____P  .o 6
POLICY RESEARCH WORKING PAPER   2069
Logit Analysis  in  a                                           Does the timing of the
transition into informal self-
Rotating  Panel Context                                         employment follow a life-cycle
and  an  Application  to                        pattern where credit-
and  an APPli*cation to
constrained workers first
Self-Employment Decisions                                       accumulate capital in the
formal sector? Or does self-
employment correspond
Patricio Aroca Gonzdlez                                         better to a traditional "safety-
William F. Maloney                                              net' for the unemployed?
The World Bank
Latin America and Caribbean Region
Poverty Reductioni and Economic Management Sector Unit                  U
February 1999



| POLICY RESEARCH WORKING PAPER 2069
Summary findings
Gonzalez and Maloney derive a methodology for                  They model the decision to move as a stopped Markov
analyzing logit models in a rotating panel context. They     process in which, in each period, the worker compares
then apply the technique to test two theories of why and     accumulated savings with the target level for switching
when salaried workers enter the informal self-employed       sectors dictated by the forecasted stream of discounted
sector.                                                      utility arising from employing labor and cal ital in each
In the traditional view, workers fired from formal jobs    sector.
queue in the informal sector to reenter the formal sector.     They test and find support for the model using the
Gonzalez and Maloney argue that for many, self-              new logit methodology and rotating panel data from
employment is a desirable goal, but that credit              Mexico.
constraints often dictate that they work in the formal
sector until enough start-up capital is accumulated.
This paper - a product of the Poverty Reduction and Economic Management Sector Unit, Latin America and Caribbean
Region - is part of a larger effort in the region to understand the functioning of developing country labor markets. Copies
of the paper are available free from the World Bank, 1818 H Street NW, Washington DC 20433. Please contact Tania
Gomez, room 18-102, telephone 202-473-2127, fax 202-522-2119, Internet address tgomez@worldbank.org. Policy
Research Working Papers are also posted on the Web at http://www.worldbank.org/html/dec/Publications/Workpapers/
home.html. The authors may be contacted at wmaloney@worldbank.org or paroca@socompa.cecun.uncn.cl. February
1999 (19 pages).
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about
development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The
papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this
paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the
countries they represent.
Produced by the Policy Research Dissemination Center



Logit Analysis in a Rotating Panel Context
and an Application to Self-Employment Decisions*
Patricio Aroca Gonzalez
Universidad Cat6lica Del Norte, Chile
William F. Maloney
The World Bank
Our thanks to Roger Koenker and to the participants at the Eighth Annual Conference on Panel Data
for helpful discussions. We also thank the Mexican National Institute of Statistics, Geography and
Information (INEGI) for the use of the data. INEGI is in no way responsible for any incorrect
manipulation of the data or erroneous conclusions drawn from it. Contacts:wmaloneyai)worldbank.org,
parocac4)socompa.cecun.uncn.cl



I. Introduction
A substantial literature exists on limited dependent models in a panel context (see
or Maddala 1983 or Baltagi, 1996 for overviews) and continuous dependent variables in
an incomplete or rotating panel context (Bjom and Jansen 1983, Nijman, Hsiao 1986,
Verbeek and van Soest 1991). This paper derives a methodology for estimating logit
models in a rotating panel. It then uses the technique to examine an unresolved problem
in development economics: the role of self-employed workers unprotected by labor
legislation in the LDC labor force. In particular, we are interested in the determinants of
the decision to leave protected (formal) work to enter self-employment. An alternative
theoretical model to the dualistic view generally accepted is offered. Both views are
tested using rotating panel data set from Mexico and the alternate view supported.
II.   An Alternate View of Informal Self-employment
Much of the literature on the informal self-employed sector in LDCs beginning
with Harris and Todaro(1970) has seen self-employed workers unprotected by labor
legislation as those rationed out of protected or "formal" salaried jobs sector jobs by
above market clearing remuneration in the protected sector. Transitions should be largely
unidirectional, from the informal and presumably very low capitalized micro-enterprises,
to the formal sector except in the event of downturns in which case laid off workers will
be thrown back on the informal safety net.
However, there is little reason to suppose that the expanding literature on self-
employment in the industrialized world that views self-employment as a desirable and
I



