Are Bitcoin Prices Rational Bubbles *

Transcription

1 The Empirical Economics Letters, 15(9): (September 2016) ISSN Are Bitcoin Prices Rational Bubbles * Hiroshi Gunji Faculty of Economics, Daito Bunka University Takashimadaira, Itabashi, Tokyo, Japan hgunji@ic.daito.ac.jp Abstract: This paper investigates whether Bitcoin prices are rational, in that as Bitcoin has no fundamental value, its prices appear like bubbles. We test the Bitcoin prices of the Chinese Yuan, US Dollar, Euro, and Japanese Yen. Results suggest that Bitcoin prices are indeed rational bubbles. Keywords: Bitcoin, Random Walk, Rational Bubble, Unit Root Tests JEL Classification Number: G12, G14, O16 1. Introduction Bitcoin is a form of private electronic money (Nakamoto, 2009 and Böhme et al., 2015). Its users trade completely peer-to-peer, so the Bitcoin system involves no intermediaries or governmental institutions. Figure 1 plots the recent exchange rates of Bitcoin to the Chinese Yuan, US Dollar, Euro, and Japanese Yen. As shown, these exchange rates have been increasing recently, and this has the appearance of a bubble. In practice, Baek and Elbeck (2015) estimate the volatility of Bitcoin returns and suggest that the Bitcoin market is highly speculative. Importantly, Bitcoin has no dividends and no interest rates. This implies that Bitcoin has no fundamental value. However, Figure 1 shows that all of the Bitcoin prices have a positive value, and in the context of economics, this infers a bubble. Now, we theoretically consider the Bitcoin price. The no arbitrage condition is: E t p t+1 p t = 1 + r where p t is the Bitcoin price at t, r is the (-invariant) interest rate, and E t ( ) is the expectation operator at t. Substituting Eq. (1) recursively, we have: p t = E t lim τ p t+τ 1 + r τ. If lim τ p t+τ is finite, then p t = 0. Hence, p t has no fundamental value. Put differently, Eq. (1) holds when p s = 0 for any s > 1. In fact, Bitcoin rates are positive, p t > 0, so Bitcoin could have bubbles. Are these Bitcoin bubbles then rational or reflective of animal * The author would like to thank seminar participants at the Institute of Economic Research, Daito Bunka University for their many helpful comments.

2 EUR/BTC JPY/BTC CNY/BTC USD/BTC The Empirical Economics Letters, 15(9): (September 2016) 820 spirits? There are many bubble sequences where Eq. (1) holds. We consider the sequence of p t with a bubble b t and no fundamental value as follows: p t = b t. Figure 1: Bitcoin exchange rates in levels For instance, let us suppose an explosive bubble, such as that in Diba and Grossman (1988): b t = E t b t+1 1+r In this case, it is straightforward to obtain the result that Eq. (1) holds. Therefore, any sequence of Eqs. (3) and (4) is a rational bubble. 2. Methods To test whether Eq. (1) holds in the Bitcoin market, we take the natural logarithm of both sides of Eq. (1) to obtain: ln p t+1 = r + ln p t + ε t+1 where ln 1 + r r and ε t+1 = ln E t p t+1 ln p t+1. This implies that the Bitcoin price

3 The Empirical Economics Letters, 15(9): (September 2016) 821 is a random walk with drift if the investors are rational. Therefore, we conduct unit-root tests. We conduct two types of unit-root tests. First, we use the augmented Dickey--Fuller (ADF) test (Dickey and Fuller (1979); Said and Dickey (1984)), which has a null hypothesis of nonstationarity. The equation we estimate is: ln p t = r + ρ ln p t 1 + ε t. The null hypothesis is ρ = 1 and the alternative is ρ < 1. However, as the power of the ADF test is weak, we also use the DF-GLS test in Elliott et al. (1996). Second, we use the KPSS test proposed by Kwiatkowski et al. (1992), which has a null hypothesis of stationarity. The equation we consider is: Δ ln p t = r + ε t e t = ln p t α β t, where ε t = μ t + v t, μ t = μ t 1 + u t, E(u t ) = E(v t ) = 0, and E(u 2 t ) = σ 2 u. hypothesis is σ 2 u = 0 and the alternative is σ 2 u > 0. The null There are several empirical studies on rational bubbles, especially in stock markets. In these, the earnings from stock consist of capital and income gains. Consequently, the noarbitrage condition is [E t p t+1 + d t ]/p t = 1 + r. In this case, we need to consider the relation between p t and d t. If the pricing is rational and these are random-walk processes, they are cointegrated. Conversely, as Bitcoin provides no income gain, we can investigate only the stationarity of Bitcoin prices. 3. Data The data we use are from Bitcoin charts. We consider four daily Bitcoin rates. The rates for the Chinese Yuan (BTC China) are from June 17, 2011 to December 26, 2013, those for the US Dollar (Bitstamp) are from December 18, 2011 to December 26, 2013, those for the Euro (Mt. Gox) are from September 5, 2011 to December 26, 2013, and those for the Japanese Yen (Mt. Gox) are from September 14, 2012 to December 26, Table 1 provides basic statistics for the Bitcoin prices in natural logarithms. Table 1: Basic Statistics CNY USD EUR JPY Mean Std. Dev Min Max

