-' AD-AII2 ,2 CENTER FOR NAVAL ANALYSES ALEXANDRIA VA NAVAL STUDIE--ETC F/6 12/1TISTIN FOR STRUCTURAL CHANG BY D-NETHOOS IN SWITCHIN* SIULTA--ETC(U)

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PROFESSIONAL PAPER 342 / February 1982

TESTING FOR STRUCTURALCHANGE BY D-METHODS INSWITCHING SIMULTANEOUSEQUATIONS MODELS

Lung-Fei LeeG. S. MaddalaR. P. Trost

A

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CENTER FOR NAVAL ANALYSES

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The ideas expressed in this paper are those of the author.The paper does not necessarily represent the views of eitherthe Center for Naval Analyses or the Department of Defemse.

PROFESSIONAL PAPER 342 / February 1982

TESTING FOR STRUCTURALCHANGE BY D-METHODS INSWITCHING SIMULTANEOUSEQUATIONS MODELS

Lung-Fei Lee

c. Maddala

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PROFESSIONAL PAPER 34. February 1982

TESTING FOR STRUCTURALCHANGE BY D-METHODS INSWITCHING SIMULTANEOUSEQUATIONS MODELS

Lung-Fei LeeG. S. MaddalaR. P. Trost

Naval Studies Group

CENTER FOR NAVAL ANALYSESMO Nenh Baureed SOW, Ahmwafl, ViqinWl 22311

I - -

TESTING FOR STRUCTURAL CHANGE BYD-METdODS IN SWITCHING SIMULTANEOUS EQUATIONS MODELS*

Lung-fei Lee, University of MinnesotaG. S. Maddala, University of Florida

and R. P. Trost, Center for Naval Analyses

1. Introduction

Goldfeld and Quandt (1972) consid- There have been previous discus-ered a switching regression model with sions of structural change by Heckmantwo regimes: (1976b) in models with endogenous switch-

ing. However, the type of model we dis-Yi = a1 + blXi + Uli (1) cuss has not been considered by him.

What he considers is a shift in the con-Yi= a2 + b2xi + u21 (2) stant term (a structural shift) and not

Uli % IN (0, a12) and u22i IN (0,022 shifts in all the coefficients as we do.

What the D-method does is to com-They assume that there is an identifiable bine the equations in the two regimes in-variable Zi such that if Zi < Z0 then the to a single equation. This will enable

observations are generated by (1) and if us to apply tests for equality of speci-t eobservations are genera- by (1) and if fic coefficients in the two regimes. The

zi Z0 then the observations are genera- alternative to this procedure would be toted by (2). They then define the step use a likelihood ratio test every timefunction the significance of a particular coeffi-

cient needs to be tested. Thus, the D-D (Zi) = 1 if Zi < Z method has the advantage of computational

(3) simplicity. In this paper we suggest= 0 if Zi . Z0 instrumental variable methods to estimate

the parameters of the combined equation.

Later, instead of considering the step Following the methods used by Amemiyafunction, they consider D(Zi) to be a (1978), we derive the asymptotic covari-cumulative normal given by ance matrix (in the appendix) and present

an empirical example illustrating the useD(Zi ) - 1 if Zi < Z0 + 0i of the suggested method.

= 0 if Z. 0 + E 2. The Switching Simultaneous Equation1 Z 0 LiSystem

where c. IN (0,I). The general model we consider in

The model considered by Goldfeld and this paper is the following:Quandt is an exogenous switching model I x16 - 0 (4)because Ei is assumed to be independent =

of uii and u2i. Goldfeld and Quandt Regime 1: BY 11 + rIZli(1973) give an extension of the switch- if I > 0 (5)ing regression model to simultaneous 1i iequations systems but the extension is Regime 2: B2Y2i + 2Z2istill in the framework of exogenous

=switching. C2i if Ii o (6)

Maddala and Nelsotf (1975) discussextensions of the Goldfeld-Quandt switch- The residuals £1, c2, and e' have a multi-regression modil to the case of endogen- variate normal distribution with meanous switching. Many of the practical vector 0 and covariance matrixproblems one encounters - disequilibriummodels, selectivity models etc., are all Iswitching regression models with endoge- 1 12

nous switching. The present paper ex- E -tends the D-method suggested by Goldfeld 2and Quandt to switching simultaneous 22 2*equations systems with endogenous switch-ing.

Define the dummy variable

*Financial support from the National Di - 1 if I > 0Science Foundation under grant Soc.T$-09473 to the University of Floridaa gratefully acknowledged. - 0 if 11 0

]-J

Without loss of generality, let us con- Equation (10) can now be written assider the first equation in the two re-gimes and assume that the endogenous var- yi - DiY~ i (y112 - 0212)+iables occurring in both are the same.Let the observed endogenous variables be + D ('ll - 021p) + Y +

y y ifD = i+2y+]i yiji i (7) + +oZi(yli-72l) + Zi21

= y2ji if Di = 0 + 4i (a 21 - ll*) + Ili (12)

Let us write the first equation in the where E(§ i) 0.two regimes as

3. Estimation Under D-MethodsYlli -

8ll2Y12i + 8l13Yl3j + "'"

(8) valiTo estimate equation (12) and develop+ 811pylpi + Yflzi + 1 i(8 valid statistical tests we will extend

and the two-stage estimations method thatHeckman (197§a) suggested. First, we getan estimate 6 of 6 by probit maximum like-

Y212i + 8213Y23i + lihood applied to equation (4).

+ 821py2p i + 72lzi + '21i (9) Second, we estimate the reduced form equa-

tions for the endogenous variables in (8)These two equations can be combined into and (9) by the following two stagea pingle equation and written as: method.

y i -Diyji(8 1 1 2 - 8212) + "'" The reduced form equation for Yjican be written as

" D iyp~i (Op - '21p) Ylii = ZiYlj + vlj i (13)Y2i212 + -+ *~ 8 21 Sne (

+ •2 2 "'" Ypi y 1p Since E(V1 J I ii > 0) + - oij .

