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Relationship of Threshold Effect among Gold, Oil, and
Exchange Rate
Chien-Chung Nieh, Chia-Yen Yeh
In this study, we employed smooth transition autoregressive (STAR) models to
investigate the nonlinear dynamic behavior of gold price, oil price, and the exchange
rate of the dollar/euro. The empirical results indicated that these variables exist in a
nonlinear relationship. We considered that the change rate of the dollar/euro is the
threshold value that represents the dynamic LSTAR-type process, whereas the change
rate of oil price is the threshold value that represents the smooth symmetric
ESTAR-type process. Our results provided the indexes of the threshold values, Et 8
> −0.0028 and Ot  7 > 0.2768. The results also presented the significant effect that
exists between the change rate of gold price and the change rate of the dollar/euro
above the two threshold values.
Keyword: gold, oil, exchange rate, threshold effects, STAR
1. Introduction
In recent years, gold and oil have attracted considerable attention from investors,
traders, portfolio managers, and policy makers because of the relatively higher price
than that observed through their historical trend and dynamic correlation. Gold and oil
are being used as hedge by investors and traders to fight the increasing risk after the
financial crisis of 2008. However, using gold and oil as hedge has further increased
gold and oil prices.
Gold plays a crucial role as a store of value and has the feature of risk avoidance,
particularly during political and economic uncertainty. Historical records provide a
reasonable example wherein investors moved funds to invest in the market for gold
and other precious metals to avoid dollar depreciation during political conflicts,
particularly those that occurred in the Middle East.
Sari et al. (2010) stated that oil and precious metals (gold, silver, platinum, and
palladium) are denominated in the US dollar, and thus, the exchange rate of the dollar
may influence both of them simultaneously. The exchange rate of the dollar/euro is
chosen in this paper because these currencies are interchangeably used in investment
portfolios that include precious metals and oil. A gold trader, for example, would be
interested in whether the exchange rate of the dollar/euro in a high volatility regime
would have more influence than that in a low volatility regime. This information is
useful for the trader while pricing gold and determining the right time for investing in
the commodity market.
2. Literature Review
Larry (2008) found that, after the dissolution of the Bretton Woods International
Monetary System, floating exchange rates among the major currencies have resulted
in price instability in the world gold market, and the US dollar bloc have strong
effects on gold price. Hammoudeh et al. (2009) found that the exchange rate of the
US dollar has a prominent role in affecting and being affected by the price of oil and
precious metals. The result of the ARDL model indicated that gold can influence oil
return in the short-term but not in the long-term. Shafiee and Topal (2010) found that
gold and oil prices were highly correlated at approximately 85% and remained stable
over a long-term. Zhang and Wei (2010) indicated a significant positive correlation
coefficient between crude oil and gold prices from January 2000 to March 2008;
therefore, there was a long-term equilibrium. Nevertheless, no significant nonlinear
Granger causality was observed between the two markets. On the contrary,
considering the influence of return, Sari et al. (2010) found that the relationship
between the return of gold and oil prices was weak and asymmetric because gold is
the least volatile commodity, whereas oil is a very volatile commodity.
Many factors affect crude oil price, such as the change in global oil demand and
supply conditions in response to geopolitics, institutional arrangements (such as
OPEC), and the dynamics of the financial markets. The most crucial factor is the
exchange rate of the US dollar, whose depreciation has played a key role in the
soaring oil price. Amano and Norden (1998) found that a stable link exists between
oil price shocks and the US real effective exchange rate during the post–Bretton
Woods period. Chen and Chen (2007) used monthly panel data for the G7 countries
and showed that real oil prices may have been the dominant source of real exchange
rate movements and that there is a cointegrating relationship between real oil prices
and real exchange rates. Zhang et al. (2008) found a significant long-term equilibrium
cointegrating relationship between the exchange rate of the dollar and oil markets.
The crucial reason for the volatility of exchange was the fluctuation in crude oil price,
but not vice versa. Hammoudeh et al. (2009) indicated that when other commodities
and interest rates are present, oil price has a forcing relationship with the exchange
rate of the US dollar in the long-term. Sari et al. (2010) indicated the existence of
comovements and information transmission among the spot prices of the four
precious metals, oil price, and the exchange rate of the US dollar/euro, as well as a
weak long-term equilibrium relationship but strong feedback in the short-term.
Samanta and Zadeh (2012) examined the variables of gold price, stock price, the real
exchange rate of the dollar, and oil price. They found the existence of comovements
by tracking the cointegration, common trend factor, and spiller index.
This paper attempts to study several different aspects. We examined whether there
is a nonlinear relationship among gold price, crude oil price, and the exchange rate.
Then, we used the ESTAR and LSTAR models to determine the threshold relationship
among the three variables.
3. Methodology
We follow the methodology proposed by Teräsvirta (1994) known as the Smooth
Transition Autoregressive (STAR). Teräsvirta (2005) defined the STAR model as
follow:
yt   ' wt   ' wt G( yt  d , h ; , c)   t
(1)
W here wt  (1, yt 1,..., yt  p )' consi st s of an i nt ercept and p lags of yt ,
  (0 ,1,..., p )' and   (0 ,1,..., p )' are parameter vectors, and  t ~ IID (0, 2 ) .
In general, the transition function G ( yt  d , h ;  , c) is a bounded function of yt  d , h ,
continuous everywhere in the parameter space for any value of yt  d , h . The regime
that occurs at t  d is determined by the observable variable yt  d , h and the
associated values of G ( yt  d , h ;  , c) . Different choices for the transition function
G ( yt  d , h ;  , c) generate different types of regime-switching behavior. We consider
two alternative definitions of the transition function, G (..) . First, the logistic function
1
k



