Transformative Policy Change
Matt W. Loftis and Peter B. Mortensen
mattwloftis@ps.au.dk peter@ps.au.dk∗
Aarhus University
April 14, 2015
Abstract
Theories of policy change vary in their concepts and explanations. Nevertheless,
they all depict a process in which decision makers’ weighting and evaluation of what
matters to policy decisions changes over time. That is, relationships between inputs
to the policy process and eventual outcomes are not static. Despite the theoretical
consensus on dynamic relationships, this expectation has not been systematically empirically tested. This paper sets out to do so by using state space models to estimate
time-varying relationships between inputs to the policy process and outcomes. We
apply this econometric technique to data spanning several decades across a variety of
contexts. The results indicate that many relationships are, indeed, time-varying and
suggest a variety of implications for future empirical research.
Most researchers studying public policy and public spending decisions with time series
analysis assume that the explanatory factors - whether political, economic, or institutional
have a time-invariant influence. Yet, this assumption about constant effects through time has
little grounding in substantive theories of public policy. Whereas the empirical approach has
been dominated by a search for stable estimates, important theories of policy change, developed over the past decades, imply time-varying effects. The Advocacy Coalition Framework
proposed by Sabatier (1987, 1988) and later developed in collaboration with Jenkins-Smith
(Sabatier & Jenkins-Smith 1993) is one. A second is the theory of social learning and policy
paradigms introduced by Hall (1989, 1993), and a third theory is the Punctuated Equilibrium
Theory developed by Baumgartner & Jones (1993).
∗
Working paper, please do not cite without permission.
1
The three theoretical approaches differ in their conceptualization, description and explanation of policy change, but they all depict a policy-making process in which the decisionmakers’ weighting and evaluation of what is important to policy decisions is likely to change
over time. Such change may be slow or fast, gradual or punctuated, and it may reflect changes
in core beliefs, policy paradigms or policy images. Nevertheless, a common implication of
the different theoretical approaches to policy change is that the importance of explanatory
factors varies over time. This means that scholars of policy change need to redirect focus
from estimating stable relationships between X and Y to a higher order level of theorizing
and modeling why and when the relationship between X and Y may change. Inspired by
(Lieberman 2002, 700) we denote the latter type of change transformative change.
The aim of this paper is to integrate the empirical study of public spending with the
dominant theories of temporal instability in policy and spending decisions. First, we review
traditional time series studies of public policy change. Second, we introduce the concept of
transformative change and show how it relates to the dominant theoretical accounts of the
policy-making process. Third, we discuss what the concept of transformative change implies
for time series studies of changes in public policy. Fourth, inspired by recent developments
in public opinion studies, we motivate our choice of method to study transformative change
empirically. Fifth, using U.S. public spending time series compiled by the Policy Agendas
Project we demonstrate how this approach can be implemented and what it can teach us
about spending change. An important aim of the paper is to argue for a new research
agenda that links the empirical approach to policy change closer to the theoretical emphasis
on transformative change, and it concludes by highlighting some of the next questions that
such a research agenda may focus on.
2
Time-series studies of public policy change
For decades public spending has been a widely used measure of public policy. The validity
of this policy indicator can be discussed, but to illustrate the non-dynamic character of
most time-series studies of public policy it suffices to review a wide range of time-series
studies using public spending as the dependent variable. Thus, we focus on research using
spending time series as the dependent variables, but claim that the general point about
a mismatch between the empirical approach and dominant theories of policy change is of
broader relevance to time series studies including those using other indicators of public policy
besides spending.
Davis, Dempster & Wildavsky (1966) were some of the first scholars to utilize spending
time series data. Based on Wildavsky’s (1964) qualitative studies of incremental decisionmaking in the U.S. Congress, they derived the implication that: “Since the budgetary process
appears to be stable over periods of time, it is reasonable to estimate the relationships in
budgeting on the basis of time series data” (Davis, Dempster & Wildavsky 1966, 531).
More particularly, they argued that year-to-year changes in budget appropriations could be
accounted for by a set of simple, linear equations. Of special interest to the argument of this
paper is, however, the conclusion they draw based on the simple regression analyses (ibid.,
542): “... the budget process is only temporally stable for short periods... scholarly efforts
would be better directed toward knowledge of why, where, and when changes in the process
occur.”
In 1974 the authors followed up on this conclusion and approached the temporal instabilities as an omitted variable problem (Davis, Dempster & Wildavsky 1974). Adding eighteen
exogenous variables to the simple time series models they hoped to be able to account for the
temporal instabilities. However, the impression is that the extended models do not explain
away the shifts in coefficients identified in the 1966-article. Surprisingly, given the focus of
the two seminal articles by Davis, Dempster, and Wildavsky, the identification of epochs and
shifts in the estimated parameters had very little impact on the field of time series analyses
3
of public spending. The part about stability and order had a much more lasting influence.
