1
Introduction
This paper compares the ability of OTC options and time-series models to forecast future realized correlations in the foreign exchange market. Options data is used as a forward-looking indicator to test whether the market’s “col-lective wisdom” provides any relevant information to the forecasting process when compared to time-series models. To estimate correlations from option prices, implied volatilities from three different option contracts are combined in order to determine correlations between any two currencies.
The estimation of foreign exchange correlations is an important compo-nent of risk management and portfolio allocation for financial institutions. Foreign exchange trading desks at major investment banks that deal with several currency pairs require accurate forecasts of correlations to minimize overall risk exposures. Also, trading desks need to determine which cross-currency hedges are safest to undertake, especially for currencies in which liquidity is an issue.
Asset managers that pursue currency overlay strategies also require ac-curate forecasts of overall risk exposures and specific co-movements between different currency pairs. Their value added comes from determining which currency pairs should be left unhedged and which pairs should be hedged.
Hedge Funds dedicated to carry trading, a strategy that is prone to high profit volatility and occasional heavy losses, need better correlation estimates to diversify their underlying exposures and to better assess the risk contribu-tion of each individual posicontribu-tion to the overall portfolio. The same applies to funds that pursue trend following and macro strategies in which currencies can play a major role as a source of risk and return.
Another area where currency correlation forecasts are playing an increas-ing role is in the management of foreign currency reserves by central banks. As the currency reserves of some countries grow, central bankers are forced to rethink the underlying currency structure in order to diversify the inherent sovereign risk and also to maximize returns.
This paper complements the works of Siegel (1997), Campa and Chang (1998), Lopez and Walter (2000), Brooks and Chong (2001), and Castren and Mazzotta (2005) by including a wider variety of developed market and emerg-ing market currencies, and two additional time-series models in the compari-son that had not been covered in any of these papers. Of all the papers men-tioned above, only Castren and Mazzotta (2005) included two pairs of emerg-ing market currencies in their analysis, the PLN/EUR-USD/EUR and the CZK/EUR-USD/EUR. This paper broadens the study of Implied Correlation in emerging markets, by including a wider variety of currency pairs, namely, the BRL/EUR-USD/EUR, KRW/EUR-USD/EUR, RUB/EUR-USD/EUR
and ZAR/EUR-USD/EUR. The emerging market currencies included in the analysis where chosen because they represent a fairly significant part of Emerging Markets Indexes and also because of availability of historical data for implied volatilities. The two other models under consideration are the Dynamic Conditional Correlation model developed by Engel and Sheppard (2001) and Engel (2002) and the RiskMetrics 2006 methodology.
The DCC model, introduced by Engel and Sheppard (2001) and Engle (2002), is a generalization of Bollerslev’s (1990) constant correlation model with the main difference being that it allows correlations to be time-varying. The main advantage of the DCC model over other multivariate GARCH mod-els is its flexibility and computational efficiency, allowing for the estimation of very large covariance matrices with a fairly small number of parameters. The model is estimated in two steps. In the first step, conditional variances for the different assets are estimated using different univariate GARCH pro-cesses. In the second step, the time varying covariance matrix is estimated using the standardized residuals estimated in the first step.
The RiskMetrics 2006 methodology attempts to correct some of the short-comings inherent in traditional GARCH based volatility models. One of the main critiques attributed to this family of models is that it fails to cor-rectly capture volatility clustering, the influence of past volatility on future volatility. Clustering is measured as the lagged correlation of squared returns plotted against time with the observed decay factor quantifying the memory process. Observations of empirical data have shown that this decay happens very slowly, resembling a logarithmic decay. GARCH based models, however, have a correlation of squared returns that decays exponentially fast. The new methodology called Long-Memory-ARCH is an extension of the original I-GARCH methodology with the effective volatility being obtained as a sum of historical volatilities, with weights that decay logarithmically.
The other methods under consideration are the simple moving average method and the original RiskMetrics 1996 model. The simple moving average method gives equal weight to all observations in the past up to a cutoff point, while the original RiskMetrics 1996 model uses declining weights based on a parameter, λ, which gives more emphasis to current data.
The forecasting accuracy of the correlation models is tested using linear regressions. For individual and combined forecasts, the model or combination of models with the lowest Root Mean Squared Error (RMSE) is considered the best forecast. RMSE is calculated individually for each currency pair studied and then is added for all currency pairs to determine the best method overall, as suggested by Engle (2002).
In a second layer of analysis, individual forecasts are compared using Diebold and Mariano (1995) tests in order to determine if two competing
forecasts are statistically different. Rejecting the null hypothesis of equal forecasting ability implies that one method is superior to the other. For combined estimates, Wald tests are performed to infer if combining implied forecasts with time-series forecasts and vice-versa adds any significant value to the overall process. A rejection of the null hypothesis implies that a specific model adds value to a combined estimate.
The results of this analysis do appear to support the use of Implied Cor-relation especially in developed market corCor-relations, since this model outper-forms all others. When it comes to forecasts of emerging market correlations, time-series models have the upper hand over Implied possibly due to less developed option markets for these currency pairs. The difference in perfor-mance between competing methods is not significant however, especially if the naive or Historical method is excluded from the comparison.
To determine the economic significance of the topic being studied, the performance of cross-currency hedges is tested. The risk incurred in the investment of one unit of a currency is compared to a portfolio consisting of a long position in one unit of that same currency and a short position in USD/EUR multiplied by an estimated beta. If the risk of the portfolio is lower than the risk incurred in the unhedged investment, then the forecasting models are adding value to the investment.
The results of cross-currency hedges vary substantially depending on the currency pair. For some currencies, hedges actually increase risk while for others there is a certain degree of risk reduction. However, even in the best hedge, the reduction in risk is around 20%, which is not substantial.
The remainder of this paper is organized as follows: Section 2 is a brief bibliographical review of other papers studying Implied Correlation as a fore-casting tool, section 3 discusses the data that will be used in the analysis, section 4 gives a definition of realized correlation against which models will be compared, section 5 provides an explanation of the different models tested, section 6 defines the methodology used to evaluate forecasts, section 7 com-pares the results and section 8 concludes.
2
Literature Review
Most of the studies up to this point focus on the ability of implied volatility to predict future realized volatility. The bibliography on Implied Correlation is somewhat scarce, and the results are not always conclusive. However, most of the studies find that there is valuable information contained in option prices not contained in time-series models.
USD-DEM-JPY trios. Implied Correlation data is tested against Historical fore-casts in order to determine which method does a better job at hedging cur-rency risk. Siegel finds that Implied Correlation is not only better at hedging currency risk for both trios tested but it also reacts faster to market changes. Another advantage of Implied over Historical is the lack of sampling bias of the former.
Campa and Chang (1998) gathered OTC data on the USD-DEM-JPY trio. Their study found that forecasts based on Implied Correlation outper-formed both GARCH and RiskMetrics methodologies for the one-month and three-month time horizons. For one-month estimates, Implied Correlation was the only methodology that consistently added value to the forecasting process, either on its own or combined with other methodologies. The op-posite was not always true, of the other methods reviewed, none was able to consistently add value to an Implied Correlation forecast.
Walter and Lopez (2000), included two currency trios in their analysis, the USD-DEM-JPY and the USD-DEM-CHF. According to the authors, Implied Correlation estimates do not incorporate all the information available in the historical data. Also, in most cases, other time-series methodologies, specifi-cally GARCH-based methods, can be very useful in forecasting and provide additional information not contained in options data. They found that the performance of Implied Correlation is not uniform for all the different pairs and it also changes significantly depending on the time period used. Implied Correlation does a better job in forecasting the USD-DEM-JPY trio than the USD-DEM-CHF trio. Also, for both currency trios reviewed, GARCH-based correlations appear to be the most consistent evidencing some forecasting ability. The authors suggest that the poor performance of implied correla-tion can be attributed to either a misspecified opcorrela-tion pricing model (i.e., the Garman-Kohlhagen (1983) model) or to a lack of pricing efficiency in the OTC market.
Regarding the misspecification of the Black-Scholes option pricing model, Scott (1992) finds that the model should contain useful information for the forecasting process since the traditional Black-Scholes option pricing model is a first order approximation of a more complex model that allows volatility to change randomly.
Even though the Black-Scholes model assumes that volatility is fixed for the duration of the contract, empirical studies have shown that the approxi-mation errors for Black-Scholes are small percentages of the correct random variance price and therefore, the Black-Scholes model should provide a rea-sonably accurate approximation of the expected volatility under the risk adjusted volatility process.
for two currency trios; USD-DEM-JPY and USD-DEM-GBP. The number of models tested is quite extensive, including several GARCH-based models, an Exponentially Weighted Moving Average model (EWMA) as well as Implied Correlation. The best model according to the authors is the simple EWMA which outperforms more complex GARCH models and Implied Correlation. Even tough Implied Correlation does a good job at estimating volatility and correlations, it fails to accurately forecast hedge ratios.
Castren and Mazzota (2005) include a wider variety of currencies in their study including two emerging market currency trios, the EUR-USD-PLN and EUR-USD-CZK. They conclude that Implied Correlation shows predic-tive power for all currency pairs examined except for one. Regarding the emerging market pairs, Implied Correlation was the top performing model for the USD-EUR-CZK trio and the second best performing model for the USD-EUR-PLZ trio. However, for all the currency pairs, GARCH based methods and RiskMetrics also show substantial predictive power. According to the authors, the best forecasts are provided by encompassing regressions in which Implied estimates are combined with GARCH and RiskMetrics es-timates.
This paper extends the previous studies done on this matter by increasing the number of currencies under analysis and dividing them into two distinct groups of developed and emerging market currencies. The other distinguish-ing feature is the introduction of two new models that had not been previously analyzed in any of these papers, the DCC model and RiskMetrics 2006.
3
Data
All correlations are calculated between the USD/EUR and Foreign
Cur-rency/EUR1 . The reason for using the USD is that it is the world’s most
liquid currency and therefore its option prices should contain the most ac-curate information about market expectations. Another consideration is the fact that for several emerging market currencies, trading in other currency pairs is done via the USD.
