This paper describes Robinson's (1988) double residual semiparametric regression estimator and Hardle and Mammen's (1993) specification test implementation in Stata. Some simple simulations illustrate how this newly coded estimator outperforms the already available semiparametric plreg command.Semipar; Semiparametric estimation

Nicolas Debarsy

Vincenzo Verardi

English

Research Papers in Economics

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ROBINSON’S !n-CONSISTENT SEMIPARAMETRIC 
REGRESSION ESTIMATOR IN STATA 
VINCENZO VERARDI & NICOLAS DEBARSY 
WP 1110 
DEPARTMENT OF ECONOMICS  
WORKING PAPERS SERIES Robinson’s
 
n-consistent semiparametric
regression estimator in Stata
Vincenzo Verardi
University of Namur
(Centre for Research in the Economics of Development)
Namur, Belgium
and Universit´ e Libre de Bruxelles
(European Center for Advanced Research in Economics and Statistics
and Center for Knowledge Economics)
Brussels, Belgium
vverardi@fundp.ac.be
Nicolas Debarsy 
University of Namur
(Center of Research in Regional Economics and Economic Policy)
Namur, Belgium
ndebarsy@fundp.ac.be
March 17, 2011
Abstract
This paper describes Robinson’s (1988) double residual semipara-
metric regression estimator and Hardle and Mammen’s (1993) speci-
ﬁcation test implementation in Stata. Some simple simulations illus-
trate how this newly coded estimator outperforms the already available
semiparametric plreg command.
Keywords: semipar, Semiparametric estimation
JEL Classiﬁcation: C14, C21
1I n t r o d u c t i o n
The objective of this paper is to present the implementation in Stata of
Robinson’s (1988) double residual semiparametric regression estimator. Also,
to check if the nonparametric part of the relation may be approximated by
 Corresponding author
1ap o l y n o m i a lf u n c t i o n a lf o r m ,w ei n t r o d u c eH a r d l ea n dM a m m e n’s (1993)
speciﬁcation test as an option in the programmed estimator. We also brieﬂy
describe this test.
The structure of the paper is the following: in Section 2, Robinson’s
(1988) semiparametric regression estimator and Hardle and Mammen’s (1993)
speciﬁcation test are described. In Section 3, the implemented Stata com-
mand (semipar)i sp r e s e n t e d . S o m es i m p l es i m u l a t i o n sa s s e s s i n gt h ep e r -
formance of the estimator and of the test are performed in Section 4. In
Section 5, we illustrate the use of the semipar command with an empirical
application. Section 6 concludes.
2E s t i m a t i o n m e t h o d
2.1 Robinson’s (1988) semiparametric regression estimator
Consider a general model of the type
yi =  0 + xi  + f(zi)+ i,i =1 ,...,N (1)
where yi is the value taken by the dependent variable for individual i, xi
is the row vector of characteristics for individual i,  0 is a constant term and
 i is the disturbance assumed to have zero mean and constant variance  2
 .
Variable z is an explanatory variable that enters the equation nonlinearly
according to a non-binding function f.T h i sm o d e lc a nb ee s t i m a t e du s i n g
Robinson’s (1988) double residual methodology that starts by applying a
conditional expectation to both sides of (1). This leads to
E (yi|zi)= 0 + E (xi|zi)  + f(zi) i =1 ,...,N (2)
By subtracting (2) from (1), we have
yi   E (yi|zi)=( xi   E (xi|zi))  +  i i =1 ,...,N (3)
If the conditional expectations are known, parameter vector   can easily
be estimated by ﬁtting (3) by ordinary least squares. If they are unknown,
they have to be estimated by calling on some consistent estimators yi =
my(zi)+ 1i and xki = mxk(zi)+ 2ki,w h e r ek =1 ,...,K is the index of the
explanatory variables entering the model parametrically. Robinson’s (1988)
double residual estimator is hence the OLS estimation of model
yi   ˆ my(zi)=( xi   ˆ mx(zi))  +  i i =1 ,...,N (4)
where xi   ˆ mx(zi)i st h er o w - v e c t o ro ft h ed i  erences between each ex-
planatory variable xki and the ﬁtted conditional expectation of xki given
zi.
2The estimated coe cients vector is therefore
ˆ   =
 
 
i
(xi   ˆ mx(zi))
  (xi   ˆ mx(zi))
  1  
i
(xi   ˆ mx(zi))
  (yi   ˆ my(zi))
(5)
with variance (if errors are i.i.d)
V ar
 
