA(3)-VECM(2,1)
MSIAH(3)-VECM(1,1)
MSIAH(3)-VECM(2,1)
2526.89
2533.38
2557.05
2564.59
2615.82
2622.14
2677.60
2609.53
2646.60
2654.31
Linear-VECM (1,1)
Linear-VECM (2,1)
MSIA(2)-VECM(1,1)
MSIA(2)-VECM(2,1)
MSIAH(2)-VECM(1,1)
MSIAH(2)-VECM(2,1)
MSIA(3)-VECM(1,1)
MSIA(3)-VECM(2,1)
MSIAH(3)-VECM(1,1)
MSIAH(3)-VECM(2,1)
2657.90
2659.15
2699.55
2708.57
2730.81
2734.30
2730.42
2741.40
2750.01
2753.92
Linear-VECM (1,1)
Linear-VECM (2,1)
MSIA(2)-VECM(1,1)
MSIA(2)-VECM(2,1)
MSIAH(2)-VECM(1,1)
MSIAH(2)-VECM(2,1)
MSIA(3)-VECM(1,1)
MSIA(3)-VECM(2,1)
MSIAH(3)-VECM(1,1)
MSIAH(3)-VECM(2,1)
2607.70
2610.47
2642.06
2656.81
2690.45
2693.13
2695.32
2704.56
2706.46
2708.23
Linear-VECM (1,1)
Linear-VECM (2,1)
MSIA(2)-VECM(1,1)
MSIA(2)-VECM(2,1)
MSIAH(2)-VECM(1,1)
MSIAH(2)-VECM(2,1)
MSIA(3)-VECM(1,1)
MSIA(3)-VECM(2,1)
MSIAH(3)-VECM(1,1)
MSIAH(3)-VECM(2,1)
2696.41
2700.75
2745.89
2755.21
2767.99
2771.53
2761.18
2787.82
2793.42
2807.64
418
417
836
834
836
834
836
834
836
834
11
15
21
29
24
32
33
45
39
51
38.00
27.80
39.81
28.76
34.83
26.06
25.33
18.53
21.44
16.35
-11.93
-11.93
-11.93
-11.88
-12.17
-12.11
-11.94
-11.86
-12.10
-11.99
-12.04
-12.08
-12.13
-12.16
-12.40
-12.42
-12.26
-12.30
-12.48
-12.49
-12.00
-12.02
-12.05
-12.05
-12.31
-12.30
-12.13
-12.13
-12.33
-12.29
Panel D: The selection of models using precious metals futures index
Model 4
2 States
3 States
418
417
836
834
836
834
836
834
836
834
11
15
21
29
24
32
33
45
39
51
38.00
27.80
39.81
28.76
34.83
26.06
25.33
18.53
21.44
16.35
-12.56
-12.54
-12.61
-12.57
-12.72
-12.65
-12.59
-12.50
-12.59
-12.47
-12.66
-12.68
-12.82
-12.85
-12.95
-12.96
-12.91
-12.93
-12.97
-12.96
-12.62
-12.62
-12.74
-12.74
-12.86
-12.84
-12.78
-12.76
-12.82
-12.77
Panel E: The selection of models using agriculture futures index
Model 5
2 States
3 States
418
417
836
834
836
834
836
834
836
834
11
15
21
29
24
32
33
45
39
51
38.00
27.80
39.81
28.76
34.83
26.06
25.33
18.53
21.44
16.35
-12.32
-12.30
-12.34
-12.32
-12.53
-12.45
-12.42
-12.32
-12.39
-12.25
-12.42
-12.45
-12.54
-12.60
-12.76
-12.76
-12.74
-12.76
-12.76
-12.74
Panel F: The selection of models using livestock futures index
Model 6
2 States
3 States
418
417
418
836
834
836
834
836
834
836
11
15
21
29
24
32
33
45
39
51
38.00
27.80
19.90
28.83
34.75
26.13
25.27
18.58
21.38
16.39
29
-12.74
-12.74
-12.84
-12.79
-12.90
-12.83
-12.73
-12.72
-12.80
-12.73
-12.85
-12.88
-13.04
-13.08
-13.13
-13.14
-13.05
-13.16
-13.18
-13.22
Table 4: MSIAH(2)-VECM(1,1) model estimates for CPI and total commodity index
This table presents the results of the estimation of an MSIAH(2)-VECM(1,1) model for the CPI and
commodity futures index, using monthly series during the period from January 1983 to December 2017.
