Download Re-thinking Equilibrium Presidential Approval: Markov-Switching Error-Correction 1 Long-Run Equilibria in Presidential Approval

Re-thinking Equilibrium Presidential
Approval: Markov-Switching Error-Correction
Simon Jackmany
July 18, 1995
1 Long-Run Equilibria in Presidential Approval
If a week is a long time in politics, then surely some forty odd years is a very long time.
From the mid-1950s to 1992 the United States had eight dierent presidents, has been in
and out of wars, recessions, economic booms, and political crises of one form and another.
Technological developments have radically altered the way political information is generated
and communicated, to say nothing of the content of that information over time.
Given these changes in the political and economic context in which presidents operate,
how likely is it that a single statistical regime will adequately capture the relationship between
presidential approval and its determinants for the whole time series? Put dierently, is there
any good reason to expect that the eects of the variables structuring presidential approval
will be constant over the entire time series, relative to one another? One can well imagine
circumstances in which ination might receive a heftier weight than other circumstances in
determining presidential approval? Or unemployment, business expectations, or even the
Prepared for the 12th Annual Political Methodology Summer Conference, Indiana University, Bloomington, Indiana, July 27-30, 1995. I thank Larry Bartels and Rob McCulloch for useful comments and
discussion. Errors and omissions remain my own responsibility.
y Department of Political Science, University of Chicago, 5828 S. University Ave, Chicago IL 60637.
([email protected])
1
extent to which presidential approval exhibits inertia? Are rally events disruptions to a longrun equilibrium relationship between approval and economic conditions, or do rally events
prompt a switch to a dierent statistical regime? Are returns to equilibria symmetric? That
is, does over-approval return to equilibrium levels as quickly as under-approval? In short, how
likely is it that the same vector of parameters will adequately tap the relationship between
approval and its determinants over presidents of dierent partisan orientations, through war,
peace, booms and recessions, and generational turnover in the electorate?
2 Markov-Switching Time Series Models
Here I investigate these possibilities with what I call a \Markov switching error correction
model". The mathematics of Markov chains provide a way of modelling changes in discrete
states, and so the application of Markov chains to regime shifts is relatively straightforward.
Markov chains are a simple way of modelling a discrete-valued random variable as a time
series, where here the discrete-valued random variable is a latent indicator of the state the
substantive process is in. At any time point t, t = 1; 2; : : : ; T , the state variable s takes on a
integer value labelling the states (i.e., 8 t; s 2 f1; : : : ; N g, where N is the number of states.
The Markov switching model I employ here considers just two states, though theoretically
more are possible.
Formally, a two-state Markov-switching model of a time series, y , is
t
t
1
y
t
8
>
<
t
X0 + Y 0 + u
=>
: X0 + Y 0 + u
t
1
t
1
1t
t
2
t
2
2t
if s = 1
if s = 2
(1)
t
t
where X is a vector of exogenous variables (possibly including a unit vector), Y is a vector
t
t
I restrict my attention to two states since this seems fairly standard in the econometric
and statistics literature on these models, and increasing the number of states raises tough
questions of interpretation and parsimony. Standard likelihood tests break down here since if
the true process is characterized by N , 1 states, since the parameters of a N state model are
unidentied under the null hypothesis. Hamilton (1994, 698) elaborates. See also Hansen
(1992). In addition, my experience is that the programming and computational burdens
increase apparently exponentially with the number of states.
1
2
of lags of y , fu g are sequences of random iid Gaussian disturbances with mean zero and
variances < 1 and fs g is a Markov chain with two states f1,2g such that
t
it
2
t
i
P (s
t
= 2js , = 1) = p
t
1
1
and
P (s
t
= 1js , = 2) = p :
t
1
2
Note that the two series of disturbances are independent of one another, and of the Markov
process. Given the denition of the transition probabilities above, a more complete statement
of the Markov process is
2
3 2
6 P (s = 1) 7 = 6 1 , p1
4
5 4
p2
t
P (s
t
= 2)
32
3
7 6 P (s ,1 = 1) 7 :
54
5
t
1,p
p1
2
P (s ,1 = 2)
(2)
t
The rst term on the right-hand side of (2) is referred to as a transition matrix, which
is dened in terms of the transition probabilities p and p (note that the columns of the
transition matrix sum to one, consistent with the rules of probability). For simplicity, these
transition probabilities are assumed constant over the sample (xed transition probabilities,
or FTP) though time-varying transition probabilities (TVTP) have been recently considered
in the literature. In the two-state, FTP case I consider here, p and p completely characterize the Markov chain, and are parameters to be estimated in addition to each state's
structural parameters, = (0 ; 0 ); i = 1; 2.
Estimation of this model occurs via iterative procedures described in the Appendix (EM,
or a Gibbs sampler). In addition to estimating parameters these procedures also yield estimates of the time-varying mixing probabilities { the quantities on the left-hand side of (2) {
which are simply optimal estimates of the probability that the process is in a particular state
at a given point in time. If, as one would hope, the estimates of the states reect meaningful distinctions in the data, then these probabilities in turn convey substantive information
1
2
2
1
i
i
2
i
Filardo (1994) models transition probabilities as a function of exogenous variables, Durland and McCurdy (1994) model transition probabilities as duration-dependent, and Ghysels
(1994) considers periodicity and seasonality in the Markov process.
2
3
about the serie's determinants at various time points. In an ideal world, an analyst would
create interactions or transformations on the right-hand side of a regression equation to capture subtle uctuations, non-linearities and interdependencies in the structural component
of a time-series. Here, the unobserved mixing probabilities optimally weight the estimated
states, and so might be thought of as proxying for an \omnibus" variable, unobserved by
the analyst, but shaping the way independent variables aect the dependent variable under
study.
2.1 Labelling States
And here lies a danger with this approach. In allowing statistical estimation procedures
to tease out dierences between (unobserved) regimes, and allowing the Markov process to
evolve independent of known dynamics in the data, one is faced with a problem of ex post
interpretation. What do the states mean? What does it mean for the process to be in state
\1" with high probability? An analogy here may be useful; I see this problem as akin to
producing ex post substantively meaningful interpretations of the dimensions produced by
exploratory factor analysis. As I show below, carefully interpreting each state's parameter
estimates allows one to make some conclusions about what the states mean; alternatively,
one might attempt to impose some structure on the Markov process directly via explanatory
variables, though such work is still in its infancy, and doesn't pin down the issue of stateidentication.
