1. Regression Specification Error Test
Can Xu

2. Functional Form Misspecification
• 𝑦 = 𝛽0 + 𝛽1 ∙ 𝑥 + 𝜖
• 𝑦 = 𝛽0 + 𝛽1 ∙ 𝑥 + 𝛽2 ∙ 𝑥2
+ 𝜖

3. Idea behind RESET
• If the original model 𝑦 = 𝛽0 + 𝛽1 ∙ 𝑥 + 𝜖
satisfies the zero conditional mean assumption,
then nonlinear functions of the independent
variable 𝑥: 𝑦2
should be insignificant when
added to the above equation.

4. Steps of RESET
• 𝑦 = 𝛽0 + 𝛽1 ∙ 𝑥
• 𝑦 = 𝛽0 + 𝛽1 ∙ 𝑥 + 𝛿1 ∙ 𝑦2
+ 𝜖
o 𝐻0: 𝛿1 = 0 (no functional form problem)
o 𝐻1: 𝛿1 ≠ 0 (functional form problem)

5. Example of RESET Application
• Karagas (1996) assessed the relationship
between toenail arsenic concentrations and
ingestion of arsenic-containing water.
• Twenty-one participants were interviewed
regarding use of their private (unregulated)
wells for drinking and cooking, and each
provided a sample of water and toenail
clippings.

6. Example of RESET Application
• Karagas MR, Morris JS, Weiss JE, Spate V,
Baskett C, Greenberg ER. Toenail Samples as
an Indicator of Drinking Water Arsenic
Exposure. Cancer Epidemiology, Biomarkers
and Prevention 1996;5:849-8

7. Original Model in Karagas (1996)
• log10 nails = β0 + β1 ∙ log10 water + ϵ
• The effect of a 10-fold increase in well water
concentrations of arsenic on toenail
concentrations would be 10β1 = 100.3257
=
2.12, about 2-fold increase.

8. RESET of Original Model
• log10 nails = β0 + β1 ∙ log10 water + 𝛿1 ∙ log10 𝑛𝑎𝑖𝑙𝑠
2
+ ϵ
o 𝐻0: 𝛿1 = 0 (no functional form problem)
o 𝐻1: 𝛿1 ≠ 0 (functional form problem)
• log10 𝑛𝑎𝑖𝑙𝑠 is OLS fitted values from the original regression

9. _cons .8369798 .3443204 2.43 0.032 .08677 1.58719
lnailshatsq 1.253439 .6573278 1.91 0.081 -.1787556 2.685633
lwater .787318 .2477668 3.18 0.008 .2474804 1.327156
lnails Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 2.30452848 14 .164609177 Root MSE = .20731
Adj R-squared = 0.7389
Residual .515737563 12 .04297813 R-squared = 0.7762
Model 1.78879092 2 .89439546 Prob > F = 0.0001
F( 2, 12) = 20.81
Source SS df MS Number of obs = 15
. reg lnails lwater lnailshatsq
. gen lnailshatsq = lnailshat^2
(option xb assumed; fitted values)
. predict lnailshat

10. Updated Model
• log10 nails = β0 + β1 ∙ log10 water + β2 ∙ log10 𝑤𝑎𝑡𝑒𝑟 2 + ϵ
• The effect of a 10-fold increase in well water concentrations
of arsenic on toenail concentrations is
10β1+2∙β2∙log10 𝑤𝑎𝑡𝑒𝑟
= 10β1 ∙ 102∙β2∙log10 𝑤𝑎𝑡𝑒𝑟
= 10β1 ∙ 10log10 𝑤𝑎𝑡𝑒𝑟2∙𝛽2
= 10β1 ∙ water2∙β2
= 100.981 ∙ water0.266
it depends on the level of Arsenic Water.

11. • At the mean value of the variable 𝑤𝑎𝑡𝑒𝑟 which equals to
0.0228, a 10-fold increase in well water concentrations of arsenic
will cause
100.981 ∙ 0.02280.266 = 3.5-fold
increase on toenail arsenic concentrations.

14. Conclusion
• There is functional form misspecification
problem in Kagras(1996).
• The model with quadratic term turned to be a
better choice.

15. Thank you!