The time series approach to public spending has not been confined to studies of incrementalism. In particular, we have seen many studies trying to model time series of U.S. defense
spending, partly reflecting a genuine interest in this policy field during the Cold War, partly
reflecting availability of rather reliable and long data time series within this policy field. A
broad range of drivers of U. S. defense and military spending have been examined in these
time series studies including election cycles (Nincic & Cusack 1979, Cusack & Ward 1981, Zuk
& Woodbury 1986, Kamlet & Mowery 1987), measures of unemployment (Griffin, Wallace & Devine 1982, Cusack 1992, Su, Kamlet & Mowery 1993, Majeski 1992, Kiewiet &
McCubbins 1991), arms races (Ostrom 1977, Correa & Kim 1992, Ostrom 1978, Ostrom &
Marra 1986), inertia (Ostrom 1977, Majeski 1983, Correa & Kim 1992), and public opinion
(Hartley & Russett 1992, Wlezien 1996, Higgs & Kilduff 1993). As good time series data
has become available within other policy domains, we have seen an expansion of time series studies of non-defense spending changes (see e.g. Jones & Baumgartner 2005, Soroka &
Wlezien 2010).
The U.S. defense spending studies disagree with respect to many model assumptions,
but common to all of them is the assumption about invariant relationships over time. Apparently, this assumption is so uncontested that it is not given any serious consideration.
Nevertheless, the focus on ordered and stable relationships is difficult to justify based on the
shift points already identified in the first time series regressions produced by Davis, Dempster
& Wildavsky. Furthermore, as argued in the next section, the time invariance assumption is
not consistent with dominant theories of policy change. In other words, there are good reasons to expect that the relevance and weight of the various determinants of public spending
vary over time.
4
Dynamic Models of Transformative Policy Change
In the late 1980s and early 1990s, three important theories about policy change were developed. One is the theory about Advocacy Coalition Frameworks proposed by Sabatier
(1987, 1988) and later developed in cooperation with Jenkins-Smith (Sabatier & JenkinsSmith 1993). A second is the theory of social learning and policy paradigms introduced by
Hall (1989, 1993) and the third theory is the Punctuated Equilibrium Theory developed by
Baumgartner & Jones (1993). The three theoretical approaches differ in their conceptualization, description and explanation of policy change (for comparative reviews, see Cairney &
Heikkila 2014). Nevertheless, all three theoretical approaches share an important implication
about politics, namely that the logic of public policy making may change over time.
According to the Advocacy Coalition Framework, every coalition of policy makers contains secondary aspects, policy core beliefs, and so called deep core beliefs. Secondary aspects
comprise a large set of narrower beliefs of the policy makers concerning, for instance, their
evaluation of relevant problems and performances within a policy subsystem as well as their
preferences regarding regulations or budgetary allocations (Sabatier 1998, 104). Policy core
beliefs refer to a coalition’s basic normative commitments and causal perceptions across an
entire policy domain, and deep core beliefs include basic ontological and normative beliefs
operating across almost all policy domains (Sabatier 1998, 103). When it comes to the core
beliefs, these are assumed to be very stable, but they are not time-invariant. Even if changes
mainly involve secondary aspects, this may, for instance, have implications for the relative
importance of various causal factors within a given policy domain (Sabatier 1998, 104).
In Hall’s approach to policy change, a distinction is made between first-order, secondorder, and third-order changes (Hall 1989, Hall 1993). First-order changes regard changes
in the instrument settings, whereas second-order change is when the instruments as well as
their settings are altered even though the overall goals of policy remain the same (Hall 1993,
278-9). Third-order change occurs when all three components of policy change simultaneously: the instrument setting, the instruments themselves, and the hierarchy of goals behind
5
policy. Hall’s prime example of a third-order change is the paradigmatic change in British
macroeconomic policy in the 1970s and 1980s from a Keynesian mode of policy making to
one based on monetarist economic theory.
Just as changes in deep core beliefs are assumed to be very rare according to the ACF,
paradigmatic third-order changes are very rare according to Hall’s theory. It is important
to note, however, that even second-order changes or, in the parlance of ACF, changes in
policy core beliefs will most likely be reflected in time-varying relationships in time-series
models of public policy. A second-order change, for instance, reflects an alternation of the
hierarchy of goals behind policy (Hall 1993, 282), and a change in the policy core beliefs of
policy makers may involve a change of their causal perceptions of a policy domain (Sabatier
1993, 103). Put differently, if the decision makers’ causal understanding of the relationship
between problems and solutions changes within policy domains such as defense, welfare,
transportation, or environmental policies, then the relationship between X and Y probably
changes as well.