1 i( 11 - 21) + ziY 2 1 where aiS . = Cov(vlj,*)

+ Dic lli + (l-Di) £ 21i (10) We can write (13) as

Let us define D iylji =i zi 7lj - Ol9* Oi + nlji

1i= O(xjS) (14)

where E(n )ji 0.and (11)li

4. = D(xi6) We estimate equation (14) by 9rdinary1 1 least squares substituting *i and Oi for

where x. is the ith observation of i and #i respectively.1^

X Oi the standard normal density Here 0i and ti are the same as #ii=l, 2..N i 1t1

evaluated at x16 and 4i is the cumulative and with 6 substituted for 6. Thisnormal: two-stage estimation gives us consistent

xV5 -u2/2 du estimates ; j and 0 1j , and we can con-_e struct the instrumental variable

Note that E(cli II1> 0)- - % *ii 0DiYlji = Oizi lj - 0 1j,$i (15)

N t From the reduced form equations for Ylji

where aii.= Cov(C illc) and one has

II Y3i - iiwlj (l-*i)Z i 2j +

Similarly E ( 211 I 0i . 0) ' ac 21, -4 ( 2j-_a j, )* i + Vji

where 021* - Cov(c21, C) where E(V i) = 0 (16)

Hence, the composite residual in (10) has

the expectation We estimate equation (16) by ordinaryleast squares substituting *t and j for

*[Di CI i + (l-Di) t21i ] #i and si to get consistent istimatis iiSf

(a c21* OeIl*) 41 12j and 0 2j. - lj,.

-2-

Now we construct instrumental vari- TA.ENT data. More detailed discussion ofables for y'. in (12) as follows the data and their use in this problen

* - can be found in Kenny (1977) and Kenny et.y~2~ l j.z ; +1) al. (1979). We will here summarize allJ *+(l-.. 2 i i'72.1 (17) the salient features of the data and the

With these instrumental variables con- model relevant for our discussion of thestructed, one can estimate the structural D-method.coefficients from the following modifiedequations: Let S = years of college educationyi = Diyi1+ E = earnings

i 2i 12-'212) + X, z1, z2 are sets of (possibly

" D aoverlapping) exogenous variablesdefined in Table 1.+ *~8 l + + "

Y10212 + Ypi 2lp The switching simultaneous equation sys-+ i Z i(Yll-2l) + ZiY 21 (18) tem we consider is the following:

+ i(0c21,-O11*) + 'li I = XS - (23)

where li = li + (Di-9i) Zi(yl-y 2 1 ) S = Z 101 £ 11 (24)+ 1 . 12 ff10

(+i-$.) Zi(Tll-y2) (19) E = YS + Z2 82 + E12 (25)

+ (2-1)( -25

and li = Di£ 1 1 i + (l-Di)£2 1i = 01 1 otherwise (26)

(2l*-all*i(20)E = Z2 83 + £22

In matrix notation, let I is an indicator function such that ifI > 0, the individual goes to college

W = [Dy,Dy3,... • Dy, and if I < 0 the individual does not goP ~to colleg e.y! .... yh,OZ, Z,61 (21)

pand 0' = (8-B The variables in the choice function

112-212'''11p'21p'212' and the schooling equation, i.e., vari-ables in the sets X and Z are% Rural,

" 21p'¥ll'2l' Split, MATH, MOCWAG, EDMAL, EDFEM, CHIL.

Y21'a 2l* - a El(defined in Table 1)

Also let • ,p•2 , The variables affecting the earnings3 .. 2.... functions, i.e., the variables in the

set Z are- Rural, MATH, Marital Statusy •*,OZ, and (defined in Table 1).

Then, the instrumental variable estima-tor of e is The variable S has been coded as fol-

((,w - lows:e W w)Wy ~ (22)0 for those with no college education

It can be shown that in this case 4.0 for Bachelor's degreegiven the way the instrumental variables 5.5 for Master's degreeare constructed as defined by equations 6.0 for six years college(17) and (15), the instrumental variable 8.0 for Ph.D.estimator is the same as the two-stagesquares estimator so that with numbers between 0 and 4 for those

0 (W'W)-I , without bachelor's degree depending on1 (22a) when they dropped out of the 4 year pro-

gram.The instrumental variable

estimator

can be easily shown to be consistent. We will briefly discuss the reason-The asymptotic covariance matrix of this ing behind the choice of the variablesestimator is derived in the appendix. It in the sets X, Zi, Z2.is the last expression in the appendix.Since it involves the definitions of sev- First, regarding the variables X anderal expressions, it is not repeated here. Z1 affecting the choice of whether or not

to go to college and the years spent in4. An Empirical Illustration college, high school students coming from

rural areas are less likely to go to col-The example we consider is the esti- lege or spend more years in college.

mation of the effects of college educa- Hence we included the 'Rural' as a dummytion on earnings based on the project variable. If children are viewed as a

-3-

good whose returns are specific to mar- practical instances where the use ofriage, then divorce would lower the de- maximum likelihood procedures would bemand for child's educational attainment, ruled out because of the large size ofHence we included the variable 'Split' the model (one example is the model con-as an explanatory variable and its coeffi- sidered by Nelson and Olson (19781). Incient is expected to be negative. The such cases the D-method suggested in thisvariable MATH measures both inherited and paper would be useful to apply tests ofachieved cognitive skills, hose with significance for equality of coefficientshigher MATH scores may learn more rapidly between the two regimes.than others and hence m~y find it optimalto get a higher degree. The variables 5. ConclusionsMOCWAG, EDMAL and EDFEM need no justifi-cation. We included the variable CHIL The paper presents an extension ofbecause an increase in the number of the Goldfeld-Quant D-method to switchingchildren in the parents' household simultaneous equation systems with endo-increases the total cost of schooling for genous switching. The method is usefulthe family leading to a reduction in the to apply tests for equality of coeffi-number of years spent in school. cients across regimes in switching simul-

taneous equation systems with selectivityAs for the variables Z2 affecting The paper derives the valid instrumental

the earnings functions, the variable Rural variables to use in this situation, the

is a crude proxy for cost of living, asymptotic covariance matrix of the sug-

Individuals who grew up in rural areas, gested estimator and gives an empirical

where cost of living (and thus wages) is illustration.low, are more likely to live in ruralareas than others. The variable MATH REFERENCESmeasures cognitive achievement and henceshould be positively correlated with Amemiya, T. (1978) - "The Estimation ofwages. The variable Marital Status is Simultaneous Equation Generalized

expected to be positively correlated Probit Model," Econometrica, Vol. 46,

with wages if we argue that married No. 5, pp. 1193-1205.

