ECO 321 Summer 2021
Practice Problem 2
1. Assume the true population regression model is
Yi = β0 + β1 X1i + β2 X2i + ui , i = 1, · · · , n
There are two independent variables X1 and X2 . The researcher, however,
makes a mistake that assuming there is only one independent variable,X1 , in
population regression model and omitting X2 . This is a misspecified model as
following
Yi = β0 + β1 X1i + vi , i = 1, · · · , n
Then, the researcher estimates β1 by the wrong model. Is the estimator, β̂1 , of
the wrong model biased? If yes, what is the bias?
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2. A researcher plans to study the causal effect of police on crime using data from
a random sample of U.S. counties. He plans to regress the county’s crime rate
on the (per capita) size of the county’s police force.
(a) Explain why this regression is likely to suffer from omitted variable bias.
Which variables would you add to the regression to control for important
omitted variables?
(b) Use your answer to a) and the expression for the omitted variable bias to
determine whether the bias will be positive or negative. (That is, do you
think β̂1 > β1 or β̂1 < β1 ?)
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3. Suppose that a researcher collects data on houses that have sold in a particular
neighborhood over the past year and obtains the regression results in table 1
below.
The dependent variable is ln(P rice), the log of the price of the house; size is the
total size of the lot (in square feet); Bedrooms gives the number of bedrooms
in the house; Recreation indicated whether the house has a recreational room;
Garage is an indicator for garage; P ref er is a dummy equal to 1 if the house
is located in a preferred neighborhood, and 0 otherwise.
(1) (2)
Size 0.00007
(0.00001)
ln(Size) 0.41153
(0.03377)
Recreation 0.17558 0.15323
(0.03230) (0.03218)
P ref er 0.17365 0.17570
(0.02847) (0.02713)
Garage 0.08822 0.08080
(0.01669) (0.01654)
Intercept 10.22418 7.12141
(0.04988) (0.27275)
SSR 44.54111 42.52150
R2 0.40937 0.43615
Table 1: Dependent Variable ln(Price). Std. Errors in parentheses below coefficients.
(a) Using the results in column (1), what is the expected change in price of
building a 500-square-foot addition to a house?
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(b) Using the results in column (1), construct a 95% confidence interval for
the percentage change in price.
(c) Comparing column (1) and (2), do you think it is better to use Size or
ln(Size) to explain house prices? What indicators are you using to reach
your conclusion?
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(d) Using column (2), what is the effect of having a recreational room on
price? Construct a 95% confidence interval for this effect.
4. Consider the regression model,
Yi = β0 + β1 Xi1 + β2 Xi2 + β3 (Xi1 × Xi2 ) + Ui
Use either calculus or first differences to show:
(a) ∆Y /∆X1 = β1 + β3 X2 (effect of changing X1 while holding X2 constant).
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(b) ∆Y /∆X2 = β2 + β3 X1 (effect of changing X2 while holding X1 constant).
(c) If X1 changes by ∆X1 and X2 changes by ∆X2 , then ∆Y = (β1 + β3 X2 ) ∆X1 +
(β2 + β3 X1 ) ∆X2 + β3 (∆X1 ) (∆X2 ).
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We compute the p-value from the table below using the computed statistic, say 1.96,
and the coordinate in the table below from the 1.9 row and the 0.06000 column.
This value is 0.9750. The p-value is then equal to twice one minus this value: 2x(1
- 0.9750)=2x0.0250=0.05. Note that you have to multiply by two and the corre-
sponding row and column stem from the t-statistic (1.9 row and 0.06 column)! Any
t-statistic larger than 3 leads to a p-value which is less than 0.0027.
Rx
Table: Area under the Normal Curve from 0 to X= −∞ normal pdf.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
