Quantile Regression

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21 Aug, 2024

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Quantile regression refers to a statistical technique applied to ascertain the approximate quantiles of a response variable among a group of predictor variables. It serves to analyze income inequality, set up risk-based targets, and determine the value of risk in finance.

Investors use it to forecast possible ranges of values in the future related to unemployment rates, stock prices, and future sales. It also identifies the impact of various factors on different quantiles of the resulting distribution. Governments use it to know the impact of their policies on various income groups. Furthermore, it comes in handy when handling data that defies the presumptions of conventional linear regression.

What Is Quantile Regression?

Quantile regression is a statistical technique used to estimate response variable quantiles. It has applications in analyzing income inequality, setting risk-based targets, and determining value at risk in finance.
Moreover, it helps in predicting financial crisis risk, setting premiums for risk-based insurance, identifying factors of poverty and inequality, and facilitating targeted marketing campaigns.
The method has the advantage of robustness towards outliers, but it also comes with the disadvantage of computational complexity.
These offer flexible analysis, robustness to outliers, and larger sample sizes, while linear and OLS regression is simpler but sensitive to outliers and assumptions.

Quantile Regression Analysis Explained

Quantile regression is a statistical technique that extends traditional linear regression by estimating different quantiles of the conditional distribution of the response variable. It differs from the traditional regression technique in that it offers a more in-depth understanding of the connection among variables. Quantile regression does it by estimating various quantities of the response variable. Its working is based on the minimization of the loss function, which calculates the difference between the estimated and observed quantities.

Here, it works by determining a separate regression line per quantile of the dependent variable. It happens by reducing the weighted sum of squared residuals in which quantile function give their weights. Then, the quantile function uses a quantile as input plus returns back the value of the dependent variable related to that quantile.

It has multiple implications in numerous fields. Hence, it gives way to more nuanced data analysis encompassing the widely spread heterogeneity throughout the distribution. This model also estimates the conditional distribution of the dependent variable to evaluate the effects of different factors on distinct quantiles of the distribution, such as the median, 10th percentile, and 90th percentile. Since it is not dependent on the homoscedasticity and normality assumptions, it is helpful for examining data that has anomalies or non-normal distributions.

Furthermore, quantile regression analysis can answer various queries in multiple fields. These include policy impacts, financial crisis prediction, risk-based insurance premiums, identifying poverty and inequality factors, targeted marketing, financial risk management, healthcare cost analysis, and dealing with non-normal data.

Thus, in the financial sector, it is essential to determine capital requirements for financial institutions. Therefore, it helps in estimating possible losses in investment portfolios and understanding wealth distribution and labor market dynamics. Finally, it is a valuable tool in finance for asset pricing model creation and portfolio optimization.

Examples

Let us use a few examples to understand the topic.

Example #1

For instance, consider a scenario where an investor wants to assess the impact of market fluctuations on the returns of a particular asset. Traditional mean-based approaches overlook the differential effects of market volatility at various percentiles of the return distribution. Quantile regression, however, allows for a more granular analysis by estimating how changes in market conditions affect portfolio returns at different quantiles.

Hence, this is particularly useful for identifying extreme events and tail-risk exposures, providing investors with insights into the potential downside risks during periods of market stress. By incorporating quantile regression in financial analysis, investors can make more informed decisions, tailor risk management strategies to specific segments of the return distribution, and enhance portfolio resilience in the face of unexpected market movements.

Example #2

Let's consider an analysis of the relationship between a stock's performance and macroeconomic indicators, such as gross domestic product (GDP) growth. Traditional mean regression might obscure how this relationship changes across different levels of returns. Quantile regression, however, enables a more nuanced exploration by estimating conditional quantiles of the stock returns distribution. Therefore, this approach is especially pertinent in the finance sector, where tail events and extreme market conditions can significantly influence investment outcomes.

By applying quantile regression, financial analysts can gain insights into how the relationship between macroeconomic factors and stock returns varies, providing a robust framework for risk assessment and tailoring investment strategies to specific percentiles of the return distribution. Hence, this flexibility is crucial for investors seeking to navigate the complexities of financial markets and optimize their portfolios in the face of diverse risk exposures.