more flexible alternative to wage work may not also be relevant in LDCs. In particular,
the debate over the dynamics underlying patterns of worker transitions into self-
employment is likely to be relevant. Johnson (1978), Jovanovic (1979) and Miller(1984)
argue that younger individuals are better able to bear the risk involved and hence should
be heavily represented among entrants into self-employment. However, as Evans and
Jovanovic note, this is inconsistent with Evans and Leighton's (1989) finding of the
hazard into self-employment being constant in age which they attribute to liquidity
constraints that dictate that workers require time to build up the capital needed to start a
business.
We argue that this phenomenon may be exacerbated in the developing world
where credit markets are poorly developed. The problem can be seen as a Stopped
Markovian Decision Process (SMDP)1 where workers, faced with uncertainty about
future streams of income as salaried and self-employed workers must decide the optimal
savings and switching strategies. Their behavior can be seen as similar to that of workers
who, perhaps with the idea of opening a business upon their return, migrate to a country
that offers the possibility of accumulating wealth more quickly, and return home only
when they reach their target level of savings (See Piore 1979). This problem has been
analyzed in detail by Berninghaus and Seifert-Vogt (1993) and we adapt their work to our
problem as a way of generating predictions to be tested in the empirical work.
We assume the worker will open his own business at time X and plans on operate
it for T-r years where T is the end of his planning horizon. He has subjective
' See Eckstein and Wolpin(1989) for a review of the specification and estimation of dynamic stochastic
discrete choice models.
2



expectations 7t on the return to his invested accumulated real wealth x, in the business.
Upon starting his business, the worker will choose a sequence of consumption bundles, c,
such as to maximize
T
E  U(C r)
s.t. O<ct<xt   xt+1= 7(xt- ct )  t = ,...,T, where u(.) represent the continuous per period
utility function. We abstract from the discount factor since it will be the same in both
sectors and we assume it unchanged. This reduces to a standard dynamic programming
problem where V. (x) is the value function at switch time r, the maximal value of the sum
of per period utility from being an entrepreneur from , to T.
While in the salaried sector from t = 1, ..., (X-1) the worker earns yt where {YtJ,
is a stochastic process whose probability law is known to the worker. In each period, the
worker chooses a consumption bundle c, subject to the condition c1 < (x, + y). Any
surplus can be saved at a real interest rate, it+, which is the realized value of a stochastic
process {IJ, of real interest rates. In each period, the worker must decide whether to work
in the salaried sector for another period, or start his business and receive V,(x~.
The optimal policy for this problem is a sequence of consumption strategies (c(.)
and the stopping (switching) time T such that the total expected reward E
E(xo io,yo)( U(C/(.)) + Vr(xr)1
t=O
is maximized given the initial state(xo, io,yo). The optimal stopping time is associated
3



with each "state history" (xp it,y) and because these are realizations of a stochastic
process, T too, is a random variable.
From the framework, several predictions emerge. First, there exists a critical level
of target savings below which the worker will prefer to continue to stay salaried. For the
case of a two period model with logarithmic utility, Berninghaus and Seifert-Vogt show
that the target level of savings falls with a rise in the subjective return to self-
employment, x, rises with an increase in the opportunity cost of savings, i, and rises if a
higher wage in the salaried sector raises the required comparable stream of income
resulting from self-employment, and hence the start up capital required. The first two
also have a predictable effect on the switch time, Tr. In the last case, however, the overall
impact of current income on T is ambiguous since higher incomes both increase the level
of target savings as well as increase the possible rate of savings accumulation. Given two
workers with identical savings, the one with higher income may find himself below the
target rate of income and stay one more period to earn another period wage.2
In sum, the probability of a move into self-employment at a particular moment:
Pr(move) = P( 7c,i,y)
ai'    al      aP
->0, -<O, -=?
as      ai    ay
It is worth comparing these predictions to those from the standard dualistic view
where an above market clearing formal sector remuneration, y, rations workers into the
informal sector where the returns fall to absorb those in the queue. A fall in relative
2By the same token, a worker who suddenly loses his job, y goes to 0, will suddenly see the target lev,el of
saving decline and is more likely to move. In this way, the common vision of the informal sector as the
reserve army of the unemployed can be seen in somewhat different light.
4



