What is regression analysis?

Regression analysis is a statistical method widely used in finance, investing, and other disciplines to determine the strength and characteristics of the relationship between a dependent variable and one or more independent variables.

Linear regression is the most common form of this technique. Linear regression, also known as simple regression or ordinary least squares (OLS), establishes a linear relationship between two variables.

Linear regression is graphically represented by a line of best fit, with the slope defining how changes in one variable affect changes in the other variable. The y-intercept of a linear regression relationship represents the value of the dependent variable when the independent variable is zero. Nonlinear regression models also exist, but they are generally more complex.

Key Points

• Regression is a statistical technique that relates a dependent variable to one or more independent variables.
• Regression models show whether changes in a dependent variable are associated with changes in one or more independent variables.
• It is mainly achieved by determining the best fit line and observing the distribution of data around this line.
• Regression analysis helps economists and financial analysts in everything from asset valuation to forecasting.
• In order to correctly interpret regression results, several assumptions about the data and the model itself must be met.

In economics, regression analysis is used to help investment managers assess the value of assets and understand the relationship between the prices of commodities and the stocks of companies that work in those commodities.

Although regression analysis is a powerful tool for revealing associations between variables in data, it cannot easily specify cause and effect. Regression as a statistical technique should not be confused with the concept of "regression to the mean", i.e. regression to the mean.

Understand regression analysis

Regression analysis captures the correlations between variables in a data set and quantifies whether these correlations are statistically significant.

The two basic types of regression are simple linear regression and multiple linear regression, although there are also nonlinear regression methods for more complex data analysis. Simple linear regression uses one independent variable to explain or predict the outcome of the dependent variable Y, while multiple linear regression uses two or more independent variables to make predictions. Analysts can examine each independent variable included in a linear regression model through stepwise regression.

Regression analysis can help financial and investment professionals. For example, a company might use it to predict sales based on weather, past sales, gross domestic product (GDP) growth, or other conditions. The capital asset pricing model (CAPM) is a regression model commonly used in finance, mainly used for asset pricing and discovering the cost of capital.

Econometrics is a set of statistical techniques used to analyze financial and economic data. An example of an application of econometrics is the study of income effects using observable data. Economists might assume that as an individual's income increases, his spending will increase.

If the data indicate such a relationship, a regression analysis can be performed to understand the strength of the relationship between income and consumption and whether the relationship is statistically significant.

It should be noted that there can be multiple independent variables in the analysis, for example, changes in GDP, inflation and unemployment can all be used to explain stock market prices. When more than one independent variable is used, it is called multiple linear regression. This is the most commonly used tool in econometrics.

Econometrics is sometimes criticized for relying too much on interpretations of regression outputs without linking them to economic theory or searching for causal mechanisms. It is crucial that the findings revealed in the data can be rationally explained by theory.

Calculate regression analysis

Linear regression models typically use the least squares method to determine the line of best fit. The least squares method works by minimizing the sum of squares produced by a mathematical function. The square is calculated by squaring the distance between the data point and the regression line or the mean of the data set.

Once this process is completed (often done using software these days), a regression model is constructed. The general form of each type of regression model is as follows:

Simple Linear Regression: 1

Multiple linear regression: 2

Examples of application of regression analysis in finance

Regression analysis is often used to determine how specific factors (for example, the price of a commodity, interest rates, a specific industry or sector) affect an asset's price movements. The CAPM mentioned above is based on regression analysis and is used to predict a stock's expected return and calculate the cost of capital. A stock's return is regressed against the return of a broader index, such as the S&P 500, to generate a beta value for the stock.

Beta value represents the risk of the stock relative to the market or index and is reflected in the slope of the CAPM. The return rate of the underlying stock is the dependent variable Y, while the independent variable X is the market risk premium.

Additional variables such as market capitalization, valuation ratios, and recent earnings can be added to the CAPM to obtain better earnings forecasts. These additional factors are called Fama-French factors, named after the professor who developed the multiple linear regression model to better explain asset returns. [3]

Why does Return get its name?

Although there is some debate about the origin of the name, the above statistical technique was most likely created by Sir Francis Galton in the 19th century to describe the statistical characteristic of biological data (such as the height of people in a population) regressing to some mean level. In other words, while there are shorter and taller people, only outliers will be very tall or very short, while most people cluster around some mean or "regress" toward that mean. [4]

What is the purpose of regression analysis?

In statistical analysis, regression is used to identify associations between variables that appear in some data. It can show the size of this association and determine its statistical significance. Regression is a powerful statistical inference tool that attempts to predict future outcomes based on past observations.

How to interpret the regression model?

The output of the regression model might appear as Y = 1.0 + (3.2)X1 - 2.0(X2) + 0.21.

Here is a multiple linear regression related to two explanatory variables X1 and X2. In this model, we can read this as: Whenever X1 changes by one unit, Y will change by 3.2 (if X1 increases by 2, Y increases by 6.4, etc.), assuming other things remain unchanged. This means that, controlling for X2, X1 has this observed relationship. Similarly, controlling X1 unchanged, for every unit increase in X2, Y will decrease by 2 units. We can also notice that the y-intercept is 1.0, meaning that when X1 and X2 are both zero, Y is 1. The error term (residual) is 0. [2]1. [2]

What assumptions must the regression model satisfy?

In order to correctly interpret the output of a regression model, the following key assumptions need to be true about the underlying data process you are analyzing:

• The relationship between variables is linear;
• Homoscedasticity must exist, that is, the variances of the variables and error terms must remain constant;
• All explanatory variables are independent of each other;
• All variables follow a normal distribution. [5]

Summary

Regression is a statistical method designed to determine the strength and characteristics of a relationship between a dependent variable and a range of other variables. It is used in finance, investing and other disciplines.

