# Interpreting Cross-Correlation Function betwen residuals of ARIMA-SARIMA models

I've seen several posts this week around this topic, but I cannot find a right answer, probably because time series is something very artisan! I'm doing a task about finding causality between surface temperature and sea level in a specific zone. The thing is to explain the relation of the process through a CCF and Transfer Function. I have modelled my series with much care as I could. In R, I took 1 difference and logarithms for $$x_t$$ and only logarithms for $$y_t$$ (this one is really seasonal - sea level!) The best models arisen from auto.arima had been :

for $$x_t$$

ARIMA(2,0,1) with zero mean

Coefficients:
ar1     ar2      ma1
0.4138  0.2498  -0.9517
s.e.  0.0526  0.0511   0.0231

sigma^2 estimated as 0.006168:  log likelihood=513.04
AIC=-1018.09   AICc=-1018   BIC=-1001.6


and $$y_t$$

ARIMA(1,0,1)(0,1,2) with drift

Coefficients:
ar1     ma1     sma1    sma2   drift
0.5607  0.4494  -0.8262  0.1173  -5e-04
s.e.  0.0493  0.0545   0.0508  0.0490   1e-04

sigma^2 estimated as 0.0009074:  log likelihood=922.19
AIC=-1832.38   AICc=-1832.19   BIC=-1807.81


Then, I got this graph from ccf(x$residuals, y$residuals), which I am triyng to interpret: Questions:

a) In this example, at $$k$$ = 4, $$x_{t + k}$$ is significantly correlated with $$y_t$$? My hypotesis is that there is backfeed but I'm not sure of that.

b) How I can calculate and express the transfer function with the information above? (e.g with MTS::tfm1)