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甜宝 · 2023年11月16日

t-test

  1. CFA II
  2. Quantitative

NO.PZ2015120204000015

问题如下：

Based on past research, Hansen selects the following independent variables to predict IPO initial returns:

Underwriter rank = 1–10, where 10 is highest rank

Pre-offer price adjustment (Expressed as a decimal) = (Offer price – Initial filing price)/Initial filing price

Offer size ($ millions) = Shares sold × Offer price

Fraction retained (Expressed as a decimal) = Fraction of total company shares retained by insiders

He also believes that for each 1 percent increase in pre-offer price adjustment, the initial return will increase by less than 0.5 percent, holding other variables constant. Hansen wishes to test this hypothesis at the 0.05 level of significance.

Hansen collects a sample of 1,725 recent IPOs for his regression model.

\Hansen’s Regression Results Dependent Variable: IPO Initial Return (Expressed in Decimal Form, i.e., 1% = 0.01)

Selected Values for the t-Distribution (df = ∞)

The most appropriate null hypothesis and the most appropriate conclusion regarding Hansen’s belief about the magnitude of the initial return relative to that of the pre-offer price adjustment (reflected by the coefficient bj) are:

选项：

Null Hypothesis
Conclusion about bj(0.05 Level of Significance)
A.
H0: bj=0.5
Reject H0
B.
H0: bj≥0.5
Fail to reject H0
C.
H0: bj≥0.5
Reject H0

解释：

C is correct.

C To test Hansen’s belief about the direction and magnitude of the initial return, the test should be a one-tailed test. The alternative hypothesis is H1: bj<0.5b_j<0.5, and the null hypothesis is H0:bj≥0.5b_j\geq0.5 . The correct test statistic is: t = (0.435-0.50)/0.0202 = -3.22, and the critical value of the t-statistic for a one-tailed test at the 0.05 level is -1.645. The test statistic is significant, and the null hypothesis can be rejected at the 0.05 level of significance.

老师，第一个表里面的t-test那一列数字有什么具体含义吗？ 比如21.53是怎么理解的？ 我以为直接用这个数字去和critical value去比较就可以呢。

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1 个答案

星星_品职助教 · 2023年11月16日

同学你好，

表格中给出的t-statistic或p-value，都是基于原假设为0时的对应值。

如21.53，就是当原假设为H0：pre-offer price adjustment的系数=0时，所得到的t-statistic。此时相当于在t-statistic的公式里的“B1”代入的是0。

所以，如果题目问的就是和0相关，如该系数是否显著，就可以直接使用这个t-statistic。大部分的题目都是这种情况。


对于本题而言，问的是该系数和0.5的关系，此时就不能直接使用给出的t-statistic。而是要在t-statistic的公式里的B1代入0.5后重新计算。然后用新算出来的t-statistic再去和critical value进行比较。

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