__p 3sw 2q3
POLICY RESEARCH WORKING PAPER 2459
Short-Lived Shocks with In theory it is possible that a
vulnerable household will
Long-Lived Impacts? never recover from a
sufficiently large but short-
Household Income Dynamics lived shock to its income-
which could explain the
in a Transition Economy persistent poverty that has
emerged in many transition
Michael Lokshin economies. But this study for
Martin Ravallion Hungary shows that, in
general, households bounce
back from transient shocks,
although not rapidly.
The World Bank
Development Research Group
Poverty and Human Resources H
October 2000i
POI fcy RESEARCH WORKING PAPER 2459
Summary findings
In theory it is possible that the persistent poverty that has To test the theory, Lokshin and Ravallion estimate a
emerged in many transition economies is attributable to dynamic panel data model of household incomes with
underlying nonconvexities in the dynamics of household nonlinear dynamics and endogenous attrition. Their
incomes-such that a vulnerable household will never estimates using data for Hungary in the 1990s exhibit
recover from a sufficiently large but short-lived shock to nonlinearity in the income dynamics.
its income. This happens when there are multiple The authors find no evidence of multiple equilibria. In
equilibria in household incomes, such that two general, households bounce back from transient shocks,
households with the same characteristics can have although the process is not rapid.
different incomes in the long run.
This paper-a product of Poverty and Human Resources, Development Research Group-is part of a larger effort in the
group to understand household-level vulnerability to shocks. Copies of the paper are available free from the World Bank,
1818 H Street NW, Washington, DC 20433. Please contact Patricia Sader, room MC3-556, telephone 202-473-3902, fax
202-522-1 153, email address psader@worldbank.org. Policy Research Working Papers are also posted on the Web at
www.worldbank.org/research/workingpapers. The authors may be contacted at mlokshiniCworldbank.org or
mravallion@worldbank.org. October 2000. (26 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 conclusionts 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
countnes they represent.
Produced by the Policy Research Dissemination Center
Short-Lived Shocks with Long-Lived Impacts?
Household Income Dynamics in a Transition Economy
Michael Lokshin and Martin Ravallion'
World Bank, 1818 H Street NW, Washington DC
Keywords: Income dynamics, poverty, multiple equilibria, Hungary
JEL: C23, 132, P20
' Much of the work on this paper was done while the second author was an academic visitor at
the Universite des Sciences Sociales, Toulouse; the hospitality of UT is gratefully acknowledged. The
support of the World Bank's Eastern Europe and Central Asia region and the World Bank's Research
Committee (under RPO 681-39).
1. Introduction
Consider a household that suffers a transient income shock, by which we mean an unexpected but
short-lived drop in income. With limited access to credit, or other forms of (formal or informal)
insurance, such a shock will cause a spell of hardship. For example, a family that was not poor
before suddenly finds that it cannot secure its basic consumption needs. But could such a shock also
cause a previously non-poor family to become poor, and stay poor, indefinitely? Or could it cause
a moderately poor family to fall into persistent destitution?
If the answer is "yes" to these questions then there will be large long-term benefits from
institutions and policies that effectively protect people from transient shocks. If the answer is "no"
then the (still potentially important) gains from such social protection will also be transient; lack of
a safety net may well cause hardship, but it would not be a cause of persistent poverty.
The answer depends on properties ofthe dynamic process determining incomes at household
level. And they are properties of income dynamics about which we currently know very little. If the
process by which household incomes evolve over time can be represented well by the simplest type
of linear (first-order) autoregression then a household that experiences a transient shock will still see
its income bounce back in due course. The family may well stay poor for a longer period than the
duration of the shock. This can happen because incomes do not adjust instantaneously but do have
some serial dependence; low current income may reduce future income such as by eroding a family's
physical and human asset base. But the household will recover from just one draw from a
distribution of serially independent income shocks. (The same is true of a broad class of commonly
assumed stationary linear autoregressive and moving average dynamic processes.)