Spot volatility prices for all currency pairs are based on at-the-money con-tracts in which the strike price equals the forward rate, both for 1-month and 3-month options. The implied volatility data for developed market currencies (USD, GBP, CHF and JPY) is derived from Bloomberg, using Bloomberg’s generic volatility quotes (BGN). These quotes are consensus estimates of the current traded level of implied volatility for FX options. The estimates are
1USD/EUR and Foreign Currency/EUR denotes the amount of USD and Foreign
based on implied volatility quotes from several banks and brokers and are combined to produce a smoother, more accurate estimate of the current pre-vailing market. For emerging market currencies (BRL, KRW, MXN, RUB and ZAR) and for the AUD, volatility quotes are provided by Citibank.
The historical spot exchange rate data comprise daily closing prices dis-tributed by Bloomberg (5PM New York Time) for the following currency pairs trading against the EUR: AUD, BRL, CHF, GBP, JPY, KRW, MXN, RUB, USD and ZAR. The dataset starts June 1, 2001 and ends December 31, 2009 yielding 2240 observations in total for each pair. Daily returns are then constructed as the first difference of logarithmic prices.
Basic statistics for currency returns data are presented in Table 1 and for Implied Volatilities in table 2. It is noticeable from the skewness value that all currency pairs appear to have asymmetric distributions. The kurto-sis statistic is also far higher than what should be expected from a normal distribution, implying that currency returns have higher peaks and higher mass concentrated in the tails. Once skewness and kurtosis are combined to create the Jarque-Bera statistic, the null hypothesis of normality is rejected at the 5% level for all currency returns.
To test for independence between squared returns, the Ljung-Box auto-correlation test statistic was calculated with 25 lags. When the test was performed for currency returns, the null hypothesis of randomness in the data is rejected at the 5% level for all currency pairs suggesting that there is in fact volatility clustering.
4
Realized Correlation
The performance of the different models will be tested against the realized correlation verified in the specific time period under analysis. For 1-month forecasts, correlations will be compared with the realized correlation observed over the last 22 days of trading data (i.e., 22-day realized correlation between
t = T−22 and T ) and for 3-month forecasts, the last 66 days of trading data
(i.e., 66-day realized correlation between t = T−66 and T ). The correlations
are calculated between the price of one unit of the domestic currency D (which in this case is the EUR) for a foreign currency F (all the currencies listed in section 3, except for the USD) and a price to exchange one unit of the domestic currency D for a second foreign currency G (in this case the USD). The formula is shown below:
ρ∗(D/F,D/G),t,T =
!T
i=1(rD/F,i− ¯rD/F,t,T)(rD/G,i− ¯rD/G,t,T)
5
Correlation Models
5.1
Historical Correlation Forecast
The historical correlation forecast between t−h and t is defined as the realized
correlation observed over a number of days prior to t. All the observations in the calculation have equal weight. The 1-month forecast equals the realized correlation observed over the previous 22 trading days (i.e., historical forecast
for the next 22 days, t equals the realized correlation observed between t−22
and t). For 3-month forecasts, the same principle applies, the historical forecast is set as the realized correlation over the previous 66 trading days (i.e., historical forecast for the next 66 days, t, equals the realized correlation
observed between t−66 and t). Again, the correlations are calculated between
the price of one unit of the domestic currency D (which in this case is the EUR) for a foreign currency F (all the currencies listed in section 3, except for the USD) and a price to exchange one unit of the domestic currency D for a secon foreign currency G (in this case the USD). The formula follows below:
ˆ
ρH(D/F,D/G),t,T = ρ∗(D/F,D/G),t−h,t
5.2
Implied Correlation
The methodology used below was first illustrated by Siegel (1997). Implied Correlation is calculated from implied volatilities on foreign exchange op-tions. The model used is the Garman-Kohlhagen (1983) pricing model which modifies the traditional Black Scholes model to account for foreign interest rates. In addition to the model, it is also assumed that the triangular par-ity condition between cross rates also holds, i.e., that there are no arbitrage opportunities in the foreign exchange options market.
To calculate Implied Correlation, three different option prices are needed: a price to exchange the domestic currency D (which in this case is the EUR) for a foreign currency F (all the currencies listed in section 3, except for the USD); a price to exchange the domestic currency D for a third foreign currency G (in this case the USD); and finally, to exchange F for G.
The Garman-Kohlhagen gives the price of a European-style call option to exchange the domestic currency (D) for a foreign currency (F) as
C = (XD/F, K, τ, rD, rF, σD/F)
= XD/Fe−rF τN(ζ)− Ke−rDτN(ζ− σD/F
√ τ )
with ζ = ln[(XD/Fe
−rF τ)/(Ke−rDτ) + σ2
D/Fτ /2
σD/F√τ
where XD/F is the spot price of the foreign currency mentioned above, K
is the exercise price, τ is the time to maturity of the option contract, rD
is the domestic interest rate, rF is the foreign interest rate, and σ2
D/F is the instantaneous variance of the rate of return. To estimate a value for volatility, different values for σ are plugged until C equals the market price of the option.
The model assumes that the exchange rate XD/F follows the Itˆo process:
dXD/F
X1 = (rD − rF)dt + σD/FdZ1
where ZD/F is assumed to be a standard Wiener process.
Similar Itˆo processes hold for the other two exchange rates, XD/G and
XF/G. By substituting the identity for the cross-option XF/G = XD/G/XD/F
into de Itˆo process
dXF/G
XF/G
= (rF − rG)dt + σF/GdZF/G
it is possible to obtain an expression for the correlation dXF/G
XF/G = (rF − rG)dt +
"
σ2
D/F + σD/G2 − 2σD/FσD/GρD/F,D/GdZF/G
where ρD/F,D/G is the correlation between the currency pairs. Therefore, the
volatility σF/G is related to the volatilities of the ordinary options and the
correlation between the exchange rate changes:
σF/G =
"
σ2
D/F + σD/G2 − 2σD/FσD/GρD/F,D/G
Solving the previous equation for ρD/F,D/G, shows how the three volatilites imply a value for the exchange rate correlation:
ˆ
ρI(D/F,D/G),t,T = σ 2
D/F + σ2D/G− σF/G2
2σD/FσD/G
To hedge the risk of an investment in currency F by creating a short posi-tion in currency G, an equaposi-tion can also be derived from the three volatilites:
β(F,G)= ρD/F,D/G σD/F σD/G = σ2 D/F + σD/G2 − σF/G2 2σ2 D/G
The formula above is used in section 7 to calculate the effectiveness of the cross-currency hedges.
5.3
Dynamic Conditional Correlation
5.3.1 Model
The DCC model, described in great detail in Engle and Sheppard (2001) and Engle (2002), is a multivariate GARCH model that assumes that a
portfolio containing n assets with returns rt = (r1t, r2t, ..., rnt)# is
condition-ally multivariate normal with zero expected value and covariance matrix
Ht= E[rtr#
t| ψt−1], where rt can be defined as
rt= Ht1/2(t
and rt is conditional on the information set ψt−1
rt| ψt−1 ∼ N(0, Ht)
where (t = ((1t, (2t, ..., (nt)# ∼ N(0, In), In is the identity matrix of order n
and
Ht≡ DtRtDt
where Dt is the n× n diagonal matrix of time varying standard deviations
from univariate GARCH models with √hit denoting the ith element on the
diagonal of D. These univariate GARCH models can include other predeter-mined or exonegeous variables and can include any GARCH processes with
normally distributed errors. Rt in the equation above is the time-varying
correlation matrix.
Once the estimates of√hitare obtained, return residuals are transformed
by their estimated standard deviations calculated in the first stage, i.e., µit=
(it/
√
hit. In the second stage, µit is used to estimate the parameters of the
conditional correlation. The covariance is given by the following formula:
Qt= (1− M # m=1 αm− N # n=1 βn) ¯Q + M # m=1 αm((t−m(#t−m) + N # n=1 βnQt−n
where Qt = (qijt) is the n× n time-varying covariance matrix of µt, ¯Q =
E[µtµ#t] is the n× n unconditional covariance matrix of µt, and α and β
are nonnegative scalar parameters that satisfy the condition α + β < 1. To
obtain a proper correlation matrix Rt, Qt must be scaled by:
Rt= Q∗−1t QtQ∗−1t
Q∗t is composed by the diagonal elements of Qt.
Q∗t = √q 11 0 0 · · · 0 0 √q22 0 · · · 0 ... ... ... ... 0 0 0 · · · √qkk
The elements of Rt will have the form ρijt = qijt
√q
iiqjj.
5.3.2 Estimation of the Model
The DCC model is estimated in two steps. In the first step, a series of univariate GARCH models are estimated and in the second step, the residuals obtained in the first step are transformed by their standard deviation to estimate the dynamic correlation parameters. The likelihood function used in the first step replaces Ik, a k sized identity matrix, by Rt. The parameters
of the model, θ, are divided into 2 groups (φ1, φ2, ..., φk, ) and (ψ) in which
the elements φi correspond to the GARCH parameters for series i, φi =
(ω, α1i, ..., αPii, β1i, ..., βQii). This results in the first stage likelihood function
shown below: QL1(φ | rt) = −1 2 T # t=1
(klog(2π) + log(| Ik|) + 2log(| Dt |) + r#tD−1t IkDt−1rt)
= −1 2 T # t=1 (klog(2π) + 2log(| Dt |) + r# tD−2t rt) = −1 2 T # t=1 [(klog(2π) + k # t=1 log(hit+ r 2 it hit)] = −1 2 k # t=1 T log(2π) + T # t=1 [log(hit+ r 2 it hit)]
The previous function can be described as a sum of the log-likelihood functions of the individual GARCH models. In a second stage, to estimate correlations the log-likelihood function must be conditioned in function of the parameters calculated in the first step:
QL2(ψ | ˆφ, rt) = −1
2 T
#
t=1
(klog(2π) + 2log| Dt | +log(| Rt|) + r#tDt−1R−1t Dt−1rt)
= −1
2 T
#
t=1
(klog(2π) + 2log| Dt | +log(| Rt|) + (#tRt−1(t)
Because of the conditioning constraints imposed above, the first two terms are irrelevant in the estimation process, therefore, the previous equation can be rewritten as follows: QL∗2(ψ | ˆφ, rt) =− 1 2 T # t=1 (log(| Rt |) + (#tR−1t (t)
5.3.3 Multi-Step Ahead Forecasting
Contrary to most GARCH models, the evolution process for the DCC hap-pens in a non-linear way, such that
Qt+r = (1− α − β) ¯Q + α((t+r−1(#t+r−1) + βQt+r−1
If it is assumed that ¯R = ¯Q and E[Rt+i | ψt]≈ E[Qt+i| ψt] for i = 1, ..., k,
a similar approach to GARCH(1,1) can be used to derive the formula for
E[Rt+i | ψt], since E[(t+r−1(#t+r−1 | ψt] = E[Rt+r−1 | ψt]. So, to forecast k
steps ahead, the formula from Engel and Sheppard (2001) is used: Et[Rt+k] =
k−2
#
i=0
(1− α − β) ¯R(α + β)i+ (α + β)k−1Rt+1
The formula above suggests that the long run forecast will eventually converge to the unconditional correlation. The time it takes to converge depends on the ratio (α + β), the closer it is to 1, the higher its persistence, and therefore a slower decay towards its long term average.
