Download Evidence on multiple equilibria in the United States unemployment rate Don Harding

Evidence on multiple equilibria in the United
States unemployment rate
Don Harding
La Trobe University
September 25, 2011
Abstract
I …nd evidence of multiple equilibria in United States unemployment rates using a partially linear autoregressive (AR) model is that
provides a parsimonious extension of the linear model.
The non-linear component is estimated using the semi parametric
p
estimator developed by Peter Robinson. This estimator achieves T
convergence for the parametric component and therefore holds the
hope of achieving clear resolution regarding the nature of the dynamics
in the partially linear AR.
Application to theUnited States yields precise estimation of dynamics in the monthly unemployment rate. Con…dence bands are
su¢ ciently tight as to permit rejection of alternative hypotheses such
as the existence of a range of equilibria.
Three equilibria are found (6.0%, 9.3% and 10.25%). The lowest
of these is a candidate for naming as the ’natural rate’. However, the
dynamics make the economy slow to reach this ’natural rate’.
These …ndings provide a framework for understanding the di…culties faced by US policy makers. The …ndings also provide insights as
to why it is that DSGE models which are log linearized about a steady
state fail to …t the data in stuations where the economy is a long way
from the low unemployment equilibrium.
Key Words: Natural rate; Unemployment, Semi-parametric; Nonparametric; Kernel regression
JEL Code E24, C14, C22
1
1
Introduction
In his 1968 presidential address to the American Economic Association Milton Friedman stated that:
the ’natural rate of unemployment’is the level that would be
ground out by the Walrasian system of general equilibrium equations, provided there is imbedded in them the actual structural
characteristics of the labour and commodity markets, including
market imperfections, stochastic variability in demands and supplies, the cost of gathering information about job vacancies and
labour availabilities, the costs of mobility, and so on (Friedman
1968, p. 8)
A key feature of Friedman’s de…nition of the natural rate is the assumption that the equilibrium is unique. Several years after Friedman de…ned the
natural rate, Debreu [1974], Sonnencschein [1974] and Mantel [1974] proved
that in economies where there are more consumers than commodities, any
continuous function that satis…es Walras law can be an excess demand function for some set of preferences and technology.1 Put more starkly, the assumption that the economy comprises consumers and …rms that optimize
over convex sets has no aggregate implications other than that the value of
excess aggregate demand must be zero. In short we cannot a priori rule out
the possibility that there exist multiple equilibria.
The theoretical considerations sketched above motivate this paper which
seeks to assemble nonparametric evidence on the nature of equilbria in the
United States unemployment rate. Most work in macroeconomics proceeds
by ignoring the possibilty of multiple equilibria and invokes the assumption
of linear dynamics either directly or through log linearization about a steady
state.
In this paper I show that using semi-parimetric techniques it is possible to
assemble strong evidence of multiple equilibria in the US unemployment rate.
One important implication of this …nding is that macroeconomic modelling
procedures that involve log linearizing about a single unique steady state
must produce poor and misleading results. This is particularly so when the
economy is near the high unempoyment equilibrium which is what I …nd for
the US at present. Another important …nding is that the model can explain
1
See Shafer and Sonnenschein (1982) for a discussion and detailed introduction to the
literature. Mas Collell (1995) provides a textbook treatment.
2
the slow adjustment to the low unemployment equilibria without assuming
near unit root behaviour or hysteresis.
1.1
Literature
Phelps (1967, 1968) and Friedman (1968) presented the natural rate as a
unique equilibrium about which the economy ‡uctuated. Later Phelps (1994)
advanced the structuralist extension to the natural rate hypothesis in which
a range of factors such as labour market regulation, labour productivity,
the real exchange rate, real interest rates, energy prices, commodity prices
and general changes in the structure of the economy occasionally caused
permanent changes to the natural rate. After such a permanent shock the
economy ‡uctuated about its new ‘natural rate’. Papell et al (2000) provide
tests of the structuralist hypothesis.
Evidence that causes doubts about the natural rate is shown in Figure
1. It is di¢ cult to discern any particular unemployment rate that could be
identi…ed as ‘the natural rate’to which the United States economy returns
quickly or regularly. It is also di¢ cult to discern large shifts in unemployment
that would be seen in the structuralist interpretation.