Another approach to this problem is motivated by Bayesian ideas: typically the researcher
will have some hunches about what distinguishes the states, and this prior information ought
to be incorporated in any modelling exercise. Examples of prior information might include
beliefs that variability in one state is greater than others (i.e., have the belief that > be
reected in a prior, see Albert and Chib 1993) or that the eect of one parameter is greater
than in another (e.g., specify a prior that > , see McCulloch and Tsay 1994). Given
the diculties of a formal, classical test of the number of states, this Bayesian approach
seems attractive. Sensitivity analysis casts further light on the problem. By easing the
1
1;k
4
2;k
2
priors on the parameters that distinguish the states a researcher can monitor the extent to
which the states become less distinct (McCulloch and Tsay 1994, 524). Increasingly diuse
priors might lead to estimates of the structural parameters that are constant across states
or estimates of state probabilities that hover close to .5 (in the two state setup). In either
case it would seem that at least on the basis of the data alone are insucient in say, by the
parameter estimates
3 A single-state error-correction model of presidential
approval
I begin by presenting estimates of a single-regime, single-equation error correction model
of presidential approval series in Table 1. Data sources are listed in the Appendix. Since
my focus here is on the Markov-switching model, I will pause only briey to explain the
error-correction set-up. A longer elaboration appears in Jackman (1995). Where one has
strong beliefs over the direction of causality in an error-correction context (in particular,
that re-equilibration can be conned to movement in just one variable) the typical two-stage
estimation process can be collapsed into a single regression (e.g., Beck 1992, 243). Formally,
the equilibrium relationship can be written as
y
t
=! +! x + ;
t
(3)
= y , !^ , !^ x
(4)
0
1
t
with
^
t
0
t
1
t
as the estimate of disequilibrium component of y to be used as an additional regressor in
the second-stage regression in dierences.
If the analyst can be condent that x is (weakly) exogenous to y then re-equilibration
occurs via changes in y , and one can proceed directly to the estimation of the regression in
dierences. That is, the analyst could estimate the \second" stage of the error-correction
t
t
t
t
5
Table 1: Single-Equation Error Correction Regression Analysis of Presidential Approval
Coecient Variable
Estimate Std Error
,:82
4:78
7:35
1:03
4:60
2:36
,3:16
1:83
,:18
:04
,:11
:19
,:68
:36
:046
:028
:12
:04
4:24
:66
:90
Turning Points Correctly Predicted (%)
70:7
DW
1:96
n.b., dependent variable is APP . Seven unreported indicator variables \dummy out" rst observation of each presidency.
N=139.
0
Intercept
1
Rally
2
Gulf War
3
Vietnam
APP ,1
, !1
INFL ,1
, !2
Unemp ,1
, !3
S& P 500 ,1
, !4
BEXP ,1
r2 (dierences)
r2 (levels)
t
t
t
t
t
t
Structural Parameters, Equilibrium Relationship,
Bootstrapped Sampling Distributions, Summary Statistics
Parameter Variable
Point Estimate Median 5{95 percentiles pr > 0
!
INFL ,
,:60 ,:58 [-2.41, 1.35]
:71
!
Unemp ,
,3:85 ,3:85 [-7.41, -.56]
:03
!
S& P 500 ,
:26
:24
[-.002, .62]
:95
!
BEXP ,
:68
:68
[.30, 1.22]
:997
1
1
t
2
t
1
3
4
t
t
1
1
6
system,
y = + z + ^ , + ;
0
t
1
t
t
1
(5)
t
in one-step with
y = + z + y , , ! x , + ;
t
1
0
t
1
t
1
t
1
t
(6)
where z are sources of short-term uctuation in y . I denote the intercept in the \one-step"
equation as to distinguish it from ; the former collapses the intercepts from both the
levels and dierences equations, while the two-step procedure yields unique estimates of each
intercept (see Beck 1992, n7). Also note that in this compact form the coecient on the
lagged exogenous \levels" variable, x , , is , ! ; i.e., the equilibrium structural parameters
are not recovered directly from this \reduced form" error-correction model. One approach is
to estimate equation (6) via non-linear least squares; however, since is estimated separately
when using a least-squares routine, I recover an estimate of ! by dividing the OLS estimate
of , ! by -1 times the OLS estimate of , ,^, and bootstrap to derive the sampling
distribution of this quotient of two regression estimates.
Since I have condensed the usual two-step procedure into one equation some explanation
of the model is warranted. I hypothesize that equilibrium levels of presidential approval are
driven by the real economy (ination, unemployment, and changes in the stock market), plus
expectations about business conditions (the BEXP variable). Short-term changes in approval
are a function of lagged out-of-equilibrium approval, Rally events, an additional indicator
for the Gulf War, and the Vietnam battle deaths variable. All variables are described more
fully in the Appendix. My classication of the candidate regressors into the \equilibrium" or
\short-term" types closely follows the approach used by other analysts modelling presidential
approval in an error-correction framework (e.g., Ostrom and Smith 1992). In the second
part of Table 1 I report the results of a re-sampling procedure used to recover the structural
coecients from the equilibrium relationship, ! ; : : : ; ! .
These estimates by and large repeat the ndings of MacKuen, Erikson, and Stimson
(1992). The equilibrium relationship between approval and the economic variables is fairly
robust, though the business expectations variable washes out the eects of ination; in the
t
t
0
0
t
1
1
1
1
1
7
4
aggregate, it seem that the prospective information contained in expectations about the
economic future is more consequential for presidential approval than a seemingly concrete
indicator like the current rate of ination. The error correction mechanism is statistically
signicant but does not speedily return approval to levels we would expect on the basis of
prevailing economic conditions and economic expectations; only 18% of \out-of-equilibrium"
approval is corrected in each quarter, implying that (^ + 1) = :82 20% of out-ofequilibrium approval persists some two years ahead. The contemporaneous eects of \irrational" expectations, rally events and other determinants of approval are but a modest
portion of the cumulative long-run eects of these variables; for any given contemporaneous
eect, Z , the cumulative eect is Z=(,^) where ^ is the estimate of the lagged approval in
equation (6). The estimated dynamics here suggest that approval is fairly \sticky", with the
contemporaneous eects accounting for only 18% of the total long-run eect of an input to
the presidential approval error-correction system.
8
8
4 Two states of presidential approval
I turn now to a two-state Markov-switching error-correction model. Each state's parameter estimates are reported in Table 2, along with estimates of the transition probabilities. In
each state an error-correction mechanism is supported by the data; the parameter estimate
on the lagged level of presidential approval is unambiguously within the negative unit interval
in both cases. However, in State 1, the error correction eect is almost three times as strong
as in State 2. In State 1 over 30% of the previous quarter's out-of-equilibrium presidential
approval is dissipated, while only 11% is corrected by the State 2 error-correction mechanism. In State 1 then, the past carries far less weight than contemporaneous inuences in
shaping presidential approval. State 2 though is characterized by a high degree of inertia
in presidential approval; out-of-equilibrium presidential approval lingers for a much greater
degree than in State 1.
Figure 1 displays the dierence between State 1 and State 2 in these patterns of per8
Table 2: Markov-Switching Error-Correction Model of Presidential Approval
Coecient Variable
State 1
State 2
Intercept
,13:91
10:03
(5:14)
(4:22)
Rally
8:87
7:08
(1:07)
(:93)
Gulf War
,5:55
6:31
(3:49)
(1:87)
Vietnam
,4:74
,2:30
(2:16)
(1:53)
APP ,
,:30
,:10
(:05)
(:04)
, !
