Regression error propagated over time

Question

I am looking for a reference request on the following problem related to regression.

In statistics, we learn about the model for simple linear regression in which we fit $y_i = ax_i + b + \epsilon_i$ where $\epsilon_i \sim N(0,\sigma^2)$. We then find estimators for $a,b$ denoted by $\hat{a},\hat{b}$. If $\sigma^2 \neq 0$, we know that $\hat{a},\hat{b}$ (which are random variables do not equal the model parameters $a,b$. We can then find statistics about the discrepancy between the predicted $\hat{y}_i$ (which uses $\hat{a},\hat{b}$) and the model $y_i$, etc.

I am wondering if some results are available for "time-dependent" regression. That is, the model looks something like $x_{i+1}=ax_i + b + \epsilon_i$. Suppose that $\hat{a},\hat{b}$ are obtained from available measurements up to a time index $K$, i.e. $i = 1,\dots,K$. I am particularly interested in how the errors in the estimators $\hat{a},\hat{b}$ propagate when we do prediction for time beyond $K$. Are there any references on this topic?

Answer 1

The model you constructed is the same as the AR model. It is the special case of a more general ARIMA model from the time series analysis. I give you the Wikipedia link: https://en.wikipedia.org/wiki/Autoregressive_model. The AR model is well established already and the various aspects of the model have been researched including the forecasting.