STUDIES ON THE LINEAR REGRESSION ANALYSIS WITH ERROR IN VARIABLES

STUDIES ON THE LINEAR REGRESSION ANALYSIS WITH ERROR IN VARIABLES

by

Koukyu KAwABATA*

(Received October 1, 1974)

SYNOPSIS

This paper treats various type of error in variable models. First, we shall con- sider the bivariate linear regression model having three cases, that is, (i) the case having non-stochastic explanatory variables and deterministic relation between real variables, (ii) the case having non-stochastic explanatory variables and sto- chastic relation between real variables, and (iii) the case having stochastic expl- anatory variables. Next we shall consider two convenient method based on com- putational technique and leading efficiency. Finally we shall examine some com- plicated regression models with lagged exogenous and endogenous variables.

1. INTRODUCTORY REMARKS

Econometricians are usually troubled by having to considering linear regression models where variables are measured with error, It is known that the OLS estimates of the coefficients of the variable will be biased. To insure this fact, we shall consider following bivariate simple regression model.

Suppose that

(1) X--X*+u (2) Y-= Y*+v (3) Y* - ct+BX*

where X, Y indicate observed values, X*, Y* the true values, and u, v the measure- ment errors. In this case it is assumed that the measurement errors are mutua- 11y and serially independent with constant variances, and also to be independent of the true values in the model. Substituting eqs. (1) and (2) into (3) yields

(4) Y=-a+BX+w

where w== v-Bu. Then,

E{w[X-E(X)]} -- E[(v-Bu)u] -- -Ba.2.

Therefore, there should be a significant covariance between the error term and

* The Department of Computer Science.

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the explanatory variable. Then, by applying the least squares method to (4), we

obtain :

(s) piim A=i+.e2/o.,2 ,

that is, plim b Års?B. As we see in the above equation, b yields an underestimate of B. To consider the above results, Johnston [1] andlakeuchi [2] suggestsome solutions which will be discussed in a later section. However, there does not seem to be any complete treatment in the econometric literature of the problem of errors in variables.

Throughout this paper, we consider various types of error in variable models and some relations among the estimation methods. In Section 2, we examine three theoretical approaches without lagged variables. Sections 3 and 4 treat the In- strumentalVariable Method and the Grouping of Observation Method in a simple computational approach. Section 4 treats some special cases of error in variable models with finite or infinite lagged explanatory variables. The conclusion of the paper are given in Section 6.

2. THREE THEORETICAL APPROACHES

In this section we shall examine three theoretical cases:

( i) the case having non-stochastic explanatory variables and deterministic rela- tions between real variables

(ii) the case having non-stochastic explanatory variables and stochastic rela- tions between real variables

(iii) the case having stochastic explanatory variables.

Initially, we examine case (i). In addition to the previous bivariate model and assumptions, suppose that u and v are normally and independently distributed va- riables with zero means and the variances o.2, a,2. Then we can obtain the follow- mg equatlon

(6) Y-a+BX*+v

and next the likelihood function for sample observations

(7) Loc ol.e--"XLtX2;.-2X;)2vEFI:,Tele(Vi2i,cr,'BY')2.

By taking logarithms to the base e, we obtain the following logarithmic likelihood function :

(8) L* = log L === const -glog a.2 -g log a.2

b 2al.2 ]2i] (Xt -Xt *)2- 2i2 \ ( Yi -a-BXt *)2

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To obtain estimates of a and B, we must form maximum-likelihood estimates of X*, thus:

(g) g.Li = - 2 ".., + 2i.., :l] (x, -x, *+ )2 -o ,

(10) aa i -=-2i, +2i,\(Y,-ct-BX,*)2-:o,

(n) aa ^xL .- .i ., (x,-x,*)+ .E, (y,-ct-Bx,*)==o i-i, 2, ..., n,

aL* 1 (12) aa == a,2\(Yt'Ct-BXt*)==O

and

aL* 1 (13) aB = .,2 ]Il]Xt"(Yi-a-BX,") -O.