However, there is no obvious a priori reason why incomes would behave this way. It has
been argued that economies as a whole have a "corridor of instability," meaning that they are stable
with respect to small shocks but not large ones (Leijonhufvud, 1973). Nonlinear dynamic models
with multiple equilibria have been widely used in explaining why seemingly similar aggregate
shocks can have dissimilar outcomes. In macroeconomics, examples can be found in models of the
business cycle (Chang and Smyth, 1971; Varian, 1979) and certain growth models (Day, 1992;
Azariades, 1996). Similar ideas have been employed in modeling micro poverty traps (Dasgupta and
2
Ray, 1986; Banerjee and Newman, 1994; Dasgupta, 1997) and in understanding famines (Carraro,
1996; Ravallion, 1997).
It is not difficult to construct theoretical models that generate a type of nonlinear dynamics
at individual level whereby short-lived shocks have long-lived effects. We give examples later.
However, while it is theoretically possible that transient shocks have persistent effects, whether they
do or not remains an empirical question. And it is a difficult question. We clearly need to observe
incomes of the same households over time; panel data appear to be essential. Even so, there is a
concern about whether we will be able to observe an unstable equilibrium. This will depend on the
speed of adjustment relative to the survey data frequency and whether shocked households stay in
the panel. Possibly the households who receive large negative shocks will drop out ofthe survey. For
example, sufficiently large shocks may entail breakup of the family, un-planned migration and/or
homelessness, and (hence) a high probability of dropping out of the panel survey. One clearly needs
to allow for endogenous attrition. There are also econometric issues about estimated dynamic effects
in panels of relatively short duration. Tests exist in the literature for determining whether a time
series with white-noise properties is stochastic (i.i.d.) or deterministic (chaotic) (Brock and Potter,
1993; Liu, Granger and Heller, 1992). However, these call for large samples over time; 600 would
be considered adequate, but not six! Furthermore, the question of interest here is not so much
whether the economic dynamics is complex, but rather whether it exhibits low-income
nonconvexities.
This paper tests whether persistent poverty can arise from sufficiently large but short-lived
income shocks at the household level. We first look at the income dynamics using simple but
flexible non-parametric methods. We then estimate a parametric model of income determination,
incorporating nonlinear dynamics and endogeneous attrition arising from a nonzero correlation
between the error term of the equation for incomes and an equation for the probability of staying in
the panel. We also test for nonlinearity in the way initial incomes influence panel attrition.
Our choice of setting was dictated in part by the fact that we require household panel data.
Of course, this would be of little use for our purpose if there had not been (unfortunately) large
income shocks at household level. We chose a six-year household-level panel data set for Hungary.
The data are close to ideal for our purposes, since the panel was designed for studying income
3
dynamics. And the setting is of substantive interest in this context. The collapse of central planning
and transition to a market economy in the 1990s entailed sizable income shocks to Hungarian
households. The shocks clearly hurt; for example, there was a rapid increase in the incidence of
poverty. A crucial question for policy in this setting is whether these income shocks had long-lasting
consequences. A further reason for choosing Hungary is that there exists a sizable safety net; we will
test how much impact this might have had on the income dynamics.
As in micro studies for other settings, past work for Hungary has shown that differences in
the long-term characteristics of households (such as asset holdings and human capital) and certain
events interpretable as shocks (such as unemployment and illness) increase the risk of poverty. (We
review this literature later.) While agreeing to the importance of such factors in determining current
household incomes, in this paper we focus on the different question of whether transient income
shocks might cause persistent poverty. Do households bounce back from such shocks? What are the
reasons for differences in household income dynamics? Are there any household characteristics that
contribute to the vulnerability of the family to income shocks? Why does it take much longer for
some households to recover from a transient shock? These questions require a rather different
approach to that found in the literature on poverty and income dynamics.
The following section gives examples of models that can yield the type of nonlinear income
dynamics whereby short-lived shocks can have permnanent consequences. Section 3 then discusses
the literature on income dynamics in Hungary and elsewhere. Our data are described in section 4.
