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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?

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  • $\begingroup$ 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. $\endgroup$
    – IrishStat
    Feb 11 '20 at 15:51
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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 enter image description hereoriginal series being here having 60 values . Now create a supporting causal series (pseudo x) using a 3 period delay enter image description here . 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 enter image description here

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.

enter image description here

enter image description here

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    $\begingroup$ Not quite: it is a constrained model in which the immediately preceding $l-1$ values of $y$ are not used for the prediction. $\endgroup$
    – whuber
    Feb 6 '20 at 23:32
  • $\begingroup$ 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 $\endgroup$
    – IrishStat
    Feb 7 '20 at 0:45
  • $\begingroup$ 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 $y_4 = b_1y_1+e_t$ $\endgroup$ Feb 7 '20 at 6:11
  • $\begingroup$ 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. $\endgroup$
    – whuber
    Feb 7 '20 at 16:53
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    $\begingroup$ 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. $\endgroup$
    – Chris Haug
    Feb 8 '20 at 1:56

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