people have greater responsibilities and Goldfeld, S.M. and R. E. Quandt (1972) -hence are likely to work harder. Another olnear Method in Eontricsarguent s tat te mrriae mrketmayNonlinear Methods in Econometricsargument is that the marriage market may (Amsterdam, North Holland Publishing

recognize that some men are better 'pros-

pects' (have higher expected wages) than Co.).

others. Finally the variable HC is used Goldfeld, S. M. and R. E. Quandt (1973) -as a proxy for on the job experience. "The Estimation of Structural ShiftsAs the number of jobs held (i.e., HC) by Switching Regressions," Annals ofincreases, the amount spent with the cur- Economic and Social Measurement, Vol.rent firm is likely to decrease. Hence 2, No. 4, pp. 475-486.HC is expected to be negatively corre-lated with earnings. Heckman, J. J. (1976a) - "The Common

Table 2 presents estimates of the Structure of Statistical Models of

parameters 6 in equation (23). All the Truncation, Sample Selection and

coefficients have the expected signs. Limited Dependent Variables and a

Table 3 presents the maximum likelihood Simple Estimation for Such Models."estimates of the parameters in equations Annals of Economic and Social Measure-

(24), (25) and (26). Table 4 presents ment, Vol. 5, No. 4, 475-492.estimates of the parameters of the earn-ings functions obtained by the D-method Heckman, J. J. (1976b) - "Simultaneousdescribed in this paper. The standard Equations Models with Continuous anderrors have been computed from the formula Discrete Endogenous Variables andin the appendix. Structural Shifts," in Goldfeld and

Quandt (Eds), Studies in Non-linear

Our objective in using the D-method Estimation (Ballinger Publishing Co.).is to test which coefficients in theearnings functions are significantly dif- Kenny, L. W. (1977) - "The Demand forferent between the two groups - those Child Quality and for Educational

Inputs, the Production of Child Qualitywith college education and those without, and Related Topics," unpublished doc-The equations in Table 4 suggest that it tonal dissertation, University of

is only variables HC and MATH that are

significantly different; 'Marital Status' Chicago.

and 'Rural' are not. Kenny, L. W., Lung-fei Lee, G. S. Maddalaand R. P. Trost (1979) - "Returns to

The example we have used here as an College Education: An Investigationillustration for the D-method is one of Self-Selection Bias Based on thewhere maximum likelihood estimation is oect-Talen Da Intentn

feasble Thee wuldhowverbe anyProject Talent Data," Internationalfeasible. There would, however, be many Fconomic Review, Vol. 20, No. 3.

-4-

REFERENCES - nl(continued) = (WWV1 ,_

Maddala, G. S. and F. D. Nelson (1975) - + ell* - oc2l,)(;-€] (10')"Switching Regression Models with Exo-genous and Endogenous Switching" - here ,,oAmerican Statistical Association Pro- w D, are all column

ceedings of the Business and Economic vectors with their i-th elements §1i' Di,Statistics Section, December 3975, ti' fi' fi ti respectively.pp. 423-426.

Nelson, F. D. and L. Olson f1978) - Expanding *. and *i up to th first

"Specification and Estimation of a power of (6-6) we getSimultaneous Equations Model withLimited Dependent Variables," Inter- 1national Economic Review, Vol. 19, . i +[--i (6)fi+0ix! (6-6)No. 3, 695-709. 1

Hence ( -$)= CX (6-6) (11')APPENDIX Similarly

In this appendix we will derive the ; -f = - FCX (6-6) (12')

covariance matrix of the two-stage least Hence we have 2 from equation (10')squares estimator (22a). A -

We have 0 -- (W,'W) [1 + H(D-)

(W-0) = (,)I nl - HCX (1-a)-( Cll.-GC21.)FCX(6-6)]

where nlIV, the ith component of ni is Now what we will do is express (6-6) ingiven by equation (19). Let us first de- terms of (D-0) so that all we have to getfine the N X N diagonal matrices is Var (§l), Var (D-0) and Cov(§l,D-0).

B = Diag (41f20... X1) Following Amemiya (1977) we can write

C = Diag [tif2 ... NJ (2')

F = Diag [x,x?..6 x;6] (3') (6-6)A(x.,NX)- X'/^(D-f) (14')

H = Diag [zly 1-y2l) ....22#11 "y2 1)J (4) Substituting this in (13') we have

V = Diag [01(1-4k),...fN(1-0N)] (5') ( ) [1 G, (

= Diag N (6')a Dia ll1) ( N (6) where G is defined in W).

and fI N^1= Diag N (7') From equation (20) in the text, we

l~l ~ N ~ C"N) 1 have

Actually/= C2V -1 andA = CV-1 but § = E21i+D i (Elii-'21i)

we have defined these two diagonal matricesseparately for convenience. - (c£2l"-oll, 1)i

Hence Var (§ ) ( 2Also let X be the N X L matrix of el i)

observations on xi, i.e. = 221 +(02 1102 21) i

xi, + (a21"-O211 N

1 x (8') 2 2

or £21,-o (§ 2 2 2

Finally define Or Var (E1) = a2l IN + (0l- £21) B

G H- [+ (ll* -2l*)F] (9') + (,2 2CX- r2.,_ 0Ell*) FC

C X 1 - (o21, - oe11,)2C2 (16')

Now from equation (19), we have Var (D-) - V where V is defined in (5')

and hence Var [G(D-f)1 - GVG' (17')

-5-

"inally, E(§5 D--)' - C TABLE 1 (cont.)1 0cll*

MOCWAG = (mean occupational wage) Log of+ (aEll* a 21) BC (18') mean earnings (as of 1960) of

full-time workers in father's

Hence, collecting the expressions (16'), occupationHC - Number of jobs held between 1965

(17'), and (18') we have and 1970N = The total sample size = 1373N1 = Number of observations in the

Var (6-0) first group (those who went to- (21 IN college = 909)

*(o2 1 + 2 2 B N2 = Number of observations in the21 N (11 12 second group (those that did not

go to college - 464)2 2+ (a 21*011,) FC TABLE 2

PROBIT EQUATION TO EXPLAIN THE

(a -a) 2 C2 GVG DECISION OF WHETHER OR NOT TO GO-21*cll,0 + G TO COLLEGE

Variable Coeff. Est. S.E.

