10.2: Autoregressive (AR) Models
A time series is a set of points that are ordered in time. Each time point is usually assigned an integer index to indicate its position within the series. For example, you can construct a time series by measuring and computing an average daily temperature. When the outcome of the next point in a time series is unknown, the time series is said to be random or "stochastic" in nature. A simple example would be creating a time series from a sequential set of coin flips with outcomes of either heads or tails. A more practical example is the time series of prices of a given stock.
When the unconditional joint probability distribution of the series does not change with time (it is time-invariant), the stochastic process generating the time series is said to be stationary. Under these circumstances, parameters such as the mean and standard deviation do not change over time. Assuming the same coin for each flip, the coin flip is an example of a stationary process. On the other hand, stock price data is not a stationary process.
This unit aims to use your knowledge of statistics to model time series data for random processes. Even though the outcome of the next time point is unknown, given the time series statistics, it should be possible to make inferences if you can create a model. The concept of a stationary random process is central to statistical model building. Since nonstationary processes require a bit more sophistication than stationary processes, it is important to understand what type of time series is being modeled. Our first step in this direction is to introduce the autoregressive (AR) model. This linear model can be used to estimate current time series values based on known past time series values. Read through this article which introduces the idea behind AR models and additionally explains the autocorrelation function (ACF).- This article delves a bit deeper into the mathematics behind AR models. You may notice a common theme developing where, as with linear regression, the least squares approach is used (in this case, to identify the model coefficients from empirical time series data).
This tutorial introduces time series analysis and concludes with coding the AR model using statsmodels. Follow along with the programming example for practice. Note that statsmodels.tsa.AR has been deprecated in favor of statsmodels.tsa.AutoReg due to processing improvements within statsmodels.