Tilburg University
An alternative derivation of the k-class estimators
Neeleman, D.
Publication date:
1970
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Link to publication in Tilburg University Research Portal
Citation for published version (APA):
Neeleman, D. (1970). An alternative derivation of the k-class estimators. (EIT Research Memorandum). Stichting
Economisch Instituut Tilburg.
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4
An Alternative Derivation of the k-Class Estimators.
1. Introductíon
In many books on econometrics, for instance, Christ 1966, Dhrymes 1970, Goldberger 1964 and Johnston 1963 the k-Class estimators are introduced with the sid of a system of equations. Only Malinvaud 1966 gives a de-rivation of this equation system.
In our study another derivation of this system is given which differs from the derivation of Malinvaud on two points.
1. Introducing the Kronecker product of matrices it is possible to trans-form the simultaneous equation model into a well-known model viz. the general linear model with restrictiona.
2. Using the theorems on extrema of quadratic forms it is possible to give a solution of the problem without introducing a normalization rule.
Afterwards one can choose an appropriate normalization rule.
2. Some Theorems on the Extrema of Quadrate Forms
These theorems, on which the derivations in ~ 4 are based, are given here without proof. The proofs can be found in Rao 1965.
Theorem
Let x be an n-vector of real elements.
x1 be an m-vector of the first m(~ n) elements of x. x2 be an (n-m)-vector of the remaining elements of x. B be an n x k matrix of real elements
2
-in`' x~ x~ - c' D~ c B'x-c
and the infizum is attained at x~ - B~D~'c, x~ - D2c provided that.the linear restrictions B'x - c are consistent.
Theorem 2
Let x be ar. ~:--~actcr of real elements
A be an -: x:. ~;~mmetric matrix of real eïer..cr.ts
B be ar. r. x:. symmetric positive definite r..s~rix of real elements
a~ ~ a,L ~...~ ', the roots of IA -?~ B~ - 0 Th en x 'r.x i:.f ~'Bx À Theorem 3
Let x be ar. -.-v.-.'.er of real elerer.ts y be s :-: -~ c- -.r cf real eler.:ents
A be ar. ... r. .. ~ ~.r-x of real eïe~-:en..~, ... , ran~ ., -B be an :- x:ti vector ~ of real elenents '
Then
in- ~ x'A'B Y t x~A~A x -- Y~B~A(A'A)-~ A'B Y
x
and this infin~ ï~ attained at xx --(A'A)-~ A'B y
3. The Model
For the sake of completeness the hypotheses by which the model is decribed, are briefly formulated below
~pothesis 1
The model is linear and contains m equations and n variables viz. m endo-genous varíables y~, y2,....,ym and k exoendo-genous variables z~, z2,...,zk
(m t k - n).
3
yt B -Byt t CZt - Et t - t,2...T ~b11 b12 ... blml b21 b22 " ' b2m bm1 bm~ ... b~~ C -Hypothesis 2The model is complete i.e. B 1 exists
rypothesis ~
Et
-c11 c12 ... c1k' c21 c22 " " c2k
~ cm 1 cm2 " '
The et (t - 1,~...T; are independent identical distributed random vari-ables with expectatio:. U and non-singular covariance matrix E.
Hypothesis 4
The equation to be estimated is identified.
Hypothesis 5
The exogenous variables are uniformly bounded i.e. there exists a number K such that for every i(i - 1,2...k) and every t(t - 1,2...,T) Izitl ~ K holds
Hypothesis 6
The vector of exogenous variables
(z11' z12,..., z1T),...~ (zkl' zk2~..., zkT) are linearly independent
b
-rypothesis 7
plim T Z'Z exists and is non-singular.
T-~~
4. The k Class Estimators
The reduced form of the model is
with
yt - iï zt f nt
(t - ~,2...T)
nt - B-~ et (2)
Starting from the hypotheses 1,2 and 3 given in section 3 or. car. derive the following properties for the error terms c: the reduced form n:
The nt (t - 1,2...i) are indeper.dent identical distributed ra::dom variables with expectation 0 and non-singular covar~:.r.~ e~!atrix (B ~ ) E (3 ~ )' - ~c.