G( yt  d , h ;  , c)  1  exp    ( yt  d , h  ck )  ，   0, c1  ...  ck
 k 1


(2)
With d  1 , and where the restrictions on the slope parameter and on the location
parameters c  (c1 ,..., ck )' are identifying restrictions. Eq. (1) and (2) jointly define
the LSTAR model. And second, the exponential function
k


(3)
G( yt  d , h ;  , c)  1  exp    ( yt  d , h  ck )2  ，   0, c1  ...  ck
 k 1

Eq. (1) and (3) jointly define the ESTAR model. Where  is a smoothness parameter
that reflects the transition speed, and c can be interpreted as the threshold between
the two regimes.
Teräsvirta (1994) indicated the estimation of STAR models consists of three
stages：
1. Specification of a linear autoregressive model. We estimate AR models to use an
order selection criterion like AIC (Akaike 1974) or SBC (Rissanen 1978; Schwarz
1978) to select a proper subset of lags. Furthermore, consider the omitted
autocorrelation may cause rejection of the linearity hypothesis, we use the
Ljung-Box statistic (1978) for autocorrelation.
2. Testing linearity for different values of the delay parameter, d. The next step is to
test for the presence of nonlinearities. The null hypothesis of the model is setting
H 0 :   0 in the STAR models. To carry out this test, we estimate the auxiliary
regression as
 t   0   1' wt  1' w( zt d )   2' w( zt d ) 2   3' w( zt d ) 3  t
(4)
Where  t is the residual of the AR model. The linearity test the joint restriction
that all nonlinear terms are zero as
H 0  1'   2'   3'  0
(5)
Under the null hypothesis of linearity, rejecting the null hypothesis that implied
the nonlinear model hypothesis is accepted.
3. Choosing the appropriate model between LSTAR and ESTAR model by the
following sequence of nested tests：
H 02 :  3'  0
(6)
H 03 :  2'  0  3'  0
(7)
H 04 : 1'  0  2'   3'  0
(8)
We would select the LSTAR model if H 02 is rejected. If H 02 is not rejected
but H 03 is rejected, we would accept the ESTAR model. If both H 02 and H 03
are not rejected but H 04 is reject, we would accept the LSTAR model.
4. Empirical Results
We used daily time series data for gold spot prices, oil spot prices, and the
exchange rate of the US dollar/euro. The exchange rate is the value of the US dollar to
one euro, and thus, a rise in this rate implies a devaluation of the dollar. The West
Texas Intermediate crude oil price is chosen to represent the oil spot market and is
expressed in US dollars per barrel. The gold spot is traded at Commodity Exchange in
New York, and its price is measured in US dollars per troy ounce. In this paper, we
used daily data between May 17, 2001 and April 26, 2011, with 2594 price
observations. Descriptive analysis suggests that sample observations are neither
symmetric nor mesokurtic. The J-B test statistics imply that probability distributions
are also not normal.
We plan to use the daily data for several years and employ the appropriate
econometric techniques to identify and detect new information regarding the
relationship among these variables. To avoid spurious regression, empirical research
based on time series data required the underlying time series to be stationary. We used
the linear ADF, NP, and KPSS tests for the unit roots to examine stationarity. Because
gold and oil prices are nonstationary, we adopted difference stationary processes to
resolve the unit root problem in the regression models. The result of the three unit root
tests, ADF, NP, and KPSS, are summarized in Table 1. ADF and NP show the null of
nonstationary. On the contrary, KPSS shows the null of stationary. Therefore, we
concluded that all variables considered in this paper are the I(1) type series.
