For convenience we repeat the model (7.1)
y . = x!3 + u. if RHS > 0 l i l
= 0 if RHS < 0 , i = 1,2,...,N .
To derive our test we begin by assuming that the density of u^ , g(u^) , is a member of the Pearson family. We order the N observations so that the first M have y. > 0 , and the second N - M have y. = 0 .
l l
Then we can write the log-likelihood for the N observations as
2 M N
M3,a
,c ,c ,a) = £ ln{g(u )} + £ ln{l -i=l i=M+l - x ^ 3
where g(u_^) = exp[ijj(u^)]/ exp [ij> (u^) ] du_^ and
1'(ui ) [(c^-u^)/ (c -c^u^+c2u ^ ) ]du^ . As in the previous chapter (see
page 147) we introduce heteroscedasticity by replacing by a + z^a , where z
^
is a q x 1 vector of fixed variables satisfying theconditions set out in Amemiya (1977a), and a is a unknown parameter vector.
Our interest is to test H : c_ = c_ = 0, a = 0 , i.e.,
o 1 2
disturbance
NH
. For this we use the LM test statistic as defined in2
equation (4.7) but now we set 0^ = (3',a )' and 0^ = (c^^c^/a')’ . After some computations we obtain (full derivations are in Appendix 7.1)
LM
NHiTOBIT)
21 11 12JI 1I ]
(7.3)where
i l [ui/Q2-u^/(3ö4)] - ^ ^ . / ( l - F i ) ] [Gil/52-Si3/(354 )]
^1 [-3/4+u4/(4ö4 )] - i 2 [Fi/ d - F i )] [-3/4+u±4/(454 )]
^1 [-l/(2G2)+u2/(254 )]zi -
l 2
[F./(1-F.)] [-1/(2^2)+u.2/(2G4 )]z.b n = yx.x![F.{u. /o4 }+G.{u2. /a4 }] 11 L l l l i2 x ll
b = £x.[F.{-u /(2a4 )+u /(2g~6°)}+G.{-u /(2a*)+u u /(2c°)}]~ 4 % ~ ~ , , ~6,
b 13 = Ix-l ^ i{ui2/a4-ui4/(3g6 ) }+Gi(u^1/a4-uilu i3/(3a°) }]~6,
b = £x. [F.{-3u /(4g2 )+u . /(4o6 ) } + G .{- 3 u . / (4aZ )+u u ~2X ~ ~ / (4o°)}],, ~6,
b 15 = Ixiz| [Fi(-uil/ (2g4 )+u_. ^ / (2G6 ) }+G_. {-u. 1 /(2a4 )+ u.^u,. 0/ (2a6 ) }]
i il il i2
b 22 = ItFi{l/(454 )-u.2/(206 )+ui4/(4G8 )}+Gi{l/(454 )-ui2/(256 )+u22/(458 )}]
~4
b 23 = 51 [Fi(-uil/ (25^) + 2 u i3/ (356 ) -u ±5 / (6a8 ) }+gA -u±1/(2a^)+u±3/~4x ~ , ,(6oö ) ~6,
+UiiUi2 /(2a6 )-u i 2U i 3/(6g8 )}]
b „, = 51[F.{3/(832 )-3u /(854 )-u /(8q6 )+u /(858 )}+G.{3/(8g2 )-3u /(8g4 )
-ui4/(8a6 )+ui2u i4/(8a8 )}]
b 25 = Izi[Fi{l/(454 )-ui2/(256 )+ui4/(458 )}+Gi{l/(454 )-ui2/(256 )+u22/(458 )}]
b 33 = I[F^{u^2/g4-2u^4/(3a8 )+u_^8 /(9°8 )^+^i^u ii/a4~ 2uiiu i3/(3g6 )+u23/(9g8 )}] b 34 = I[Fi{-3uil/(452 )+ui3/(4G4 )+ui5/(4G6 )-ui7/(12G8 )}+Gi{-3uil/(4G2 )
+u i3/(4G4 )+uilu i4/(4G6 )-u.3u i4/(12G8 )}]
i3y i3'
b 35 = IzitFi(-uil/(2G4 )+2ui3/(3G6 )-ui5/(6G8 )}+Gi{-u.1/(2G4 )+u.^/(6G6 ) ~ ~ . _ ~8,
+U ilU i2 /(2G )"U i2U i3/(6a )}1
b 44 = ^ [Fi(9/16-3ui4/ (8a4 )+ui8/ (1658 ) } + £ . . { 9 / 1 6 - 3 ^ / (S^4 ) + u ^ / (16a8 ) }]
b 45 = y2 : [Fi(3/(852)-3u. 0/(8g4 )-u ./,/(856 )+u .c./(858 )}+G,{3/(s5 2 )-3u .0/(8g4 )
i2' i4 i2
and
b 55 = ^Ziz ' [Fi{l/(4ö4 )-ui2/(2a6 )+ui4/(458 )}+Gi{l/(4ö4 )-ui2/(2a6 )+u^2/(4S8 )}] .