INFL ,
:30
,:41
(:20)
(:17)
, !
Unemp ,
,2:04
:10
(:43)
(:30)
, !
S&P 500 ,
,:11
:12
(:03)
(:02)
, !
BEXP ,
:36
,:04
(:05)
(:04)
2:89
2:89
Transition Probabilities
:65
:33
n.b., dependent variable is APP . Seven unreported indicator variables \dummy
out" rst observation of each presidency. constant across states (identifying
constraint). Transition probabilities are probability of changing to other state at
t + 1, conditional on being in given state at t. Standard errors in parentheses.
N=139.
Structural Parameters, Equilibrium Relationships,
Bootstrapped Sampling Distributions, Summary Statistics
Parameter Variable
Point Est. Median 5-95 % prop > 0
State 1:
!
INFL ,
:97
:97 [-.11, 2.34]
:93
!
Unemp ,
,6:70 ,6:70 [-9.28,-4.62]
0
!
S&P 500 ,
,:34 ,:34 [-.60,-.16]
:001
!
BEXP ,
1:18
1:18 [.89, 1.61]
1
State 2:
!
INFL ,
,3:96 ,3:95 [-9.16,-1.39]
:008
!
Unemp ,
:93
:93 [-3.49, 9.03]
:63
!
S&P 500 ,
1:14
1:14 [.63, 2.69]
:998
!
BEXP ,
,:41 ,:40 [-1.66, .17]
:13
0
1
2
3
1
t
1
1
t
2
1
t
3
1
t
4
t
1
t
1
1
t
2
t
1
3
4
1
1
t
1
1
t
1
t
2
t
1
3
4
t
t
1
9
100
Figure 1: Dierences in Presidential Approval's Durability, State 1 vs State 2.
0
20
Shock (%)
40
60
80
State 1
State 2
0
5
10
15
Time (Quarters)
sistence, for an identical, nominal shock. After twelve quarters about 25% of the shock is
retained in the State 2 regime. However, in State 1, after just four quarters the eect of
the nominal shock has already dropped to roughly this 25% level. New information exerts
a considerably more powerful inuence in State 1 than in State 2, the latter exhibiting considerable persistence in presidential approval. Figure 2 displays the actual period-specic
error-correction eects, bounded by the \pure" State 1 error-correction eect (-.31) and the
\pure" State 2 eect (-.10); estimated period-specic eects between these \pure" types
reects the estimated probability of mixing between the two states.
The dierence across the two states in the coecients for lagged business expectations
(BEXP) is also noteworthy and large. In State 1 business expectations are an important
determinant of presidential approval; the bottom panel of Table 2 reports estimates of the
structural parameters of the equilibrium part of the error-correction model, for both states.
Business expectations in State 1 pick up a large positive coecient. Each point of the
Survey of Consumer's 200-point BEXP scale translates into roughly 1.1 points of presidential
10
Figure 2: Estimated Period-Specic Error-Correction Eects
Eisenhower
JFK
LBJ
Nixon
Ford
Carter
Reagan
Bush
-0.10
-0.15
-0.20
-0.25
-0.30
1956
1958
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
approval. Contrast State 2. Both reduced form and structural parameters for BEXP are
indistinguishable from zero in State 1.
These two dierences alone represent quite a departure from standard stories about business expectations and presidential approval. The relationship between presidential approval,
its past, and economic expectations is fairly subtle; the evidence here indicates that the
eects of lagged approval and business expectations on current approval is not constant, but
is stronger at some times than at others. And it is only in State 1 that I nd a strong,
signicant relationship between economic expectations and presidential approval.
In State 2 there is no relationship between economic expectations and presidential approval. The process characterized by the State 2 parameter estimates might be termed a
\steady-state" or \business-as-usual" type of presidential approval, characterized by high
11
1.00
0.75
0.50
0.25
0.0
1958
1960
Eisenhower
1956
LBJ
1966
1968
1970
Nixon
1972
1974
Ford
1976
1980
Carter
1978
1982
1986
Reagan
1984
Figure 3: Estimated Probability of State 1.
1964
JFK
1962
12
1988
1992
Bush
1990
inertia (a low rate of error-correction), and responsiveness to the usual \objective" economic
culprits (with the exception of unemployment) and rally events. And moreover, State 2 is
the more prevalent state in these data (see Figure 3).
In Figure 4 I graph the predicted values of change in presidential approval implied by
State 1, State 2, and the overall model (the latter quantity being the weighted sum of the
state-specic predicted values). By and large, State 1 is associated with falls in presidential
approval; about seventy percent of the data points in the rst panel of Figure 4 are below zero.
Given the faster error-correction mechanism implied by the State 1 estimates, it seems that
State 1 brings presidential approval back down to levels we would be more likely to expect
on the basis of prevailing economic conditions, with economic expectations also contributing
signicantly. In contrast, State 2's contribution to the mixture-model span a larger range
Probability of being in State 1
13
\
t
1956
-10
0
10
20
1956
-5
0
5
10
15
20
25
-15
1956
-10
-5
0
5
1958
Eisenhower
1958
Eisenhower
1958
Eisenhower
1960
1960
1960
1962
JFK
1962
JFK
1962
JFK
1964
1964
1964
1966
LBJ
1966
LBJ
1966
LBJ
1968
1968
1968
1974
1976
1972
Nixon
1974
1976
Ford
1978
1970
1972
Nixon
1974
1976
Ford
1978
1980
Carter
1980
Carter
1980
Carter
1978
State 2, Predicted Effects
1972
Ford
Complete Model, Predicted Effects
1970
1970
Nixon
State 1, Predicted Effects
1982
1982
1982
1986
1986
1984
1986
Reagan
1984
Reagan
1984
Reagan
1988
1988
1988
1992
1992
1990
1992
Bush
1990
Bush
1990
Bush
Figure 4: Predicted Change in Presidential Approval (APP ), Markov-Switching Error-Correction Model
Change in Presidential Approval
than State 1's contributions, in part a reection of the slower error-correction mechanism in
State 2. And since the overall process tends to gravitate towards State 2 (recall the transition
probabilities reported in Table 2), it seems that State 1 seems to operate as a short, sharp
corrective to out-of-equilibria approval around rally events. Prominent examples of State 1
period-specic corrections in my data include
the Watergate revelations (73:2), associated with a decline of 15 points in approval of
Nixon;
an estimated 6.5 point rebound in Ford's approval after his pardon of Nixon (75:2);
poor economic conditions under Carter, leading to period-specic declines of 6.5 points
of approval in both 79:1 and 79:2;
uctuations in Carter's approval through the Iranian hostage crisis (a estimated 4 point
boost to approval in 79:4, but a more-than-osetting 5 point decline in 80:2);
the Iran-Contra revelations under Reagan (86:4, but with the correction specic to
State 1 coming in 87:1, estimated to be on the order of -11 points of approval); and
three, large, negative impacts on Bush' approval rating in 90:4, 91:4, and 92:1, estimated to be -11, -7, and -11 points, respectively.
