AD-A106 193 CENTER FOR NAVAL ANALYSES ALEXANDRIA VA INST OF NAVAL--ETC F/6 5/10

ANOTE ON ESTIMATING CONTINUOUS TIME DECISION MODELS.(U) N

SEP 81 R P TROST. P LURIE, E BERGER

UNCLASSIFIlED CNA-PP-320I

,@, LEYVELPROFESSIONAL PAPER 320 / September 1981

A NOTE ON ESTIMATINGet CONTINUOUS TIME

DECISION MODELS

R. P. TrostPhilip LurieEdward Berger DTIC

S ELECTEOCT 2 7 1981

B

SDISTRIBUTION STATEMENT A

Appzoved fat public zelaq IDistribution Un, , .i4

CENTER FOR NAVAL ANALYSES

N\ 8 110 22

/ ,i PROFESSIONAL PAPE5 k20 September 1981

/ - NOTE ONSTIMATINGCONTINUOUSJIME

VECISION MODELS

R. P~lrositPhilipfkurie

I Edward/Berger

/ I

a Institute of Naval Studies

CENTER FOR NAVAL ANALYSES

2000 Nprth BeAuregard Street, Alexandria, Virginia 22311

• I,.

TABLE OF CONTENTS

Page

I ntroduction....................................... ............. 1

A source of bias: censored observatins ........................ 2

A method for estimating continuous time models when thedata are censored ............................................... 4

Two empirical examples with censored observations ...............5

Duration of unemployment ....................................6

Childspacing equation ...................................... 9

Conclusions............................. o....................... 14

Refarences .......................... ............. .............. 15

Acces~on Fo

A NOTE ON ESTIMATING CONTINUOUS

TIME DECISION MODELS

by

R. P. Tiost, Philip Lurie and Edward Berger

1. INTRODUCTION

Continuous time decision models have a long history in economics. In

fact, the early work of Mincer [19621 implies that a woman's labor force

participation decision is made with a continuous time horizon in mind.

More recently, Mincer and Polacheck [1974 and 1978], Mincer and Ofek [1980),

and Sandell and Shapiro [1978] analyze the length of time women remain out of

the labor force. Sandell [1977] examines the determinants of the number of

years women work after the birth of their first child, and in a stock adjustment

model of fertifility decisions, Hyman [1980] examines the length of time

between births. There are also several studies that seem suited to continuous

time analysis but are estimated with discrete time choice models. For example,

Shapiro and Mett [19791 estimate a discrete time choice model of women's

post child labor force participation rates. One could also analyze post

child labor force participation, as Mincer and Polacheck 11974] do, by looking

at a woman's interval of nonlahor force participation following the birth of a

child.

All of the above studies choose (or imply) as the dependent variable of

interst some measure of continuous time, whether it be in percentage terms

or actual years. In some of these papers (e.g., Mincer and Polacheck [1974]

and Hyman [19801) this dependent variable is studied by looking at its mean

We would like to thank Robert Dorfman, Jacob Mincer and Solomon Polacheck

for comments on earlier versions of this paper. Any remaining errors areour own responsibility.

value (actual or predicted) in various age and education subgroups. In

other studies (e.g., Mincer and Ofek [1980] and Shapiro and Mott [19791)

the determinants of the dependent variable are estimated with regression

analysis. While the estimation of continuous time models is a step in the

right direction (away from discrete time choice models), the purpose of our

note is to point out that the empirical results in these otherwise excellent

theoretical papers may be biased. The source of this bias lies in the way

censored observations are handled in the empirical analysis. The rest of

the paper is divided into the following sections. In Section Two we discuss

a potential source of bias in previous studies. In Section Three we propose

an alternative method for estimating time decision models and in Section

Four, we consider two different applications of this method. Section Five

contains the conclusions.

II. A SOURCE OF BIAS: CENSORED OBSERVATIONS

All the studies cited in the introduction derive and estimate life

cycle decision models. One variable of interest in these papers is some sort

of "length of time" variable. For example, Mincer and Polacheck [19741

study, among other things, the length of time women remain out of the labor

force following the birth of the first child. Mincer and Ofek [1980] estimate

the determinants of a woman's duration of unemployment. Hyman (19801

estimates a stock adjustment fertility model where one variable of interest

is the length of time between children (i.e., childspacing). The authors'

IAlthough Hyman's theoretical interest include the decision regarding thedesired spacing between children, his actual empirical analysis only looksat the proportion of respondents who had one child during the followingyear, given that they desired to have one more child. However, child spacingin the usual meaning of the word is implied in his paper and is worthinvestigating.

