Commodity Currencies and Commodity Prices

0 downloads 0 Views 4MB Size Report
Aug 3, 2016 - (2011) examine relationships among currency and commodity futures .... Canadian and New Zealand dollar and Norwegian krone (in particular ...
Commodity Currencies and Commodity Prices: Modelling Static and Time-Varying Dependence 3 August 2016

Katja Ignatieva a,∗ , Natalia Ponomareva b a School

of Risk and Actuarial Studies, Business School,

University of New South Wales, Sydney, Australia b Department

of Economics, Faculty of Business and Economics,

Macquarie University, Sydney, Australia

Abstract This paper employs the copula approach to study the relationship between exchange rates and commodity prices for large commodity exporters. Using data for the nominal exchange rates of four commodity currencies (Australian, Canadian, and New Zealand dollars, and Norwegian krone) against the US dollar and the relevant country-specific commodity price indices, constructed on a daily basis, we find: (i) a positive dependence between the values of commodity currencies and commodity indices, i.e. a commodity index increases when a respective currency appreciates, and provide several explanations for this finding; (ii) no major asymmetries in the tail dependence for most pairs of exchange rates and commodity indices; (iii) a pronounced increase in the time-varying tail dependence following the global financial crisis. Key words: independently floating exchange rates, commodity prices, dependence modelling, copulas

∗ Corresponding author. Email address: [email protected] (Katja Ignatieva).

1 Electronic copy available at: https://ssrn.com/abstract=2853052

1

Introduction

This paper uses the copula approach to examine the dependence structure between pairs of exchange rates and the relevant country-specific commodity indices. In our analysis we construct daily commodity price indices for the four commodity-exporting OECD countries: Australia, Canada, New Zealand and Norway, for the period from 2001 to 2015. We study co-movements between those indices and the exchange rates between the four commodity currencies and the US dollar. In addition, we explore if there are any asymmetries in the tail dependence (that is, occurrence of joint extreme movements in the tail of the distribution) between the pairs of the exchange rates and commodity indices, and analyse how the dependence and tail dependence vary over time.

There has been some interest in the empirical literature dealing with behaviour of commodity currencies and their relationship with commodity prices. A commodity currency is a name given to currencies of countries with a large proportion of primary commodities in their exports, and generally refers to the Australian dollar, Canadian dollar, New Zealand dollar, Norwegian krone, South African rand, Brazilian real, and the Chilean peso. One strand in the literature investigates the relationship between prices of primary commodities and values of the real and nominal exchange rates of the commodity-exporting countries. For instance, Amano and van Norden (1995) look at the relationship between the Canadian/US dollar real exchange rate and the terms of trade, and find a cointegration relationship, with causality runing from the terms of trade to the exchange rate, and that the terms of trade are useful in forecasting the exchange rate. The authors also report a surprising result that increases in energy commodity prices tend to weaken the Canadian dollar. This finding is challenged by Issa et al. (2008) who apply structural break tests and document a break point when this relationship changes from negative to positive in the early 1990s. They argue that the timing of the break is consistent with major changes in Canada’s energy policies and in energy-related cross-border trade and investment. Swift (2001) uses multivariate cointegration techniques to examine the pricing of Australian metal exports of three refined metals (aluminium, copper and lead) with an emphasis on the degree and timing of the pass-through of exchange rates. The author finds two cointegrating vectors for each metal, one of which represented the complete absorption of exchange rate changes in domestic export prices. Chen and Rogoff (2003) analyse correlations between real exchange rates and commodity price 2 Electronic copy available at: https://ssrn.com/abstract=2853052

indices in three commodity-exporting OECD economies: Australia, Canada and New Zealand. The authors document a strong and stable effect of commodity prices on the real exchange rate for Australia and New Zealand (with commodity price elasticity estimates ranging between 0.5 and 1). The results for Canada are found to be different with a presence of a long-run cointegrating relationship between commodity price index and real exchange rate, but relatively weak comovement is evident in a shorter run. Clements and Fry (2008) examine whether currency values are determined by commodity prices or vice versa for Australia, Canada and New Zealand. The authors jointly estimate the prices of currencies and commodities allowing for spillovers between them. They find that the commodity returns are more affected by the currency factor than vice versa, which indicates that the considered commodity-exporting countries possess some market power. Chan et al. (2011) examine relationships among currency and commodity futures markets. In particular, they consider four commodity currencies: Australian, Canadian and New Zealand dollar, and South African rand. The authors find contemporaneous commodity/currency relationships but no Granger-causality in either direction. They attribute these results to the informational efficiency of futures markets.

Another strand in the literature analyses whether commodity prices are useful in forecasting exchange rates and vice versa. Starting from the studies of Meese and Rogoff (1983a,b), the general convention is that economic variables are of limited use in exchange rate forecasting. Chen et al. (2010) explore the dynamic relationship between commodity price movements and exchange rate fluctuations of several commodity currencies, namely, the Australian, Canadian, and New Zealand dollars, as well as the South African rand and the Chilean peso. After controlling for time-varying parameters, they find a robust relationship and demonstrate that exchange rates are very useful in forecasting future commodity prices but not the other way around. The authors offer an explanation based on the fact that exchange rates are strongly forward looking, whereas commodity price fluctuations are typically more sensitive to short-term demand. In addition, the authors document that exchange rates of commodity currencies predict primary commodity prices both, in-sample and out-of-sample; however, the out-of-sample predictive ability in the reverse direction (namely, the ability of the commodity price index to predict nominal exchange rates) is not strong at the quarterly frequency that they consider. Ferraro et al. (2015) explore whether oil prices have a reliable and stable out-of-sample relationship with the Canadian/US dollar nominal exchange rate. While there is little evidence of systematic relationship between oil prices and the 3

exchange rate at the monthly and quarterly frequencies, the relationship is found to be robust at the daily frequency. The effects of changes in oil prices are immediately translated into changes in exchange rates, and appear to be very short-lived. It is shown that oil prices contain valuable information for predicting exchange rates out-of-sample in Canada, which is a significant oil exporter. This finding seems to overturn an important conventional result of unpredictability of the nominal exchange rate in the literature (see e.g., Rossi (2013) for a literature review on exchange rate predictability). It also emphasises the importance of investigating the relationship between variables at different frequencies including daily frequencies.

There is also a small separate strand in the literature studying the dependence structure between the exchange rates and oil prices using copulas. Copula is a multivariate distribution function which allows full description of the (possible non-linear) dependence structure between two or more variables (see Nelsen (1998) for introduction and Patton (2002) for applications). One of the advantages of using copulas is that the marginal distributions of underlying variables (in this case, univariate time series of exchange rates and commodity indices in each of the countries) can be modelled separately from the dependence structure between the marginals, which is captured via copula. Furthermore, the copula approach allows modelling the possibly non-linear dependence structure and in particular, many copula families allow for tail dependence. Although the copula methodology has been increasingly popular in statistics and finance, there has been only limited use of this approach in economics literature. This is due to the fact that most macroeconomic variables are only available at low frequencies (annual, quarterly or monthly) with only a few variables available at higher frequencies. Exchange rates and commodity prices are among these variables available at daily frequency. Patton (2006) explores dependence between the Deutsche mark and the Japanese yen, and finds that the mark/dollar and yen/dollar exchange rates are stronger correlated when they are depreciating against the dollar than when they are appreciating. Benediktsdottir and Scotti (2009) extend the results of Patton (2006) to several pairs of currencies and also find asymmetry in the tail dependence. Our study is in line with Aloui et al. (2013), Reboredo (2012) and Wu et al. (2012) who examine co-movement between oil prices and exchange rates using copulas. In particular, Aloui et al. (2013) report significant and symmetric dependence between daily oil prices and the exchange rates between the US dollar and five most traded currencies (euro, pound, Swiss franc, Canadian dollar and yen). They find that the rise in the oil price is associated with the depreciation of the US dollar and that this 4

link is stronger during the period of the global financial crisis (GFC). Reboredo (2012) reports similar results for a slightly different set of currencies (euro, pound, yen, Australian and Canadian dollar, Norwegian krone and Mexican peso). The dependence is found to be more intense for oil exporting countries. Wu et al. (2012) consider dependence between the US exchange rate index and oil prices and find that the rise in oil price is associated with the depreciation of the US dollar and that there is no tail dependence. Our paper has several distinctive features compared to the above studies. First, we exclusively focus on the four commodity currencies: Australian, Canadian and New Zealand dollar and Norwegian krone (in particular, none of the above studies looked at the case of New Zealand). Second, we construct country-specific commodity indices instead of looking at a single commodity such as oil. Third, we consider mixture copula models to capture asymmetric dependence in the tails of the distribution. In addition, when appropriate we model time-varying dependence with the mixture copulas. 1 This study is interested in the relationship between the pairs of exchange rates and the corresponding commodity indices at a daily frequency. Given that commodity indices are unavailable at this frequency, we construct our own indices for the four countries using data on trade volumes and commodity prices. We model marginal distributions of commodity indices and exchange rates returns using an AR(1)-GARCH(1,1) model, choosing the distribution of the error-term from the family of Symmetric Generalised Hyperbolic Distributions. In order to model dependence between commodity indices and exchange rates we consider six popular copulas: Gaussian, Student-t, Gumbel, survival Gumbel, Clayton, survival Clayton as well as the following mixture modes: Clayton & survival Clayton, Gumbel & survival Gumbel, Clayton & Gumbel and survival Gumbel & survival Clayton. Our results indicate that for a static case the Student-t copula is the optimal choice for Australia and New Zealand, and the mixture Gumbel & survival Gumbel copula is optimal for Canada and Norway. These optimal copulas are then used in a time-varying setting, which allows modelling dependence over time. We find that the commodity index and the value of the respective currency move in the same direction, and exhibit low to moderate tail dependence. The results from the time-varying setting indicate a significant increase in dependence during and following the global financial crisis. We also find that the tail dependence is relatively small and almost symmetric, but again we observe a pronounced increase in the tail 1

Reboredo (2012) also considers time-varying dependence but does this only for the Student-t and

normal copulas that do not generate any asymmetric tail dependence.

