# What is the natural progression from discrete AR models into continuous time?

Lets say we want to predict a single target variable and we have 10 regressors/features. Assume we would like to predict 30 days ahead (daily predictions up to 30 days ahead) and our data is a daily time-series where we have NO future regressor values.

In this case, the only way we can use a discrete model such as ARIMAX is to lag the regressors by our forecast horizon (in our case 30 days) in order to be able to pass some regressor values for future predictions. However this means we are using outdated information to predict the future - instead of using yesterday's regressor information, we are using information from 30 days ago.

From my understanding continuous time models such as Gaussian Processes, Ornstein-Uhlenbeck (which is supposedly the continuous time version of an AR process), CARMA process etc, would allow us to use ALL the data that we have to predict n-steps ahead as they are continuous time models.

Is my logic sound or is there something I am missing here?

I disagree that we are using outdated information to predict the future. At time $$t$$, if you want to predict $$y_{t+30}$$, you cannot use information from $$t+1,t+2,\dots,t+29$$ because you do not have it. You are using the freshest information available: that from time $$t$$. Therefore, I do not think that this is a problem. Consequently, I do not think it needs a solution in terms of a continuous time model. Even with a continuous time model, you will not have future data at time $$t$$, so you will have to keep the 30 day lag to be able to produce a forecast.
• I understand your point, however if we assume that current time is $t$ and if our regressors are available up to $t$, then when we shift them by 30 lags, to predict tomorrow $t+1$, we are using regressor values up to 29 days ago and the other 29 days are reserved for predicting $t+2$ until $t+30$. This means that predicting tomorrow will not use regressor data from yesterday which is probably the most relevant one. Unless I am missing something here? Commented Feb 8 at 10:54