-2-

justification for studying these dei.ondent variables is well founded in

economic theory, and we have no quarrel with their theoretical models. Our

concern is with the empirical section and in particular with the manner in

which censored observations are handled in the analysis. A simple example

will demonstrate our point.

Suppose we want to estimate the average length of time women wait

before returning to work following the birth of their first child. For

simplicity, assume we have a sample of ten women who all gave birth on the

same day. Let the observations y. be:

1, 1, 2, 2, 3, 3, 4, 4, 5, 5

where y is the number of years elapsed before the women return to work.

If we analyze these data five years after the children are born, we can simply

average the ten observations yi to get an unbiased estimate of the mean.

Suppose, as is often the case with panel data, we do not have "completed"

values of y. For all the observations. This would be the case, for example,

if we have data for only the first three years following the birth of these

children. In this case we -ll have six observations where the length of time

is observed (1,1,2,2,3,3) and four where the length of time is censored

(tht, values 4, 4, 5, 5 whichof course are unobserved) at 3 years. Here the

term censored means that all we know is these four women wait at least three

years before returning to work. If we simply aveirage the six uncensored

observations, we will underestimate the mean of Yi" Even if we include the

censored observations and enter a value of "3" for them, we will still

underostimate the mean value of v.. These same problems exist if we wish

to use regression arialysis to measure the impact of an exogenous variahly X.

t -3-

on Yt, i.e., we will underestimate E(YilXi). Despite these obvious biases, it

is surprising that none of the previously cited papers discuss (or take

account of) this "censored data" problem. As we show in the next section,

there is a simple method for handling this problem, that to our knowledge has

never been applied to economic data.

11r. A METHOD FOR ESTIMATING CONTINUOUS TIME MODELS WHEN THE DATA ARECENSORED

The estimation of continuous time models have a long history in the

biostatisticat literature. They have been used extensively in the biomedical

sciences in the general area of patient survival. Here the problem has been

to estimate the probability that a patient survives beyond time T. A plot of

these probabilities as a function of time, i.e.,

S(t) = P(T > t), (1)

is called a "survival curve."

Kaplan and Meir 11958] were the first to derive a nonparametric maximum

likelihood estimate of the true survival function in the presence of

censoring. However, the Kaplan-Meir method only estimates "unadjusted"

survival curves. That is, the survival curves ire not adjusted for exogenous

characteristics. It was not until 1972 that a nonparametrtc method became

available for handling censored observations while adjusting for factors

(i.e., exogenous variables) which may affect the probability of survival.

Thi method was first proposed by Cox [1972] and is known as the Cox

Regression model. The Cox model expresses a hazard function 2 as

2The hazard function h(t), is defined as the conditional probability of afailure in the !nterval (t,t+dt), given survival to time t. That is,

h(t)dt - P(t<T<t+dtlT>t).

-4-

h Ct) h (t)e

where Z is a vector of exogenous variables, is a vector of unkno.,n

coefficients and h (t) il; assurmed fixed and independent of Z, but otherwise0

completely unspecified. Note that h (t) corresponds to the hazard function

for the situation when Z = 0. The survival function S 0(t) refers to the

case where the exogenous variables Z = 0 and is expressed as

So(t)= e 1h ( X) d X

Since the model assumes proportional hazards i.e., the hazard ratio for

any two values\0f Z is independent of time, then

8'Z1

S1 (t) = (So(t))e'

where S1 (t) is the survival curve for an individual with exogenous variables

Z I' Cox 11972] shows how to estimate the vector S and the function h0 (t)

with a maximum likelihood approach.

In the next section we demonstrate the usefulness of the Cox Regression

technique by estimating an unemployment duration equation and a childspacing

ecs,,at ion.

IV. TWO ENPIRICAL E.AMPIES WITH CENSORED OBSERVATIONS

In this section we demonstrate the feasibility of the Cox model with

two empirical examples. The first application estimates an equation where the

dependent variable is a woman's duration of unemployment following the bicth ot

her first child. The second application concerns the length of time a family

waits before having their first child.

-5.

A. Duration of Unemployment

As we noted earlier, many have studied the labor force participation

decisions of women. Some, like Nelson [1977) and Heckman and Willis [1977]

study it in the context of discrete time, while others like Mincer and

Ofek (19801 analyze it in continuous time. In the present paper we show

how the Cox regression model can be used to estimate the probability that a

woman returns to work (following the birth of her first child) after one

year, two years, three years, etc. This method takes account of the censored

observations in the sample and will yield an unbiased estimate of the

mean duration. We use the 1973 wave of the Parnes NLS data on young women

to estimate the unemployment duration equation. Table 1 gives mean values

for the six exogenous variables and the mean value of unemployment duration

(the dependent variable). Note that by ignoring the fact that 79 of the

duration observations are censored, one would underestimate the mean duration

of unemployment for these women by 16 percent. A consistent , estimate of mean

duration is 3.25 and is easily calculated from E(T) = i S(t)dt.0

Table 2 gives the maximum likelihood estfmates for the Cox model. To

see how these coefficients are interpreted, recall that the Cox model expresses

a "survival" curve as:

S(t) = (S0(t)) "xp(B'X) (2)

Consequently, a positive sign on the coefficient for exogenous variable Z

means that as Z is larger, the women will decide to go back to work sooner.

A negative sign means that as Z is larger, the women will wait lonjger beforc

returning to work.

-6-

TABLE 1

DEFINITION AND MEAN OF VARIABLES

Variable Mean Definition

Dependent variable 3.25* Defined in table 2

EDUC 11.95 Wife's education

HUSINC $9274.55 Husband's income

DSMSA .39 Dummy = 1 if in SMSA

DRACE .245 Dummy = 1 if nonwhite

UN EMP 7.7 Unemploy7nent rate of

county

Age 19.54 Age of women when childwas born

*This mean takes account of the censored values. It is calculated

as the area under the survival curve. If we calculate a simple mean ofthe 212 observations we get a value of 2.72. Hence, by ignoring the censoringproblem we underestimate the duration of unemployment by 16 percent.

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TABLE 2

COEFFICIENT ESTIMATES IN THE COX REGRESSION MODEL

(Dependent Variable = Length of time tOre-entering the labor force after having first child)

Standard 2,Variable Coefficient deviation 1

EDUC .1354 .0674 4.036

HUSINC -. 0000265 .0000206 1.654

DSMSA .1326 .1878 .499

DRACE .0586 .2092 .078

UNEMP -. 0127 .0155 .667

AGE -. 02195 .0465 .223

Log likelihood (all Betas = 0) =- 650.25Log likelihood (all MLE) = -647.09Number of observations = 212Number of censored values = 79

L2*Using a significance level of .05, any x1 value greater than 3.84

is considered significant.

-8-

For the coif f ic iouit -; in Ta'. : 2 then, * we e thnat ;i t, - wm:l

education Lncrcaqes, she will go :ack to work. soon.-r. As htr husband

income , tier age or the unemplo.yDnt rate increase howo;vver, she will wait

longer before returning to work. The rnly signffcafnt cotfficient at tue

.05 level is the education variable.

To see what effect education has on the time at -..hich a wnmaok returns

to the labor force, recall that "survival" in our model means a woman did not

go back to work by time t. Coiumn 2 of Table 3 gives the survi,'aL prebifit

for exogenous variables Z1 = (1) IUDUC = 12, (2) HKSING = $9200, (3) -s:sA 0,

(4) DRACE = 0, (5) UNE II 77, and, (6) ACE = 19.5. Column 3 of Table 3

gives the survival probabilities for exogenous variables Z2 where 7", is

the same as 7 I except EDUC 16 rather than 12. Table 3 tells us that the

probability of a woean with characteristics? of not going back to worV

within 2 years is .63. For a woman with characteristics 7 2' this prcihilit,

is .5

B. ,Thi ldspacrn; Fiuation

A second exarple that demonet -ate; the usefulnoss of the Cox t.caicue

in thie estiintin of economic decision equations ia found in the' work of

Hvi:-in [ P80j . One variable of interest. in Hvman's paper is childspacing,

whre 'i, doffins "childspacin" as the proportion: of- respondent a who P ow ,Iloc

child during the following year, giv.n that they desired to hare one no child.

in our papor w, rief ine ch i Isp cing as the length of ti:ie between children ind

estimate an equation where the dependent variable is the length oft tnc wv -u,

wait before having their first child. We again uiie the 1973 wave of the

Parne , NLS data on young women. The dependent variable will be censored tor

tho.e couples who did not have their first child by 1973. To handle this

-9-

TABLE 3

SURVIVAL* PROBABILITIES EVALUATED AT Z AND Z2

Time Survival at-ZI Survival at z2

0 1.00 1.00

1 .75 .61

2 .63 .45

3 .51 .32

4 .44 .24

5 .38 .19

6 .34 .16

7 .29 .12

-I: (1) EDUC = 12, (2) HUSINC= $9200, (3) DSMSA 0,(4) DRACE 0, (5) UNEMP 77, and, (6) AGE 19.5

Z2 : Same as Z1 , except EDUC = 16.

*Surviving to time t is defined as not returning to work by (i.e.,to and including) time t.

-10-

cen;,ig problem, the Co:.: rt-gres-ion model is a natural choice of estimation

techniques and i; the- one we use. Table 4 gives the maximum likelihood

est imates.

For the coefficients in Table 4 we see that as the wife's age, educatir.,

or IQ increases, or as husbands income goes up, the couple will wait longer

before having the first child. Also, Table 1 tells us that couples who live

in rural regions will wait longer than similar couples who live in urban

areas before having the first child. However, the only significant

coefficients are the education and income coefficients.

For this application "survival" means the family did not have a baby

by any given time t. Column 2 of Table 5 gives the "survival" probabilities

for exogenous variables Z1: (1) Age = 20, (2) Dummy Rural = 0, (3) IQ = 100,

(4) Educ. = 12, and, (5) Income = $5,000. Column 2 tells us that probability

of not have a child by two years is .425. A similar interpretation holds

for the rest of column 2 in Table 5.

Column 3 of Table 5 gives the "survival" probabilities fc exogenous

variables Z :Z,, is the same as Z1, except income = $10,000 rather than $5,000.2>1

Notice that the probabilities are uniformly (because of proportional hazard)

higher for Z This means that as income goes up (from $5,000 to $10,000),

couples wait longer before having the first child. The probability of not

having a child by any given year is greater for couples with husband's inccome

of $10,000 than it is for similar couples with husband's income of $5,000.

A similar interpretation holds for Column 4 of Table 5.

In this section we gave two examples of how the Cox model can be used

to estimate economic time decision models. Our purpose in this empirical

suction was not to re-do previous studies, but to demonstrate the applicabilitv

and feasibility of the Cox regression technique. It is also hoped that we make

others aware of the censored data problem in future analyses.

-It.-

Table 4

Coefficient Estimates in theCox Regression 1odel

(Dependent Variable = Number of Years BeforeHaving First Child)

Variable Coefficient Standard Deviation X2_*

Age at marriage -.000218 .0267 0.00

Dummy (=I if Rural) -.1072 .0873 1.51

IQ of Wife -.0014 .0031 .21

Education of Wife -.08899 .0308 8.37

Husband's Annual Income -.000064 .000015 17.73

Log Likelihood (all Betas = 0) = -3692.19

Log Likelihood (at MI.E) = -3671.59

Number of observations 681

Number of couples having a child 620

Number of censored values = 61

Since we only had annual data, our dependent variables took on discretevalues 1, 2, 3, etc. Of course, the Cox model can easily handle a continuousdeperd>nt variable.

** 2All X are with I degree of freedom. Using a significance level of .05,any X2 value greater than 3.84 is considered significant.

-12-

Table S

Survival Probabilities Evaluated at

7V z and Z3

Time Survival at Z Survival at Z2 Survival at Z3

0 1 1 1

1 .589 .681 .667

2 .425 .537 .519

3 .276 .393 .373

4 .194 .304 .285

5 .156 .260 .241

6 .119 .213 .196

7 .080 .160 .144

8 .073 .150 .135

9 .064 .136 .122

10 .044 .104 .091

11 .044 .104 .091

Z (1) Age = 20, (2) Dutmy Rural : 0, (3) IQ 100, (4) Education 12,

(5) Husband's Annual Income $5,000.

S2: Samae as Z' except husbands annual incom = $10,000.

Z3 : Same as Z 1 ' except Education - 15.

,*

"Surviving" to time t is defined ,n ; not having a bcabv by (i.e. , up to and

includin.) time t.

-13-

V. CONCLUSIONS

In this paper we pointbAd out a potential source of bias in the

estimation of continuous time decision equations. This bias will exist

whenever there are censored observations in the data and estimation techniques

such as least squares are used. To correct for this bias one has to use an

estimation technique, such as the Cox regression model, which takes censored

observations into account. We demonstratetthe usefulness of the Cox

model by estimating an unemployment duration equation and a childspacing

equation. We think that the Cox model performs adequately and yields

reasonable estimates.

-14-

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PP 243 PP 260Stepson, Willim Rt., "A Probabilistic Formulation of Murphy Mongol, Waee- S. aid Coe, Davis K., 8Deecton Arte endDynamics as Applied to te Analysis of Operational Research Sweep Width In Visual Search," 14 pp., Nov 1979, AD 4077 834Problems,- IS pp., Dec 1975, to A063 761

pp 281pp 244 tile, Carlos L., Zvljoc, David J. aid Ross, Jahn, *Fe-anck-

Shersen, Allen and Ifeowltz, Stanley A., "Ilatenenceo Costs Condon Theory of Chemical Dynamics. VI. Angular Distrlbu-of Cosplak Equipment.- 20 pp., Doc 1978 (Peublished By The tinse of Reaction Prodocts." 14 pp., Nov 1979 (ReprintedAmeican. Society of Navel Engineers, Naval Engineers fe-ce Journal Chemical Phys. 70012), iS Jun 1979),Journal, hol. 91, No. 6, Dec 1979) AD A071 473 AD 4076 287

PP 249 PP 262Slopson, William A., *The Acoirmasr Methods of Obtaining Petersen, Charles C., "Th ird Wrld Military Elites In SovietAircraft Pee-foremo. fe-om Filot Test Data (Dyneeic Per- Perspective," 90 pp.. Nov 1979, AD A077 835fmamnce Taiting),- 403 pp., Jun 1979, to A075 226

PP 263PP 246 Robinson. Kath I., "U~sing Corcial TeNkers and Container-

lmemoil 9,. fe-aoe, wleyoffs and lhomployent Insurance." 35 Ships for Wavy Underway Aspesishoont," 25 pp., Nov 1979,pp., Feb 1979 (Presenited at the Aber Conference on -Low AD A077 a36lncm Low kats," Chicago, Jun 1978), AD 4098 629

111 24Thmas, Jame A., Jr., *The Transport Properties of DilutGem In Applied Fields," 183 pp., Nor- 1978, 40 A096 464

-2-

PP24PP 27heU94~

Studies, Th. roig Institutionanth air hbu. WiSehnM.CoslIfecsadteUeofFc:ATdtyo, 16-20 Oct 1978) AD A066 837 Critique of Force Without Wr,' 50 pp., may 1IWO.

AD A660 097PP 265

Poiia Imlctoso the Chngn Military Situation In Goidberg. Lawrenc.Itlat fte fet fASi'Nlorthuern Eurp., IS1 pp.. Nov 1979 (Peae fo tmnon+eFttoReofIsEupw:Anplispresentation to the Conferencen of the Nri sac n fo fEocnrcAayi. - p.Ar16,A M0

Perspetive: The Chnging Military an PoiclSituation.' Conter S trategic and Interunational Studies, PP 281Georgetown 5.rstJu 916, 19781 AD A077 836 MIlreahl. Maurice M.6 m'omnt on Ol1scretlzetlon Problems of

PP 266 pbihd I Pyia ve - o. 2 19)Utgoif. Kathy Claime, and Brchi Ing. Fak tn"ad A 0 9lunflstlon,* 25 pp., Nov 1979, D A681 194

pp 26? Dlanuks. Bradford, "Expected Dmand for the U.S. favy toTrost. Robert P., and Vogel, Robert C., "The Response of Serve as An Instrumnt of U.S. Foreign Policy: ThindkingState Goverimunt Receipts to Econceic Fluctuations and the About Pot itcal and MIItIary Environmental Factors,- 30 pp.,Allocation of 0ounsr-CyclIcel Revenue Sharing Grents.' Apr 1960. AD A6665 09912 pp., Dec 1979 (Reprinted fro the Review of Economics andStatistics, Vol. LXI, he. 3, August 1979) pp 294

J. Kellamn,' W. Nunn, and U. Sualte.- 'The Laguerre Trans-pp 260 form,' 119 pp., Way 1980, AD 6605 100

Thomson. Jms S.. * Sepot Dependence and Istar-State -The Graduate School of Managent. University of RochesterCoopetion: The Case of Sub-Saluaren Africa.- 1d1 pp.. aid the Canter for Naval AnalysesJan 1980. AD 6681 193 -The Grauajte School of Mwnegsnet. University of Rochester