5

dependence during the GFC. These results are consistent with Aloui et al. (2013) and Reboredo (2012) despite the fact that these studies analyse oil prices rather than commodity indices. We also consider a multivariate copula linking all four commodity indices and four bilateral exchange rates. We find positive dependence and moderate symmetric tail dependence with a pronounced increase in dependence during the crisis period. Given that marginal volatilities are high during the crisis, our results are in line with the findings documented in the literature on asset returns and volatility spillovers (see e.g, King and Wadhwani (1990), Ramchand and Susmel (1998) as well as Longin and Solnik (2001), Diebold and Yilmaz (2012), and Silvennoinen and Thorp (2013)). Another possible explanation for the increase in dependence during the crisis period is the fact that in periods of a greater uncertainty about the future economic indicators (which can be important for the exchange rate determination), the current commodity prices may have a larger effect on the exchange rates. The rest of the paper is organised as follows. Section 2 describes data used in our empirical analysis and discusses the construction of commodity price indices. Section 3 provides a brief overview on copula models and reviews the main concepts on dependence modelling. Section 4 presents the estimation methods for the marginal time series and describes the results for univariate time series of all of the exchange rates and commodity indices. Section 5 discusses the results of fitting static and time-varying copulas to the data. Section 6 concludes.

2

Data

We consider the following commodity countries: Canada, Norway, Australia and New Zealand. For each of these countries we construct a commodity price index on a daily basis for the time period from January 2001 to December 2015. The list of products exported by each country is based on the statistics from the United Nations Commodity Trade Statistics Database (UNcomtrade) available from Index Mundi. 2 For each country we sort exported commodities by their shares in the value of the country’s total commodity exports and apply a cut-off of 1%, i.e. we generally exclude commodities whose average share over the period 2001–2015 is below 1% and does not exceed 1% mark in any of the years considered. The remaining commodities account for 2

http://www.indexmundi.com

6

more than 95% of the total commodity exports in each country. Also the considered commodities account on average for 70% of the total Australian exports, 42% of the total Canadian exports, 78% of the total Norwegian exports, and 65% of the total New Zealand’s exports. Table 1 summarises the major commodity exports for three years corresponding to the beginning (2001), the middle (2008) and the end (2015) of the sample period for each of the considered countries. Commodity prices are obtained from the Datastream Thomson Financial. i of commodity c in the exports of country i at time t as a share of We calculate the weight wt,c

the export value of this commodity in the total export value of the major commodity exports. P

c∈Ci

i wt,c = 1, where Ci is the set of commodities for country i listed in Table 1. Since the export

values are only available on an annual basis, the weights are assumed to be constant throughout the year. We use the following formula to compute the commodity index return for each country: IndRetit =

X

i wt,c Retct ,

(2.1)

c∈Ci

where Retct denotes the log-return on commodity c at time t. In addition to commodity indices, we consider the exchange rates rates (FX) between the USD and the currencies of the considered countries (CAD, NOK, AUD, NZD). The FX rate data covering time period from January 2001 to December 2015 are obtained from Datastream Thomson Financial. Figure 1 shows the commodity indices in the left panel and the corresponding FX rates in the right panel. 3 All commodity indices and exchange rates series appear to be highly persistent and possibly non-stationary. Therefore, the empirical section of the paper analyses log-returns. One can also observe the apparent positive correlation between commodity prices and various exchange rates: when a commodity price index increases, the corresponding FX rate (expressed as the price of a particular currency in US dollars) increases, i.e. the currency appreciates against the US dollar. We confirm this finding by looking at the linear correlation between commodity indices and FX rates reported in Table 2. These preliminary results indicate the presence of a 3

Note that for the purpose of graphical representation of all FX rates in one figure we scale the exchange

rate USD/NOK by factor 0.2.

7

Table 1 Major commodity exports by year (2001, 2008, 2015) for Australia, Canada, New Zealand and Norway. Australia coal iron ore/iron aluminium/aluminium oxide/ores gold crude oil natural gas meat bovine, sheep and goat/live bovine copper/copper ores and concentrates wheat/ muslin oil not crude wool wine zinc/zinc ores and concentrates nickel/nickel ores lead/lead ores and concentrates cotton wood milk barley cheese/curd paper Total New Zealand milk products, butter meat bovine, sheep and goat and live bovine wood cheese, curd, casein aluminium fish wool iron pulp paper leather, skin malt gold Total

2001 0.1655 0.0778 0.1249 0.0683 0.0861 0.0490 0.0816 0.0368 0.0573 0.0354 0.0388 0.0257 0.0224 0.0238 0.0094 0.0268 0.0142 0.0207 0.0129 0.0131 0.0094 1 2001 0.2485 0.1993 0.1144 0.1401 0.0623 0.0473 0.0420 0.0235 0.0283 0.0374 0.0425 0.0029 0.0115 1

2008 0.2838 0.1896 0.0780 0.0870 0.0630 0.0635 0.0425 0.0466 0.0231 0.0210 0.0134 0.0153 0.0142 0.0108 0.0106 0.0027 0.0094 0.0079 0.0076 0.0055 0.0046 1 2008 0.3236 0.1977 0.0928 0.1092 0.0607 0.0300 0.0247 0.0408 0.0246 0.0263 0.0210 0.0265 0.0220 1

2015 0.2093 0.2640 0.0658 0.0756 0.0330 0.0920 0.0722 0.0469 0.0313 0.0110 0.0156 0.0116 0.0168 0.0045 0.0108 0.0057 0.0088 0.0056 0.0093 0.0047 0.0055 1 2015 0.3312 0.2336 0.1291 0.0958 0.0373 0.0260 0.0268 0.0197 0.0274 0.0175 0.0169 0.0239 0.0147 1

Canada crude oil natural gas wood paper oil not crude iron/iron ore aluminium/aluminium oxide/ores gold pulp wheat/muslin nickel copper/copper ores and concentrates seeds coal meat swine/live swine meat bovine/live bovine zinc/zinc ores and concentrates silver

2001 0.1192 0.2072 0.1416 0.1431 0.0513 0.0594 0.0613 0.0164 0.0544 0.0288 0.0167 0.0178 0.0131 0.0135 0.0170 0.0275 0.0086 0.0030

2008 0.2964 0.1594 0.0432 0.0622 0.0785 0.0628 0.0520 0.0359 0.0307 0.0307 0.0314 0.0286 0.0259 0.0271 0.0119 0.0126 0.0075 0.0032

2015 0.3115 0.0571 0.0728 0.0524 0.0713 0.0638 0.0515 0.0770 0.0372 0.0385 0.0256 0.0363 0.0379 0.0169 0.0161 0.0179 0.0072 0.0093

Norway crude oil natural gas fish oil non-crude aluminium nickel iron ore/iron/steel

1 2001 0.6276 0.1770 0.0712 0.0417 0.0574 0.0098 0.0153

1 2008 0.4979 0.3243 0.0480 0.0572 0.0448 0.0141 0.0137

1 2015 0.3403 0.3995 0.1153 0.0665 0.0470 0.0157 0.0157

1

1

1

Table 2 Pearson linear correlation between commodity indices and FX rates computed using time period from January 2001 to December 2009. USD/CAD USD/NOK USD/AUD USD/NZD

Index Canada 0.9488 0.8805 0.9189 0.8891

Index Norway 0.8946 0.8708 0.8206 0.8347

Index Australia 0.9498 0.8257 0.9329 0.8591

Index New Zealand 0.7256 0.5464 0.7829 0.8287

co-movement between commodity prices and exchange rates. We will further examine this comovement using copulas. This will allow us to model a possibly non-linear dependence structure between the time series. The analysis will be carried out in both static and time-varying settings. 8

USD/CAD USD/NOK USD/AUD USD/NZD

0.8

FX rates

200

1.0

250

Canada Norway Australia New Zealand

50

0.4

100

0.6

150

Commodity indices

FX rates 1.2

300

Commodity indices

2001

2003

2005

2007

2009

2011

2013

2015

2001

2003

2005

year

2007

2009

2011

2013

2015

year

Fig. 1. Commodity indices for Canada, Norway, Australia and New Zealand (left panel) constructed on a daily basis and daily FX rates for USD/CAD, USD/NOK, USD/AUD and USD/NZD (right panel) for the time period from January 2001 to December 2015.

3

Methodology for Copulas

This section focuses on the copula methodology, the main concepts of dependence modelling and techniques for copula estimation. Copulas are multivariate distribution functions connecting d one-dimensional uniform-(0,1) marginals to a joint cumulative distribution. According to Sklar’s theorem, if F is a d-dimensional distribution function with marginals F1 . . . , Fd , then under some general conditions there exists a copula C with F (x1 , . . . , xd ) = C{F1 (x1 ), . . . , Fd (xd )}

(3.1)

for every x1 , . . . , xd ∈ R. The survival copulas C ∗ corresponding to C is defined as: F (x1 , . . . , xd ) = C ∗ {F 1 (x1 ), . . . , F d (xd )} where F (x1 , . . . , xd ) = P (X1 > x1 , . . . Xd > xd ). 9

(3.2)

Copulas provide a flexible method of modelling the dependence structure between random variables which can be non-linear. In particular, the copula approach allows measuring extreme dependence in the tails of a multivariate distribution. In a bivariate case, let us denote by (U1 , U2 ) a pair of uniform variables on the unit square [0, 1]2 . The lower tail dependence coefficient λl ∈ [0, 1] is defined as C(u, u) . u→0+ u

λl = lim P (U1 ≤ u|U2 ≤ u) = lim u→0+

(3.3)

Similarly, the upper tail dependence coefficient λu ∈ [0, 1] is defined as C ∗ (u, u) , u→1− 1 − u

λu = lim P (U1 > u|U2 > u) = lim u→1−

(3.4)

where C ∗ is the survival copula. If λl = 0 or λu = 0, then U1 and U2 are said to be asymptotically independent in the lower tail or the upper tail, respectively. Throughout the paper we will concentrate on two popular copula families: the elliptical copulas family and the Archimedean copulas family. Elliptical copulas have a dependence structure generated by elliptical distributions such as normal or Student-t. The Gaussian copula generates the dependence structure given by the multivariate normal distribution. In the case of the normal marginals, that is, if Xj ∼ N (0, 1) and X = (X1 , . . . , Xd )> ∼ Nd (0, Ψ), where Ψ denotes a correlation matrix, the explicit expression for the Gaussian copula is given by CΨGa (u1 , . . . , ud ) = FX {Φ−1 (u1 ), . . . , Φ−1 (ud )}.

(3.5)

Thus, combining normal marginals by using the Gaussian copula leads to the multivariate normal distribution. Note that the Gaussian copula can be used with any other marginal distribution (in which case the resulting multivariate distribution will not be normal). Furthermore, the Gaussian copula does not generate any tail dependence. The Student-t copula generates the dependence structure from the multivariate Student-t distribution. If X = (X1 , . . . , Xd )> ∼ td (ν, µ, Σ), i.e. X has a multivariate Student-t distribution with ν degrees of freedom, mean vector µ and positive-definite covariance matrix Σ, the Student-t copula is given by t −1 Cν,Ψ (u1 , . . . , ud ) = tν,Ψ {t−1 ν (u1 ), . . . , tν (ud )},

(3.6)

where t−1 is the quantile function from the univariate t-distribution and Ψ is the correlation ν 10

matrix associated with Σ. 4 The Student-t copula generates the symmetric tail dependence with the tail dependence coefficients defined by 

 q

λu = λl = 2tν+1 − (ν + 1)(1 − ρ)/(1 + ρ) ,

(3.7)

where tν is the Student-t distribution function with ν degrees of freedom, and ρ is the correlation coefficient. Modelling dependence by using elliptical distributions can be found among others in Hult and Lindskog (2002), Fang et al. (2002) and Frahm et al. (2003), Breymann et al. (2003), McNeil et al. (2005) and Dias and Embrechts (2008). In our empirical analysis we will also use the Clayton and Gumbel copulas that belong to the family of Archimedean copulas. The Clayton copula with the dependence parameter θ ∈ (0, ∞) is defined by Cθ (u1 , . . . , ud ) =

−1/θ   d   X −θ   . − d + 1 u j  

(3.8)

j=1

As the copula parameter θ goes to infinity, the dependence becomes maximal and as θ goes to zero, we have independence. The Clayton copula can mimic lower tail dependence with the tail dependence coefficient λl = 2−1/θ but no upper tail dependence, that is, λu = 0. The Gumbel copula with the dependence parameter θ ∈ [1, ∞) is given by 

Cθ (u1 , . . . , ud ) = exp − 

 d X 

1/θ    (− log uj )θ . 

(3.9)

j=1

For θ > 1 this copula generates upper tail dependence with the tail dependence coefficient λu = 2 − 21/θ but no lower tail dependence, that is, λl = 0. For θ = 1 it reduces to the product copula (i.e. independence): Cθ (u1 , . . . , ud ) =

Qd

j=1

uj . Maximal dependence is achieved when θ

goes to infinity. In addition to the Archimedean and elliptical copulas discussed above we also consider some mixture models of Archimedean copulas as introduced in Joe (1993). Mixture copulas take the form of a convex combination of two or more copulas. Denoting C A and C B copulas with dependence 4

Since copula functions remain invariant under strictly increasing transformations of X (e.g. stan-

dardisation of the marginal distributions), see Nelsen (1998), the copula of a td (ν, µ, Σ) distribution is identical to that of a td (ν, 0, Ψ).

11

parameters θ1 and θ2 , respectively, the mixture model takes the following form: B A (u1 , . . . , ud , θ2 ), (u1 , . . . , ud , θ1 ) + (1 − θ3 )CX CX (u1 , . . . , ud , θ) = θ3 CX

(3.10)

where 0 ≤ θ3 ≤ 1. Empirical applications of the mixture copulas can be found e.g. in Dias (2004) for FX rates, or in Hu (2006), where the mixture copulas are used for modelling the dependence across international financial markets. In our empirical analysis we will consider four mixture models of copulas, namely the Clayton & survival Clayton, Clayton & Gumbel, survival Clayton & survival Gumbel and Gumbel & survival Gumbel mixture copulas. Note that the literature suggests different approaches to the copula estimation, see e.g. Joe (1997); Cherubini et al. (2004). In our empirical analysis, we estimate copulas using the inference for marginals (IFM) method, which is a sequential two-step maximum likelihood method. According to the IFM, marginal parameters are estimated in the first step and substituted into a copula to obtain the pseudo log-likelihood function, which is then maximized with respect to the copula dependence parameter θ. The two-step procedure employed by this method makes it computationally efficient. For details on the IFM and the alternative estimation techniques refer to McLeish and Small (1988) and Joe (1997).

4

Empirical Methodology for the Marginals

As indicated in the previous section, before estimating copula models we need to specify marginal distributions. It has been documented in the literature 5 that financial data returns usually do not follow normal distribution since they exhibit heavy tails and excess kurtosis. In order to capture these characteristics of the data, we consider several alternative distributions from the family of Symmetric Generalised Hyperbolic (SGH) distributions: the Student-t, the Normal Inverse Gaussian (NIG), the Hyperbolic (HYP) and the Variance Gamma (VG) distributions.

5

For instance, see Hurst and Platen (1997), Platen and Rendek (2008), Hu and Kercheval (2007),

Ignatieva and Platen (2010) and Ignatieva et al. (2011)

12

4.1

Symmetric Generalised Hyperbolic Distributions

The family of generalised hyperbolic distributions was introduced in Barndorff-Nielsen (1977) and discussed in its general form in Jørgensen (1982), Barndorff-Nielsen and Stelzer (2005), and McNeil et al. (2005). For the purpose of our analysis we concentrate on the symmetric representation which assumes that the location of the distribution and the skewness parameter are set equal to zero. Thus, we consider the symmetric generalised hyperbolic (SGH) density of the form: 1 fX (x) = δσKλ (¯ α)

s



2

 1 (λ− 1 )

x  α ¯ 1+ 2π (δσ)2

2

2

Kλ−

 v u u  1 α ¯ t1 + 2



x2  (δσ)2

(4.1)

where α 6= 0 if λ ≥ 0 and δ 6= 0 if λ ≤ 0. Kλ (·) denotes a modified Bessel function of the third kind with index λ, see Abramowitz and Stegun (1972). The parameters λ and α ¯ can be interpreted as the shape parameters for the tails of the distribution. Varying λ and α ¯ allows one to specify special cases of the SGH distribution. In particular, we will investigate the following important special cases: the Student-t distribution (¯ α = 0 and λ < 0, see Praetz (1972)), the Normal Inverse Gaussian (λ = −0.5, see Barndorff-Nielsen (1994)), the Hyperbolic distribution (λ = 1, see Eberlein and Keller (1995)) and the Variance Gamma distribution (¯ α = 0 and λ > 0, see Madan and Seneta (1990)). For the Student-t density we only consider λ ≤ −1 in which case the number of degrees of freedom equals ν = −2λ ≥ 2. 6 Note that in the Student-t case the parameter σ is not the standard deviation of the random variable X, which is given by σX = σ

q

ν . ν−2

When the number of degrees of freedom ν decreases, we observe an increase

in the tail heaviness of the density, which implies a larger probability of extreme values. In addition, as the number of degrees of freedom ν goes to infinity, the Student-t density converges asymptotically to the Gaussian density. Further details on the representation of the density functions can be found in Platen and Rendek (2008). In order to choose the distribution from the SGH family that fits the data best, we use the

6

We do not consider the case when −1 < λ < 0, since it corresponds to ν < 2 for which the normali-

sation constant diverges, see Platen and Rendek (2008).

