ARMA and GARCHA: Bitcoin’s Dependency Analysis On The Exchange Rate Of Global Fiat Currencies


Throughout the past few years, the value of bitcoin, and a handful of other cryptocurrencies, has skyrocketed. Moreover, bouts of uncertainty in regards to the fiat economy have been usually accompanied by sharp bitcoin price surges during the past couple of years which urged a large number of researchers to study the dependency between bitcoin and some of the world’s fiat currencies.

Throughout a paper that was published on the latest issue of the “Dynamic Econometric Models”, a group of researchers tried to find the relationship between bitcoin price and some of the world’s leading fiat currencies including the US Dollar, Sterling Pound, Euro, Polish zloty and Chinese Yuan. The researchers utilized the ARMA and GARCH models to analyze the values of variance and conditional means of bitcoin and those Fiat currencies which led to some interesting results. So, let’s summarize the most important points in this paper:

Methods Used In The Dependency Analysis:

To analyze the dependency between the exchange rates of bitcoin and selected fiat currencies during specific time windows, the researchers utilized vector auto-regressive models that were first used in 1980. The problem is that analyzing the relationship between time series of different currency exchange rates represents a complex and daunting process. Dependent variables are formulated using a group of explanatory variables. Vector auto-regressive models are ideal for simultaneous analysis of relationships as such. These models are perfect for this type of study because they lack restrictions regarding categorizing variables into “endogenous” and “exogenous”. VAR also permits analysis of bidirectional relationships, i.e. when two variables can influence each other which aids in the examination of interdependence between the EURUSD exchange rate and a group of eight variables.

VAR is based on two models; ARMA and GARCH including all their modifications and generalizations. VAR is usually used to formulate return rates of various forms of assets. ARMA aids in formulating the conditional mean, while GARCH is utilized to formulate the conditional variance. In literature, combining these two models has been proven efficient in analyzing the exchange rates of various fiat currencies.

ARMA (p, q) represents a process that combines two processes:

1- An auto-regressive order p : AR (p)

2- Moving average of order q: MA (q)

Now, we can borrow Montgomery’s formula to calculate the auto-regressive process of the order p:

= δ + + +…… + +

where represents a white noise and ,…, represent parameters.

Process realization at a specific point in time t is related to p previous observations and the value of white noise at the same point t . On the other hand, if process realization at t time is related to q prior random terms, then this process would represent a moving average of the order q. Such dependency can be calculated by the following formula:

= + +…+ =

where represents a white noise and ,…., are parameters

The ARMA model represents a combination of the auto-regressive process of order p along with a moving average of order q and can be calculated using the following formula:

= δ + + +

ARCH models rely on the assumption that process residuals’ variance are not constant with reference to different points of time. ARCH models consider the error’s term conditional variance at point t of time is determined by the previous error’s term p:

= +


where represents variance, ,…., represent parameters, is a constant and represent the error’s term.

The GARCH model represents a generalization of the ARCH model and can be represented by the following formula:

= + +

where represents variance at time point t and ,…., and ,…., are parameters.

Results of the Dependency Analysis:

As regards the conditional mean, that was represented by the ARMA model, the researchers concluded that bitcoin price is independent from the influence of all the currencies which were analyzed in the study; US Dollar, Sterling Pound, Euro, Chinese Yuan and Polish Zloty. However, in terms of the GARCH model, the researchers concluded that bitcoin price is related to the logarithmic rate of exchange of US Dollar, Euro, Chinese Yuan to Polish Zloty, but independent from the exchange rate of Sterling Pound to Polish Zloty. The researchers also concluded that exponential GARCH models are the best to determine the conditional variance of bitcoin.

The results of the study show that the price of bitcoin, in Polish Zloty, is not totally independent from the external influences of fiat currencies’ exchange rates, which can open the door to control by third parties and speculators who can make enormous profits.

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