# Forecasting with after x lags values

I like to build a forecasting model where am allowed to use only l lagged values.

That means the model should forecast only l lagged values like $$y_{t}$$ can be only predicted using values $$y_{t-l}$$, $$y_{t-l-1}$$, etc.

$$y_{t}=b_{t-l}y_{t-l} + b_{t-l-1}y_{t-l-1} + {...} + b_0y_0 + e_t$$

Could you suggest which model should work for such problem?

• I suggested a transfer function where one could include lags of the pseudo x variable that was generated . This allows you to form the model that you are after where lags BEFORE l are precluded. If so then please upvote and accept or explain to me why my answer does not work for you using ANY software of your choice. Feb 11 '20 at 15:51

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.

Q.E.D.

• Not quite: it is a constrained model in which the immediately preceding $l-1$ values of $y$ are not used for the prediction.
– whuber
Feb 6 '20 at 23:32
• that is still an arima model where the # of auto-regressive coefficients is NOT equal to # . For example the model y(t)= .2 *y(t-1) +.3*y(t-4)+ .2*y(t-9) is an ARIMA model , Generalized arima modelling is not restricted to (p,d,q)(P,D,Q) like auto.arima Feb 7 '20 at 0:45
• not convinced. we cannot use preceding $l-1$ values for predicting $y_t$ as @whuber mentioned. For example, if $l=3$, to forecast $y_4$, we can only use $y_1$. so the model can be $y_4 = b_1y_1+e_t$ Feb 7 '20 at 6:11
• How does creating a "supporting causal series" work for providing correct estimates of variances, prediction intervals, and so on? I'm not saying it doesn't; I'm only saying you have a burden of demonstrating this approach is correct and by simply asserting it's a solution you haven't met that burden.
– whuber
Feb 7 '20 at 16:53
• This answer seems to be focused on the specific contortions required to get AUTOBOX to fit a specific model which, aside from the fact that it doesn't appear to be the one that was asked about, is completely off topic here. Feb 8 '20 at 1:56