PP 269 pp 295Weiss, Kensnt" G., .me Soviet Invoiveeu In the Ogaden Reensk. Richard B.. 'Superpower Security Interests In +heWar,' 42 pp., Joen 1960 (Presented at the Southern Conference Inile Ocean Area.* 26 pp.,* Jun 1960. AD A681 113on Slavic Studies In October. 1979), AD A662 219

PP 296PP 270 Murhi. Rolurice Mf., 'On the WS Approximation to the

Ramd, Richard, 'Soviet Policy In the Korn of Africa: The Propagator for Aritrary Hmanltontans.' 25 pp.. Aug I900Decision +a Intervene,* 52 pp.. Jon 11060 (To be published In (Published In Journal of Math. Phys.. 221t) Jan 1981).MTh Soviet Union In the Third World: Success or Fat lure." AD A091 307

ed. by Robert H. Donaldson, Wstylen Press, Boulder. Co..Sumer 1990), AD 6681I 9 PP 281

Cope, Davis, 'Lieit Cche Solutions of Reaction-Olt fusionPP 271 Equuations.- 35 pp.. Jun 1980, AD 6681 114

McConnell, Jams, "Soviet and American Strategic Doctrines:One Moe Ties,' 43 pp.. Jan 1980, AD 6681 192 PP 28

Uoluun, Waiter. 9Oonlt Let Your Slides Flip You: A PainlessPP 272 Guide to Visuals That Really Aidi 28 pp.. Oct 1990.

Weiss, Kenneth 9., 'The Azores Is DIphonacy and Strategy. AD A092 7321940-1945. 46 pp.. War 1960, AD A065 094

PP 273pp 289PP 13RobI son, Jack. 'Adequate Classification ghuidance - A

Iluaa Michael K., -Laor Supply of Wives with Wtsbanto Solution and a Problem,' 1 pp., Aug 1960. AD A091 212CEployad Either Full Tim or Pert Tie."' 39 pp., Her 1980,AD AM82220 pp 29

Watson, Gregory H., 'Evaluation of Oceuter Softwere In auPP 275 Operational Enviroumit,' 17 pp., Aug 1990. AD AM9 213

Goderg, Lawrence, 'Recruiters Advertiling and Navy Enlist-mete,' 34 pp.. Mar I9S0, AD AM0 221 PP 291

Meddale, U. S.' and Troat. R. P., *'Som Emtenslons of thepp 276 Nerlos. Press Mbda,' I1 pp.. Oct 1960, AD A091 946

Godberg. Lawrence. 'Delaying an Overhaul and Ship's Equip- *UnIversity of Floridamet.' 40 pp., y 190, AD AM6 095

PP 292PP 277 Thses, Jame A., Jr., 'fTh Transport Properte @4m ofInary

NWel l"re, -11il Flwctuuat n is etos; vith Multiple Gas Mixtures In oppled Wageetc Fielde.' 10 pp., Sept 1900Lisdt Cyes,'- 19 pp. liar 19410 (Publiahed fin SIAM J. Appl. (Pulished Is Journal of Chmilcal Physics 72(101,Matti.. Vol. 38. Me. 1. Feob t961 At) AM6 229 i9 May 1980

PP 293 PP 307Theme. Jese A . Jr.. "Eneluetion of Kinetic Theo" Cot - Thomason, Jeas, -Dspndance. Risk, and tulnerab1ilty,"sloe Integrals Using the Gseeral lead Phase Shift Approach," 43 pp., Jun 199112 pp.. Sept 1919 (Printed In Journal of Choo.eeal Physics72(10). 15 May 1980 PP 304

Mlzrahl. MN.. -Corepondence Rles and Path Integrals,"pp 294 Jul 1981. Published In NIfu0v Cidnto 61, V'ol. 6? (1901)

Roberts, Stephen S., "French Navel Policy Otside ofEurope,- 30 pp., Sept 1960 (Pr'esented at the Conference of pp 309the Section on Mlitary Studies. International Studies Weland, Robert G., *An (Tho?l Explaeation of the SovietAesociation Klmush Island. S.C.1, AD A091 306 invasion of Afoonlstan,- 44 pp., MAY 1961

