Testing for a Functional Form of Mean Regression in a Fully Parametric Environment

Stanislav Anatolyev

doi:10.1515/jem-2016-0013

Published by De Gruyter February 10, 2018

Testing for a Functional Form of Mean Regression in a Fully Parametric Environment

Stanislav Anatolyev

From the journal Journal of Econometric Methods

https://doi.org/10.1515/jem-2016-0013

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Abstract

We develop a test for a restricted functional form of a mean regression when a complex distributional model for all variables is estimated. The test statistic is an average squared deviation from the estimated hypothesized function of the form implied by the estimated parametric model, and is asymptotically distributed as a mixture of χ² distributions. The test is easy to implement using numerical derivatives, and it performs well in samples of typical size. We illustrate the test using data on labor market characteristics of US young men.

Keywords: mean regression; functional form; specification test; wage regression

JEL Classification: C12; C21

Acknowledgement

I am grateful to the Co-Editor and two anonymous referees for useful suggestions that significantly improved the presentation. I also thank Nikolay Kudrin for excellent research assistance.

A Appendix

Proofs

Proof of Lemma 1

Consistency and asymptotic normality of ϑ^ follow from Newey and McFadden (1994, theorems 2.5, 2.6, 3.3 and 3.4) using Assumption 1 and Assumption 2. □

Proof of Lemma 2

Note that

E [ ‖ ∂ E ( u | v , θ 0 ) ∂ ϑ ‖ 2 ] = E [ ‖ ∫ − ∞ + ∞ u ∂ f ( u | v , θ 0 ) ∂ θ d u ‖ 2 ] = E [ ‖ E [ u ∂ ln ⁡ f ( u | v , θ 0 ) ∂ θ | v ] ‖ 2 ] < ∞ ,

which follows from Assumption 3. Next,

E [ ‖ ∂ E ( u | v , θ 0 ) ∂ ϑ ∂ ψ ( v , β 0 ) ∂ ϑ ′ ‖ ] ≤ E [ ‖ ∂ E ( u | v , θ 0 ) ∂ ϑ ‖ ‖ ∂ ψ ( v , β 0 ) ∂ ϑ ′ ‖ ] ≤ E [ ‖ ∂ E ( u | v , θ 0 ) ∂ ϑ ‖ 2 ] 1 / 2 E [ ‖ ∂ ψ ( v , β 0 ) ∂ ϑ ′ ‖ 2 ] 1 / 2 < ∞ ,

which follows from the previous and Assumption 2(f). Finally, Mψψ is finite by Assumption 2(f). This shows finiteness of Δ.

Now,

E [ sup θ ∈ N θ , β ∈ N β ‖ ( ∂ E ( u | v i , θ ) ∂ ϑ − ∂ ψ ( v i , β ) ∂ ϑ ) ( ∂ E ( u | v i , θ ) ∂ ϑ − ∂ ψ ( v i , β ) ∂ ϑ ) ′ ‖ ] ≤ E [ sup θ ∈ N θ , β ∈ N β ( ‖ ∂ E ( u | v i , θ ) ∂ ϑ ‖ + ‖ ∂ ψ ( v i , β ) ∂ ϑ ‖ ) 2 ] ≤ 2 E [ sup θ ∈ N θ ‖ E [ u ∂ ln ⁡ f ( u | v , θ ) ∂ θ | v ] ‖ 2 ] + 2 E [ sup β ∈ N β ‖ ∂ ψ ( v i , β ) ∂ β ‖ 2 ] < ∞

by Assumption 2(d) and Assumption 3. Then, by Lemma 4.3 of Newey and McFadden (1994), Δ^→pΔ. □

Proof of Theorem 1

Take a second-order stochastic expansion of nD^ around the true parameter value ϑ⁰:

n D ^ = ∑ i = 1 n ( E ( u | v i , θ 0 ) − ψ ( v i , β 0 ) ) 2 + n ∂ D ^ ∂ ϑ ′ | ϑ 0 ζ ^ ϑ + ζ ^ ϑ ′ 1 2 ∂ 2 D ^ ∂ ϑ ∂ ϑ ′ | ϑ 0 ζ ^ ϑ + O P ( 1 n ) ,

where

ζ ^ ϑ = n ( ϑ ^ − ϑ 0 ) → p ζ ϑ = d N ( 0 , V ϑ ) ,

and Vϑ=H−1ΩH−1 is the asymptotic distribution of ϑ^. Under H₀, the leading term is zero. Next, under H₀,

