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Multiple linear regression (MLR), simply called multiple regression, is a statistical technique that uses multiple explanatory variables to predict the outcome of a response variable.
Multiple linear regression (MLR), simply called multiple regression, is a statistical technique that uses multiple explanatory variables to predict the outcome of a response variable. The goal of MLR is to establish a linear relationship between the explanatory variable (independent variable) and the response variable (dependent variable). In fact, multiple regression is an extension of ordinary least squares (OLS) regression because it involves multiple explanatory variables.
Simple linear regression is a function that allows an analyst or statistician to make predictions about one variable based on information known about another variable. Linear regression can only be used when you have two continuous variables - an independent variable and a dependent variable. Independent variables are parameters used to calculate the dependent variable or outcome. Multiple regression models are extended to multiple explanatory variables.
The MLR model is based on the following assumptions:
IMPORTANT NOTE: MLR assumes that there is a linear relationship between the dependent variable and the independent variables, that there is no strong correlation between the independent variables, and that the variance of the residuals is constant.
The coefficient of determination (R-squared) is a statistical measure of how much of the variation in an outcome can be explained by variation in the independent variable. Even though the predictor variables are not related to the outcome variable, R² increases when adding more predictors to the MLR model.
Therefore, R² by itself cannot be used to identify which predictors should be included in the model and which should be excluded. R² ranges from 0 to 1, with 0 indicating that the outcome cannot be predicted by any of the independent variables and 1 indicating that the outcome can be predicted without error by the independent variables.
When interpreting multiple regression results, beta coefficients are valid holding all other variables constant ("ceteris paribus"). The output from multiple regression can be presented as an equation or displayed vertically as a table.
For example, an analyst might want to know how market changes affect Exxon Mobil's (XOM) stock price. In this case, the linear equation would have the value of the S&P 500 as the independent variable, or predictor, and the price of XOM as the dependent variable.
In fact, a variety of factors can predict the outcome of an event. For example, ExxonMobil's stock price changes based on more than just the performance of the overall market. Other forecast factors such as oil prices, interest rates and crude oil futures prices also affect the price of Exxon Mobil (XOM) and the stock prices of other oil companies. To understand relationships involving more than two variables, MLR is used.
MLR is used to determine the mathematical relationship between multiple random variables. In other words, MLR examines the relationship between multiple independent variables and one dependent variable. Once the predictive power of each independent variable on the dependent variable is determined, information from these multiple variables can be used to accurately predict the degree of influence on the outcome variable. The model formally creates a best-fit linear relationship as close as possible to all individual data points.
Referring to the MLR equation above, in our case:
Least squares estimates - B0, B1, B2...Bp are usually calculated by statistical software. And you can include multiple variables in the regression model, where each independent variable is distinguished by a number - 1, 2, 3, 4...p.
Tip: Multiple regression can also be non-linear, in which case the dependent variable and the independent variable do not follow a straight-line relationship.
Multiple regression models enable analysts to predict an outcome based on information from multiple explanatory variables. However, the model is not always completely accurate as each data point may differ slightly from what the model predicts. The residual value E, the difference between actual results and predicted results, is included in the model to account for this small change.
We ran the XOM price regression model in statistical computing software. It returned the following output:
Analysts would use this output to conclude that when market oil prices increase by 1%, the price of XOM will increase by 7.8%, assuming other variables remain constant. The model also shows that for every 1% increase in interest rates, the price of XOM will decrease by 1.5%. R² shows that 86.5% of ExxonMobil's stock price changes can be explained by changes in interest rates, oil prices, oil futures, and the S&P 500 Index.
Ordinary least squares (OLS) regression compares the response of a dependent variable when some explanatory variable changes. However, the dependent variable is rarely explained by just one variable. In this case, analysts use multiple regression, which attempts to explain the dependent variable through multiple independent variables.
Multiple regression can be linear or nonlinear. MLR is based on the assumption that there is a linear relationship between the dependent variable and the independent variable. It also assumes that there are no major correlations between the independent variables.
Multiple regression considers the impact of multiple explanatory variables on an outcome of interest. It evaluates the relative effect of these explanatory variables (independent variables) on the dependent variable while holding constant all other variables in the model.
The dependent variable is rarely explained by just one variable. In this case, analysts use multiple regression, which attempts to explain the dependent variable through multiple independent variables. The model assumes that there are no major correlations between the independent variables.
This is unlikely because multiple regression models are complex, and the complexity increases as more variables are included in the model or as the amount of data to be analyzed increases. To run multiple regression, you may need to use specialized statistical software or functions in programs such as Excel.
In multiple linear regression, the model calculates a line of best fit that minimizes the variance of each variable related to the dependent variable. Since it fits a straight line, it is a linear model. There are also nonlinear regression models involving multiple variables, such as logistic regression, quadratic regression, and probabilistic unit models.
Any econometric model involving more than one variable is likely to be a multiple regression. Factor models compare two or more factors to analyze the relationship between variables and their resulting performance. The Fama-French three-factor model is such a model. Based on the capital asset pricing model (CAPM), it adds scale risk and value risk factors, thereby expanding the market risk factors. By including these two additional factors, the model adjusts for excess returns, making it a better tool for assessing manager performance.
MLR is a statistical tool used to predict variable outcomes based on two or more explanatory variables. If only one variable affects the dependent variable, a simple linear regression model is sufficient; conversely, if multiple factors affect the variable, MLR is required.
The classic example is the drivers that influence a company's valuation on the stock market. Typically, a company's share price is affected by a variety of factors. In this case, the dependent variable will be the stock price, which is what we are trying to predict, and the independent variables, the explanatory variables, will be the factors that affect the stock price.
