Example of garch formula

(Comments)

The example

Image by Freepik

\[ \begin{align*}
Y_t &= \mu + \varepsilon_t \\
\varepsilon_t &= \sigma_t \cdot z_t \\
\sigma_t^2 &= \omega + \alpha \cdot \varepsilon_{t-1}^2 + \beta \cdot \sigma_{t-1}^2
\end{align*} \]

Certainly! Here's the LaTeX formula for a GARCH(1,1) model, which is one of the simplest forms of a Generalized Autoregressive Conditional Heteroskedasticity model:

```latex
\begin{align*}
Y_t &= \mu + \varepsilon_t \\
\varepsilon_t &= \sigma_t \cdot z_t \\
\sigma_t^2 &= \omega + \alpha \cdot \varepsilon_{t-1}^2 + \beta \cdot \sigma_{t-1}^2
\end{align*}
```

In this formula:

- \(Y_t\) represents the observed time series data.
- \(\mu\) is the mean of the time series.
- \(\varepsilon_t\) is the residual or innovation at time \(t\).
- \(\sigma_t\) is the conditional standard deviation (volatility) at time \(t\).
- \(z_t\) is a white noise error term with mean zero and unit variance.
- \(\omega\), \(\alpha\), and \(\beta\) are the model parameters to be estimated. \(\omega\) is the constant term, \(\alpha\) is the autoregressive term for squared residuals, and \(\beta\) is the autoregressive term for conditional variances.

Now, let's provide an example. Suppose we have daily stock returns and want to estimate a GARCH(1,1) model for volatility forecasting. We have the following data:

- \(Y_t\) represents the daily stock returns.
- \(\mu = 0.001\) (mean return).
- We estimate \(\omega = 0.0001\), \(\alpha = 0.2\), and \(\beta = 0.7\) as model parameters.

You can use these values in the GARCH(1,1) model formula to forecast the conditional standard deviation (\(\sigma_t\)) for future time periods. This can be done recursively, starting from an initial estimate of \(\sigma_0\).

Please note that in practice, model parameters are estimated from historical data using statistical software or programming languages like R, Python, or specialized econometric software.

Certainly! The GARCH(1,1) model is a way to understand and forecast volatility (how much a value fluctuates) in time series data, like stock prices. Here's a simpler explanation:

1. **Mean:** Suppose you have some data, like daily stock returns. You start by estimating the average (mean) return, which is represented by \(\mu\). This is just the typical return.

2. **Volatility:** Next, you want to know how much the returns vary from day to day. This is where the GARCH model comes in. The idea is to estimate the "volatility" of the returns, represented by \(\sigma_t\), for each day.

3. **How Volatility Changes:** The GARCH(1,1) model says that the volatility today (\(\sigma_t\)) depends on three things:
- A constant term \(\omega\) (like a baseline volatility).
- The squared error from yesterday (\(\varepsilon_{t-1}^2\)) which measures how much the returns deviated from the mean yesterday.
- The previous day's estimated volatility (\(\sigma_{t-1}^2\)), which accounts for how volatile it was yesterday.

4. **Parameters:** There are three numbers to estimate (\(\omega\), \(\alpha\), and \(\beta\)) that control how these factors affect today's volatility. \(\omega\) is like a starting point for volatility, \(\alpha\) measures how sensitive today's volatility is to past errors, and \(\beta\) measures how persistent past volatility is.

So, in practice, you estimate these three numbers from historical data to create a model. Then, you can use this model to forecast how volatile the data is likely to be in the future.

For example, if you estimate \(\omega = 0.0001\), \(\alpha = 0.2\), and \(\beta = 0.7\), you can use this model to predict future volatility based on past data.

Currently unrated

Comments

Riddles

22nd Jul- 2020, by: Editor in Chief
524 Shares 4 Comments
Generic placeholder image
20 Oct- 2019, by: Editor in Chief
524 Shares 4 Comments
Generic placeholder image
20Aug- 2019, by: Editor in Chief
524 Shares 4 Comments
10Aug- 2019, by: Editor in Chief
424 Shares 4 Comments
Generic placeholder image
10Aug- 2015, by: Editor in Chief
424 Shares 4 Comments

More News  »

Fixing the issue in assumption of OLS step by step or one by one

Recent news

Hi, I want to raise the issue related to know whether your OLS is ok or not. 

read more
3 weeks, 1 day ago

Meaning of 45 degree in economics chart

Recent news

The **45-degree line** in economics and geometry refers to a line where the values on the x-axis and y-axis are equal at every point. It typically has a slope of 1, meaning that for every unit increase along the horizontal axis (x), there is an equal unit increase along the vertical axis (y). Here are a couple of contexts where the 45-degree line is significant:

read more
1 month, 3 weeks ago

hyperinflation in hungary

Recent news

The **hyperinflation in Hungary** in the aftermath of World War II (1945–1946) is considered the worst case of hyperinflation in recorded history. The reasons behind this extreme economic event are numerous, involving a combination of war-related devastation, political instability, massive fiscal imbalances, and mismanagement of monetary policy. Here's an in-depth look at the primary causes:

read more
2 months ago

what is neutrailty of money

Recent news

**Neutrality of money** is a concept in economics that suggests changes in the **money supply** only affect **nominal variables** (like prices, wages, and exchange rates) and have **no effect on real variables** (like real GDP, employment, or real consumption) in the **long run**.

read more
2 months ago

Japan deflationary phenomenon

Recent news

Deflation in Japan, which has persisted over several decades since the early 1990s, is a complex economic phenomenon. It has been influenced by a combination of structural, demographic, monetary, and fiscal factors. Here are the key reasons why deflation occurred and persisted in Japan:

read more
2 months ago

What the tips against inflation

Recent news

Hedging against inflation involves taking financial or investment actions designed to protect the purchasing power of money in the face of rising prices. Inflation erodes the value of currency over time, so investors seek assets or strategies that tend to increase in value or generate returns that outpace inflation. Below are several ways to hedge against inflation:

read more
2 months ago

Long and short run philip curve

Recent news

The **Phillips Curve** illustrates the relationship between inflation and unemployment, and how this relationship differs in the **short run** and the **long run**. Over time, economists have modified the original Phillips Curve framework to reflect more nuanced understandings of inflation and unemployment dynamics.

read more
2 months ago

How the government deal with inflation (monetary and fiscal) policies

Recent news

Dealing with inflation requires a combination of **fiscal and monetary policy** tools. Policymakers adjust these tools depending on the nature of inflation—whether it's **demand-pull** (inflation caused by excessive demand in the economy) or **cost-push** (inflation caused by rising production costs). Below are key approaches to controlling inflation through fiscal and monetary policy.

read more
2 months ago

More News »

Generic placeholder image

Collaboratively administrate empowered markets via plug-and-play networks. Dynamically procrastinate B2C users after installed base benefits. Dramatically visualize customer directed convergence without