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This paper discusses the asymmetric momentum threshold effect of copper futures returns on spot returns volatility in the London Metal Exchange. Referring the Threshold Autoregressive (TAR) and Momentum Threshold Autoregressive (MTAR) models, this study utilizes a Hybrid MTAR-GARCH model to test the asymmetric momentum threshold effects of LME copper futures returns on spot returns volatility. It is revealed that there are indeed asymmetric momentum threshold effects of LME copper futures returns on spot returns volatility. This finding would be beneficial to financial decision-making concerning copper price hedging, arbitrage and investment amidst high volatility market conditions.

Copper is a malleable metal with high thermal and electrical conductivity. It is widely used in infrastructure projects such as power electronics, architecture and transportation. As a result, the copper market has become an investment instrument and entity for related industries and financial professionals, and the impact of copper price fluctuations on the global economy is often discussed.

Gao and Wang [

Marbrouk [

Based on cost of carry principles, the price variance between spots and futures could be regarded as non-equilibrium adjustment conditions. Deduced from Fama and French [

Traditionally, the GJR-GARCH model is used to investigate the asymmetric residual effects on conditional volatility [

By using HMTAR-GARCH model, this study intends to test the asymmetric effects and nonlinear momentum threshold effects of London’s copper futures returns on spot returns volatility under highly volatile market conditions. The findings would improve the forecast of copper spot volatility and arbitrage decision-marking under high volatility. The rest of paper is organized as follows. Section 2 describes data and research methodology, including linear and non-linear unit root tests, ARCH effect tests, GJR-GARCH, and HMTAR-GARCH model development and estimation procedure. Section 3 shows the empirical research findings. Section 4 summarizes the conclusion and suggestions.

The daily data were London Metal Exchange (LME) copper spots and futures closing prices. The sample period ranges from Jan. 2, 1997 to Dec. 31, 2018. A total of 5605 samples (including from the period of financial crisis) from LME were collected.

Firstly, the returns of the LME copper spots and futures are defined as follows:

R E T t = ln ( S P R t / S P R t − 1 ) = ln ( S P R t ) − ln ( S P R t − 1 ) (1)

F R t = ln ( F P R t / F P R t − 1 ) = ln ( F P R t ) − ln ( F P R t − 1 ) (2)

SPR_{t} represents the copper spot prices at time t, RET_{t} represents the copper spot returns. FPR_{t} represents the copper futures prices, FR_{t} represents the copper futures returns. Referring to the research of Baur and McDermott (2010), the period of financial crisis is defined from Oct. 29, 2007 to Nov. 20, 2008, which is the ending day when the International Monetary Fund (IMF) signed the USD 2.1 billion Economic Stabilization Program with Iceland.

In 1981, Dickey-Fuller [

1) The random walk model without drift or trend components

Δ y t = λ y t − 1 + ∑ i = 1 k δ i Δ y t − i + ε t (3)

2) The random walk model with drift but no trend component

Δ y t = α 0 + λ y t − 1 + ∑ i = 1 k δ i Δ y t − i + ε t (4)

3) The random walk model with drift and trend components

Δ y t = α 0 + γ t + λ y t − 1 + ∑ i = 1 k δ i Δ y t − i + ε t (5)

The null hypothesis is H 0 : λ = 0 (there is a unit root, time series data are not stationary).

The alternative hypothesis is H 1 : λ ≠ 0 (there is no unit root, time series data are stationary).

In 1988, Phillips and Perron [

y t = α + σ ( t − T 2 ) + φ y t − 1 + u t (6)

The null hypothesis is H 0 = y t − 1 + u t .

Where the α, σ and φ are regression coefficients test when α = σ = 0 and φ = 1 , if there is a unit root, the time series data are not stationary; otherwise the time series data would be stationary and the null hypothesis would be rejected.

ADF and PP unit root tests are based on the assumptions that the series is linear, and therefore excludes the nonlinear series which might cause low statistical power. In light of this, Kapetanios et al. [

Δ y t = γ y t − 1 [ 1 − exp ( − θ y t − 1 2 ) ] + ε t (7)

The null hypothesis is H 0 : θ = 0 (There is a unit root, the time series data are not stationary). Δ y t is the rate of change of parameter coefficient, ε t is the error term, θ is the conversion rate of ESTAR model. Since γ could not be verified under KSS unit root null hypothesis, the formula (7) is re-calculated using Taylor series expansions. After first asymptotic expansion, the model is as follows:

Δ y t = υ y t − 1 3 + error (8)

In response to the error series correlation, the lagged variable Δ y t is added into the following model:

Δ y t = υ y t − 1 3 + ω 1 Δ y t − 1 + ω 2 Δ y t − 2 + ⋯ + ω p Δ y t − p + error (9)

The null hypothesis is H 0 : υ = 0 , there is a unit root, the time series data are not stationary; otherwise the time series data would be stationary, and the null hypothesis would be rejected.

