回归分析的基本条件假设（SLR, MLR）

作者: 王叽叽的小心情 | 来源:发表于2020-02-29 17:55 被阅读0次

对回归分析进行参数估计时，有三种估计方法，最小二乘法（OLS， ordinary least squares），广义矩估计（GMM， general moment method）以及最大似然估计（MLE, maximum likelihood estimation），最为常用的方法即是最小二乘法，即采用的是高斯马尔科夫定理。

参考伍德里奇的计量经济学导论，但在采用最小二乘法进行估计时，针对SLR（一元线性回归），其变量需要满足如下的条件假设

模型的线性
Assumption SLR.1 (LINEAR IN PARAMETERS)
In the population model, the dependent variable y is related to the independent variable x and the error (or disturbance) u as
$y = \beta_0 + \beta_1 x +u$
where $\beta_0$ and $\beta_1$ are the population intercept and slope parameters, respectively.
变量的随机性
Assumption SLR.2 (RANDOM SAMPLING)
We can use a random sample of size $n, {(x_i,y_i): i =1,2,…,n}$ , from the population model.
零条件均值假设
Assumption SLR.3 (ZERO CONDITIONAL MEAN)
$E(u|x) = 0$
自变量方差不为零
Assumption SLR.4 (SAMPLE VARIATION IN THE INDEPENDENT VARIABLE)
In the sample, the independent variables $x_i, i = 1,2,…,n$ , are not all equal to the same constant. This requires some variation in x in the population.

而如果进行的是 MLR（多元线性回归），其假设条件为：

模型的线性
Assumption MLR.1 (LINEAR IN PARAMETERS)
The model in the population can be written as
$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + u$
where $\beta_0, \beta_1, \beta_k$ are are the unknown parameters (constants) of interest, and u is an unobservable random error or random disturbance term.
变量的随机性
Assumption MLR.2 (RANDOM SAMPLING)
We have a random sample of n observations, ${(x_{i1}, x_{i2},…,x_{ik}, y_i): i =1,2,…,n}$ , from the population model described by (3.31).
零条件均值假设
Assumption MLR.3 (ZERO CONDITIONAL MEAN)
The error $u$ has an expected value of zero, given any values of the independent variables. In other words,
$E(u|x_1, x_2,... x_k) = 0$
变量无完美共线性
Assumption MLR.4 (NO PERFECT COLLINEARITY )
In the sample (and therefore in the population), none of the independent variables is constant, and there are no exact linear relationships among the independent variables.
方差齐性
Assumption MLR.5 (HOMOSKEDASTICITY)
$Var(u|x_1, x_2,...x_k) = \sigma^2$
6.正态性
Assumption MLR.6 (NORMALITY)
The population error $u$ is independent of the explanatory variables $x_1, x_2, …, x_k$ and is normally distributed with zero mean and variance $\sigma^2$ : u ~ Normal(0, $\sigma^2$ )