I have a question regarding the following model for the short rate in two currencies and the exchange rate, [$]x(t)[$],[$]\begin{align}dr_d(t) &= \left[ \nu_d(t) - \kappa_d r_d(t) \right] dt + \sigma_d(t) dW_d(t) \\dr_f(t) &= \left[ \nu_f(t) - \kappa_f r_f(t) \right] dt + \sigma_f(t) dW_f(t) \\dx(t) &= r_{d,f}(t) x(t) dt + \sigma_x(t) x(t) dW_x(t)\end{align}[$]Here, [$]W_d,W_f,W_x[$] are correlated Brownian motions under domestic risk neutral measure with instantaneous correlation matrix[$]\rho=\begin{bmatrix}1 & \rho_{fd} & \rho_{xd} \\\rho_{fd} & 1 & \rho_{fx} \\\rho_{xd} & \rho_{fx} & 1 \\\end{bmatrix}[$]If we assume a form for the diffusion coefficients in each SDE, e.g. piecewise constant, the IR models can be calibrated separately to swaption prices and the FX model to FX option prices.Assuming that there are no market instruments available to calibrate the instantaneous correlation parameters, does anyone have any hints or references for estimating the instantaneous correlation parameters from historical time series?I have seen similar questions asked in other posts but there is not much detail on how to do the historical estimation. Is there an example of what market instruments (or derived rates) to use to form the IR time series in each currency and what the estimate is for that choice of market instruments?Any hints or approximations on this are much appreciated.

Since the one-factor mean reverting behavior in each currency's interest rates is badly mis-specified in (at the very least) the post-crisis world, I would not try to estimate the correlations from short rate moves. I would rather pick a maturity most relevant to the problem you are trying to solve (e.g. somewhere in the 10-30 year range if looking at PRDCs), calculate the correlation of basis point change in rates for that maturity with each other and with log FX returns, and back into what instantaneous correlations would produce those term correlations. Minor caveat: I haven't seriously looked at these kind of problems since 2002.