Price of bitcoin
- The figure below shows a trend line for the average price of bitcoin in US dollars.
The price of bitcoins shows an increasing trend between 2015 and mid-2017. Although the trend in the prices of bitcoins is increasing, the increase is at a slow rate. The highest value reached by the prices is recorded in the period between 2017 and 2018 when the prices suddenly shoot up. This region shows the presence of an outlier that was caused by occurrences in the financial environment. After the period there is a significant change in the direction of the trend as it starts decreasing. The decreasing trend continues up to mid-2018 when the increasing trend starts again.
The histogram does not display any characteristics of normality in the data. Frequencies of the sales of bitcoins vary according to the price. As the price increases, the number of bitcoins sold decreases. This is shown by the continued cumulative frequencies of the histogram and supported by the Jacques_bera test statistics. The test statistic is greater than the alpha level of confidence leading to the rejection of the null hypothesis which states that the data is normally distributed. Data displaying normality would have the highest frequencies in the middle and decreasing frequencies on both sides.
The behavior of the prices of bitcoins over time is relatively persistent. The number of flukes displayed by the autocorrelation plot indicates that the data has cases of autocorrelation. This implies the behavior of the prices of the bitcoins is relatively stable.
ADF intercept
The null hypothesis of the ADF test is based on the presence of one or more unit roots in the data. The ADF test with the only trend returns an ADF test statistic with the value -1.537790 which is greater than the critical value at 5% alpha level of significance. For this data, we fail to reject the null hypothesis indicating the presence of a unit root. According to the ADF test when considering the intercept, the series for bitcoins is not stationary. This is supported by the probability which is 0.5143 is greater than the alpha level of significance. The ADF test which includes both the intercept and trend produce almost similar results as the probability is greater than 0.05 which is the alpha level of significance. The ADF test statistic is also greater than the critical values at 1%.5% and 10% alpha levels of confidence.
| t-Statistic | Prob.* | |||
| Augmented Dickey-Fuller test statistic | -1.537790 | 0.5143 | ||
| Test critical values: | 1% level | -3.433795 | ||
| 5% level | -2.862948 | |||
| 10% level | -2.567567 | |||
ADF intercept and trend
| t-Statistic | Prob.* | |||
| Augmented Dickey-Fuller test statistic | -2.491938 | 0.3322 | ||
| Test critical values: | 1% level | -3.963169 | ||
| 5% level | -3.412317 | |||
| 10% level | -3.128094 | |||
Phillips Pearson intercept
The Phillips-Perron test is very sensitive to structural breaks and is non-parametric making it different from the ADF test. Like the ADF test, the Phillips-Perron’s test null hypothesis is also based on the presence of a unit root in the data. The probability in the Phillips-Perron test for the intercept is greater than 0.05 leading to the failure in the rejection of the null hypothesis. The rejection of the null hypothesis is also supported by the Phillips-Perron test-statistic value being greater than the tests critical values at 1%,5% and 10%. The Phillips-Perron using both the trend and intercept also displays similar results indicating a conclusion from the Phillips-Perron test supporting the presence of a unit root test in the data.
| Adj. t-Stat | Prob.* | |||
| Phillips-Perron test statistic | -1.462690 | 0.5525 | ||
| Test critical values: | 1% level | -3.433793 | ||
| 5% level | -2.862947 | |||
| 10% level | -2.567566 | |||
| *MacKinnon (1996) one-sided p-values. | ||||
Phillips Pearson intercept and trend
| Adj. t-Stat | Prob.* | |||
| Phillips-Perron test statistic | -2.400195 | 0.3793 | ||
| Test critical values: | 1% level | -3.963166 | ||
| 5% level | -3.412315 | |||
| 10% level | -3.128094 | |||
KPSS intercept
Performing the KPSS test for stationarity yields a KPSS test statistic with a value of 3.598 which is greater than the critical value at 5% when the test is performed using the intercept only. Using both the trend and intercept produces similar results leading to the rejection of the null hypothesis for the KPSS test. The null hypothesis for the KPSS unit root test states that the series being analyzed is stationary.
| LM-Stat. | ||||
| Kwiatkowski-Phillips-Schmidt-Shin test statistic | 3.598714 | |||
| Asymptotic critical values*: | 1% level | 0.739000 | ||
| 5% level | 0.463000 | |||
| 10% level | 0.347000 | |||
| *Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1) | ||||
kiss intercept and trend
| LM-Stat. | ||||
| Kwiatkowski-Phillips-Schmidt-Shin test statistic | 0.266412 | |||
| Asymptotic critical values*: | 1% level | 0.216000 | ||
| 5% level | 0.146000 | |||
| 10% level | 0.119000 | |||
| *Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1) | ||||
The presence of a unit root test indicates that the series is non-stationary. This implies that there are random changes which are statistically significant in the bitcoin market. The non-stationarity means that the process is ever-changing and technical analysis of the market may not lead to abnormal profits for investors. This supports the weak-form efficiency of the bitcoin markets.