- C1l* (CG'+GC) Constant -5.259 2.108Rural (Dummy) - .164 .093Split - .078 .145

+ l ( 11,-oc 2 1*(BCG'+GCB)1 MATH .015 .001MOCWAG .922 .589

(W'W) EDMAL .044 .017EDFEM .029 .019

where the expressions B, C, F, G etc. are CHIL - .057 .021all defined in equation (1') to (8').

TABLE 4FOOTNOTES: EARNINGS FUNCTIONS WITH D-METHOD1Richard Quandt kindly pointed out to us Variables Equation 1 Equation 2 Equation 3

that there is an algebraic error in equa- Constant 5.72467 5.69896 5.70961tion (3.8), p. 425 of the Maddala-Nelson (0.12918) (0.08953) (0.09773)paper. The two denominators in t should Marital 0.14620 0.15656 0.15511be 2 2 21/2 2 2 2 1/2 Status (0.06281) (0.01298) (0.01808)(y22 + 403) and (y12b1 2+03)

1/

(a 2a 2 1/2 a 2 a 2 1/2 Rural -0.07161 -0.05352 -0.05030instead of 2 and 1 + 3 (0.07239) (0.02185) (0.02416)

respectively. MATH -0.00359 -0.00377 -0.00336(0.00231) (0.00206) (0.00193)

2In the discussion that follows, the nota- HC 0.03196 0.03184 0.03234tion A means that the two expressions have (0.01304) (0.01275) (0.01315)

the asymptotic distributions. 0.74827 0.82369 0.66080(0.48986) (0.35637) (0.20604)

TABLE 1 4 -0.12980 -0.15768LIST OF VARIABLES (0.34249) (0.27731)

E = Natural logarithm of the hourly O*Marital 0.01454wage (in cents) in 1971 (0.08599)

S - Years of college education in 1971MATH - Score on a composite of mathe- C*Rural 0.03258

matics achievement tests in 1960 (0.10696)SPLIT - 0 if children living with both VMATH 0.00560 0.00589 0.00507

mother and father in 1960 (0.00306) (0.00262) (0.00226)- I Otherwise

EDMAL - Years of father's education O*HC -0.06444 -0.06335 -0.06671

EDFEM - Years of mother's education (0.02026) (0.01920) (0.01894)CHIL - Number of living children (in S 0.05141 0.05774 0.04202

1960) in the family in which the (0.04710) (0.03536) (0.02590)respondent grew up

RURAL - 1 if pupils in grades 9-12 came (Table 3 is available from the authorsfrom an area primarily small upon request.)town (under 5,000 people) orrural farm

a 0 Otherwise.-6-

=4 IUWEU"aV. pols - 4717 MW

PP 211 PP 222Nlarabi. Nol N., nba Approtlesrtlng thle Circalar Cocerege M4treal, Mourlos N.. 00orreeeondfe Rules end PathFunction,' 14 pp., Feb 1970, I A054 429 lntegsls,* 30 pp.. Jun 1978 (invited pqer presented at tile

iNRS metIng an 41111thaetical Problem In Fayima's PethPP 212 Integrels,' Ibreellle, France, Ne 22-36. 1971) (Pa611sil

Monol. Moc. 4on Singular Characteristic Initial Volvo In Springer Veriag Lectere Notes In Physics. 106, (19M.)Problem witha Unique Solution.* 20 pp., Jun 1978, 234-253) AD AD" 536

PP 223PP 213 Ningel. Marc. 'Stochastic Mechaanics of Ibleculelon Nbleoule

Hompel, Wac. "Pluctutica In Systems with Multiple Steady Roectloas.0 21 pp., Jun 1978, AD AM 227States. ApplIication t Lanhaseter Equations.' 12 pp.,Feb 78 (Presente at theg First Annual Norks"q on thme PP 224Information LUabep Befuses Applied Notaoems ad Meanger, lhrc, A*A W~ealc., itarcatlaa, ad Eatlactca, InIndustry. Navel Pe Schocol. Feb 23-25, 1976). 10 AD71 472 Exploted Anlml Populations*," 46 pp., We 1970.

AD AM 536PP 214 "Portlons of tis wrlg wne started at tile Institute of

Wslalood, Roert Q., 'A ~Smewat Differn View of The Applied Nthotlm and Statistics. University of irtlshOtlml Newel Posture,' 37 pp., Jun 1978 (Presented at the 00luebis, Vancouver. S.C.. Come"1976 onventiona of the Awrices Political Scinc teAsocis-tion (APWA/US Panel on 'Oaening Strate Raquiremets wad PP 225Mili1tary Posture"). Chicago. il.. Septmer 2, 1976). Mengel. Marc. 'bciletons, Fluctuations, and tile Hopt

10 A036220SliarcatioO0,0 43 pp., Jun 1976. 00 AWO 537'P1ortions of tis work owe coiete at te inetitul, of

PP 215 Applied )btaetiag and Statistics, University of WitlshCoils, bsseli C., 'Cmnmto on: Principles of Information Coluabia, Vancouaver, Canada.Reotrieval by Manfred Kodaen' 10 pp., Mar 78 (Published as aLette to tile Editor, Journal of Domomtation, Vol. 31. PP 226No. 4, paes. 290-301). Dcser 1979), AD A054 428 Ralston. J. M. ad J. V. gan.- -TMetare ad Carreat