We then present our econometric model in section 5. Section 6 presents our results, and our
conclusions are summarized in Section 7.
2. Nonlinear dynamics in household incomes
Probably the simplest model that can generate the type of nonlinear income dynamics we are
interested in testing for assumes that a family cannot borrow or save and derives income solely from
labor earnings, but with a nonconvexity at low earnings. We can suppose that the worker's expected
productivity and (hence) wage rate depends on consumption, as in the classic Efficiency Wage
Hypothesis (Mirrlees, 1975; Stiglitz, 1976). This assumes that labor productivity and earnings are
4
zero at a low but positive level of consumption; only if consumption rises above some critical level,
YZ">O, will the worker be productive. In the efficiency wage literature, Ym' is usually interpreted as
the nutritional requirements for basal metabolism, which represent two-thirds or more of normal
nutritional requirements (Dasgupta, 1993). There are other interpretations. One can assume that a
minimum expenditure level is necessary to participate in society, including getting a job. The
expenditure is required for housing (or at least an address) and adequate clothing. Thus one can say
that consuming below this point creates "social exclusion." Higher consumption permits social
inclusion, but there are presumably diminishing income returns to this effect. For example, earnings
rise but at a declining rate until after some point the productivity effect of consumption vanishes.
Nonlinear dynamics can be introduced into this model by simply assuming that the wage rate in any
period is contracted at the beginning of the period. Finally we assume that this dynamic process of
income determination has at least one date for which incomes have risen.
Combining these assumptions, the process generating current income (Y,) can be written as
the nonlinear difference equation: Y,=f (Y,-,), where the functionf is continuous withf (Y)=0 for
YYmt2. An equilibrium of this model is a
steady-state solution such that Y=f (Y). It is evident that the model must have at least one such
equilibrium, and if there are two, the one with lower income will be unstable2.
Alternatively, we can think of a liquidity constrained household that faces the choice of
investing in (physical or human) capital accumulation or consuming all income in a given period.
Suppose that the household is only willing to forgo current consumption in order to invest if its
income exceeds a critical level Ym'i. The investment yields an income at time t off(Y,-,) where this
function has the same properties as above.
The recursion diagram in Figure 1 illustrates the case of two equilibria. The equilibrium at
Y** (>Ynif) is stable, but Y* is not. Consider a household at Y**. With any shock exceeding
Y** Y*, the household will be driven beyond the unstable equilibrium, and will then see its income
decline steadily (even precipitously). Persistent poverty will be the inevitable result.
2 Note that the assumption that Y,>Yt-l for some t assures that there is at least one unstable
equilibrium (by the intermediate value theorem, given continuity off and the assumption that Yn"n>O).
5
One can propose more complicated models than this one. For example, on can allow for some
positive lower bound to incomes. Assuming that this lower bound is below Y**jin Figure 1 there will
now be three equilibria, with the extra (stable) equilibrium at the lower bound. Again, with a
sufficient negative income shock, a household at its high (stable) income will see its income then
decline until it reaches the lower bound.
This type of model has a powerful policy implication. A transfer payment T Y$* will
eliminate the low-income unstable equilibrium. The family will be fully protected from the
possibility of a transient shock having an adverse long-term effect. The transfer will not only help
protect current living standards, but will also generate a stream of future income gains. The safety
net could be a long-term investment, and with a high return.3
Later we will see whether the empirical dynamics of household incomes in Hungary looks
like Figure 1, such that sufficiently large short-lived shocks can have long-lived impacts.
3. The setting and literature
The last decade has seen a sharp decline in Hungary's GNP (by nearly a fifth of its 1989 value in the
first four years of transition), large scale unemployment, declining real wages and household
incomes, and a sharp increase in income poverty. Between 1990 and 1994 the number of employed
people decreased by 1.4 million, and by 1995 formal employment had dropped by more than a
quarter of its pre-transitional level. Unemployment increased by approximately 500 thousand people
for that period (Galasi, 1998; Forster and Toth, 1998). The proportion of the population living below
the subsistence minimum was about 50 percent higher in 1996 than in 1992 (Speder, 1998).