Applications

Its ability to estimate the median is one of its primary uses, which makes it resistant to outliers. There are several important applications and consequences of it, as shown below:

In Multiple Fields: A wide range of queries can be answered in the multiple fields.
Assessing Policy Impacts: Assessing various impacts of government policies on different income classes.
Financial Crisis Prediction: It helps in the prediction of financial crisis risk.
Risk-Based Insurance Premiums: It aids in setting premiums for risk-based insurance.
Identifying Factors of Inequality: It exposes the factors responsible for poverty and inequality.
Marketing: It is applied in marketing to analyze consumer behavior and purchasing patterns. Hence, it allows marketers to understand how the impact of advertising or pricing strategies varies across different quantiles of the sales distribution.
Financial Risk Management: It plays a greater role in the estimation of Conditional Value at Risk (CVaR) and Value at Risk (VaR), which is vital for financial risk management.
Healthcare Cost Analysis: Researchers in health economics use quantile regression to investigate the impact of various factors on health outcomes at different levels of the health distribution.
Education: In education research, quantile regression can be used to examine the impact of socioeconomic factors on student performance across different percentiles of the achievement distribution.

Furthermore, in various programming languages, such as quantile regression in r, quantile regression python, and quantile regression stata, it is widely implemented besides the listed applications above. Moreover, analysts also use these languages to build quantile regression models, allowing for a more comprehensive analysis of the relationship between variables. Thus, it provides insights into different quantiles of the response variable's distribution.

Advantages And Disadvantages

Quantile regression is quite important in every field, but certain limitations are also there. Let's look at some of its advantages and disadvantages using the table below:

Advantages	Disadvantages
It has robustness towards outliers.	There is computational complexity in it.
These can handle non-normal distributions.	Model selection for the process or regression becomes difficult.
Here, it has flexibility in the modeling relationship.	Many software does not support it or do not have inbuilt functions to work on it.
Moreover, it provides results that have higher interpretability.	Interpretation of the results becomes challenging.

Quantile Regression vs Linear Regression vs OLS (Ordinary Least Square) Regression

All three are related to the regression analysis of data, albeit with differences. Let's look at their differences using the table below:

Quantile Regression	Linear Regression	OLS Regression
It measures conditional quantities of dependent variables, leading to the analysis of the relationship by estimation of different quantities.	The relationship is modeled on the linear equation.	A particular kind of linear regression that estimates coefficients using least squares.
No requirements for the assumption of linearity or normality	Assumes independent observations, normally distributed residuals & constant variance.	Reduces squared residuals.
Heteroscedasticity is allowed	Homoscedasticity is used.	Homoscedasticity is allowed.
Outliers have a limited impact.	Here, it can be implemented and interpreted in a straightforward manner	Assumes linearity, independent observations, normality and constant variance.
Analyzes asymmetric relationships.	Provides coefficient estimates.	Simple to implement and easier to interpret
Requires larger sample sizes	Sensitive towards influential observations and outliers.	Sensitive towards outliers and influential observations
These becomes computationally intensive.	Its linearity assumption may not hold.	If assumptions are met, then it gives an efficient estimation.
Difficult to interpret the results.	Bias may occur when assumptions are violated.	These give coefficient estimates.

Frequently Asked Questions (FAQs)

1. Why should one care about quantile regression?

It is helpful because, in comparison to conventional mean-based regression, it offers insights into different aspects of data distribution, providing a more thorough knowledge. In domains such as finance, economics, and medicine, evaluating various data quantiles is crucial.

2. Is quantile regression biased?

When working with skewed or heteroscedastic data, it is less biased than mean-based regression (OLS) as it is less susceptible to outliers. Although not wholly biased, it should only be applied sparingly.

3. What is the alternative to quantile regression?

A variety of strategies, such as robust regression approaches like the Huber loss function or ordinary least squares (OLS) regression, which focuses on calculating the conditional mean, can be used as an alternative to it. The decision is based on the properties of the data and the research question.

4. Is quantile regression parametric or nonparametric?

Since the conditional quantiles are assumed to have a particular functional form, it is commonly regarded as a parametric technique. However, other nonparametric quantile regression techniques offer greater freedom in situations when the underlying data distribution is not well understood.