ˆ  
 
=  2
 
 
 
i
(xi   ˆ mx(zi))
  (xi   ˆ mx(zi))
  1
(6)
where  2
  is the variance of the error term. If errors are non i.i.d., stan-
dard sandwich and cluster variance formulas can be used.
Having estimated parameter vector  ,i ti sn o wp o s s i b l et oﬁ tt h en o n l i n -
ear relation between zi and yi by simply estimating equation (7) presented
below nonparametrically.
yi   xiˆ   =  0 + f(zi)+ i,i =1 ,...,N (7)
2.2 Hardle and Mammen’s (1993) test
It is sometimes suggested that nonparametric functions may be approxi-
mated by some parametric polynomial alternative. To test fort h ea p p r o -
priateness of such an approximation, Hardle and Mammen (1993) propose
as t a t i s t i cw h i c hc o m p a r e st h en o n p a r a m e t r i ca n dp a r a m e t r i cr e g r e s s i o nﬁ t s
using squared deviations between them. The test-statistic is:
Tn = N
 
h
N  
i=1
 
ˆ f(zi)   ˆ f(zi, )
 2
 (·)( 8 )
where ˆ f(zi)i st h en o n p a r a m e t r i cf u n c t i o ne s t i m a t e di n( 7 ) , ˆ f(zi, )i sa n
estimated parametric function and h is the bandwidth used.  (·)i saw e i g h t
function. To obtain critical values for the test, Hardle and Mammen (1993)
suggest to call on simulated values obtained by wild bootstrap. Obviously,
an absence of rejection of the null (i.e. “accepting” the parametric model)
means that the polynomial adjustment is at least of the degreet h a th a sb e e n
tested.
We implemented this estimator in Stata under the command semipar
which is described below.
3T h e s e m i p a r c o m m a n d
The semipar command ﬁts Robinson’s double residual estimator in the case
of a unique variable entering the model nonparametrically. The default ker-
nel regression used for all stages is a gaussian kernel weighted local polyno-
mial ﬁt. This kernel is of order 2. The optimal bandwidth used minimizes
the conditional weighted mean integrated squared error.
3The general syntax for the command is:
semipar varlist
 
if
  
in
 
,n o n p a r ( string)
 
generate(string)
partial(string)n o g raph ci level(#)t i t l e ( string)y t i t l e ( string)
xtitle(string)r o b u s tc l u s t e r ( varname)t e s t ( #)n s i m ( #)
weight(varname)d e g r e e ( #)
 
The ﬁrst option, nonpar,i sc o m p u l s o r ya n dn e c e s s a r yt od e c l a r ew h i c h
variable enters the model nonparametrically. All other choices are optional.
The ﬁrst one, generate,r e p r o d u c e st h e“ n o n p a r a m e t r i c a l l y ”ﬁ t t e dd e p e n -
dent variable. The user chooses the name of this new variable by deﬁning it
in parentheses. Similarly, the partial option is needed to generate a new
variable that contains the parametric residuals (i.e. the left-hand side of
equation (7)). The option nograph will be used if the user does not want
the graph of the nonparametric ﬁt of the variable set in nonpar to appear.
The ci option allows to visualize the conﬁdence interval around then o n p a r a -
metric ﬁt, while the level option sets the level of conﬁdence for inference
(by default set to 95%). The options title, ytitle and xtitle are used to
indicate respectively the title and the labels of the axes of the graph illus-
trating the nonparametric relation between the dependent variable and the
variable deﬁned in the nonpar option. The robust and cluster options call
for standard errors of the estimated parameters that are respectively resis-
tant to heteroskedasticity and clustered errors. The test option implements
Hardle and Mammen’s (1993) statistic to test whether the nonparametric
ﬁt could be approximated by a polynomial ﬁt, the order of whichm u s tb e
set by the user. For the sake of clarity, we rescaled the statistic in such a
way that it can be compared with the quantile of a Normal distribution.
Note however that the test is not normally distributed. The nsim option
deﬁnes the number of bootstrap replicates used to get inference. Its default
value is set to 20. The weight option allows to introduce a weighting vector
for the test (i.e.  (·)i ne q u a t i o n( 8 ) ) . B yd e f a u l t ,t h i sw e i g h t i n gv e c t o r
is set to  N/N with  N au n i tv e c t o ro fd i m e n s i o nN.F i n a l l y ,t h e degree
option allows the user to specify the degree of the local polynomial ﬁt used
to nonparametrically estimate the regressions. By default,i ti ss e tt o1 .
To assess the performance of the programmed estimator, in then e x t
section we present some simple simulations in which we compare this esti-
mator with the already available plreg command. The latter implements
Yatchew’s (1998) di erence estimator where the nonparametric part in (1)
is partialled out by di erencing rather than by removing the conditional
expectations. Since the highest e ciency of Yatchew’s (1998) estimator is
attained by a di erencing of order 10, we will use this di erencing order as
ab e n c h m a r k .
4Table 1: Comparison between semipar and plreg
Bias x2 Bias x3 MSE x2 MSE x3
plreg -0.4695 -0.1039 0.2208 0.0111
semipar -0.0435 -0.0183 0.0022 0.0007
4S i m u l a t i o n s
The simulation setup is the following. To begin, we generate (for a sample of
300 observations) two explanatory variables x2 and x3 from two independent
N(0,1). An additional random variable x1 is generated from a discrete
Uniform distribution on [ 10,10]. This sample design remains unchanged
for all simulations. Then, for each replication, we generatea ne r r o rt e r m
e from a standard normal and create variable y according to DGP y =
x1 + x2
1 + x2 + x3 + e.W er u nt h esemipar and plreg estimators for each
replicate and Table 1 reports both the bias and MSE of the coe cients
associated with x2 and x3. We carry out 1000 simulations. The variable that
enters the equation nonparametrically is generated from a discrete Uniform
distribution on purpose to illustrate the fragility of plreg with respect to
this kind of data. Robinson’s (1988) estimator, that is basedo np a r t i a l l i n g
out the nonparametric part removing conditional expectations rather than
by di erencing, behaves much better.
In this setup, Robinson’s (1988) estimator leads to smaller biases than
Yatchew’s (1998) di erencing estimator. From equation (7), this also implies
that the nonparametric ﬁt is better estimated by semipar than by plreg.
To illustrate the ﬁtting performance of the proposed estimation proce-
dure, we generate four samples according to the following DGPs:
a) y = x2 + x3 + e
b) y = x1 + x2 + x3 + e
c) y = x1 + x2
1 + x2 + x3 + e
d) y = x1   x2
1   x3
1 + x2 + x3 + e.
In Figure (1), we present the scatter plots, the non-parametric ﬁt (thick
plain line) and the true DGP (red dashed line) related to the four DGPs
described above. As expected, the results are unambiguous.
In the absence of any relation between x1 and y (panel a), no clear pat-
tern emerges and the non-parametric curve lies close to the horizontal line
(the true DGP). In the three other cases (panels b, c and d), then o n p a r a -
metric estimation of the relation matches the true functional form quite
well.
As mentioned in the previous section, the Tn statistic assesses the ade-
quacy of a polynomial adjustment compared to a nonparametricﬁ t .T a b l e
5−
4
−
2
0
2
4
−4 −2 0 2
x1
a
−
5
0
5
−4 −2 0 2 4
x1
b
−
5
0
5
1
0
−2 −1 0 1 2 3
x1
c
−
2
0
−
1
0
0
1
0
−4 −2 0 2
x1
d
Figure 1: Non-parametric ﬁt of the four DGPs
6Table 2: Performance of the comparison Test Tn
Order tested
0123
0 0.053 0.06 0.055 0.039
True 1 10 . 0 6 4 0 . 0 5 5 0 . 0 2 1
Order 2 11 0 . 0 6 0 . 0 6 2
3 111 0 . 0 6 6
Figures correspond to rejection rates of the
test.
2p r e s e n t st h ep e r f o r m a n c eo ft h et e s tf o rt h eD G P sd e s c r i b e dabove. The
rows indicate the order of the generated polynomial while thec o l u m n ss p e c -
ify the order of the polynomial that has been tested. Thus, thed i a g o n a l
(and the upper triangle) elements are the simulated sizes of the test while
elements below the main diagonal are some measure of power. Toc o n s t r u c t
this table we replicated the DGPs 1000 times. Each time a new error term
is randomly drawn and a new dependent variable is generated (the design
space remains unchanged). Inference for the test is based on 100 bootstrap
replications. We observe that the test has good rejection rates when the
order of the polynomial adjustment tested is lower than the generated one.
Besides, the size of the test (whose theoretical value is set at 5%), is very
close to its nominal value.
5E x a m p l e
To illustrate the usefulness of this semiparametric model ine m p i r i c a la p -
plications, we call on a dataset used by Wooldridge (2002) that studies the
e ects of an incinerator location on housing prices. The data are for houses
that were sold during the year 1981 in North Andover, MA; 1981 was the
year construction began on a local garbage incinerator. The dependent vari-
able is the log of price of houses (lprice)a n dt h ev a r i a b l eo fi n t e r e s ti st h e
distance from the house to the incinerator measured in feet and expressed
in logs (ldist). To control for confounding e ects, the author suggest to
include the log of interstate distance (linst), the log of the square footage of
the house (larea), the log of the lot size in square feet (lland), the number
of rooms (rooms), the number of bathrooms (baths), and the age of the
house (age)a sa d d i t i o n a lc o v a r i a t e .H o w e v e r ,h ea l s oa s s e r t st h a tt h ee ect
of the log of the interstate distance is not linear and proposes to consider
it squared. In this application we carry out this exercise again but do not
impose any functional form to the log of interstate distance and estimate
the model semiparametrically. We then check if the square approximation
is appropriate. More precisely, we run the following commandl i n e s :
7.u s eh t t p : / / f m w w w . b c . e d u / e c - p / d a t a / w o o l d r i d g e / H P R I C E 3
.s e m i p a rl p r i c el d i s tl a r e al l a n dr o o m sb a t h sa g e ,n o n p a r ( l inst) xtitle(linst) ci
Number of obs = 321
R-squared = 0.4437
Adj R-squared = 0.4331
Root MSE = 0.2646
lprice Coef. Std. Err. t P>|t| [95% Conf. Interval]
ldist .1083941 .0640184 1.69 0.091 -.0175636 .2343519
larea .4887243 .0668208 7.31 0.000 .3572527 .6201959
lland .0866459 .036037 2.40 0.017 .0157423 .1575495
rooms .0436451 .0221781 1.97 0.050 9.12e-06 .087281
baths .0806555 .0335251 2.41 0.017 .014694 .146617
age -.003481 .0005436 -6.40 0.000 -.0045506 -.0024114
The results of the parametric part (see Stata output above) show that the
distance from the incinerator does not seem to be signiﬁcant (the t-stat
associated with the coe cient is smaller than the critical value of 1.96).
