为什么size要用ln100，ln200，而不是直接代入？题目里哪里可以看到用ln？

问题如下：

Anna Gabor is an analyst with an investment management firm. She is investigating the impact of leverage on annual ROE in the firm's Eurozone equities fund ("the fund").She collects data on the 72 stocks in the fund, including company size as an independent variable.

Exhibit 1 shows the results from estimating the model

R_i = b₀ + b₁(Size_i) + b₂(D/E_i) + ε_ι

As part of her analysis Gabor compares the predictions of her model with the actual ROE for two stocks in the portfolio which have the same debt ratio: A and B, which have market capitalizations EUR 100 million and EUR 200 million respectively.

Based on Gabor's regression, the amount that ROE for Company A exceeds that of Company B (in percentage points) is closest to:

选项：

A.

0.83

B.

1.2

C.

6.36

解释：

A is correct because this reflects the right application of the results of the regression.

The coefficient of the independent variable Size_i is b₁ = -0.012. This implies a negative relationship between the log of market capitalization and ROE.

The prediction of ROE, according to the regression, is given by the equation: R_i = b₀ + b₁(Size_i) + b2(D/E_i)

Or, reading the coefficients from Exhibit 1: R_i = 0.2248 - 0.012(Size_i) - 0.0172(D/E_i)

We are told that the debt/equity ratio D/E is the same for both companies, therefore 0.2248 - 0.0172(D/E_i) is the same amount for both companies A and B; let us say that 0.2248 - 0.0172(D/E_i) = K (a constant) for both A and B

Then R_i = K - 0.012(Size_i)

The market cap of Company A is EUR 100 million, therefore Size_A = ln (100)

The market cap of Company B is EUR 200 million, therefore Size_B = ln (200) = ln (2) + ln (100)

ROE for Company A is K - 0.012*ln(100) and the ROE for Company B is: K - 0.012*ln(100) - 0.012*ln(2)

So ROE for Company A exceeds that for Company B by: 0.012 * ln (2) = 0.008318 or approximately 0.83 percentage points.