Transfer function from experimental time domain data
assuming your sets of time-domain data begin with $n=0$, Z transform both the input data, $x[n]$ and the output data, $y[n]$.

$$ X(z) \triangleq \mathcal{Z}\{x[n]\} = \sum\limits_{n=0}^{\infty} x[n] \ z^{-n} $$
$$ Y(z) \triangleq \mathcal{Z}\{y[n]\} = \sum\limits_{n=0}^{\infty} y[n] \ z^{-n} $$

divide the Z transform of the output by the Z transform of the input.  the result of that division is your transfer function.

$$ H(z) \triangleq \frac{Y(z)}{X(z)} = \frac{\sum\limits_{n=0}^{\infty} y[n] \ z^{-n} }{\sum\limits_{n=0}^{\infty} x[n] \ z^{-n} } $$

i cannot guarantee that, if you were to factor numerator and denominator, there will not be identical poles with zeros that could cancel and make the transfer function simpler, but it's a transfer function.