We have used £ , ^ 2 , and £ to denote, respectively, summation from
to M; i = M + 1 to N; and i = 1 to N . We also have
- x^3 , and use
U ij to denote the estimated j-th moment of a truncated normal variable; more formally we have [see equations
(A7.2)] üi2 = 52 [i-(x:ß)fi/Fi] ü±3 = 52fi [252+(x^ß)2]/F. Si4 = ö2 [352-352 (x^ß)fi/Fi-(x^ß)3fi/Fi) ü = 52fi [8ä4+4ö2 (x’ß)2+(x|ß)4 ]/F U i6 u i7 and u i8 o 2 [15g4-15Q4 (x|3)fi/Fi-5ö2 (xjB)3fi/Fi-(x^ß)5f /F ] o 2f [48a6+24a4 (x^3) 2+6o2 (x |ß) 4+ (xM3)6 ] ö 2 [105a6-105a6 (x !ß)fi/Fi-35ö4 (x M3)3f . j / F ^ a 2 (xM3)5f / F . - (xMS)?f /F ] ~ ~2 2
In the previous expressions, 3 and G denote the MLEs of 3 and o
~2
under H : u. ^ fl/H and G. = F./(l-F.) > where F. is the integral from
O 1 1 1 1 1
~ ~2 . ~2
-oo to x^3 of a N(0,o ) . In addition, f^ is the value of N ( 0 ,g ) at x!3 .
The null hypotheses would be rejected - for large samples1 - if the computed value of LM exceeded the appropriate significance
Nti(1UdII ) 2
point of a Xg+2 • The lengthy expressions that appear in {TOBIT) are not indicative of computational difficulty. The values f and F_^ are easily determined (standard FORTRAN functions exist for these), and, apart from simple products and summations, we only require inverting one (k+l)x(k+l) and a (q+2)x(q+2) matrices. Hence, the computa
tional details for L M ^ (TOBIT) ma^ ^e rea^ ^ incorporated into computer programmes for the Tobit model.
Particular cases of the above procedure are, firstly, a test for N given H . Here the null hypothesis would be H : c. = c = 0 .
o 1 2
Application of the LM procedure would give a test statistic, denoted by LM.7,monTm. and defined as LM.,,. except that the "row" correspond-
N{TOBIT) NH{TOBIT) * *
ing to a in d^ , and the "fifth row and column" in I would be omitted. Given H and under N , LM r would be asymptotically
/V v 1 UO-L i )
2
distributed as x2 • A second case refers to a test for H given N . Here we would remove, in LM , the first two rows in d„ , and
il/n \1Ud11 ) Z
the third and fourth rows and columns in I obtaining, say, LMn .„nDT„. .
n [1 UdI 1 )
This is equal to the test suggested in Jarque (1981) and under H would be asymptotically distributed as x
q
We now briefly consider the truncated model, where y. = x!(3 + u.
i l l
is restricted to, say, non-negative values, but does not have a number of values clustered at a particular point [see Amemiya (1973)]. In this case, u. > - x!ß , and a test for disturbance truncated NH is obtained by
l l
proceeding as above, but assuming
In applications of the Tobit model, particularly in cross-sectional studies, large samples have been typically available, e.g., N = 735, 2798 and 6366 respectively in the studies of Tobin (1958) , Quester and Greene (1978) and Fair (1978).
g(u ) = exp[ip (ui ) ] / -x! 3
l
exp [ij; (u^) ] du_^
that is, g(u^) is a member of the truncated Pearson family. The log-
likelihood for this model is Z (.) = ln{g(u_^)} which under the null hypothesis reduces to 2 (7.4) 2 N f £(3,g ) = £ p i n 2tf - In i=l ^ (Y.-x!3) a --- — - m F. 2a
Application of LM test to the above truncated model gives equation (7.3) but now in d^ all summations are from i = 1 to N and F./(l-F.) has
2 l i
to be replaced by + 1 ; also, in b^_. (but not in u _ ) we would replace
~ ~2
F_^ and respectively by + 1 and - 1 . Here, 3 and a would be the MLEs obtained by maximizing (7.4). The resulting statistic could be denoted by LM,7r7 , from which "one-directional" tests
1
NH(TRUN)
LMN(TRUN)
andLMH(TRUN)
Can be obtained b¥ deleting appropriate rowsand columns. As expected, if we set f = 0 , M = N and allow to
tend to unity, LM/ l W
(TOBIT) '
LM/'/(TOBIT)
and [or' indeed'LHNH(TRUN) ' ™N(TRUN)
and “ W W ”°Uld redU°e ' resPectively- toLM , LM^ and LM^ , where the latter are LM tests for
NH
,N
given
H
, andH
givenN
in the ordinary regression model (see page 149). In the previous chapter we noted that LM = LM . + LM .Din DJ n
However, for the LDV models this "additivity" relation no longer holds good.
Statisticians have recently been interested in testing if a set
of observations y^ come from a particular truncated distribution. Until
now, the only available test for this was the Kolmogorov-Smirnov test with
Koziol and Byar (1975)]. Our statistic LM may be used to test
N \1RUN)
observations for truncated N by setting the number of regressors equal to one and x_^ = 1 (i=l, 2,. . . ,N) , thus providing a solution to this statistical problem.
The exact finite sample properties of our test statistics are
2
analytically intractable. In practice we have to use the asymptotic x critical values. Also we cannot use simulated critical values since our test statistics are not invariant of any scale transformation. But this should not be viewed as a serious drawback of our tests. For non-linear models only asymptotic results are available. Since our models are non linear even under the null hypothesis it is not possible to have "exact" tests. However from the computational viewpoint the LM tests are quite simple.