13

Anderson-Darling (AD) distance for the log-returns as a goodness-of-fit statistic: AD =

supx∈R |Fs (x) − Fˆ (x)| q , Fˆ (x)(1 − Fˆ (x))

(4.2)

where Fs (x) denotes the empirical sample distribution and Fˆ (x) is the estimated distribution. We prefer the AD statistic to another popular goodness-of-fit statistic: the Kolmogorov-Smirnov (KS) distance 7 , due to its ability to capture the deviations around the median of the distribution, as well as the discrepancies in the tails.

4.2

An AR - GARCH Model for the Marginals

After specifying the best performing distribution family, we estimate parameters for the marginals assuming that the conditional mean is constant and the volatility is time-varying. We use the common specification of the AR(1)-GARCH(1,1) process that can be written as follows 8 : Xt = a0 + a1 Xt−1 + σt ut ,

(4.3)

2 σt2 = α0 + α1 σt−1 + β1 ε2t−1 ,

(4.4)

where (Xt )t≥0 is the process describing log-returns of commodity indices or FX rates, (ut )t≥0 is a sequence of i.i.d. random variables with zero mean and unit variance, ut is independent of (Xs )s≤t , α0 > 0, α1 ≥ 0, β1 ≥ 0, α1 + β1 < 1. Note that innovations εt = σt ut in Eq. (4.3) have by definition zero mean and conditional variance V ar(εt |Ft ) = σt2 which is modeled via Eq. (4.4). We estimate the parameters by the maximum likelihood for each marginal series assuming that the fitted residuals ubt = εbt /σbt are i.i.d. and follow one of the distributions from the SGH family. The preferred distribution of ut is chosen according to the AD goodness-of-fit statistic. 7

The KS distance, defined as KS = supx∈R |Fs (x) − Fˆ (x)|, is more sensitive closer to the centre of the

distribution and fails to capture the tails. 8 GARCH(1,1) is a parsimonious model providing a good fit to the data (even if time series are volatile and exhibit jumps), see e.g. Morana (2001), Lin and Tamvakis (2001), Garcia et al. (2005), Misiorek et al. (2006), Worthington et al. (2005), and Ketterer (2012). In particular, Andersen and Bollerslev (1998) and Wang and Wu (2012) argue that the GARCH model can often deliver more precise estimates compared to more complex models.

14

Table 3 Anderson-Darling distance for the FX returns (top panel) and commodity index returns (bottom panel) using different marginal distributions. Estimates refer to daily observations for the period from January 2001 to December 2015. The best performing model is indicated in bold. Anderson-Darling Canada Norway Australia New Zealand

Normal 8.0453 8.1921 40.5184 24.5387

Student-t 0.1902 0.3513 0.3388 0.1629

Anderson-Darling Canada Norway Australia New Zealand

Normal 11.9506 24.3157 39.883 133.8952

Student-t 0.1098 0.2483 0.2518 0.3877

5

FX rates NIG 0.1631 1.2784 4.5220 0.3936 Commodity indices NIG 0.2136 7.539 0.3227 0.1910

HYP 0.7571 2.0966 8.1422 0.4285

VG 0.9144 2.9101 8.1408 0.5066

HYP 0.4258 3.7839 8.1957 7.8073

VG 0.4056 3.7405 0.2718 1.5020

Empirical Analysis

5.1

Analysis of the Marginals

We use the AD goodness-of-fit statistic to compare the performance of alternative marginal distributions for the log-returns of FX rates and commodity indices. Table 3 summarises the results for FX rates (top panel) and commodity indices (bottom panel) obtained using the entire sample from January 2001 to December 2015. We observe that all distributions from the SGH family provide a considerably better fit compared to the normal distribution. Assuming the Student-t distribution for ut in Eq. (4.3) results in the smallest values for the AD statistics for nearly all the cases, with an exception of the FX rate USD/CAD and the commodity index for New Zealand, where the Student-t is slightly outperformed by the NIG distribution. Since the Student-t distribution is ranked first in six out of eight cases and is the second-best in the remaining two cases, we favour it over the competing distributions from the SGH family, as well as the normal distribution. In the next step, we use the AR-GARCH model (Eq. (4.3)-(4.4)) with Student-t innovations to estimate time-varying volatilities σbt , which will be used for modelling dependence across log-returns via copulas. The Student-t-AR(1)-GARCH(1,1)-fitted volatility at time t is computed using a 9 c t moving window of log-returns {X} The s=t−n+1 of size n = 252 scrolling in time for t = n, ..., T .

9

We use the initial sample of n = 252 observations corresponding to one year to obtain the first

15

100 50 0

0

50

100

150

t-GARCH-fitted annualized volatility for commodity index (Norway)

150

t-GARCH-fitted annualized volatility for commodity index (Canada)

2002

2005

2008

2011

2014

2002

2008

2011

2014

100 50 0

0

50

100

150

t-GARCH-fitted annualized volatility for commodity index (New Zealand)

150

t-GARCH-fitted annualized volatility for commodity index (Australia)

2005

2002

2005

2008

2011

2014

2002

2005

2008

2011

2014

Fig. 2. Annualized Student-t-GARCH(1,1)-fitted volatilities for commodity indices of Canada, Norway, Australia and New Zealand for the time period from January 2002 to December 2015.

average annualised volatilities of the commodity indices for Australia, New Zealand, Canada and Norway are 8.0%, 11.6%, 13.4% and 22.7% respectively (see Figure 2). The average annualised volatility for all FX currencies ranges from 7.7% for USD/CAD to 10.9% for USD/NZD (see Figure 3).

estimate of volatility on 1 January 2002. Please note that using approximately one year of data as a moving window is a standard choice in the literature that uses time-varying copulas for dependence modelling (see e.g. Aussenegg and Cech (2009), Ignatieva et al. (2011), Kim (2014), Fengler and Okhrin (2016) and Ignatieva and Trueck (2016)).

16

60 40 20 0

0

20

40

60

80

t-GARCH(1,1)-fitted annualized volatility for FX USD/NOK

80

t-GARCH(1,1)-fitted annualized volatility for FX USD/CAD

2002

2005

2008

2011

2014

2002

2008

2011

2014

60 40 20 0

0

20

40

60

80

t-GARCH(1,1)-fitted annualized volatility for FX USD/NZD

80

t-GARCH(1,1)-fitted annualized volatility for FX USD/AUD

2005

2002

2005

2008

2011

2014

2002

2005

2008

2011

2014

Fig. 3. Annualized Student-t-GARCH(1,1)-fitted volatilities for FX rates between USD and the respective currency of the considered countries (CAD, NOK, AUD, NZD) for the time period from January 2002 to December 2015.

5.2

Copula Analysis

In this section we analyse the dependence structure between the FX rates (USD /CAD, USD/NOK, USD/AUD and USD/NZD) and the corresponding commodity indices for Canada, Norway, Australia and New Zealand. Based on the results in Section 5.1, we assume the Student-t marginals for log-returns of all commodity indices and FX rates. We use these marginals to estimate the dependence structure via copulas. The IFM method is applied to estimate different copulas in static and time-varying settings. 17

5.2.1

Fitting Static Copulas

A static copula is used to estimate the global (average) dependence parameter using log-return data from the time interval covering the whole sample from January 2002 to December 2009. 10 To compare the performance of the fitted models we use the Akaike Information Criterion (AIC): AIC = −2l(α; x1 , . . . , xT ) + 2q,

(5.1)

where l(α; x1 , . . . , xT ) is the maximised value of the log-likelihood and q is the number of parameters in the model. Smaller values of the AIC indicate a better fit. Table 4 reports copula parameters estimated using different one-parametric families of copulas and the mixture copula models for pairs of commodity indices and the corresponding FX rates. Parameter θ1 represents the dependence parameter in the case of one-parametric models and the dependence parameter for the first term in the mixture models; θ2 represents the dependence parameter for the second term in the mixture models and θ3 is the parameter representing the proportion of the first term in the mixture models. The standard errors are reported in parenthesis. We also report the lower and the upper tail dependence coefficients denoted by λl and λu , respectively. Note that Gaussian copula does not generate tail dependence, that is λl = λu = 0, whereas Student-t copula generates symmetric tail dependence i.e. λl = λu . The last column reports the model ranking based on the AIC. As can be seen from Table 4, all models demonstrate positive dependence between commodity indices and the corresponding exchange rates indicating that the commodity index increases when the respective currency appreciates and vice versa. There are several possible explanations for this finding. First, an increase in commodity price index of a certain country often indicates an increase in demand for commodities of this country, which in turn leads to higher demand for domestic currency (more of which is needed to buy larger amounts of commodities for higher prices) and, as a result, to its appreciation. If only future demand changes (but not the current one), there will be a similar effect because the current exchange rate depends on the future exchange rate. Second, given that an increase in commodity prices is often associated with an increase in investment in the relevant sectors of the economy and therefore, their growth, there 10

Note, that the first year of data (2001) corresponding to 252 observations is used in a moving window

procedure to estimate time-dependent volatilities from the GARCH model.