Coefficient estimates under the two regimes are reported, with the corresponding t-statistics shown in
the brackets below. The transition matrix between regimes, as well as implied durations and ergodic
probabilities of each regime are also provided.
Parameter
Const.
∆CPI(t-1)
∆COMM(t-1)
EC
SE
Regime 1:
∆CPI(t) ∆COMM(t)
0.002
-0.001
[5.616]
[-0.093]
0.134
-2.073
[1.791]
[-0.920]
0.037
0.332
[7.624]
[2.628]
-0.002
-0.105
[-1.005]
[-1.915]
0.003
0.079
Transition matrix
0.759
0.241
0.089
0.911
Implied durations
Regime 1
4.150
Regime 2 11.280
Ergodic prob.
Regime 1
0.269
Regime 2
0.731
30
Regime 2:
∆CPI(t) ∆COMM(t)
0.002
0.010
[12.224]
[1.760]
0.228
-0.361
[4.528]
[-0.164]
0.015
-0.002
[7.895]
[-0.025]
-0.002
0.008
[-3.924]
[0.456]
0.001
0.041
Table 5: Estimates of MS-VECM models for CPI and prices of commodity subindexes
This table presents the results of the estimation of MS-VECM models using various subindexes of
commodity futures, including energy (Panel A), industrial metals (Panel B), precious metals (Panel
C), agriculture (Panel D) and livestock (Panel E). The sample contains monthly time series during the
period from January 1983 to December 2017. Coefficient estimates under the two regimes are reported,
with the corresponding t-statistics shown in the brackets below. The transition matrix between regimes,
as well as implied durations and ergodic probabilities of each regime are also provided.
Panel A: CPI and the energy subindex
Parameter
Const.
∆CPI(t-1)
∆EN(t-1)
EC
SE
Regime 1:
∆CPI(t)
∆EN(t)
0.003
-0.025
[7.734]
[-1.300]
0.028
-3.664
[0.433]
[-1.126]
0.051
0.436
[14.073]
[2.485]
0.003
-0.275
[1.722]
[-2.551]
0.001
0.083
Transition matrix
0.746
0.254
0.024
0.977
Implied durations
Regime 1
3.930
Regime 2
42.640
Ergodic prob.
Regime 1
0.085
Regime 2
0.916
Regime 2:
∆CPI(t)
∆EN(t)
0.002
0.016
[16.888]
[2.321]
0.112
-3.350
[2.808]
[-1.444]
0.011
0.120
[11.385]
[2.296]
-0.002
-0.015
[-5.455]
[-0.662]
0.001
0.083
Panel B: CPI and the industrial metals subindex
Parameter
Const.
∆CPI(t-1)
∆IM(t-1)
EC
SE
Regime 1:
∆CPI(t)
∆IM(t)
0.001
0.019
[1.738]
[1.684]
0.381
0.340
[3.449]
[0.141]
0.012
0.533
[2.079]
[3.780]
-0.006
-0.162
[-1.572]
[-1.745]
0.004
0.082
Transition matrix
0.646
0.354
0.089
0.911
Implied durations
Regime 1
2.830
Regime 2
11.230
Ergodic prob.
Regime 1
0.201
Regime 2
0.799
Regime 2:
∆CPI(t)
∆IM(t)
0.002
0.000
[10.071]
[-0.060]
0.265
1.116
[4.357]
[0.615]
0.004
-0.193
[2.705]
[-3.365]
-0.003
0.057
[-3.976]
[2.353]
0.001
0.049
Panel C: CPI and the precious metals subindex
Parameter
Const.
∆CPI(t-1)
∆PM(t-1)
EC
Regime 1:
∆CPI(t)
∆PM(t)
0.001
0.012
[2.845]
[1.915]
0.439
-3.640
[4.722]
[-3.054]
0.021
-0.092
[2.791]
[-0.853]
-0.003
0.006
Transition matrix
0.836
0.164
0.060
0.940
Implied durations
Regime 1
6.100
Regime 2
16.570
Ergodic prob.
Regime 2:
∆CPI(t)
∆PM(t)
0.002
-0.005
[8.804]
[-0.844]
0.247
3.273
[3.552]
[1.395]
0.001
-0.136
[0.632]
[-2.125]
-0.002
0.023
Continued on next page
31
Table 5: — continued from previous page
SE
[-1.617]
0.004
[0.171]
0.046
Regime 1
Regime 2
0.269
0.731
[-4.570]
0.023
[1.592]
0.045
Panel D: CPI and the agriculture subindex
Parameter
Const.