The latter implication also follows from the depiction of the policy-making process provided by the Punctuated Equilibrium Theory (PET). According to the original formulation
of the theory, the policy-making process is dominated by policy monopolies based on a dual
foundation of an institutional structure that limits access to the policy process and supported by a powerful policy image (Baumgartner & Jones 1993, 7). From time to time,
these monopolies are successfully challenged by outsiders and replaced by a new institutional structure supported by a new dominant policy image. Later, Jones and Baumgartner
introduced the model of disproportionate information processing, where the concepts of images and subsystems are generalized to a generic decision-making theory about information
processing and friction in the policy-making process (Jones & Baumgartner 2005). Yet, the
dynamic picture about how the understanding of policy issues changes over time remains a
central characteristic to the model.
Similar to the two other theories about policy change, Jones and Baumgartner’s model
6
is founded on a micro level of bounded rationality. The same was claimed by scholars of
incrementalism, but according to Jones and Baumgartner the incremental approach failed to
see that the serial processing capacities of the decision makers, which in periods of stability
serve to prevent policy change, also ensure increased focus on new issues and perspectives
to the exclusion of others once the focus of the policy makers shifts. Thus, government
attention is not only selective, it also shifts among perspectives and objectives of attention.
As True (2002, 177-180) argues in his application of the PET to the domain of U.S. national
security policy: “The relationship between Soviet capabilities and U.S. defense spending was
a complicated one, and it appears to have been episodic [...] At some times Soviet spending
may have been important to U.S. budget decision making, and at other times it was not [...]
In understanding national security and in explaining policy decisions over time, no one logic
and no single frame of reference lasts forever.” Over time, according to PET, most policy
areas may be subject to such a change in focus.
From the perspective of this paper, the three theoretical approaches share an important
and theoretically rather uncontested implication about the time-varying importance of causal
factors. Examples of such time-varying relationships can be found in qualitative applications
of the theories (for overviews, see Baumgartner et al. 2014; Jenkins-Smith et al. 2014), but
the implication has not been examined in time-series analyses of policy change (two partial
exceptions are True 2002, Jones, Baumgartner & True 1998). None of the theories offers
point predictions about when the parameters will change, and across theories there is no
consensus about whether such change will be gradual or punctuated. A crucial first step to
approaching these questions, however, is to begin to model such transformative changes. In
the next section, we motivate the adoption of a flexible approach to time-series analysis that
can uncover time-varying relationships.
7
Studying dynamic relationships
This study is the first to attempt a broad and systematic empirical test for the presence
of dynamic relationships in macro-level policy outcomes. However, there is a large research
literature examining such time-varying relationships in other contexts. The econometric
problem of estimating changing relationships between variables is challenging and has inspired a variety of solutions. Ideally, we might estimate a model that allows the parameter
for the effect of x on y to vary by time periods:
yt = β0 + xt β1t + This model, however, is non-identifiable without additional assumptions. For example,
with a large enough cross-section and interaction terms or hierarchical levels we could estimate how β1 changes with t. The assumption required is that the cases in the cross-section
are sufficiently similar that β1 describes the relationship across cases. For a single time series, the required assumptions become still more demanding. One option is to split the single
series into meaningful time periods and estimate β1 in separate regressions by time period
or via interactions for time periods. This approach has been used in the only other work
to previously test for the presence of dynamics in relationships between macro-level policy
outcomes (True 2002).
True’s (2002) results are suggestive that dynamics are present, as theorized. However,
this approach to estimating variation in effect parameters suffers two weakness that are
common to several other approaches that might also be applied. The first, and lesser, issue
relates to statistical power. Subdividing the time series necessarily weakens the strength of
our inferences by working with smaller samples, increasing the risk of both Type I and Type
II errors. Low statistical power is also a difficulty for CUSUM and CUSUMSQ plots as tests
for parameter stability (Wood 2000, Baltagi 2011). These plot functions of cumulative sums
of residuals in order to identify breaks that may signal structural shifts in the parameters.
8
The passage of time and therefore the lengthening of our data on policy change ameliorates
this issue to some degree though it cannot eliminate it altogether.
The second, and potentially more challenging, difficulty is that subdividing a time series
requires the analyst to identify and motivate the choice of relevant time periods a priori.
Although a principled method of selecting time periods may be available for a given test, it
does not translate into a methodology that can be more widely applied. Furthermore, this
still forces the analyst to assume that effect parameters are constant within periods. The
same difficulty with motivating a priori information applies to the Chow test, which can
reveal if a shift in an effect parameter occurs at a particular point (Wood 2000).
