A Kernel Test for Neglected Nonlinearity

by

Ralph Bradley

Robert McClelland

Room 3105

US Bureau of Labor Statistics

2 Massachusetts Ave. NE

Washington, DC 20212

(202)-606-6573

The authors would like to thank Tim Erickson, Jeff Wooldridge and participants of seminars at the American Statistical

Association, the Bureau of Labor Statistics, George Mason University and the 1994 Summer Econometric Society meetings. The

views expressed in this paper are solely those of the author, and do not necessarily reflect the policy of the Bureau of Labor

Statistics (BLS) or the views of other staff members of BLS.
page 1

1. Introduction

Recent tests of the nonlinearity of the functional form a regression model have focused on attempts to develop consistent

tests that do not specify a parametric alternative and encompassing forms. These tests are a special case of the broader

conditional moment tests for misspecification, such as that of Bierens (1990), that have greater power by restricting the

specification to the linear form. Examples of this are the tests of Lee, White and Granger (1993) (LWG), which is based on the

Bierens test and which detects omitted nonlinearity through the use of a 'neural net', and Wooldridge (1992), which is based on

the Davidson-Mackinnon test and uses a Fourier expansion. Like the test of Bierens, these tests are consistent against deviations

from the null.

In this paper we adapt the consistent test of Bradley and McClelland (1994) to a test of nonlinearity in a manner

analogous to the adaptation of the Bierens test by LWG. Our test is based on the idea of Newey (1985) that if the model is not

properly specified then the vector of explanatory variables, x, can better predict the residuals from a linear regression than the

residual's mean. Thus, while many x-measurable functions, such as a neural net, can be used to detect neglected nonlinearities

an appealing function to use is the conditional expectation of the residuals given x. We implement this idea in the first stage of

our test by estimating the residuals as a function of the explanatory variables. Because we obtain our estimate with a kernel

regression, our test is nonparametric. The estimate is then a component of the second stage, which is similar to the neural net

test with the kernel estimate in the first stage replacing the net.1 By directly estimating the nonlinearity with a conditional

expectation function, the limit of our test over the sample size is the largest of all conditional moment tests. Because the kernel

regression is an estimate of the misspecification, the test can be used to better understand the nature of any misspecification.

Another advantage to our test is the potential for increased finite sample power through a constrained cross-validation of

the window width parameter. Cross-validating with a quadratic loss function is particularly appealing in our case because the

expected value is minimized by predicting the residuals with their conditional expectation. This connects with the idea that the

regressors can better predict the residuals than the residual's unconditional mean in a misspecified model, so that procedures that

increase this predictability increase the power of our test. Lewbel (1993), Wooldridge (1992), and Bierens (1990) discuss the

 1For an example of a two-stage procedure for semi-parametrically estimating a binary choice model under uncertainty

using estimated conditional expectations, see Manski (1991).
page 2

problem that a cross-validation type mechanism can overfit the alternative model under the null hypothesis, forcing their tests to

acquire undesirable or unknown distributions. Thus, none of these studies use cross-validation.

In our study we cross-validate but address the problem of overfitting by resampling and limiting the window width to a

compact set of permissible values. This resampling also solves the problem caused by the dependence inherent in the residuals

produced by a regression that includes a constant term. This, plus the absence of bias for the kernel regression under the null

hypothesis also allows the distribution of our statistic to converge to a χ2(1) distribution.

Finally, we compare Monte Carlo simulations of the results of using our test and the neural net and Ramsey RESET test.

Because we use the same models studied in Wooldridge (1992), we can also make limited comparisons with the sieve test

developed in that paper. These simulations show that the kernel test is about as powerful as the neural net test in many

simulations and more powerful than it or the RESET test in others. We then choose one simulation at random under this

alternative hypothesis and show how the kernel estimation in the first stage our test can be used to describe the nature of the

misspecification.

The remainder of our paper is organized as follows. Section 2 describes the test of Bradley and McClelland (1994) and

then shows how this can be adapted to a test for the detection of nonlinearity in a regression. Section 3 discusses results from

Monte Carlo experiments that compares our test to the neural net and Ramsey RESET test and conclusions are in Section 4. The

assumptions on the random vectors of dependent and independent variables (y,x) are listed in Appendix A and the proof of

theorem 1 is in Appendix B.

