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This study applies threshold regression model in a bivariate framework to explore the nonlinear long-term relationship among Bitcoin and gold prices over the period 2010-2018. Results are threefold: first, we show that gold is a significant predictor of Bitcoin prices. Second, we find evidence of a non-linear relationship between Bitcoin and gold prices characterized rather by a two-regime relationship with a structural break occurring in October 2017. Third, before the break, there is significant, negative but weak causality indicating that Bitcoin is a speculative asset. After the break, the relationship becomes significantly positive revealing diversifier and hedge properties of Bitcoin.

Bitcoin has been the focus of many recent studies in the literature, mainly because of noticeable price development and market capitalization since 2016. Launched in November 2008 by [

Its popularity has been keeping on growing particularly during the Great Recession in 2008-2010. Since then, different lines of enquiry of Bitcoin have been dominating in previous literature. There is especially a significant strand of the literature focusing on price formation ( [

Such analyses about the abilities of Bitcoin led to a more recent strand in the literature that aims to compare Bitcoin with gold properties [

Our paper aims to assess the predictive power of gold prices for Bitcoin prices and brings two main contributions to the previous literature. Firstly, it brings a methodological contribution since we use a threshold regression to take into consideration potential structural breaks, after testing for the structural stability of the parameters based on recursive residuals and the cumulative sum of square (CUSUM) statistic. The paper appears therefore to complement the recent wave of studies using mainly GARCH models [

The rest of the paper is organized as follows. Section 2 provides a data description. In Section 3, we present the methodology, while in section 4 we discuss empirical findings in light with the recent literature. Section 5 concludes.

We use daily data covering the period from the 19^{th} of July 2010 to the 31^{st}of December 2018 (3088 observations for each series). Bitcoin price data are collected from Coindesk^{1} Price Index while gold price data is sourced from World Gold Council. All data are labeled in U.S. dollars.

Mean | St. Dev. | Skewness | Kurtosis | Min. | Max. | Obs. | |
---|---|---|---|---|---|---|---|

Bit_pr | 1525.1 | 2981 | 2.6 | 6.94 | 0.05 | 18,960.5 | 3088 |

Gold_pr | 1354 | 185 | 0.9 | −0.24 | 1049.4 | 1895 | 3088 |

Source: Authors’ computation. Notes: St. Dev. = Standard Deviation, Min. = Minimum, Max. = Maximum, Obs. = Number of observations.

the data series trend in the beginning of 2017. The year 2017 appears a year of significant and rapid development of the Bitcoin. In August, the Bitcoin for the first time, traded above $4000. On the 18^{th} of December, it reached $19,000, before starting to drop and to reach roughly an average of $8000 during 2019. The main drivers of this rise are to find in the sudden rising Asian demand for the cryptocurrency, especially from China and Japan [

The evolution in the gold price is volatile over the entire period but to a less extent. We distinguish a turning point in 2013. Before this turning point, the regime is characterized by a high mean, while after that we note a downward slope associated with a low mean. From pictorial analysis, it is reasonable to expect 1) the presence of a deterministic trend for both series on one hand and 2) one or more structural breaks on the other.

Most of the macroeconomics temporal variables could inhibit structural breaks and an ignorance to the same could results into misleading results [

We conduct a two-step approach. As a first step, we estimate the linear relation between Bitcoin and gold prices, and continue by testing for its structural stability based on recursive residuals. Recursive residuals are defined as the difference between BIT and the first t − 1 observations and can be used both to test for non-linearity and to test for structural breaks. The CUSUM (cumulative sum of squares) test [

W t = ∑ j = k + 1 T w t σ ^ (1)

with

σ ^ 2 = ∑ j = k + 1 T ( w t − w ¯ ) 2 T − k − 1 (2)

and

w ¯ = ∑ j = k + 1 T w t T − k (3)

where W t is the cumulative sum of recursive residuals, T represents the number of observations, k represents the minimum sample size for which we can fit the model, w t is the t^{th} standardized recursive residual.

The CUSUM statistic is calculated for each t and is plotted under the null hypothesis of model stability, given by the following form:

H 0 : E ( W t ) = 0 (4)

If the sum goes outside a critical bound, it advocates that there is a structural break at the point at which the sum begins its movement toward the bound. Therefore, the null hypothesis is rejected and the alternative one is accepted, suggesting that there are structural breaks in the series.

