Detrended fluctuation analysis (DFA) and detrending moving average (DMA) methods are standardly used for fractional differencing parameter d estimation. Recently, the DFA and DMA based estimators of standard regression parameters have been proposed. The estimators possess some desirable properties with regards to long-range dependence, trends, seasonalities and heavy tails. We study properties of both estimators beyond the general fractional cointegration framework, i.e. we examine a simple model yt =α+β xt + ut, where xt ~ I(d) and ut ~ I(d-b), which implies yt ~ I(\max[d,d-b]). The fractional cointegration requires b>0, while the standard cointegration CI(1,1) assumes xt,yt ~ I(1) and ut ~ I(0). We are interested in various combinations of d and b parameters (0 ≤ d,b ≤ 1, i.e. we cover not only the fractional cointegration framework). We provide a broad Monte Carlo simulation study focusing on different time series lengths, combination of d and b parameters, and on possible spurious relationships. Specifically, we compare the estimators based on DFA and DMA with the standard OLS procedure under true and spurious relationships β=0 and β ≠ 0). Based on the bias, standard error and mean squared error of the estimators, the new procedures outperform OLS for various settings (e.g. with d=1 and b<0.5).