Large State 2 estimated contributions include
a 6.5 point increase in Nixon's approval in 72;2, associated with the Moscow U.S.{
Soviet summit, and a 5 point boost in 73;1, associated with Nixon's re-inauguration
and the end of draft for the Vietnam War;
the Watergate revelations, associated with estimated 5 point declines in Nixon's approval in both 73:3 and 73:4;
Ford's pardon of Nixon (roughly -5 points in 75:1);
14
the rally in support for Carter during the early stages of the Iran hostage crisis (9
points in 79:4, 5 points in 80:1, but a 7 points decline in 80:2);
a rally in approval of Reagan after the killing of U.S. Marines in Beirut and the invasion
of Grenada (83:4, 6 points);
a 5 point fall in approval coinciding with the Iran-Contra revelations (86;4); and
a mammoth 25 point boost to George Bush's approval in 91:1 at the time of the Gulf
War.
Figure 5 is a slightly dierent interpretation of these data: a scatter-plot of the mixture
model's predicted change in presidential approval by the probability of State 1, along with
a least squares regression t (and 95% pointwise condence bounds) and a loess t. This
graphical representation of the data repeats the story of Figure 4 quite starkly. The more
\pure" instances of State-1-type-approval are associated with the more precipitous predicted
falls in presidential approval. The labelling of the more extreme observations also reveals
the volatility of the approval-generating mechanism under George Bush's tenure: 90:2, 90:4,
91:1, 91:4, and 92:1 are among the extremes in both the probability of either state and the
predicted change in presidential approval.
I am condent that the dierences between States 1 and 2 here are statistically signicant.
The single-state error-correction model of presidential approval has a log-likelihood -389.61,
an estimated standard error (^) of the regression of 4.24, and an r of .66. The mixture
model I present here has a log-likelihood of -368.10, an estimated standard error of 2.89
(which is held constant across the states; see the Appendix for more details), and an r of
.78. Minus twice the dierence in the log-likelihoods is the test statistic for a likelihood
ratio test; put simply, the likelihood ratio test here tests whether the dierence in the loglikelihoods is signicant given the extra degrees of freedom consumed in estimating the extra
parameters of the mixture model. Twice as many regression parameters are estimated in
the mixture model than in the single-state model, and two transition probabilities are also
estimated, for a total of 11 extra parameters. Accordingly, the likelihood ratio test statistic
2
2
15
\
Figure 5: Estimated Change in Presidential Approval (APP ) and Probability of State 1 (P [S = 1]).
t
t
74:4
91:1
10
75:2
0
55:3
79:1
-10
16
Predicted Change in Presidential Approval
20
81:1
79:291:4
90:4
87:1
92:1
90:2
80:2
73:2
0.0
0.2
0.4
0.6
Probability of State 1
0.8
1.0
here is ,2 (,389:61 + 368:10) 43:02 which asymptotically follows a distribution
with 18 degrees of freedom (16 structural parameters per state { including the seven dummy
variables for change of presidencies { plus the two transition parameters). In this instance
the null hypothesis can be rejected fairly overwhelmingly; the critical p = :05 value for a distribution with 18 degrees of freedom is 28.87. The improved t of the Markov-mixing
error correction model is not a clever statistical sleight of hand, or simply a consequence of
estimating more than twice as many parameters as the single-state ECM. A model which
allows switching between two statistical regimes quite clearly is a better characterization of
the dynamics of presidential approval than the single-state model.
My ndings suggest that past ndings about the relationship between economic expectations and presidential approval require revising. In particular, economic expectations matter
in structuring presidential approval, but only when State 1 mixing probabilities are non-zero.
Whenever State 2 dominates, my estimates are that economic expectations have no eect
on presidential approval. Economic expectations carry large weight only in State 1, and a
standard single-regime time-series analysis of presidential approval that nds large eects for
economic expectations might be picking up on the eects of economic expectations in the
relatively few cases where State 1 mixing probabilities are close to one, in much the same
way as outlying observations can cause a linear regression to yield a signicant relationship
between two variables. A standard single-equation analysis in eect aggregates across States
1 and 2, nding an fairly impressive overall eect for economic expectations.
The switching-regime model tells a more realistic story. Economic expectations are part
of the mix of considerations which comes into play to help drive approval back towards more
\realistic" levels after the shock of a rally event. Economic expectations are not a constant
determinant of presidential approval, but exert powerful eects on presidential approval from
time to time, helping to speed the return of presidential approval to levels more in line with
economic circumstances after the intrusion of a rally event. The variability in the eect of
economic expectations on presidential approval is for all practical purposes equivalent to the
mixing probabilities plotted over time in Figure 3. Since economic expectations count for
2
2
17
nought in State 2, at any given point in the time series the eect of economic expectations
on presidential approval is the equal to the the coecient on economic expectations in State
1 times the probability that the process is in State 1.
The dierences across states in other independent variables are of interest as well:
3
Ination diminishes presidential approval in State 2 (t ,2:5, reduced form coecient, and the structural parameter for the equilibrium relationship in levels is -3.96
and highly signicant). But there is also some weak evidence to suggest that ination
modestly enhances approval in State 1 (t 1:46 for the reduced form coecient, and
the structural parameter is small but signicantly positive).
Unemployment has a statistically signicant eect on presidential approval only in
State 1, and in the anticipated negative direction. This eect is large relative to the
ination results; a one-time additional percentage point of unemployment in State 1
leads to roughly a one-time two point decrease in presidential approval, while the eect
in the equilibrium relationship is much larger: a sustained increase in unemployment
would lead to a new equilibrium level of presidential approval 7 points lower than the
prior equilibrium level (see the estimates of the structural parameters in the bottom
panel of Table 2).
The variable tapping changes in the stock-market, S&P 500, has a small, negative
eect in State 1, but has a reasonably impressive structural parameter in State 2
(approximately 1.14, and clearly non-zero).
The Gulf War variable is at rst glance an odd case, since it picks up a statistically
signicant positive coecient in State 2 (t 3:4), but a marginally signicant negative
coecient in State 1 (t ,1:6). Two caveats are in order here. First, the Gulf War
eect can not be interpreted without also considering the eect of the Rally variable.
The predicted eects plotted over time in Figure 4 are generated in this fashion: i.e., form
y^ by multiplying the observed data on X by the estimated coecients for each state, but
condition on the probability of being in either state by multiplying y^ by that probability.