From eqs. (9) and (10) we obtain maximum-likelihood estimates of the error va- riances, which are written as follows:

(14) '.".2=-ili-IEi.](X,-.Sl",*)2

and

(is) 3.2 :;i-\, (y,-ZII-R .it,*)2

where tv designates the maximun-likelihood estimate of a parameter. In a similar manner for eq. (11):

(i6) y, - ,El -' BV x"" ,* =- - Na.v.2 (x, -r,*)

B o.2

Then, we adopt the following assumption between the error variances:

(17) o.2/o.2 = k

where k is a given known value. Substituting eq. (17) into eq. (16) yields

]Yli d Ei! -NB XN i" == - .vl (Xi "Xfi*) '

Bk

ko,2(y,-,il-BN i,*) -- at2 (x,-.:y',*) , B

(X,-X,*)+kB(Y,-at-l9 X,*) =- O and

(18) .i}l',*..Xt+kBYt..-ka-:B

1+k B2. •

From eqs. (12) and (13), we also obtain maximum-likelihood estimates of a, B,

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that is,

l-J - tv tSt (19) a-- Y-3Xi*

where-designates the mean value of a variable.

]E) .jsl,*(y,-a-b .xi•,*) -= o ,

s

which can be rewritten as

tV d--t tv tv tu (20) Z (X,*-X*) ( IYI, - ct -B X, *) - O

l

Substituting for a from eq. (19), we have

tV tV - tV t'-- A. .V

Z (X,*-X*) ( Y', - Y+ B X*-B X,*) - O l

\ (Åí*-Sii*) [ (y,- ii ') -(sli•,*-x'v*)i'gv] -- o

lv rh- - iv i-V iv

Z (X,* ---- X*) ( Yl, - Y) =- Z (X,*-X*)2 B

.., Z(X,*-X*)(Y,-Y)

(21) B==` .. - -

Z (X,*-X*)2 i

It should be noted that this is similar to the OLS estimate of B c;xcept that Xi 74 Xi*.

Therefore, the maximum-likelihood metnod corresponds to the least-squares regression of the observed Yvalues on the estimated value .Xii*. From eq. (18), we obtain r,*-Xi* == iN [(x,-:g)+ka(y,--y)].

1+kB2

Hence,

z (.iiili',*- fie'*) ( y, --y) - i .v [= (x, -:jil) ( y, --- 'i7) +k 'l`?' ( y, --y)2]

i 1+k B2 i

and

z (.Sl-,*- liir*)2= 1 A. [Z (x, -Xi )2+2 k B""Z (X, -X-)

i (1 +k B2)2 i i

( }tr, - Y) +k2 B2 2] ( ]Y7, - Y) 2] .

i

Introducing the notation,

mxy = Z xiyi/n , mxx = Z xi 2/n , myy = Zyi2/n ,

where xi == Xi-X, yi == Yt-Y, we can rewrite eq. (21) as follows:

(22) BN -- (1+ k B2 N) (mxy+k BN m..)

myy+2 3 mxy+k2 B myy

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and further,

(23) kg2 m..-g(km..--m..) --m.. == o . From this, we obtain the result

(24) bd == ny Å} viv2+ 1/k J' ---- 1, 2 where v -= MYV - imtf.k) MXX

Therefore, BN depends on k, mxx, myy, and mxy. In addition to this, bj,we obtain maximum-likelihood estimates of a by means of the followingprocess,: from eq.

(19) we obtain .v - tv -V ct -- Y•-BX*

and substituting this value for a into eq. (18), we obtain

- Nm NF-. NN

jS}. .= X+k B Y--k B(Y-B X*) 1+k B2

l*=x- Thus, we have

IV - N- (25) a- Y- B.X .

