主题：【原创】问个问题，统计中自由度为什么要减一？ -- baiqi

我猜你的意思是说非线性的模型是否比线性的更好。这可以通过简单的R2来看是否显著上升（当然也不一定，因为你多了一个或几个解释变量），也可以看二次方项的系数T检验是否显著。但是如果你要检验整个模型在线性和非线性的条件下那个更优，可以用阶梯式的F检验。

Hübler 2009年有篇文章：

The nonlinear link between height and wages in Germany, 1985–2004

Economics & Human Biology

Volume 7, Issue 2, July 2009, Pages 191-199

Olaf Hübler

就是用F检验来一次，二次还是三次更显著， 你可以看一下，用STATA等软件很简单就可以做。

Table 1

Estimates of height coefficients – dependent variable: log wages per hour.

Note: The standard errors in parentheses are estimated by the cluster approach, which takes into account intragroup correlation, but the observations are

independent across groups (individuals). Wages per hour are measured in real terms and euros. Control variables in specifications (1)–(5): SCHOOLING,

TENURE, TENURE2, EXPERIENCE (actual experience), EXPERIENCE2, NBULA (dummy; =1, if new German federal state (Bundesland)), 20 TIME DUMMIES (for 1985–2004).

F tests of joint significance of the height variables (H0: height effects onwages are zero;H1: height effects are linear (columns (1) and (2)), of third order (columns (3) and (4)) or of second-order (column (5)) are presented in the last but two line (F_1), and the F tests of nonlinear height effects (H0: height effects on wages are linear;H1:height effects are of third-order (columns (3) and (4)) or of second-order (column(5)) are presented inthepenultimate line (F_2).

F_3 presents the F test statistic where the second-order approach (H0) is tested against the third-order approach (H1). Source: GSOEP 1985–2004.

*Significance at the 0.10 level.

** Significance at the 0.05 level.

*** Significance at the 0.01 level.

不过我更推荐用SEEMINGLY UNRELATED ESTIMATION中的CH2检验更好些，可以允许两个方程的残差相关。好像是乌特勒支大学的一个人1999年在STATA BULLINTIN 中有篇相关的文章，你可以去查一下。