Testing additivity

By assuming an additive structure of ϕ one might reduce the effect of dimensionality of the regressors on the convergence rate of an estimator (cf. Chen and Pouzo [2012] in case of instrumental quantile regression). Applying this structure leads, however, to inconsistent estimators in general if the function ϕ does not obey an additive form. Our aim in the following is to test whether

H₀add:there exist functions ϕ1, ϕ2∈ B such that P(Y 6 ϕ1(Z0) + ϕ2(Z00)|W ) = q.

3.4 Monte Carlo simulation 121

Similarly as above we obtain the test statistic

S_nadd:= mn X j=1  n−1 n X i=1 1 Yi 6ϕb1n(Z 0 i) +ϕb2n(Z 0 i) − q
f_jτ(W_i)  2

where the estimator (ϕb1n,ϕb2n) of ϕ = (ϕ1, ϕ2) is given by 3.13. The next asymptotic

normality result is a direct consequence of Theorem 3.2.1 and hence its proof is omitted.

Corollary 3.3.3. Given the conditions of Theorem 3.2.1 we have under Hadd

0 √ 2ς_mn
−1 n q(1 − q)S add n − µmn  _d → N (0, 1).

3.4 Monte Carlo simulation

In this section, we study the finite-sample performance of our test by presenting the results of a Monte Carlo simulation. The sample size is 1000 and there are 1000 Monte Carlo replications in each experiment. Results are presented for the nominal levels 0.05. Let Φ denote the cumulative standard normal. Throughout this simulation study, re- alizations (Z, W ) were generated by Z = Φ ζω + (1 − ζ2_)ε

and W = Φ(ω) where

ω, ε ∼ N (0, 1), Here, the constant ζ > 0 measures the correlation of Z to W and is varied in the experiments. Realizations of Y were generated from

Y = ϕ(Z) + cUU

where U = ϑ ε +√1 − ϑ2_{ε with ε ∼ N (0, 1) and where the constants c}

U > 0, ϑ > 0

are varied in the experiments. As basis {fj}j>1we choose cosine basis functions given by

fj(t) =

√

2 cos(πjt) for j = 1, 2, . . .

Testing Exogeneity The realizations (Y, Z, W ) are generated as described above with

cU = 0.5 and structural effect ϕ1(z) = P∞j=1(−1)j+1j−2sin(jπz). For computational

reasons we truncate the infinite sum at K = 100. The resulting function is displayed in Figure 1. We estimate the structural relationship using Lagrange polynomials. Note that

ϑmeasures the degree of endogeneity of Z and is varied among the experiments. The null hypothesis H0holds true if ϑ = 0 and is false otherwise.

In Table 2 we depict the empirical rejection probabilities when using either no smooth- ing or additional smoothing with τj = j−.25 or τj = j−.5, j > 1, which we denote

ζ ϑ Empirical Rejection probability using S_n0e S_n0.25e S_n0.5e 0.4 0.0 0.048 0.045 0.037 0.2 0.137 0.175 0.208 0.25 0.254 0.287 0.346 0.3 0.387 0.446 0.508 0.35 0.565 0.627 0.690 0.7 0.0 0.034 0.031 0.032 0.2 0.248 0.298 0.376 0.25 0.492 0.548 0.652 0.3 0.764 0.818 0.876 0.35 0.941 0.956 0.984

Table 3.1: Empirical Rejection probabilities for testing exogeneity

Pmn

j=1τj ≈ n1/3. Hence, without additional smoothing the number of basis functions

used is mn = 10 (= n1/3). On the other hand, with additional smoothing in case, of

τj = j−.25we have mn= 20whereas if τj = j−.5we let mn= 100. We like to emphasize

that, especially in the case of additional smoothing, the results of our test statistic are not sensitive to the choice of the number of basis functions. As we see from Table 2, our test becomes slightly with additional smoothing.

Testing a Nonparametric Specication In case of nonparametric specification, we consider the structural function ϕ2(z) = P∞j=1 j−4cos(jπz). Again, for computational

reasons we truncate the infinite sum at K = 100. The resulting functions are displayed in Figure 1. To estimate the structural function we apply the procedure of Chen and Pouzo [2012] given in (3.12) with b-splines as approximation basis functions. That is, for the sieve space Bkn we use b-splines of order 2 with 5 knots (hence kn= 5) and for

b

T we use b-splines of order 6 with 11 knots (hence ln= 15).

If H0 is false, then P(Y 6 ϕ(Z)|W ) = q + ξ(W ) for some function ξ. In our exper-

iments, we consider ξ(W ) = −P(ϕ(Z) < Y 6 ϕ(Z) + ρ(Z)|W ) for some function ρ which we specify below. The definition of ξ implies P(Y 6 ϕ(Z) + ρ(Z)|W ) = q + ξ(W ). Consequently, when H0 is false we generate realizations of Y from

Y = ϕ(Z) + cjρj(Z) + U

3.4 Monte Carlo simulation 123 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z ϕ1 ( z ) 0.0 0.2 0.4 0.6 0.8 1.0 −1.0 −0.5 0.0 0.5 1.0 1.5 z ϕ2 ( z )

Figure 3.1: Graph of ϕ1and ϕ2

ρ1(z) = 1 − (2z − 1)2,

ρ2(z) = z1{z 6 1/2} + (1 − z) 1{z > 1/2},

ρ3(z) = exp(2z)1{z 6 1/2} + exp(2(1 − z)) 1{z > 1/2} − 1, ρ4(z) = exp(4z)1{z 6 1/2} + exp(4(1 − z)) 1{z > 1/2} − 1,

and cj > 0is a normalizing constant such that

R1

0 ρj(z)dz = 0.5for j = 1, 2, 3, 4.

In Table 2, we depict the empirical rejection probabilities when using Snnp with either

no smoothing or additional smoothing τj = j−0.25, j > 1, or τj = j−5, j > 1, which we

denote by Sn0np, Sn0.25np, or Sn0.5np, respectively. The number of cosine basis functions fj

to construct our test statistic is exactly the same as in the setting of the test of exogeneity as described above.

The results of the experiments are shown in Table 2. We see that the empirical rejec- tion probability increases as the function ρ becomes more and more irregular. Interest- ingly, although ρ1 is a smooth function we reject in this case the null hypothesis more

often than in every second Monte Carlo iteration. The reason is that adding ρ1 on the

structural function ϕ2 increases the nonlinearity and hence the number of spline basis

Model Empirical Rejection probability S_n0 S_n0.25 S_n0.5 H0true 0.041 0.052 0.056 ρ1 0.664 0.657 0.699 ρ2 0.789 0.805 0.847 ρ3 0.829 0.839 0.879 ρ4 0.871 0.885 0.909

Table 3.2: Empirical Rejection probabilities