# Autoregressive model

Autoregressive model | |
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**Autoregressive models** (AR) are linear models that assume a linear relationship between lagged values of the same variable over time. This relationship is used to predict future values of the same variable. Specifically, an AR model assumes that the current value of a time series is a linear combination of its past values plus some random noise. For example, an AR(1) model is given by the formula

\( y_t = \phi_1 y_{t-1} + \epsilon_t\)

where $y_t$ is the current value of the time series, /phi&_{1} is an autoregressive parameter, and /epsilon;_{t} is white noise. A higher order AR model can be specified by increasing the order of the lag, such that an AR(2) model is given by the formula

\(y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \epsilon_t\)

The autoregressive parameters /phi&_{1} and /phi&_{2} can be estimated using least squares regression or maximum likelihood estimation. Autoregressive models are useful for forecasting, as they can capture the underlying dynamics of a time series and make accurate predictions. They have been widely used in economics and finance, as well as in other fields such as meteorology and engineering.

## Example of Autoregressive model

One example of an autoregressive model is the Autoregressive Integrated Moving Average (ARIMA) model. ARIMA models are a type of autoregressive model that include an additional component known as an integrated component. The integrated component accounts for the non-stationary nature of some time series. The general form of an ARIMA(p,d,q) model is given by the formula

\((1-\phi_1 B - \phi_2 B^2 - ... - \phi_p B^p)(1 - B)^d y_t = (1 + \theta_1 B + \theta_2 B^2 + ... + \theta_q B^q)\epsilon_t\)

where B is the backshift operator, /phi&_{i} are autoregressive parameters, /theta&_{i} are moving average parameters, and /epsilon;_{t} is white noise. The autoregressive and moving average parameters can be estimated using maximum likelihood estimation. ARIMA models can be used to identify seasonal patterns in time series data, as well as to forecast future values of the time series.

In summary, autoregressive models are linear models that assume a linear relationship between lagged values of the same variable over time. They are useful for forecasting and have been widely used in economics and finance, as well as in other fields. An example of an autoregressive model is the ARIMA model, which can be used to identify seasonal patterns and forecast future values of a time series.

## Formula of Autoregressive model

Autoregressive models are linear models that assume a linear relationship between lagged values of the same variable over time. This relationship is specified by the formula\[y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \ldots + \phi_p y_{t-p} + \epsilon_t\]

where y_{t} is the current value of the time series, /phi;_{1} to /phi;_{p} are the autoregressive parameters, and /epsilon_{t} is white noise. The autoregressive parameters can be estimated using least squares regression or maximum likelihood estimation.

## When to use Autoregressive model

Autoregressive models are best suited for predicting future values of a time series when there is a clear linear relationship between past values of the same variable. Specifically, they are most effective when the current value of a time series is strongly correlated with past values of the same variable. They are also more effective when the data is stationary, meaning that the mean, variance and covariance are constant over time.

In addition, autoregressive models are well suited for forecasting when there is no trend or seasonality in the data, as these effects can be difficult to capture with an autoregressive model.

Overall, autoregressive models are useful for forecasting when there is a clear linear relationship between past values of the same variable, the data is stationary, and there is no trend or seasonality in the data.

## Types of Autoregressive model

Autoregressive models come in a variety of forms, each with different assumptions and assumptions about the data. The three most common types are:

**Autoregressive (AR) models**: These models use lagged values of a single variable to predict future values of that same variable.**Autoregressive Moving Average (ARMA) models**: These models use lagged values of both the dependent and independent variables to predict future values of the dependent variable.**Autoregressive Integrated Moving Average (ARIMA) models**: These models use differenced lagged values of both the dependent and independent variables to predict future values of the dependent variable.

In each case, the autoregressive parameters are estimated using least squares regression or maximum likelihood estimation. Autoregressive models are versatile tools for forecasting, as they are able to capture the underlying dynamics of a time series and make accurate predictions. They have been widely used in economics and finance, as well as in other fields such as meteorology and engineering.

## Steps of applying Autoregressive model

**Step 1**: Specify the model order (p). This is the number of lags used in the model and is typically determined using statistical tests such as the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC).**Step 2**: Estimate the autoregressive parameters. This can be done using least squares regression or maximum likelihood estimation.**Step 3**: Evaluate the model fit. The fit of the model can be evaluated using metrics such as the root mean squared error (RMSE) or the mean absolute error (MAE).**Step 4**: Make predictions. Once the model is fitted, it can be used to make predictions about future values of the time series.

## Advantages of Autoregressive model

Autoregressive models have several advantages over other forecasting models. Firstly, they can capture both short-term and long-term patterns in the data. This makes them better suited to forecasting than other models such as moving average models. Secondly, they are simple to implement and interpret, as the coefficients are easy to interpret and interpretable. Lastly, they are easy to extend to higher order models, allowing for more accurate predictions.

## Limitations of Autoregressive model

Despite their widespread use and usefulness, Autoregressive models are limited in their ability to capture non-linear relationships between time series. Additionally, Autoregressive models tend to overestimate extreme values and can be difficult to interpret. As such, they should always be used with caution, and in conjunction with other forecasting models such as exponential smoothing or ARIMA.

There are several other approaches that are related to the Autoregressive model, such as:

**ARIMA (AutoRegressive Integrated Moving Average)**: An ARIMA model is an extension of an AR model that incorporates both autoregressive and moving average components. It is used to capture both short-term and long-term trends in a time series.**ARCH/GARCH (Autoregressive Conditional Heteroskedasticity/Generalized Autoregressive Conditional Heteroskedasticity)**: The ARCH/GARCH models are used to capture the time-varying volatility of a time series. They assume that the variance of the time series is a function of its past values and that the variance is not constant over time.**Vector Autoregressive (VAR)**: Vector Autoregressive models are used to model the relationship between multiple time series. They assume that each series is a linear combination of its own past values and the past values of the other series in the model.

These related approaches are all useful for time series analysis, and can be used to gain a better understanding of the dynamic behavior of a time series. They can also be used to make more accurate predictions than an Autoregressive model alone.

In summary, Autoregressive models are linear models that are used to capture the dynamics of a time series and make predictions about its future values. There are several related approaches, such as ARIMA, ARCH/GARCH, and VAR, that can be used to gain a better understanding of the time series and make more accurate predictions.

## Suggested literature

- Akaike, H. (1998).
*Autoregressive model fitting for control*. Selected Papers of Hirotugu Akaike, 153-170. - Kokoszka, P., & Reimherr, M. (2013).
*Determining the order of the functional autoregressive model*. Journal of Time Series Analysis, 34(1), 116-129. - Davis, R. A., Huang, D., & Yao, Y. C. (1995).
*Testing for a change in the parameter values and order of an autoregressive model*. The Annals of Statistics, 282-304.