This article gives a simple introduction to GARCH, its fundamental principles, and offers an Excel spreadsheet for GARCH(1,1).Scroll down to the bottom if you just want to download the spreadsheet, but I encourage you to read this guide so you understand the principles behind GARCH.
Least squares is a fundamental concept in statistics, and is widely used across many fields, including engineering, science, econometrics, and finance. Least squares determines how a dependent variable changes in response to the variation of another variable (call the independent variable).
The difference between the actual and the predicted value is known as the residual. Fitting a modeling involves minimizing the sum of the squares of the residuals.
The least squares approach assumes that the squared error has the same magnitude across the entire data set. This assumption is known as homoskedasticity. But financial data (known as a time series) has periods of high and low volatility, with periods of high volatility often clustering together. This is known as heteroskecadicity.
In reference to modeling fitting, this means the residuals vary in magnitude. Volatility clustering means the data is auto correlated. GARCH is a statistical tool that helps predict the residuals in k data
ARCH means Autoregressive Conditional Heteroskedasiticy and is closely related to GARCH. The simplest method to predict stock volatility is an n day standard deviation, and let’s consider a rolling year with 252 trading days. If we want to predict stock prices for the next day, the mean is usually a safe starting point.
But the mean treats each day with the same weight. Giving the recent past more significance is more logical, with perhaps an exponential weighted average being a better method to predict tomorrow’s stock price.
However, this method does not capture any data older than a year, and the weighting is rather arbitrary. The ARCH model, however, varies weights on each residual such that the best fit is obtained. The GARCH (General Autoregressive Conditional Heteroscedasiticy) is similar, but gives recent data more significance.
The GARCH(p,q) model has two characteristic parameters; p is the number of GARCH terms and q is the number of ARCH terms. GARCH(1,1) is defined by the following equation.
h is variance, ε is the residual squared, t denotes time. ω, α and β are empirical parameters determined by maximum likelihood estimation. The equation tells us that tomorrow’s variance is a function of
- today’s squared residual,
- today’s variance,
- the weighted average long-term variance
GARCH(1,1) captures only once square residual and one square variance.
This is not a magic wand, and financial analysts should be use the approach with a high degree of caution. Given the appropriate circumstance, the predicted variance can greatly differ from the actual variance. Techniques such as the Ljung box text are used to determine if any autocorrelation remains in the residuals.
Several researchers have highlighted deficiencies in GARCH(1,1) models, including its failure to predict the volatility in the S&P500 more accurately than other methods.
GARCH in Excel
This Excel spreadsheet models GARCH(1,1) on time series data. You can use your own data, but the spreadsheet uses the GBP/CAD exchange rate between May 2007 and October 2011 (data obtained using this Forex data downloader spreadsheet). The spreadsheet uses Excel’s Solver for the maximum likelihood estimation, but full instructions are given on its use.