∆CPI(t-1)
∆AGR(t-1)
EC
SE
Regime 1:
∆CPI(t)
∆AGR(t)
0.001
0.006
[3.993]
[0.695]
0.462
-3.416
[6.067]
[-2.039]
0.015
-0.060
[3.755]
[-0.692]
-0.002
-0.024
[-1.718]
[-0.850]
0.003
0.070
Transition matrix
0.857
0.143
0.077
0.923
Implied durations
Regime 1
7.010
Regime 2
12.960
Ergodic prob.
Regime 1
0.351
Regime 2
0.649
Regime 2:
∆CPI(t)
∆AGR(t)
0.002
-0.010
[9.287]
[-1.941]
0.212
4.498
[2.911]
[2.271]
0.001
0.062
[0.296]
[0.891]
-0.001
-0.003
[-3.990]
[-0.251]
0.001
0.037
Panel E: CPI and the livestock subindex
Parameter
Const.
∆CPI(t-1)
∆LS(t-1)
EC
SE
Regime 1:
∆CPI(t)
∆LS(t)
0.001
-0.009
[2.875]
[-1.616]
0.419
1.718
[4.312]
[1.919]
0.011
0.003
[0.911]
[0.029]
-0.003
-0.031
[-1.493]
[-1.438]
0.004
0.033
Transition matrix
0.813
0.187
0.060
0.940
Implied durations
Regime 1
5.340
Regime 2
16.670
Ergodic prob.
Regime 1
0.243
Regime 2
0.757
32
Regime 2:
∆CPI(t)
0.002
[8.566]
0.298
[4.433]
0.000
[-0.060]
-0.001
[-3.787]
0.001
∆LS(t)
0.000
[-0.040]
2.334
[1.419]
0.017
[0.290]
-0.008
[-0.746]
0.043
33
SE
EC
∆Bond(t-1)
∆Stock(t-1)
∆IM(t-1)
∆CPI(t-1)
Const.
Parameter
SE
EC
∆Bond(t-1)
∆Stock(t-1)
∆COMM(t-1)
∆CPI(t-1)
Const.
Parameter
∆CPI(t)
0.001
[1.024]
0.436
[3.671]
0.010
[1.677]
0.017
[1.969]
0.008
[0.389]
-0.002
[-0.720]
0.004
∆CPI(t)
0.002
[3.812]
0.223
[2.588]
0.034
[8.336]
0.004
[0.727]
-0.006
[-0.491]
0.000
[-0.149]
0.003
Regime 1:
∆IM(t)
∆Stock(t)
0.017
-0.030
[1.390]
[-2.727]
-1.342
0.171
[-0.619]
[0.090]
0.603
0.274
[4.884]
[2.549]
0.212
0.201
[1.167]
[1.314]
-0.651
0.466
[-1.485]
[1.292]
-0.052
0.073
[-1.187]
[1.822]
0.067
0.060
Regime 1:
∆COMM(t)
∆Stock(t)
0.002
-0.023
[0.191]
[-2.460]
-1.199
0.348
[-0.465]
[0.178]
0.367
0.012
[3.037]
[0.130]
0.132
0.326
[0.755]
[2.437]
-0.429
0.446
[-1.108]
[1.561]
-0.152
0.115
[-1.776]
[1.642]
0.076
0.058
Ergodic prob.
Regime 1:
0.240
Regime 2:
0.760
Implied durations
Regime 1:
4.250
Regime 2:
13.480
Transition matrix
0.764
0.236
0.074
0.926
∆Bond(t)
0.014
[2.689]
-1.143
[-1.580]
-0.007
[-0.136]
-0.145
[-2.379]
-0.045
[-0.310]
-0.002
[-0.089]
0.023
Ergodic prob.
