Given input and output time series, obtaining filter design
If your system is linear, then you can find the impulse response $h(n)$ of the system using the convolution sum $y(n)=\sum_{k=0}^{N} h(k)x(n-k)$, which can be written in matrix-vector form as $y=Xh$, where $X$ is a convolution matrix of the input data and $N$ is the order of the system, say $N=10$ for example. Since you know $X$ and $y$, you can find $h$ via the Pseudo inverse, assuming you have created an over-determined system $y=Ah$. The pseudo-inverse is $h=(X^T X)^{-1}X^T y$. After finding $h$, find the FFT of $h$ or take the Laplace transform of $h$ to get the transfer function. I hope this helps you.