RESET (Regression error specification test)
In multiple regression the population regression equation could be nonlinear in any or
all the explanatory variables. Suppose we were estimating
Y = α1 + α2 · X1 + α3 · X2 + ε
(1)
but suspected that the true regression line was nonlinear. Estimating (1) would yield
Ŷ = α̂1 + α̂2 · X1 + α̂3 · X2 .
(2)
The square of Ŷ obtained from (2) depends both on the squares of X1 and X2 and
on their cross–product X1 X2 . Similarly higher powers of Ŷ will be functions of higher
powers and cross–products of X1 and X2 . Hence, instead of attempting to estimate a
multiple regression equivalent to
E(Y ) = α1 + α2 · X1 + α3 · X2 + α4 · X12 + α5 · X22 + α6 · X1 · X2 + . . . ,
we can estimate the multiple regression equivalent
Y = α1 + α2 · X1 + α3 · X2 + δ · Ŷ 2 + ε.
(3)
That is, we added to equation (1), as extra explanatory variable, the power of Ŷ obtained from (2). Values for the coefficients of the power of Ŷ that prove significantly
different from zero could then be regarded as evidence of a nonlinear population regression equation.
For the household data we have
Ŷ = 36, 9150 + 0, 331374 · X1 + 0, 125786 · X2 .
So we can estimate model (3) and obtain a t ratio for δ of 0,850. The critical t value
in this case (n − k = 25 − 4 = 21 df) is t0,05 = 1, 721. Since δ = 0 is not rejected, the
linear model may be preferred.
We have now seen that significant coefficients on the powers of Ŷ can be regarded
as indicating either omitted relevant variables or mis–specified functional form. Thus
RESET test can be regarded as a test of general mis–specification. When we apply it,
the null hypothesis is that of correct specification but we have no definite alternative
hypothesis in mind. Rejection of the null hypothesis merely indicates that the equation
has been mis-specified in some way or other.
1
Model (3): OLS estimates using the 25 observations 1–25
Abhängige Variable: cons
Variable
const
income
assets
Ypred2
Koeffizient
Std. Fehler
47,5997
0,424189
0,0444514
0,000000
21,4766
0,204911
0,102956
0,000000
Mittelwert der abhängigen Variable
S.D. of dependent variable
Summe der quadrierten Residuen
Standardfehler der Residuen (σ̂)
Unkorrigiertes R2
Korrigiertes R̄2
F (3, 21)
Log-Likelihood
Akaike Informations-Kriterium
Schwarz’ Bayes-Kriterium
Hannan–Quinn-Kriterium
2
t-Statistik
P-Wert
2,2164
2,0701
0,4317
0,8498
0,0378
0,0510
0,6703
0,4050
163,294
81,3144
31431,3
38,6876
0,801931
0,773635
28,3412
−124,68
257,364
262,240
258,716