Figure 1: Monthly and annual Unemployment rates, United States: Annual
1890 to 2009 top panel; Monthly 1948.1 to 2010.4 bottom panel
Persistence in annual unemployment rate 1890 to 2009
Annual unemployment rate 1890 to 2009 (per cent)
30
100
90
80
70
Unemployment this year
Annual unemployment rate (per cent)
25
20
15
10
5
60
50
40
30
20
10
0
0
1880
0
1900
1920
1940
1960
1980
2000
20
30
40
50
60
70
80
90
100
Unemployment last year
Monthly unemployment rate January 1948 to April 2010
Persistence of the monthly unemployment rate
16.0
14.0
14.0
12.0
Unemploymenr this month
Unemployment rate (per cent)
10
2020
12.0
10.0
8.0
6.0
10.0
8.0
6.0
4.0
4.0
2.0
2.0
0.0
0.0
1947-Jan
0.0
1957-Jan
1967-Jan
1977-Jan
1987-Jan
1997-Jan
2007-Jan
2.0
4.0
6.0
8.0
Unemployment Last month
3
10.0
12.0
14.0
The hysteresis hypothesis advanced by Blanchard and Summers (1986,
1987) led many authors to model the unemployment rate as if it were well
approximated by an I(1) process. There is a vast literature that tests whether
or not the unemployment rate has a unit root. This literature focuses attention away from the important questions which are how good is the I(1)
approximation and is it inferior or superior to feasible alternative speci…cations of the DGP for the unemployment rate.
1.2
Plan of paper
Section 2 addresses the question of how good is the I(1) approximation to
the DGP for the unemployment rate.
The partially linear model set out in section 3 represents a feasible alternative to the I(1) approximation. A semi parametric procedure for estimating
this model is discussed in section 4. Results from applying this estimation
procedure to the monthly unemployment rate for the United States are discussed in section 5. Implications of the …ndings for economic theory and
policy are discussed in section 6. Conclusions are presented in section 7.
2
Implications of the I(1) approximation
Recently, Beyer and Farmer (2007) expressed doubts about the natural rate
hypothesis. They study the Phillips curve within framework where unemployment is I(1): This literature can be traced back to King and Watson’s
(1994) revisionist interpretation’of the data in which they argued that the
unemployment rate should be modelled as if it were I(1). To see the implications of this assumption assume that zt has the following Beveridge-Nelson
decomposition2
zt = z0 +
Where
(1) =
1
P
j=0
(1) ("1 + ::: + "t ) +
j;
t
=
1
P
j "t j ;
j=0
t
j
"t ~N (0; )
0
=
1
P
i=1
j+i
and
1
P
j=0
(1)
j j j < 1.
Constructed in this way zt is the sum of initial conditions, a random walk
and a stationary process. If (1) 6= 0 the random walk determines the long
run properties of zt . In the literature the unemployment rate is related to zt
by a monotonic transformation. The simplest transformation which appears
frequently in the literature is,
2
Here I specify a gaussian process but nothing of importance turns on this speci…cation.
4
(2)
ut = zt
Models such as Staiger Stock and Watson (1997, 2001) have the representation (1) as do many other models of the unemployment rate. An objection
to the setup in (1) and (2) is that the unemployment rate is bounded above
and below and these processes will eventually violate those bounds. This has
led to suggestions that the unemployment rate should be modelled as arising
from monotonic transformation of zt that respects the bounds. For example,
100
1 + g (z)
u (z) =
(3)
and its inverse
z (u) = g
1
100 u
u
(4)
where g (z) is a monotonic function with the property that g (z) > 0 .3
Implications of the model (1) and (2) are discussed in section 2.1. While
implications of the model (1) and (3) are discussed in section 2.2.
2.1
If United states unemployment is an I(1) process
how quickly would it exceed its bounds?