We must be very careful to consider two interesting results from this model; one is that the maximum-likelihood estimates of the parameter B are given by the least-squares regression of Y on X if a.2=O,,and the other is that it is also given by the reciprocal of the slope of the regression of X on Y if o,2=O. From the above consideration, we can understand that the OLS estimate is equal to the ex- treme estimates of this error in variables model. Secondly, we examine case (ii). Now, we have following model equations and assumptions:

(1) Xi = Xi*+ut (2) Yi = IY'i*+vi (26) IYi" =: ct+BXi*+Ei

where Ei is a stochastic-disturbance term.

E(u) - E(v) - E(E) === O

cov (u,v) = cov (u, E) = cov (v, E) = O

E(uu') --- a.2 , E == (vv') == a,2 , E(EE') == oE2 ,

and each error term is assumed to be normally and independently distributed with zero expectations and variances o.2, o.2, and oE2. Using these descriptions, we can easily deduce that

Yi = a+BXi*+Ei+vi

= a+BX,*+gi

where 6i = E,+vi and ei also have normal distributions with variance o,2 and zero means, where oe2 =r= aE2+o,2. From eq. (1), (2) and (26) we have

i 2tl] (x, --x)2 - -ili- ) (x,* - .-2kr*)2+ i\ (u,- a) 2 +l ] (x,*-'x*) (u,- fi)

-L--Z] ( Y, -MY)2 =- !it' :l] (X,*--X*)2 + i:i ](e,-g-)2 + -l}\(x,*-x-*) (e,-S)

- !li' ÅrI] (X, --X*)2 + i 2? (E, -e) 2

+ -:- Ei) (vi - J) 2 + cross-product terms

l:l](Yi -Y) (Xi - X-) = -Bi \(X,*- X-")2 + cross-product terms

Since, the expected values of all cross-product terms are zero, we take the ex- peted value of all equations :

(27) E[l:l.l (X,-hX)2] =- -lt2 (x,*-r*)2+ nii1 a.2 (2s) E[ i :i]( y, - My)2] - -B.e ] (x, *- I*)2 + n:i o,2

(29) E[i :l] ( Y, -NY) (X, --X) ] === e \ (X,*-x-*)2 .

By replacing the expected values on the left-hand side of these equations by the sample second-orber moments mxx, myy, and mxy, and then solve for the unknown parameters, we have, from eqs. (27) and (29),

(30) j =" m..- Ii/inX-Y 1) a.2

n and

ev - tNf- (25) a-- Y-BX

where o.2 is assumed to be known, and from eqs. (27)-(29), myy == Bmxy + n i; 1 (3,2 + 3.2)

== bm..+ n-1 'a",2 n

and

(31) 3e2= .Zl (Myy-bMxy)

= nli (myy- Miyli

)

Ou2

mxx N ij

n

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and furthermore, if o,2 is assumed to be known, we also have (32) 3E2 = nn" 1 (M'y- mx. -M XnYll l a.,) - Ov2 '

n

Since the sample second-order moments mxx, myy, and mxy will c6nverge in probability to the limit of their expectations, the estimates described by eqs. (30),

(25), (31) and (32) are seen to be consistent.

Finally, we examine case (iii) which have stochastic explanatory variables.

First, we consider

(1) X-- X*+u

(2) Y-Y*+v

(33) Y" -= ct+BX*

and assume that the stochastic variables X*, u, and v have independent normal distributions with zero means and variances ax.2, a.2 and o.2, respectively. Accord- ing to these descriptions, we obtain

ax2 == ax*2+a.2 , ay2 == B2ox,2+o.2

and

axy = B ax.2.

Since X and Y have bivariate normal distributions, the sample moments mxx, myy and mxy are equal to the maximum-likelihood estimates of ax2, ay2 and oxy, res- pectively, and furthermore, if we take previous assumption

ou2/ov2 = k , then we have Mxx == ox.2+k o,2 , m.. = B'V2 a'V..2+No.2

and

Mxy == S ox.2 .