Enders and Granger [

y t = β 0 + β 1 X t + u t (10)

Δ u t = I t ρ 1 u t − 1 + ( 1 − I t ) ρ 2 u t − 1 + ∑ i = 1 k γ i Δ u t − 1 + ε t (11)

where ε t is white noise, β 0 , β 1 , ρ 1 , ρ 2 and γ i are regression coefficients, τ is the unknown threshold value simulation:

I t = { 1 if u t − 1 ≥ τ 1 0 if u t − 1 < τ 1 (12)

MTAR is used to adjust the process. The first residual difference series is:

Δ u t = M t ρ 1 u t − 1 + ( 1 − M t ) ρ 2 u t − 1 + ∑ i = 1 k γ i Δ u t − 1 + ε t (13)

M t = { 1 if Δ u t − 1 ≥ τ 2 0 if Δ u t − 1 < τ 2 (14)

The conventional approach uses Formula (11) and (13) to test separately, thus it is unable to simultaneously test two effects. The following section would use the hybrid MTAR-GARCH model to examine when the residuals and differences are lower than thresholds, whether there are asymmetric and nonlinear incremental effects.

Finally, this study refers to the hybrid MTAR-GARCH model adopted by Goo and Shih [

R E T t = ζ 0 + ζ 1 R E T t − 1 + ζ 2 F R t − 1 + ζ 3 ( I t F R t − 1 ) + ζ 4 ( M t F R t − 1 ) + ζ 5 ( D 1 F R t − 1 ) + ζ 6 ( D 1 I t F R t − 1 ) + ζ 7 ( D 1 M t F R t − 1 ) + ε t (15)

h t = θ 0 + ν 1 h t − 1 + θ 1 ε t − 1 2 + η 1 F R t − 1 + η 2 ( I t F R t − 1 ) + η 3 ( M t F R t − 1 ) + η 4 D 1 F R t − 1 + η 5 ( D 1 I t F R t − 1 ) + η 6 ( D 1 M t F R t − 1 ) (16)

where,

I t = { 1 if ε t − 1 ≥ τ 1 0 if ε t − 1 < τ 1 (17)

M t = { 1 if Δ ε t − 1 ≥ τ 2 0 if Δ ε t − 1 < τ 2 (18)

where R E T t is the copper spot returns, F R t − 1 is the copper futures returns, D 1 is the variable of simulation during the financial crisis. Coefficients η 2 and η 5 test whether the residuals have positive or negative asymmetric TAR effect; coefficients η 3 and η 6 test whether the residual differences have a nonlinear MTAR effect. The residual term ε t does not have autocorrelation and has white noise feature. Moreover, when τ 1 = τ 2 = 0 , Models (15) and (16) are similar to the GJR-GARCH model.

When a stock market is under overwhelming sell-off pressure (i.e. under high volatility), short-term futures would be under higher pressure to sell, and traders could sell the futures prior to the upcoming stock price drop. As a result of the increasing pressure to sell short-term futures, the futures prices would drop more rapidly, leading to augmented price fluctuations. Speculators and arbitrage investors could sell spots and buy futures to obtain additional premiums.

Therefore, if the previous residuals ( ε t − 1 ) are lower than certain thresholds, it suggests that the rate of futures price drops was faster than expected and caused the following trading day to have larger volatility; and if the previous residual differences ( Δ ε t − 1 ) are lower than certain thresholds, it reveals that the market is becoming highly unexpected, volatility is likely to increase even more. This study refers to the assertions of Gao and Wang [

Hypothesis 1: During the Financial crisis, when the pre-Financial crisis residual is lower than its threshold τ 1 , the impact of future returns on spots volatility is greater (The TAR effect, i.e., η 5 < 0 ).

Hypothesis 2: In the financial crisis period, when the pre-financial crisis residual difference is lower than the momentum threshold τ 2 , the impact of future returns on spots volatility is greater (The MTAR effect, i.e., η 6 < 0 ).