2. A Misspecification Test

In this section we describe a test in the set of conditional moment tests that use a direct estimate of the misspecification

as the weighting function on the estimated residuals. Relying upon the work of Bierens (1982, 1987, 1990) and described in

detail in Bradley and McClelland (1994), this test is consistent against all deviations from the null hypothesis and has the

greatest asymptotic power of all tests in set of conditional moment tests in the sense that the probability limit of the kernel

estimate is the function that maximizes plim (W/n), where W is defined as an element in the set of conditional moment tests and

n is the sample size.

We also increase the finite sample size power of our test over a fixed window width by using a cross-validated window

width. Only two parameters need to be set before using the test: bounds on the window width and the size of the bootstrap
page 3

sample. Lastly, we can use the estimate of the misspecification to gather insight into the nature of any problem detected by the

test. This general specification test can also be modified to test the more restrictive null of linearity. By following the same

intuition as that guiding LWG, we construct a Lagrange multiplier test of linearity that can be constructed from a simple R2

statistic.

To focus our discussion upon the use of a direct estimate of the misspecification as a weighting system and to make our

test as compatible as possible with the sieve test of Wooldridge, we restrict ourselves to an independently and identically

distributed (IID) random sample {yi

, xi}, i=1,...,n from a distribution F(y,x) on R×Rk. Further, we assume that E(y2) < ∞ and

that:

y = f(x,θ) + u.

E(y|x) = f(x,θ).

Suppose we use the functional form f(x,θ) to estimate E(y|x) and define ûi

 = yi - f(xi, θ

ˆ). A direct estimate of the

misspecification is then g(x) ≡ E(û|x). Under Ho, we assume that the true value of θ which we denote as θο satisfies:

(1.) θο = argminθ∈θ E([yi - f(xi

,θ )]2 ) and E([yi-f(xi,θο )]|xi) = 0.

Let θ

ˆ be a consistent estimate of θo. This study is based on assumptions outlined in Appendix A. If Ho in (1.) is correct, then

(2.) plim E(û*h(x)) = 0

for all h(x) ∈H(X), the set of x-measurable functions on X, the domain of x. Under H1, there exists some function

 (3.) plim E(û*h(x)) > 0.

A reasonable set of tests for misspecification would then be

{ } W n

uhx

s

h hx HX

j j j

n

=

























∈





























∑ =

1

1

2

$ ( )

$ () ( ) .

The Rao-Blackwell theorem tells us that under H1 :

(4.) lim E(û-g(x))2 < lim E(û - E(û))2

which implies lim E(ûg(x))=lim E(g(x)2) > 0. Under Ho, since plim θ

ˆ=θo , the inequality in (4.) becomes an equality so that

lim E(ûE(û|x)) = 0. The function g(x) therefore satisfies both conditions (2.) and (3.) and our test uses a nonparametric estimator

of g(x). The features in (1.) through (4.) suggest the following four steps to test the null hypothesis in (1.):

(i) Estimate θ and denote this estimate as θ

ˆ.
page 4

(ii) Generate ûi

 = yi - f(xi,θ

ˆ) for i = 1,..,n.

(iii) For each observation i, estimate g(x)=E(ûi

|xi) with a kernel regression by selecting a random subsample with

replacement of size n’=int(n3/2γn

k/2) from the existing sample.

Specifically, for the original sample of size n let N={1,...,n} index the sample. We then choose for each sample point i a

random subsample with replacement from N of size n' which we denote as Ni

'. We then define the cardinal ‘bootstrapping’

variable S(A) as the number of occurrences of event A, choose for each sample point i a random subsample with replacement

from N of size n' which we denote as Ni

' and calculate for each i a kernel estimator denoted as gˆi(xi, γi) where:

(5.) $

$ ( )

( )

g (x , )

S(j Ni

' )u K

x x

S(j Ni

' )

x x i i

j

i j

'

i j

γ γ

γ

i

i

i

=

∈ −

∈ −

∑

∑ Κ

and γi is the bandwidth of the kernel function K(·) with the following properties:

(P.1) ∫ K(u) du = 1

(P.2) ∫ uK(u)du = 0

(P.3) u du j

i

j

k

j

j

∫ ∏ = 0 for i m j j

k

∑ < , where m is the kernel order such that nγn

k

→∞ and nγn

2m+k/2→0.

(P.4) 0 = argmax ( ) u

Ku.

(iv) Calculate the statistic:

(6.) W( , ) $ $ γ θ ∗ = n Tˆ 2(γ∗ ,θ

ˆ)/sˆ2(γi

),

where

(7.) $ ( *, $ T n ug x ) (/ ) $ $ (, ) i

i

n

iii γθ γ = 



 



 =

1 ∑

1

,

γ* = {γ1,γ2,...,γn}

and sˆ2(γi

) is a consistent variance estimate.

To increase the predictability of gˆ(x,γ) in finite samples, we choose γi

 as the cross validated window width by solving:

γ γ

γ

i

C B n n

= − u g





 



 < < <∞ −

∈

arg max ∑ $ $

$ ( , ))

'

k kk

k Ni

x

2

where
page 5

$

$ ( ,)

$ ( )

( )

',

',

g

u K

x x

K

x x

j

k j

j Ni j k

k j

j Ni j k

−

∈ ≠

∈ ≠

=

−

−

∑

∑

k k x γ

γ

γ

and the bound Bn to have two properties:

(P.5) Bn ≤ Bn+1

(P.6) lim sup Bn = B < ∞ .

(P.7) Cn = O(nγn

2m+k/2) for Cn< Bn for all n.

The limit, B, on the window width γi

 is the only parameter we must choose a priori and has another important feature

under Ho. The bound B along with the property (P.4) for the kernel K(⋅) guarantees that sˆ(γ*)2 is always positive as long as the

variance of x is greater than zero. It also prevents the window width from becoming so large that the test statistic degenerates to

a constant. The lower bound Cn is used to prevent overfitting under the null.

Wˆ (γ∗ ,θ

ˆ) is essentially a Wald Test for the restriction that E(û[Eû|x]) = 0. The following theorem is the main result of

Bradley and McClelland (1994):

Theorem 1 (Bradley and McClelland (1994)): Let the assumptions in Appendix A hold. The statistic Wˆ (γ∗ ,θ

ˆ) generated in

equation (6.) is asymptotically distributed as χ2(1) under Ho.

This theorem shows that the limiting distribution of Wˆ (γ∗ ,θ

ˆ) under the null is a χ2 distribution with one degree of freedom.

Given the properties (P.1) through (P.4) Bradley and McClelland (1994) test is consistent against all alternative hypotheses and

additionally that Wˆ (γ∗ ,θ

ˆ) is the most asymptotically powerful test in the set {Wh}. The reason for this is that E(û|x) is the

function h(x) in the set of bounded functions which maximizes E{ûh(x)}. In essence, Tˆ in equations (6.) and (7.) is an estimator

of E{û E(û|x)} and therefore the numerator in equation (5.) uses an estimator that maximizes the value of E{ûh(x)}.

We now turn to a possible complication. First, gˆi

(xi,γi) is not an IID process even though the errors in the true model are

IID under Ho, partially because there is finite sample dependence among observations of ûi

 arising from the first moment

restriction, u$ i i

n = ∑ = 0

1 when there is a constant term. We solve this problem in step three by using a bootstrap from the

original sample. This bootstrapping allows us to use a projection theorem from the literature on U-statistics that shows that the

moments of {ûi

 , gˆi(xi,γi)}, i=1,2,..n converge in probability to the moments of an IID random process. The only parameter we

must choose is the size of the bootstrap, which can be chosen according to standard bootstrap criteria, as long as the bootstrapped

sample is of size O(n).
page 6

3. The Kernel Test

Given the statistic Wˆ (γ∗ ,θ

ˆ), we can construct a test for neglected nonlinearity in the same manner that Lee, White and

Granger (1993) use the results of Bierens (1990) to construct the neural net test for nonlinearity. The null that we wish to test is

now more restrictive than (1). For the linearity case, the assumptions in Appendix A. ensure that under Ho plim θ

ˆ-θo= 0 and

under H1 plim θ

ˆ exists.

(8.) Ho: there exist θ ∈ Θ ⊂ Rk such that Pr(E(y|x)=x'θ) = 1

(9.) H1: Pr(E(y|x)=x'θ) < 1 for all θ ∈Rk

Here the null is that the true conditional expectation function is linear, which allows us to offer a Lagrange Multiplier

version of the statistic Wˆ (γ∗ ,θ

ˆ). To form this statistic, we derive θˆ from an ordinary least squares (OLS) regression of yi

 on xi.

We calculate the estimated residuals, ûi

, and we estimate the kernel density gˆi(xi,γi) as outlined in (5.). Finally, we regress ûi on

xi

 and gˆi(xi,γi).

In order to use the statistic, we need the following theorem for x-measurable functions h(x):

Theorem 1)

Under the linear null hypothesis in (8), and homoskedasticity, define the statistic Wn as follows:

Wn = n [ ] h u

u h I x(x(x x x h u

i

i i

' $

$ ' ( ' ) )' $

2

1 − − .