∂ D ^ ∂ ϑ ′ | ϑ 0 = 1 n ∑ i = 1 n ∂ ∂ ϑ ( E ( u | v , θ 0 ) − ψ ( v i , β 0 ) ) 2 = 2 1 n ∑ i = 1 n ( E ( u | v , θ 0 ) − ψ ( v i , β 0 ) ) ∂ ( E ( u | v , θ 0 ) − ψ ( v i , β 0 ) ) ∂ ϑ = 0.

Finally,

1 2 ∂ 2 D ^ ∂ ϑ ∂ ϑ ′ | ϑ 0 = 1 2 1 n ∑ i = 1 n ∂ 2 ∂ ϑ ∂ ϑ ′ ( E ( u | v i , θ 0 ) − ψ ( v i , β 0 ) ) 2 = 1 n ∑ i = 1 n ∂ ∂ ϑ ′ [ ( E ( u | v i , θ 0 ) − ψ ( v i , β 0 ) ) ( ∂ E ( u | v i , θ 0 ) ∂ ϑ − ∂ ψ ( v i , β 0 ) ∂ ϑ ) ] = 1 n ∑ i = 1 n ( ∂ E ( u | v i , θ 0 ) ∂ ϑ − ∂ ψ ( v i , β 0 ) ∂ ϑ ) ( ∂ E ( u | v i , θ 0 ) ∂ ϑ − ∂ ψ ( v i , β 0 ) ∂ ϑ ) ′ + 1 n ∑ i = 1 n ( E ( u | v i , θ 0 ) − ψ ( v i , β 0 ) ) ∂ ∂ ϑ ′ ( ∂ E ( u | v i , θ 0 ) ∂ ϑ − ∂ ψ ( v i , β 0 ) ∂ ϑ ) = H 0 1 n ∑ i = 1 n ( ∂ E ( u | v i , θ 0 ) ∂ ϑ − ∂ ψ ( v i , β 0 ) ∂ ϑ ) ( ∂ E ( u | v i , θ 0 ) ∂ ϑ − ∂ ψ ( v i , β 0 ) ∂ ϑ ) ′ = Δ + o P ( 1 )

by the law of large numbers (see the proof of Lemma 2) and because

E [ ‖ ( ∂ E ( u | v , θ 0 ) ∂ ϑ − ∂ ψ ( v , β 0 ) ∂ ϑ ) ( ∂ E ( u | v , θ 0 ) ∂ ϑ − ∂ ψ ( v , β 0 ) ∂ ϑ ) ′ ‖ ] ≤ E [ ‖ ∂ E ( u | v , θ 0 ) ∂ ϑ − ∂ ψ ( v , β 0 ) ∂ ϑ ‖ 2 ] ≤ 2 E [ ‖ ∂ E ( u | v , θ 0 ) ∂ ϑ ‖ 2 + ‖ ∂ ψ ( v , β 0 ) ∂ ϑ ‖ 2 ] < ∞ .

Summarizing, we have that under H₀,

n D ^ = ζ ϑ ′ Δ ζ ϑ + o P ( 1 ) .

Now, using Lemma 3.2 from Vuong (1989), we get that

n D ^ → p ∑ j = 1 dim ⁡ ( ϑ ) λ j ζ j 2 ,

where {λj}j=1dim⁡(ϑ) are eigenvalues of VϑΔ=Λ, and {ζj2}j=1dim⁡(ϑ)∼IIDχ(1)2. □

Proof of Theorem 2

It follows from the proof of Theorem 1 that

n D ^ = ∑ i = 1 n ( E ( u | v i , θ 0 ) − ψ ( v i , β 0 ) ) 2 + O P ( n ) .