3
t
t
t
18
In 90:4 both Rally and Gulf War take on the value 1; in 91:1 they both take on the
value 2. The eect of the Gulf War rally is thus partitioned into two components;
the part due to a general \rally" eect, and an additional component picking up to
the extent to which the Gulf War is an extraordinary rally event. Second, the eects
reported in Table 2 have to be multiplied by the period-specic probabilities of being in
either state in order so as to derive the actual eect for a specic period. The massive
boost in approval for Bush associated with Gulf War comes in 91:1, when the process
appears to be in State 2, thereby ignoring the negative coecient on this rally in State
1. The \rally" eects of the Gulf War at time t can be expressed formally as
y^
= p(s = 1) y^ + p(s = 2) y^ ;
h
i h
i
0
0
^
^
^
^
= P [s = 1] X + 1 , P [s = 1] X ;
t
t1
t
t
t2
t
t
1
t
t
2
where P^ [s = 1] is the estimated probability that the system is in State 1 (and since
there are only two states here, one minus this quantity is the probability that the
system is in State 2), and ^ denotes a vector of the relevant parameter estimates from
State 1 (and similarly for State 2), and X is a vector of the relevant variables (Rally
and Gulf War) measured at time t. Substituting the relevant parameters estimates
from Table 2, state probabilities, and values of Rally and Gulf War for 90:4 and 91:1,
I obtain the following predicted values for the Gulf War rally:
t
1
t
90:4
= 1) 1 , P^ (s = 1) Rally Gulf War y^
y^
1.00
0.00
1
1
3.32 0.00
91:1
0.00
t
P^ (s
t
t
t
1.00
2
t
2
st =1
st =2
y^
t
3.32
0.00 26.78 26.78
As this table makes clear, since presidential approval switches from \pure" State 1 in
90:4 to \pure" State 2 in 91:1, the calculations here are somewhat simplied. Combining the \generalized" rally eect and the incremental boost due to the Gulf War with
the estimated state-probabilities produces a small \rally" eect in 90:4 (on the order
of three points in approval). In 91:1 this eect is enormous { over eight times as large
{ worth roughly 27 points of approval to Bush.
19
5 Volatility between states and approval of Bush
Recalling Figure 3, it is apparent that State 1 is not at all prevalent these data; presidential
approval enters State 1 with probabilities equal to one in only four instances. Of these four
instances of \pure" State 1 presidential approval, two occur in the Bush presidency. But,
these cases aside, the dynamics of approval for George Bush more closely resemble those
captured by State 2.
Table 3: Summary Statistics, Probability of State 1, by Presidents
President Mean Median Std Dev N
Eisenhower
:29
:22
:27 14
JFK
:36
:35
:19 12
LBJ
:28
:25
:22 17
Nixon
:37
:36
:22 23
Ford
:34
:33
:22 9
Carter
:38
:33
:26 16
Reagan
:34
:34
:27 32
Bush
:40
:31
:40 16
Table 3 summarizes the smoothed state probabilities by presidencies, and this information
is also represented graphically in Figure 6. The median probability of approval being in State
1 is just .25 under George Bush; the second lowest median tendency towards State 1 I observe
in the 8 presidencies I analyze. But at the same time the smoothed state probabilities exhibit
their greatest variability under George Bush; while the median probability of \State-1-typeapproval" is comparatively low under George Bush, the mean probability is the highest among
the 8 presidencies (indicating a quite skewed distribution of state probabilities under Bush),
and the standard deviation for the Bush-specic state probabilities is almost twice as large as
the next largest presidency-specic standard deviation. Finally, it is also during Bush's tenure
that I nd 50% (three out of six) of the few instances where the system enters State 1 with
probabilities greater than .95. This volatility in the state probabilities (observable in the timeseries plotted Figure 3) stems in no small measure from the volatility in presidential approval
itself (see Figure 7). The mixture-model here provides a way of addressing the volatility in the
approval series by providing estimates of the state probabilities, which appear quite volatile
20
0.0
0.2
0.4
0.6
0.8
1.0
Figure 6: Estimated Probabilities of State 1, by Presidents
Eisenhower
JFK
LBJ
Nixon
Ford
Carter
Reagan
Bush
themselves under Bush's tenure. The rapidly varying state probabilities observed under the
Bush presidency in turn imply rapidly varying eects of the right-hand side variables on
presidential approval.
In short, these data strongly suggest that there is something unique about the translation
of economic perceptions into political sentiments during Bush's tenure. The low median
tendency towards \State-1-type" presidential approval observed under George Bush, coupled
with high volatility in the state probabilities in turn suggests that the mix of considerations
brought to bear in assessing George Bush was itself volatile. Recall that lagged values of the
business expectations variable and unemployment carry more weight in State 1 than in in
State 2, whereas approval is much \stickier" in State 2 and responsive to ination and recent
changes in the stock market. But the exceptional features here are the abrupt transitions in
21
and out of the two states under George Bush.
Doesn't this characterization strikes something of a chord when we recall the presidency
of George Bush? The pattern of volatility I report above bears a resemblance to the changing
perceptions of Bush as president: a sound, economic manager inheriting the reins of power
from a popular two-term president, stumbling in the summer 1990 budget negotiations with
Congress as the economy weakened, triumphant commander-in-chief and leader of a new
world order six months later, and then seemingly helpless to halt economic recession and a
growing sense of national malaise that lingered long into 1992, if not up until the election
itself.
Looking closely at the smoothed state probabilities, approval for George Bush enters
\pure" State 1 on three occasions; 90:4 (p 1.00), 91:4 (p .97), and 92:1 (p 1.00). All
three quarters involve some of the three most precipitous declines in Bush's approval: -10.67
points in 90:4, -11.33 points in 91:4, and -15.67 points in 92:1. In these same three quarters
the Survey of Consumers business conditions expectations measure, BEXP, was 75, 100 and
107.7, respectively, below or close its median under Bush of 105.3, and even further below
the median of BEXP for the whole time series of 111.3. The other variable that carries
large weight in State 1, unemployment, was rising through this period, further depressing
approval for Bush.
4
In fact, this last reading on BEXP in 92:1, the highest of the three here, asks respondents
to imagine the business conditions of 93:1 relative to 92:1, and may possibly reect an
optimism as to the state of the economy under a dierent President. I am reluctant to push
this point too hard, since while I found a cycle in BEXP corresponding to the presidential
electoral calendar, I have no evidence that this cycle is associated with (aggregated) beliefs
about the probability of the incumbent president being re-elected. Nonetheless, the monthly
data on BEXP from early 1992 reveal a denite bounce from low readings in late 1991.
Between November and December 1992, after the election, BEXP jumps 20 points (116
to 136), the single largest positive increase under Bush's presidency save for the 33 points
increase between February and March 1991 after the conclusion of the Gulf War (98 points
to 131).
4
22
6 Modelling Approval, more generally
The estimates I report here suggest a recasting of some common understandings about
presidential approval and its determinants, and perhaps especially economic expectations.
