Time Series Model Building Process,

Introduction
Linear time series models, e.g., ARIMA models for univariate time series, is a popular tool for modeling dynamics of time series and predicting the time series. The methodology is not popular only in statistics, econometrics and science, but also in machine learning business applications. The key question is how to build the model. We have to choose the form of the model, in particular number of lags for so-called autoregressive part (usually denoted as p) and moving average part of the model (usually denoted as d). The common guides usually provide a ‘cook book’ for model selection, see for example ARIMA Model – Complete Guide to Time Series Forecasting in Python. This goal of this simulation is to compare four basic methods for selection of lags (p) for the autoregressive part.

Problem Definition
The autoregressive process has form

(math formula that is difficult to format in this editor -_-")

However, we observe only the realized value and we do not know the hyperparameter p. Therefore, we must estimate it. The considered methods are following.