Say I have 5 minutes of input and output audio. One method I know is to do FFT on windows from input and output. Then divide FFT output into bins and find average energy in various bands for all the windows. After this, take ratio of energy bands from output windows with corresponding energy bands from input windows. And then take average of all these ratios. This will give an approximation to FFT(Y)/FFT(X) i.e. transfer function of the filter.
Are there any other methods to find transfer function given input and output time series?