4 EUR/BTC in log difference JPY/BTC in log difference CNY/BTC in log difference USD/BTC in log difference The Empirical Economics Letters, 15(9): (September 2016) 822 Figure 2 illustrates the log differences of the Bitcoin exchange rates in percentage terms. As shown, each series exhibits quite large fluctuations, anywhere from -40% to 40%. However, the means do not seem to change much over. Likewise, as depicted in Figure 1, while the Bitcoin prices appear to be increasing, the growth rates appear stationary. Accordingly, we hypothesize that the Bitcoin prices are integrated of order one or I(1). That is, they are rational bubbles. Figure 2: Bitcoin Exchange Rates in log Differences Results Table 2 presents the results. Maximum lags are set by p max = [12 T/100 1/4 ], where we select the lags using the modified Akaike information criterion (MAIC). As discussed, we conduct two types of unit-root tests. First, we use the ADF test (Dickey and Fuller (1979); Said and Dickey (1984)), which has a null hypothesis of nonstationarity. All the Bitcoin prices in levels are unable to reject the null hypothesis, unlike those in first differences. This suggests that the Bitcoin prices are random-walk processes. Second, we use the KPSS test proposed by Kwiatkowski et al. (1992), which has a null hypothesis of stationarity. In this case, each of the level series rejects the stationary null, but three are unable to reject the null in first differences and JPY rejects the null only at the 10 percent significance level.

5 The Empirical Economics Letters, 15(9): (September 2016) 823 Table 2: Unit-Root Tests Level First difference Variable Statistic Lag Statistic Lag ADF Test CNY *** 1 USD *** 4 EUR *** 1 JPY *** 1 KPSS Test CNY 2.12 *** USD 1.07 *** EUR 1.49 *** JPY 0.65 *** * 5 Note: ***: p < 0.01; **: p < 0.05; *: p < Thus, although the result for the Bitcoin--Japanese Yen exchange rate is ambiguous, almost all of the test results support the finding that the Bitcoin prices represent random walks with drift. In other words, Bitcoin prices are rational bubbles. 5. Concluding Remarks This paper tests the hypothesis that Bitcoin prices are rational bubbles. By virtue of having no dividends, the Bitcoin price theoretically has no fundamental value. This feature allows us to use standard unit-root tests. The results showed that all Bitcoin prices follow an I(1) process. In other words, the Bitcoin price is a rational bubble. References Baek, C., and Elbeck, M., 2015, Bitcoins as an investment or speculative vehicle? A first look, Applied Economics Letters, 22, Böhme, R., Christin, N., Edelman, B., and Moore, T., 2015, Bitcoin: Economics, Technology, and Governance, Journal of Economic Perspectives, 29: Diba, B.T., and Grossman, H.I., 1988, Explosive rational bubbles in stock price? American Economic Review, 78, Dickey, D.A., and Fuller, W.A., 1979, Distribution of the estimators for autoregressive series with a unit root, Journal of the American Statistical Association, 74, Elliott, G., Rothenberg, T. J., and Stock, J. H., 1996, Efficient tests for an autoregressive unit root, Econometrica, 64 (4),

6 The Empirical Economics Letters, 15(9): (September 2016) 824 Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., and Shin, Y., 1992, Testing the null hypothesis of stationarity against the alternative of a unit root, Journal of Econometrics, 54, Nakamoto, T., 2009, Bitcoin: A Peer-to-Peer Electronic Cash System, Said, S.E. and Dickey, D.A., 1984, Testing for unit roots in autoregressive-moving average models of unknown order, Biometrika, 71,