Regime 1:
0.190
Regime 2:
0.810
Implied durations
Regime 1:
2.540
Regime 2:
10.800
Transition matrix
0.606
0.394
0.093
0.907
Panel B: The industrial metals futures model
∆Bond(t)
0.013
[3.367]
-0.750
[-1.022]
-0.045
[-1.184]
-0.080
[-1.693]
-0.144
[-1.305]
-0.022
[-0.719]
0.022
Panel A: The commodity futures model
∆CPI(t)
0.002
[11.915]
0.224
[4.058]
0.003
[1.919]
0.001
[0.651]
-0.006
[-1.310]
-0.002
[-4.397]
0.001
∆CPI(t)
0.002
[13.643]
0.103
[1.959]
0.018
[9.135]
0.002
[0.797]
-0.001
[-0.313]
-0.003
[-4.787]
0.001
Regime 2:
∆IM(t)
∆Stock(t)
0.007
0.020
[1.245]
[6.100]
1.511
-1.573
[0.827]
[-1.477]
-0.114
-0.063
[-1.852]
[-1.872]
-0.280
-0.062
[-3.277]
[-1.260]
0.139
0.088
[0.878]
[0.941]
0.028
0.001
[2.084]
[0.162]
0.054
0.033
Regime 2:
∆COMM(t)
∆Stock(t)
0.023
0.020
[4.888]
[6.030]
-3.808
-0.126
[-2.459]
[-0.114]
-0.026
-0.102
[-0.451]
[-2.398]
-0.197
-0.202
[-2.476]
[-3.831]
-0.151
0.138
[-1.127]
[1.435]
0.009
0.012
[0.530]
[0.999]
0.042
0.031
∆Bond(t)
0.004
[2.135]
0.128
[0.208]
-0.007
[-0.321]
-0.037
[-1.255]
0.038
[0.695]
-0.017
[-3.537]
0.019
∆Bond(t)
0.005
[2.668]
-0.432
[-0.657]
0.022
[0.795]
-0.074
[-2.022]
0.108
[1.816]
-0.031
[-4.127]
0.019
This table presents the results of the estimation of an MSIAH(2)-VECM(1,1) model that investigate the inter-dependence between commodity futures, CPI, common
stocks and government bonds, using monthly series from January 1983 to December 2017. The commodity futures index used in Panel A is total commodity futures
index. In Panel B, we examine one subindex of commodity futures, i.e., the industrial metals futures index. Coefficient estimates under the two regimes are reported,
with the corresponding t-statistics shown in the brackets below. The transition matrix between regimes, as well as implied durations and ergodic probabilities of
each regime are also provided.
Table 6: Estimates of an MSIAH(2)-VECM(1,1) model with common stocks and government bonds
Commodity
Energy
12,000
5,000
10,000
4,000
8,000
3,000
6,000
2,000
4,000
1,000
2,000
0
0
1985
1990
1995
2000
2005
2010
2015
2020
1985
1990
1995
Industrial Metals
2000
2005
2010
2015
2020
2010
2015
2020
2010
2015
2020
Precious Metals
5,000
5,000
4,000
4,000
3,000
3,000
2,000
2,000
1,000
1,000
0
0
1985
1990
1995
2000
2005
2010
2015
2020
1985
1990
1995
Agriculture
2000
2005
Livestock
5,000
5,000
4,000
4,000
3,000
3,000
2,000
2,000
1,000
1,000
0
0
1985
1990
1995
2000
2005
2010
2015
2020
1985
1990
1995
2000
2005
Fig. 1. Commodity futures prices. Notes: All futures prices are in US dollar. X-axis plots the time
period in months during 1983-2017. Y-axis denotes the value of commodity futures prices. All data
are extracted from Thomson Reuters DataStream.
34
1.00
Probabilities of Regime 1
0.75
0.50
0.25
1.00
1985
1990
Probabilities of Regime 2
1995
2000
2005
2010
2015
1985
1995
2000
2005
2010
2015
0.75
0.50
0.25
1990
Fig. 2. Smoothed state probabilities of a MSIAH(2)-VECM(1,1) for CPI and commodity futures index.
Notes: X-axis plots the time period in months during 1983-2017. Y-axis denotes the probability value
ranging from 0 to 1.
35
36
1995
2000
2005
2010
2015
1995
2000
2005
2010
2015
1995
2000
2005
2010
2015
2000
2005
2010
2015
1995
2000
2005
2010
2015
1995
2000
2005
2010
2015
1990
1995
1995
(e) Livestock subindex
1985
1985
1990
Probabilities of Regime 2
Probabilities of Regime 1
2000
2000
2005
2005
2010
2010
2015
2015
1990
1995
1995
2000
2000
1985
1990
1985
1990
Probabilities of Regime 2
Probabilities of Regime 1
1995
1995
2000
2000
(f) Commodity, stocks, and bonds
0.25
0.50
0.75
1.00
0.25
0.50
0.75
1.00
(c) Precious metals subindex
1985
1985
1990
Probabilities of Regime 2
Probabilities of Regime 1
2005
2005
2005
2005
2010
2010
2010
2010
2015
2015
2015
2015
Fig. 3. Smoothed state probabilities of Markov-switching vector error-correction models. Notes: X-axis plots the time period in months during 1983-2017. Y-axis
denotes the probability value ranging from 0 to 1.