returns of self-employment for salaried work poccurs in the contp,xti of economic
downturns where the informal return must fall to absorb displaced workers. Similarly, to
the degree that increased interest rates are associated with recession and the loss of
salaried jobs, again, we may expect more movement of the displaced into the informal
sector: movement into self-employment would be counter-cyclical. In both cases, the
predicted signs would be the opposite of those postulated by the model above.3
The next section offers a method for using logit methods in a rotating panel
context to estimate the determinants of the worker's decision to move, and hence to test
between these two views.
HII.   Logit Analysis in a Rotating Panel Context
Selection of Individuals
In the relatively common case that we address, individuals are selected according
to a "rotating" scheme in the following manner. In period 1 of a total of T periods, the
first sample is selected of N individuals who will remain in the sample for z periods: yll,
Y21      YNI. In the second period, the first m = NIz individuals are retired and the first
place until the Nth place are occupied by the individuals who follow individual m: Y12, Y22
. . -,YN2. The process of retiring and replacing continues for each period t with a new
sample: Y,t Y2t ,--YNt.
The combination of data obtained by this process is called a Rotating Panel and
we
can considered it ordered as:
3 See Maloney (1997) for a discussion of the relative merits of formal vs informal work and the
procyclicality of the latter in Mexico.
5



Y115Y2154tulx Y(m+1)13 ...             ...-YNI
Y12'Y22 ...Ym2 Y(m+1)2 ...       .YN2
Y13 .. -  Ym3- ...  *.-.-YN3
In this manner, H = (T-I)m + N individuals are partially observed across T periods. In
our example, y represents whether the worker moves in that period (y=l) or stays in
salaried employment (y=O).
It is useful to reframe the problem as a T X H fixed panel:
Y'11 Y21.       YHI
Y 12 Y22 .    Y  2
Y IT Y2T ..........H
where Yjt denotes the position of individual j in time t, whether there is an entry or, not.
For example: Y 12 Y 22 ........Y m2 are positions that do not have entries.
Observation: Each position in the fixed panel corresponds to one in the rotating sample:
Y jt= Yj (t l)m t    ( I )
However, an individual is included in the rotating sample, and has an entry in the fixed
panel in time t only if:
I < j - (t - l)m < N
In the analysis, we will only be concerned with individuals in the sample for a full
z periods which can be shown to be the case for individuals entering the panel in t
E {1,2,...,T-z+l }. Several results pertaining to this group are described in appendix I.
6



Definition: For t e { 1,2,...,T-z+1 }, if individual j enters the sample of size N in period
t, we define   as a vector representing the sequence of the z consecutive entries.
= (Yit, Yi(t+,.   Yj(ezl ))
In the present application, this is the sequence of moves that individual j is observed to
make across the z periods in the sample. In theory, there could be multiple moves or
none.
Probability Function
We are interested in understanding what determines the timing of the individual's
decision to change state. To be consistent with the theory above, we assume the
individual moves only once into self-employment, and that the decision to move in each
period is independent of the previous decisions. The vector
e'k (0, 1,0 O.0) E RZ
t    position k
permits us to identify the period in which the individual changes state.
We define the probability that an individual j that has been k periods in the
sample, changes its state in period t, and that it changes only once as:
pj,(k) = p rvt = ek                ijj(tk-1)  k=1,2,3,...,z
To calculate pjt(k), we first find an expression for:
7