Under these conditions maintaining a social safety net has become an important concern of
the Hungarian government. Both Hungarian and international scholars have been involved in the
debate about the reforms ofthe social support system to avoid the emergence of massive poverty and
to make the current system of social protection fiscally sustainable.
3 A similar point is made by Keyzer (1995) in his analysis of a generalized version of the
Dasgupta and Ray (1986) model.
6
The dynamics of poverty and the performance of the safety net in Hungary have been a themC
of past research.4 Dynamic aspects of poverty in Hungary were studied by Ravallion et al. (1995)
based on two rounds of data from the Household Budget Survey conducted by the Central Statistical
Office for 1987 and 1989. They constructed the joint distribution of household welfare over time,
in which the panel structure was exploited to show how households moved between welfare groups.
The results showed considerable transient poverty over the period of the survey. The safety net did
help protect vulnerable households from falling into poverty.
Further research on poverty dynamics has been facilitated by the Hungarian Household Panel
Survey (HHPS). This was conducted by Hungary's Social Research Informatics Center (Tarki) and
began in 1991, with the purpose of providing researchers with data for further investigating
household income dynamics.5 Several recent papers have used the HHPS to analyze the dynamic
aspects of poverty in Hungary (Galasi 1998; Speder 1998; Forster and Toth 1998). Using income
transition matrices, Galasi (1998) studied the dynamics of poverty incidence, the chances of escaping
from and reentering poverty, and the characteristics that distinguish households who stay in poverty
from those who escape. The results suggest considerable income mobility from one year to the next.
Most of the initially poor escaped poverty within two years, but a high proportion of the households
who escaped poverty were found to be poor again withing three years. However, the majority of
households move to neighboring quintiles, and households in the middle of the income distribution
experience the most income mobility. The income level of households in the top and bottom
quintiles tends to be more stable.
Applying a similar method, Speder (1998) examined the effects of certain life cycle events
on the long-term income status of Hungarian households. Changes in household composition and
size were found to have an impact on household incomes. Childbirth, dissolution of the household
(divorce and widowhood) as well as changes in economic-activity status were found to increase the
' While there is a large and recent literature on poverty in Hungary, here we focus on panel data
studies. The composition of absolute poverty was examined by Kolosi et al., (1995). Relative poverty
was studied by Andorka (1992) and Andorka and Speder (1993a, b). Work by Toth et al., (1994) and
Andorka et al., (1995) looked at the composition of poverty using various measures.
5 Information on the sample design, sample weights and representability can be found in Toth
(1994) and Sik and Toth (1993, 1996, 1997).
7
risk of being poor. Analysis of household income components indicated that wages and joint
incomes ofthe household members were mainly responsible for the dynamics of poverty in Hungary.
Forster and Toth (1998) found that the durations of poverty spells in Hungary depended on
characteristics of the individual and the household. Persons with lower education were less likely
to escape poverty than persons with higher levels of education. Persistent poverty is rare among
persons with a university diploma. Children and the elderly have fewer chances of escaping poverty.
None of this past work has tested whether the dynamic process determining incomes is such
that transient income shocks can create persistent, long-term, poverty. Indeed, we know of no tests
for any other setting. Although much has been learnt about the processes determining poverty in the
present setting, past work cannot answer the question in our title. The following sections propose
and implement a method of testing for nonlinearity in the income dynamics consistent with the
existence of multiple equilibria.
4. Data and descriptive results
We use six waves (1992-1997) of the HHPS. The first wave of the survey was designed to include
a nationally representative sample of Hungarian households. The aim was for all persons living in
households selected for the first wave to be re-interviewed at one-year intervals. Originally (in
1992) the panel included 2668 households. The household response has been around 85 percent at
each round of the survey, so that by the sixth wave (1997) only 52 percent (1385) of the initially
selected households remained in the sample. Attrition is clearly a concern with this survey.
