Multiple regression analysis
Multiple regression analysis
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DOI：
10.53347/rid-35429
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发表时间：
2015-04
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通讯作者：
H. Knipe
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文献类型：
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作者：
H. Knipe
假设我们感兴趣的是小时工资和教育之间的关系，但我们认为，工资实际上不仅取决于教育水平，还取决于经验。也就是说，小时工资的真实模型是：现在，我们感兴趣的仍然是系数，1美元，保持固定的所有其他因素影响工资。(That是，我们想知道教育对工资的影响，ceteris parabus.）我们可以把这个模型写成一个更一般的形式，其中我们有k个自变量：其中$0是截距，$1是与x 1相关的参数，依此类推。请注意，由于有k个自变量和一个截距，因此该方程包含k+1个（未知）总体参数。注：在多元回归模型中，如果模型在参数中是线性的（即$），则该模型被认为是“线性“的。这意味着我们可以让自变量进入模型，但我们想-平方，立方，对数形式等关键假设：多元回归分析的一个关键假设是误差项之间的关系，和自变量，x1. xk.我们将假设以下关系，它可以被描述为一个条件期望：也就是说，给定x，误差项的期望值为零。未观测误差项与任何自变量都不相关。估计的OLS方程由下式给出：其中B 0是$0的OLS估计，依此类推。普通最小二乘法与二元模型的方法完全相同。也就是说，估计值是通过最小化平方误差之和得到的：
Suppose that what we are interested in the relationship between hourly wage and education, but we believe that wage is, in fact, determined not solely by education level, but also by experience. That is, the true model of hourly wage is given by: Now, what we are interested in is still the coefficient, $ 1, holding fixed ALL OTHER FACTORS affecting wage. (That is, we want to know the effect of education on wage, ceteris parabus.) We can write this model in a more general form where we have k independent variables as: where $ 0 is the intercept, $ 1 is the parameter associated with x 1 and so forth. Note that since there are k independent variables and an intercept, this equations contains k+1 (unknown) population parameters. NOTE: in multiple regression models, a model is considered " linear " if it is LINEAR IN THE PARAMETERS (those are the $'s). That means we can have the independent variables enter into the model however we would like – squared, cubed, log form, etc. Key Assumption: A key assumption for multiple regression analysis is the relationship between the error term, ,, and the independent variables, x1...xk. We will assume the following relationship, which can be described as a conditional expectation: That is, GIVEN THE x's, the expected value of the error term is zero. The unobserved error term is uncorrelated with any of the independent variables. The estimated OLS equation is given by: where b 0 is the OLS estimate of $ 0 , and so forth. The method of ordinary least squares is exactly the same as for the bivariate model. That is, the estimates are found by MINIMIZING the sum of squared errors: