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Terryzxw · 2022年01月21日

关于t统计量的计算问题

  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.

pre-offer price adjustment是in decimal,在计算假设检验的t统计量（H0:b<=0.5%）时，是不是要用(0.4350-0.005)/对应标准误？

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星星_品职助教 · 2022年01月21日

同学你好，

表格上方的说明针对的是表格自身的数字如何显示。和假设检验是不相关的。

具体而言，这里面的in decimal指的是“表格”里的数字都是小数。以“pre-offer price adjustment ”的系数为例，如果表格要求用in percentage形式，显示就是43.5，不用再带百分号，而现在要求用in decimal表示，就直接显示实际数字0.435。

-------------------------------

而假设检验是基于题干中的信息的。根据“.....each 1 percent increase in pre-offer price adjustment, the initial return will increase by less than 0.5 percent”，可知X增加1%，Y增加不到0.5%，这就等价于X增加1，Y增加不到0.5，百分号在这个过程中自然消掉。这个和表格里用percentage还是decimal的数字显示方法是不冲突的。

所以最后的关系是用0.435-0.5，而不是0.435-0.005

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