The questionnaire includes detailed questions about the incomes of every adult member of
the household. Income components that cannot be directly allocated to any individual household
member are registered separately in the questionnaire. Total household income is calculated as a sum
of wages and salaries of individual members of the household, social security transfers, private
transfers, in-kind income, and income from home production, with imputed values when necessary.
Table 1 provides some descriptive results on household recovery times following a negative
income shock. We selected all households who experienced a decline in their real total income
between 1992 and 1993 and categorized these households according to the time it took them to get
8
back to at least 98% of their income in 1992. More than one third (37.5 percent) of households that
had a negative income shock recovered their income loss within one year. However, 47 percent of
Hungarian households had not recovered within five years after a shock.
The time it takes for a household to recover after a decline in income clearly depends on the
size of the shock. Among households that experienced a decline in real income of less than 10%
between 1992 and 1993,47% recovered within the first year after the shock. Among the households
that lost more than 30% of their income between 1992-93, only 15% recovered in the first year and
73% had not recovered after five years.
These calculations might be interpreted as indicating that two types of income dynamics exist
amongst Hungarian families. For the first type, an initial income shock leads to only a temporary
drop in household income. However, it seems that for almost half of the households in Hungary, the
income shock was more devastating, and appears to have put them on a declining income path
leading to chronic poverty.
That interpretation is questionable however. There are other ways one might explain Table
1. Possibly the households that had not recovered within five years experienced other shocks in the
intervening period. Or possibly the first shock was not transient, and lasted for many years. Or the
shock may have been transient, but the recursion process is linear with a slow speed of adjustment
due to sizable lagged effects of past incomes on current incomes. One cannot conclude from Table
1 that short-lived shocks have long-lived impacts.
Quite generally we can postulate that a household has its own stable equilibrium income Y*
which is a function of the household's characteristics. The time it takes for the household to reach
its equilibrium state depends on the size of the income shock, the level of pre-shock income, and the
characteristics of the household. However, conditional on household characteristics there may well
be more than one steady state.
It is instructive to first examine some graphs of the relationship between income changes and
initial incomes to see if there is any sign of multiple equilibria. Figure 2 shows a smoothed plot (a
Lowess running-line smoother) based on the pooled sample of observations for all six years of the
survey. On the vertical axis we graph the difference between current and last-year's income. The
horizontal axis gives last-year's income. The intersections with the horizontal axis represent
9
equilibria at mean values of all other factors influencing incomes. There is only one stable
equilibrium Y* in the positive quadrant. For all households that had last-year's income less than Y*
the difference Y(,) -Y(,,) is positive. Over time, the income of such households will increase until it
reaches Y*. Households with income in the previous year greater than Y* will experience a decline
in income over time, and their income will stabilize at Y*.
It can be seen from Figure 2 that the relationship is quite flat in a neighborhood of the
equilibrium. Consider the interval between the median and 2 Y minus the median (i.e., a symmetric
interval around Y*). On the lower side of Y* (with rising incomes), the slope is about
-0.15, equivalentto an autoregression coefficient of0.85. The slope is abouttwice as high onthe side
with falling incomes, implying an autoregression coefficient of 0.70. The slope tends to rise at low
incomes, implying lower serial correlation.
Figure 2 suggests considerable stickiness (high serial correlation) in household incomes in
a neighborhood ofthe steady state equilibrium. Modest transient shocks from equilibrium could thus
entail quite long-lasting effects, given this pattern in the income dynamics. This does not arise from
multiple equilibria, but rather serial dependence of incomes in a region of the equilibrium. Consider
a one-year only income loss at year 1 for a household at the steady state value indicated in Figure 2.
With an autoregression coefficient of 0.7, about half this shock will still be evident in year 3 and one
quarter in year 5.
The pattern in Figure 2 was also found when we stratified the sample into various household
types. Figure 3 shows a non-parametric estimation of income dynamics for male and female headed
households. While income trajectories for these two types of households look similar, the point of
a stable equilibrium for the households headed by females is associated with the lower level of
household income.