Hypothesis 3: The MTAR effect has a greater negative impact compared to the TAR effect during volatile market condition, i.e., H 3 -1 : η 6 < η 5 . Another way to verify the hypothesis is to include non-volatile period, i.e., H 3 - 2 : η 3 + η 6 < η 2 + η 5 .

In Baur and McDermott’s [

The descriptive statistical summary of futures returns (FR) is shown in

Compared to other periods, FR has a larger standard deviation during the Financial crisis, with a negative average returns. Then, the ADF, PP and KSS unit root tests on RET series are examined. In

The study then adopts a 3-tier computational model. Model 1 is the ARCH model without threshold. Model 2 is the GJR-GARCH (1, 1) model with the TAR threshold (τ_{1}) and the MTAR threshold (τ_{2}) set as 0. Model 3 is the HMTAR model with random τ_{1} and τ_{2}. The goodness of fit test of the models adopts AIC principles [

Hypothesis 1 and Hypothesis 2 are supported, since η 5 and η 6 are negative significantly. The WALD tests show that the null hypothesis H 3 -1 : η 6 < η 5 , Chi-square = 0.67915 (p = 0.4099) fails to reach the significant level; while the null hypothesis H 3 - 2 : ( η 3 + η 6 ) < ( η 2 + η 5 ) , Chi-square = 4.5936 (p = 0.0321) reaches the significant level. Hence Hypothesis 3 is supported.

Period | Trading Days | |
---|---|---|

Pre-Financial Crisis | January 2, 1997-October 26, 2007 | 2779 |

Financial Crisis | October 29, 2007-November 20, 2008 | 270 |

Post-Financial Crisis | November 21, 2008-December 31, 2018 | 2555 |

Overall Observed Time | January 2, 1997-December 31, 2018 | 5604 |

Mean | Standard Deviation | Minimum | Maximum | |
---|---|---|---|---|

Pre-Financial Tsunami | 0.002 | 0.644 | −4.961 | 4.77 |

Financial Tsunami | −0.124 | 1.14 | −4.517 | 5.16 |

Post-Financial Tsunami | 0.009 | 0.682 | −3.406 | 3.882 |

Overall Observed Time | 0.694 | 0.008 | −4.961 | 5.16 |

lags | ADF | PP | KSS |
---|---|---|---|

t-Statistic | t-Statistic | t-Statistic | |

5 | −80.2347*** | −80.2347*** | −14.1395*** |

10 | −21.1022*** | −80.1512*** | −10.8130*** |

20 | −14.6596*** | −80.0157*** | −8.7144*** |

Notes: *, ** and *** indicate 10%, 5% and 1% significant level.

Heteroscedasticity Test(lags) | F-Statistic Test | Chi-Square Test |
---|---|---|

ARCH(1) | 430.1581 *** | 399.6207 *** |

ARCH(5) | 199.369 *** | 846.9598 *** |

ARCH(10) | 120.7771 *** | 994.9198 *** |

Notes: *, ** and *** indicate 10%, 5% and 1% significant levels.

RET | Model 1 | Model 2 | Model 3 | ||||||
---|---|---|---|---|---|---|---|---|---|

(ARCH) | (GJR-GARCH) | (HMTAR-GARCH) | |||||||

(No Threshold) | ( τ 1 = τ 2 = 0 ) | ( τ 1 = − 1.4202 ; τ 2 = 1.1655 ) | |||||||

Coeff. | t-stat | Coeff. | t-stat | Coeff. | t-stat | ||||

Mean Equation: | |||||||||

ζ_{0} | 0.0041 | 0.5877 | 0.0044 | 0.6355 | 0.0049 | 0.7071 | |||

ζ_{1} | −0.4212 | −26.7077 | *** | −0.4232 | −26.9314 | *** | −0.4093 | −26.0514 | *** |

ζ_{2} | 0.5819 | 38.8885 | *** | 0.5904 | 30.0805 | *** | 0.4208 | 5.662 | *** |

ζ_{3} | −0.2164 | −3.4728 | *** | 0.0094 | 0.3615 | 0.1793 | 2.4352 | ** | |

ζ_{4} | −0.0309 | −1.1802 | −0.2639 | −4.8958 | *** | ||||

ζ_{5} | 0.0322 | 0.5865 | 0.0164 | 0.2747 | |||||

ζ_{6} | −0.0792 | −0.6917 | −0.2375 | −2.9419 | *** | ||||

ζ_{7} | −0.1343 | −1.1317 | 0.2789 | 1.6334 | * | ||||

Variance Equation: | |||||||||

θ_{0} | 0.0025 | 5.5391 | *** | 0.0025 | 5.4008 | *** | 0.002 | 4.8943 | *** |

ν_{1} | 0.9405 | 217.289 | *** | 0.9412 | 211.21 | *** | 0.9466 | 235.8294 | *** |