18

Table 4 Copula dependence parameter estimates for different bivariate copula models for the returns of commodity indices and FX rates (USD per units of currency). In the case of the mixture models, θ1 and θ2 are dependence parameters for the first and the second terms in the mixture, respectively, and θ3 gives the proportion of the first term in the mixture model. The lower and the upper tail dependence coefficients are denoted by λl and λu , respectively. Copula model Clayton surv.Clayton Gaussian Gumbel surv.Gumbel Student-t Clayton & surv.Clayton Clayton & Gumbel surv.Clayton & surv.Gumbel Gumbel & surv.Gumbel Clayton surv.Clayton Gaussian Gumbel surv.Gumbel Student-t Clayton & surv.Clayton Clayton & Gumbel surv.Clayton & surv.Gumbel Gumbel & surv.Gumbel Clayton surv.Clayton Gaussian Gumbel surv.Gumbel Student-t Clayton & surv.Clayton Clayton & Gumbel surv.Clayton & surv.Gumbel Gumbel & surv.Gumbel Clayton surv.Clayton Gaussian Gumbel surv.Gumbel Student-t Clayton & surv.Clayton Clayton & Gumbel surv.Clayton & surv.Gumbel Gumbel & surv.Gumbel

θˆ1 (s.e.) θˆ2 (s.e.) θˆ3 (s.e.) 2-dim portfolio (Canada, USD/CAD) 0.309 (0.024) 0.345 (0.026) 0.269 (0.015) 1.202 (0.015) 1.184 (0.018) 0.272 (0.016) 0.341 (0.108) 0.548 (0.226) 0.570 (0.140) 0.484 (0.243) 1.199 (0.053) 0.331 (0.138) 0.732 (0.243) 1.159 (0.027) 0.286 (0.086) 1.370 (0.144) 1.134 (0.038) 0.379 (0.119) 2-dim portfolio (Norway, USD/NOK) 0.191 (0.023) 0.183 (0.024) 0.166 (0.017) 1.108 (0.013) 1.104 (0.013) 0.166 (0.017) 0.151 (0.030) 0.840 (0.365) 0.831 (0.069) 0.143 (0.067) 1.276 (0.308) 0.727 (0.238) 0.953 (0.436) 1.079 (0.016) 0.132 (0.062) 1.324 (0.266) 1.073 (0.024) 0.214 (0.146) 2-dim portfolio (Australia, USD/AUD) 0.431 (0.025) 0.396 (0.027) 0.342 (0.014) 1.259 (0.016) 1.260 (0.015) 0.351 (0.015) 0.522 (0.073) 0.693 (0.139) 0.617 (0.063) 0.556 (0.146) 1.305 (0.081) 0.503 (0.088) 0.864 (0.207) 1.253 (0.026) 0.240 (0.057) 1.418 (0.125) 1.239 (0.038) 0.312 (0.077) 2-dim portfolio (New Zealand, USD/NZD) 0.023 (0.013) 0.045 (0.018) 0.049 (0.016) 1.010 (0.010) 1.259 (0.011) 0.049 (0.001) 0.030 (0.047) 0.058 (0.049) 0.326 (0.608) 0.022 (0.024) 1.031 (0.076) 0.727 (0.735) 0.045 (0.019) 1.010 (0.001) 0.990 (0.012) 1.015 (0.010) 1.010 (0.000) 0.990 (0.015)

19

λl

λu

AIC

rank

0.106 0.000 0.000 0.000 0.204 0.021 0.074 0.079 0.129 0.097

0.000 0.134 0.000 0.219 0.000 0.021 0.121 0.145 0.111 0.129

-202.98 -204.36 -253.77 -245.01 -235.92 -267.06 -261.67 -265.24 -263.22 -266.62

(10) (9) (6) (7) (8) (1) (5) (3) (4) (2)

0.026 0.000 0.000 0.000 0.126 0.000 0.008 0.005 0.085 0.072

0.000 0.022 0.000 0.131 0.000 0.000 0.073 0.076 0.063 0.066

-83.270 -64.517 -92.462 -83.380 -91.025 -92.780 -98.445 -97.121 -98.278 -97.251

(9) (10) (6) (8) (7) (5) (1) (4) (2) (3)

0.200 0.000 0.000 0.000 0.266 0.013 0.163 0.144 0.198 0.172

0.000 0.174 0.000 0.265 0.000 0.013 0.141 0.148 0.107 0.115

-383.20 -272.62 -433.69 -361.58 -426.87 -449.34 -451.15 -447.59 -450.64 -448.44

(8) (10) (6) (9) (7) (3) (1) (5) (2) (4)

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.021 0.000 0.000 0.000 0.000 0.000 0.021

-1.15 -4.52 -7.99 -0.61 -1.30 -4.99 -0.99 2.26 -0.46 3.42

(5) (3) (1) (7) (4) (2) (6) (9) (8) (10)

might be an increase in demand for assets of commodity-exporting firms. This would lead to an appreciation of domestic currency. Finally, given that portfolio investment in emerging economies, such as China, can be risky, commodity-exporting countries with strong links to these economies can be considered as their safer proxies. For instance, increased growth of Chinese economy or expectations of the future growth (which create higher demand for commodities and an increase in commodity prices) would provide an incentive to invest in Australia, as a safe growth proxy for China, which would lead to an appreciation of the Australian dollar. In the case of New Zealand we observe that the Gaussian copula outperforms all competing models for the pair (New Zealand, USD/NZD). The dependence parameter, which is equivalent to the correlation coefficient for the Gaussian copula, is low at 0.049 and there is no tail dependence. 11 For Canada, the Student-t copula is ranked number one overall, indicating that for the pair (Canada, USD/CAD) the dependence tends to be symmetric in both tails. However, the small tail dependence coefficients (equal to 0.021) suggest low probabilities of the simultaneous occurrence of extreme events. The best performing copula model for the pairs (Australia, USD/AUD) and (Norway, USD/NOK) is the mixture Clayton & survival Clayton copula. The lower and the upper tail dependence coefficients for (Norway, USD/NOK) are both pretty low (λl = 0.008 and λu = 0.073). However, a slightly stronger dependence in the upper tail suggests that there is some tendency for a large increase in Norwegian commodity price index and a large appreciation of the NOK to occur simultaneously. In the case Australia, the lower and upper tail dependence coefficients are similar (λl = 0.1635 and λu = 0.141) indicating that the dependence structure is close to symmetric. Also both upper and lower tail dependence coefficients for the pair (Australia, USD/AUD) are the highest across all other pairs formed by the commodity index and the respective exchange rate. We also consider the 8-dimensional copulas formed by the four returns of FX rates and the 11

This result is also consistent with the estimated number of degrees of freedom for the Student-t copula

of the pair (New Zealand, USD/NZD) - it corresponds to 168. A large number of Student-t degrees of freedom indicates that the distribution approaches Gaussian. For the other copula models the number of degrees of freedom of the Student-t copula corresponds to 11, 28 and 15 for (Canada, USD/CAD), (Norway, USD/NOK) and (Australia, USD/AUD), respectively.