5.4
RiskMetrics (1996)
RiskMetrics uses an exponential moving average model of historical obser-vations that attributes higher weights to the latest obserobser-vations. The main advantage of this model is that volatility reacts faster to shocks in the mar-ket as recent data carry more weight than data in the distant past. Also, following a shock (a large return), the volatility declines exponentially fast as the weight of the shock observation falls. The main disadvantage of this model is that without a fixed termination point in the past there is no way of telling when data becomes irrelevant to the forecasting process.
This model depends on the decay factor λ (0 < λ< 1). This parameter determines the relative weights that are applied to the observations and the effective amount of data used in estimating correlation. The lower the weight of λ, the faster the rate of decay.
Since the data used is based on daily returns, λ will be fixed at 0.94 as suggested by the RiskMetrics Technical Document (1996) and the forecast for h-day covariance becomes
ˆ ρRM 96
5.5
RiskMetrics (2006)
The new methodology introduced in RiskMetrics (2006) incorporates some of the properties observed in financial time series, in particular volatility clustering and fat tails. One of the main goals of the new methodology was to maintain the simplicity of RiskMetrics (1996) by limiting the number of parameters used and also setting universal values for the parameters that could be used for all assets.
RiskMetrics (2006) attempts to deal with volatility clustering by mea-suring and modelling the lagged correlation of volatility. The decay of the lagged volatility is an important component in measuring the memory of the process. By being able to measure clustering, the new methodology attempts to quantify the impact of past volatility on future volatility. Through em-pirical observations, it has been determined that the lagged correlation of volatility tends to decay very slowly resembling a logarithmic function.
The new methodology, called Long-Memory-ARCH (LM-ARCH) attempts to capture the long memory of volatility by using a multi-scale extension of the original RiskMetrics (1996) model, the I-GARCH model. Historical volatility is measured as a series of exponential moving averages under dif-ferent time horizons. The historical volatilities gathered are then summed to obtain an effective volatility using weights that decay logarithmically. In this process, since there is no mean volatility parameter, the components with the longest time horizons are used to smooth the evolution of the model.
As mentioned above, the new methodology is an extension of RiskMetrics
(1996) in which a series of historical volatilities σkare measured on geometric
time horizons τk where δt equals 1 day. In this specific case, it is applied to
covariances:
τk = τ1ωk−1 k = 1, ..., kmax
µk = exp(−δt/τk)
ˆ
ρRM 06D/F,D/G,t,h = µkρˆRM 06D/F,D/G,t−1+ (1− µk)(rD/F,t−1− ¯rD/F)(rD/G,t−1− ¯rD/G) In the next step, to calculate the effective covariance, a new set of weights with logarithmic decay is introduced that adjusts the forecast for the long memory behavior observed in financial data:
ˆ ρRM 06ef f,t = k#max k=1 wkρˆRM 06D/F,D/G,k wk = 1 C * 1− ln(τk) ln(τ0) +
The constant C introduced above enforces the constraint that weights
must equal 1, !kwk = 1. The model has three main parameters; the log
decay factor τ0 with a value estimate in the range of 3 to 6 years, the lower
cut-off point τ1 which should be in the order of 1 to a few days, and the upper
cut-off kmax with a range of a few months to a few years. It is worth pointing
out, that for τ1, by assuming a low cut-off point, the estimator will be most influenced by the most recent returns, resulting in a noisier estimator. The
parameter ω is set at √2 as in RiskMetrics (2006).
According to Zumbach (2009), the appropriate values for the parameters
are the following: τ0 = 1560 days; τ1 = 4 days and kmax = 15 days.
5.6
Adjusted RiskMetrics (2006)
Even tough parameters are set to be universal for all time horizons, a slightly different version of the model is also tested. Since the purpose of this paper is to study correlation forecasting methods at 1-month and 3-month time
horizons, the lower cut-off point, τ1 is increased to 12 days.
This change should reduce the noise generated by using fewer observations and seek information content from a longer time horizon. This version of the model is referred to as Adjusted RiskMetrics 2006.
Several values for τ1 were tested on a trial and error basis, starting with
6 days and going up to 15 days. As the value of τ1 kept increasing, the Root
Mean Squared Errors (RMSE) in linear regressions kept getting smaller. As the number of days was increased to 15, RMSE started increasing again, so
the decision was to fix τ1 at 12 days.
Therefore, the original RiskMetrics 2006 methodology is tested with the
following parameters: τ0 = 1560; τ1 = 12 and kmax = 15.
6
Forecast Evaluation Methodology
To test the forecasting ability of each model, the following different methods are used:
6.1
Linear Regressions
The regressions are done in-sample for both 1-month and 3-month correlation estimates. The first step is to perform simple linear regressions, in which realized correlation, the independent variable is regressed against the different correlation forecasts, the dependent variables. The formula for ordinary least
squares is presented below:
ρ∗D/F,D/G,t = α + β ˆρfD/F,D/G,t+ (t
where ˆρfD/F,D/G,t stands for one of the correlation forecasts previously
de-scribed.
Also, bivariate regressions are run in order to determine if time-series methods add any value to the forecasting ability of Implied Correlation and vice-versa. The formula used is:
ρ∗D/F,D/G,t = α + βf 1ρˆD/F,D/G,tf 1 + βf 2ρˆf 2D/F,D/G,t+ (t,f 1,f 2
where βf 1ρˆf 1D/F,D/G,t and βf 2ρˆf 2D/F,D/G,t stand for two of the correlation
fore-casts previously described.
To generate a consistent estimate of the OLS covariance matrix, regres-sions are run using the Newey-West (1987) methodology to correct for het-eroskedasticity and serial correlation in the data.
6.2
Root Mean Squared Forecast Error (RMSE)
RMSE is used to quantify the difference between the estimated value of correlation and the true value of the ex-post realized correlation. The lower the RMSE, the better the estimator predicts the true value of the correlation. RMSE is evaluated on a currency by currency basis, and will also be added across all currencies to determine the best overall method.
RMSE = , -.1 n n # n=1 (ˆρ∗ i,t − ρfi,t)2
6.3
Diebold-Mariano Test
The test introduced by Diebold and Mariano (1995) compares the forecasting accuracy of two competing models. The null hypothesis is that the two different methods have equal forecasting capabilities. If the null hypothesis is rejected, then one of the forecasting methods is statistically superior to
the other. The test involves the derivation of a loss function dt defined as
dt= E[g((i,t)− g((j,t)] = 0
where (i,t and (j,t are the residuals of the two competing methods, i and j,
respectively, when performing h-step forecasts ahead. The formula for the DM statistic is
DM = d¯
ˆ
In this equation, ¯d represents the mean of the loss function mentioned above
and ˆV ( ¯d) is the standard error of d. The standard error is calculated as
ˆ V ( ¯d) = 1 n[ˆλ0+ 2 h−1# k=1 ˆ λk] where ˆ λk = 1 n n # t=k+1 (dt− ¯d)(dt−k− ¯d)
If the DM value is statistically significant at a 5% confidence level, then one of the models is superior to the other.
6.4
Wald Test
Wald tests are performed to test the significance of each coefficient in a multi-ple linear regression model. The test uses restricted and unrestricted models to determine if the parameters associated with a group of explanatory vari-ables are zero. If for a particular explanatory variable, or group of varivari-ables, the Wald test is significantly different from zero, then the parameters associ-ated with these variables should be included in the model. If the Wald test is not signicant then these explanatory variables can be omitted from the model. The formula to determine the Wald statistic is the following:
W = (R 2− R2 restricted) (1− R2)/N ∼ χ 2 where R2
restricted represents the model containing only one of the correlation
forecasts (implied or other) and R2represents the model containing combined
forecasts (implied with other). The asymptotic distribution is χ2, with the
number of restrictions imposed equaling the degrees-of-freedom.
7
Results
As previously stated, the forecasting accuracy of the correlation models was tested using linear regressions. The models were tested individually and in combinations of Implied Correlation with other methods. To generate a consistent OLS covariance matrix, regressions were run using the Newey-West (1987) methodology to correct for heteroskedasticity and serial correlation in the data. The number of lags applied to the Newey-West OLS covariance matrix was 22 for 1-month forecasts, as in Castren and Mazzotta (2005), and 66 for 3-month forecasts. For Diebold-Mariano tests, the number of lags was 21 for 1-month forecasts and 65 for 3-month forecasts.
7.1
Individual Forecasts
The results in tables 3 (1-month forecasts) and 4 (3-month forecasts) show predictive ability at both time horizons for all models tested. The perfor-mance improves for 3-month forecasts, possibly due to mean reversion in correlations.
Regression coefficients are all significantly different from 0 at the 1% level, except 3-month forecasts for the CHF/EUR-USD/EUR pair. For this specific currency pair, both Historical forecasts and RiskMetrics 1996 fail to be significant at the 5% level, with only Implied and the DCC being significant at the 1% level.