p (YJ7  = e,k )   k = 1,2,...z
For example:
p(Yj, = el) = p(yjt = 1)p(yj,,, = 0) ...p(yj+-l = 0)
We assume that the probability that an individual changes state follows a logit
distribution, and this probability is a function of a set of the environment and individual's
characteristics (X). This can be shown equal to:
exp[8Xj-(,-l)mt]          
(I + exp[Ej fX)j    t ]) (1 + exp[Yj ,,m,t+l])  (1 + eXP[Xj-(+z-2)m,t+z-1])
exp [,iX j-(t-l)m,t ]
M j (6 )
where Mjt( 6) is the product of the denominators above. Generalizing, we obtain that the
probability of changing states exactly once in the kth period of observation is:
p (Yj, = ek    exp [8Xj-(+k-2)m,t+k-]
The probability of moving in the kth period of observation, given that the individual will
move exactly once in one of the k periods is:
Pj,jt =ke))+ p(Yj, = e2k+.+p(Yit = e:)
exp[fXiXt+k-2)m,t+k-l I
exp[iXj3t_)m,t I + eXp[L8jtmt+l ]. .* + eXP[jXt+z-2)m,t+z-1 ]
And finally, the conditional probability that individual j who enters in the sample of size
N in period t and remains z periods counting from t and who changes state in exactly one
of the z periods is:
8



pj  Pj ,(l) j Pj (2)Yit+... Pj (z),"+'
As Yij takes a value of 0 or 1, and in our example there is uniquely one non-zero entry,
this expression effectively selects which Pjt determines Pjt.
Likelihood Function
Before writing the likelihood function we establish some definitions to simplify
the presentation.
Definition: Let M(K*J) be the set of the matrix of rank (K*J). If 1 < u < K, we can
define:
ir.: M(K* J) -RJ
)r.(A) = g.u(akj)1,5kK = (aulu2)-...aKJ)
which states that ir projects the row u of the matrix A. This ensures the condition that
each individual moves only once or:
,Yj, +zr2Yj,+ .+C,'Yj, = 1
Definition: Let Bt = {j such that 'lyKt + 'T2 �,i +...+;r:Yj, = 1 }, that is the set of j where
the individual moves only once.
Definition: Let B = {Bt such that 1 < t < T - z + 1}, that is, the set of Bt such that the
individual is in the sample for exactly five periods.
Definition: Let Xj, be the z*p matrix of independent explanatory variables for each
individual (See appendix for more detail).
With these definitions, we can rewrite Pj, (k) as:
9



Pj, (k) = z      1             V  1<k<z
E exp[Q(, - TOk)XiJt]
and finally:
Pi,t =j 7Pj(k) kyl
k=1
Therefore the log of the likelihood function for all individuals in the panel through all
periods will be:
2= EX lnpj, = -Z    zrkYj, lnzexp(f7T, - k]xIfi))
teB jeB,      teB jeB, k=1    /=1
Employing the Newton-Raphson algorithm, the first and second derivative of the
likelihood function with respect to the parameters are calculated as:
Vi(/) (O         E           {eXpV1 - 'k W]t /6)}T[  - ;k W}jl >i
tEB jeB, k=1    1=1
V22 ,(/J) = -_   EEEEZ  gkyJt Pl exp([r, +,T, - 2,, ]jxj1f) *[V1 -r];XjJ If [1 -)r Jjty]
leB jEB, k=1 1=1 r>l
which permits us to estimate /3, and V(,B).
IV. Data:
The National Urban Employment Survey (NUES) conducts extensive quarterly
household interviews in the major metropolitan areas and is available from 1987 to 1993.
It is structured as a rotating panel where in each quarter, a fifth of the sample is dropped
and replaced by individuals who will be interviewed for each of the next five quarters. In
24 overlapping panels spanning 1987-1993, individual workers can be followed as the
10