Dependence of Deradation In Pad-Edtting GaP LEas,* 34 pp..PP 216 Jun 1971 (Publishaed In Journal of Appliled Physics, 50. 3630,

Coll% Russell C., 'Lofteas Frequency Distribution of Nay 1979) AMI 5OScientific Productivity,* 16 pp., Feb 1973 (Plished In tile 'Del I Telephocne Laoratories, Inc.Journal of tile Amrican Societ for Inforeeatlon Science,Vol. 3 N&. 6, pp. 366-370, Novowbr 1977). AD A04 425 PP 227

Mengel, Marc. 'Uniform Treatst of Fluctuations at Criticalpp 217 Points," 50 pp., May 1978, AD AW 539

Coile, ftaaeei C.. 111ibiloonIic Studies of ScientificProductivity.* 17 pp., lbr 73 (Presented at tile Annual PP 226meting of 11am Alricon Society for Informatlin Science held Mongoi. Marc, "Relegation at Critical Points; DeterinisticIn Sen Francisco California, October 1976), AD A04 442 and Stochastic Theory,' 54 pp.. Jun 1970, AD A098 540

pp 216 - Caeif led PP 229Nanglf Marc. 'Diffusion Theory of Reaction Rlates, It

pp 219 Formulation and Elnstin-Soladacusfi Approaimtion.*A!nager, R. LeVar, Vbrke Analysis witha Rational Expec 90 pp., Jan 1970.,A MI 541tations: Thaca and Estition.1 60 pp., Apr 7.0, A05W4 422

pp 230PP 220 Mangel, more. "Diffusion Theory GE Reaction Rates, 11

Nearer, Donald 16, IDiagonalization by a7u Mtrice,' Ornatein-hihlenbedi Approitiloton,' 34 pp., Feb 1973,2S pp.,W 6.1 A09I4 443 AD AD" 542

pp 221 pp 231veianmd, NobertI 0. 91uperpoeer Naval Dilomaecy In the IlIson, Deand P., Jr., wRevel Projection Forms: The CaseOctebers 1973 Ara-srooi War. 76 pp., Jun 1973 CMablsaed fcr a Rasponsive OF.' Aug 1970. MI A034 543In Usoooer Is til Mditerraean: Political utility adil"r Cootreiatsr The lesilinv Papers Ofs. S1. flevarly PP 232

Mill1s end Landens aW Pubictiene, 1919) 00 AM 54 Jecebeon, Louis, wDen policy Changee So Made Aceptable toLabor?' Aig 1970 (Subvitted for paliicetion In oniaftrlet

and Labor Relations Reaviw), M I MII 2

AMS Preoeaal Papas Vit a 10 w~e may be obtained iron the Notlontal Technical informations Service, U.16 Department ofCoesve dypiaill. Virginia 22151. Oftaer papes are available iron file Mebimnt Informilo Pu., Denter for NaMa

Analyses,3N Ner ooevled Steet, Aoeadriae Wirgials M2il. An ladmKat oSelected Pahlllcetlone Is elso syllable onieqaaeet. Tiles Wnho ielefte a Listing of Proeseional Popersa vith distracts Issued fron 1996 to June 1901.

PP 233 PIR 249Jacobson. Louis. *An Aternative EVlanstion of the Cycical Glasser, Kenneth S., "A Secretary Probiem with a ftenoPattern of quite," 23 pp.. Sep 1978 Number at Oholces,0 23 pp.. star 1979

pp 234 - ev Ised pp 250Jonro. Jne end Levy, PobeErt A., Moss Federal Eigoadi- Mangel. Marc. %bdol ing Fluctuations In Macroscopic *a-tura Displace State end Local Expandituret The Caae of tame." 26 pp..* Jun 1979Construct ion grant*." 25 pp., Oct 1979, AD A061 529

PP 251pp 235 Trost, PAjrt P.. *"The Estimation end Interpretartion of

Mizrohl, Maurice N., *The Semiclassical Expansion of the Several Selectivity Ibdeis,* 37 pp., Jun 1979, AD A070 941Anhwenl-OsiiitarPropagietor," 41 pp.. Oct 1973 (Pub-

lished In Journal at Mathematical Physics 20 (1979) pp. 844 PP 292885S), ADl AO1I 536 Nujnn. Walter R., -Position Finding with Prior Knolodq* of

Coveriance Poraestors,* 5 pp., Jun 1979 1PublIshed In IEEEPP 237 Transect ions on Aerospace & Electronic Sstm, Vol. AES-1S,

Nowrr Donald. "A Matrix Criterion tor Nourmal Intagral No. 3, Nor 1979Base." 10 pp., Jan 1979 tPublished In the Illinois Journalof Mathematics, Vol. 22 (1970), pp. 672-681 PP 253

Glaser. Kenneth S., 'The d-Coiace Secatery Pieso.PP 236 32 pp., Jun %979, 00 A075 225

Otgot?, Kathleen Classes, %reemloyent Insurance and TheEployet Rate," 20 pp., Oct 1976 (Presented at the Con- PP 254

kfarance on EnoIc indicators and Performance: The Current Nongol, Marc end goanbedi. Devid B., flntevatlon of aOl loo Facing government and Business Leaders, presented by Bivariate Normal Over an Offset Circle," 14 pp6, Jun 1979,Indiana University Graduat School of Business). AD A061 827 AD AD96 471

pp 239PP 255 - Classified, AD 8D51 441LTrost, R. P. end Warner, J. T., *The Effects of MilitaryOccpational Training an Civilian Earnings- Ail Incom PP 256Selectivity Approach.* 38 pp., Nov 1979k, AD A077 631 Mourer, Donald E., "Using Personnel Olstrioutlon Wbdels,"

PP 24027 pp., Feb 1980, AD A062 218

Powers. Bruce, "Gloals of the Orntor for Navel Antalyses," PP 2571) pp., Dec 1976, AD A063 759 Thaler, A., "Discounting end Fiscal COnstraint,: Why Ois-

counting Is Always Right.- 10 pp., A" 1979, AD A075 224PP 241

Magei, Marc, "Fluctuations at Chemical Instabilities,' PP 2824 pp., Doe 1978 (Published In Journal of Chemical Physics, Mengel, Marc S. end Thomas, Jons A., Jr., "AnelyticalVol. 69, ft. 8, Oct 1S, 1918). AD AD03 787 Methads In Search The"r," 86 pp., Nov 1979, AD A0778032