Because E(u|v,θ0)≠ψ(v,β0) almost surely, we have that nD^ tends to +∞ as n → ∞ as it is positive by construction. □

B Appendix

Details on Simulation Experiments

Consider the setup of the first experiment. Because E(u|v)=μu+ρ(v−μv) and ψ(v,β)=a+bv, we compute that

∂ E ( u | v ) ∂ ϑ − ∂ ψ ( v , β ) ∂ ϑ = [ 1 − ρ v − μ v − 1 − v ] .

Note that there are only two non-collinear elements. Hence,

Δ = [ 1 − ρ 0 − 1 − μ v − ρ ρ 2 0 ρ ρ μ v 0 0 1 0 − 1 − 1 ρ 0 1 μ v − μ v ρ μ v − 1 μ v 1 + μ v 2 ] ,

which, expectedly, has a rank of 2.

The logdensity is

ln ⁡ f ( u , v | θ ) = − ln ⁡ 2 π − 1 2 ln ⁡ ( 1 − ρ 2 ) − ( u − μ u ) 2 − 2 ρ ( u − μ u ) ( v − μ v ) + ( v − μ v ) 2 2 ( 1 − ρ 2 ) ,

and its derivatives are

∂ ln ⁡ f ( u , v | θ ) ∂ θ = [ 1 1 − ρ 2 ( ( u − μ u ) − ρ ( v − μ v ) ) 1 1 − ρ 2 ( ( v − μ v ) − ρ ( u − μ u ) ) ρ 1 − ρ 2 − ρ ( 1 − ρ 2 ) 2 ( ( u − μ u ) 2 + ( v − μ v ) 2 ) + 1 + ρ 2 ( 1 − ρ 2 ) 2 ( u − μ u ) ( v − μ v ) ] .

Then

E [ ∂ 2 ln ⁡ f ( u , v | θ ) ∂ θ ∂ θ ′ ] = [ − 1 1 − ρ 2 ρ 1 − ρ 2 0 ρ 1 − ρ 2 − 1 1 − ρ 2 0 0 0 − 1 + ρ 2 ( 1 − ρ 2 ) 2 ] .

The derivatives of the hypothesized regression function are

∂ ψ ( v , β ) ∂ β = ( − 1 − v ) ,

and hence

E [ ∂ ψ ( v , β ) ∂ β ∂ ψ ( v , β ) ∂ β ′ ] = [ 1 μ v μ v 1 + μ v 2 ] .

So, the (minus) inverted Hessian is

− H − 1 = [ 1 ρ 0 0 0 ρ 1 0 0 0 0 0 ( 1 − ρ 2 ) 2 1 + ρ 2 0 0 0 0 0 1 + μ v 2 − μ v 0 0 0 − μ v 1 ] .

Next we compute

E [ ∂ ln ⁡ f ( u , v | θ ) ∂ θ ∂ ln ⁡ f ( u , v | θ ) ∂ θ ′ ] = [ 1 1 − ρ 2 − ρ 1 − ρ 2 0 − ρ 1 − ρ 2 1 1 − ρ 2 0 0 0 1 + ρ 2 ( 1 − ρ 2 ) 2 ] , E [ ( u − ψ ( v , β ) ) 2 ∂ ψ ( v , β ) ∂ β ∂ ψ ( v , β ) ∂ β ′ ] = ( 1 − ρ 2 ) [ 1 μ v μ v 1 + μ v 2 ] , E [ ( u − ψ ( v , β ) ) ∂ ln ⁡ f ( u , v | θ ) ∂ θ ∂ ψ ( v , β ) ∂ β ′ ] = [ 1 μ v − ρ − ρ μ v 0 1 ] .

Hence, the matrix of expected cross-products of the elements of the score vector is

Ω = [ 1 1 − ρ 2 − ρ 1 − ρ 2 0 1 μ v − ρ 1 − ρ 2 1 1 − ρ 2 0 − ρ − ρ μ v 0 0 1 + ρ 2 ( 1 − ρ 2 ) 2 0 1 1 − ρ 0 1 − ρ 2 ( 1 − ρ 2 ) μ v μ v − ρ μ v 1 ( 1 − ρ 2 ) μ v ( 1 − ρ 2 ) ( 1 + μ v 2 ) ] .

Then the asymptotic variance matrix is

V ϑ = [ 1 ρ 0 1 − ρ 2 0 ρ 1 0 0 0 0 0 ( 1 − ρ 2 ) 2 1 + ρ 2 − μ v ( 1 − ρ 2 ) 2 1 + ρ 2 ( 1 − ρ 2 ) 2 1 + ρ 2 1 − ρ 2 0 − μ v ( 1 − ρ 2 ) 2 1 + ρ 2 ( 1 − ρ 2 ) ( 1 + μ v 2 ) − μ v ( 1 − ρ 2 ) 0 0 ( 1 − ρ 2 ) 2 1 + ρ 2 − μ v ( 1 − ρ 2 ) 1 − ρ 2 ] ,

and, consequently,

V ϑ Δ = 2 ρ 2 1 − ρ 2 1 + ρ 2 [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 μ v 0 − μ v 0 0 − 1 0 1 ] .

For the second experiment, we extend the method of Anatolyev and Gospodinov (2010) of constructing a joint distribution of mixed discrete and continuous marginals to the cases of the cardinality of the discrete marginal’s support higher than two. The joint CDF/CMF is

F ( u , v ) = C ( F ( u ) , G ( v ) ) ,

so the PDF/PMF is a derivative with respect to the continuous argument and a difference with respect to the discrete one:

f ( u , v ) = ∂ C ∂ u ( F ( u ) , G ( v ) ) − ∂ C ∂ u ( F ( u ) , G ( v − 1 ) ) = f u ( u ) f ∂ ( u , v ) ,

where the second term is

f ∂ ( u , v ) = [ ∂ C ∂ w ( w , G ( v ) ) − ∂ C ∂ w ( w , G ( v − 1 ) ) ] w = F ( u ) ,

f ∂ ( u , − 1 ) = [ ∂ C ∂ w ( w , q − 1 ) ] w = F u ( u ) , f ∂ ( u , 0 ) = [ ∂ C ∂ w ( w , 1 − q + 1 ) − ∂ C ∂ w ( w , q − 1 ) ] w = F u ( u ) , f ∂ ( u , 1 ) = 1 − [ ∂ C ∂ w ( w , 1 − q + 1 ) ] w = F u ( u ) .

For the FGM copula,

∂ C ∂ w 1 ( w 1 , z ) = z + ρ ( 1 − 2 z ) w 2 ( 1 − w 2 ) ,

implying the distorted success probabilities

q − 1 C ( z ) = q − 1 + ρ ( 1 − 2 z ) q − 1 ( 1 − q − 1 ) , q 0 C ( z ) = 1 − q − 1 − q + 1 + ρ ( 1 − 2 z ) [ q + 1 ( 1 − q + 1 ) − q − 1 ( 1 − q − 1 ) ] , q + 1 C ( z ) = q + 1 − ρ ( 1 − 2 z ) q + 1 ( 1 − q + 1 ) .

The joint density/mass is

f ( u , v ) = f u ( u ) q − 1 C ( F u ( u ) ) 1 { v = − 1 } q 0 C ( F u ( u ) ) 1 { v = 0 } q + 1 C ( F u ( u ) ) 1 { v = + 1 } ,

and the result follows.

C Appendix

Details on Empirical Illustration

We omit the parameters during the derivations. In the case of only one discrete component, the joint PDF/PMF is

f ( u , v ) = ∂ C ∂ u ( F u ( u ) , G v ( v ) ) − ∂ C ∂ u ( F u ( u ) , G v ( v − 1 ) ) = f u ( u ) f C ( u , v ) ,

where the last term is

f C ( u , v ) = [ ∂ C ∂ w ( w , G v ( v ) ) − ∂ C ∂ w ( w , G v ( v − 1 ) ) ] w = F u ( u ) .