Then W nR n p − → 2  0, where R2 is the coefficient of determination of ûi

 regressed on x and h(x) where h is the

column vector of h(xi).

Proof See Appendix B.

Because gˆi

(xi,γi) is an x-measurable function, we can apply this theorem to it. We then have a test statistic that avoids

explicit computation of Wˆ (γ∗ ,θ

ˆ) because

(10.) nR2 d 1  → χ2( )

where R2 is the uncentered squared multiple correlation from an ordinary least squares regression of ûi

 on xi and gˆi(xi,γi).

If H1 is true, then the probability limit of the regression coefficient on gˆi

(xi,γi) should be one. If one rejects the null that

the coefficient is equal to one then there is evidence that gˆi

(xi,γi) is not picking up the misspecification E(u|x). Since gˆi(xi,γi) is an

estimator for E(û|xi

), it can provide direction in which the investigator can go to better specify the model if the coefficient is not

significantly different from one. Although not proved in this article, under misspecification the OLS coefficient on gˆi

(xi,γi)
page 7

converges in probability to one. If the coefficient is not significantly different from one, then one can look at the confidence

intervals of derivatives of gˆ(x, γi

) to determine the additions to the linear model that need to be added to achieve a correct

specification.

As with the more general test we need to select two parameters: bounds on the bandwidth and the bootstrap sample size.

For the simulations in the next section, we set the upper bandwidth bound to 19.5 + 1/n and the lower bound to 1.5n-1/δ

 where δ

satisfies the inequality, k<δ<m-k/2. These bounds on the cross validation serve to prevent overfitting under the null.

Additionally, these limits force the cross validated bandwidth to converge to zero at a rate where the asymptotic bias of the kernel

estimate will go to zero. The bootstrap sample size is trunc(.85n) and the estimated variance in the denominator equals the

order of our statistic in the numerator so that the statistic does not degenerate under the null. In addition, we use the

Epanechnikov kernel, which satisfies the properties P.1, P.2 and P.4..

4. Monte Carlo Experiments

We now briefly consider two Monte Carlo experiments discussed in Wooldridge (1992), allowing us to make some

comparisons with his sieve test. In each, we compare Wn with the neural net test of LWG using ten nodes and the Ramsey

RESET test using three powers of the estimated conditional expectation of y. We consider sample sizes of 50, 100 and 200 and a

variety of misspecifications. We also show how the function gˆ(x, γi

) can be used to gather information about the nature of the

misspecification.

Both the sieve test and the neural net test are also robust tests. Our test is more closely related to the neural net test in

that both are variations of the Bierens 1990 test and both are based on the asymptotic distribution of nR2 of the residuals

regressed on the X values and an additional non-linear X-measurable function. Wooldridge uses the Davidson-Mackinnon test

framework with a linear regression and a nonparametric sieve regression. In order to achieve consistency under the null his

truncation parameter has to expand at a rate that is an inverse function of the rate at which yˆ converges to y. Because

Wooldridge uses the entire sample for his sieve estimation, this bound on the expansion rate prevents him from cross-validating

his truncation parameters.

The neural net test takes the inner product of the column vectors of x and q independent random k-vectors. These q

random vectors are also known as "nodes". The q products x ̃ are then transformed by the logistic cumulative density function

(1+e-x ̃ )-1. To reduce correlation among the q transformed products, only the first two principle components of the transformed
page 8

products are taken. These components are then regressed on û. The resultant test statistic formed by the R2 multiplied by the

number of observations has an asymptotic χ2(2) distribution.

The RESET test is designed to detect misspecifications of E(y|x) by modeling the misspecification as a power expansion

of E(y|x). While the test is easy to calculate, its power depends upon the degree to which the misspecifications are well

approximated by a low-order expansion. We implement the RESET test by checking the joint significance of α2

 , α3 , α4 in a

regression of the form

yi = xiθ + ∑j=2

4 αi

(yi

ˆ |xi)j + ei.

Under the null hypothesis of a correct specification, all αi

's should equal zero. We use the LWG version of the test so that it has a

χ2(3) distribution under the null.

The data generating process for both experiments is essentially the same. Let v1 and v2 be two independent random

variables with uniform distributions on [0, 2π] and define

z1 = v1

z2=0.5v1 + 0.5v2.

Then the vector of independent variables are defined as

x=[1, z1, z2]

and the coefficient vector is

θ=[1,1,1]'.

Given homoskedastic residuals u drawn from a standard normal distribution, the data generating process under the null

hypothesis is:

yi

 = xiθ + ui.