The practice of modelling the unemployment rate as an I(1) process, as in (1)
and (2), is based on the proposition that for the length of data considered the
probability that the unemployment rate exceeds its bounds is su¢ ciently low
as to make the approximation reasonable for empirical work. As t becomes
large the distribution of zt is well approximated as N z0
(1)2 2 t so
0;
the probability that at date t the unemployment rate is within its bounds of
zero to 100 percent is
100 u0
p
(1)
t
Pr (ut 2 [0; 100] ju0 ) ' 1
u0
p
(1)
t
Clearly, this probability goes to zero as t goes to in…nity. But the more
interesting question relates to the rate at which it goes to zero. To establish
this I obtain estimates of (1) and 2 by estimating an AR(12) on the …rst
di¤erence of the monthly unemployment rate for the United States. The
estimated coe¢ cients are in Table 1..
3
A special case is the logistic transformation suggested by Wallis (1987) in which g (x) =
e andg 1 (y) = ln y:
x
5
Table 1: Estimated coe¢ cients for AR(12) in change in unemployment rate
Estimate
Standard Error
t_statistic
ut 1
0.006
0.037
0.16
ut 2
0.234
0.037
6.39
ut 3
0.170
0.037
4.53
ut 4
0.095
0.038
2.49
ut 5
0.116
0.038
3.03
ut 6
-0.005
0.038
-0.13
ut 7
-0.023
0.038
-0.59
ut 8
0.017
0.038
0.46
ut 9
-0.004
0.038
-0.10
ut 10
-0.111
0.037
-2.98
ut 11
0.024
0.037
0.65
ut 12
-0.116
0.037
-3.17
b (1) = 1:675
R2 = 0:18
b2 = 0:195
Using the logistic transformation, the estimated parameters in table 1
and starting from an unemployment rate of 3:4 percent (the rate in January
1948), the the model (1) and (2) implies that there would have been a 40 per
cent chance that the unemployment rate would have exceeded it bounds in
December 1999.
Simulation of the model 10,000 times each with a path 120 years in duration yielded the statistic that 88 per cent of paths exceeded the observed
minimum and maximum unemployment rates of 1.2 and 21.3 per cent respectively. A basic requirement of any model is that when simulated it generates data with properties akin to those on which the model was estimated.
Clearly, the model in which unemployment is I(1) does not meet this basic
requirement and therefore cannot be considered a plausible approximations
to the DGP of the unemployment rate.
2.2
What are the implications of modelling transformations of the unemployment rate as I(1)?
The …nding of the last section does not rule out the possibility some monotonic
transformation of the unemployment rate such as zt in (3) and (4) can be
unbounded and have an DGP that is I(1).
Suppose that zt has the representation (1). Using the notation that capitals denote random variables and lower case denote realizations, Zt is approximately N z0
(1)2 2 t . The transition probability distribution
0;
6
of the unemployment rate is de…ned as P (t; u0 ; B)
Pr (Ut 2 BjU0 = u0 )
and that for Zt is Q (t; z0 ; D) = Pr (Zt 2 DjZ0 = z0 ) for intervals B and D
respectively. Using the normality assumption
Z
1
z z0
p
p dz
Pr (Zt 2 DjZ0 = z0 ) '
t
(1)
t
z2D
Now since u is a monotonic transformation of z the intervals B and D are
closely related. Speci…cally, let B (u; u) = [u; u] then D (u; u) = [z (u) ; z (u)] :
Thus the probability that Ut lies in the interval [u; u] is
Pr (Ut 2 [u; u] jU0 = u0 )
Pr fZt 2 [z (u) ; z (u)] jZ0 = z (u0 )g
z (u) z (u0 )
z (u) z (u0 )
p
p
'
(1)
t
(1)
t
Estimating an AR(12) on zt where zt
ln 100ut ut yields estimates
of b (1) = 1:626 and b = 0:0377: Thus the probability that starting at an
unemployment rate of 3.4 per cent in 1948.1 the unemployment rate is in the
observed range of 1.2 to 21.3 per cent is 0.51. Again this model fails the test
of being able to generate data akin to that on which it was estimated.