PP 242 PP 289Simpson. Whilia R., 'The Analysis of Dynenically Inter- Glass, David V.. Hw, Ih-Ching; Nunn, Wolter At.. end Paris,active Systems (Air Cmat by the Numbers)." 160 pp., David A., "A Class of Coinatative Morliow Mtrices, 17 pp.Dec i979, AD A063 76 Nov 1979, AD AD77 833

PP 243 PP 260StPeon, W1 Ilen R., *A Arbillstic Formlation of Murphy Mongel. Morc S. end O~., Davis K.. etection flat andDynics as Applied to the Anaiysis wt Operational Resarch Sinap Width In Visual Search." 14 pp., Nov 1979. AD A077 634Probiefs.1 18 pp., Dec 1978, AD A063 761

PP 261PP 244 Vila, Carlos L.; Zvljoc. David J. and flaes, John. "French-

Shamn, Allan and Abs Itz, Stanley A., 'Maintenance Costs Condon Theory of Chemical Dnmca. V1. Angular Oitrlbi'of omplex Euipmnt," 20 pp., De 1978 (Published By The tions of iReaction Products," 14 pp.. lov 1979 (ReprintdAnerican Society of Navel Engineers, Navel Engineers from Journal Checal Phys. 70(121, 1S Jun 1979),Journal. Vol. 91, No. 6, Dec 1979) AD A071 473 AD /076 287

pp 248 PP 262Sipom, WIllIsm R., "he Acceiscmm ae Methods of Obtaining Ptersen, Charies C., "Third World Mltary gilt"e In SovietAircraft Performance from Flight Test Daea (Dynamic Par- Perspective.a %0 pp., Nov 1979, AD A077 639formene Teeting)," 403 pp., Jun 1979, AD A075 226

PP 263PP 246 Roblnson, Kathy I., 'Using Denamrciel Taigirs end container-

Brcialing, Frees,, "Layef s and Uniarent Insurance," 35 ships tar Navy Underway Replenlshmnt, 2 pp., low 1979.pp., Feb 1979 (Presented at the Nber Contarenc on 'Low AD A077 836Incm Labor Parkets," Uvlcago, Jun 1978), AID A096 629

pp 216Tmas, Jane A., Jr., "fT Tranegert PfA"rtIms of Diloftegaes in Applied Fields,- 110 pp., Mar 1979, AD /096 464

pp 264 PP 277Wlftiaad Robert 6. "The U.S. Navy In the Pacific Past, Wangal, Mmrc. 01mli Fluctuations In Systems *Ith MultipleMrasait. and 9Ollpaas of the Nue.* 31 pp., Nov 1979 Limit C~clos, 19 pp., War 1960 (Pubisahed io SIAN4 J. Appi.(Delivered at the International Symposium on the se,. Math., Vol. 36, No. 1. Fab 196W AD A006 229sponsored by the International hnstituta for ratagic

Studies$ The Grockngs Institution and tha YooiurI 3hleban, pp 279iiTcyo, 16-20 Oct 1976) AD 1066 837 NiwaohI, aricb *A Targating Problm3 Exact vs. Empocted.Valua Aproaches." 23 pp., for 1960. AD M9U 096

PP 269%mWinland, Robert r.. 9fr and Pas"e In the Nortw ca PP 279political Implications of the Changing Military UtuatilM In ltt Stephen N., "Causal inferenoas and the Useat Foram ANort1hern birope." 1S pp.. Nov 1979 (Prepared for 0,itlque of 0bre Without War," 90 pp., Way 190o

presentetion to the (bnferenca of the ibrdlc blenca In AD ADIS 097perspective; The Changing ilitary and PoliticalStuation.- Ointer for trategic and Intersetlonal Sfwlies, pp 260Georgotm University, Jon 15-16. 1970 AD AD77 636 Goleuarg, Laurence. Eatlsetlon of the Effects of A Ship's

Steselfg on the AmIlure Noe of Its liulpmets I A pplicePP 266 tlon of Econmtric Analyal *.1 23 pp., Apr 1960, AD AM6 09

Utgof I. Kathy Classes, and Greclilg, Frak.L "Taxes endInflation.- 25 pp., Nov 1979, AD 1061 194 PP 261

Iaahi, Warlca N.. "Consent on -Di scfetizatlen Probisd ofPP 267 Functional Integral* In these licol,* 2 pp., Way 12410,

Trost. Robert P.. and Vogel. Robert C., "The Response of publlshed In "Phyulcal Review D", Vol. 22 (196W.*at* government Raceipts to Economic Fluctuations and the AD 1094 9094Allocation of Gounter-Cyclical Revenue Soawing Grants,"12 pp., Doc 1979 111sr inted from the flviwls of Econmidcs and PP 263Statistics, Vol. LXI, No. 3. August 197% Diseuhas, lbedford. 'Exeacted Oand for the U.S. Wavy to

Serve as An Instrument of U.S. fbeige Policy: Thlalsbgpp 266 About Pal itical and MilI Itay Environmental Factors.* 30 pp.,

Thaws*% Jews S.,* Isaport Dependence and inter-State Apr 1960, AD ADS 099(Imperstionm lbe Ca of S-1haran Nrlca," 141 pp.,Jan 1910, AD A0111 193 PP 264

J. Kai lsos9 W. Nunn. and U. Sobse, "The Laguerre Trees-PP 269 form.- 119 pp., Way 1960. AD MDS 100

Weiss. Kenneth G., "The Soviet Involvement In the Ogaden Meh Greduate school of Wanegoment, University of RochesterWar." 42 pp, Jan 1960 (Presented at the Soithern (bonfarence end the (bnter for Nevel nalyseson Slavic Studies In October, 1973, AD A082 219 gre aduate Schooi of Waonant, University of Rochester