The first experiment considers the model:

(DGP1) yi

 = xiθ + γ(xiθ)2 + ui.

Table 1 shows that all three tests are effective in detecting the misspecification. With 50 observations all three tests reject almost

100% of the iterations with γ equal to 0.20 or -0.20. With 100 observations the three tests reject when γ is equal to 0.15. Because

the RESET actually nests the true misspecification, it should and does have the greatest power. Comparing the kernel test and

the neural test to the RESET is worthwhile here because it represents the upper bound on power. Both the kernel and neural net
page 9

tests seem to reject almost exactly the same number of cases although a cursory examination of the rejection patterns shows that

different simulations are being rejected. As the sample size increases, the rejection rates increase towards 1.000 rejection rate.

One difficulty with the kernel test appears to be its high rejection rate under the null hypotheses (γ=0). This rate reflects

the finite sample bias of the kernel estimate. As the sample size increases, however, the rejection rate decreases somewhat and

should continue to decrease to the proper rejection rates.

These results may also be compared with the results of Wooldridge (1992) using the sieve test. With 100 observations

and γ equal to 0.1, the sieve test rejects 0.576 and 0.418 of the cases at the ten and five percent points, respectively. With the

same number of observations and γ equal to -0.1 the test rejects 0.999 of the cases at both the ten and five percent points. This

shows that for this model the power of the sieve test is more sensitive to the direction of the misspecification than any of the three

tests considered here.

The second experiment considers the model:

(DGP2) yi

 = (xiθ)λ + ui.

For values of λ we consider a range of values from 0.5 to 1.5 and for -0.5. The results in table 2 are somewhat

ambiguous. For values of λ between 1.2 and 1.5 the kernel and neural tests move together, although the neural test decreases in

power from 1.2 to 1.3 more rapidly than the kernel test. As expected, the RESET test performs at a progressively greater rate as

λ rises. When λ is equal to 1.5 with 100 observations we can make a direct comparison with the sieve test. Comparing the

nearly uniform rejection from all three tests against the rejection rates of 0.804 and 0.693 at the 10% and 5% points shows tha t

other nonparametric tests have greater power against this alternative.

For values between 0.8 and 0.5 there is little change in power either as the sample size increases or as λ decreases. The

neural net and RESET tests appear to have essentially no power to detect the misspecification while the kernel test shows some

power. When λ is equal to 0.5 with 100 observations we can make another comparison with the sieve test. Comparing the low

power of all three tests against the rejection rates of 0.24 and 0.166 at the 10% and 5% points shows that the sieve test performs

approximately as well as the kernel test and outperforms the neural and RESET test. It would be of interest to know how the

sieve test exhibits the same constancy of rejection rates over the λ range of 0.8 to 0.5 as the three tests show in table 2.

Finally, the negative value of -0.5 for λ reveals nearly the same rejection rates as for 0.5. Again the neural and RESET

test appear to have little power for any of the sample sizes. Interestingly, the RESET test could not be performed for sample sizes

of 100 because of repeated failures of the matrices to invert at the second stage of the test. This appears to be due to the creation
page 10

of a variable yi

ˆ that has no essentially no variation. In contrast, the sieve test has much greater power than any of the tests,

rejecting 0.443 and 0.307 percent of the cases at the 10% and 5% points and this power appears to grow with the sample size,

rejecting 0.646 and 0.489 percent of the cases when the sample size increases to 300.

Although these simulations shed some light on the relative advantages of the tests for the misspecifications in DGP1 and

DGP2, an additional advantage of the kernel test is that an estimate of the misspecification is preserved. We can show how this

information, obtained with the estimation of gˆ(x, γi

), can be used to help the researcher understand the nature of the

misspecification. To do this, we plot the first regression from DGP2 with 200 observations and λ equal to 0.6. Figure one

shows the kernel estimate and the true conditional expectation ûi

 given x1 and holding x2 constant at the mean. We also include

the 95% confidence intervals to show how closely gˆ(x, γi

) follows ûi.

As the figure shows, û1 has a concave form consistent with modeling (xi

θ)0.6 as (xiθ). The kernel estimate gˆ(x, γi) also

have the same general shape, although it is much less smooth than û1. Still, this example shows how given only gˆ(x, γi

) and the

confidence limits, a researcher may obtain information about the manner in which the proposed model differs from the true

relationship.