The performance of the models varies depending on whether the currency pairs are developed market or emerging market pairs. Implied Correlation has the lowest overall RMSE for developed market currency pairs at both time horizons while for emerging market currency pairs, Adjusted RiskMet-rics 2006 provides the best overall forecasts.
The difference between the best model and the remaining top models is not substantial. For 1-month developed market correlations, the DCC model and Adjusted RiskMetrics 2006 also provide fairly good forecasts, while for emerging market correlations adjusted RiskMetrics 2006 barely outperforms RiskMetrics 2006.
For 3-month forecasts, there is a wider dispersion of results for developed market correlations with Implied having a more comfortable advantage over the second best model Adjusted RiskMetrics 2006. For emerging market correlations, the dispersion in results is now much smaller when compared to 1-month forecasts, with Adjusted RiskMetrics 2006 slightly outperforming RiskMetrics 2006.
Implied Correlation provides the worst forecasts for emerging market cur-rency pairs at both time horizons, but its performance improves slightly for 3-month estimates. The only exception is MXN/EUR-USD/EUR, where at both time horizons, Implied has the lowest RMSE.
The naive Historical model is the worst forecaster for developed market correlations and second worst for emerging market correlations. It is also worth pointing out that both RiskMetrics 2006 methodologies tested outper-form the original RiskMetrics 1996 methodology.
These results are also supported by Diebold-Mariano tests shown in table 5. At the 1-month time horizon, Implied is a superior forecaster to all other models for the JPY/EUR-USD/EUR pair with a 99% confidence level. For the GBP/EUR-USD/EUR pair, Implied is also a better forecaster than all other models with a 99% confidence level, except when compared to the DCC model. For the DCC model, the test is not conclusive about which model is
superior.
Despite providing one of the worst forecasts for CHF/EUR-USD/EUR, the test cannot determine if there are any superior forecasting methods to Implied. For the AUD/EUR-USD/EUR, Implied is better than the Histor-ical method with a 95% confidence level, but the null hypothesis of equal forecasting accuracy cannot be rejected for the remaining models.
At the 3-month horizon, Implied forecasts are statistically better with a 99% confidence level than all other models for GBP/EUR-USD/EUR. For all other pairs, according to Diebold-Mariano tests, the models appear to provide equal forecasts.
The picture changes somewhat for emerging market currencies. At the 1-month horizon, Implied is inferior to all time series models for RUB/EUR-USD/EUR at a 1% significance level. For KRW/EUR-RUB/EUR-USD/EUR, the null hypothesis of equal forecasting ability is also rejected with a 99% confidence level for all models, except when compared to Historical forecasts. The test is not conclusive about this specific comparison.
As previously stated, the only correlation in which Implied outperforms is MXN/EUR-USD/EUR, where it is a superior forecaster to Historical with a 99% confidence level. The test is not conclusive for the other two pairs, BRL/EUR-USD/EUR and ZAR/EUR-USD/EUR.
The performance of Implied improves for 3-month forecasts, despite hav-ing the highest RMSE. It is still a statistically worse forecaster for the KRW/EUR-USD/EUR than time series models with a 99% confidence level, however, it is now a better forecaster for the MXN/EUR-USD/EUR than most of the other models. It is significant at the 1% level versus the His-torical model and at the 5% level versus all RiskMetrics models. The test fails to provide significant information for the other three currency pairs, BRL/EUR-USD/EUR, RUB/EUR-USD/EUR and ZAR/EUR-USD/EUR.
7.2
Combined Forecasts
In a second stage, combinations of Implied Correlation with time-series mod-els were tested in order to determine if there is any improvement in forecast-ing accuracy compared to individual estimates and if so which combinations work best. At first glance, there is a benefit of combining Implied Correla-tion with time-series methods since overall RMSE tends to decrease, at both time horizons. The results for multiple regressions are shown in table 6 for 1-month forecasts and table 7 for 3-month forecasts.
The results for combinations are somewhat better but it seems that they are dominated by the top performing model. Again, as in the simple linear regressions, the difference between the best combination and the worst, as
measured by RMSE, is fairly small. At both time horizons, the best com-binations are Implied with the DCC for developed market correlations and Implied with Adjusted RiskMetrics 2006 for emerging market correlations.
When Implied is regressed with the DCC, it dominates the regressions. For 1-month forecasts, its coefficients are always significant at the 1% confi-dence level while DCC coefficients fail to be significant for the GBP/EUR-USD/EUR and JPY/EUR-GBP/EUR-USD/EUR. The exception to this rule is the CHF/EUR-USD/EUR pair in which the DCC is significant at the 1% confi-dence level, while Implied is not significant.
The same happens for 3-month forecasts, Implied is always significant except for the CHF/EUR-USD/EUR pair. The DCC also fails to be signif-icant for this specific currency pair, even though this combination provides the best forecast.
None of the regression coefficients for time series models are significant for GBP/EUR-USD/EUR and JPY/EUR-USD/EUR at both time horizons. Implied is amongst the worst individual forecasters for CHF/EUR-USD/EUR, and its coefficients in combinations for this pair are only significant when it is combined with Historical, significant at the 1% level, and with Adjusted RiskMetrics 2006, significant at the 5% level. For AUD/EUR-USD/EUR, all coefficients (time series models and Implied) are significant with a 99% confidence level.
For emerging market correlations, Adjusted RiskMetrics 2006 dominates the regressions with Implied. At the 1-month horizon, it is always significant at the 1% level while Implied is significant for the MXN/EUR-USD/EUR and RUB/EUR-USD/EUR at the 1% level and at 5% for the BRL/EUR-USD/EUR. Despite being a statistically worse individual forecast according to the Diebold-Mariano test for the RUB/EUR-USD/EUR, its coefficient is significant when combined.
For 3-month forecasts, it is now the only significant coefficient (5% level) for BRL/EUR-USD/EUR and maintains its significance for the other two pairs, MXN/EUR-USD/EUR and RUB/EUR-USD/EUR. The coefficient for Adjusted RiskMetrics 2006 is significant at the 1% level for the KRW/EUR-USD/EUR, RUB/EUR-USD/EUR and ZAR/EUR-KRW/EUR-USD/EUR, and it fails to be significant for the other two currency pairs.
Regression coefficients for Implied are always significant for MXN/EUR-USD/EUR and KRW/EUR-MXN/EUR-USD/EUR at both time horizons. Time-series coefficients are never significant in combinations for MXN/EUR-USD/EUR and are always significant for KRW/EUR-USD/EUR. The results vary for the other currency correlations.
The Wald tests performed and shown in table 8 confirm most of the results discussed so far and the fact that all models show some predictive ability
for most currencies. The combination of models adds value to individual forecasts either when Implied is added to a time-series forecast or vice-versa. For developed market correlations, adding a time-series forecast to an Im-plied forecast will result in better results for the AUD/EUR-USD/EUR and CHF/EUR-USD/EUR at both time horizons. For the GBP/EUR-USD/EUR pair, it appears that adding an Historical, Adjusted RiskMetrics 2006 and RiskMetrics 1996 forecast does not improve the results from an Implied 1-month estimate and the DCC and RiskMetrics 2006 are significant at the 5% level.
For 3-month estimates of GBP/EUR-USD/EUR, combinations offer the most value. As for the JPY/EUR-USD/EUR, the only time-series model that adds value to 1-month Implied forecasts is RiskMetrics 2006 at a 5% significance level. All other models fail to add value for 1-month forecasts and for 3-month estimates, none of the time-series models adds any value.
For emerging market combinations, adding time-series forecasts to Im-plied forecasts always results in better estimates, except for Adjusted Risk-Metrics 2006 in 3-month forecasts for the MXN/EUR-USD/EUR pair. For this specific combinaton, there is no value added to the forecast.
Adding Implied to time-series forecasts always results in better estimates for developed market correlations regardless of the time horizon examined. For emerging market correlations, and despite its poor performance in in-dividual forecasts, Implied improves time-series estimates for all currencies with the exception of KRW/EUR-USD/EUR. At the 1-month horizon, it fails to provide any value to Adjusted RiskMetrics 2006 and is significant at the 5% level when combined with the DCC, but for 3-month forecasts, it only adds value to an Adjusted RiskMetrics 2006 forecast.
7.3
Risk Reduction Through Cross-Currency Hedges
As a test for forecasting accuracy, portfolios combining two different positions in currencies were created. The purpose of this exercise is to determine if it makes economic sense to reduce the risk of an investment in a certain currency pair by taking a corresponding short position in USD/EUR. The portfolios are static, i.e., the positions created during a certain day are maintained for a full month (1-month correlation forecasts) and for 3 months (3-month correlation forecasts). The results are shown in table 9.
The results of this experiment were somewhat disappointing despite the fairly good performance of correlation forecasts. Of all the portfolios cre-ated, four for developed market currency pairs and five for emerging market currency pairs, only five in total result in a reduction in risk for 1-month port-folios and four for 3-month portport-folios. For all other cases, the hedges actually
increase the risk of the position. In general, the models provide fairly similar results with the exception of the Historical method, which underperforms all others.
The reductions in risk are not significant either, with the best performance being 1-month portfolios for RUB/EUR-USD/EUR. It is also worth pointing out that the reductions in risk achieved are better for 1-month portfolios despite the fact that correlation estimates improves for 3-month forecasts. It appears that static hedges work best for shorter time periods and that for longer time horizons hedges must necessarily be adjusted dynamically to optimize results.
For 1-month developed market correlations, the DCC provides the best results, followed by Implied. The DCC is the only model that reduces the risk of an investment in CHF/EUR, although only negligibly, and it also provides the best results for investments in GBP/EUR and JPY/EUR. The most successful of the hedges is achieved in JPY/EUR, in which having a USD/EUR short position reduces the risk of the investment by 18% for the DCC and 16% for Implied. With regards to the AUD/EUR, Implied is the only hedge that does not increase the risk of the position, although it does not reduce it either.
Implied provides the best hedges for 1-month emerging market invest-ments of all the models tested. For KRW/EUR, it is the only model that reduces the risk of the position by around 5%, while all others increase it. The only currency pair that does not benefit from a hedge is ZAR/EUR, regard-less of the model used. The best results of the entire exercise are obtained with two emerging market currency pairs, the MXN/EUR and RUB/EUR. For both of these currency pairs, Implied provides the best reductions of risk, 20% and 22%, respectively followed closely by the DCC and Adjusted Risk-Metrics 2006. Implied also provides the best hedge for the other currency pair BRL/EUR, but the result is a more modest, with a reduction of around 6%.