The Markov-switching model suggests fairly strongly that a single-state model is an inadequate approximation of the process generating presidential approval, at least over the long,
quarterly time series I analyze here. Put quite blandly, \dierent things carry dierent
weights at dierent times." Specically, economic expectations and unemployment are not
a \constant" in the mix of considerations brought to bear in assessing a president. The
role of economic expectations and unemployment is actually more limited than conventional
single-state analyses suggest. Economic expectations and unemployment appear to help reequilibrate presidential approval after rally events, as part of a State 1 \corrective" or even
\counter-shock". Recall that the set of parameter estimates specic to State 1 give heavy
weight to unemployment levels and economic expectations in the error-correction component
of the model. Furthermore, the transition probabilities show that State 1 occurs infrequently
and is not at all durable; as a result, State 2 dominates the mixture of state probabilities.
State 1 then appears as something of a \short, sharp, shock" to presidential approval,
hastening the return to normalcy after a rally event has driven approval to levels higher
than we would expect given economic conditions. This short, sharp, (counter) shock is
characterized by the aggregate public paying more attention to sources of possible economic
insecurity than is typical: unemployment and economic expectations. In the more normal
State 2 these variables count for nought in shaping equilibrium levels of approval; approval
appears to stand in an equilibrium relationship with ination and changes in the stock
market, and reverts only slowly to this equilibrium after a rally event. Cast in this light,
State 1 seems somewhat peculiar: economic considerations that appear to matter little for
most of the time abruptly become germane in assessing a president, and often, hot on the
heels of a rally event. These data appear to suggest a bitter and perhaps even ironic political
reality for presidents. Short term boosts in presidential approval quickly dissipate in response
23
to the aggregate public's attention turning to economic insecurities that do not appear to
drive assessments of presidents before or even during the rally. Unemployment and economic
expectations quickly claw approval back down into equilibrium, even though unemployment
and economic expectations do not form part of a major component of the typical (State 2)
equilibrium.
In short, while approval may rally from time to time, presidents can apparently rest
assured that the public will quickly nd something to grumble about. One interpretation
of these results is that after being distracted from economic concerns by a rally event, the
aggregate public's attention turns to quite weighty economic matters|the state of the job
market and the medium-term economic future|before settling back into a more \normal"
focus on ination and short-term changes in the stock market.
Pushing interpretations like this too far here is somewhat risky. Embedded in the model I
present here is the assumption that the Markov process governing the transitions from state
to state is independent of the other dynamics in the model, namely, the error correction
mechanism linking presidential approval to economic conditions. Nothing here necessarily
links occurrences of State 1 to out-of-equilibrium presidential approval, although, ex post it
does appear that this is a reasonable characterization of the switching process. In future
elaborations I aim to model the transition probabilities as time-varying, with a response to
dis-equilibrating rally events and other shocks an explicit part of the switching process. Other
extensions might be to compare how this switching model does compares with an ARCH
setup, given what appears to be increasing volatility in approval under Bush's tenure.
A Appendix
A.1 Survey Data
The Survey of Consumers is a periodic survey of the consumer attitudes and expectations,
conducted by the Survey Research Center of the University of Michigan. The Survey was
initiated in 1946, and provides a roughly quarterly series of economic expectations and
24
retrospections since the mid 1950s. Since 1978 the Survey has been administered monthly.
The measure of business expectations I use (BEXP) is derived from responses to the
following item in the Survey of Consumers:
And how about a year from now, do you expect that in the country as a whole,
business conditions will be better, or worse than they are at present, or just about
the same?
A.2 Aggregation
The dierence between the percentage of respondents replying \better" and \worse" (a
\balance statistic") is added to 100, to yield an aggregate score with theoretical bounds of
0 and 200. I average the monthly relative scores to from the post{1978 surveys to quarters,
so as to maintain comparability with the pre-1978 data.
The method of aggregation employed here is not as ad hoc as it might rst seem. The
key quantity is the dierence between the proportions of respondents reporting \higher" or
\better" expectations and respondents reporting \lower" or \worse" expectations; adding
one hundred to the result is an arbitrary scaling factor introduced for convenience. This
technique has been in use since at least the early 1950s (Pesaran 1987, 212) and was given
an explicit econometric foundation by Theil (1952).
A.3 Sources for Economic and Political Variables
All economic data are from CITIBASE, and the Rally measures and Vietnam measures
I employ follow fairly standard usages in the literature. The Rally measure is simply an
indicator, scored 1 for the following rallies: the Geneva Summit (55:3), the Cuban missile
crisis (62:4), the Moscow summit (72:2), the Paris peace talks (73:1), the Mayaguez incident (75:2), the Camp David accords (78:4), the assassination attempt (81:2), the Grenada
invasion (83:4), and the Gulf War (90:4, 91:1). Negative events (scored -1) include Nixon's
pardon (74:4) and the Iran{Contra controversy (86:4). Quarters 73:2{73:4 are coded -1 (Watergate), and the Iran{hostages crisis is coded 2 for 79:4, 1 for 80:1, and -1 in 80:2 (see
25
Figure 7: Gallup Presidential Approval, by quarters.
Eisenhower
JFK
LBJ
Nixon
Ford Carter
Reagan
Bush
80
70
60
50
40
30
1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992
MacKuen, Erikson, and Stimson 1992, 609). Vietnam is measured as tens of thousands of
U.S. battleeld deaths, by quarter. The Gallup presidential approval series (see Figure 7) is
originally from Edwards (1990), aggregated to quarters or months by simple averaging, and
is part of the approval.asc data set Neal Beck contributed to the maxlik archive, available
from Statlib (anonymous ftp to lib.stat.cmu.edu) in the general sub-directory. Pre-1953
Gallup approval numbers come from Gallup (1980).
A.4 Switching Regime Models in Time Series
The discussion here closely follows that in Hamilton (1994, ch22). Models with regime
shift are not uncommon in econometrics, but what is distinctive here is that the regime shifts
26
are not discrete. Econometric models with regime change typically involve shifts from one
regime to others, and often the shifts are once-and-for-all, and one might use a Chow test
or some recursive estimation procedure to nd where a structural break takes place. The
Markov switching model I consider here is also a \mixing" model, in the sense that at any
given time, all regimes (or \states") have non-zero contributions to the structural component
of the model. At a given time point the dependent variable is a modelled as mixture of the
various regimes, and the sum over the weights of the regimes at any time point is one.
Accordingly, the Markov switching model is a varying parameter model of sorts, where
although the structural parameters associated with any one regime are constant, the relative
contributions of the regimes vary over time. The contribution of a given independent variable
varies over time, since the contribution of an independent variable in the overall model is
equal to the sum over the regimes of the relevant parameter estimate times the time-specic
probability of being in each of the states. Further, the mixing weights (state probabilities)
evolve via a Markov process, which I specify as independent of the dynamics linking past
observations of the dependent variable to current values.
This model is known in econometrics as \the Hamilton model" after Hamilton's (1989,
1990) use of this type of model to analyze U.S. GDP growth (e.g., Hansen 1992, Lam 1990).