To further understand the problems with this speci…cation note that as t
)
z(u) z(u0 )
p 0 and
p
both go to zero so that
becomes large z(u)(1) z(u
t
(1)
t
lim Pr (Ut 2 [u; u] jU0 = u0 ) = 0
t!1
so that as t goes to in…nity the probability that Ut is in the interval
[u; u] becomes vanishingly small. Clearly this is unattractive in a model of
the unemployment rate. To fully appreciate how unattractive this is as a
model of the unemployment rate it is instructive to calculate the limit of the
transition probability distribution. De…ne P (t; u0 ; u) as the probability that
Ut is in [0; u]; where u < 100, conditional on u0 2 (0; 100) follows
P (t; u0 ; u)
Pr fZt 2 [z (u) ; 1] jZ0 = z (u0 )g
Z1
1
z z0
p
p
=
dz
t
t
z(u)
= 1
z (u)
7
z (u )
p 0
t
The long run distribution is denoted F (uju0 ) is de…ned as
F (uju0 ) = lim P (t; u0 ; u)
t!1
1(u<100)
0)
In this case lim z(u) pz(u
= 0 so that F (uju0 ) = 12
: Thus the long
t
t!1
run distribution does not depend on the initial value u0 : The long run density
is discrete so that Pr (U = 0) = 12 and Pr (U = 100) = 12 : It is evident that
the long run distribution is also the stationary distribution. Clearly, these
long run and stationary distributions are unsuitable for modelling the United
States unemployment rate which is observed to have a minium of 1.2 per cent
and a maximum of 21.3 percent over the 120 year period from 1890 to 2009.
3
A partially linear model of the unemployment rate
An attractive option that achieves much of the parsimony of the linear model
but respects the bounded nature of the unemployment rate is the partially
linear model
k
X
ut =
(5)
i ut i + ' (ut 1 ) + t
i=1
Where t is a martingale di¤erence process.
This model is rich in its implications. For example if
' (ut 1 ) = 0
> 0
< 0
ut
ut
ut
1
1
1
2 [u; u],
2 [0; u)and,
2 [u; 100)
Then we have a bounded random walk with a range of equilibria in the
interval [u; u]. Models with this type of equilibria have been discussed by
Early papers were Woglom (1982), Bhaskar (1990) and Akerlof, Dickens and
Perry (1996).
If on the other hand there exist r points f0 u1 < u2 ::: < ur 100g such
that
' (ui ) = 0
' (ut 1 ) > 0
' (ut 1 ) < 0
ui 2 fu1 ; u2 :::; ur g ,
ut 1 < u1 and,
ut 1 > u r
8
Then the model has r equilibria and the equilibria at u1 and ur are stable.
Moreover, if over some range ua ; ub the distance between ' (ut 1 ) and
ut 1 is small relative to the distance between u1 and ut 1 then adjustment
to equilibria will be slow.
Thus the partially linear model can capture many of the features considered important in discussions of the unemployment rate.
4
Semi parametric estimation
To explore these various possibilities the model (5) can be estimated semiparametrically using Robinson’s(1988) estimator. This involves two steps.
The …rst step estimates and the second step estimates ' (ut 1 ) :
4.1
Estimating
Take the expectation of (5) conditional on ut
E ( ut jut 1 ) =
k
X
iE
i=1
1
to obtain
( ut i jut 1 ) + ' (ut 1 )
(6)
Subtract (6) from (5) to obtain
ut
E ( ut jut 1 ) =
k
X
i
[ ut
E ( ut i jut 1 )]
i
i=1
Replacing the expectations with sample analogs allows estimation of . Form
u
et
i
Where
b ( ut i jut 1 ) =
E
T
k
T
and the matrix of regressors
1
k
T
X
uj
K
uj
i
j=k+1
1
fb(ut 1 ) =
et = [( ut
X
T
X
1
K
uj
j=k+1
u
et 1 ) ; :::; ( ut
ut
h1
1
1
fb(ut 1 )
ut
1
i = 0; :::; k
1
h1
k
u
et k )]
yet = ut
u
et
Note the random denominator fb(ut 1 ) causes some technical di¢ culties that
are discussed in Li and Racine (2007, p224). These technical di¢ culties can
9
be addressed by trimming out observations where fb(ut 1 ) < bT for some
bT that is small and has the property that bT ! 0 as T ! 1: As Li and
Racine (2007, p225) observe in empirical applications one can ignore trimming (because for an appropriate choice of bT ; fb(ut 1 ) > bT for all t within
the sample). The estimators set out below ignore trimming for that reason.
et .
