The domain of finance and economics uses the GARCH model frequently. The GARCH model is a time series model that helps in the analysis of different data points collected over certain periods. The GARCH model is a time series model. This model is trendy because it calculates the correlation between the data points, some underlying patterns, and the underlying conduct of the data points. The GARCH model is an extension of the ARCH model. In this article, we will see the details of the GARCH model and some implementation examples. Now, let’s get started!

## Introduction to the GARCH Model

GARCH means Generalized Autoregressive Conditional Heteroskedasticity, which is based on the statistical time series model. This model was proposed in 1982 by Robert F. Engle, and it is an extension of the ARCH model. ARCH stands for Autoregressive Conditional Heteroskedasticity. The basic functionality of this model involves finding out the pattern in the data points, which helps track future data points and results.

### Mathematical Formula of GARCH Model

The mathematical formula to represent the GARCH model is represented using two components, i.e., p and q, where p is the order of the autoregressive component and q is the moving average component. Let’s see the formula.

*σt*2 is the conditional variance, *σt*−*j*2 is past conditional variance, *εt*−*i* is past squared residuals, w is the constant term in the conditional variance equation, and *αi* and *βj* are the estimated parameters.

## Components of the GARCH model

The GARCH model has different types like ARCH, EGARCH, and TGARCH. Let’s see these types one by one.

### ARCH Model

The ARCH model, i.e., Autoregressive Conditional Heteroskedasticity is the previous version of the GARCH model. This model is also used to determine the risks in the finance sector, which is used for risk management. This model can find the relationship between two different data points, i.e., the previous data point and the current one. So, it is easy to predict future conditions. ARCH can provide more accurate estimates and predictions. Robert F. Engle proposed the ARCH model in 1982, which is the predecessor to the GARCH model. This model captures the relationship between the conditional variance of a time series and the squared residuals from the past.

### EGARCH Model

The EGARCH model, which Kevin Sheppard and Robert Engle introduced in 2001, is an expansion of the GARCH model that takes into account the unequal effects of positive and negative shocks on volatility. Instead of the conditional variance, it models the logarithm of the conditional variance. The mathematical formula for the EGARCH model is simple as follows:

### TGARCH Model

Tim Bollerslev proposed the TGARCH model, which is an extension of the GARCH model with a threshold parameter, in 1986. This parameter is important because it allows the conditional variance to react differently depending on whether the shock is positive or negative. This model is particularly helpful when there are varying volatility responses during extreme events. The mathematical formula for the TGARCH model is as follows:

## Implementation of the GARCH Model in Python

To use this model effectively, we need packages like Numpy, Pandas, and Arch. The GARCH model is fitted to the provided time series data using the arch model. But first, we will see the installation procedure for this model and then start the actual implementation.

For the implementation of the GARCH model, we need to import the ARCH module from Python. This library comes with some functions like arch_model() that help get the results. Along with this, the Numpy and Pandas libraries are also required for implementation.

```
pip install arch
```

After the installation of the Arch Module, Let’s implement the code!

```
import numpy as np
import pandas as pd
from arch import arch_model
np.random.seed(0)
returns = np.random.normal(loc=0.001, scale=0.01, size=1000)
date_range = pd.date_range(start='2023-01-01', periods=len(returns), freq='B')
data = pd.DataFrame(data={'Returns': returns}, index=date_range)
p = 1
q = 1
model = arch_model(data['Returns'], vol='Garch', p=p, q=q)
results = model.fit()
print(results.summary())
forecast = results.forecast(horizon=1)
print("Forecasted volatility for the next day:", forecast.variance[-1:].values[0, 0])
```

In this code of the model, after importing all the required modules for calculations and models, we will create random data for seeds and create a pandas dataframe for the same. Then, specify the values of p and q that are required to calculate the GARCH model in Python. p is the lag order, and q is the moving average order in this case. In the end, we are printing the summary of the results. Let’s see the result for more clarification.

Now, we can see the forecasted volatility for the next day, which is around 92%. In this way, you can fit your data points into this code and implement the model.

## Comparison of GARCH and ARCH Model

We know that the ARCH model is the base of the GARCH model. The GARCH model is more precise and has some extra components of the autoregressive model. The ARCH model has a moving average component. For time series data and analysis, the GARCH model is extensively used. The ARCH model is less accurate as compared to the GARCH model. You can also notice the mathematical difference between the formulas of both models.

## Application of the GARCH Model in the Finance And Economics Sector

The GARCH model is mostly used by the finance and economics sectors. As this model is based on estimating the results based on previous and current outcomes, it is best for stocks, bonds, and commodities and their risk management. This model is used in decision-making for option pricing and hedging strategies. Price fluctuations and the patterns used to decide future data are the most important things in this sector that can be achieved using this model.

When estimating economic relationships, the model is utilized in economics to account for the presence of varying variance in time series data. This becomes particularly crucial when working with financial and economic data that display varying variances over time.

## Limitations of the GARCH Module

Data points are important in the GARCH model as everything depends upon them. So, if we don’t have enough data points to make predictions, then it will lead to inaccuracy in the predictions. In short, we need a large database. If we compare, then the model shows less precision in smaller databases compared to large databases. Large amounts of outliers also affect the accuracy of the overall model. sometimes, the model fails to predict the accurate results. So, the best use of this model is for short-term predictions.

## Summary

This model will give a brief idea of the GARCH model. There are different sectors, like finance and economics, where this model is widely used for forecasts. The base of this model is the time series model, and the model is an extension of the ARCH model. There is a nominal difference between each model, which is reflected in the formula of the models. Modules like Numpy, Pandas, and Arch are used to implement this method in Python. A detailed explanation of the model’s implementation is also given. There are different applications and boundaries of the model, which are also explained in detail.

## References

You can read more about the arch module here.