From these equations, we obtain

(34) kB2 mxy-B(k myy-mxx)-mxy=O •

Notice that this is quite similar to eq. (23.) Therefore, in a Similar manner,

we can obtain the maximum-likelihood estimates aN , B7 X,N*, o.rw2 andoN,2. In other words, this is quite similar to case (i). Next, we intend to add a stochastic-dis-

turbace term to this model, thus,

X == X*+u, y== y*+v

and

Y* = a + BX* +E .

Since X*, u, vand E all have independent and normal distributions, stochastic variables X*, Y*, X and Y have a bivariate normal distribution, and so, we have ox2 == ax.2+au2 ,

oy2 == B2 ax.2+aE2+a,2 and

axy =B ox.2 ,

and furthermore, by replacing the elements on the left-hand side of the above equations by their maximum-likelihood estimates mxx, myy and mxv and assuming a.2 to be known, we obtain

mxx == 3x.2+a.2 , lv tv tv

Myy = B2 ax.2+ae2 and

mxy == BN2 3x.2 ,

where Je2==3E2+'aV.2. We, then, obtain the maximum-likelihood estimate of B, which is written :

(35) BN=m.llliYa.2.

This is quite similar in form to eq. (30) mentioned in case (ii).

3. INSTRUMEINr]rAL VARIABIES METHOD

We, again, assume the assumptions:

(1) X == X*+u ,

(2) Y-Y*+v, (3) Y*=-a+BX*

and

(4) Y--a+BX+w,

where w==v-Bu. As was seen in Section 1, generally speaking, the OLS estimate does not give the BLUE nor even consistent estimates. Now, we introduce the instrumental variable z, which is independent of u and v, and consider,

(36) a=. ]211 yi zi ;E] xi zi ' i

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where each variable is measured as a deviation from the mean. Therefore, from eq. (4), we obtain

b=-

X(Bxi + (wi - w) )zi Z X, Z, i

i

BZxi zi + Zzi(wi-w)

ii ^{Z x, z,}

and we take the probability limit. Thus, we have A (37) plimB-B

so,we have a consistent estimate of B. At this point, it should be noted that the estimate clearly fails when Zxi zi==O and that z should be chosen so as to have a fairly i

high correlation with X, which is understood from eq. (37). This method, howe- ver, has some demerits, vii., (i) there is the arbitrary nature of the variables cho- senas instrumental (ii) it is impossible to check the assumption, cov(z, u) ==cov(z, v)

==o and (iii) this method raises consistency to a position of extreme importance.

Therefore, we will not be able to advance it vigorously, but from a computational technique it is often employed conveniently and is similar to the grouping of the observation method which is mentioned in the next section. In the multivariate regression model, we can also solve the problem by importing instrumental vari- ables which correspond to the number of explanatory variables. Takeuchi [2] has reexamined these multivariate cases in greater depth, that is, he defined the follow- ing statistical variables:

xy,2 zy,x.••••••]z]y,x,, nH= X YjXrj ZXij2""""'XXijXpj

--t---t---e--ee--t--

2] Y,Xp, XX,,X,,••••••ZX,2,

b--

ao2 ---•--- O

l• oi2 l

l -!---...L.. IttT l ---L-L- l

: '.-L.1

where Yj(J'---1, ..., n) is a dependent variab!e and Xij(i---1, ..., p, j=1, ..., n) is an

explanatory variable in which the first suMx i indicates the number of explanatory variables and the second suMx j indicates the number of'observations, and o,2(i=

O, 1, •••, p) is the variance of each variable. Then, the required estimates, g,, are

obtained through the following process, that is, (i) solving the characteristic pol-

ynomial I H-Zbl == o, (ii) obtaining aminimum value of the eigenvalues and eigen-

vectors P*== (Bi •••, Bp*) corresponding totheminimum eigenvalue and (iii) calcul-

h iv ating Bi==Bi*/Bi" (see also Takeuchi [2], pp. 87-93). However, notice that this procedure is only limited to the cases that provide special information concerning the variance ratio.

