what you propose is an ARIMA model where the past of y is used to predict the next y but expressly ignoring lags 1 , 2 and 3 .
Consider the original series being here having 60 values . Now create a supporting causal series (pseudo x) using a 3 period delay . You know have y(t) as a function of y(t-3) with 57 pairs of y and x. If you disable ARIMA structure and enable possible lags of the pseudo x series you will restrict the model to not employ lags 1,2 and 3 in the originally observed series.
Even St. Thomas needed proof ... here goes ..
I took the original series and computed the acf .. notice acf lag 3 is -.155
I then took the bivariate data set where the x variable was lag 3 of the output series and estimated an OLS regression model and specified ( no arima structure , no latent deterministic structure , no test for constantcy of parameters , no test for constacy of error variance and no stepdown AND obtained a regression coefficient of -.161 which is might darn close to -.155.
The program simulated via montecarlo bootstrapping a family (1000) of forecasts for x AND then simulated 100 values for the causal model y=f(x) and then combined them to obtain prediction limits for y taking into account the uncertainty in the future x.