The results of the hedges are worse for 3-month than for 1-month port-folios. For developed market investments, the hedges are only successful for the JPY/EUR, failing to decrease risk for all other currency pairs. The best hedge is provided by Implied, which achieves a reduction in risk of around 17% while the next best two are the DCC and Adjusted RiskMetrics 2006, which achieve reductions of 13,5% and 13,2% respectively.
For emerging market currency pairs, reductions are also smaller than for 1-month portfolios. There are now two currency pairs for which the hedges actually increase risk, KRW/EUR and ZAR/EUR. For the BRL/EUR, Im-plied reduces risk by around 5% while adjusted RiskMetrics 2006 and the DCC reduce by 1,5%. Again the best results are achieved for MXN/EUR
and RUB/EUR, although the decrease in risk is now in the range of 11-14%. The DCC provides the best result for the MXN/EUR followed very closely by Implied, 14,6% and 14,1%, respectively. Implied takes the top spot for the RUB/EUR with 11,9% with Adjusted RiskMetrics 2006 coming in second with 11,6%.
8
Conclusion
This paper compares the performance of Implied Correlation and relevant time-series models in forecasting correlations for a series of developed and emerging market currency pairs. The data used covers ten different currency pairs for nine correlation forecasts. A total of 2240 daily observations are used covering the period of June 1, 2001 to December 31, 2009.
The currency correlations are divided into two distinct groups, one com-posed of developed market correlations and the other comcom-posed of emerging market correlations. Emerging market currency pairs are chosen depending on their weight on emerging market equity indexes and also on the existence of an historical time series of implied volatility data.
The main evaluation benchmark for the performance of the different mod-els is the calculation of root mean squared errors. For individual forecasts, Diebold-Mariano tests are conducted to assess if the individual forecasting accuracy of Implied Correlation versus time-series models is statistically su-perior.
Combinations of Implied with time-series models are also estimated and evaluated according to root mean squared errors. In addition, Wald tests are performed in order to determine if the inclusion of one additional model to an individual forecast adds any significant value to the forecasting process.
All models tested in the analysis perform better at the 3-month horizon than at the 1-month horizon, possibly due to mean reversion in correlations. Also, the difference in forecasting accuracy between the relevant models is not substantial, especially if the Historical or naive method is excluded. It could be that market expectations of volatilities and correlations are based on estimates provided by time series models, which then works as a deterrent for these two different measures to be substantially out of line with each other. For developed market correlations, the best method at both time hori-zons is Implied. From the analysis, it does seem that currency managers focusing exclusively on G10 currencies could benefit from using covariances determined by market expectations to evaluate portfolio risk.
Implied fairs significantly worse for emerging market correlations, with the exception of the MXN/EUR-USD/EUR correlation. Overall it was the
worst model for emerging markets, even worse than Historical forecasts re-gardless of the time horizon. The model introduced in this paper, Adjusted RiskMetrics 2006 provides the best forecasts for this specific set of currencies although the DCC and RiskMetrics 2006 also provide good forecasts.
The relative underperformance of Implied in emerging market correla-tions is worse for 1-month forecasts than for 3-month forecasts. This might happen due to less liquidity in 1-month option prices for emerging market currency pairs. The improvement in performance for 3-month forecasts could be because this market segment is more liquid and therefore contains more information about expectations.
There is also an improvement in using the new RiskMetrics 2006 method-ology that incorporates long-term dependence in estimates. Both RiskMet-rics 2006 methodologies tested are superior to the original RiskMetRiskMet-rics 1996 methodology at both time horizons. Also, and even though, the goal of RiskMetrics is to have a set of parameters that can be used for all assets and at all time-horizons, it seems that for long-term estimates, increasing
the number of days of the τ1 parameter to account for longer-term horizons
and reduce short-term volatility will result in a more accurate forecast, at least for currency correlations.
The combination of individual time-series models with Implied Correla-tion results in better predicCorrela-tions at both time-horizons, suggesting that there is information in both methodologies that is relevant to the forecasting pro-cess. For developed market currency pairs, the best combination is with the DCC model while for emerging market currency pairs it is with Adjusted RiskMetrics 2006.
The results from hedging investments in the analyzed currencies with a short position in USD/EUR are somewhat disappointing. For several cases the hedges increase the risk of the underlying positions, while for others in which risk reduction does occur, it is fairly small. The currency investments that benefit from a short position in USD/EUR are for the most part man-aged floats, i.e., currencies in which the central bank monitors carefully the value of the local currency versus the USD.
