Thank you sir for your clarification.

Shihab,

the common part to both model classes are the autoregressive AR terms. The main difference between the NARMA and the NARX, is that the latter adds variables from an eXogenous input (that is the typical case with input-output systems), whereas the former adds Moving Average terms, which are "noise" terms and therefore unmeasured (that is the typical case with time seires, where we have no inputs). In practice the noise terms are replaced by the residuals. So, the NARMA model can be interpreted as an "input"-output model where the "input" is an unmeasured stochastic process (residuals) responsible for the variability in the output. NARX models are input-output models where the input is measured. Hence, practical estimation of NARX models is typically easier.