Figure 4 gives the results stratified by the educational level of the household head. Again,
there is only one point of stable equilibrium in the positive quadrant of (Y(,), Y(,-,)) space for each type
of household. The equilibrium level of income almost coincides with the median income for
households whose head has a high-school-only level of education. For such households one would
expect to observe both downward and upward income mobility. For households with higher levels
of education, the equilibrium levels of income exceed the median incomes, and this difference is
10
larger for the households where the head holds a university degree. More than half of these
households experience upward income mobility in the absence of income shocks.
5. Econometric model
To further investigate household income trajectories with a broader set of controls, and to allow for
attrition, we need an econometric model. Total household income Y(,) at time t is assumed to be a
smooth non-linear function J(Y(),p X,) of income Y(,l) at time t-l and the set of household
characteristics (X,), both permanent and time-variant, at period t. The simplest form of the non-linear
relationship between Y(,) and Y(,,) that can allow two equilibria in a positive quadrant as a general
case is a third degree polynomial. That is what we assume.
Numerous consistent estimators for dynamic panel data models have been proposed in the
literature, including IV type estimators (Balestra and Nerlove, 1966; Sevestre and Trognon, 1992;
Anderson and Hsiao, 1982), FIML estimators (Bhargava and Sargan, 1983) and GMM estimators
(Arellano and Bond, 1991; Arellano and Bover, 1995). However, none of these methods controls for
panel attrition, which is clearly an important feature of the data, and may well be endogenous to the
shocks and household characteristics. We estimate a dynamic panel data model of income dynamics
with a control for panel attrition bias, treating lagged income as endogenous.
The system of equations for the six-year (1992-1997) panel of Hungarian data consists of five
simultaneous equations of income dynamics for the years after the first, namely:
3
Yi(t) = Yo + E amYi(t-i) + Xi(tp + Ei(t) (t = 1,...,5) (1)
m=1
where Yi, is the total income of household i in year t, Yi(, ,) is total income of household i in year
t- 1, X, is a vector of exogenous variables, and the a's and O's are unknown parameters. The error
terms are allowed to be serially dependent and correlated with lagged incomes. Following Bhargava
and Sargan (1983), we also have an instrumenting equation that determines initial income (1992)
as a function of the exogenous variables for all six years of the survey:
6
Yio =o + E Xk(i)bk + EiO (2)
k=O
11
where the bk's are the vectors of coefficients on all exogenous variables.
To control for attrition bias, we estimate equations (1-2) simultaneously with the equation
that determines whether the households that were selected in the sample in the first wave of the
survey stayed in the panel until the end. The equation that controls for attrition has the form:
Z; = X,i7 + Si Di = 1 if Zi > 0
Di = 0 otherwise
Pr(D, = 1) = Pr(191 > -X1i7r) = T(X,i7r) (3)
where Ziis a continuous latent variable that determines whether the household was in the sample in
rounds 1 through 6 and D, is an indicator variable that has value 1 if the household stayed in the
sample all six years and has the value 0 otherwise, Xli is the vector of explanatory variables from the
first wave of the data and T is the cumulative normal distribution function.
To estimate the system of simultaneous equations (1)-(3) we use a Semi-Parametric Full
Information Maximum Likelihood method (Heckman and Singer, 1984; Mroz and Guilkey, 1992;
Mroz, 1999). A five-factor specification is used to approximate an unrestricted error structure for
equations (1)-(3). The Appendix describes our estimation method in detail.
The set of exogenous variables includes: household size, number of children under 7 years
of age, number of children 7-16 years, number of elderly people, type of locality where the
household resides, gender and educational level of the household head, and some household asset
indicators. Endogenous variables consist of the polynomial of lagged income. Values of the
exogenous and endogenous variables are normalized to be in the [0,1 ] range.
For comparison, we also estimate (1)-(2) without the correction for attrition. The econometric
specification is then a simplification of the model described above (see Appendix).