The Gaussian copula is C(w,y)=Φ2(Φ−1(w),Φ−1(y)), where Φ₂ is CDF of the standard bivariate normal, and Φ⁻¹ is inverse to the standard normal CDF. Note the important property:

∂ Φ 2 ( x 1 , x 2 ) ∂ x 1 = ∂ ∂ x 1 ∫ − ∞ x 1 ∫ − ∞ x 2 ϕ 2 ( t 1 , t 2 ) d t 1 d t 2 = ∂ ∂ x 1 ∫ − ∞ x 1 ∫ − ∞ x 2 ϕ ( t 2 | t 1 ) ϕ ( t 1 ) d t 1 d t 2 = ∂ ∂ x 1 ∫ − ∞ x 1 ϕ ( t 1 ) ( ∫ − ∞ x 2 ϕ ( t 2 | t 1 ) d t 2 ) d t 1 = ∂ ∂ x 1 ∫ − ∞ x 1 ϕ ( t 1 ) Φ ( x 2 | t 1 ) d t 1 = ϕ ( x 1 ) Φ ( x 2 | x 1 ) .

This leads to

∂ C ( w , y ) ∂ w = ∂ Φ 2 ( Φ − 1 ( w ) , Φ − 1 ( y ) ) ∂ w = ∂ Φ 2 ( x 1 , x 2 ) ∂ x 1 | x 1 = Φ − 1 ( w ) , x 2 = Φ − 1 ( y ) ⋅ ∂ Φ − 1 ( w ) ∂ w = ϕ ( x 1 ) Φ ( x 2 | x 1 ) | x 1 = Φ − 1 ( w ) , x 2 = Φ − 1 ( y ) ⋅ 1 ϕ ( x 1 ) | x 1 = Φ − 1 ( w ) = Φ ( Φ − 1 ( y ) | Φ − 1 ( w ) ) .

Then,

f C ( u , v ) = Φ ( Φ − 1 ( G v ( v ) ) | Φ − 1 ( F u ( u ) ) ) − Φ ( Φ − 1 ( G v ( v − 1 ) ) | Φ − 1 ( F u ( u ) ) ) .

Note that because Φ₂ is bivariate standard normal with correlation coefficient ϱ, we have, by normality of the conditional distributions under joint normality, that

Φ ( Φ − 1 ( y ) | Φ − 1 ( w ) ) = Φ ( Φ − 1 ( y ) − ϱ Φ − 1 ( w ) 1 − ϱ 2 ) ,

and hence

f C ( u , v ) = Φ ( Φ − 1 ( G ( v ) ) − ϱ Φ − 1 ( F ( u ) ) 1 − ϱ 2 ) − Φ ( Φ − 1 ( G ( v − 1 ) ) − ϱ Φ − 1 ( F ( u ) ) 1 − ϱ 2 ) .

In the case of two discrete components, the joint PDF/PMF is

f ( u , v 1 , v 2 ) = ∂ C ∂ u ( F u ( u ) , G 1 ( v 1 ) , G 2 ( v 2 ) ) − ∂ C ∂ u ( F u ( u ) , G 1 ( v 1 − 1 ) , G 2 ( v 2 ) ) − ∂ C ∂ u ( F u ( u ) , G 1 ( v 1 ) , G 2 ( v 2 − 1 ) ) + ∂ C ∂ u ( F u ( u ) , G 1 ( v 1 − 1 ) , G 2 ( v 2 − 1 ) ) = f u ( u ) f C ( u , v 1 , v 2 ) ,

where the last term is

f C ( u , v 1 , v 2 ) = [ ∂ C ∂ w ( w , G 1 ( v 1 ) , G 2 ( v 2 ) ) − ∂ C ∂ w ( w , G 1 ( v 1 − 1 ) , G 2 ( v 2 ) ) − ∂ C ∂ w ( w , G 1 ( v 1 ) , G 2 ( v 2 − 1 ) ) + ∂ C ∂ w ( w , G 1 ( v 1 − 1 ) , G 2 ( v 2 − 1 ) ) ] w = F u ( u ) .

Consider the three-dimensional Gaussian copula

C ( w , y 1 , y 2 ) = Φ 3 ( Φ − 1 ( w ) , Φ − 1 ( y 1 ) , Φ − 1 ( y 2 ) ) .