Regression analysis reveals associations between variables in the data but cannot easily indicate cause and effect.

References

[1] Margo Bergman. "Quantitative Analysis for Business: 12. Simple Linear Regression and Correlation." University of Washington Pressbooks, 2022.

[2] Margo Bergman. "Quantitative Analysis for Business: 13. Multiple Linear Regression." University of Washington Pressbooks, 2022.

[3] Fama, Eugene F., and Kenneth R. French, via Wiley Online Library. "The Cross-Section of Expected Stock Returns." The Journal of Finance, vol. 47, no. 2, June 1992, pp. 427–465.

[4] Stanton, Jeffrey M., via Taylor & Francis Online. "Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors." Journal of Statistics Education, vol. 9, no. 3, 2001.

[5] CFA Institute. “Basics of Multiple Regression and Underlying Assumptions.”

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TheQuantHackers is a cutting-edge community composed of a group of people from the world's top universities and quantitative finance practitioners, dedicated to exploring the infinite possibilities in the fields of artificial intelligence (AI) and quantification (Quant). Our team members come from world-renowned universities such as the University of Cambridge, Oxford University, Harvard University, ETH Zurich, Peking University, and the University of Science and Technology of China. Our external consultants come from first-class companies such as Microsoft, HSBC, Citadel, Man Group, Citi, Jump Trading, and top domestic private equity companies.$\begin{aligned}&Y = a + bX + u \\\end{aligned}$

多元线性回归：2

$\begin{array}{rl}{\displaystyle }& {\displaystyle Y=a+b_1{X}_1+b_2{X}_2+b_3{X}_3+\mathrm{.}\mathrm{.}\mathrm{.}+b_t{X}_t+u}\\ {\displaystyle }& {\displaystyle \mathbf{\text{其中:}}}\\ {\displaystyle }& {\displaystyle Y=\text{您试图预测或解释的因变量}}\\ {\displaystyle }& {\displaystyle X=\text{解释性的（自变量）}}\\ {\displaystyle }& {\displaystyle \text{变量，您用它来预测或与Y相关联}}\\ {\displaystyle }& {\displaystyle a=\text{y截距}}\\ {\displaystyle }& {\displaystyle b=\text{（beta系数）是解释性变量的斜率}}\\ {\displaystyle }& {\displaystyle }\end{array}$ 回归分析是一种统计方法，广泛应用于金融、投资及其他学科，旨在确定因变量与一个或多个自变量之间关系的强度和特征。

线性回归是这一技术中最常见的形式。线性回归也被称为简单回归或普通最小二乘法（OLS），它建立了两个变量之间的线性关系。

线性回归通过最佳拟合直线图形化展示，斜率定义了一个变量变化如何影响另一个变量的变化。线性回归关系的y截距表示当自变量为零时因变量的值。非线性回归模型也存在，但它们通常更为复杂。

关键要点

• 回归是一种将因变量与一个或多个自变量相关联的统计技术。
• 回归模型能够显示出因变量的变化是否与一个或多个自变量的变化相关联。
• 其主要通过确定最佳拟合线，观察数据围绕该线的分布情况来实现。
• 回归分析帮助经济学家和金融分析师进行从资产评估到预测等各个方面的工作。
• 为了正确解读回归结果，必须满足关于数据及模型本身的多个假设。

在经济学中，回归分析被用于帮助投资经理评估资产的价值，并理解商品价格与从事这些商品的企业股票之间的关系。

尽管回归分析是揭示数据中变量间关联的有力工具，但它不能轻易指明因果关系。回归作为一种统计技术，不应与“回归中值”概念混淆，即平均回归。

理解回归分析

回归分析捕捉数据集中变量之间的相关性，并量化这些相关性是否具有统计显著性。

回归的两种基本类型是简单线性回归和多元线性回归，尽管对于更复杂的数据分析也有非线性回归方法。简单线性回归利用一个自变量来解释或预测因变量Y的结果，而多元线性回归则使用两个或更多自变量来进行预测。分析师可以通过逐步回归来检查线性回归模型中包含的每一个自变量。

回归分析可以帮助金融及投资专业人士。例如，一家公司可能会使用它根据天气、以往销售额、国内生产总值（GDP）增长或其他条件来预测销售情况。资本资产定价模型（CAPM）是金融中常用的一种回归模型，主要用于资产定价和发现资本成本。

计量经济学是一套用于分析金融和经济数据的统计技术。计量经济学的一个应用示例是使用可观察数据研究收入效应。经济学家可能会假设，随着个人收入的增加，其支出也会增加。

如果数据表明存在这样的关联，则可以进行回归分析，以了解收入与消费之间关系的强度以及该关系是否具有统计显著性。

需要注意的是，在分析中可以有多个自变量，例如，GDP、通货膨胀和失业率的变化都可以用来解释股市价格。当使用一个以上的自变量时，称为多元线性回归。这是计量经济学中最常用的工具。

计量经济学有时因过于依赖回归输出的解释而受到批评，而没有将其与经济理论联系或寻找因果机制。至关重要的是，数据中揭示的发现能够通过理论得到合理解释。

计算回归分析

线性回归模型通常采用最小二乘法来确定最佳拟合线。最小二乘法是通过最小化由数学函数产生的平方和来实现的。平方的计算是通过对数据点与回归线或数据集均值之间的距离进行平方得出的。

完成这一过程后（如今通常使用软件进行操作），便构建了回归模型。每种类型的回归模型的一般形式如下：

简单线性回归：1