4. Conclusion

In this paper, we adapt the test of Bradley and McClelland (1994) to a test of nonlinearity. Using the idea of Newey

(1985) that if the model is not properly specified then the vector of explanatory variables can better predict the residuals from a

linear regression than the residual's mean we use the conditional expectation of the residuals given the regressors. We implement

this idea by estimating the residuals as a function of the regressors with a kernel regression. By directly estimating the

nonlinearity with a conditional expectation function, the limit of our test over the sample size should be the largest of all

conditional moment tests.

Our test also allows researchers to increase power through cross-validation of the window width. We avoid the problems

of cross-validation discussed by other authors by resampling. This resampling also solves the problem caused by the dependence

inherent in the regression residuals.

In several Monte Carlo simulations we compare our kernel test to that of the neural net test, the Ramsey RESET test and

the sieve test. Although the choice of models essentially guarantees that the RESET test will outperform others in most

simulations, our test compares favorably with the neural net test in most simulations and appears to be somewhat more powerful
page 11

that either the neural net or the RESET tests in others. The sieve test does not appear to have as much power, with one the

notable exception. Lastly, we use the results from an arbitrary regression to show how the kernel regression estimated for the

test can be used by the researcher to gather information about the nature of any specification errors detected.
page 12

TABLE I

Simulations of DGP1 for Various Levels of γ

Percent of 1,000 Trials Rejected at the Asymptotic 10 per cent and 5 per cen

Significance Level

Sample Size

Significance Level

50

10% 5%

100

10% 5%

200

10% 5%

γ=0.20

 Kernel 0.970 0.957

 Neural 0.952 0.932

 RESET 1.000 1.000

γ=0.15

 Kernel 0.900 0.872 0.998 0.998

 Neural 0.912 0.870 0.985 0.973

 RESET 0.997 0.994 1.000 1.000

γ=0.10

 Kernel 0.669 0.614 0.916 0.899 0.997 0.996

 Neural 0.757 0.654 0.916 0.892 0.980 0.972

 RESET 0.922 0.868 0.998 0.995 1.000 1.000

γ=0.05

 Kernel 0.334 0.251 0.525 0.472 0.742 0.681

 Neural 0.372 0.243 0.553 0.421 0.795 0.706

 RESET 0.454 0.327 0.712 0.594 0.952 0.901

γ=0.00

 Kernel 0.163 0.101 0.157 0.099 0.151 0.09

 Neural 0.109 0.053 0.087 0.044 0.095 0.048

 RESET 0.119 0.051 0.113 0.058 0.098 0.049

γ=-0.05

 Kernel 0.302 0.218 0.483 0.417 0.748 0.694

 Neural 0.351 0.232 0.528 0.413 0.815 0.727

 RESET 0.372 0.249 0.658 0.543 0.954 0.909

γ=-0.10

 Kernel 0.648 0.578 0.922 0.901 0.994* 0.994

 Neural 0.740 0.629 0.922 0.883 0.980* 0.972

 RESET 0.860 0.785 0.992 0.987 1.000* 1.000

γ=-0.15

 Kernel 0.893 0.869 0.991 0.989

 Neural 0.915 0.875 0.974 0.963

 RESET 0.997 0.989 1.000 1.000

γ=-0.20

 Kernel 0.976 0.969

 Neural 0.953 0.933

 RESET 1.000 1.000

*Only 500 iterations are used in this simulation
page 13

TABLE 2

Simulations of DGP2 for Various Levels of λ

Percent of 1,000 Trials Rejected at the Asymptotic 10 per cent and 5

percent Significance Level

Sample Size

Significance Level

50

10% 5%

100

10% 5%

200

10% 5%

λ=1.5

 Kernel 0.896 0.875 0.995 0.995 1.000 1.000

 Neural 0.921 0.884 0.980 0.971 0.998 0.992

 RESET 0.993 0.987 1.000 1.000 1.000 1.000

λ=1.4

 Kernel 0.642 0.579 0.896 0.882 0.996* 0.994

 Neural 0.717 0.596 0.908 0.853 0.976* 0.970

 RESET 0.857 0.763 0.989 0.980 1.000* 1.000

λ=1.3

 Kernel 0.334 0.269 0.589 0.525 0.794 0.777