PP 270 PP 263Reahk Richard, "Soviet Policy In the Horn of Africa The Reanek, Richard S.. "Superpower Security interests In theUcision to Intervene.* 32 pp., Jan 19110 (To be piblashed In Indian Omean Ams." 26 pp., Jun 1960, AD ADS? 113"The Soviet Union In the Third World: Success or Faluvre."ad. by Robert H. onaidson, Wstvywlresss, Bolder. Co., PP 286Summesr 1900. AD A06l 195 Miahi, Wurice N., "On the USI Approuiaatlon to the

Propagator for AbItrary Hamiitoniae, 25 pp. Avg 19610pp 2?; (Published In Journal Of Wth. Phys., 2211) Jan 1991),

McConnellI, Jewes# "Soviet and American Strategic Doctrines AD 1091 307* One Obra Time," 43 pp., Jan 1960, AD /08 192

PP 267PP 272 Cape, Oewls. -Limit Cycle Solutions of Reactlonolftfusion

Wles, Kenneth 6., "The Atores in Dipioncy and Strategy. Sluatlons," 35 pp., Jun 1960. AD M? 1141940-1949, 46 pp., Ibr 1960, AD MO6 094

PP 266pp 273 Dolmn. Wetlter, "Don't Let Your s1Ides FIlp You: A Painless

Wakeds, Michael Koo "Labo Suppiy of Wives with Iisbans (bids to Visuals That Roal ly Aid." 28 pp., Oct 1960.Employed Either FullI Tine or Part Tin*," 39 pp., Msr 19110, AD A092 732AD A082 220

PP 269PP 274 Robinson, Jack, *" eute Classification Guldenos - A

Wnoo, Waiter R., "A Result In the Theory of Spirai Search," Solution and a Frobins," 7 pp., Aug 1960, AD 091 2129 pp., War 1960

pp 290pp 273 Wts% Gregory Mi.. alveiuetiont of Dag~te Soft si an

Goiebrg, Laurence. "leriters Advertising and Wavy Eni st- Wational Favromet, 17 pp., Augi IO. AD MI 213ments," 34 pp., Mir 1960, AD 1052 221

top291PP 276 Waddele, 9. S.0 and Tost. R. P., *Iwme Extension of the

Gold*arlb Larnce. "Delaying an Overhaul and Ship's Equip- Sowso Waess 5451," 17 pp., Oct 19610, AD 1091 946smat," 40 pp., Way 1960, AD =9 09 9"Nereity af Florida

-3-

PP 292 PP 305Thaws. Jewes A.. Jr.. 'The Transport Propertiee s of neay *ann LeOWa M.. 'Am Introduction to Mhe LI teratua'e of Seareb

Me ixtures IN Applied Mane Fltds,' 10 pp., Sept 1900 Theory" $2 pp., Jun 1931(Published In Journal of Chemical Physics 72110).is play 1960 P 0

Anger, Thorn. C. "Iat good Pro lare.. Obols?' 7 pp.,pp 293 Noy 2961

Thaws, Jews A.. Jr.. 'lEvelustion of Kintetic Theory Cal I I P 0sion SntuaI* thing the bnealiid Phae. Sutif Aproch, PP3022 pp.. Sept 1960 (Printed In Journal of Chemical Physics Thaftt* Jaw.. 'bd916000% Risk, and VInArubllty,'

72110). 13 Vey 19e0 43 pp., hAn 1961

PP 294 pp 306Poberts, Stephen S., Wrench Novel Pblicy Qatnlde of Mashl N.M.. 60rrspondence hale, ead Path Integral,'Grap." 30 pp., Sept 1960 (Presented at the Coofereace of Jul 1201. P6llehed In 'iavo Clwnto 6-, Vol. 61 (1981)the Section on Ill Itaai Studlea, International *vtesioAssociationa Klamah Islead, S.C.), AD 1091 306 pp309

Velnlen4 Robert 6.. 'An (The?) Explantion of the Sovietpp 295 Invasion of Aftnlstan.1 44 pp., Play 1901

Roberts. Stephen S.. "A Indicator of Infaml OpIrsPattern* of UWS. Nevy Oulsing on Overeas Rations. 109- PP 5101897,' 40 pp..* Sept 19100 (Presented at Fourth Novel MOM9~ Stenford, Jenette H. and Tel Ta *A.* "A Predictive Nothod

Symoeiu, US Newel Academy. 26 October 1979. AD 0091 316 for Determining possilie 11ree-dismnsonal hldings ofImmnogtlbal In Ilackbones Around Autlbop Combining Sites.*

PP 296 29 pp., Jun 1911 timllsed to J. theor. Nol. (1981) Be,DimAnues, Wedford end Potersefl 0,aorle C.. 4brltha 421-430hactas Altectno Searlen Secrity.- lthares lbritlaf Wae 'Wthensternt University. Evenstoo, IL

MAfcten Ls Seaarided lbelci 14 pp., Oct 2960. AD A092 733 p 1

PP 297 - Cassifiled games. Morienn% Grochlng, Fpso P. R.. end Utgotf,Vathleen P. Olessen, I valuation of U1 Foad.* 13 pp.,

pp 296 Nay 1911 (Published In National Omaslon on UnemploymentNIlaehi.o Nour ice 01,0 *A Morkov Approach to Large Misle Compensation's Unemployennt Coqeneeloas Studies endItacks.' 31 pp.. Jon 1961. AD AM.139 Reseach. Volume 2, July 191101

pp 299 PP 312loraoa, Jies 14. end Levy, flolert A., 'Aege Leadership In Jandra, Jies; bues, Nerlenne end Levy, Robert. 'TheSenttention, 19 pp., Jan 1981, AD) 194 797 Optne Ieed Limit.* 23 pp., May 1901

PP 300 PP 313Jodr, Je end Scheldt. Per,' *Ca the Estleetlon of Roberts, Stephen S., 'The U.S. Ma" In the I10.. 36 pp.,lohalcal Inefficiency In the Stochastic Wontler Production Jul 1911Function Modl.' "2I pp.. Jan 1961. AD A096 239elichige ate 8 h0UlensltY PP 114