The best results in risk reduction are achieved for 1-month portfolios, even though the best correlation forecasts are for 3-month time horizons, and are for the JPY/EUR, MXN/EUR and RUB/EUR. It is also worth pointing out that for emerging market hedges, Implied provides the best results at the 1-month horizon even tough it is the worst method at predicting correla-tions. Also, for developed market hedges, the DCC provides the best hedges for 1-month portfolios, even though it is not the top performing correlation forecaster. For 3-month developed market portfolios, Implied does regain the top spot followed by the DCC model.
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T able 1: T im e S e rie s D a ta -B a si c S ta ti st ic s E U R A U D E U R B R L E U R C H F E U R G B P E U R J P Y E U R K R W E U R MX N E U R R U B E U R Z A R E U R U S D M ea n 0 ,0 0 % 0 ,0 1 % 0 ,0 0 % 0 ,0 2 % 0 ,0 1 % 0 ,0 2 % 0 ,0 4 % 0 ,0 3 % 0 ,0 2 % 0 ,0 2 % M ed ia n -0 ,0 3 % -0 ,0 2 % 0 ,0 1 % 0 ,0 0 % 0 ,0 5 % 0 ,0 1 % 0 ,0 2 % 0 ,0 1 % -0 ,0 2 % 0 ,0 2 % M a x im u m 6 ,5 6 % 6 ,3 6 % 3 ,3 0 % 3 ,1 5 % 7 ,0 0 % 9 ,8 6 % 6 ,6 5 % 5 ,5 8 % 1 4 ,2 7 % 3 ,4 7 % M in im u m -6 ,9 6 % -9 ,3 4 % -2 ,0 3 % -3 ,1 3 % -5 ,6 6 % -1 3 ,6 0 % -9 ,2 4 % -5 ,8 7 % -1 3 ,9 7 % -2 ,4 3 % S td . D ev 0 ,7 4 % 1 ,2 3 % 0 ,3 1 % 0 ,5 0 % 0 ,7 5 % 0 ,9 4 % 0 ,8 5 % 0 ,6 0 % 1 ,2 2 % 0 ,6 3 % T es t fo r n o rm a li ty S k ew n es s 0 ,4 6 9 0 ,0 5 0 0 ,3 8 5 0 ,2 0 2 -0 ,1 4 5 -0 ,7 8 3 -0 ,0 5 9 0 ,1 5 8 0 ,5 8 9 0 ,0 5 4 K u rt o si s 1 2 ,4 0 0 4 ,6 7 4 1 3 ,3 4 0 4 ,3 8 3 9 ,1 7 7 2 9 ,0 4 6 1 0 ,2 5 3 1 2 ,6 7 4 1 8 ,9 1 5 1 ,6 6 4 J a rq u e-B er a 1 4 .3 6 3 ,8 3 8 2 .0 2 8 ,9 3 9 1 6 .5 8 4 ,3 7 0 1 .7 9 8 ,2 2 0 7 .8 2 9 ,2 5 9 7 8 .6 0 5 ,5 0 3 9 .7 6 3 ,4 3 6 1 4 .9 2 9 ,0 2 8 3 3 .3 6 5 ,1 6 9 2 5 7 ,4 8 0 P ro b a b il it y 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 0 ,0 0 1 T es t fo r a u to co rr el a ti o n o f sq u a re d re tu rn s L ju n g -B ox (2 5 ) 5 6 ,7 3 0 5 5 ,3 3 8 1 0 5 ,4 5 6 5 8 ,6 5 4 6 7 ,4 5 6 1 5 0 ,3 9 3 5 1 ,1 1 3 8 4 ,7 4 1 5 3 ,0 4 1 3 0 ,6 5 8 P ro b a b il it y 0 ,0 0 0 0 ,0 0 0 0 ,0 0 0 0 ,0 0 0 0 ,0 0 0 0 ,0 0 0 0 ,0 0 2 0 ,0 0 0 0 ,0 0 1 0 ,2 0 1
T able 2: Basic S ta tist ics -Implied V olat ilit ies E U R A U D E U R B R L E U R C H F E U R G B P E U R J P Y E U R K R W E U R MX N E U R R U B E U R U S D E U R Z A R 1 -M o n th M ea n 1 0 ,4 7 1 8 ,5 3 4 ,4 4 7 ,4 2 1 1 ,0 3 1 1 ,9 6 1 3 ,7 2 9 ,2 2 1 0 ,0 5 1 7 ,3 4 M ed ia n 1 0 ,7 5 1 6 ,2 5 3 ,8 0 6 ,5 5 9 ,3 8 1 0 ,5 0 1 2 ,0 0 1 0 ,0 0 9 ,5 3 1 6 ,7 0 M a x im u m 3 6 ,3 6 6 5 ,7 5 1 9 ,3 3 2 3 ,6 2 4 8 ,4 2 7 1 ,6 9 7 5 ,0 0 3 6 ,0 0 2 8 ,8 9 6 6 ,2 5 M in im u m 5 ,0 7 7 ,1 0 2 ,3 5 3 ,5 0 4 ,6 5 4 ,8 0 7 ,4 5 3 ,9 0 4 ,6 5 9 ,2 5 S td . D ev 3 ,7 4 8 ,3 7 2 ,1 6 3 ,4 0 5 ,2 7 8 ,4 5 6 ,7 4 3 ,7 5 3 ,3 3 6 ,0 3 3 -M o n th M ea n 1 0 ,6 8 1 8 ,5 3 4 ,3 3 7 ,5 1 1 0 ,8 9 1 1 ,5 2 1 3 ,8 7 9 ,9 7 1 0 ,2 2 1 6 ,7 0 M ed ia n 1 1 ,3 8 1 6 ,6 0 3 ,9 5 6 ,8 0 9 ,4 5 1 0 ,5 0 1 3 ,0 0 1 0 ,0 0 9 ,8 8 1 6 ,2 5 M a x im u m 2 8 ,0 3 5 0 ,6 0 1 4 ,5 5 2 1 ,8 6 3 3 ,9 6 5 5 ,5 4 4 8 ,5 0 3 6 ,0 0 2 4 ,7 0 4 6 ,5 0 M in im u m 5 ,4 2 8 ,9 0 2 ,4 8 3 ,6 5 5 ,2 0 5 ,0 0 7 ,7 8 4 ,0 5 5 ,0 0 1 0 ,0 0 S td . D ev 3 ,1 7 7 ,1 9 1 ,7 1 3 ,1 5 4 ,5 0 6 ,5 7 5 ,6 0 4 ,3 0 3 ,0 4 4 ,5 2 U S D A U D U S D B R L U S D C H F U S D G B P U S D J P Y U S D K R W U S D M X N U S D R U B U S D Z A R 1 -M o n th M ea n 1 2 ,0 3 1 7 ,1 2 1 0 ,5 0 9 ,2 8 1 0 ,5 9 1 0 ,2 4 1 0 ,4 9 6 ,8 0 1 8 ,6 8 M ed ia n 1 0 ,6 5 1 5 ,1 5 1 0 ,3 4 8 ,2 3 9 ,6 3 7 ,4 5 8 ,2 5 5 ,3 0 1 7 ,7 5 M a x im u m 4 4 ,5 3 6 7 ,5 0 2 5 ,0 0 2 9 ,6 2 3 8 ,4 2 7 5 ,0 0 7 5 ,0 0 3 9 ,0 0 6 7 ,0 0 M in im u m 5 ,9 3 6 ,0 0 5 ,2 0 4 ,4 5 5 ,8 5 3 ,4 5 4 ,8 5 2 ,5 0 8 ,1 0 3 -M o n th M ea n 1 1 ,9 0 1 7 ,0 9 1 0 ,6 0 9 ,3 7 1 0 ,3 0 9 ,8 4 1 0 ,5 5 7 ,9 2 1 8 ,0 3 M ed ia n 1 0 ,8 0 1 5 ,1 5 1 0 ,5 5 8 ,4 0 9 ,5 5 7 ,8 0 8 ,7 5 6 ,4 5 1 7 ,5 0 M a x im u m 3 3 ,9 4 5 0 ,0 0 2 3 ,8 6 2 4 ,9 0 2 7 ,3 9 5 8 ,0 0 4 8 ,5 0 3 9 ,0 0 4 7 ,0 0 M in im u m 6 ,3 5 7 ,9 5 5 ,5 0 4 ,9 0 6 ,1 0 3 ,6 0 5 ,0 5 3 ,9 5 9 ,0 0 S td . D ev 4 ,3 7 7 ,0 7 2 ,4 0 3 ,3 3 2 ,9 0 7 ,7 2 5 ,8 7 4 ,8 8 4 ,6 0
T able 3: Linear Regressions -1 M on th F orecasts D e v e lo p e d Ma r k e t C u r r e n c ie s E m e r g in g Ma r k e t C u r r e n c ie s A U D C H F G B P J P Y T o ta l B R L K R W MX N R U B Z A R T o ta l R M S E 0 .2 1 9 8 0 .3 0 1 9 0 .1 7 6 9 0 .2 0 3 6 0 .9 0 2 2 0 .2 6 9 9 0 .1 8 0 9 0 .1 9 5 1 0 .1 8 8 3 0 .3 1 8 5 1 .1 5 2 7 Im p li ed C o effi ci en t 0 .9 9 1 0 .5 0 1 0 .9 9 5 0 .7 9 4 0 .6 1 8 0 .3 3 2 0 .9 6 9 0 .8 3 1 0 .5 6 3 t-st a t 1 2 .7 3 0 * 3 .9 8 5 * 1 1 .2 1 6 * 6 .4 4 3 * 5 .0 4 4 * 4 .9 0 2 * 8 .2 8 3 * 6 .0 3 2 * 4 .4 7 8 * R M S E 0 .2 2 6 2 0 .3 0 0 5 0 .1 8 7 9 0 .2 1 1 8 0 .9 2 6 4 0 .2 5 1 2 0 .1 5 3 8 0 .2 1 2 0 0 .1 7 9 2 0 .3 0 1 2 1 .0 9 7 4 DCC C o effi ci en t 0 .9 9 9 1 .4 4 1 0 .9 7 5 1 .0 0 2 0 .7 8 4 0 .8 0 3 0 .7 4 0 0 .8 0 8 0 .7 1 9 t-st a t 1 1 .2 4 0 * 4 .9 0 1 * 7 .7 3 8 * 4 .9 3 8 * 9 .0 4 6 * 1 0 .0 6 3 * 7 .8 8 5 * 8 .3 8 4 * 7 .2 3 0 * R M S E 0 .2 4 6 9 0 .3 0 6 1 0 .1 9 9 5 0 .2 1 6 4 0 .9 6 8 9 0 .2 6 1 6 0 .1 6 5 9 0 .2 1 9 8 0 .1 8 3 7 0 .3 0 9 9 1 .1 4 1 0 H is to ri ca l C o effi ci en t 0 .5 8 8 0 .2 7 2 0 .4 3 7 0 .2 5 0 0 .5 2 1 0 .5 1 3 0 .5 0 3 0 .7 0 6 0 .4 0 0 t-st a t 9 .9 7 5 * 4 .1 2 4 * 7 .0 8 8 * 3 .6 9 5 * 8 .2 6 7 * 8 .5 0 6 * 6 .3 4 3 * 7 .7 4 5 * 5 .7 6 2 * R M S E 0 .2 3 