But the model has been known to statisticians for some time as a \hidden Markov" model
(Pesaran and Potter 1992) or a \doubly stochastic" model (Tjostheim 1986) and can also
be thought of as special type of random coecient model (Tjostheim 1994; Nicholls and
Quinn 1982), sometimes referred to as a \suddenly changing autoregressive model" (SCAR)
(Tyssedal and Tjostheim 1988; Karlsen and Tjostheim 1990; Granger and Terasvirta 1993,
18{9, 144{5). The more common deterministic multiple regime setup can be considered
a special case of the Markov-switching model if one (or more) of the states in the Markov
process is an absorbing state (i.e., once the chain got to one or more of the states it remains
in that subset of states with probability one).
5
I am grateful to Nick Polson for enlightening me as to the history of this class of model,
and referring me towards the Tjostheim articles in particular.
5
27
A.4.1 Markov Chains
Formally, a Markov chain is just a simple mathematical characterization of how a discrete
variable like s changes over time. If s takes on integer values f1; 2; : : : ; N g then a fairly
simple specication of its dynamics might be to assume that the probability that s equals
some particular value j depends only on s , :
t
t
t
t
P (s
t
1
= j js , = i; s , = k; : : :) = P (s = j js , = i) = p :
t
1
2
t
t
t
1
(A1)
ij
In words, we say that s is an N -state Markov chain and p is the \probability that the
chain is in state j given that it was in state i". p is a transition probability. Transition
probabilities sum to one at any given time:
t
ij
ij
p 1 +p 2 +:::+p
i
i
iN
= 1:
(A2)
As I show in the text, transition probabilities are usually gathered in a transition matrix:
P
2
6
6
6
= 666
6
4
p11
p21
p12
p22
p1
p2
3
::: p 1 7
7
::: p 2 7
7:
7
.
: : : .. 7
7
5
N
...
...
N
(A3)
N
::: p
N
NN
In my application this reduces to a 2 by 2 matrix, since I have only a 2-state model.
A.4.2 Mixing Models
The form of the mixing model I use here is a special case of the following generic case
presented by Hamilton (1994, 690). Let y be an endogenous variable, and X be a k vector
of observations on exogenous variables. Let Y = (y ; y , ; : : : ; y ; X ; X , ; X ) be a vector
containing all observations up to time t. If switching between regimes follows the simple
Markov process described above, then the conditional density of y is
t
t
t
t
t
1
1
t
t
1
1
t
f (y js
t
t
= j; X ; Y , ; );
t
28
t
1
(A4)
where is a vector of parameters characterizing this conditional density. With N dierent
states in the process, there will be N dierent densities (i.e., j = 1; 2; : : : ; N in (A4). These
densities can be collected in an N by 1 vector .
For my error-correction setup, there are the unknown regression parameters to be
estimated for each state ( ; ; : : : ; ; ! ; ! ; : : : ; ! ), error-correction parameters
( ; ; : : : ; ), as well as state-specic error variances ( ; ; : : : ; ). For state j then,
t
1
1
2
2
1
N
2
N
2
1
N
2
2
2
N
y = Z + y , , ! X , + ;
t
t
j
j
t
1
j
j
t
1
(A5)
tj
which can be written more compactly (without loss of generality) as
y =
t
j
X + ;
(A6)
t
t
with N (0; ); 8 t.
Given the assumption of Normality,
2
tj
j
2
f (y js = 1; X ;
6
=4
3 2
6
1 ; 1 ) 7
5=6
4
3
p21 exp ,(y ,2 X ) 7
(A7)
, X ) 7
5;
,
(
y
1
p2 exp
f (y js = 2; X ; ; )
2
for the two-state model I employ. Note that the conditional densities depend on only the
current state, and not on what state the process happened to be in at t , 1. Hamilton
(1994, 691) considers relaxations of this, as does Lam (1990); it is also possible to impose
some structure on the Markov process via independent variables (e.g., Filardo 1994), but
here I prefer to keep the specication relatively simple, keeping the structural dynamics of
the model conned to an error-correction framework, and having the mixing be between two
regimes, each co-integrated.
The unconditional density of y is just a weighted sum of the state-specic densities in
, where the weights are simply the probabilities that the process is in each of the states, at
a particular time point. These probabilities are unobserved by the analyst in any interesting,
non-deterministic process, but here evolve according to a simple Markov process. The full
log-likelihood for the model is thus
t
t
t
t
t
t
t
2
2
t
2
1
1
t
t
2
t
2
2
2
2
1
2
t
t
ln L() =
XX
T
N
p(s
t
t=1
= j ) f (yjs = j; X ; ; );
t
j =1
29
t
2
t
j
j
(A8)
P
where = ( ; )0 = ( ; ; : : : ; ; ; ; : : : ; )0. Note again that
p(s = j ) =
1; 8 t.
Since this is a mixture model it is necessary to constrain the across the N mixtures: this
is because the log-likelihood function in (A8) has no global maximum, since a singularity
arises whenever one of the N mixtures is imputed to be equal to its mean with no variance.
Note that this also occurs if the algorithm attempts to t one data point as a separate regime
(Filardo 1994, 301). At this singularity the log-likelihood is innite. This problem is inherent
in all mixture models and my experience is that unconstrained optimization algorithms will
typically ounder on this problem. Setting all the variance parameters equal across the
states (i.e., dropping the j subscript on in (A8)) is a common solution to this problem
when working with mixture distributions. See Jackman (1994, 352) and the references
there. Preliminary work with alternative estimation strategies permitting a relaxation of
this constraint (e.g., an unreported implementation of a Gibbs sampler ) suggests that for
these data the equality constraint on the regime-specic variances is reasonable.
2
1
2
N
2
1
2
2
2
N
j =1
N
t
j
6
2
j
7
A.4.3 Inferences about Mixing Probabilities
Where do the p(s ) = j in (A8) come from? This is where the Markov process comes
into play. Let P (s = j jY ; ) be the best guess about s based on the information in the
sample data up through time t and the parameters . These conditional probabilities, for
j = 1; 2; : : : ; N can be stacked in a vector ^ j . The \one-step-ahead" forecast of this vector
is written ^ j , which is just the probability that the process will be in state j at time t + 1,
given sample information and the parameters up through time t. Optimal inferences and
forecasts result from a ltering algorithm that iterates on the following pair of equations:
^ ^ j = 0 ^j ,
;
(A9)
1 ( j , )
t
t
t
t
t t
t+1 t
t t
1
t
t t
t t
1
t
The researcher typically waits as one of the variance parameters shrinks towards zero
and the log-likelihood starts to increase rapidly, after having apparently settled down at a
local maximum. The EM algorithm employed here gives only linear convergence but has the
same diculty, just more slowly and less dramatically.
7 See McCulloch and Tsay (1993, 1994) and Albert and Chib (1993) for examples.
6
30
^
j
t+1 t
= P ^ j ;
(A10)
t t
where is the N by 1 vector of conditional densities dened (A4), P is the N by N
transition matrix dened in (A3), 1 is a N by 1 unit vector, and the symbol denotes
element-by-element multiplication. Thus the denominator in the expression (A9) is just the
weighted sum of the conditional densities in , where the weights are just the t , 1 forecasted
probabilities of being in the state corresponding to a particular element of . ^ j itself is just
a vector of normalized, weighted, conditional densities, where the weights are the forecasted
state-probabilities calculated at t , 1 and the normalization is with respect to the sum of
these quantities, ensuring that the probabilities sum to unity.