θ_{1} | 0.053 | 13.6055 | *** | 0.0525 | 13.2126 | *** | 0.0488 | 13.2596 | *** |

η_{1} | −0.0014 | −0.6734 | −0.0040 | −1.1717 | −0.1201 | −4.5597 | *** | ||

η_{2} | −0.0646 | −3.4636 | *** | −0.0025 | −0.4837 | 0.119 | 4.5505 | *** | |

η_{3} | 0.0089 | 1.3767 | * | −0.0012 | −0.0748 | ||||

η_{4} | −0.0242 | −0.7131 | 0.0321 | 0.7343 | |||||

η_{5} (H_{1}) | −0.0814 | −1.2785 | −0.0668 | −1.9357 | ** | ||||

η_{6} (H_{2}) | 0.0297 | 0.6768 | −0.1366 | −1.6298 | * | ||||

AIC | −516.4206 | −498.8721 | −553.2718 | ||||||

SBC | −525.7901 | −524.2416 | −578.6419 | ||||||

LL | 267.2103 | 266.436 | 293.6359 |

Notes: *, ** and *** indicate 10%, 5% and 1% significant level.

Recent research mostly focuses on the correlation between LME’s copper prices with oil, metals, overall economic indicators and price volatility. For example, Gao and Wang [

This paper integrates the TAR and MTAR GARCH model as the Hybrid TAR/MTAR-GARCH model to forecast the volatility of copper returns. Meanwhile, it introduces the assessment on the simulated interaction of TAR and MTAR into average and variance formulas so as to prevent the econometrics model from generating errors. Then, it designs two random threshold values to augment the testing method of GJR-GARCH model with zero-threshold value, enabling the variance formula to test two random threshold values simultaneously.

By estimating the asymmetric threshold parameter, it is revealed that the HMTAR-GARCH model has higher explanatory power than the GJR-GARCH model. This study verifies three hypotheses. Firstly, it verifies that during highly volatile period, when the lagged residuals are lower than their thresholds, the impacts on copper returns volatility are greater. Secondly, it verifies that during highly volatile period, when the lagged residual differences are lower than momentum thresholds, the impacts on copper returns volatility are greater. Finally, results show that during highly volatile period, the MTAR effect is even greater than the TAR effect.

In conclusion, the above findings suggest that the Hybrid HMTAR-GARCH model has better level of fitness than the GJR-GARCH model, and the former is able to capture the asymmetric and nonlinear nature of copper returns volatility. In particular, it reveals the asymmetric and nonlinear momentum threshold effects of London’s copper futures returns on spot returns volatility under highly volatile market conditions; hence it could facilitate the forecasting of copper spots volatility and arbitrage decision-making during a period of high volatility.

Moreover, in terms of supply and demand of copper, the HMTAR-GARCH model could help clients and suppliers set up better risk prevention strategies and obtain additional earnings from financial practices on top of operating profits. For example, global electricity and electronics sector, architectural sector, transportation industry, industrial manufacturing, retail and consumers, copper mining, copper recycling professionals and governments’ infrastructure administrative departments could apply the HMTAR-GARCH model to risk management, thus enable traders and manufacturers to develop better investment and risk prevention strategies.

Although this paper attempts to derive a better threshold GARCH model for exploring better volatility forecasting model, there are some possible modifications. Firstly, the double thresholds model could be extended to multiple thresholds in mean and variance equations. Secondly, the discrete threshold model could be extended to continuous ESTAR or LSTAR-type model, which could measure continuous unexpected information threshold shocks to conditional volatility.

The authors declare no conflicts of interest regarding the publication of this paper.

Goo, Y.J. and Chen, C.C. (2020) Asymmetric Momentum Threshold Effect of Copper Futures Returns on Spot Returns Volatility in London Metals Exchange under High Volatility. Modern Economy, 11, 51-61. https://doi.org/10.4236/me.2020.111006