(d) Agriculture subindex
0.25
0.25
1990
0.50
0.50
1985
0.75
0.75
1.00
0.25
1995
0.50
0.25
1.00
0.50
1.00
1990
(b) Industrial metals subindex
0.75
1985
1990
Probabilities of Regime 2
Probabilities of Regime 1
0.75
1.00
(a) Energy subindex
1985
0.25
0.25
0.25
1990
0.50
0.50
0.50
1985
0.75
1.00
0.75
1985
1990
Probabilities of Regime 2
0.75
1.00
0.25
0.25
0.25
1985
1990
Probabilities of Regime 2
0.50
1.00
0.75
1.00
0.50
Probabilities of Regime 1
0.75
1.00
0.50
Probabilities of Regime 1
0.75
1.00
1.00
Probabilities of Regime 1
0.75
0.50
0.25
1.00
1985
1990
Probabilities of Regime 2
1995
2000
2005
2010
2015
1985
1995
2000
2005
2010
2015
0.75
0.50
0.25
1990
Fig. 4. Smoothed state probabilities of a MSIAH(2)-VECM(1,1) for CPI, industrial metals, stocks, and
bonds. Notes: X-axis plots the time period in months during 1983-2017. Y-axis denotes the probability
value ranging from 0 to 1.
37
Table A1: Univariate time series AR order selection
The table presents the Akaike and Schwarz information criterion of each series for selecting autoregressive order. COMM stands for the aggregate index, EN for energy, IM for industrial metals, PM for
precious metals, AGR for agriculture, LS for livestocks. All sample statistics are on a monthly basis.
The minimum is displayed in bold fonts.
Series
CPI
Bonds
Stocks
COMM
EN
IM
PM
AGR
LS
Lag
AIC
SIC
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
-9.169
-9.344
-9.373
-9.377
-4.891
-4.892
-4.896
-4.896
-3.408
-3.410
-3.405
-3.402
-2.887
-2.918
-2.914
-2.909
-2.019
-2.053
-2.047
-2.042
-2.643
-2.647
-2.640
-2.651
-3.234
-3.282
-3.286
-3.281
-3.056
-3.049
-3.052
-3.045
-3.519
-3.514
-3.508
-3.510
-9.150
-9.315
-9.335
-9.329
-4.872
-4.863
-4.857
-4.847
-3.389
-3.381
-3.367
-3.354
-2.868
-2.889
-2.875
-2.861
-2.000
-2.024
-2.009
-1.993
-2.624
-2.618
-2.601
-2.603
-3.215
-3.253
-3.248
-3.233
-3.037
-3.020
-3.013
-2.996
-3.500
-3.485
-3.470
-3.462
38
Table A2: Results of the ADF stationarity test
The table shows results of the ADF stationarity test. I - ADF test with a trend and an intercept, II - ADF
test without trend and with an intercept, III - ADF test without trend and without intercept. COMM
stands for the aggregate index, EN for energy, IM for industrial metals, PM for precious metals, AGR
for agriculture, LS for livestocks. The bottom panel reports the critical values at 1%, 5%, and 10%.
Series
Lag
I
Level series
II
III
I
First differences
II
III
CPI
Bonds
Stocks
COMM
EN
IM
PM
AGR
LS
2
0
0
1
1
0
1
0
0
-1.110
-1.751
-2.149
-1.293
-1.607
-1.336
-1.936
-2.035
-1.840
-4.001
-2.488
-1.319
-2.231
-2.181
-1.561
-0.028
-1.555
-2.703
8.744
5.390
3.805
0.723
0.458
1.685
1.251
-0.300
0.925
-13.438
-19.388
-18.607
-16.913
-16.792
-18.432
-23.490
-20.649
-19.751
-12.652
-19.224
-18.605
-16.768
-16.702
-18.410
-23.435
-20.541
-19.426
-6.015
-17.996
-18.002
-16.741
-16.694
-18.281
-23.385
-20.564
-19.395
-3.980
-3.421
-3.133
-3.446
-2.868
-2.570
-2.571
-1.942
-1.616
-3.980
-3.421
-3.133
-3.446
-2.868
-2.570
-2.571
-1.942
-1.616
1%
5%
10%
39