6. Results
Table 2 gives our estimates of equation (1). Household composition, characteristics of the locality,
and individual characteristics of the household members affect total income. The estimated
parameters on the Xvariables have the signs one would expect. Larger families tend to have higher
income, households with children are significantly poorer than households with no children,
12
households from Budapest are better off than households in other rural and urban areas of Hungary
and in rural Hungary. Households for which the head has a university degree have higher income,
families with access to land and households that own a car are better off. The presence of people
aged 60-69 has a negative impact on the level of total household income.
Table 3 gives the equation for attrition. There some significant demographic, life-cycle and
geographic effects. Households with a middle-aged head were less likely to drop out, as were smaller
households, and those not living in Budapest. However, the most notable feature is that initial
income is not a significant predictor of attrition. We also tried adding squared and cubed terms in
initial income, but these were individually and jointly insignificant.
We also tested whether negative income shocks lead to households dropping out ofthe panel.
To test this we used the second year as the base, namely 1993, and added a variable for the change
in income between 1992 and 1993. The coefficient on this variable was allowed to vary according
to whether income increased or decreased between 1992 and 1993. There was no significant effect
of an income change in either direction on the probability of staying in the panel; for an income
decline, the z-score was 0.27, while for an income increase it was 1.77 which is not significant at the
5% level (though it does make it at the 8% level).
To interpret the income dynamics implied by the parameters of our cubic specification,
letq = I, - Ia 2; r = I (aB - 3y) - a2 whereZ+a Z2 + , Z+y = 0 . Ifq3+r2>0 there will be one
3 9 6 27
real root and two complex conjugate roots, if q3+r2>0 all roots are equal and at least two will be
equal, and q3+r2<0 the equation will have three real roots. Given the values of the estimated
coefficients, the cubic polynomial equation has three real roots. However, we find that in the positive
quadrant there is only one point of equilibrium when setting all exogenous variables at their mean
points. This equilibrium is stable.
The income paths are different for households with different characteristics, though the
property of a single stable equilibrium still holds. For example, Figure 5 presents the simulated
income dynamics for the households categorized by educational level of the head interacted with
13
whether the household lives in Budapest or not.6 For each household category there is only one point
of stable equilibrium in the positive quadrant. (This was true for other combinations of
characteristics.) The equilibrium level of income for the households where the head holds a
university degree and lives in Budapest is the highest. It is almost five times higher than the income
level of households for which the head has no more than a high school diploma and does not live in
Budapest.
Would there have been a low level unstable equilibrium without the safety net? We repeated
this set of calculations setting all government transfers to zero. This ignores behavioral responses
to the safety net, though if anything one would expect that they would make it even less likely that
there is a nonconvexity at low levels, because pre-intervention incomes will probably not be as low
as simply subtracting transfers would suggest. Again only one root was found in the range of the
data.
7. Conclusions
Economic theory offers little support for the common assumption of linear income dynamics,
whereby households inevitably bounce back in time from a transient shock. Indeed, one can construct
theoretical models that exhibit nonlinear income dynamics, with low-level nonconvexities, such that
a short-lived uninsured shock can have permanent consequences. Whether this exists in reality, and
so might explain the seemingly persistent poverty that has emerged in many transition economies,
is an open empirical question.
We have offered what we believe to be the first test. This entails estimating a dynamic model
of incomes, allowing current income to be a nonlinear function of lagged income with endogeneous
attrition from the survey. On implementing the test on household panel data for Hungary in the
1990s, we find evidence of nonlinearity in the dynamics of household incomes. However, we find
6 The variables are scaled to be between 0 and 1 to minimize the likelihood of overflow and
underflow and to improve the convergence properties of the optimization algorithm (see for example
Judd, 1998).
14
no evidence in these data of low-level non-convexities. The data are not consistent with the existence
of an unstable equilibrium at low incomes.
Our results suggest that households in this setting tend to bounce back from transient shocks.
The adjustment process is clearly not rapid. Transient shocks can have relatively long-lasting impacts
due to the evident stickiness of incomes. However, it does not appear likely that a short-lived shock
can create permanent destitution.