The coe¢ cients are obtained via OLS regression of yet on X
b=
X
1
et X
e0
X
t
X
et yet
X
The asymptotic distribution of e is given in Li and Racine (2007) as
p
d
1
T b
! N 0; 1
1
1
A consistent estimator of
is b 1 b b 1 where b and b are de…ned
in exercise 7.2 of Li and Racine (2007) That is,
X
et X
et0
b= 1
X
T k
b=
4.2
1
T
bt =
k
ut
X
et X
e0
b2t X
t
u
et
Estimating ' (u)
et b
X
De…ne yt as
yt =
ui
k
X
i=1
e E ut
i
i
Li and Racine (2007) Robinson (1988) suggests a feasible estimator of
' (ut 1 )
XT
K ut h12 u yt
t=1
'
b (u) = XT
K ut h12 u
t=1
The asymptotic distribution of '
b (u) is set out in chapters 2, 7 and 18 of
Li and Racine (2007). Speci…cally,
Where =R
and 2 =
p
R
T h2 '
b (u)
' (u)
d
h22 B (u) ! N
2
0;
(u)
f (x)
K ( )2 d ; 2 (u) = E (vt2 jut 1 = u) and B (u) =
2
K ( ) d see Li and Racine (2007 p62-63).
10
2
2
f2f 0 (u) '0 (u) + f (u) '00 (u)g
5
Evidence on natural rate for the United
States
Robinson’s semi-parametic estimator is be applied to study the DGP of the
United States annual and monthly unemployment rates shown in Figure 1.
Through out a Gaussian kernel is used with a plug in estimator of h =
1
1:06 std(ut ) T 5 :
When applied to the monthly United States unemployment rate data
from 1948.1 to 2010.4 the …rst stage yields the estimates of shown in Table
2.
Table 2: Estimated AR coe¢ cients for partially linear model of United States
unemployment rate 1948.1 to 2010.4
Estimate
Standard Error
t_statistic
ut 1
0.003
0.073
0.04
ut 2
0.227
0.044
5.21
ut 3
0.164
0.044
3.72
ut 4
0.0908
0.035
2.61
ut 5
0.113
0.047
2.39
ut 6
-0.007
0.050
-0.13
ut 7
-0.025
0.036
-0.67
ut 8
0.015
0.039
0.38
ut 9
-0.008
0.041
-0.18
ut 10
-0.117
0.039
-2.99
ut 11
0.016
0.039
0.42
ut 12
-0.124
0.039
-3.18
The estimate of ' (u) obtained from the second stage of the Robinson
estimator is shown in Figure 2 together with con…dence bands constructed
with 95% coverage probability. Clearly, ' (u) is estimated with considerable
precision as the con…dence bands are very tight.
Several features are evident in Figure 2. First, there are three equilibria
and a ’near equilibrium’.4 The equilibrium at an unemployment rate of 6.0
per cent is stable and is a candidate for the name ’the natural rate’. The
equilibrium at an unemployment rate of 9.3 per cent is unstable. Clearly
once unemployment rates exceed 9.3 per cent the dynamics of the unemployment rate push it towards higher values. There is a stable equilibrium at
4
I use the term ‘near equilibrium’in the sense that it is so close to being an equilibrium
that a small change in the structure of the economy could make it an equilibrium
11
Figure 2: Semi parametric estimate of '(u) for United States monthly unemployment rate: 1948.1 to 2010.4
0.15
Upper 95% conf idence band
0.1
Semi-parametric estimate of non linear component
Change in unemployment rate from
non linear term (per centage point)
0.05
0
-0.05
Lower 95% conf idence band
-0.1
-0.15
-0.2
0
2
4
6
Unemploy ment rate last month
8
10
12
10.25. Once the unemployment rate exceed this level the dynamics of the
unemployment rate push strongly towards a lower unemployment rate.