Note the property

∂ Φ 3 ( x 1 , x 2 , x 3 ) ∂ x 1 = ∂ ∂ x 1 ∫ − ∞ x 1 ∫ − ∞ x 2 ∫ − ∞ x 3 ϕ 3 ( t 1 , t 2 , t 3 ) d t 1 d t 2 d t 3 = ∫ − ∞ x 2 ∫ − ∞ x 3 ( ∂ ∂ x 1 ∫ − ∞ x 1 ϕ 3 ( t 1 , t 2 , t 3 ) d t 1 ) d t 2 d t 3 = ∫ − ∞ x 2 ∫ − ∞ x 3 ϕ 3 ( x 1 , t 2 , t 3 ) d t 2 d t 3 = ∫ − ∞ x 2 ∫ − ∞ x 3 ϕ 2 ( t 2 , t 3 | x 1 ) ϕ ( x 1 ) d t 2 d t 3 = ϕ ( x 1 ) ∫ − ∞ x 2 ∫ − ∞ x 3 ϕ 2 ( t 2 , t 3 | x 1 ) d t 2 d t 3 = ϕ ( x 1 ) Φ 2 ( x 2 , x 3 | x 1 ) ,

which leads to

∂ C ( w , y 1 , y 2 ) ∂ w = ∂ Φ 3 ( Φ − 1 ( w ) , Φ − 1 ( y 1 ) , Φ − 1 ( y 2 ) ) ∂ w = ∂ Φ 3 ( x 1 , x 2 , x 3 ) ∂ x 1 | x 1 = Φ − 1 ( w ) , x 2 = Φ − 1 ( y 1 ) , x 3 = Φ − 1 ( y 2 ) ⋅ ∂ Φ − 1 ( w ) ∂ w = ϕ ( x 1 ) Φ 2 ( x 2 , x 3 | x 1 ) | x 1 = Φ − 1 ( w ) , x 2 = Φ − 1 ( y 1 ) , x 3 = Φ − 1 ( y 2 ) ⋅ 1 ϕ ( x 1 ) | x 1 = Φ − 1 ( w ) = Φ 2 ( Φ − 1 ( y 1 ) , Φ − 1 ( y 2 ) | Φ − 1 ( w ) ) .

Then,

f C ( u 1 , v 1 , v 2 ) = Φ 2 ( Φ − 1 ( G 1 ( v 1 ) ) , Φ − 1 ( G 2 ( v 2 ) ) | Φ − 1 ( F u ( u ) ) ) − Φ 2 ( Φ − 1 ( G 1 ( v 1 − 1 ) ) , Φ − 1 ( G 2 ( v 2 ) ) | Φ − 1 ( F u ( u ) ) ) − Φ 2 ( Φ − 1 ( G 1 ( v 1 ) ) , Φ − 1 ( G 2 ( v 2 − 1 ) ) | Φ − 1 ( F u ( u ) ) ) + Φ 2 ( Φ − 1 ( G 1 ( v 1 − 1 ) ) , Φ − 1 ( G 2 ( v 2 − 1 ) ) | Φ − 1 ( F u ( u ) ) ) .

As a computational matter, we use the fact that

( y 1 y 2 ) | x ∼ N ( μ ϱ x , Ω R ) ,

where

μ R = ( ϱ 1 ϱ 2 ) , Ω R = [ 1 − ϱ 1 2 ϱ 0 − ϱ 1 ϱ 2 ϱ 0 − ϱ 1 ϱ 2 1 − ϱ 2 2 ] ,

and that

Φ 2 ( y 1 , y 2 | x ) = 1 2 π det Ω R ∫ − ∞ y 1 ∫ − ∞ y 2 exp ⁡ ( − 1 2 ( ( z 1 z 2 ) − μ ϱ x ) ′ Ω R − 1 ( ( z 1 z 2 ) − μ ϱ x ) ) d z 1 d z 2 .

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Published Online: 2018-02-10

Testing for a Functional Form of Mean Regression in a Fully Parametric Environment

Abstract

Acknowledgement

A Appendix

Proofs

Proof of Lemma 1

Proof of Lemma 2

Proof of Theorem 1

Proof of Theorem 2

B Appendix

Details on Simulation Experiments

C Appendix

Details on Empirical Illustration

References

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