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Journal of Econometrics 34 (1987) 349-354. North-Holland
A NOTE ON SARGAN DENSITIES
Y.K. TSE*
National University of Singapore, Singapore 051 I
Received August 1985, final version received April 1986
This note reconsiders the general class of Sargan densities studied by Goldfeld and Quandt (1981) and Missiakoulis (1983) in the context of approximating normal densities in regression models. Some errors in these papers are pointed out and the Sargan densities are recomputed. To the contrary of Missiakoulis’ conclusion, the first two order densities are not the best choice. Higher-order densities are better approximations, though the best choice appears to be the second-order density based on the Goldfeld-Quandt criteria.
1. Introduction
In a very interesting article Goldfeld and Quandt (1981) investigated the use of Sargan densities in some econometric models where it is desirable to approximate the normal densities by some densities whose distribution functions are computable in closed form. The problem was pursued by Missiakoulis (1983), who suggested a particular family of the Sargan densities as approximations to the normal. Within the family of densities suggested, Missiakoulis (1983, p. 227) made a somewhat unexpected claim that ‘only the first and second order cases are good approximations to the normal distribu- tion’. In this note we point out that his results are based on a wrong formula for calculating the scaling parameter to standardize the Sargan densities.
The density proposed by Goldfeld and Quandt was the second-order case with unit variance and the same density at the origin as the standard normal. However, the implied parameters given in their paper are based ‘on an incorrect formula for calculating the variance.
In the next section we give correct formulae for the moment generating function and the moments of the Sargan densities. The approximations suggested by Goldfeld-Quandt and Missiakoulis are re-calculated. It turns out that higher-order cases in the Missiakoulis family are better approximations than lower-order cases, though the second-order density based on the Goldfeld-Quandt criteria appears to be the best.
*I am indebted to an associate editor and an anonymous referee for several useful suggestions, which materially improved the paper. Any errors are of course mine only.
0304-4076/87/$3.5001987, Elsevier Science Publishers B.V. (North-Holland)
350 Y. K. Tse, A note on Sargan densities
2. The Sargan densities and their distributional properties
The Sargan densities, with definition and notations given by Missiakoulis (1983) have the form
where
y()=l, a>o, yj20, j=l >...> P,
and
(4
For all values of P greater than or equal to one, the first derivative of f(u) is continuous if and only if y1 = 1, which is henceforth assumed. These densities have an advantage that their distribution functions are obtainable in analytical form, as given by eqs. (3) and (4) of Missiakoulis (1983). However, eqs. (5) and (6) of Missiakoulis (1983) for the moment generating function and the moments of the Sargan densities are incorrect. The correct formulae are
M(B)=: $j! ( &+lyj &+lyj
,_o (a+e)j+l+ (a-qj+l ’ I and
CL,=0 if r isodd,
= z,$oYj(i+r)! if r is even. (4)
Missiakoulis suggested approximating the standard normal density by treating the P th-order Sargan density as the density of the arithmetic mean of P + 1 Laplace variables with mean zero and variance 2( P + l)‘/(u*. The scaling parameter (Y is then chosen so that the Sargan density has unit variance. The correct formula for ar is [cf. eq. (12) of Missiakoulis (1983)]
. (5)
Goldfeld and Quandt (1981) considered a second-order Sargan density with variance 1 and the same value at the origin as the standard normal density.
Y. K. Tse, A note on Sargan densities 351
Assuming y1 = 1, the value of (Y and yz are solved simultaneously. However, the variance formula used in their calculation is incorrect. The appropriate moment generating function and variance are [cf. eq. (2.3) of Goldfeld and Quandt (1981)]
a 2 2
-+- 2Y2cu3 2Y2N3
CY+8
M(8) =
a” 0 + (Jr@2 + (a?j)’ + (a + e)’ + (a - l9)’
20
+Yl+ 2Y2) 9
(6) and
1 02=-
a2 (7)
The implied correct parameter values are (Y = 2.6950 and y2 = 0.6888. These parameter values were also given by Kafaei and Schmidt (1985) in a paper on studying the inconsistency of the maximum likelihood estimates when the Sargan distribution is used as an approximation to the normal. However, examining the first-order derivative of the second-order Sargan density, it is found that the density is unimodal if and only if y2 I 0.5. Indeed, the Goldfeld-Quandt density has a local minimum at u = 0 and two local maxima at u = +0.2034. If we retain the condition of unit variance, the value of the second-order Sargan density at the origin is (1 + 3~,)‘/~/(2(1+ Y~)~/~), which is a decreasing function of y2. This value is 0.4303 when y2 = 0.5. As the standard normal density at the origin is 0.3989, the best choice for a unimodal second-order Sargan density with unit variance is probably given by y2 = 0.5, with the corresponding value of (Y being 2.5820.’ The performance of this approximation will be investigated below. As for the Missiakoulis family, the first derivative of f(u) for u > 0 is
fj( U) = Feea’ y.j+l{ (j + l)~~+~ - y.} d - ap+ly,up .
