What are autoregressive models?

Autoregressive (AR) models are machine learning (ML) models that use previous inputs to predict the next component in a sequence. This kind of analysis is used when there is a correlation between the time series values and their previous and subsequent values. Autoregression is a statistical approach used in time-series analysis that assumes that a time series’ current value is determined by its previous values. For example, an autoregressive model might try to forecast a stock’s future price based on its past performance.

Basics of autoregressive models

Let’s discuss the basic concepts behind autoregressive models:

1. Time series data: It studies sequential observations across different periods to predict events such as stock prices and temperature readings by analyzing the data’s patterns, trends, and relationships.

2. Concept of autoregression: The theory behind autoregression is that a time series variable’s current value depends linearly on its past values plus a random error component. This concept operates under the assumption that a variable’s past behavior influences its behavior in the present.

3. Mathematical representation: An autoregressive model of order $n$ may be represented mathematically as follows:

• $X_t$: It represents the current value or predicted value.

• $c$: It is a constant term.

• $w_1$: It is the model parameter, representing the weight associated with past values.

• $\epsilon_t$: It is the error term, commonly considered white noise, representing unexplained variance or randomness at time t.

From the above equation, we can observe that the current value of the time series $X_t$ is a linear combination of its past value $X_{t-1}$, multiplied by an autoregressive parameter $w$​, with an added constant term $c$ and random error $\epsilon$.

Characteristics of autoregressive models

The following are the properties of autoregressive models:

• Stationary: Autoregressive models often assume stationarity, meaning the time series’ statistical features, such as mean and variance, stay constant all the time. This guarantees that the relationships recorded by the model are consistent.

• Order: The order of autoregression refers to the number of lagged values of the dependent variable in the model. It specifies how far back in time the model searches to forecast the present value.

• Coefficients: Autoregressive models estimate coefficients to represent the relationship between past and current values of a variable. Coefficients are determined through statistical methods like ordinary least squares (OLS) or maximum likelihood estimation (MLE).

Types of autoregressive models

These are the various kinds of autoregressive models mentioned below:

• AR(1) model: These are the simplest autoregressive models in which the variable’s current value is predicted based on its prior value. The model’s equation is:

• AR(n) model: These model uses $n$ number of lagged values for prediction. The equation for an AR(n) model is:

• Moving average (MA) model: These models represent the relationship between an observation and the resulting error of a moving average model applied to lagged observations. The equation for the MA model is:

Several more autoregressive models, such as autoregressive integrated moving average (ARIMA), seasonal ARIMA (SARIMA), etc are used in time series analysis and forecasting. Each has unique traits and serves a certain purpose.

Pros and cons of autoregressive models

In conclusion, we learned about autoregressive (AR) models and how they utilize past inputs to predict the next component in a sequence. Autoregressive models are flexible tools with many applications in various disciplines, such as NLP, time series forecasting, econometrics, etc.