 Neural 0.381 0.272 0.645 0.529 0.877 0.813

 RESET 0.476 0.345 0.752 0.654 0.955 0.922

λ=1.2

 Kernel 0.267 0.198 0.290 0.237 0.418 0.339

 Neural 0.192 0.118 0.254 0.155 0.408 0.304

 RESET 0.221 0.134 0.291 0.194 0.486 0.368

λ=0.8

 Kernel 0.209 0.141 0.221 0.151 0.223 0.153

 Neural 0.142 0.077 0.135 0.070 0.125 0.072

 RESET 0.141 0.067 0.141 0.073 0.130 0.072

λ=0.7

 Kernel 0.196 0.139 0.205 0.142 0.280 0.202

 Neural 0.119 0.049 0.135 0.068 0.147 0.064

 RESET 0.115 0.051 0.141 0.070 0.145 0.083

λ=0.6

 Kernel 0.214 0.143 0.225 0.158 0.243 0.161

 Neural 0.122 0.066 0.143 0.081 0.147 0.064

 RESET 0.132 0.068 0.135 0.065 0.148 0.078

λ=0.5

 Kernel 0.200 0.126 0.222 0.147 0.229 0.167

 Neural 0.114 0.052 0.111 0.063 0.140 0.077

 RESET 0.116 0.063 0.108 0.054 0.138 0.073

λ=-0.5

 Kernel 0.189 0.122 0.218 0.153 0.230 0.165

 Neural 0.114 0.051 0.106 0.061 0.101 0.057

 RESET 0.118 0.058 0.111 0.066

*Only 500 iterations are used in this simulation.
-0.15

-0.1

-0.05

0

0.05

0.1

0.15

A Comparison of the Conditional Expectation of the Residual and the Kernel Estimate and 95% Confidenc

Kernel Estimate

Expected Residual

Figure 1
REFERENCES

Bradley, R. and R. McClelland: “An Improved Nonparametric Test for Misspecification of Functional Form,”

manuscript, (1992)

Bierens, H.J.: "Consistent Model Specification Tests," Journal of Econometrics, 20 (1982), 105-134.

—————: "Kernel Estimators of Regression Functions," in Advances in Econometrics, Fifth World Congress, vol. 1,

ed. T. R. Bewley, (1987).

—————: "A Consistent Conditional Moment Test of Functional Form," Econometrica, 58 (1990), 1443-1458.

Lee, T.H., H. White, and C.W.J. Granger: "Testing for Neglected Nonlinearity in Time Series Models", Journal of

Econometrics, 56 (1993), 269-290.

Lewbel, A.: "Consistent Tests with Nonparametric Components", manuscript (1993), University of Brandeis

Manski, C.: "Nonparametric Estimation of Expectations in the Analysis of Discrete Choice Under Uncertainty" in

Nonparametric and Semiparametric Methods in Econometrics and Statistics. Cambridge, Massachusetts: Cambridge

University Press, 1991.

Newey, W.K.: "Maximum Likelihood Specification Testing and Conditional Moment Tests", Econometrica 53 (1985),

1047-1070

Ramsey, J.B.: "Tests for Specification Errors in Classical Least Squares Regression Analysis," Journal of the Royal

Statistical Society Series B, 31 (1969), 350-371.

Robinson, P.: "Root-N Consistent Semiparametric Regressions", Econometrica, 56 (1988), 931-954

White, H.: Asymptotic Theory for Econometricians. Orlando, Florida Academic Press, 1984

Wooldridge, J.M.: "A Test for Functional Form Against Nonparametric Alternatives", Econometric Theory, 8 (1992),

452-475
page 16

APPENDIX A

Assumptions for the random vector (y,x)

(A.1) {yi,xi}, i=1,...,n are a simple random sample from a continuous probability distribution on R×Rk with

E(yi

2) < ∞ .

(A.2) Under Ho, the parameter space Θ is a compact and convex subset of Rm and f(x,θ ) is a Borel measurable real

function on real k and for each k vector x is a twice continuously differentiable real function on θ. E[supθ∈Θ f(xi

,θ )2] <

∞ and for i1,i2 = 1,..,m,

E fx fx i i

sup {( / ) ( , )}{( / ) ( , )} θ

∂ ∂θ θ ∂ ∂θ θ

∈





 



 < ∞

Θ 1 2 1 1

E y fx fx fx i i

sup { ( , )} {( / ) ( , )}{( / ) ( , )} θ

θ ∂ ∂θ θ ∂ ∂θ θ

∈ − 



 



 < ∞

Θ 1 1

2

1 1 1 2

E y fx fx i i

sup 

θ

θ ∂ ∂θ ∂ ∂θ θ

∈ − 



 



 < ∞

Θ { ( , )}( / )( / ) ( , ) 11 1 1 2

(A.3) E(y1 - f(x1,θ)) takes on a unique minimum on Θ at θo. Under Ho, the parameter vector θo is an interior point of 

Θ.