2 8 0 .2 9 9 9 0 .1 9 7 7 0 .2 1 1 1 0 .9 4 1 5 0 .2 5 4 3 0 .1 5 5 0 0 .2 0 7 7 0 .1 6 6 0 0 .3 0 2 6 1 .0 8 5 6 R M 0 6 C o effi ci en t 0 .7 8 3 0 .4 5 4 0 .5 7 3 0 .4 5 5 0 .7 6 1 0 .6 4 8 0 .7 2 0 0 .8 8 0 0 .6 5 1 t-st a t 1 1 .7 0 3 * 5 .0 0 7 * 7 .2 4 5 * 5 .2 0 7 * 8 .6 8 9 * 1 1 .3 2 8 * 7 .8 3 3 * 8 .6 2 9 * 7 .0 7 3 * R M S E 0 .2 2 6 7 0 .3 0 1 1 0 .1 9 0 9 0 .2 1 0 1 0 .9 2 8 8 0 .2 5 7 6 0 .1 5 0 5 0 .2 0 8 3 0 .1 6 5 3 0 .3 0 1 8 1 .0 8 3 4 R M 0 6 12 C o effi ci en t 0 .9 2 8 0 .5 4 5 0 .7 9 0 0 .5 9 7 0 .9 0 6 0 .7 4 6 0 .8 0 3 0 .9 7 3 0 .8 2 7 t-st a t 1 0 .3 6 2 * 4 .3 8 2 * 7 .3 0 8 * 5 .2 7 6 * 7 .3 3 7 * 1 0 .4 1 7 * 6 .6 4 7 * 7 .7 9 1 * 6 .5 4 8 * R M S E 0 .2 3 2 1 0 .3 0 3 1 0 .1 9 4 9 0 .2 1 2 2 0 .9 4 2 3 0 .2 5 2 1 0 .1 5 6 2 0 .2 1 1 1 0 .1 7 6 0 0 .3 0 1 2 1 .0 9 6 7 R M 9 6 C o effi ci en t 0 .7 0 3 0 .3 6 4 0 .5 3 6 0 .3 7 5 0 .6 3 3 0 .6 1 8 0 .6 2 4 0 .7 6 0 0 .5 4 5 t-st a t 1 1 .7 9 3 * 4 .4 1 4 * 7 .6 4 4 * 4 .7 3 7 * 9 .0 3 4 * 1 0 .8 8 8 * 7 .6 3 1 * 7 .7 4 4 * 7 .0 8 3 * ∗ S ig n ifi ca n t a t th e 1 % le v el ∗∗ S ig n ifi ca n t a t th e 5 % le v el
T able 4: Linear Regressions -3 M on th F orecasts D e v e lo p e d Ma r k e t C u r r e n c ie s E m e r g in g Ma r k e t C u r r e n c ie s A U D C H F G B P J P Y T o ta l B R L K R W MX N R U B Z A R T o ta l R M S E 0 .1 8 6 3 0 .2 1 8 2 0 .1 2 2 1 0 .1 3 9 8 0 .6 6 6 4 0 .2 2 6 1 0 .1 5 9 0 0 .1 7 9 6 0 .1 5 0 9 0 .2 2 9 3 0 .9 4 5 0 Im p li ed C o effi ci en t 1 .0 4 0 0 .3 7 3 1 .0 5 4 0 .8 3 7 0 .6 5 2 0 .3 3 5 0 .8 9 0 1 .0 5 3 0 .5 4 5 t-st a t 1 0 .5 6 8 * 2 .3 7 7 * 9 .8 2 2 * 5 .4 9 1 * 3 .8 8 9 * 4 .4 4 1 * 4 .4 8 6 * 7 .9 3 0 * 3 .4 4 1 * R M S E 0 .1 9 0 3 0 .2 2 0 3 0 .1 4 2 7 0 .1 5 4 7 0 .7 0 8 0 0 .2 2 2 6 0 .1 2 5 2 0 .1 9 1 2 0 .1 7 7 9 0 .2 2 2 0 0 .9 3 8 9 DCC C o effi ci en t 0 .9 2 1 0 .5 0 8 0 .8 8 0 0 .6 6 4 0 .5 4 1 0 .7 8 5 0 .5 8 2 0 .7 3 8 0 .5 4 4 t-st a t 8 .8 8 6 * 2 .1 9 3 * 5 .6 5 0 * 2 .8 6 4 * 4 .5 7 5 * 9 .1 7 0 * 5 .6 7 0 * 6 .7 6 6 * 4 .4 8 2 * R M S E 0 .2 0 2 3 0 .2 2 2 1 0 .1 4 5 6 0 .1 5 0 5 0 .7 2 0 5 0 .2 3 4 8 0 .1 2 4 8 0 .1 9 7 0 0 .1 5 5 1 0 .2 2 4 9 0 .9 3 6 6 H is to ri ca l C o effi ci en t 0 .6 7 2 0 .2 1 7 0 .5 6 7 0 .3 3 5 0 .3 7 9 0 .6 4 6 0 .4 7 0 0 .8 4 1 0 .4 1 1 t-st a t 7 .6 5 8 * 1 .6 3 4 5 .3 3 8 * 3 .4 0 1 * 3 .1 5 8 * 8 .4 9 3 * 4 .0 3 9 * 1 0 .3 7 7 * 3 .1 0 7 * R M S E 0 .1 9 7 3 0 .2 2 0 6 0 .1 4 9 4 0 .1 5 0 2 0 .7 1 7 5 0 .2 2 5 3 0 .1 2 7 0 0 .1 8 9 2 0 .1 5 5 4 0 .2 2 4 0 0 .9 2 0 8 R M 0 6 C o effi ci en t 0 .7 1 9 0 .2 4 2 0 .5 5 1 0 .3 4 2 0 .5 1 3 0 .6 3 1 0 .5 5 5 0 .8 5 4 0 .4 7 6 t-st a t 8 .9 1 2 * 2 .1 5 4 * * 5 .5 5 4 * 3 .9 1 5 * 4 .2 9 3 * 1 0 .3 0 4 * 4 .4 2 4 * 8 .0 1 3 * 4 .0 6 9 * R M S E 0 .1 9 1 3 0 .2 1 9 5 0 .1 4 3 8 0 .1 4 8 6 0 .7 0 3 3 0 .2 2 8 0 0 .1 2 3 8 0 .1 8 9 1 0 .1 4 5 9 0 .2 2 5 0 0 .9 1 1 7 R M 0 6 12 C o effi ci en t 0 .8 5 1 0 .3 2 0 0 .7 3 3 0 .4 6 3 0 .5 9 8 0 .7 1 4 0 .6 2 3 0 .9 8 2 0 .5 8 9 t-st a t 8 .4 4 9 * 2 .1 3 4 * * 5 .4 7 6 * 3 .9 5 4 * 3 .7 0 9 * 9 .4 4 2 * 3 .8 0 8 * 8 .7 2 5 * 3 .5 1 9 * R M S E 0 .1 9 8 3 0 .2 2 2 9 0 .1 4 7 8 0 .1 5 0 4 0 .7 1 9 4 0 .2 2 4 0 0 .1 2 7 7 0 .1 9 1 5 0 .1 6 6 9 0 .2 2 3 2 0 .9 3 3 4 R M 9 6 C o effi ci en t 0 .6 3 8 0 .1 7 7 0 .5 0 1 0 .2 9 1 0 .4 2 8 0 .6 0 6 0 .4 7 9 0 .7 3 1 0 .3 9 8 t-st a t 9 .2 1 3 * 1 .6 8 0 5 .8 8 3 * 3 .6 1 1 * 4 .4 2 0 * 9 .5 9 8 * 5 .0 1 6 * 7 .9 8 7 * 4 .1 2 2 * ∗ S ig n ifi ca n t a t th e 1 % le v el ∗∗ S ig n ifi ca n t a t th e 5 % le v el
T able 5: D ie b o ld M a ri a n o T es t -S ig n ifi c a n c e m e a su re d a g a in st Im p li e d fo re c a st s 1 Mo n th F o r e c a st s 3 M o n th F o r e c a st s D C C H ist o r ic a l R M 0 6 R M 0 6 1 2 R M 9 6 D C C H ist o r ic a l R M 0 6 R M 0 6 1 2 R M 9 6 D M C u r r e n c ie s E U R A U D -0 .4 7 4 -2 .1 3 8 * * -0 .9 8 7 -0 .4 0 2 -0 .8 7 8 -0 .1 8 4 -1 .1 6 1 -0 .8 7 4 -0 .3 3 8 -0 .6 3 6 E U R C H F 0 .3 8 8 -0 .2 8 6 0 .7 9 2 0 .3 7 7 -0 .1 2 7 -0 .7 1 6 -0 .8 3 3 -0 .6 5 5 -0 .3 2 2 -1 .1 5 4 E U R G B P -1 .7 4 3 -3 .5 7 3 * -3 .7 6 5 * -2 .6 0 8 * -3 .4 4 5 * -2 .5 6 0 * -2 .4 1 2 * -3 .3 7 5 * -2 .8 4 9 * -3 .0 5 7 * E U R J P Y -2 .5 0 1 * -3 .2 6 3 * -2 .8 8 8 * -2 .5 5 9 * -2 .7 2 9 * -1 .0 2 6 -0 .9 7 3 -1 .0 7 2 -0 .9 4 5 -0 .9 6 9 E M C u r r e n c ie s E U R B R L 1 .4 3 7 0 .6 5 4 1 .1 3 2 0 .7 2 4 1 .2 5 4 0 .1 3 1 -0 .8 9 8 -0 .1 8 7 -0 .4 3 0 -0 .0 1 3 E U R K R W 3 .1 3 2 * 1 .9 4 1 3 .3 2 8 * 3 .6 4 4 * 2 .8 9 9 * 2 .3 4 7 * 2 .2 3 2 * 2 .3 7 5 * 2 .4 8 5 * 2 .1 9 0 * E U R M X N -1 .8 6 7 -2 .9 1 7 * -1 .3 4 9 -1 .4 8 0 -1 .7 3 8 -1 .9 4 6 -2 .3 2 4 * -2 .0 7 9 * * -1 .9 5 7 -2 .0 6 9 * * E U R R U B 3 .2 6 3 * 2 .7 3 2 * 4 .2 0 4 * 3 .8 0 1 * 3 .4 9 3 * 0 .2 3 7 1 .2 4 6 1 .1 9 1 1 .5 2 0 0 .6 9 0 E U R ZA R 1 .4 3 5 0 .6 1 8 1 .5 2 7 1 .7 8 2 1 .5 2 8 0 .5 8 4 0 .1 3 9 0 .5 6 9 0 .5 5 9 0 .6 7 7 ∗ S ig n ifi ca n t a t th e 1 % le v el ∗∗ S ig n ifi ca n t a t th e 5 % le v el