move among sectors of work. Individuals are matched by position in an identified
household, sex, level of education, and age to ensure against generating spurious
transitions. The analysis restricts itself to men aged 16-65 with a high school education or
less. It also focuses on formal salaried workers and the "informal" self-employed,
including owners of firms under 16 employees who do not have social security or
medical benefits and are therefore not protected.4 Only those who begin in formal salaried
employment and move only once over five quarters into self-employment are retained,
yielding a sample of 1087 workers. In the estimations, we employ predicted earnings in
each sector as a measure of the "own" and "alternate" earnings, given the standard human
capital variables, experience, experience squared, education, education squared. The
return to accumulated capital (the opportunity cost of using savings to open a business) is
the real 30-60 day deposit rate as calculated from the International Financial Statistics of
the IMF deflated by growth of the consumer price index. We also test state dependence
through introducing the lag of the independent variables in the regressions.
V. Results:
Table 1 presents the results of the estimation of the model set out in section II.
Table 1: Results from Rotating Panel Logit Regression
COEF.          S.E            COEF           S.E.
Wage (Salaried)    -.347           1.34           -.357          1.28
Wage (-1)           .132           1.41                X                       l
Earnings (Self)     4.02           .249           4.03            .235
Earnings (-1)       -.115          .354                                        l
Interest Rate      -3.39 e-3      2.45 e-5        -2.71 e-3      1.92 e-5
Interest Rate(-1)   -2.68 e-3      1.93 e-5       -3.83 e-3      1.65 e-5
Nobs=1078, Sample includes 24 complete panels of 5 quarters each spanning 1987-1993
4 It is often the case that the informal sector is defined as firms with five or less workers. As we are
ocusing on informality defined as being unprotected by social security or other legislation, we loosen the
size limit to the next category tabulated. In practice, the vast majority of fimns are under 3 workers.
11



The results are supportive of the model. The first and second columns present the
complete specification and show that for only the interest rate are lagged values
significant. This suggests the absence of state dependence. The second specification
presents only the significant coefficients. Here, self-employed earnings appear very
strongly and of the correct sign reflecting that as opportunities improve in the informal
sector, workers are more likely to open their own businesses. The current wage in. the
formal sector still enters ambiguously, again, as predicted, and is not significant. This is
to be expected given that a rise both increases the attractiveness of formal sector
employment, and raises the savings rate making a move into self-employment possible.
Finally, the interest rate is strongly significant and of the predicted sign suggesting that a
rise in the opportunity cost of the capital used for start up discourages opening lup a
business. In all cases, the sign is the opposite of that predicted by conventional dualistic
views of informal self-employment.
Appendix II derives the cross section marginal effects and Table 2 calculates them
for the regression above. In each panel of the table, k represents the period in which the
individual moved and h the period corresponding to the variables observed. Of greatest
importance, the diagonals of the tables are both relatively stable and of the sign foundL in
table 1. Calculating the marginal effects has not reversed the effect as is sometimes
found and the theoretical framework remains supported. The off-diagonal elements
(symmetric) are less intuitive. In every case the impact of the variable one period
forward or backward has the reverse impact of the contemporaneous effect.
12



Table 2: Cross Section Marginal Effects
Self-Employed Earnings
dPjt(k)/dX1h
k\ h        1          2         3              4
1     0.6978200 -0.1989130 -0.2283948 -0.2705122
2                0.6957832 -0.2274624 -0.2694078
3                          0.7651952 -0.3093380
4                                     0.8492580
Formal Sector Wage
dPjt(k)/dXVh
k\ h        1          2         3              4
1     -0.0617104 0.0175905  0.0201977  0.0239222
2                -0.0615302 0.0201152  0.0238246
3                          -0.0676686 0.0273557
4                                     -0.0751025
Interest Rate
dPjt(k)/dXh
k\ h        1          2         3              4
1     -0.0004693 0.0001338  0.0001536  0.0001819
2                -0.0004680 0.0001530  0.0001812
3                          -0.0005147 0.0002081
4                                     -0.0005712
Interest Rate Lagged
dPjt(k)/dX"h
k\ h        1          2         3              4
1     -0.0006628 0.0001889  0.0002169  0.0002569
2                -0.0006609 0.0002160  0.0002559
3                          -0.0007268 0.0002938
4                                     -0.0008066
13