Some insight into the justication for this algorithm comes by noting that since X is
exogenous with respect to s , the j th element of ^ j , is equivalent to P (s = j jX ; Y , ; ).
Also, recall that the j th element of is f (y js = j; X ; Y , ; ). Thus the j th element of the
numerator in (A9) is the product of these two quantities, which in turn is the conditional joint
density of y and s , since the conditioning arguments are identical in the separate marginal
densities. Thus the \conditional" probability in (A9) is just the joint density divided by the
\marginal" density of y (Hamilton 1994, 693).
As in most Kalman-lter applications, there is also a smoothing step involved here,
which is used to derive estimates of the elements of P , the matrix of transition parameters.
In estimating these transition probabilities it is optimal to exploit all sample information
through time T . Smoothing algorithms provide a way of calculating the ^ conditional on the
sequence obtained by iterating on the equations in (A9) and (A10). For the simple rst-order
Markov chain considered here, the smoothing algorithm recommended by Hamilton (1994,
694) is
^ j = ^ j P 0 [^ j ^ j ]
(A11)
t
t
t
t t
t
t
8
t t
t
t
t
t
1
t
t
t
t
t
1
1
t
t
t
t T
t+1 T
t t
t+1 t
Here exogeneity is taken to mean that there is no information in X about s beyond
that contained in Y , . But in my application y y , which includes the lagged level of
y , , but this does not seem to violate the exogeneity condition, since y , is in Y , in any
event. Alternatively, the error-correction regression could be trivially re-parameterized as a
regression in levels with a lagged dependent variable.
8
t
t
t
t
1
t
1
t
t
31
1
t
1
where the sign denotes element-by-element division, and ^ j is calculated (and stored)
when iterating on (A9) and (A10). This smoothing algorithm starts with t = T , 1 and
with ^ j from (A10) with t = T . With the full set of smoothed probabilities, ^ j ; t =
T , 1; T , 2; : : : ; 1 one has a T by N matrix of probabilities with generic row-column element
(t; j ) equal to P (s = j jY ; ^ ). With this matrix maximum likelihood estimates of the
transition probabilities are simply
t+1 t
T T
t T
t
T
p^
ij
=
P
T
P (s = j; s ,1 = ijY ; ^ )
;
P (s ,1 = ijY ; ^ )
=2
P
t=2
t
t
(A12)
T
T
t
t
T
which is simply the number of times state i is estimated to have been followed by state j ,
divided by the number of times the process was in state i (Hamilton 1994, 695), and where
^ is the maximum likelihood estimate of .
A.4.4 Estimation
With an arbitrary (but reasonable) vector of starting values for ^ j and P one can start
the algorithm dened by (A9) and (A10). These starting guesses are only used in the rst
iteration of the estimation algorithm and the consequences of wildly implausible choices
of starting values seem rarely serious. Starting values are also required for the structural
parameters I have gathered here in . For these I simply use the results of the single-state
ECM, replicated across the N states (here N = 2); in this case one needs to be sure that some
asymmetry between the states is captured in the choice of starting values for the transition
parameters and/or ^ j otherwise the algorithm fails to improve on the (perfectly symmetric)
starting values.
With starting values of , say , the conditional densities dened in (A4) can be
used to cycle forwards through the data on (A9) and (A10), and then back through the
data using (A11) to obtain smoothed estimates of P (s = j jY ; ^ ). One then estimates
each of the N regressions in (A6) via GLS, with weights for the tth observation in the j th
regression equal to the square root of the smoothed estimate of probability P (s = j jY ; ^ ).
This application of GLS yields an updated estimate of = ( ; ; : : : ; )0. An updated
10
10
(0)
t
t
t
t
1
32
2
N
t
1.00
0.75
0.50
0.25
0.0
Prob of State 1
Prob of State 1
1.00
0.75
0.50
0.25
0.0
1.00
0.75
0.50
0.25
0.0
Prob of State 1
Prob of State 1
1.00
0.75
0.50
0.25
0.0
JFK
JFK
Eisenhower
JFK
Eisenhower
Eisenhower
JFK
Eisenhower
LBJ
LBJ
LBJ
LBJ
Ford
Ford
Carter
Carter
Ford
Carter
Ford
Carter
Iteration 12 - change in llh = 0.06312
Nixon
Iteration 7 - change in llh = 4.415
Nixon
Iteration 3 - change in llh = 0.02193
Nixon
Iteration 1
Nixon
Reagan
Reagan
Reagan
Reagan
Bush
Bush
Bush
Bush
Prob of State 1
JFK
JFK
Eisenhower
JFK
Eisenhower
Eisenhower
JFK
Eisenhower
LBJ
LBJ
LBJ
LBJ
Ford
Carter
Ford
Carter
Ford
Carter
Ford
Carter
Iteration 20 - change in llh = 0.004621
Nixon
Iteration 9 - change in llh = 1.497
Nixon
Iteration 5 - change in llh = 4.171
Nixon
Iteration 2 - change in llh = 0.5121
Nixon
Reagan
Reagan
Reagan
Reagan
Figure 8: Iterative History, State Probabilities, Markov-Switching Model of Presidential Approval
1.00
0.75
0.50
0.25
0.0
1.00
0.75
0.50
0.25
0.0
1.00
0.75
0.50
Prob of State 1
Prob of State 1
Prob of State 1
0.25
0.0
1.00
0.75
0.50
0.25
0.0
33
Bush
Bush
Bush
Bush
estimate of is obtained by simply taking (1=T ) of the combined sum of squares from the N
separate regressions. Together, the updates of and form ^ . Updates of the transition
probabilities are obtained by applying (A12) to the current round of smoothed probabilities.
The smoothed estimate ^ j replaces the starting values used to start the iterations on (A9)
and (A10) and another iteration commences.
This estimation procedure is an application of the EM algorithm (Dempster, Laird, and
Rubin, 1977). Convergence is linear but occurs reasonably quickly with modest computing
power. In S+ on a HP 715/64 workstation I obtained convergence (dened by the loglikelihood increasing by less than 10, ) in roughly 10 minutes; since convergence of the EM
algorithm is linear, I obtained convergence much more rapidly with less strenuous convergence criteria. Additional overhead was consumed monitoring convergence graphically: at
each iteration I plotted the estimates of the state probabilities. These estimates at selected
iterations are displayed in the panels in Figure 8. The algorithm appears to settle on estimates of the state probabilities quickly: for instance, its the distinctiveness of successive
observations during the Bush presidency becomes apparent after only four or ve iterations.
Like most optimization algorithms, a good deal of time is spent in the neighborhood of what
is at least a local optimum.
2
2
1
(1)
T
8
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