As far as the e ect of the log of the interstate distance is concerned,
Figure (2) shows that it is clearly nonlinear.
4
4
.
5
5
5
.
5
6
7 8 9 10 11
linst
Figure 2: Nonlinear link between the price and interstate distance (in logs)
8Indeed, when the interstate distance increases, the e ect of house prices
ﬁrst increases and then decreases. When we check if the quadratic approxi-
mation proposed by Wooldridge (2002) is appropriate, it turns out that this
assumption is clearly rejected by Hardle and Mammen’s (1993)t e s t( s e eb e -
low). However, when we compare it with a polynomial adjustment of degree
3, the null is no longer rejected which means that instead of a semipara-
metric model, a pure parametric model with a polynomial ﬁt of degree 3 of
linst could be used.
The two Stata outputs below summarize results of the Hardle and Mam-
men (1993) test when the polynomial adjustment tested is of order 2 or 3
respectively. These outputs do not present the results concerning the para-
metric part since they are the same as in the output presented above.
.u s eh t t p : / / f m w w w . b c . e d u / e c - p / d a t a / w o o l d r i d g e / H P R I C E 3
.s e m i p a rl p r i c el d i s tl a r e al l a n dr o o m sb a t h sa g e ,n o n p a r ( l inst) nograph test(2)
>n s i m ( 1 0 0 )
Simulation the distribution of the test statistic
bootstrap replicates (100)
1 2 3 4 5
.................................................. 50
.................................................. 100
H0: Parametric and non-parametric fits are not different
-------------------------------------------------------
Test statistic T: 2.762337
Critical value (95%): 1.959964
P-value: 0
.u s eh t t p : / / f m w w w . b c . e d u / e c - p / d a t a / w o o l d r i d g e / H P R I C E 3
.s e m i p a rl p r i c el d i s tl a r e al l a n dr o o m sb a t h sa g e ,n o n p a r ( l inst) nograph test(3)
>n s i m ( 1 0 0 )
Simulation the distribution of the test statistic
bootstrap replicates (100)
1 2 3 4 5
.................................................. 50
.................................................. 100
H0: Parametric and non-parametric fits are not different
-------------------------------------------------------
Test statistic T: .9705025
Critical value (95%): 1.959964
P-value: .3
6C o n c l u s i o n
In econometrics, semiparametric regression estimators have become stan-
dard tools for applied researchers. In this paper, we presentR o b i n s o n ’ s
(1988) double residual semiparametric regression estimator and Hardle and
Mammen’s (1993) speciﬁcation test. We then present the Statac o d e sw e
created to implement them in practice. Some simple simulations and an
empirical application to illustrate the usefulness of the procedure are also
9shown.
7A c k n o w l e d g m e n t s
We would like to thank all our colleagues at CRED, ECARES, and CERPE
as well as Emmanuel Flachaire, Marjorie Gassner, Whitney Newey, Christo-
pher Parmeter and S´ ebastien Van Bellegem. Vincenzo Verardii sa na s s o c i -
ated researcher of the National Science Foundation of Belgium and gratefully
acknowledges their ﬁnancial support. Nicolas Debarsy is a research fellow
of the National Science Foundation of Belgium and gratefullya c k n o w l e d g e s
their ﬁnancial support.
References
Hardle W., E. Mammen. 1993. Comparing nonparametric versus parametric
regression ﬁts, Annals of Statistics 21: 1926-1947.
Robinson P.M. 1988. Root-N-consistent semiparametric regression. Econo-
metrica 56: 931-954.
Wooldridge J.M. 2002. Econometric analysis of cross section and panel data,
MIT Press, London.
Yatchew A. 1998. Nonparametric regression techniques in economics, Jour-
nal of Economic Literature 57: 135-143.
10

Comparing nonparametric versus parametric regression ﬁts,

Econometric analysis of cross section and panel data,

Nonparametric regression techniques in economics,

Root-N-consistent semiparametric regression.

Robinson's Squareroot-of-n-consistent Semiparametric Regression Estimator in Stata

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Robinson's Squareroot-of-n-consistent Semiparametric Regression Estimator in Stata

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