T able 6: Mult iple Regressions -1 M on th F orecast s D e v e lo p e d Ma r k e t C u r r e n c ie s E m e r g in g Ma r k e t C u r r e n c ie s A U D C H F G B P J P Y T o ta l B R L K R W MX N R U B Z A R T o ta l R M S E 0 .2 1 2 4 0 .2 9 7 1 0 .1 7 6 7 0 .2 0 3 6 0 .8 8 9 8 0 .2 4 7 9 0 .1 5 3 6 0 .1 9 4 2 0 .1 7 0 1 0 .2 9 8 3 1 .0 6 4 1 Im p li ed C o effi ci en t 0 .6 2 4 0 .2 8 5 0 .8 9 2 0 .7 6 1 0 .2 3 0 0 .0 4 5 0 .8 2 7 0 .4 1 2 0 .2 3 5 t-st a t 5 .2 2 9 * 1 .9 2 9 5 .9 2 0 * 4 .9 8 3 * 1 .9 2 8 0 .6 5 7 5 .7 0 7 * 2 .7 4 9 * 1 .9 2 6 DCC C o effi ci en t 0 .4 7 3 0 .9 4 0 0 .1 3 5 0 .0 7 2 0 .6 3 0 0 .7 6 9 0 .1 6 0 0 .5 4 0 0 .6 0 2 t-st a t 3 .6 3 1 * 2 .7 2 3 * 0 .6 4 9 0 .2 9 3 5 .8 6 6 * 8 .5 5 6 * 1 .4 8 1 5 .9 0 8 * 5 .3 1 7 * R M S E 0 .2 1 6 4 0 .3 0 0 2 0 .1 7 6 9 0 .2 0 3 5 0 .8 9 6 9 0 .2 5 4 7 0 .1 6 2 9 0 .1 9 4 9 0 .1 7 1 4 0 .3 0 4 8 1 .0 8 8 7 Im p li ed C o effi ci en t 0 .8 0 4 0 .3 7 5 1 .0 0 0 0 .8 4 6 0 .3 2 5 0 .1 6 6 0 .9 0 8 0 .4 6 2 0 .3 1 1 t-st a t 7 .4 9 8 * 2 .3 5 0 * 9 .8 9 9 * 6 .0 1 4 * 2 .6 1 0 * 2 .3 7 8 * 6 .6 4 3 * 3 .0 3 3 * 2 .5 5 8 * H is to ri ca l C o effi ci en t 0 .1 8 4 0 .1 2 7 -0 .0 0 4 -0 .0 4 0 0 .3 7 1 0 .4 4 1 0 .0 5 5 0 .4 4 8 0 .3 0 8 t-st a t 2 .5 1 4 * 1 .5 8 8 -0 .0 6 9 -0 .5 8 8 5 .2 3 8 * 6 .3 1 9 * 0 .6 2 7 5 .3 7 9 * 4 .1 4 2 * R M S E 0 .2 1 4 8 0 .2 9 7 9 0 .1 7 6 7 0 .2 0 3 4 0 .8 9 2 8 0 .2 5 0 0 0 .1 5 4 8 0 .1 9 4 2 0 .1 6 1 1 0 .2 9 9 5 1 .0 5 9 5 Im p li ed C o effi ci en t 0 .7 0 2 0 .2 4 8 1 .0 8 0 0 .7 1 4 0 .2 5 9 0 .0 5 4 0 .7 9 9 0 .3 0 1 0 .2 4 2 t-st a t 6 .1 6 6 * 1 .5 9 6 9 .3 1 1 * 4 .6 9 9 * 2 .1 7 0 * * 0 .8 4 8 5 .3 5 2 * 2 .4 1 9 * 1 .9 2 1 R M 0 6 C o effi ci en t 0 .3 0 8 0 .3 0 0 -0 .0 8 3 0 .0 7 8 0 .5 9 0 0 .6 1 5 0 .1 6 9 0 .6 7 7 0 .5 3 9 t-st a t 3 .2 3 5 * 2 .7 9 9 * -0 .8 6 7 0 .7 5 6 5 .5 4 5 * 9 .1 5 9 * 1 .5 8 6 8 .1 0 3 * 5 .1 9 3 * R M S E 0 .2 1 3 7 0 .2 9 9 8 0 .1 7 6 9 0 .2 0 3 4 0 .8 9 3 9 0 .2 5 3 7 0 .1 5 0 5 0 .1 9 4 8 0 .1 6 0 4 0 .2 9 9 6 1 .0 5 9 0 Im p li ed C o effi ci en t 0 .6 3 5 0 .3 3 2 1 .0 2 2 0 .7 1 4 0 .2 5 8 0 .0 0 6 0 .8 5 6 0 .2 9 8 0 .2 0 8 t-st a t 4 .8 8 4 * 1 .9 8 5 * * 6 .4 3 8 * 4 .0 4 3 * 2 .0 0 1 * * 0 .0 8 5 5 .5 0 3 * 2 .4 1 1 * 1 .4 9 0 R M 0 6 12 C o effi ci en t 0 .4 1 8 0 .2 4 1 -0 .0 3 0 0 .0 9 4 0 .6 8 5 0 .7 4 2 0 .1 2 2 0 .7 5 2 0 .6 9 7 t-st a t 3 .0 3 7 * 1 .3 6 9 -0 .1 6 7 0 .5 9 4 4 .3 8 1 * 8 .7 9 1 * 0 .8 4 7 7 .1 9 9 * 4 .6 3 7 * R M S E 0 .2 1 4 2 0 .2 9 9 4 0 .1 7 6 9 0 .2 0 3 6 0 .8 9 4 1 0 .2 4 8 9 0 .1 5 5 5 0 .1 9 4 6 0 .1 6 8 9 0 .2 9 8 5 1 .0 6 6 5 Im p li ed C o effi ci en t 0 .6 9 1 0 .3 1 9 1 .0 4 2 0 .7 6 5 0 .2 2 8 0 .0 8 4 0 .8 5 6 0 .3 7 3 0 .2 2 9 t-st a t 5 .8 8 4 * 2 .0 4 2 * * 8 .8 7 5 * 4 .9 1 1 * 1 .9 0 3 1 .3 0 1 5 .4 0 3 * 2 .4 7 4 * 1 .8 0 4 R M 9 6 C o effi ci en t 0 .2 8 8 0 .1 9 8 -0 .0 4 0 0 .0 2 4 0 .5 0 6 0 .5 7 2 0 .1 0 3 0 .5 3 0 0 .4 5 7 t-st a t 3 .2 3 8 * 2 .0 4 7 * * -0 .4 4 6 0 .2 4 8 5 .7 7 4 * 8 .7 6 2 * 0 .9 8 7 5 .9 0 8 * 5 .1 2 5 * ∗ S ig n ifi ca n t a t th e 1 % le v el ∗∗ S ig n ifi ca n t a t th e 5 % le v el
T able 7: Mult iple Regressions -3 M on th F orecast s D e v e lo p e d Ma r k e t C u r r e n c ie s E m e r g in g Ma r k e t C u r r e n c ie s A U D C H F G B P J P Y T o ta l B R L K R W MX N R U B Z A R T o ta l R M S E 0 .1 8 0 0 0 .2 1 6 6 0 .1 2 1 5 0 .1 3 9 8 0 .6 5 7 8 0 .2 1 7 6 0 .1 2 5 2 0 .1 7 9 0 0 .1 4 5 8 0 .2 1 7 2 0 .8 8 4 9 Im p li ed C o effi ci en t 0 .6 2 8 0 .2 7 3 1 .2 3 2 0 .8 5 0 0 .3 5 3 -0 .0 2 9 0 .7 6 1 0 .8 2 9 0 .2 9 2 t-st a t 3 .1 6 1 * 1 .7 5 7 8 .5 6 2 * 4 .8 4 7 * 1 .9 4 6 -0 .3 8 9 2 .3 4 7 * 4 .4 8 0 * 2 .0 7 7 * * DCC C o effi ci en t 0 .4 4 2 0 .2 7 6 -0 .2 0 8 -0 .0 3 8 0 .3 6 3 0 .8 0 5 0 .1 2 6 0 .2 5 9 0 .4 0 8 t-st a t 2 .3 2 9 * 1 .2 8 5 -0 .9 8 4 -0 .1 6 2 2 .8 7 8 * 7 .1 3 3 * 0 .7 5 5 1 .9 6 4 4 .0 0 4 * R M S E 0 .1 8 3 8 0 .2 1 8 2 0 .1 2 1 6 0 .1 3 9 8 0 .6 6 3 4 0 .2 2 5 2 0 .1 2 4 8 0 .1 7 9 3 0 .1 3 4 5 0 .2 2 1 5 0 .8 8 5 4 Im p li ed C o effi ci en t 0 .8 0 4 0 .3 5 4 1 .1 9 1 0 .8 1 9 0 .5 3 7 0 .0 1 7 1 .0 0 1 0 .6 4 2 0 .2 7 5 t-st a t 4 .2 0 6 * 2 .7 2 6 * 9 .0 9 4 * 4 .3 5 4 * 2 .4 1 6 * 0 .2 2 5 3 .3 5 6 * 3 .2 8 0 * 2 .3 3 8 * H is to ri ca l C o effi ci en t 0 .2 0 5 0 .0 1 9 -0 .1 1 2 0 .0 1 5 0 .1 1 5 0 .6 3 8 -0 .0 9 2 0 .4 6 4 0 .2 8 9 t-st a t 1 .4 9 0 0 .1 5 6 -0 .9 1 5 0 .1 3 1 0 .8 1 5 6 .6 3 7 * -0 .6 7 2 3 .8 0 3 * 2 .3 3 3 * R M S E 0 .1 8 1 4 0 .2 1 7 5 0 .1 2 1 8 0 .1 3 9 8 0 .6 6 0 6 0 .2 1 9 2 0 .1 2 7 0 0 .1 7 9 3 0 .1 3 5 5 0 .2 1 8 9 0 .8 7 9 9 Im p li ed C o effi ci en t 0 .7 2 2 0 .2 8 4 1 .1 4 6 0 .8 3 6 0 .3 9 0 -0 .0 1 8 0 .7 7 7 0 .6 4 2 0 .3 0 5 t-st a t 4 .7 3 1 * 1 .8 4 5 1 1 .0 7 4 * 4 .6 7 5 * 2 .1 1 9 * * -0 .2 6 2 2 .4 0 6 * 3 .5 7 3 * 2 .0 6 0 * * R M 0 6 C o effi ci en t 0 .2 8 9 0 .0 9 6 -0 .0 8 2 0 .0 0 0 0 .3 2 2 0 .6 4 1 0 .0 9 6 0 .4 6 5 0 .3 4 5 t-st a t 2 .6 1 2 * 1 .0 1 8 -0 .9 9 9 0 .0 0 6 2 .6 3 4 * 7 .8 6 1 * 0 .6 2 3 3 .1 8 8 * 3 .4 0 7 * R M S E 0 .1 8 1 0 0 .2 1 7 7 0 .1 2 1 3 0 .1 3 9 8 0 .6 5 9 9 0 .2 2 1 7 0 .1 2 3 2 0 .1 7 9 6 0 .1 2 9 7 0 .2 2 0 6 0 .8 7 4 9 Im p li ed C o effi ci en t 0 .6 4 8 0 .2 6 6 1 .2 5 0 0 .8 5 7 0 .4 1 7 -0 .0 8 2 0 .8 4 8 0 .5 6 8 0 .2 9 6 t-st a t 3 .3 8 9 * 1 .8 3 3 9 .3 0 8 * 4 .0 4 0 * 2 .0 9 5 * * -1 .0 2 0 2 .6 4 3 * 2 .9 5 7 * 1 .7 9 8 R M 0 6 12 C o effi ci en t 0 .3 8 6 0 .1 2 1 -0 .1 9 5 -0 .0 2 0 0 .3 3 1 0 .7 6 8 0 .0 3 9 0 .6 0 0 0 .4 1 5 t-st a t 2 .3 0 0 * 0 .8 3 1 -1 .2 0 4 -0 .1 2 8 1 .8 9 6 7 .2 9 2 * 0 .2 0 4 3 .4 8 8 * 2 .5 2 6 * R M S E 0 .1 8 1 2 0 .2 1 8 0 0 .1 2 1 9 0 .1 3 9 8 0 .6 6 0 9 0 .2 1 8 9 0 .1 2 7 6 0 .1 7 9 5 0 .1 4 2 2 0 .2 1 8 4 0 .8 8 6 5 Im p li ed C o effi ci en t 0 .7 2 9 0 .3 3 6 1 .1 3 5 0 .8 4 3 0 .3 6 6 0 .0 2 3 0 .8 1 0 0 .7 4 0 0 .2 9 9 t-st a t 4 .4 4 1 * 2 .1 6 6 * 1 0 .0 4 5 * 4 .3 3 6 * 1 .9 5 2 0 .3 3 6 2 .3 9 1 * 3 .8 4 4 * 2 .0 4 8 * * R M 9 6 C o effi ci en t 0 .2 5 7 0 .0 3 8 -0 .0 6 3 -0 .0 0 4 0 .2 7 6 0 .5 9 5 0 .0 6 2 0 .3 2 3 0 .2 9 2 t-st a t 2 .4 1 7 * 0 .4 2 1 -0 .7 6 8 -0 .0 4 5 2 .5 9 3 * 7 .3 2 5 * 0 .4 1 2 2 .5 1 0 * 3 .4 6 5 * ∗ S ig n ifi ca n t a t th e 1 % le v el ∗∗ S ig n ifi ca n t a t th e 5 % le v el