i J (8)
j=l 1 Due to eq. (10) of Missiakoulis (1983), (j + l)y,+r - y, < 0 for j = 1,. _ . , P - 1, so that f’(u) is always negative. A similar conclusion holds for u < 0. Hence, Missiakoulis’ densities are always unimodal.
Tables 1 and 2 compare the density function and distribution function of the standard normal with its various approximations. GQ is the second-order
‘This density was suggested to me by an anonymous referee, who also pointed out the unimodality of the Missiakoulis densities.
Table 1
Comparisonofstandard
normal,Laplace
and Sargandensities.
l4
Nor
mal
GQ
R
L
apla
ce
SI
s2
s3
s4
S5
0.0
0.39894
0.39894
?!
0.43033
0.70711
0.50000
0.45928
0.44194
0.43234
0.42625
h
0.1
0.39695
0.40206
0.42931
0.61386
0.49124
0.45475
0.43843
0.42927
0.42342
3
0.2
0.39104
0.40472
0.42359
0.53290
0.46922
0.44176
0.42816
0.42023
0.41507
f
0.3
0.38139
0.40147
0.41147
0.46263
0.43905
0.42177
0.41187
0.40574
0.40162
0.4
0.36827
0.39076
0.39314
0.40162
0.40440
0.39650
0.39062
0.38658
0.38371
$
0.5
0.35207
0.37307
0.36973
0.34865
0.36788
0.36771
0.36561
0.36368
0.36214
o
0.6
0.33322
0.34986
0.34272
0.30267
0.33131
0.33693
0.33806
0.33806
0.33778
0.7
0.31225
0.32284
0.31356
0.26276
0.29592
0.30548
0.30911
0.31073
0.31155
k
0.8
0.28969
0.29369
0.28356
0.22811
0.26247
0.27439
0.27976
0.28262
0.28432 2
0.9
0.26609
0.26382
* 0.25378
0.19802
0.23142
0.24441
0.25085
0.25454
0.25687
p
1.0
0.24197
0.23437
0.22505
0.17191
0.20300
0.21609
0.22300
0.22715
0.22988
s
1.5
0.12952
0.11414
0.11073
0.08476
0.09957
0.10689
0.11135
0.11436
0.11652
EJ
2.0
0.05399
0.04805
0.04798
0.04179
0.04579
0.04758
0.04867
0.04942
0.04998
2.5
0.01753
0.01845
0.01914
0.02061
0.02021
0.01974
0.01938
0.01912
0.01891
3.0
0.00443
0.00665
0.00721
0.01016
0.00868
0.00779
0.00721
0.00680
0.00650
3.5
0.00087
0.00229
0.00260
0.00501
0.00365
0.00296
0.00255
0.00227
0.00207
4.0
0.00013
0.00076
0.00091
0.00247
0.00151
0.00109
0.00086
0.00072
0.00062
l4
Tab
le 2
Com
pari
son
of
sta
nda
rd n
orm
al,
Lap
lace
an
d S
arga
n d
istr
ibu
tion
fu
nct
ion
s.
0.0
0.50
000
0.5w
Oo
0.50
000
0.50
000
0.50
000
0.5O
OO
o 0.
