This is a practitioners exercise and statistical theory will be kept to a minimum for easier reading but embedded links will give further background for the interested reader.
Originally used by climatologists to study the effect of structural shifts in climate change, a statistical structural analysis is performed on Bitcoin to see whether historical price evolution was impacted by structural shifts or shocks. Those shifts are then used as intervention variables to help find a better ARIMA model forecast for Bitcoin. Also, over the last 800 days, 46 individual outliers were detected across the 4 different types of outliers examined (see method by Chen and Liu (1993) ). Their net effect on Bitcoin price is approximately a negative impact of -$7065, implying that without these multiple shocks, a “natural evolution” of Bitcoin price would currently be around $15,380 (or about +85% higher than current levels of ~$8300!)
Time series data often undergo sudden changes that alter the dynamics of the data temporarily or sometimes permanently. These changes are typically non-systematic and cannot be captured by standard time series models. That’s why they are known as “outlier effects”. Detecting outliers is important because they have an impact on the selection of the model, and consequently, on forecasts.
For example, the linear regression below can be better estimated by employing a structural break and analyzing the data points as 2 separate segments.
Identifying Structural Breaks
In our exercise, we use 2 years of aggregated daily Bitcoin price data taken from Cryptocompare.com with USD prices ranging from $500 to $20,000 (for about a 40X gain) as seen below:
We first analyse if there are any statistically significant structural breaks using the method from Bai & Perron (2003) , which allows for the estimation of multiple breakpoints in time series data by using OLS regression employing dynamic programming to evaluate multiple breakpoint models. The key aim is to find m potential breakpoints with the best model that minimizes the RSS or BIC criteria.
The model used on Bitcoin price data suggests 5 breakpoints (minimized BIC) as can be seen from the RSS/BIC plot :
The 5 breakpoints and their corresponding level shifts (in blue) are indicated below, along with the 95% confidence intervals for each break point (in red).
The confidence intervals are surprisingly tight/accurate for each of the breakpoints and correspond quite well to significant events around Bitcoin’s timeline, as well as key price breakouts and also the last big correction in late Jan 2018:
An Improved ARIMA Forecast Model
We can now explore an improved ARIMA model to help forecast for Bitcoin price incorporating the structural changes we have found above. (In short, an ARIMA model takes into account autoregressive (AR) /lagged data points, as well as a moving average (MA) component of random shocks/error terms that propagate directly into the future values of the time series).
The fitted multiple level shifts (as determined by the structural breaks analysis) can be used as an intervention regressor variable to help fit a better ARIMA model, as shown below:
An ARIMA (2,1,2) model was chosen after examining ACF, PACF plots to determine the order of autoregressive and moving average lags in conjunction with examining an automated forecast (auto.arima). In particular, the level shift regressor was statistically significant at the 0.1% level (z-test). The 2 shaded areas in the chart above represent a 60 day forecast at the 80% & 95% confidence levels and are likely wide given the volatility of the underlying. The forward 60 day point forecast (end of July 2018) is: $8715 with a 80% confidence interval of $4850 ~ $12,592 and 95% confidence interval of $2900 ~ $14,643
Outliers and Intervention Analysis
We now consider a model that assimilates the effects of outliers or observations, that are highly unusual relative to normal behavior. Intervention analysis, introduced by Box and Tiao (1975), provides a framework for assessing the effect of an intervention on a time series . It is assumed that the intervention affects the process by changing the mean function or trend. Interventions can be “natural” (eg. animal populations changes due to changes in climiate) or “man-made” (eg. increasing the speed limit on a highway and its impact on car accidents).
We examine Bitcoin price series for the following 5 types of intervention outliers:
1. AO = Additive Outlier – an isolated spike/shock
2. TC = Transient Change – a spike/shock that takes several periods to gradually disappear
3. IO = Innovative Outlier – a shock in the error terms of the process that continue forward through time
4. LS = Level Shift – a key shift in the mean level of a process
5. SLS = Seasonal Level Shifts
The procedure consists of two main stages:
1. Detection of outliers upon a chosen ARIMA model.
2. Refitting the ARIMA model including the outliers detected in the previous step and remove those outliers that are not statistically significant in the new fit.
The series is then adjusted for the detected outliers and the stages (1) and (2) are repeated until no more outliers are detected or until a maximum number of iterations is reached.
A total of 46 outliers were detected: 13 AO (Additive Outliers), 11 TC (Transient Changes), 3 IO (Innovative Outliers), 19 LS (Level Shifts) but interestingly no seasonal level shifts (SLS) were found:
The cumulative net effect of these 46 outliers is a negative impact of -$7065 to the BTC price timeline. This implies a “natural” outlier-free BTC price of $15,380 which is shown below with the original BTC times series in grey, the ‘natural’ BTC price in blue and the net outlier effects in the lower plot in red.
Concluding remarks :
Bitcoin price was examined for structural breaks and 5 significant breakpoints were found that corresponded well to key events in Bitcoins history. Those level shifts were then used as an intervention variable regressor and shown to be statistically significant in deriving an ARIMA (2,1,2) model with drift to forecast Bitcoin price movements. However the (95%) confidence levels remained wide with a 60 day point forecast of $8715 and a range of ($2700 ~ $14,600). Intervention analysis was also performed and classified 46 outliers which had a net negative impact of about -$7000 to Bitcoin’s price over the last 2 years of data. Thus implying that the price of Bitcoin in a timeline free of shocks would be about ~$15,400 (or 85% higher than current levels of ~$8300).
Disclaimer: This is not investment advice and is a practical example for illustrative purposes only. Please note that historical gains may not be representative of future returns. And as always before any investment, especially in cryptoworld, please thoroughly DYOR. #DoYourOwnResearch
Great work. A beautiful mind!