Table 3 derives the marginal effects over time, which are calculated by taking the
difference between the maximum and the minimum value of each variable. As with the
cross sectional marginal effects, the signs are those predicted and expected self-employed
earnings is the most important variable to explaining the transitions change from formal
to informal sector.
Table 3: Marginal Effects Over Time
Variable Variation      Probability Variation
Formal Sector Wage                 0,3654                   -0,0315
Self-Employed Earnings            0,2411                    0,29 13
Interest Rate                      57,83                    -0,0377
Interest Rate Lagged               64,03                    -0,0429
VI. Conclusion
The paper has derived a methodology for analyzing logit models in a rotating
panel context. Using data from Mexico, it then applied the technique to test between two
theories of why salaried workers enter the informal self-employed sector. The evidence
supports a view that self-employment is a desirable destination, but one that in the
presence of credit constraints requires accumulated capital before the business is opened,
over the more traditional view of self-employment as a safety net for those losing
preferred formal sector jobs.
14



References:
Baltagi, B. H. (1985), "Pooling Cross Sections with Unequal Time Series Lengths"
Economics Letters, 18 p. 133-136.
Baltagi, B.H. (1996) Econometric Analysis of Panel Data, John Wiley and Sons (New
York).
Baltagi, B. H. and B. Raj (1992), "A Survey of Recent Theoretical Developments in the
Econometrics of Panel Data," in
Bjom, Erik,(1981) Estimating Economic Relation from Incomplete Cross Sectional/Time
Series Data, Journal of Econometrics 16:221-236.
Bjom, E. and E.S. Jansen (1983) "Individual Effects in a System of Demand Functions,
Scandinavian Journal of Economics 85:4 1983.
Berninghaus, B. and H G. Seifert-Vogt (1993), The Role of the Target Saving Motive in
Guest Worker Migration, Journal of Economic Dynamics and Control 17, p181-205.
Eckstein, Z. and K I. Wolpin (1989) "The Specification and Estimation of Dynamic
Stochastic Discrete Choice Models, a Survey, The Journal of Human Resources, 24:4
Evans, D.S. and B. Jovanovic (1989), "An Estimated Model of Entrepreneurial Choice
under Liquidity Constraints, Journal of Political Economy 97:4 pp. 808-826
Evans, D.S. and L. Leighton(1989), "Some Empirical Aspects of Entrepreneurship,"
American Economic Review 79.  
Harris, J.R. and M. P. Todaro (1970), "Migration, Unemployment, and Development: A
Two Sector Analysis," American Economic Review, 60:1, 126-142.
Hsiao, C.(1986), Analysis of Panel Data, Econometric Society Monograph no. 11,
(Cambridge University Press, Cambridge).
Jovanovic, B. (1979), "Job Matching and the Theory of Turnover." Journal of Political
Economy 87:5 972-90.
Nijman, R., M. Verbeek, and A. van Soest (1991), "The Efficiency of Rotating-Panel
Designs in an Analysis of Variance Model," Journal of Econometrics , 49:373-399.
Maddala, G.S. (1983) Limited Dependent and Qualitative Variables in Econometrics
(Cambridge University Press, Cambridge).
is



Maloney, W.F. (1997) "Labor Market Structure in LDCs: Time Series Evidence on
Competing Views," Working Paper University of Illinois.
Miller, R.A.(1984) "Job Matching and Occupational Choice" Journal of Political
Economy 92 1086-1120.
Piore, M. (1979), Birds of Passage (Cambridge University Press, New York, NY).
16