5OO
Oo
0.50
000
0.50
000
0.1
0.53
983
0.54
002
0.54
301
0.56
594
0.54
970
0.54
578
0.54
408
0.54
313
0.54
253
0.2
0.57
926
0.58
039
0.58
570
0.62
318
0.59
781
0.59
067
0.58
746
0.58
565
0.58
450
0.3
0.61
791
0.62
076
0.62
751
0.67
287
0.64
327
0.63
389
0.62
951
0.62
700
0.62
537
0.4
0.65
542
0.66
043
0.66
779
0.71
601
0.68
547
0.67
484
0.66
967
0.66
665
0.66
468
0.5
0.69
146
0.69
868
0.70
597
0.75
347
0.72
409
0.71
308
0.70
751
0.70
419
0.70
199
0.6
0.72
575
0.73
486
0.74
161
0.78
598
0.75
904
0.74
832
0.74
271
0.73
929
0.73
701
0.7
0.75
804
0.76
852
0.77
444
0.81
420
0.79
039
0.78
044
0.77
507
0.77
174
0.76
949
0.8
0.78
814
0.79
936
0.80
430
0.83
870
0.81
829
0.80
943
0.80
451
0.80
141
0.79
929
0.9
0.81
594
0.82
724
0.83
116
0.85
998
0.84
297
0.83
536
0.83
104
0.82
827
0.82
635
1.0
0.84
134
0.85
214
0.85
509
0.87
844
0.86
466
0.85
837
0.85
472
0.85
234
0.85
068
1.5
0.93
319
0.93
700
0.93
676
0.94
006
0.93
777
0.93
678
0.93
618
0.93
576
0.93
544
2.0
0.97
725
0.97
555
0.97
459
0.97
045
0.97
253
0.97
368
0.97
439
0.97
487
0.97
522
2.5
0.99
379
0.99
111
0.99
037
0.98
543
0.98
821
0.98
958
0.99
040
0.99
095
0.99
135
3.0
0.99
865
0.99
692
0.99
650
0.99
282
0.99
504
0.99
603
0.99
659
0.99
695
0.99
720
3.5
0.99
977
0.99
897
0.99
877
0.99
646
0.99
795
0.99
853
0.99
884
0.99
902
0.99
915
4.0
0.99
997
0.99
966
0.99
958
0.99
825
0.99
916
0.99
947
0.99
962
0.99
970
0.99
975
GQ
R
L
apla
ce
Sl
s2
s3
s4
X5
Tab
le 3
Mea
n a
bsol
ute
dev
iati
ons
betw
een
sta
nda
rd n
orm
al a
nd
vari
ous
appr
oxim
atio
ns
for
u =
0.0
(0.1
)4.0
Den
sity
fu
nct
ion
D
istr
ibu
tion
fu
nct
ion
GQ
R
L
apla
ce
Sl
s2
s3
s4
S.5
0.00
681
0.01
004
0.03
798
0.02
013
0.01
385
0.01
055
0.00
851
0.00
713
0.00
345
0.00
513
0.01
739
0.00
982
0.00
686
0.00
528
0.00
429
0.00
361
354 Y.K. Tse, A note on Sargan densities
Sargan density based on the Goldfeld-Quandt criteria, R is the second-order Sargan density with (Y = 2.5820 and yz = 0.5, and SZ,. . . , S5 are the Missiakoulis family of Sargan densities. It is obvious that higher-order densi- ties in the Missiakoulis family are better approximations than lower-order cases. Missiakoulis’ finding that higher-order densities do not perform well arises from the fact that the CY values he used produce densities with variance greater than 1. Indeed, since he used standardized arithmetic mean of indepen- dently identically distributed Laplace variables to approximate the normal density, we should expect the approximation to work better as P increases, by virtue of the central limit theorem.
In order to obtain a measure for comparison we compute the mean absolute deviations between the density functions and the distribution functions of various approximations and those of the standard normal, for u from 0.0 through 4.0 in steps of 0.1. The results are tabulated in table 3. The figures suggest that the Goldfeld-Quandt density is the best, while the Missiakoulis approximations are improving as P increases. R is a better approximation than Sl, S2 and S3, though it is not as good as GQ, S4 and SS.
References
Goldfeld, SM. and R.E. Quandt, 1981, Econometric modelling with non-normal disturbances, Journal of Econometrics 17, 141-155.
Kafaei, M. and P. Schmidt, 1985, On the adequacy of the ‘Sargan distribution’ as an approxima- tion to the normal, Communications in Statistics: Theory and Methods 14, 509-526.
Missiakoulis, S., 1983, Sargan densities, which one?, Journal of Econometrics 23, 223-233.