20

Table 5 Copula dependence parameter estimates for different 8-dimensional copula models for dependence between FX rates (USD per units of currency) and the corresponding commodity indices. For each of the fitted models, the last two columns provide results for the AIC and the model ranking (in parentheses). In the case of the mixture models, θ1 and θ2 are the dependence parameters for the first and the second terms in the mixture, respectively, and θ3 gives the proportion of the first term in the mixture model. The lower and the upper tail dependence coefficients are denoted by λl and λu , respectively. Copula model θˆ1 (s.e.) 8-dim portfolio (USD/CAD, USD/NOK, Clayton 0.240 (0.005) surv.Clayton 0.299 (0.006) Gaussian 0.291 (0.005) Gumbel 1.210 (0.004) surv.Gumbel 1.190 (0.537) Student-t 0.307 (0.006) Clayton & surv.Clayton 0.482 (0.034) Clayton & Gumbel 0.697 (0.026) 0.779 (0.030) surv.Clayton & surv.Gumbel Gumbel & surv.Gumbel 1.210 (0.007)

θˆ2 (s.e.) θˆ3 (s.e.) λl USD/AUD, USD/NZD, Canada, Norway, 0.055 0.000 0.000 0.000 0.209 0.012 0.513(0.037) 0.512 (0.019) 0.122 1.100(0.008) 0.486 (0.023) 0.179 1.088(0.007) 0.453 (0.022) 0.059 1.000(0.001) 0.999 (0.001) 0.000

λu AIC Australia, New Zealand) 0.000 -3324.17 0.099 -3525.92 0.000 -4635.17 0.226 -3740.31 0.000 -3690.00 0.012 -5122.83 0.126 -5168.55 0.062 -4572.52 0.186 -4495.04 0.226 -1867.54

rank (9) (8) (3) (6) (7) (2) (1) (4) (5) (10)

commodity indices. The results are presented in Table 5. We observe that the Clayton & survival Clayton copula outperforms all other models. According to this preferred model, there is some moderate symmetric tail dependence.

5.2.2

Fitting Time-Varying Copulas

While the results from fitting static copulas are informative, it is interesting to investigate how dependence between commodity indices and exchange rates varies over time. In order to estimate the dependence parameter in a time-varying setting we use a moving window of size n = 252, which corresponds to one year of data. The copula is fitted to the subsets of size n of log returns c }s {X t t=s−n+1 and the procedure is repeated for s = n, ..., T . This generates a time-series for the

dependence parameter {θbt }Tt=n . In all periods we fit the copula that was identified as optimal for a particular pair (commodity index, FX rate) in the static case. For Australia and Norway it is the Clayton & survival Clayton mixture copula, for Canada it is the Student-t copula and for New Zealand it is the Gaussian copula. From the upper panel of Figure 4 we observe that the Student-t copula parameter measuring dependence between the returns of the Canadian commodity index and the USD/CAD exchange rate increased from being close to 0 at the beginning of 2003 to more than 0.3 by 2006, where it 21

stayed until early 2007. It then dropped to almost zero before starting to increase dramatically in 2008. This increase in the copula dependence parameter is attributed to the start of the global financial crisis (GFC). The dependence reached its peak of 0.6 in 2010 and remained hight until 2013. Then there was a steep decrease that continued until late 2014, when the dependence parameter reached a nearly zero mark. From then there was another substantial increase with dependence reaching 0.4 at the end of 2015. The tail dependence coefficient (symmetric in this case) fluctuated between 0 and 0.1 before 2008, then it increased and started fluctuating substantially during the GFC phase. We can observe some spikes in late 2009, early 2011 as well as late 2012 when the maximum tail dependence reached a nearly 0.2 mark. Figures 5 and 6 show the results for Norway. Figure 5 plots the dependence parameters θ1 attributed to the first copula in the mixture (Clayton) in the top panel; θ2 attributed to the second copula in the mixture (survival Clayton) in the middle panel; and θ3 measuring the proportion of the first copula in the mixture in the bottom panel. The top and the bottom panels of Figure 6 refer to the lower and the upper tail dependence coefficients, respectively. It is difficult to interpret the results from Figure 5 because the strength of dependence is determined by three parameters and we cannot observe any clear-cut patterns indicating the direction of a change in dependence. However, Figure 6 is more informative. We can observe that initially the lower tail dependence was quite low and then increased dramatically in 2008, which is attributed to the start of the GFC. It remained at a moderate level, fluctuating between 0.10 and 0.25 from 2008 to 2012 (with a dramatic but short-lived drop in 2010). In the second half of 2012 the lower tail dependence started to decrease and remained relatively low until the end of the sample period (December 2015). The upper tail dependence coefficient fluctuated between 0 and 0.15 for most of the periods with an exception of a sharp increase in dependence in 2009-2010 period when it reached its pick of above 0.3. The results for the Australian commodity index and the USD/AUD are demonstrated in Figure 7, presenting the Clayton & survival Clayton copula parameters and Figure 8, presenting the tail dependence parameters. As in the case of Norway, it is generally hard to interpret the dependence using Figure 7 because it is determined by θ1 , θ2 and θ3 . However, we can observe a clear increase in dependence in 2011 when θ2 increased and θ3 remained nearly unchanged. From Figure 8 we can see a dramatic increase in the lower tail dependence in early 2009 which lasted until the end of 2011 (with the maximum value of dependence reaching 0.6). As far as the upper tail 22

dependence is concerned, it experienced a pronounced increase in 2008 and then had a series of dramatic drops and increases in the period 2009-2013.

Figure 9 presents the results for New Zealand. The Gaussian dependence parameter between the returns of the NZ commodity index and the USD/NZD exchange rate, shown in the top panel, demonstrates an interesting pattern: although most of the time dependence is positive, in some periods it becomes negative. For instance, in the beginning of 2015 it falls below -0.1. The maximum positive dependence of around 0.2 is observed in the period 2011-2013. Based on this evidence we can conclude that the dependence between the New Zealand commodity prices and the exchange rate is rather weak and unstable.

Overall, there are similarities in patterns for dependence and the tail dependence across most of the pairs (commodity index, USD value of the currency). In the time-varying setting, there is generally an increase in dependence and the tail dependence during and following the GFC. This finding is also observed in Figures 10 and 11 demonstrating the results of fitting the multidimensional time-varying Clayton & survival Clayton copula to the four exchange rates and the four commodity indices. In this case we can see a clear increase in both, θ1 and θ2 started in 2008, continued into 2010 and remained high until 2012. The pronounced increase in dependence during the GFC and post-GFC periods is likely to be associated with an increase in volatility of returns during this period. As documented in King and Wadhwani (1990), Ramchand and Susmel (1998) and Longin and Solnik (2001), correlations between returns increase during the volatile periods, and thus, an increase in the dependence parameters between the foreign exchange rates and commodity indices during the crisis period is consistent with the increased volatility of the marginals, see Figures 2 and 3. In addition, Diebold and Yilmaz (2012) document an increase in volatility spillovers across different asset markets during the GFC. Silvennoinen and Thorp (2013) also find that correlations between different assets increase dramatically during the periods of financial market turmoil, and higher asset volatility is associated with increased correlations between commodity returns and equity returns. Another possible explanation for the increase in dependence during the crisis is that in periods of great uncertainty about the future economic indicators (which can be important for the exchange rate determination), the current commodity prices may have a larger effect on the exchange rates. 23

5.3

Implications for Investors and Policymakers

Positive dependence structure between most of pairs of commodity indices and exchange rates (USD/currency) is intuitive for commodity-exporting countries: since commodity index increases when the respective currency appreciates, an investor can achieve diversification by constructing his/her portfolio using commodity-related assets and US dollars (or US bonds). Interestingly, in the case of New Zealand the dependence is negligible. This might be attributed to the fact that the New Zealand dollar is more closely linked to the value of the Australian dollar rather than to the value of commodities. This case deserves further investigation in the future work. Given the fact that generally the tail dependence during the calm periods is rather low, it is not possible to utilise both instruments (commodity indices and exchange rates) to protect the investor against extreme price fluctuations. Nevertheless, during the times of uncertainty attributed to financial markets turmoil (e.g. GFC and post – GFC periods), an increase in dependence and tail dependence would suggest that a well diversified portfolio could be constructed by simultaneously holding assets related to commodities and US dollars.

Our results also have some implications for policymakers. The positive dependence between commodity indices and exchange rates may lead to a situation of a two-geared economy when an increase in the commodities prices leads to a boom in the mining industry, and the associated currency appreciation leads to a slump in the tourism, manufacturing and wholesale/retail industries. This is also known as a Dutch disease. In a mirror situation with falling commodity prices and currency depreciation, the mining sector would be in a slump and tourism, manufacturing and wholesale/retail would boom. Given that in the short-run a change in the interest rate would lead to a change in the exchange rate (both nominal and real), monetary policymakers may consider decreasing the policy rate when they observe a large increase in commodity prices, thus, preventing a large appreciation of domestic currency. This would help to maintain competitiveness of the non-commodity trading sector and also help its growth by making investment more attractive due to lower interest rates. However, a decrease in the interest rate would also lead to a further boom in the commodity-exporting sector due to higher revenues in domestic currency and higher investment. This boom may result in higher domestic demand and eventually higher inflation. Given that all considered countries have inflation targeting regimes, the adjustment of 24

Student-t dependence parameter for (USD/CAD,Canada)

0.4 0.3 0.2 0.0

0.1

Dependence parameter

0.5

0.6

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

2013

2015

Time

0.20

Student-t tail tail dependence coefficient for (USD/CAD,Canada)