Figure 2 provides an informative insight into the behavior of unemployment in the United States over the period August 2009 to May 2010. During
this period the unemployment rate ranged between 9.7 and 10.1 per cent
which is the range in which ' (u) is positive and achieves a local maximum.
Thus the current unresponsiveness of the unemployment rate to economic
stimulus is part of the dynamics of the unemployment rate rather than re‡ecting additional weakness in the economy. Stimulus needs to push the
unemployment rate below the (unstable) equilibrium at 9.3 per cent before
the dynamics of the labour market will begin to push unemployment towards
the equilibrium at 6.0 per cent.
The sum of the coe¢ cients in table 2 are approximately 0:35. Since these
coe¢ cients are on the ut j terms they can be described as representing
’momentum’as they magnify the e¤ects from the non linear terms.
The conditional standard deviation of the shocks to the partially linear model are shown in …gure . The standard deviation of these shocks
are somewhat larger than the estimated non linear component ' (u) : Thus,
movements caused by the deterministic component in Figure 2 are masked
12
by random ‡uctuations of the same order. The standard deviation of these
shocks when unemployment is 2 per cent is two-thirds of the value when the
unemployment rate is 11 per cent.
0.32
Conditional standard error of shock
0.3
0.28
0.26
0.24
0.22
0.2
0.18
0.16
2
3
4
5
6
7
8
9
10
11
Unemployment rate
Finally, the density of United States unemployment rates is shown in
Figure 3. The density has a single well de…ned mode at about 5.5 per cent.
This is below the equilibrium at 6.0 per cent for two main reasons. First,
the standard deviation of the shocks is smaller below 6.0 per cent than it
is above 6.0 per cent. This means that when the unemployment rate falls
below 6.0 per cent it spends longer in that region.
6
Implications for economic theory and policy
The natural rate continues to plays an important role in macroeconomic theory and most macroeconomic models used for policy have a unique natural
rate embedded in them. Some models (eg Beyer and Farmer (2007)) do not
have the natural rate feature as they model unemployment as a transformation of an I(1) process. The results on I(1) processes presented earlier should
raise serious issues about models with this speci…cation. Most notably these
models imply that there is zero probability of observing data paths that are
close to those on which the models are estimated.
13
Figure 3: Density of monthly unemployment rate: United States 1948.1 to
2010.4
0.35
0.3
Frequency
0.25
0.2
0.15
0.1
0.05
0
2
3
4
5
6
7
8
9
10
11
Unemployment rate
Empirical results raise the question of how might theoretical models be
modi…ed to encompass dynamics closer to those estimated with the partially
linear model. Most notably what features of the economy cause the two
equilibria at 9.3 and 10.25 per cent respectively. A related question that
also has both theoretical and policy dimensions is what changes (if any) to
the structure of the economy would turn the ‘near equilibria’at 4.2 per cent
unemployment into a stable low unemployment equilibria. And would the
bene…ts from these changes exceed the costs.
From a short run stabilization perspective there is the question of what
are the costs in terms of increased in‡ation of attempting to stimulate the
economy so as to hold it at the low unemployment ‘near equilibria’
Many DSGE macroeconomic models are solved by linearizing about the
steady state. The empirical results in Figure 2 show that for unemployment
rates between 5 and 8 per cent this practice is reasonably accurate. But it has
some dangers. For unemployment rates higher than 8 per cent is encounters
di¢ culties with the two equilibria at 9.3 and 10.25 per cent respectively. For
unemployment rates below 5 per cent log linearization misses the important
feature of a low unemployment ‘near equilibria’.
14
7
Conclusions
Models in which the unemployment rate is modeled as a monotonic transformation of an I(1) process are implausible in that when simulated they cannot
generate simulated data paths with features akin to those seen in the data
on which they were estimated.
The partially linear model provides a parsimonious alternative that encompasses a rich range of dynamics. The model can be estimated using
Robinsons (1988) semi parametric estimator. The estimates are surprisingly
accurate and provide a clear resolution of unemployment dynamics. Three
equilibria can be found. The lowest of which is at 6.0 per cent and might
be named the natural rate. The resolution of the semi parametric method is
su¢ cient to make these equilibria clearly de…ned. Adjustment to the natural
rate in this model can be complicated and slow.
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