PP 293 PP 310Roberts, Stephen S.. *"An Indicator of Inforeal Epire: Stanford, Janette N. end Tel To Ni.,* "A Predictive MehttodPatterns of U.S. Navy Cruising on Oversees Stations, 1869- for Determining Posible Three-di(ons lonel folings of1897.w 40 pp.. Sept 1960 (Presented at Fourth Novel History Imnoglobulln Backhone* Around Antibody Combning SlteS."Sympos lum, US Novel Academy. 26 October 1979, AD A091 316 19 pp., Jun 1981 (Pub6lished In J. theor. Rlot. (1961) 11,

421-439PP 296 wworthsweetran iversity, Evanston, IL

Dimulkes. Bradford and Petersen, Cherles C., "arltleeFactors Affecting lherien Security,' iWactorss l'arltlos Ouo PP 31lAt ectan La Securlded Ibelca) 14 pp., Oct 1960, AC A092 133 Bowes. Marianas, Brecht 159, Frank. P. R., end Utg01t.

Kathleen P. Classen. "An Evaluation of Ut Funds.' 13 pp.,PP 291 - Classified Nat 1961 (Published In National Coalslon on Unemployment

Conpensatton's Uneploymnt Compeneetlon: Studies andPP 298 Research, toluee 2, July 1980)

Mlzrahl, Maurice M., "A Morhov Approach to Large ifsileAttacks," 31 pp.. Jan 0981. AD 409,159 PP 312

Jondrom, James; Bowes, Marianne and Levy, Robert, "Thepp 299 Optimu Speed Llait,n 23 pp.. Nay 198t

Jondro.. Jaes M. end Levy, Robeart A., "Wage Leadership InConstruction, 19 pp., Jan 1981. AD A094 791 pp 313

Roberts, Stephen S.. Men U.S. Navy Is the 1960,, 36 pp.,PP 300Jul 1981

Joedrow. Jaes end Sch~eldt, Poee," *On the Estletatlon ofTechnical Inefficiency In the Stochattl Frontier Production pp 3Function Moidel." It pp., Jan 1991, AC AD96 159 Buck, Ralph V., Capt.. "La Catastrophe by any other'Michigan State Uivlersity sane...," 4 pp.. Jul 1961

pp 301 pp M1

Jondrow, Jeaws At.; Levy, Robert A. and High**, Clalre, Roberts, Stephen S., "Wetern European end NATO Navies.-Technical Change end Employmnt In Steel, Putos, Alueinum. 1 980 ,n 20 pp., Aug 1981and Iron (re, 11 pp., Nor 1961. PD PAD9 394

pp 3fTPp 302 Roberts, Stephen S., *Sujperpower Noavel Crisis Manageeent in

Jondrow. Jeas If. end Levy, Robert A., "The Effect of the Modltorranean," 345 pp., Aug 1981Imprts on Employment Undev Rational Expectatlons," 19 pp.,Apr I9"I. ADP0AM 392 p318

tego, Mfila. N., "Yugoslavia end the Soviet Potlicy of Forcepp 303 In the Mediterranean Since 1961," 187 pp., Aug 1991

Thomison, James. "The Rarest Comodity In the CoingResource a's." 3 pp., Aug 1981 (Published In the lbshtngton pp M1Star, April 15, 1991 Smith, Michel Wf., "Antlair Warfare Defense of Ships et

See," 46 pp., Sep 1981 (Tis talk was delivered a+ the NavalPP 304 Warf are System and Technology Conference of the American

Duffy, Michel K.; Greewood, Michael J.4 end Mc~owell, John institut of' Aeroneutics end Astronautics In Washington onM..- "A Crues-Sectionat ANol of Annual Interregional Decemer 12, 1960; In Bloston on January 20, 1901; and In LosMigration and Eaployment Growth: Intertapporal Evidence of Angeles on June 12, 1981.)Structural Change, 1959-19743," 31 pp., Apr 15101, AD A099 393eUniversity of Colorado PP 320-Arliand State University Trost, R.P.; Lurie, Philip and Berger, Edivard, "A Note on

Estlnmtlng Covtlnaus Time Decision NMdIS." 13 pp. Sep 1961PP 305

Numnn. Laura N., "An Introduction to the Llterature Of SearchTheory." 32 pp., Jun 1981

PP 306Anger, Thoes E., * itat Good Pee Warfare MOdMe? I pp..tWA 196