15
Appendix: SPFIML estimation of equations (1)-(3)
Let the error terms of equations (1)-(3) have the form:
4
£i(t) P mi(t) + EP(It,v(li) + P(2i)V(2i) (4.1)
I=]
Si = ii + P(2i)V(2i) (4.2)
where Ii(,) is a normal IID random variable, Vm(1j) are components (common factors) of the error term,
which are uncorrelated with the observed exogenous variables of the model and uncorrelated with
I1i(t) but can be correlated with the lagged incomes in equations (1 )-(2), andV(2,, is a common factor
that is responsible for the correlation between the error terms arising from endogenous attrition. We
introduce five-factor specification to be able to approximate an unrestricted error structure for
equations (1)-(3).7 Conditional on the value taken by the factors v, and V2. the joint distribution of
the error terms can be written as:
5~~~~
1 £(m -E p1 P2V2
E(£O), ... (5), 1 |VI ... VI sv2 )=3 (,9 -P2V2). * p kleP=1 (5)
-,=O am Cy
where cUr'S are square roots of the variances of the error terms in equation (2), and(p is the
probability density function of a standard normal distribution. If the cumulative distribution
functions of v, is F1(v) and the cumulative distribution function of v2 is F2(v2), then the
unconditional distribution of the errors is:
f(8(Q),...,s(5),9)= f f(£(o) I.(5)1 1 1 v42W () V I' I ) dFl(V2 (6)
The cumulative distributions of the common factors v, and v2 can be approximated by a step
function. Suppose that the distributions of v, and v2 are given by:
7 For a discussion of the choice of the optimal number of factors, see Anderson and Rubin
(1956).
16
L
Pr(v' = t1O ) = p, > 0; E p, = 1 (I = l, .. . L; k = 1, ..4) (7.1)
L
Pr(v2 = y0) =In,Z 2 °;Ez1 = l(I= l,...,L) (7.2)
1=1
where 7k and yt are points of support of the approximated distributions, and k and 1 are the numbers
of points of support. Then the unconditional distribution functions are:
f(~~(o) L A B C D P9 I'21I _ P-- PI PI (8)2Y
AC(O 60), v 5)) yfF b P E d- 2r, P.I P_n Im P,IN P,mn PlV (P(dP2 (8)
1=1 -I b=l c-I d=1 Cd 0Jd ) m =O0, GCm
and the corresponding log-likelihood function for the system of simultaneous equations is:
E (L AIP. B C Y_- (m) Im 1 2 3 4 (9)
iz I 1=1 a=l b=l c=l d=l x2 = 0.0000
Log likelihood = -1515.032 Pseudo R2 = 0.0285
Note: * is significant at 10% level; ** at 5% level; *** at 1% level.
23
Figure 1: Income dynamics with a nonconvexity at low income
Yt Yt=Yt- I
y t- l
24
Figure 2: Non-parametric estimation of income dynamics in Hungary
20000
Median household income
0
0- -- - - 1__ _ _
-1000 0 _ _ _ _ _ _ _ _ ___ _ _ _ _ _ _ _
0 50000 100
Y(t-1 ) Y(t-1 )*100
Figure 3: Non-parametric estimation of income dynamics by the
gender of the household head
20000 Median income Median income
of female headed of male headed
households households
l7 \ ~ |Male headed
households
Female hea Jed
_i households
_1 oooO- I\
0 50000 100000
Y(t-1)
25
Figure 4: Non-parametric estimation of income dynamics by the
education level of the household head
20000 5 Median income Median income
high school university
\ ~~~~~~~~Median ir COI ne
_ ~~~~~~~~\ ~~~~technical
. : : '~~~~~~~~~- ~~~~ e ~~~University
0
\T~~~~c nica
-10000)
High sc
- T
0 50000 100000
Y(t-1)
Figure 5: Simulated income dynamics from the econometric model for households with
different levels of education and in Budapest versus other regions
V(t) Y(t-l)
Budapest, Universlty educated
-1=
,=