(A.4) The matrix A defined in (13.) is nonsingular.
page 17

APPENDIX B

Proof of Theorem 1:

There are three steps:

i) show that Wn - Ωn = 0

where Ωn is the Wald test for γ =0 in model Y = θX + γh(x) + ε

ii) show that Ω Λ n n p − → 0

where Λn is the Lagrange multiplier test for the above model

iii) show that Λn - nR2 = 0

i) In the linear model we may write Wn as

[ ]

[ ] W n

h u

uhI XXX X hu n = − −

' $

$' (' ) ' ' $

2

1

= [ ] [ ]n

uhh I XXX X h hu

u u

$' ' (' ) ' ' ' $

$' $

− − − 1 1

where h= [h(x1),...,h(xn)]'.

This is just a Wald test for γ=0:

γ γ $ $ Γn

−1 ,

where γˆ is an OLS estimator of γ, using x and h,

Γn R ZZ n R u u

n = − '( ' / ) $' $ 1 ,

R=[01,...,0k, 1]'

and

Z = [X, h].

ii) See White (1984)

iii) We begin with the Lagrange Multiplier statistic for the null γ=0. From the above,

E(Y|X) = θX + γh.

Let B=[θ, γ]'.
page 18

Then the null hypothesis is

RB=0,

where

R = [ 01,...0k 1].

Let

1) && [ ( ' / ) '] $ λ = − − 2 1 1 R Z Z n R RB

2) && $ (' / ) && B B ZZ n R = − / −1 λ 2

 = [$θ 0]'

The Lagrange Multiplier statistic is:

4) L n n n nn = → λλ χ && ' Λ$− &&  ( ) 1 2 1 ,

where

5) $ ( '( ' / ) ) ( ' / ) && Λ ( ' / ) '( '( ' / ) ) n n = R ZZ n R RZZ n V ZZ n R R ZZ n R −− − − −− 4 11 1 1 11 ,

where V is an unbiased estimate of the variance of (Z'u/n && 1/2).

We can simplify Ln to

6) RB = R(Z' Z / n)Z' Y / n $

We know that

7) RB&& = 0

Adding 7) to the righthand-side of 6):

8) RB = R(Z' Z / n)Z' (Y - Z' B) / n $ &&

= − RZZ n Zu n (' / ) ' $ / 1

Substituting the righthand-side of 8) into 1), we get

&&λ = [ ( ' / ) '] ( ' / ) ' $ / −− − 2 11 1 RZZ n R RZZ n Zu n .

Partitioning Z'Z-1, we have.

( ) ( ) ( )

( ) ( ) ( )

(' )

' ' ' ' (' )

' (' ) ' ' '

Z Z

X I hhh h X E Xhhh

D hXXX h I XXX X h

−

− − − −

− − − − = − −

− −





















1

1 1 1 1

1 1 1 1 ,

where
page 19

E X I h hh h hX = − − − '( '( ' ) ) ) 1 1

D h I XXX X Xh = − − − '( ( ' ) ') ) 1 1 .

Then

RZZ R h I XXX X h (' ) ' ' ' ' ( ) ( ) ( ) − − −= − 1 1 1

RZZ D XXX h I XXX X h (' ) (' ) ' ' ' ( ) ( ) ( ) −− − − −=− − 



 



 11 1 1 1

Then

( ) RZZ R RZZ h I XXX X h D XXX h I XXX X h ( ) ' ' (' ) ' ' ' (' ) ' ' ' ( ) ( ) ( ) ( ) ( ) ( ) − − − − − − − −

=− − − 



 



 1 1 1 1 1 1 1 1

= [ ] − − ( ) ( ) ( ) − − − h I X X X X hD h X X X I ' ' ' '' 1 1 1

= −[ ] ( )− hXXX I ' ' 1 .

Further,

( ) RZZ R RZZ Z h I XXX X ( ) ' ' (' ) ' ' ' ' ( ) ( ) − − − − = − 1 1 1 1 .

Then

9) ( ) ( ) &&λ= − ' '' $ / − 2 1 h I XXX X u n

= 2h u n ' $ /

Given homoskedasticity, V in (5) can be simplified as &&

10) V ZZ n && = σ&&(' / )

= (uu n ZZ n $' $ / )( ' / )

Using (10) and substituting allows us to simplify (5) as

11) $Λ ($' $ / )( ' / ) n = uu n ZZ n − 4 1

Substituting (9) and (11) into (4), then

L nu Z Z Z Z u u u nR n = = − $' (' ) ' $ / $' $ 1 2 .