Table 1. Unit root test results.
DGold
EX
ADF
NP
DOil
Intercept
-49.8164 a (0)
-50.4905a (0)
-49.0678a (0)
Int. and trend
-49.8648 a (0)
-50.4941a (0)
-49.0761a (0)
Intercept
-25.4416 a (0)
Int. and trend
-25.4451 a (0)
KPSS Intercept
Int. and trend
-3.8203a (14)
-25.4354a
(0)
-3.9240a (14)
-25.3828a (0)
0.3565 b (20)
0.1230a (13)
0.1339a (4)
0.0258a (21)
0.0400a (13)
0.0854a (4)
Note: DGold, EX, and DOil are the symbols for the first difference of gold price,
change rate of dollar/euro exchange rate and the first difference of oil price,
respectively.a, b, and c denote significance at 1%, 5%, and 10% levels, respectively.
Before employing the linearity tests, it was necessary to estimate the optimal lags to
determine an appropriate linear AR model. Following Teräsvirta (1994), we
determined the optimal lags to find the AR model using the rule of the AIC/SBC
criterion. As shown in the second column of Table 2, the appropriate AR model was
AR(1) for the change rate of oil price and the change rate of exchange rate of the
dollar/euro. After determining the optimal lags for the linear portion of the AR models,
we estimated the auxiliary regression in Eq. (4).
Table 2. The result of optimal lags and the linearity test .
d (delay)
change rate of oil price
change rate of dollar/euro
exchange rate
F statistic ( p-value)
5.0938(0.00162) b
7.1369(0.00009) b
7
8
Linear or nonlinear
nonlinear
Nonlinear
Note: b represent a rejection of null hypothesis of linear model at the 5% level.
The linearity test in Eq. (5) involves testing the joint hypothesis that all the
coefficients corresponding to Eq. (4) are zero. This test also determines the delay
length, d, which involves running the linearity tests for all plausible values of d and
selecting the value of d that minimizes the test’s p-value. If the delay length allows us
to reject the null hypothesis of linearity, the series are nonlinear; otherwise, they are
linear. Table 2 shows the result of the linearity tests using the F statistic and p-value.
Obviously, the series for these two variables were a dynamic process of nonlinearity.
Because the assumption of linearity is rejected, we determined that the appropriate
nonlinear model is an ESTAR or LSTAR model. To discriminate between the LSTAR
and ESTAR models, we used a sequence of ordinary F tests by Eq. (6), (7), and (8).
The results obtained using the LSTAR and ESTAR models are shown in Table 3. The
variable of the change rate of oil price accepted Eq. (6) and rejected Eq. (7); thus, we
chose the ESTAR model. Conversely, the variable of the change rate of exchange rate
of the dollar/euro rejected Eq. (6); thus, we chose the LSTAR model.
Table 3.
The model specification for the ESTAR Versus the LSTAR models
F statistic
( p-value)
Determination
of STAR
specifications
H 02
H 03
H 04
change rate of oil
price
1.2184
(0.2698)
13.2845
(0.0003)b
0.7736
(0.3792)
ESTAR
change rate of
dollar/euro exchange
rate
12.53949
(0.000405)b
0.1636
(0.6859)
8.6709
(0.0033)
LSTAR
Note: b represent a rejection of null hypothesis of linear model at the 5% level.
After deciding the appropriate nonlinear model, we further considered the
estimated ESTAR and LSTAR models. When the variable of the exchange rate of the
dollar/euro was the transition variable, the LSTAR model is as follows:
Gt  1O Ot  1E Et  1O Ot  1E Et   g LST ( Et  d ; , )
where g LST ( Et d ; , )  1  exp   Et d   
1
The estimates of the LSTAR models are as follows:
1


Gt  0.0911Ot  31.7156 Et   0.0103 Ot  30.5698 Et   1  exp 
.1022  Et 8  0.0028 
993
 (9)
( 5.95)
( 3.07 )
(

0
.
80
)
(

1
.
94
)
( 7.35)
 ( 0.42)
 