As a second step, once the presence of one or more structural break(s) is confirmed, we run the regression. We assume here that we identified one structural break—a two-regime regression. The two-regime threshold regression equation is therefore expressed as follows:

B I T t = ( α 1 g o l d t ) × f ( k t ≤ c 1 * ) + ( α 2 g o l d t ) × f ( k t > c 1 * ) + ε t (5)

where B I T t is the Bitcoin price in time t, g o l d t is the gold price in time t, f ( . ) is an indicator function, ε t is an error term. We assume that if the relation (.) is true, then f ( . ) equals 1, otherwise f ( . ) equals 0. k t and c 1 * represent the threshold variable and the optimal threshold value, respectively.

The optimal threshold value c 1 * in the model can be determined by choosing the smallest residual variance of Equation (1):

c 1 * = arg min σ ^ 2 ( c t ) , t = 1 , ⋯ , n (6)

c 1 * ∈ [ c , c ¯ ] (7)

where σ ^ t ( c t ) is the residual variance from Equation (1) with optimal threshold value c 1 * .

For robustness check, we run the Chow first test [

We first start by using a basic linear model to examine the relation between Bitcoin and gold prices. Results displayed in

While the F-stat points out that all the variables are significant, the Ramsey RESET test shows that we can reject the null hypothesis that there are no neglected nonlinearities in the model. It indicates a preliminary evidence of a nonlinear relation between Bitcoin and gold prices and on a bias in these linear results.

Coefficient | p-value | t-statistics | Standard errors | |
---|---|---|---|---|

Gold_pr | −0.30 | 0.00*** | −2.62 | 0.11 |

Constant | 0.004 | 0.00*** | 3.63 | 0.006 |

R-squared = 0.32 | ||||

Adj. R-squared = 0.33 | ||||

Number of observations: 3056 | ||||

F test (H_{0}: The fit of the intercept-only model equals the model) | ||||

F(1, 3054) = 6.84 | ||||

Prob > F = 0.009 |

Source: Authors’ computation. Notes: *** denotes 1% level of significance.

Going deeper into the analysis, we plot the recursive cumulative sum in

A unit root test with structural break is run to determine the date of the break (^{th} of October 2017. We saw from

Test statistic | 1% critical value | 5% critical value | 10% critical value | |
---|---|---|---|---|

Recursive | 0.830 | 1.1430 | 0.9479 | 0.850 |

Source: Authors’ computation. Note: H_{0}: No structural break. Number of observations: 3056. Sample: July 19. 2010-December 31. 2018.

Coefficient | p-value | |
---|---|---|

Wald statistics | 16.09 | 0.00*** |

Estimated break date | 05/10/2017 |

Source: Authors’ computation. Note: H_{0}: No structural break; *** denotes 1% significance level; 3056 observations. Sample: July 19. 2010-December 31. 2018.

break can be interpreted as a market correction, a movement towards a different relationship between Bitcoin and gold prices which follows a temporary upswing in market prices.

We can now conduct the threshold regression with two distinct regimes since we identified one optima structural break over the period. ^{2} (AIC) [

This study uses cross-sectional data to investigate the Bitcoin-gold prices nexus over the daily period 2010 and 2018. This study contrasts with others in the previous literature, because it uses both linear and nonlinear threshold regression models and tests the structural stability of the parameters based on recursive residuals.

Coefficient | p-value | z-statistics | Standard errors | |
---|---|---|---|---|

Threshold: 5/10/2017 | ||||

Regime 1 gold_pr constant | −0.51 0.017 | 0.00*** 0.00*** | −1.28 5.92 | 0.40 0.003 |

Regime 2 gold_pr constant | +0.265 0.018 | 0.02*** 0.07** | 2.18 1.78 | 0.12 0.001 |

Chow test for parameter stability | F(3, 3056) = 5.05 with prob. > F = 0.0017 |

Source: Authors’ computation. Note: *** denotes 1% significance level. 3056 observations with 2595 observations under regime 1 and 494 observations under regime 2.

Empirical results indicate that gold is a significant predictor of Bitcoin prices. They indicate, however, that this impact is not linear over the period. A structural break occurring on the 5^{th} of October 2017 implies in fact a two-regime relationship between Bitcoin and gold prices. Before the turning point, a significant negative and weak impact of gold on Bitcoin prices is found in line with previous literature [

Like many other studies, our research presents some room for further improvement. First, such results need to be confirmed by investigating the Bitcoin—gold prices nexus in the future, as at present the second period since October 2017 being six times shorter than the first one in our study. Second, global factors like oil prices or stocks indices which are barometers of macroeconomic and financial environment could also be included to design multivariate approaches and assess the property of safe haven of Bitcoin, beyond hedging.

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

Syed Zwick, H. and Syed, S.A.S. (2019) Bitcoin and Gold Prices: A Fledging Long-Term Relationship. Theoretical Economics Letters, 9, 2516-2525. https://doi.org/10.4236/tel.2019.97159