[1] Yale University. "Multiple Linear Regression."
[2] CFA Institute. "Basics of Multiple Regression and Underlying Assumptions."
[3] Boston University Medical Campus-School of Public Health. "Multiple Linear Regression."
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简单线性回归是一种函数，允许分析师或统计学家基于已知关于某一变量的信息来对另一变量进行预测。线性回归仅能在拥有两个连续变量一个自变量和一个因变量时使用。自变量是用于计算因变量或结果的参数。多元回归模型扩展到多个解释变量。
MLR模型基于以下假设：
重要提示： MLR假设因变量与自变量之间存在线性关系，自变量之间没有强相关性，且残差的方差是恒定的。
决定系数（R平方）是一种统计指标，用于衡量结果的变异有多少可以通过自变量的变异来解释。即使预测变量与结果变量并无关系，R²在向MLR模型添加更多预测因子时也会增加。
因此，R²本身不能用来识别模型中应包含哪些预测因子和排除哪些预测因子。R²的取值范围在0到1之间，0表示结果无法通过任何自变量进行预测，1表示结果可以毫无误差地由自变量预测。
在解释多元回归结果时，贝塔系数在保持其他所有变量不变的情况下有效（“其他条件不变”）。来自多元回归的输出可以以方程的方式呈现，也可以以表格的方式纵向展示。
例如，一位分析师可能想知道市场变化如何影响埃克森美孚（XOM）的股票价格。在这种情况下，线性方程会以标普500指数的值作为自变量或预测因子，而以XOM的价格作为因变量。
实际上，多种因素都可以预测事件的结果。例如，埃克森美孚的股票价格变化不仅仅取决于整体市场的表现。油价、利率以及原油期货价格等其他预测因子也会影响埃克森美孚（XOM）的价格及其他油企的股票价格。为了理解涉及两个以上变量的关系，使用MLR。
MLR用于确定多个随机变量之间的数学关系。换句话说，MLR考察多个自变量与一个因变量之间的关系。一旦确定每个自变量对因变量的预测能力，就可以利用这多个变量的信息准确预测对结果变量的影响程度。该模型在形式上创建一个最佳拟合的线性关系，尽量接近所有的个别数据点。
参考上面的MLR方程，在我们的例子中：
最小二乘估计值B0、B1、B2…Bp通常由统计软件计算。而且可以在回归模型中包含多个变量，其中每个自变量用编号区分1、2、3、4…p。
提示: 多元回归也可以是非线性的，此时因变量和自变量则不会遵循直线关系。
多元回归模型使分析师能够基于多个解释变量的信息预测一个结果。然而，由于每个数据点可能与模型预测的结果略有不同，该模型并不总是完全准确。残差值E，即实际结果与预测结果之间的差异，被纳入模型以解释这种微小的变动。
我们在统计计算软件中运行了XOM价格回归模型。它返回了以下输出：
分析师会根据这一输出得出这样的结论：如果其他变量保持不变，当市场油价上涨1%时，XOM的价格将上涨7.8%。模型还表明，利率每上涨1%，XOM的价格将下降1.5%。R²显示，埃克森美孚的股票价格变化86.5%可通过利率、油价、油期货和标普500指数的变化来解释。
普通最小二乘（OLS）回归比较因变量在某些解释变量变化时的响应。然而，因变量很少只通过一个变量来解释。在这种情况下，分析师使用多元回归，试图通过多个自变量来解释因变量。
多元回归可以是线性或非线性的。MLR基于因变量与自变量之间存在线性关系的假设。它还假设自变量之间没有主要相关性。
多元回归考虑多个解释变量对某一感兴趣结果的影响。它评估这些解释变量（自变量）对因变量的相对效应，同时保持模型中其他所有变量不变。
因变量很少只通过一个变量来解释。在这种情况下，分析师使用多元回归，试图通过多个自变量来解释因变量。该模型假设自变量之间没有主要相关性。
这不太可能，因为多元回归模型较为复杂，当模型中包含的变量增多或需分析的数据量增大时，复杂性更甚。要运行多元回归，你可能需要使用专门的统计软件或Excel等程序中的函数。
在多元线性回归中，模型计算出最佳拟合线，尽量减少与因变量相关的每个变量的方差。由于它拟合一条直线，因此是线性模型。还有涉及多个变量的非线性回归模型，如逻辑回归、二次回归和概率单位模型。
任何涉及多个变量的计量经济模型都可能属于多元回归。因子模型比较两个或多个因子，以分析变量之间的关系及其导致的表现。法马-法rench三因子模型即是这样一种模型，它在资本资产定价模型（CAPM）的基础上，增加了规模风险和价值风险因子，从而扩展了市场风险因子。通过纳入这两个附加因子，该模型针对超额收益进行调整，使其成为评估管理者表现的更好工具。
MLR是一种统计工具，用于预测变量的结果，基于两个或多个解释变量。如果只有一个变量影响因变量，简单线性回归模型就足够了；反之，如果有多个因素影响该变量，则需要使用MLR。
经典的例子是影响公司在股市上估值的驱动因素。通常，一家公司的股价受多种因素影响。在这种情况下，因变量将是股价，这是我们试图预测的目标，而自变量，即解释变量，将是影响股价的因素。
[1] Yale University. "Multiple Linear Regression."
[2] CFA Institute. "Basics of Multiple Regression and Underlying Assumptions."
[3] Boston University Medical Campus-School of Public Health. "Multiple Linear Regression."
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