Suppose that one wishes to estimate the first equation of the structural form of the model. Without limiting the validity of the conclusions one can state that the number of endogenous variables in this equation equals m.
Because of the order condition this structural equation is identifiable only if thé number of exogenous variables excluded from this equation is at least as great as (m-1) i.e. in the first row of C at least (m-1) zero's appear.
Rearranging the variables such that these zero's occupy the last places in the first row of C, the first equation of the structural form can be written as
o) Zt
Et(t - i,2...T)
(3)
5
-b' is the first row of B (c', 0) is the first row of C
The ordinary least squares estimators of b and c are not unbiased and even not consistent hence one has to search for another estímation me-thod. For that purpose the model is written, using the Kronecker product of matrices, as
vec Y-(Im
Z) vec II' t vec n
(1~)
where
Y - Z
-zT
n
-The restríctior. (3) can be written like
(b'~ Tk) vec 1I' - (-~)
(5)
(6)
For the'properties of the Kronecker product we refer to Neudecker 1968. The problem can be solved as follows
Step 1
Estimate, vith the sid of generalised least squares vec II' starting from known b and c taking into account the restriction (6)
In other words minimize
S-[ vec Y-(Im~ Z)vec II'j' (S2-~ ~ IT) [ vec Y-(Im ~ Z) vecll'J (7)
Step 2
Calculat~ the minimum of S in step 1. This minimum is a function of b and c. iJext calculate those values of b and c that realise the
mi-nimum oi this function
(i) The problem as it is formulated in step 1 can be solved with the aid of theorem 1 in section 2 by formulating it as follows:
Determine ti~e linear function
vec '~ f d
so that
V(q' vec Y t d) - q' ( S2 ~ IT) q
is minimum under the constraints
E(q'.vec Y f d) - p' vec II'
(b' ~ Ik) vec II' - (-~)
where p is an arbitrary vector.
The condition (9) implies that
q' ( Im ~ Z) vec 11' } d- p' vec tI'
(8)
(9)
( 10)
Then from ( 10) and (11) it follows that there exists a vector r such that
q' (Im ~ Z) t r'(b' ~ Ik) - p~
and
(12)
7
-The problem then reduces to minimizing
q' ( S2 ~ IT ) q
subject to the constraint
(Im ~ Z', b ~ Ik)(q) - p
Or taking into account that
(; ~ 1T) ' (WW' ~ I~)
where W is a nonsingular m x m matrix and taking
(~"" ~ ~,) q - s
the problem is trar.s2'or~ed intc minimizing
s's
subject to the constraint
((W 1)' ~ Z', b~ Ik~ (r) - P
Application of theorem 1 in section 2 gives
s~ -(W 1~ Z) D~ p or q~ -(52~1 ~ Z) D~ p
rx - Dr
2 P
where D1 and D2 are submatrices of the generalised inverse
-
~-ï}:e ~inimum variance unbiased estimator of p' vec it' is
p'vec ;Í' - p' D~(a2-~ ~ Z) vec Y t P' D2(-~) (22)
where vec II' is the solution of the equations
(52-~ ~ Z"L) vec :I' - (b ~ Ik) a - (-?-~ ~ Z) vec Y
(23)
(b' ~ Ik) vec II'
- (-~)
Solution of this equations gives
vec iT' -( Im~ (Z'Z)-~Z'J vec Y -[(b' ~~ b)-~ 2 bb' ~(Z'Z)-~Z'J vec Y
-[(b' S c)-~ S2 b~ Ik) (~) (2~)
(ii) Ia step 2 w~ substitute v~c P.' into (7) ~ahic! y:- 13s
S-(vec Y)' (~c-~ ~ Z' ) vec Y -( vec Y)' ( S~-~ ~ Z(Z'Z)-}Z'J vec Y t(vec Y)' [(b' Z b)-~ bb' ~ Z(Z'Z)-lZ'] vec Y t (25) t 2(c',0) [(b' S2 b)-~ b' ~ Z') vec Y t (c',0) [(b' S2 b)-~)~ Z'Z (~)
b and c has to be chosen in such xay that S becomes minimum. This
means that one can only consiàer that part of S in xhich b and c appear.