4. GROUPING OF OBSERVAT[[ON METHOD

Suppose that

X- X*+u , Y== Y*+v and

Y* =- a+BX* ,

where the error terms u and v are assumed to be serially and mutually indepen- dent. Now, if the number of observations is even, n=21, asapractical matter,the procedure becomes very simple. Consider the following time series data which is arranged sequentially,

Xi, X2, •.•Xi, Xi+i, Xt+2, •••, Xn

and consider also the next time series data which correspond to the above time series, respectively,

yl, y,, ..., y,, y,.1, yl+,, .... y. , and define

]jii1=1?l.l?lxi , ]ii2=--}-l.liit]il Xi

and

-y, .= Jli- A., yi,, Ei7i2 -- -;- ,=S., Yt

According to this notation, we have (3s) fa=.Ii-!2

Xi-X2 and

A in A- (39) a-Y-BX.

It is known that these estimates are consistent under the assumption that

(40) lim.knf -l' i.il..l, Xi"'vllT,=., Xi* ÅrO

(see Wald[3]). However, assumption (40) is not valid for normally distributed variables, and so we attempt the following revision

iiiii='il' ]i.ll, Xi ' I)il2 `= -llTi.n$k+i X`

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and

MYi--iliF S., Yi• Y2 == -l;,--.\,., Yi•

and then we have, (41) A=,.I;2--Yi X2-Xi and

A-A. (42) a-- Y-BX.

By means of this revision, we can obtain a larger usefu!ness, but it iS hard to understand how the convenient number k can be found. Batlett[4], however, A suggests that the sampling variance of B can be minimized by setting k==n/3 for

the case of equally spaced X values. But this is merely a case fitted to a par- ticular sample. Thus, strictly speaking, this method is also merely a simlifica- tion procedure derived from the computational technique and is an incomplete method because it neglects the assumption of normality.

5. FURTHER RESULTS

This section treats error in variables model with lagged explanatory variables and serially correlated variables. For concreteness, we consider following types of models :

(i) Yt == BoXt+Bi Xt-i+•••+BpXt-p+Ut ,

( ii ) y, == B 1$l] 6ixt-i+v t i==o

and

(iii) yt == ayt-i+Bxt+wt•

In each case, we assume that the exogenous variables are observed with a measure- ment error 6t, which is uncorrelated with any other variable in the model and that all variables have zero means. Throughout this section the following notational conventions will be observed:

COV(Xt Yt-d) =a., (j) , jÅrseO

=a.,, j=O

and

oxy (j) /ax ay == pxy (J')

First, we consider model (i) in which yt depends only upon the current level of xt and a possible finite number of lagges x's as well. Let us assume:

(43) yt = Bxt+ut , E(xt) = E(ut) =O , E(ut xt-j) =- O , vj ;

x, -- x,*+6,, E(g,) ==O, E(6,x,-,) == E(g,u,-,) == O, v,;

where xt* is a true value of xt. As was seen in the previous sections, the OLS estimate of B is

(44) G? -= i+ ae2 /a., == i9 (i --- o. ,ai 2a,2) = B(i-k) wh ere k == a. ,Oi 2o,, .

Thus, we have the useful result for the bivariate regression model. Now, if e, is the calculated residual from the regression, then it can be written as follows :

AA

et === yt-yt == Bxt+ut---Bxt*

=: i?xt+ut-B(1-k) (xt-6t)

== u, + i?kx, -- B (1 - k) e,

and the first order autocorrelation of et is equal to

pee(i)==B2k2aBxfi;.).it.BES2i--k2iiS/i3f(2'+).+.20.""(').

This p,,(1) designates the weighted averages of p..(1), pee(1), and p..(1) with weights B2k2a.2, B2(1-k)2ae2 and a.2, respectively. Thus, we can understand that the presence of measurement errors lead to residuals having autocorrelation.

When serial correlat;on is found in the residuals, it is a common practice to reestimate the model using p-differenced data. If B is estimated by regressing yt-pyt.i on xt*---pxt-i, where p is some number between plus and minus one, the resulting estimate is

(45) b== a.2a+p2-SSXp2.(.1(IsP)2+--a2,P2r(PIXX+(lp)2-2pp,,a)) .

