Measuring Similarity using GARCH Residuals

Oct 13, 2025 by miodeya.com 88 views

Introduction

Measuring the similarity between time series is a crucial task in various fields, including finance, economics, and data analysis. Time series data, such as swap rates of different currencies, can exhibit complex patterns and relationships, making it challenging to identify similarities. In this article, we will explore the use of GARCH residuals to measure the similarity between time series.

What are GARCH Residuals?

GARCH (Generalized Autoregressive Conditional Heteroskedasticity) is a statistical model used to analyze the volatility of financial time series. The GARCH model estimates the conditional variance of the time series, which is the variance of the series given the past values. The residuals of the GARCH model are the differences between the actual values and the predicted values of the time series.

Why Use GARCH Residuals for Similarity Measurement?

GARCH residuals can be used to measure the similarity between time series because they capture the underlying patterns and relationships between the series. By analyzing the residuals, we can identify the common features and structures that exist between the time series. This is particularly useful when dealing with financial time series, such as swap rates, which can exhibit complex and non-linear relationships.

Cross-Correlation Analysis

One common method for measuring similarity between time series is cross-correlation analysis. Cross-correlation measures the correlation between two time series at different lags. However, cross-correlation analysis has its limitations, as it only measures the linear relationship between the series. GARCH residuals, on the other hand, capture both linear and non-linear relationships, making them a more comprehensive measure of similarity.

ACF and PACF Analysis

Another method for analyzing time series is the use of Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots. ACF plots show the correlation between the time series and its past values, while PACF plots show the correlation between the time series and its past values, after controlling for the effects of intermediate lags. While ACF and PACF analysis can provide valuable insights into the structure of the time series, they do not directly measure similarity between multiple time series.

Measuring Similarity using GARCH Residuals

To measure the similarity between time series using GARCH residuals, we can use various metrics, such as:

Correlation Coefficient: Measures the linear relationship between the GARCH residuals of two time series.
Distance Metrics: Measures the distance between the GARCH residuals of two time series, such as Euclidean distance or Manhattan distance.
Clustering Algorithms: Groups time series based on their GARCH residuals, such as k-means or hierarchical clustering.

Example Use Case

Suppose we have three time series representing swap rates of different currencies: EUR-USD, USD-JPY, and EUR-GBP. We can estimate the GARCH model for each time series and calculate the residuals. Then, we can use the correlation coefficient to measure the similarity between the GARCH residuals of the three time series.

Code Example

Here is an example code in Python using the statsmodels library to estimate the GARCH model and calculate the GARCH residuals:

import pandas as pd
import numpy as np
from statsmodels.tsa.garch.model import GARCH

# Load the time series data
data = pd.read_csv('swap_rates.csv', index_col='date', parse_dates=['date'])

# Estimate the GARCH model for each time series
garch_model = GARCH(data['EUR-USD'], order=(1,1))
garch_resids = garch_model.fit().resid

# Calculate the GARCH residuals for the other time series
garch_resids_usd_jpy = garch_model.fit().resid
garch_resids_eur_gbp = garch_model.fit().resid

# Calculate the correlation coefficient between the GARCH residuals
corr_coef = np.corrcoef(garch_resids, garch_resids_usd_jpy)[0,1]
print(corr_coef)

Conclusion

Q: What is the main advantage of using GARCH residuals to measure similarity between time series?

A: The main advantage of using GARCH residuals is that they capture both linear and non-linear relationships between the time series, making them a more comprehensive measure of similarity.

Q: How do I choose the order of the GARCH model?

A: The order of the GARCH model is typically chosen based on the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC). A lower AIC or BIC value indicates a better fit of the model.

Q: Can I use GARCH residuals to measure similarity between time series with different frequencies?

A: Yes, you can use GARCH residuals to measure similarity between time series with different frequencies. However, you may need to adjust the GARCH model to account for the different frequencies.

Q: How do I handle missing values in the time series data?

A: You can handle missing values in the time series data by using techniques such as interpolation or imputation. Alternatively, you can use a GARCH model that can handle missing values, such as the GARCH-M model.

Q: Can I use GARCH residuals to measure similarity between time series with different scales?

A: Yes, you can use GARCH residuals to measure similarity between time series with different scales. However, you may need to adjust the GARCH model to account for the different scales.

Q: How do I interpret the results of the GARCH residual analysis?

A: The results of the GARCH residual analysis can be interpreted as follows:

A high correlation coefficient between the GARCH residuals of two time series indicates a strong similarity between the series.
A low correlation coefficient between the GARCH residuals of two time series indicates a weak similarity between the series.
A high distance metric between the GARCH residuals of two time series indicates a strong dissimilarity between the series.
A low distance metric between the GARCH residuals of two time series indicates a weak dissimilarity between the series.

Q: Can I use GARCH residuals to measure similarity between time series with different levels of volatility?

A: Yes, you can use GARCH residuals to measure similarity between time series with different levels of volatility. However, you may need to adjust the GARCH model to account for the different levels of volatility.

Q: How do I choose the metric to use for measuring similarity between time series?

A: The choice of metric depends on the specific research question and the characteristics of the time series data. Some common metrics used for measuring similarity between time series include:

Correlation coefficient
Distance metrics (e.g. Euclidean distance, Manhattan distance)
Clustering algorithms (e.g. k-means, hierarchical clustering)

Q: Can I use GARCH residuals to measure similarity between time series with different levels of seasonality?

A: Yes, you can use GARCH residuals to measure similarity between time series with different levels of seasonality. However, you may need to adjust the GARCH model to account for the different levels of seasonality.

Q: How do I handle non-stationarity in the time series data?

A: You can handle non-stationarity in the time series data by using techniques such as differencing or detrending. Alternatively, you can use a GARCH model that can handle non-stationarity, such as the GARCH-M model.

Q: Can I use GARCH residuals to measure similarity between time series with different levels of skewness?

A: Yes, you can use GARCH residuals to measure similarity between time series with different levels of skewness. However, you may need to adjust the GARCH model to account for the different levels of skewness.

Q: How do I choose the GARCH model to use for measuring similarity between time series?

A: The choice of GARCH model depends on the specific research question and the characteristics of the time series data. Some common GARCH models used for measuring similarity between time series include:

GARCH(1,1)
GARCH-M
EGARCH
GJR-GARCH

Q: Can I use GARCH residuals to measure similarity between time series with different levels of autocorrelation?

A: Yes, you can use GARCH residuals to measure similarity between time series with different levels of autocorrelation. However, you may need to adjust the GARCH model to account for the different levels of autocorrelation.