Two stochastic processes \(Y_{1,j}\) and \(Y_{2,j}\), \(j=1,2,\dots,n\) are observed, \[ Y_{1,j}=\mu_1(X_j+\sigma_1u_{1,j}), \] \[ Y_{2,j}=\mu_2(X_j +\sigma_2u_{2,j}). \] Here \(\mu_1\) and \(\mu_2\) are two real-valued regression functions, \(\sigma_1\) and \(\sigma_2\) are two positive numbers and the errors \(u_{i ,j}=\displaystyle\sum_{k=0}^{\infty} b_{i ,k}{\epsilon}_{i,j-k}\), \(i = 1,2\), where \(\epsilon_{1,j}\) and \(\epsilon_{2,j}\) are two independent sequences of i.i.d. standardized random variables. The common covariate process \(X_j\) is assumed to be a stationary long memory moving average process. The stated problem is to test the null hypothesis \(H_0:\mu_1(x)=\mu_2(x)\), for \(x\in [a,b]\) against the two-sided alternative hypothesis: \(H_a:\mu_1(x)-\mu_2(x) =\delta(x)\neq 0\) for some \(x\in [a,b]\), based on data \(X_j\), \(Y_{1,j}\), \(Y_{2,j}\), \(j=1,2,\dots,n\), provided from the model. One introduces a process \(U_n\) as a marked empirical process with the marks \(D_j:=Y_{1,j}-Y_{2,j}\) and one bases tests on Kolmogorov-Smirnov-type functions. Under some quite special assumptions, results on estimators, asymptotic behaviors and Monte Carlo simulations are shown.