# #StackBounty: #neural-network #statistics #recurrent-neural-net #forecast #forecasting Is an Arma model equivalent to a 1-layer Recurre…

### Bounty: 50

Given a time series $$f(t)$$ to forecast, let us consider an Arma model of the form:
$$f(t) = c + sum_{i=1}^p a_i f(t-i) + e(t) + sum_{j=1}^q b_j e(t-j)$$

where $$e(t)$$ are the forecast errors.

On the train set, if $$f(t)$$ is the ground truth, then we define its estimate obtained with this model as $$widetilde{f}(t) = f(t) + e(t)$$.

Let $$m = min(p,q)$$, we can rewrite the first equation as:
$$widetilde{f}(t) = c + sum_{i=1}^m (a_i + b_i) f(t-i) + sum_{i=m+1}^p a_i f(t-i) – sum_{j=1}^q b_j widetilde{f}(t-j)$$
Then after reparametrization can be rewritten as:
$$widetilde{f}(t) = c + sum_{i=1}^k c_i f(t-i) – sum_{j=1}^q b_j widetilde{f}(t-j)$$
Which is the equation of a 1-layer recurrent neural network (RNN) without activation function.

So, are Arma models a subset of RNNs or is there a flaw in this reasoning ?

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