# The Ecosystem Relationship

As Kelp launches as the first cryptocurrency governed by the workings of a Monetary Policy framework, the model that examines interactions among variables within the ecosystem need to be set out. The Vector Autoregressive (VAR) based Johansen cointegration analysis will be applied to all our variables, and in the absence of cointegration, we will apply a vector autoregressive model to the return of the variables. The VAR-based Johansen (1988) co-integration test is specified by the following equation:

$\text{ë}_{Trace}(r|n) = - T \sum_{i=r+1}^n log(1 - \text{ë}_i) \tag{13}$Where:

$r$ = number of cointegrating relations

The null hypothesis for the trace statistics tests that there are at most 𝑟 cointegrating relations, against the alternative of 𝑛 relations, 𝑟=0,1,…𝑛−1. The maximum eigenvalue statistics tests the null hypothesis that there are 𝑟 cointegrating relations against the alternative that there are 𝑟+1 cointegrating relations.

The dynamic interrelationship between money supply and other macroeconomic variables is established using the unrestricted VAR model. The multivariate VAR is specified as:

$Z_t = a + \Phi_1 Z_{t-1} + ... + \Phi_k Z_{t-k} + \varepsilon_t \tag{14}$Where:

$Z_t$ = a $k$ vector of endogenous variables

$\Phi_1 ... \Phi_k$ = matrices of estimated coefficients

$\varepsilon_t$ = is a vector of innovations, which may be contemporaneously correlated with each other but are uncorrelated with their own lagged values and all the right-hand side variables (Lütkepohl, 1991)

The impulse-response function from the estimated model will track the effect of the shock of current innovations on the future values of the remaining variables. This allows us to measure the sensitivity of each variable to shocks emanating from within itself as well as from other variables.

The Kelp Protocol seeks to ensure a stable ecosystem using levers available within the ambit of the Kelp ecosystem. For monetary policy to be necessary, there has to be an economy, which in this case consists of the goods and services exchanged within the Kelp ecosystem. This, we will term KelpAct, which stands for Kelp Activity, and will be calculated using the GDP value-add approach, where:

$KelpAct = P_1 Q_1 + P_2 Q_2 + P_3 Q_3 ... + P_k Q_k \tag{15}$Where:

P = Price

Q = Quantity

The value added at any particular stage is simply the difference between input cost and the cost of the final goods and services. This approach ensures we avoid the problem of double counting and the ambiguity associated with trying to isolate only final goods and services.

The rate of change in the prices of goods and services exchanged within the Kelp ecosystem will be measured by the Kelp Price Index (KPI), where:

$Rate \space of \space Inflation = \frac{KPI_{x+1} - KPI_x}{KPI_x} \tag{16}$ $KPI_x = Initial \space Kelp \space Price \space Index \tag{17}$The third macroeconomic indicator of interest is the $KELP exchange rate, which is indicative of how competitive or otherwise $KELP is compared to other cryptocurrencies. Given the volatility within the cryptocurrency space, the true $KELP exchange rate will be determined using a basket of currencies, which will include the two top cryptocurrencies by market capitalization and volume (BTC & ETH), as well as the Tether (USDT) because of its characteristic as a good store of value. At the point of construction, the weight of each cryptocurrency in the basket will be assigned by the inverse of the preceding year’s volatility, specified as:

$W_x = \frac{1}{\sigma_{BTC}} + \frac{1}{\sigma_{ETH}} + \frac{1}{\sigma_{USDT}} \tag{18}$