To compare eq. (44) with eq. (45) gives an interesting result. Suppose that we assume that past levels of x as well as current levels of x enter into the model and consider the following model

Yt =: BoXt*+BiX*t-n+ut . Then, the OLS estimates are

b, .. B [ox`(1 '- pxx2i.n.), ()1+-Oi,2.f.x.2((nl)ii Pxx(n)Pee("))]

(46)

and

a,--BOia.12[,Pff2Z)t,(nt.p.f)ek"Z/,

A

at this point Bj 7AO(j=O,1). Therefore (46) shows that the presence of measure-

ment errors will results in ,significant coeMcients of lagged x*'s even though the

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true model contains no lags as in equation (43), and thus, this can be ex- panded into the more general form, that is, if yt depends upon {xt-i, j--O, 1, .,., p}

then it will normally be the case that x*'s lagged more than p periods

will have significant coefiELcients in the regression of yt on (x*t.j, J'=O, 1, •••, p,p+

1, •••, m). Next, we consider model (ii), which is a distributed lag model in distributed lag form, viz., the model is specified with only exogenous variables.

We assume a Koyck-type lag model:

B _{(47) yt =} xt+e, 1-rL

where L is a lag operator defined by Lxt!Ext-i. Due to measurement errors, we have

B _{(48) yt==}

xt*+Et 1-rL

and define that

x,. (i) == Xl e,.(i) -- 9 x*,. - x,. (i) + e,.(7)

1-rL' 1--rL'

where lrl Åq1. In order to obtain maximum-likelihood estimates of Band r, we use x*t* instead of Xi Then, if both xt and et are uncorrelated over time, we obtain

R(7) =:B( ,i--i )(i-k) , var(x,.(:r)).= axh2

1-r2 ,

(49)

var(e,,(-r)).= ae2"

1-r2 , and

var(h) =a,2--B2a.2 (1-r-)2(1-k) (1-r r)

A where u is a calculated residual, and if the xt are serially correlated, we obtain li (i) -= (,E;i,;fi)-(i,gfir.'i F) {..,(,.ai-ii'.+.P,;(),-,7))

var(x,.(7)) = -"Ox2(1-+pr)- (1-p r) (1-r2) ' (50)

var(6,.(i)).., Oe2-

(1-r2)

and

var(h)=a,2m B2(1;r2)(1-p2rr)2a.2 fh a.2a+pi) )

(1 -r r) (1 -pr) 2(1-p2-r2) No.2(1+ p-r) + a,2(1-p7) J

where the terms in curly brackets designate the effects of the measurement errors. From eqs. (49) and (50), we can, thus, understand that the measurement errors lead to the appearances of a higher-order distributed lag than is acctually the case as the latter of model (i). Finally, we consider model (iii) in which yt depends on lagged values of the dependent variable and a set of explanatory variables. Now, the estimating equation is of the form:

AA A (51) yt = ct yt-i+Bxt" •

If thex sequence is positively correlated over time and the disturbances are either independent or positively correlated, then, OLS estimates have the follow- ing limits

."-ct===aeiO,y.2.(.P,iy21]i,2H(-ig-)+.,,O.x.i2a-"Y;l.),(i)'

(52)

and

b-B=.,2i..B-O'.2.0i2a)J-.B,g;'..(iila."i,$llllL)

and adding the assumptions that xt and ut are first-order autoregression with parameters p. and p. respectively, we have

2r -a =. oe2(IPiB:OpX.2 + lttaa"i.) + p. o.2mqL.2

ay2 a..2-axy2(1) (ay2 ax.2-axy2(1) (1-apu)

(53)

and

b -- B -= zF,, .-.,B,a-e2aO.,y,2(i-) - (a,,o.,i-a.,e(PiX)a)"iiP!Ot:Z.)(i-ap.)' .