Appendix I
Theorem: Individual j enters the rotating sample of size N in period t only if:
(z - l)m   + I < j - (t - l)m  < N          (2)
Demonstration: The last m entries of the sample of size N=zm can be written as
{ Y(z-I)m+i,t, Y(z-l)m+2, t.    YN, t}
For individual j to be one of these entries, condition (1) implies condition (2). If (2) holds
we can also show that jyj, has an entry and that at time t, the individual has just entered
the sample of size N.
Observation: Only for t e { 1,2,. ..,T-z+1 } are the last m individuals entering the sample
of size N observed for all z periods. There are
(T - z + 1)(N - (z - I)m) = (T - z + I)N
z
such individuals of the H total.
Corollary: For t E {1,2,...,T-z+l } condition (2) is necessary and sufficient for
individual j to be observed for z periods.
Appendix II
The sequence of decisions to stay or move for each individual is the vector
it= [Yj-(t-1)mJ ' Yj-tm,t+ I .. 9 *Yj-(t+z-2)m,t+z-1]
and the set of variables that determine that choice:
Xj -Q-1)m,f    j-(I-l)m,t           XJ (t-l)m t
Xlj-tm,t+l
XI                 ...    ... 
j-(t+z-2)m,t+z-1                  Xj (-+z-2)m,r+z-l
17



We distinguish two types of marginal effect, across the individuals and across time.
Cross Section Marginal Effect
The cross section marginal effect measures the change in the probability of a move due to
individual differences in the independent variables. For each individual we can write:
-pj    (%,pj,(k)    apj,(k)         apj,(k)  
ax' ix'i'ax
Ia Xj -(I 1)MI t  xj-Im,1+l     axj (t+z-2)m,i+z-1
orh= 1,2,.. . ,z
apj, (k)             sj, (k)            1         sjt (k)
aXj (I+h-2)m,+h-l   aXj (t+h 2)m,t+h-I    s, (k) aXj-(t+h-2)m,t+h-I
1   [e  p[x   X /' ferxu']]
sj, (k)    8XjQ(,+h 2)mt+h1-
I ______  [,j]'*    6ie-4[kX           +  -     l*          8e4fXjf]      1
S,(k)l                                                  r=1 aXj-(Q+h-2)m,t+h-I J
__    [    [ r   ]*  -AIkxJI,] *   [-fikXJ~]'  +] *e2tX-Jk]'  8[f ifkXJ-x                      1
S i.  r=1      ]             * j-Qt+h-2)m,I+h-1                r=1ax
sj, (k)                               x r=j-(t+h-2)m,i+h-I
we know that:
4/J7TrXJtl       8f4i6Xj-(t+r-2)m,t+r-l + fl2Xj-(t+r-2)m,t+r-I + +iPXT(t+r2)mt+r1]
aXJ --(t+h-2)m,t+h-I                       IDxji 1(t+h-2)m,t+h-I
18



81  Vr = h    Pi 
{0   Vr# 
ii  Vrh
where: Srh=o   Vr�h
therefore
ap1, (k)          1  [           *                      ]                    [ e]
j-(t+h-2)m,t+h-1                            L            ] +          *         r=[
~Pf (k)Z3,Bj(Sh -) s,        k)X )x,,r]
r=l
Marginal Effect Over Time (MEOT)
The marginal effect over time measures the change in the probability of a move with
diffferent levels in an independent variable. Given that we work in discrete time, for each
individual we can write:
Marginal Effect Over Time = (pjt (k + 1) - pjt (k)|Xi(k+) � xk),  Vk=1,2, **,z-1
We have that:
eflkk+IXPe] fl[kXjt]
Pj, (k + 1) =  z        , and  Pjt (k) = -
z                             z
:~~~~~~ Lefll"                                Eepl*j]
h=1                           h=l
Therefore:
i        I                wB~~~~~~~~,6Xrx-(t+k-2)mit+k-
[ eA' X-(I+k-1)mJ+k - efliXj-(+k-2)m,t+k-1 ]er r=l
MEOT =                                        |
h=1
19