- Application of integration techniques on the series implies the presence of a non-stationary process. The efficient market hypothesis assumes that it is not possible to gain an advantage due to the possession of any important information on the market simply because the information is already incorporated into the market prices. Any order integration will successfully suppress trend in the data making it possible to make forecasts from studying this data. It will be possible to analyze the bitcoin market and come up with forecasts for the prices. The analysis and successful determination of the trend violate all the assumptions of the Efficient Market Hypothesis (EMH). It will, therefore, be possible for investors to carry out research on the data especially past financial data and come up with successful models that will help them earn huge sums of money. The EMH supports the independence of future prices from past prices although they are closely related
- The investigation of a Johansen cointegration test on three prices i.e Eur, USD and AUD reveal the presence of an error correction factor among the prices of the bitcoins. At 0.05 alpha level of confidence, the trace test statistic for the Johansen cointegration test is greater than the critical value at 5%. This implies the three prices have forces that influence them in a certain way that is statistically significant. These forces may be aimed at correcting errors to make better models for the prices.
- The table below shows the three variable cointegration test.
| Unrestricted Cointegration Rank Test (Trace) | ||||
| Hypothesized | Trace | 0.05 | ||
| No. of CE(s) | Eigenvalue | Statistic | Critical Value | Prob.** |
| None * | 0.025164 | 63.24324 | 29.79707 | 0.0000 |
| At most 1 * | 0.008357 | 17.62346 | 15.49471 | 0.0236 |
| At most 2 | 0.001452 | 2.601575 | 3.841466 | 0.1068 |
- Running VAR models helps in the identification of the most suitable structure for the variables. The number of lags most suitable for this model is infinite so adoption of any lag should be uniform across all the three prices.
- The table below shows the statistics for the vector error correction model for the three variables.
| AVG_AUD | AVG_EUR | AVG_USD_F | |
| AVG_AUD(-1) | 1.139578 | 0.204788 | 0.123433 |
| (0.08350) | (0.05416) | (0.06367) | |
| [ 13.6473] | [ 3.78098] | [ 1.93870] | |
| AVG_AUD(-2) | -0.146669 | -0.134729 | -0.097382 |
| (0.08485) | (0.05504) | (0.06470) | |
| [-1.72855] | [-2.44795] | [-1.50522] | |
| AVG_EUR(-1) | 0.442875 | 0.966953 | 0.470682 |
| (0.12988) | (0.08424) | (0.09903) | |
| [ 3.40996] | [ 11.4781] | [ 4.75303] | |
| AVG_EUR(-2) | -0.408796 | -0.095198 | -0.470090 |
| (0.12887) | (0.08359) | (0.09826) | |
| [-3.17207] | [-1.13884] | [-4.78401] | |
| AVG_USD_F(-1) | -0.207487 | -0.100576 | 0.644023 |
| (0.11960) | (0.07758) | (0.09119) | |
| [-1.73481] | [-1.29644] | [ 7.06216] | |
| AVG_USD_F(-2) | 0.185287 | 0.110922 | 0.316905 |
| (0.11870) | (0.07699) | (0.09051) | |
| [ 1.56094] | [ 1.44064] | [ 3.50142] | |
| C | 14.11839 | 18.90027 | 13.45327 |
| (11.2676) | (7.30860) | (8.59125) | |
| [ 1.25301] | [ 2.58603] | [ 1.56593] | |
| R-squared | 0.996893 | 0.996524 | 0.996463 |
| Adj. R-squared | 0.996883 | 0.996512 | 0.996451 |
| Sum sq. resids | 1.70E+08 | 71524924 | 98832938 |
| S.E. equation | 308.5212 | 200.1188 | 235.2394 |
| F-statistic | 95510.42 | 85337.37 | 83854.75 |
| Log likelihood | -12817.75 | -12041.60 | -12331.51 |
| Akaike AIC | 14.30535 | 13.43960 | 13.76298 |
| Schwarz SC | 14.32679 | 13.46104 | 13.78442 |
| Mean dependent | 5360.804 | 3366.331 | 3890.873 |
| S.D. dependent | 5525.769 | 3388.600 | 3948.663 |
| Determinant resid covariance (of adj.) | 1.34E+12 | ||
| Determinant resid covariance | 1.33E+12 | ||
| Log likelihood | -32656.13 | ||
| Akaike information criterion | 36.44967 | ||
| Schwarz criterion | 36.51399 | ||
| Number of coefficients | 21 | ||
- The model has an r squared value of 0.99. The vector error correction is almost perfect in the representation of the three variables. The model can predict the values of the bitcoin prices with an accuracy of about 99%. The low AI and Schwarz determinant also indicated that the model perfectly fits the prices of bitcoins.