0.10 0.00

0.05

Tail dependence parameter

0.15

Time-varying parameter Global parameter

2003

2005

2007

2009

2011 Time

Fig. 4. The copula dependence parameter θˆ estimated using a bivariate Student-t copula with Student-t marginals for the pair (USD/CAD, Canada) (top panel) and the corresponding tail dependence coefficient (bottom panel). The time-varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

the policy rate, aimed at retaining competitiveness of the non-commodity trading sector, should be made with great caution. In addition, there is some scope for government actions in the countries that can be subjected to a two-geared economy (due to rising or falling commodity prices and currency values). In such countries there can be relatively large fluctuation in employment across industries and therefore it would be worthwhile for the government to introduce or enhance retraining programs for workers who lose their jobs in a slumping sector. Such programs would be helpful in decreasing unemployment and long-term unemployment. 25

7

Theta 1 from the mixture Clayton & survival Clayton copula for (USD/NOK,Norway)

0

1

2

3

4

5

6

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

10

20

30

Time-varying parameter Global parameter

0

Dependence parameter

40

Theta 2 from the mixture Clayton & survival Clayton copula for (USD/NOK,Norway)

2003

2005

2007

2009

2011

2013

2015

Time

0.2

0.4

0.6

0.8

1.0

Time-varying parameter Global parameter

0.0

Dependence parameter

1.2

Theta 3 from the mixture Clayton & survival Clayton copula for (USD/NOK,Norway)

2003

2005

2007

2009

2011

2013

2015

Time

Fig. 5. The copula dependence parameter θˆ1 (upper panel), θˆ2 (middle panel) and mixture parameter θˆ3 estimated using the Clayton & survival Clayton mixture copula for the pair (USD/NOK, Norway) (top panel) and the corresponding tail dependence coefficient (bottom panel). The time-varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

26

0.25

Low tail dependence from the mixture Clayton & survival Clayton copula for (USD/NOK,Norway)

0.00

0.05

0.10

0.15

0.20

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Upper tail dependence from the mixture Clayton & survival Clayton copula for (USD/NOK,Norway)

0.0

0.1

0.2

0.3

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Fig. 6. The lower tail dependence parameter λl (upper panel) and the upper tail dependence parameter λu (upper panel) estimated using the Clayton & survival Clayton mixture copula for the pair (USD/NOK, Norway). The time-varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

6

Summary and Concluding Remarks

This paper studies dependence between commodity indices for four commodity-exporting countries (Canada, Australia, New Zealand and Norway) and the corresponding exchange rates using static and time-varying copulas. The pairs consisting of log-returns of commodity indices and the corresponding exchange rates against the US dollar demonstrate low (in the case of New Zealand) to moderate (in all other cases) positive dependence indicating that the commodity index increases when the respective currency appreciates. Several explanations are provided for this positive dependence structure. First, an increase in the commodity price index of a certain 27

Theta 1 from the mixture Clayton & survival Clayton copula for (USD/AUD,Australia)

6 4 0

2

Dependence parameter

8

10

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Time

8

Theta 2 from the mixture Clayton & survival Clayton copula for (USD/AUD,Australia)

4 0

2

Dependence parameter

6

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Time

1.0

Theta 3 from the mixture Clayton & survival Clayton copula for (USD/AUD,Australia)

0.6 0.4 0.0

0.2

Dependence parameter

0.8

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Time

Fig. 7. The copula dependence parameter θˆ1 (upper panel), θˆ2 (middle panel) and mixture parameter θˆ3 estimated using the Clayton & survival Clayton mixture copula for the pair (USD/AUD, Australia) (top panel) and the corresponding tail dependence coefficient (bottom panel). The time-varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

28

0.6

Low tail dependence from the mixture Clayton & survival Clayton copula for (USD/AUD,Australia)

0.0

0.1

0.2

0.3

0.4

0.5

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Upper tail dependence from the mixture Clayton & survival Clayton copula for (USD/AUD,Australia)

0.0

0.1

0.2

0.3

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Fig. 8. The lower tail dependence parameter λl (upper panel) and the upper tail dependence parameter λu (upper panel) estimated using the Clayton & survival Clayton mixture copula for the pair (USD/AUD, Australia). The time-varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

country often indicates an increase in demand or expected future demand for commodities of this country, which leads to higher demand for the domestic currency and, as a consequence, its appreciation. Second, higher commodity prices may be associated with an increase in investment in the relevant sectors of the economy and therefore, their growth, so that there may be an increase in demand for assets of commodity-exporting firms. This increase in demand for domestic assets would also lead to an appreciation of the domestic currency. Finally, commodity-exporting countries with strong links to emerging market economies may be considered as safer proxies to these economies. This would also suggest a positive dependence between the currency value of the commodity-exporting country and its commodity index. As far as the tail dependence is 29

Gaussian dependence parameter for (USD/NZD,New Zealand)

0.1 0.0 -0.1

Dependence parameter

0.2

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Time

Fig. 9. The copula dependence parameter θˆ estimated using a bivariate Gaussian copula with Student-t marginals for the pair (USD/NZD, New Zealand). The time-varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

concerned, in the static case it is found to be relatively low and close to symmetric in most of the cases.

There are interesting results arising from analysing the time-varying setting. In the case of almost all bivariate copulas and the multivariate copula an increase in dependence and the tail dependence is observed during and following the GFC. Given that in this period the volatilities of the marginals are high, these results are in line with findings documented in the literature. Another possible explanation of this higher dependence following the crisis is that commodity prices may play a more important role in exchange rate determination of commodity currencies when there is an uncertainty about the future development of the country-specific economic fundamentals. There are interesting results for New Zealand in the time-varying setting where we observe small positive dependence in some periods and small negative dependence in others.

Our results have important implications for investors aiming to diversify their risks in general and during the extreme events. Also there is some scope for official policies that can soften the effect of a two-geared economy. Although monetary policy tools can be helpful to some extent, they should be used very carefully given that all considered countries have inflation targets. At 30

Theta 1 from the mixture Clayton & survival Clayton copula for (USD/CAD, USD/NOK, USD/AUD, USD/NZD, Canada, Norway, Australia, New Zealand)

1.0 0.8 0.6 0.2

0.4

Dependence parameter

1.2

1.4

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Time

1.2

Theta 2 from the mixture Clayton & survival Clayton copula for (USD/CAD, USD/NOK, USD/AUD, USD/NZD, Canada, Norway, Australia, New Zealand)

0.8 0.6 0.2

0.4

Dependence parameter

1.0

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Time

0.8

Theta 3 from the mixture Clayton & survival Clayton copula for (USD/CAD, USD/NOK, USD/AUD, USD/NZD, Canada, Norway, Australia, New Zealand)

0.6 0.5 0.3

0.4

Dependence parameter

0.7

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Time

Fig. 10. The copula dependence parameter θˆ1 (upper panel), θˆ2 (middle panel) and mixture parameter θˆ3 estimated using the Clayton & survival Clayton mixture copula for the pair (USD/CAD, USD/NOK, USD/AUD, USD/NZD, Canada, Norway, Australia, New Zealand) (top panel) and the corresponding tail dependence coefficient (bottom panel). The time-varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

31

Low tail dependence from the mixture Clayton & survival Clayton copula for (USD/CAD, USD/NOK, USD/AUD, USD/NZD, Canada, Norway, Australia, New Zealand)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

0.25

Upper tail dependence from the mixture Clayton & survival Clayton copula for (USD/CAD, USD/NOK, USD/AUD, USD/NZD, Canada, Norway, Australia, New Zealand)

0.00

0.05

0.10

0.15

0.20

Time-varying parameter Global parameter

2003

2005

2007

2009

2011

2013

2015

Fig. 11. The lower tail dependence parameter λl (upper panel) and the upper tail dependence parameter λu (upper panel) estimated using the Clayton & survival Clayton mixture copula for the pair (USD/CAD, USD/NOK, USD/AUD, USD/NZD, Canada, Norway, Australia, New Zealand). The time– varying parameter (solid line) is estimated using a moving window of length 252 corresponding to one year of observations. The global parameter (dashed line) is estimated using the entire sample period from January 2002 to December 2015.

the same time, government retraining programs for workers can be effective in reducing unemployment in countries experiencing large fluctuations in unemployment across the sectors due to large swings in commodity prices and exchange rates.

Although the copula approach that we use in this paper is very informative about contemporaneous dependence between exchange rates and commodity indices, it would also be interesting to explore if there are any causal relationships between the commodity indices and the value of commodity currencies. This opens some possible avenues for future research. One possibility 32

would be to use wavelet analysis as in Tiwari et al. (2013) and Reboredo and Rivera-Castro (2013) to analyse possible lead and lags effects of the commodity indices and exchange rates.