Policy Research Working Paper Series
Contc
Title                            Author                  Date              for paper
WPS2046 Restructuring of Insider-Dominated   Simeon Djankov       January 1999       R. Vo
Firms                                                                       33722
WPS2047 Ownership Structure and Enterprise  Simeon Djankov        February 1999      R. Vo
Restructuring in Six Newly                                                  33722
Independent States
WPS2048 Corruption in Economic            Shang-Jin Wei           February 1999      C. Bernardo
Development: Beneficial Grease,                                             31148
Minor Annoyance, or Major Obstacle?
WPS2049 Household Labor Supply,           Kaushik Basu            February 1999      M. Mason
Unemployment, and Minimum Wage Garance Genicot                              30809
Legislation                      Joseph E. Stiglitz
WPS2050 Measuring Aid Flows: A New        Charles C. Chang        February 1999      E. Khine
Approach                         Eduardo FernAndez-Arias                    37471
Luis ServAn
WPS2051 How Stronger Protection of        Carsten Fink            February 1999      L. Willems
Intellectual Property Rights Affects   Carlos A. Primo Braga                85153
International Trade Flows
WPS2062 The Macro Wage Curve and Labor   Dorte Verner             February 1999      H. Vargas
Market Flexibility in Zimbabwe                                              37871
WPS2053 Managing Foreign Labor in Singapore Elizabeth Ruppert     February 1999      A. Sperling
And Malaysia: Are There Lessons for                                         37079
GCC Countries?
WPS2054 Who Controls East Asian           Stijn Claessens         February 1999      R. Vo
Corporations?                    Simeon Djankov                             33722
Larry H. P. Lang
WPS2055 Social Security Reform, Income    Carlos Serrano          February 1999      M. de Loayza
Distribution, Fiscal Policy, and                                            38902
Capital Accumulation
WP82056 Does a Thin Foreign Exchange      Hong G. Min              February 1999     K. Labrie
Market Lead to Destabilizing    Judith A. McDonald                          31001
Capital-Market Speculation in the
Asian Crisis Countries?
WPS2057 Finance and the Sources of Growth  Thorsten Beck          February 1999      K. Labrie
Ross Levine                                31001
Norman Loayza
WPS2058 Do Depositors Punish Banks for    Maria Soledad Martinez   February 1999     A. Yaptenco
'Bad' Behavior? Market Discpline  Peria                                     38526
in Argentina, Chile, and Mexico  Sergio L. Schmukler



Policy Research Working Paper Series
Contact
Title                            Author                  Date              for paper
WPS2059 Financial Intermediation and Growth: Ross Levine          February 1999       K Labrie
Causality and Causes             Norman Loayza                              31001
Thorsten Beck
WPS2060 The Macroeconomics of Delayed     Daniel Kaufmann         February 1999       D. Bouvet
Exchange-Rate Unification: Theory   Stephen A. O'Connell                    35818
And Evidence from Tanzania
WPS2061 A Framework for Regulating        Hennie van Greuning     February 1999      A Thornton
Microfinance Institutions        Joselito Gallardo                          80409
Bikki Randhawa
WPS2062 Does Financial Reform Increase    Oriana Bandiera          February 1999     A. Yaptenco
or Reduce Savings?               Gerard Caprio, Jr.                         38526
Patrick Honohan
Fabio Schiantarelli
WPS2063 The Practice of Access Pricing:   Tommasso M. Valletti    February 1999       G. Chenet-Smith
Telecommunications in the United                                            36370
Kingdom
WPS2064 Regulating Privatized Rail Transport Javier Campos        February 1999      G. Chenet-Smith
Pedro Cantos                               36370
WPS2065 Exporting, Externalities, and     Howard Pack             February 1999      C. Bernardo
Technology Transfer              Kamal Saggi                                31148
WPS2066 Flight Capital as a Portfolio Choice   Paul Collier       February 1999      A. Kitson-Walters
Anke Hoeffler                              33712
Catherine Pattillo
WPS2067 Multinational Firms and Technology  Amy Jocelyn Glass     February 1999       L. Tabaida
Transfer                         Kamal Saggi                                36896
WPS2068 Quitting and Labor Tumover        Tom Krebs               February 1999      T. Gomez
Microeconomic Evidence and       William F. Maloney                         32127
Macroeconomic Consequences