Autoregressive models (AR models) are a class of statistical models that can be used to analyze time-series data, where the current value of a variable is predicted based on its past values. These models are commonly used in a variety of fields, including finance, engineering, and economics. In this article, we will explore autoregressive models in more detail, discussing their characteristics, advantages, and limitations. Additionally, we will provide an easy-to-understand implementation of an AR model.

## What are autoregressive models?

An autoregressive model is a time-series model that represents a dependent variable as a function of its own past values. In other words, an AR model is a linear regression model where the dependent variable is regressed on its own past values. The order of an AR model determines how many past values are used to predict the current value. For example, a first-order autoregressive model, denoted as AR(1), uses only the previous value to predict the current value, while a second-order autoregressive model, denoted as AR(2), uses the two previous values.

### Advantages of autoregressive models

AR models have several advantages over other time-series models.

- They are relatively simple to understand and implement.
- They can capture both short-term and long-term patterns in the data, making them useful for forecasting future values.
- They are efficient in terms of computational complexity, which makes them suitable for analyzing large datasets.

## Limitations of autoregression

Despite their advantages, AR models have some limitations that need to be taken into account.

- They assume that the time-series data are stationary, meaning that the mean and variance of the data remain constant over time. If the data are non-stationary, the model may not be accurate.
- They do not take into account the effect of external factors on the dependent variable. If the data are influenced by external factors, the model may not capture this effect.
- Finally, they are sensitive to outliers in the data, which can affect the accuracy of the model.

An improvement over AR model is ARIMA model. Refer to this link for more information.

## Implementing Autoregressive Models in Python

In this section, we will provide an easy-to-understand implementation of an autoregressive model. We will use the Python programming language and the Statsmodels library, which provides a wide range of tools for statistical analysis.

First, we need to import the necessary libraries:

```
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.ar_model import AutoReg
```

### Load time series data

Next, we need to load the time-series data that we want to analyze. For this example, we will use the Airline Passengers dataset, which contains the monthly number of airline passengers from January 1949 to December 1960.

```
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/airline-passengers.csv'
df = pd.read_csv(url, header=0, index_col=0, parse_dates=True)
```

### Plot and Visualize Data

We can then plot the data to visualize its pattern over time.

```
plt.plot(df)
plt.show()
```

The plot shows that the number of airline passengers increases over time, but there is also a seasonal pattern in the data.

### Create an AR Model

We can then create an AR model by specifying the order of the model and fitting it to the data.

```
model = AutoReg(df, lags=20)
model_fit = model.fit()
```

In this example, we have specified a first-order autoregressive model (lags=20), which uses 20 previous value to predict the current value. We can then print a summary of the model to see its coefficients and other statistical measures.

Using more lags can capture longer-term patterns in the data, but can also make the model more complex and computationally intensive. It is important to choose the appropriate number of lags to balance the trade-off between capturing long-term patterns and keeping the model simple.

The choice of the number of lags can also depend on the characteristics of the data ( by using auto-correlation) being analyzed and the goals of the analysis.

```
print(model_fit.summary())
```

The output of the code provides a summary of the autoregressive model, including the coefficients, standard errors, t-values, and p-values. Here I’ve shown it only for the first 10 passengers. We can use these values to evaluate the performance of the model and make predictions for future values.

To make predictions with the autoregressive model, we can use the forecast method, which takes as input the number of periods to forecast.

```
forecast = model_fit.forecast(steps=24)
```

In this example, we have specified to forecast the next 24 months. We can then plot the predicted values along with the original data to visualize how well the model fits the data and how well it predicts future values.

```
plt.plot(df, label='Actual')
plt.plot(forecast, label='Predicted')
plt.legend()
plt.show()
```

The output of the above code shows a plot of the original data and the predicted values for the next 24 months. The predicted values follow the seasonal pattern of the data, and the model captures the general trend of the data.

## Conclusion

AR models are a class of statistical models that can be used to analyze time-series data. These models are relatively simple to understand and implement and can capture both short-term and long-term patterns in the data. They have some limitations, including their assumption of stationarity and their sensitivity to outliers.

In this article, we provided an easy-to-understand implementation of an AR model using the Python programming language and the Statsmodels library. We used the Airline Passengers dataset as an example and showed how to create an AR model, make predictions, and evaluate the performance of the model.