Brian邵彬 · 2024年05月21日

为什么少了一个regressor

NO.PZ2020010801000025

问题如下：

A model was estimated using daily data from the S&P 500 from 1977 until 2017 which included five day-of-the-week dummies (n = 10,087). The $R^2$ from this regression was 0.000599. Is there evidence that the mean varies with the day of the week?

选项：

解释：

The model estimated is

$Y_i = \beta_1D_1 + \beta_2D_2 + \beta_3D_3 + \beta_4D_4 + \beta_5D_5 + \epsilon_i$ ,

where Di is a dummy that takes the value 1 if the index of the weekday is i (e.g., Monday = 1, Tuesday = 2, c). The restriction is that

$H_0:\beta_1 = \beta_2 = \beta_3 = \beta_4 = \beta_5$

so there this is no day-of-the-week effect. This model can be equivalently written as

$Y_i = \mu + \delta_2D_2 + \delta_3D_3 + \delta_4D_4 + \delta_5D_5 + \epsilon_i$ ,

therefore, here the null is

$H_0:\delta_2 = \delta_3 = \delta_4 = \delta_5$ .

In the two models, $\mu = \beta_1$ , and $\mu + \delta_i = \beta_i$ . The second form of the model is a more standard null for an F-stat.

The F-stat of the regression is

$\frac{(R^2-0)/4}{(1-R^2)/(n-5)}=\frac{0.000599/4}{(1-0.000599)/(10087-5)}=1.51$

The distribution is an $F_{4,10082}$ and the critical value using a 5% size is 2.37. The test statistic is less than the critical value, therefore, the null that all effects are 0 is not rejected.