We considered that the change rate of the dollar/euro is the threshold value, and the
estimated coefficient of  is significant at the 5% level. The transition is logistic and
the delay is eight quarters. According to our empirical result, an estimated great value
of  (  = −993.1022) would imply a sharp transition from one regime to another. It
also reflected a faster speed in regime switching. According to our result, the change
rate of gold prices would have a positive influence when the threshold value is
below –0.0028. It indicates that the low change rate of the dollar/euro would lead to a
high change rate of gold price. On the contrary, when the threshold value is
above –0.0028, the high change rate of the dollar/euro would lead to a low change
rate of gold price.
The LSTAR model distinguishes the exchange rate market between high and low
volatility regimes. Figure 1 depicts the estimate transition functions of the LSTAR
model. The two regimes would have different dynamics. The change rate of oil price
and the change rate of the dollar/euro will have a statistically positive impact on the
change rate of gold price in Regime I. In Regime II, it would be the sum of the
coefficient of statistically positive impact and the threshold effect. According to Eq.
(9), there is an indifferent relationship between the change rate of oil price and the
change rate of gold price. When the threshold value is below −0.0028, the estimated
coefficient is 0.09, which is similar to that when the threshold value is above −0.0028
and the estimated coefficient is 0.1( 1O  1O ). However, we found an obvious
threshold relationship between the change rate of the dollar/euro and the change rate
of gold price. When the threshold value is below −0.0028, the estimated coefficient is
31.71, but when the threshold value is above −0.0028, the estimated coefficient is
62.29( 1E  1E ). This suggests that the change rate of the dollar/euro is the threshold
value that has more significant effect than the change rate of oil price.
Transition Variable
1.00
Transition Function
0.75
0.50
0.25
0.00
-60
-40
-20
0
20
40
60
80
exchange rate
Figure 1. Diagrams of transition functions of the exchange rate for LSTAR model.
The variable of the change rate of oil price was the transition variable, and
therefore, the ESTAR model is represented as follows:


Gt  1OOt  1E Et  1OOt  1E Et  g EST ( Et  d ;  , )


where g EST ( Et  d ;  , )  1  exp    Ot  d   
2

The estimates of the ESTAR models are represented as follows:
2



Gt  0.1323 Ot  40.42 Et    0.174 Ot  8.2114 Et   1  exp  0.0544  Ot  7  0.2768   (10)
( 9.36)
( 9.16)
( 0.54)
( 0.59) 
 ( 2.84)
 
 (1.38) 

We considered that the change rate of oil price is the threshold value and the transition
function is the exponential. Figure 2 depicts the estimated transition functions of the
models. The figure displays the modulus of the dominant root for Et  d and Ot  d in
the sample. The transition lag is seven, and the model defines three states: the
threshold value is below 0.2768 ( Ot  7 < 0.2768) and above 0.2768 ( Ot  7 > 0.2768).
When the threshold value is below 0.2768, the estimated coefficients are 0.1323 and
40.42, respectively. It indicates that the change rate of gold price is influenced for
both the change rate of oil price and the change rate of the dollar/euro. However,
when the threshold value is above 0.2768, gold price behaves in a different manner
for the change rate of oil price and the change rate of the dollar/euro. When the
threshold value is above 0.2768, the estimated coefficient is −0.0417( 1O  1O ) and
48.6314( 1E  1E ), respectively. The result illustrates that the relationship between
the change rate of the gold price and the change rate of the dollar/euro has a more
significant effect than that between the change rate of oil price and the change rate of
the dollar/euro.
Transition Variable
1.00
Transition Function
0.75
0.50
0.25
0.00
-60
-40
-20
0
20
40
60
80
oil
Figure 2. Diagrams of transition functions for the oil price for ESTAR model.
5. Conclusions
In this paper, we followed the nonlinear procedure proposed by Granger and
Teräsvirta (1993) and Teräsvirta (1994) to examine the time series properties of the
change rates of gold price, oil price, and the exchange rate of the dollar/euro. The
empirical evidence rejects linearity for the series of these variables. We found that the
variables of the change rate of oil price and the change rate of the dollar/euro follow
the ESTAR and LSTAR of nonlinear models, respectively. This indicates that the
change rates of gold price, oil price, and the exchange rate of the dollar/euro exist in a
nonlinear relationship that always provides a better fit than that provided by linear
models.
We considered that the change rate of the dollar/euro is the threshold value that
represents a sharp transition and dynamic LSTAR-type process. Our empirical results
found that the relationship between the change rate of gold price and the change rate
of the dollar/euro has a more significant effect than that between the change rate of oil
price and the change rate of the dollar/euro above the threshold. On the contrary, the
change rate of oil price is the threshold value that represents a smooth symmetric
ESTAR-type process. The relationship between the change rate of gold price and the
change rate of oil price will have a smaller effect above the threshold. However, the
relationship between the change rate of gold price and the change rate of the
dollar/euro has a larger effect. Our results provide the indexes of the threshold values,
Et 8 > −0.0028 and Ot  7 > 0.2768, and represent the significant effect that exists
between the change rate of gold price and the change rate of the dollar/euro.
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