This part can be xritten as
b'Y'Z(Z'Z)-~ Z'Yb t 2c' Z~ Yb t c' Z~ Z~ c
Spart-b' Si b
(26)
xhere Z~ is the matrix of the first k~ columns of Z(k~ is the number
of exogenous variables belonging to the first equation). Nox it is easy to see that determiaing the minimiun of
Spart depends
9
-inf 2 c' Z~ Y b} c'Z'Z c-- b'Y'Z (Z'Z
c
1 1 1 1 1and this infinum is attained at
c-- (Z~ Z1 )-1 Z~ Y b
So the problem reduces to Determine
,
b' [Y'Z(Z'Z)-1 Z'Y - Y'Z1(Z~Z1)-1 Z~ Y] b inf
b b' S2 b
(27)
(28)
(29)
From theorem 2 in section 2 it follows immediately that the infinum
of (29) equals a~, where a1 is the smallest root of
~Y'Z(z'z)-1 Z 'Y - Y'z1(Z~Z1)-1 Z~ Y- a 52~ - 0 (30)
This means that for every 5 that is a solution of
[Y'z(z'z)-1z'Y - Y'Z1 (z~z1)-1 z~Y - al nl s- o
the infinum is attained.The equations (28) and (31) can be combined to
Y'Z(Z'Z)-1 Z'Y -~1 S21 Y'Z1 b
1 1 1
- 0
(31)
(32)
If we take as normalization rule that the first.element of b equals -1 then the system of equations (32) changes into
10
-where (b2) are the remaining unknown elements of (b) after normalization
Y1 is the first row of Y .
Y2 is the matrix of the remaining rows of Y
S„~ is the first column of S2 minus the first element c.
5222 is the matrix of the last (m-?) rows and the last (m-1) cclsms
If S2 is know~- 4 „ b,, and c can be calculated from ( 30) and ( 33). In ~
general hcw~-r..r !i will be unknown but can be ccx.sistantly estimated
by
~ - ~-~
(3~)
where
V- Y- Z(Z'Z)-1 Z'Y í35)
In this case ( 3~) becomes
r ~ Y'Z(Z'Z)-1Z'Y - Y'Z1(Z~Z1)-1 Z~ Y - a VTV ~- 0 (36) and (33) becomes Y2 Y2 -(1 t T1)V2V2 Y2Z1I Ib2' a Y2Y1 -(1 t T1) V2 V1
- I
I (37)
Z1Y2
Z1z11 I~ I
I
Z, Y,
where V2V2 en Vc'd1 are submatrices of V'V that correspond with 5222 en S2
21 ~1
It can be proved, see for instance Goldberger 1964 that plim (1t T)- 1 T-~~
and that b2 and c are consistent estimators.
a
11
-IY2Y2 - kV2V2 Y2Z1 Ib2l -I Z1Y2 Z~Z1 Ic I Y2Y - kV2V1l
,
Z1Y1I
(38)
which are the equations of the socalled k class estimators. It is easy to se that if plim k- 1 this estimators are consistent.
T -~ m
4. ,eferences
Christ C.F. ( ï966) Econometric Models and Methods, New York Wiley 1966
Dhrymes P.J. 1970 Econometrics, New York, Harper en Row 1970
Goldberger A.S. (1964), Econometric Theory, New York: Wiley 196k
Johnstc:. J. ( '963), Econometric Methods New York: McGraw-Hill 1963
ï~íaiinvauc ~. (1966), Statistica-~ Methods of F conometrics, Amsterdam:
ivcrt: :.-~anà ruGlist:i.-.~ Ccr.:~any 1966
.Tendecker :. (1968), The Ks~onecker Matrix Product and Some of its Applications in Econometrics