In each case these equations imply the inconsistency, and in the most frequently assumed case (p., p.ÅrO) they tend to increase the inconsistency. Furthermore, according to Levi[5], under the special structure which has only one explanatory variable with a measurement error (such as x!) and other explanatory variables that are measured without error, the direction of bias of the coefucient can be calculated through following process : suppose that

X- X*+U ,

y == y*+v and

y* - X*P+e

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where y, y*, P, v, and e areT-component (column) vectors ofT observations and X, X*, and U areTxPmatrix of explanatory variables and measurement errors, respectively, and that

E( U) -= O , E(U U') - a.2 I,

E(v) =rO, E(vv') =a,2I, E(e)-=O, E(ee') =- e,2I,

and that all errors are independent of the true values ofthevariables and of each other. Then

plim b =-= plim (X' X) -iX' y =- plim (X'TX) -iplim (Xi Y) -- ( E] + 2) -i Z B ,

and thus,

d(dP iilll, ,A) --(Åí+2) -i 8 8 8 (.- . .) -, 2) p.= -(x ..) -, Piii8 bi),

6o----:o 61

where 2i: and 2 are finite limiting variance-covariance matrices, and especially the 2 have following form :

OUi O""""'O 2 .= O O-•-••••O .

--

6 o••••--•6 j

From the above considerations, the direction of bias of the coefficient simply de- pends on the knowledge of the variance-covariance matrix of observations.

6. CONCLUSIONS

In this paper, the various types of error in variable models have been examined.

In the theoretical approaches, case (i) has maximum-likelihood estimates, aand

BN , which include OLS estimates, a and b, as a special case, obtained by solving the quadratic equation, kB2 mxy-B(kmyy-mxx)-mxy==O, and a lmear equation, a

=i-' B" X". Case (ii) indicates that each estimate is consistent. Case (iii) shows an interesting situation in which a model having a non-stochastic relation be- tween the real variables is quite equalto case (i) and that themodel having asto- chastic relation between the real variables is quite similar in form to case (ii).

The Instrumental Variable Method and Grouping of Observation Method are also

used as a simple method from the point of computational techniques, inspite of some

demerits. In further results, we treat some types of models with lagged ex-

planatory variables and serially correlated variables. For model (i) we have shown

that measurement errors in the exogenous variable may lead to the appearance

of spurious long lags in adjustment and that it may produce residuals which are autocorrelated. In the case of distributed lagmodels, we under- stand that, under plausible conditions, measurement errors also lead to the appea- rance of a higher-order distributed lag than is actually the case. For model (iii), we have shown that the presence of measurement errors tends to increase the inconsistency. And furthermore, for the fairly special case of only one explanatory variable with measurement error, the direction of bias on the coeMcient simply depends upon the knowledge of the variance-covariance matrix of observations.

REFERENCES

[1] Johnston, J.: Econometric Methods. New York: McGraw-Hill, 1963.

[2] Takeuchi, K.: Study of Econometrics. Tokyo: Touyou Keizai, 1973. (in Japanese) [3] Wald, A.: "The Fitting Straight Lines If Both Variables are Subject to Error," Ann.

Math. Statist., Vol. 11, pp. 284-300, 1940.

[4] Bartlett, M. S. : "The Fitting of Straight LinesIf Both Variables Are subject to Error,"

Biometrics, vol. 5,pp. 207-242, 1949.

[5] Levi, M. D.:"Errors in the Variables Bias in the Presence of Correctly Measured Variables," Econometrica, vol. 41, pp. 985-986, 1973.

[6] Goldberger, A. S.: Econometric Theory. New York: John Wiley, 1964.

[7] Theil, H.: Principles of Econometrics. New York: John Wiley, 1971.

[8] Grether, D. M., and G. S. Maddala : "Errors in Variables and Serially Correlated Distur- bances in Distributed Lag Models," Econometrica, vol. 41, pp. 255-262, 1973.

[9] Casson, M. C. "Linear Regression with Error in the Deflating Variable," Econometrica, vol. 41, pp. 751-759, 1973.

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