References Abramowitz, M., Stegun, I., 1972. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York. Aloui, R., A¨ıssa, M. S. B., Nguyen, D. K., 2013. Conditional dependence structure between oil prices and exchange rates: A copula-GARCH approach. Journal of International Money and Finance 32, 719–738. Amano, R. A., van Norden, S., February 1995. Terms of trade and real exchange rates: the Canadian evidence. Journal of International Money and Finance 14 (1), 83–104. Andersen, T. G., Bollerslev, T., 1998. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39(4), 885–905. Aussenegg, W., Cech, C., 2009. Simple time-varying copula estimation. In book: Mathematical, Econometrical and Computational Methods in Finance and Insurance, Publisher: Publisher of the Karol Adamiecki University of Economics in Katowice, Editors: Andrzej Stanislaw Barczak, Ewa Dziwok, 9–20. Barndorff-Nielsen, O., 1977. Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society London A 353, 401–419. Barndorff-Nielsen, O., 1994. Normal-inverse Gaussian processes and the modelling of stock returns. Issue 300 of Research Reports - Department of Theoretical Statistics, Institute of Mathematics. University of Aarhus. Barndorff-Nielsen, O., Stelzer, R., 2005. Absolute moments of generalized hyperbolic distributions and approximate scaling of normal nnverse Gaussian Levy-processes. Scandinavian Journal of Statistics 32, 617–637. Benediktsdottir, P., Scotti, C., 2009. Exchange rates dependence: What drives it? Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 969. Breymann, W., Dias, A., Embrechts, P., 2003. Dependence structures for multivariate highfrequency data in finance. Quantitative Finance 3, 1–14. Chan, K., Tse, Y., Williams, M., March 2011. The relationship between commodity prices and 33

currency exchange rates: Evidence from the futures markets. In: Commodity Prices and Markets, East Asia Seminar on Economics, Volume 20. NBER Chapters. National Bureau of Economic Research, Inc, pp. 47–71. Chen, Y.-C., Rogoff, K., May 2003. Commodity currencies. Journal of International Economics 60 (1), 133–160. Chen, Y.-C., Rogoff, K. S., Rossi, B., 2010. Can exchange rates forecast commodity prices? The Quarterly Journal of Economics 125 (3), 1145–1194. Cherubini, U., Luciano, E., Vecchiato, W., 2004. Copula Methods in Finance. Wiley Finance Series. Clements, K. W., Fry, R., 2008. Commodity currencies and currency commodities. Resources Policy 33 (2), 55–73. Dias, A., 2004. Copula inference for finance and insurance. doctoral thesis. Dias, A., Embrechts, P., 2008. Modelling exchange rate dependence at different time horizons. Working paper. Diebold, F. X., Yilmaz, K., 2012. Better to give than to receive: Predictive directional measurement of volatility spillovers. International Journal of Forecasting 28, 57–66. Eberlein, E., Keller, U., 1995. Hyperbolic distributions in finance. Journal of Business 1, 281–299. Fang, H., Fang, K., Kotz, S., 2002. The meta-elliptical distributions with given marginals. Journal of Multivariate Analysis 82(1), 1–16. Fengler, M., Okhrin, O., 2016. Managing risk with a realized copula parameter. Computational Statistics and Data Analysis 100, 131–152. Ferraro, D., Rogoff, K., Rossi, B., June 2015. Can oil prices forecast exchange rates? Journal of International Money and Finance 54, 116–141. Frahm, G., Junker, M., Szimayer, A., 2003. Elliptical copulas: applicability and limitations. Statistics and Probability Letters 63, 275–286. Garcia, R., Contreras, J.and van Akkeren, M., Garcia, J., 2005. Spillover effects in energy futures markets. IEEE Transactions on Power Systems 20, 867–874. Hu, L., 2006. Dependence patterns across financial markets: a mixed copula approach. Applied Financial Economics 16, 717–729. Hu, W., Kercheval, A., 2007. Risk management with generalized hyperbolic distributions. FEA ’07 Proceedings of the Fourth IASTED International Conference on Financial Engineering and Applications, 19–24.

34

Hult, H., Lindskog, F., 2002. Multivariate extremes, aggregation and dependence in elliptical distributions. Advances in Applied Probability 34, 587–608. Hurst, S., Platen, E., 1997. The marginal distributions of returns and volatility. IMS Lecture Notes - Monograph Series. Hayward, CA: Institute of Mathematical Statistics 31, 301–314. Ignatieva, K., Platen, E., 2010. Modelling co-movements and tail dependency in the international stock market via copulae. Asia-Pacific Financial Markets 17, 261–302. Ignatieva, K., Platen, E., Rendek, R., 2011. Using dynamic copulae for modelling dependency in currency denominations of a diversified world stock index. Journal of Statistical Theory and Practice 5(3), 425–452. Ignatieva, K., Trueck, S., 2016. Modeling spot price dependence in australian electricity markets with applications to risk management. Computers and Operations Research 66, 415–433. Issa, R., Lafrance, R., Murray, J., August 2008. The turning black tide: Energy prices and the Canadian dollar. Canadian Journal of Economics 41 (3), 737–759. Joe, H., 1993. Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis 46(2), 262 – 282. Joe, H., 1997. Multivariate models and dependence concepts. Chapman & Hall. Jørgensen, B., 1982. Statistical properties of the generalized inverse Gaussian distribution, Lecture Notes in Statistics. Springer-Verlag. Ketterer, J., 2012. Designing carbon and energy markets to encourage climate change mitigation. PhD thesis, Ludwig-Maximilians-Universitaet Muenchen. Kim, W., 2014. Dependence structure of korean financial markets using copula-garch model. Computational Statistics and Data Analysis 21(5), 445–459. King, M., Wadhwani, S., 1990. Transmission of volatility between stock markets. Review of Financial Studies 3, 3–33. Lin, S., Tamvakis, M., 2001. Spillover effects in energy futures markets. Energy Economics 23, 43–56. Longin, F., Solnik, B., 2001. Extreme correlation of international equity markets. Journal of Finance 56, 649–676. Madan, D., Seneta, E., 1990. The variance gamma model for share market returns. Journal of Business 63, 511–524. McLeish, D. L., Small, C. G., 1988. The theory and applications of statistical inference functions. Lecture Notes in Statistics. Vol. 44. Springer-Verlag.

35

McNeil, A., Frey, R., Embrechts, P., 2005. Quantitative risk management: Concepts, Techniques, and Tools. Princeton Series in Finance. Meese, R., Rogoff, K., 1983a. Empirical exchange rate models of the seventies. Do they fit out of sample? Journal of International Economics 14, 3–24. Meese, R., Rogoff, K., 1983b. The out-of-sample failure of empirical exchange rates: sampling error or misspecification? in J. Frenkel (ed.) Exchange Rates and International Macroeconomics, 67-105, Chicago: NBER and University of Chicago Press. Misiorek, A., Trueck, S., Weron, R., 2006. Point and interval forecasting of spot electricity prices: Linear vs. non-linear time series models. Studies in Nonlinear Dynamics and Econometrics 10(3). Morana, C., 2001. A semiparametric approach to short-term oil price forecasting. Energy Economics 23, 325–338. Nelsen, R., 1998. An introduction to copulas. Springer-Verlag. Patton, A., 2002. Modelling time-varying exchange rate dependence using the conditional copula. Working paper 01-09, University of California San Diego. Patton, A. J., 05 2006. Modelling asymmetric exchange rate dependence. International Economic Review 47 (2), 527–556. Platen, E., Rendek, R., 2008. Empirical evidence on Student-t log-returns of diversified world stock indices. Journal of Statistical Theory and Practice 2, 233–251. Praetz, P. D., 1972. The distribution of share price changes. Journal of Business 45, 49–55. Ramchand, L., Susmel, R., 1998. Volatility and cross correlation across major stock markets. Journal of Empirical Finance 5, 397416. Reboredo, J. C., 2012. Modelling oil price and exchange rate co-movements. Journal of Policy Modeling 34 (3), 419–440. Reboredo, J. C., Rivera-Castro, M. A., 2013. A wavelet decomposition approach to crude oil price and exchange rate dependence. Economic Modelling 32 (C), 42–57. Rossi, B., 2013. Exchange rate predictability. Journal of Economic Literature 51, 1063–1119. Silvennoinen, A., Thorp, S., 2013. Financialization, crisis and commodity correlation dynamics. Journal of International Financial Markets, Institutions and Money 24 (C), 42–65. Swift, R., 2001. Exchange rates and commodity prices: The case of Australian metal exports. Applied Economics 33 (6), 745–753. Tiwari, A. K., Dar, A. B., Bhanja, N., 2013. Oil price and exchange rates: A wavelet based

36

analysis for India. Economic Modelling 31 (C), 414–422. Wang, Y., Wu, C., 2012. Forecasting energy market volatility using GARCH models: Can multivariate models beat univariate models? Energy Economics 34(2). Worthington, A., Kay-Spratley, A., Higgs, H., 2005. Transmission of prices and price volatility in Australian electricity spot markets: a multivariate GARCH analysis. Energy Economics 27(2), 337–350. Wu, C.-C., Chung, H., Chang, Y.-H., 2012. The economic value of co-movement between oil price and exchange rate using copula-based GARCH models. Energy Economics 34 (1), 270–282.

37