Jahn, Orlstquharl Norwiltz, *stay A. end Lockaen,PP 301 hbart F., wIxadnlng the Draft Dete, 20 pp., Jul 1991

Jondal, Jews N.; Le"y, lobart A. end Itaghes, Clalro,'bodwnlcel Chage ad Eplomnt In Mteel. Atos, Aluadave. PP 315end Iron Or*, 17 pp, Har 1961, AD AD" 394 bauck, Ralph V., Capt., nLa Cotastraphe by eny olher

nie.'4 pp., Jul 2982pp 302

Jondrt. So 14. and Levy, Robert A., 'Th Etfect of PP 316Iqports on Eployment lier Motional Euectation,' t9 pp., flaberte, Stephen S., NWestern Giaoee end NIATO Hawles,Apr IM6, AD A09" 392 1960,' 20 pp., Aug 1961

PP 303 PP 317Thoasob Jaws, 'Te flirest Coeeodity In the Coeing fbu-te, Stephen S., 'Superpower Newel 01.1. lbnagement InResource Were.- 3 pp., Aug 1961 IfPullshed In the liehingtoo the Mdterraneen,' 35 pp., Aug 1991Star, April 13. 1962)

pp M18PP 304 Ve114 14 Ian PL. Yuagolsvloaend the Soviet Policy of Force

Duffy, Micheal K., Greenwood, Michael J.0 and Mcowl 1 John In the Md~terraneen Sauce 1961,1 287 pp.. Aug 91N4.,** 'A 0,oss-owtionel Noal of Aumel IntereloneiMigrationu and Employment 'mtluh lltertemoa'. Evidence of PP 319Structural diange, 1966-1971,' 31 pp., Apr 1961, AD 099 393 Smith. Micheel W., 'Antlelr Waritere Defenee of Voipe atQunlversity of Colorado sbe,' 46 pp., top toll2 thits talk was &Il Fared at the NMIl"A~lan.t Rte t ulrsity Vrfere System and uTh"Ohlogy abnaes of the American

Instituate of Aeronautics end Atlvemtls Is Mashlepos onDecmber 22. 19601 In beston on Jernaury M0 1"11 end In LosAstgp1es. o Ane Il. 1961.)

-4-

PIR 320 pp 339Trost. R.P.. LurIo, Philip and Bargr. Edsar4 -A Mate *a Wilson. Owmond P., The Persan~ (611 ad tI.. *tionalSbtlinting Qltiftuous Time bICISiGo MMdolS." i pp., hntereast, 1i pp., Fab 1962tap 1961

pp 321pp 340PP 321 LarIs Philip. Trost, R.,. and ftrga. Edwar4 -A Methiod

Duffy. Michael X. and Ladmn, Jerry A*,O -The Siutamme~ for thelyslng *Z.ltlpioei 91 urtion Date.* 34 pp.. Nab11terdnatlon of incons ad Emplovaht in itilted V~se- 1IlaMexico Sordar Reogion Econonles, 34 pp., Sep 1961*A&$oclot* frofessor of womemics. hSAll *ate Iimirty. pp 341

Tome. AZ. Trost. Robert P. mod Vogi, fNobert C., m~fedictlam with

pp 322 Stvdle%. 6 pp., Feb IM6

Werner, John T., -issues io NBuy Manpower Research andFbolcyt An raononlat's Airspetlve," 64 pp., Doc 1981 PP 342

1. Lang-Fahi. bddaIq, M.., and Trost. M.P.. -Testing forPP 323 tructiref &&W by D-bthod Is Sultdcl ng Slasitaeos

&aMe. R~dWidt M-. 0aratlon of Correlastad Log-No... Equations Models., 5 Pp..* Feb 192Uquences for- the Simlatlon of Clutter Edmase, 33 pp.. Doc1961 PP 343

Golueil Not-Im S.. Wrojoctlan. ths Havy 1.1 1sted ForcePP 324 Level.- 9 pp., Fab 1662

Norcoitz. ltaiW A., wguantfivlng 3paposer lbadinas.6 pp., bc 1661 (Poblisloed In gaensoe Phnalemnt Journal,Vol. 1S. No. 2)

PP 327Msmmo (biq, apt.. USN and Ora". David ft., Dr,"Ntmstlo and Ahseul of Navy S9lpballding AaogramOfisruption Coats.. 12 pp.. Nor 1962

PP 328Wslnfand. ftbert 0.. -Mrtharn Woearsi Their 1tragicSignificance.- 21 pp.. be 1960

PP 330Lodumnn Febat F., *Ailternat ive lipproachaa to Atr itioneMimagarnnt,- 30 pp.. Jan 10112

PP 333Lee. Ling-Fl ad Tr-ost. flobort P., "Estimation of IlawLimited Ospandent Variable WdeSi with /Oplication toHlousing Deand,- 26 pp., Jan 11012. PubbIished In Jounai of

oomtrica S (1976) 357-362.

PP 334Kenny. LawroenV., Lao, Lung-Fel. Maddol% (6S.. and TrostR.P.. *Fbturns to *biegs Education: An lnvestliftioa ofSoft-fealction alas flamed on the Project Talent Dat.* 15pp.. Jan 1962. Peiiiseod In International 6,ooIc lnovies.Voi. 20, 16% 3, October 1979.

PP 335ILo. Luang-Pl. O.S. HOddla6 and M.P. mroat. *Asymptotic02vrlance Obtrices of Tu-Wago "oit and Tem-Weag. bitMasthoda for Sl m I tena.s Equations Mlodels v Ith SelIect Iv Ity.13 pp., Jan 1962. Peil1shed In lbonoatrica, Vol. 46, Mo. 2

pp 336O'Neilli. hmas, 00"ility Fuels for t"a buvy.' 13 pp.,Jan 1982. "foped for poblcation In Novel Institutefocoodina.

PP 337Wrner, John T. and Gomibar Mwttbw S., 'The influoncoof

MmnPh4%ary Feltorsaon Labor Sapply.' 23 pp.. bne 1961