Read Example 7.9 on page 233 carefully.

Consider the following sample regression function:

log
⁡
(
𝑤
𝑎
𝑔
𝑒
)
^

=

𝛽
^
0
+
.2
𝑐
𝑜
𝑚
𝑝
𝑤
𝑜
𝑟
𝑘
+
.05
𝑐
𝑜
𝑚
𝑝
ℎ
𝑜
𝑚
𝑒
+
.03
𝑐
𝑜
𝑚
𝑝
𝑤
𝑜
𝑟
𝑘
⋅
𝑐
𝑜
𝑚
𝑝
ℎ
𝑜
𝑚
𝑒
+
𝑜
𝑡
ℎ
𝑒
𝑟
𝑓
𝑎
𝑐
𝑡
𝑜
𝑟
𝑠

(
.01
)
(
.02
)
(
.005
)

where the variables wage, compwork, and comphome are defined respectively as hourly wage, = 1 if an individual uses a computer at work, and = 1 if an individual uses a computer at home. Which of the following statements is correct?

Question

Read Example 7.9 on page 233 carefully.

Consider the following sample regression function:

log
⁡
(
𝑤
𝑎
𝑔
𝑒
)
^
	
=
	
𝛽
^
0
+
.2
𝑐
𝑜
𝑚
𝑝
𝑤
𝑜
𝑟
𝑘
+
.05
𝑐
𝑜
𝑚
𝑝
ℎ
𝑜
𝑚
𝑒
+
.03
𝑐
𝑜
𝑚
𝑝
𝑤
𝑜
𝑟
𝑘
⋅
𝑐
𝑜
𝑚
𝑝
ℎ
𝑜
𝑚
𝑒
+
𝑜
𝑡
ℎ
𝑒
𝑟
𝑓
𝑎
𝑐
𝑡
𝑜
𝑟
𝑠

(
.01
)
(
.02
)
(
.005
)

where the variables wage, compwork, and comphome are defined respectively as hourly wage, = 1 if an individual uses a computer at work, and = 1 if an individual uses a computer at home. Which of the following statements is correct?

BlackTom AI · Accepted Answer

The question presents a log-linear regression: log(wage) = β0 + 0.20·compwork + 0.05·comphome + 0.03·compwork·comphome + other terms. Here, compwork = 1 if the person uses a computer at work (0 otherwise) and comphome = 1 if the person uses a computer at home (0 otherwise).

Option 1: The estimates imply that the average wage of a person who uses a computer at work (but not at home) is about 20% above the average wage of a person who doesn’t use a computer at all, holding all the other factors constant.
- When compwork = 1 and comphome = 0, the predicted change in log(wage) is +0.20. To translate a log wage change into a percentage change in wage, we use exp(Δlog wage) − 1. Here, exp(0.20) − 1 ≈ 0.221, i.e., about a 22.1% increase in wage. The statement says about 20%, which is a rough, approximate interpretation. The order of magnitude is correct, but the precise translation yields slightly more than 20% when using the exact exponentiation.

Option 2: The estimates imply that the average wage of a person who uses a computer at home (but not at work) is about 5% above the average wage of a person who doesn’t use a computer at all, holding all the other factors constant.
- When compwork = 0 and comphome = 1, the predicted change in log(wage) is +0.05. Converting to a wage percentage: exp(0.05) − 1 ≈ 0.0513, i.e., about a 5.13% increase. This aligns well with the statement that it is about 5%. The approximation is fairly tight because 0.05 is small, so the exponential and the linear percent-change interpretation are close.

Option 3: The estimates imply that the average wage of a person who uses a computer at home and at work is about 28% above the average wage of a person who doesn’t use a computer at all, holding all the other factors constant.
- When compwork = 1 and comphome = 1, the interaction term adds 0.03 to the main effects, so the total Δlog wage is 0.20 + 0.05 + 0.03 = 0.28. The exact wage-change interpretation is exp(0.28) − 1 ≈ 0.323, i.e., about a 32.3% increase, not 28%. Reporting ~28% would be a modest under- or over-approximation depending on whether one uses the log-to-percentage shortcut (linear approximation) or the exact exponential transformation. The precise interpretation therefore yields around 32%, which differs notably from 28% if taken as an exact figure.

Option 4: All of the above.
- If we evaluate each statement against the exact transformation, we see that Option 1 yields about a 22% increase rather than exactly 20%, Option 2 yields about a 5% increase which matches well, and Option 3 yields about a 32% increase rather than exactly 28%. Whether one accepts these as 'about' figures depends on the tolerance for approximation when converting from log changes to percentage changes. When treated strictly, not all three statements align perfectly with the precise exponential conversion, so combining them as an exact truth set would be questionable. However, in many applied contexts, folks treat the reported coefficients as approximations to percent differences.

In sum, the core idea is that coefficients on dummy variables in a log-linear model translate into percentage changes in the dependent level via the formula exp(Δlog(wage)) − 1. The pure, exact percentages for the given combinations are approximately: working at work only ≈ 22%, home only ≈ 5%, both ≈ 32%. The statements give rounded figures: ~20%, ~5%, ~28%. The second figure is closest to the exact 5%; the first and third are approximate with small discrepancies when using the exact exponential conversion.

类似问题

Consider the following estimated regression model [math: ln⁡(yi)^=0.034−0.023xi]\widehat {\ln (y_i)}=0.034-0.023x_i where [math: ln⁡(.)]\ln (.) denotes the natural log. Which of the following is a correct interpretation?[Fill in the blank]

The following data were collected for two related variables x and y. A scatterplot indicates a non-linear association. The data is linearised using a logx transformation and a least squares line is then fitted. The equation of this line isclosest to:

An exponential model with a quarterly data forecasting equation is log( ˆ Y i)=2.0886+0.0016Xi−0.0528Q1−0.0389Q2−0.0449Q3 Where Xi represents the coded quarter, Qi is a dummy variable for quarters, Qi=1 if it is ith quarter. What is the prediction when Xi=27, and it is the fourth quarter?

The following data were collected for two related variables x and y. A scatterplot indicates a non-linear association. The data is linearised using a logx transformation and a least squares line is then fitted. The equation of this line isclosest to:

What was the world's first widely adopted biodiversity policy?

The aim of the Paris Agreement is to ensure that the earth’s preindustrial temperature is not exceeded by more than º Celsius.

The IPCC's mission is to Blank ______.

Why have most international agreements for environmental protection been based on the honor system?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单