It is worth emphasizing that a seemingly low r - squared does not necessarily mean that an ols regression equation is useless 在社会科学中,特别是在截面数据分析中,回归方程得到低的r -平方值并不罕见。
If ols chooses any value other than zero , it must be that this value reduced the ssr relative to the regression that excludes the regressor 如果ols使此解释变量取任何非零系数,那么加入此变量之后, ssr降低了。
Reason no . 1 : we may prefer to report the usual ols standard errors and test statistics unless there is evidence of heteroskedasticity 理由1 :除非有证据显示异方差存在,我们仍会偏好于常规ols的标准差及检验统计量。
Idea : because the ols estimates are chosen to minimize the sum of squared residuals , the ssr always increases when variables are dropped from the model 由于ols是用于最小化残差平方和,当有变量被从模型中舍弃时, ssr必定上升
To prove that ols estimators are asymptotically efficient , one needs to ( 1 ) present an estimator that is consistent but its variance is larger 为了证明ols估计量是渐近有效的,我们需要( 1 )给出一致的估计量但证明它有更大的方差。
In such case , the usual ols standard error is invalid . the t statistics is also invalid , and tend to be too large in the case of > 0 在这种情况下,通常的ols标准差不再正确。 t统计量也不再正确。在第二项为正的情况下, t会变大。
The results are compared with the ols ' s results ( supposing the two transition be the second order transition ) and all results are plotted in the same figure 作为比较我们把slo结果(假定两次转变都为二级转变)和我们的结果画在同一图上。
If ols happens to choose the coefficient on the new regressor to be exactly zero , then ssr will be the same whether or not the second variable is included in the regression 如果ols恰好使第二个解释变量系数取零,那么不管回归是否加入此解释变量, ssr相同。
If we are presented with an estimator that is both linear and unbiased , then we know that the variance of this estimator is at least as large as that from ols 如果有人向我们提出一个线性无偏估计量,那我们就知道,此估计量的方差至少和ols估计量的方差一样大。