For a more systematic test, then, we should favor estimation methods that are flexible
with regard to the timing of shifts and require as little a priori information as possible from
the analyst. Dynamic conditional correlations (DCC) have been proposed for this purpose
(Lebo & Box-Steffensmeier 2008). DCC dynamically estimates the covariance of two time
series using a framework adapted from generalized autoregressive conditional heteroskedasticity modeling. This is a powerful method for identifying a certain type of dynamics, but it
requires quite large data sets, the method works only for bivariate relationships, and some debate is ongoing with regard to the econometric properties of DCC (Caporin & McAleer 2013).
Therefore, we turn to a class of modeling techniques for our test that are quite flexible
with regard to detecting the timing of changes, can handle multivariate models, and demand
minimal assumptions with regard to the form of dynamic relationships. These are the methods of flexible least squares (Wood 2000) or dynamic linear models using Kalman filtering
(Shumway & Stoffer 2010, Mcavoy 2006, Kim & Nelson 2000). These approaches have been
shown to be algebraically equivalent approaches to modeling dynamic coefficients (Montana,
Triantafyllopoulos & Tsagaris 2009). For simplicity, we adopt the dynamic linear modeling
with Kalman filter framework - i.e. state space modeling.
Returning to our idealized model above, this approach can recover estimates of the timevarying effect coefficients, β1t , provided some scaffolding of additional assumptions. The key
9
one is that β1t is a realization of an observed and underlying “state” that varies as a Markov
chain over time, with each new estimate at time t diverging from its previous realization at
time t − 1 to improve the model’s predictive accuracy in the new period. Predictive accuracy
is balanced against the magnitude of divergence from β1t−1 by a cost function. This means
that the model can flexibly return a time-constant coefficient estimate where appropriate or
detect the presence of dynamics where appropriate.
The disadvantages of state space modeling are minimized in our application. One of
the chief potential drawbacks to consider is that the model weights observations equally that is, observations in the distant past have an influence on parameter estimates at time t
equal to that of the observation at time t − 1 (Lebo & Box-Steffensmeier 2008). This is true
because the logic underlying the model is essentially Bayesian. At each period, t, all of the
information about past observations is summarized in the distribution of the parameter at
time t − 1, which is updated each period as new information is introduced.
Since our largest application uses a time series of several decades of policy outputs, we
avoid a situation in which estimates are influenced by the very distant past. Furthermore,
this feature of the modeling technique means that changes in parameters must reach a
threshold of distinctiveness to be detected. This makes it less likely that potentially minor
or random fluctuations from period to period will cause us to incorrectly identify the presence
of dynamics.
A further potential drawback lies in the state space model’s assumptions regarding the
distributions of the time-varying coefficients (Wood 2000). With regard to these issues,
we are less concerned. These models have been widely applied in engineering, economics,
biology, genetics, finance and other fields with great success (Petris, Petrone & Campagnoli
2007, Shumway & Stoffer 2010). We have no reason to suspect that its assumptions are
inappropriate for our applications.
In the next section, we turn to a detailed explanation of our estimation strategy and then
to summarizing the results of our analyses.
10
State space modeling
We apply a Kalman filter and smoother to estimate dynamic linear models of the impact of
various inputs to the policy process on public budgets. The model is structured as follows:
yt = Xt βt + vt
(1)
vt ∼ i.i.d.(0, σv2 )
βt = βt−1 + wt
(2)
wt ∼ i.i.d.(0, Q)
The first line is the measurement equation. The inputs to the policy-making process in
vector Xt influence the realization of the public budget, yt , via the vector of time-varying
effect coefficients, βt . The disturbances, vt , are independent Gaussian white noise with
variance σv2 . The third line, equation 2, is the state equation. The effect coefficients, β, are
modeled as an unobserved state varying in a random walk over time. The state disturbances,
in vector wt , are also Gaussian white noise with covariance matrix Q and are uncorrelated
to vt .
The model is estimated recursively. At each time t, βt−1 is our expectation of this
period’s new value of βt , conditional on the information observed up to time t − 1. Based
on this conditional expectation for βt , we calculate a prediction for the outcome in time t:
ŷt . The error in this prediction, yt − ŷt , is used to update our final estimate of βt , such
that larger errors provoke larger shifts in the coefficient estimates. This recursive estimation
process means that at any time, t, all of the past information about the underlying state is
summarized in their point estimates and covariance matrix at time t − 1.
Shifts in the coefficient estimates each period are moderated by the ratio of uncertainty
regarding the estimates of β to the total overall estimation uncertainty in the model (including that from β). Where uncertainty about β is a larger share of overall model uncertainty,
11
this ratio - named the Kalman gain - is closer to one. Updates to the coefficients become more
responsive to that period’s prediction error as the Kalman gain approaches one. A formal
definition of the Kalman gain and detailed estimating equations are provided in appendix C.
The final step in the estimation of the model is to apply a smoother to the time-varying
state estimates. The Kalman smoother utilizes the same approach as the filter, but run
in reverse. Recall that when filtering at each time period, t, all information about the
unobserved states up to that period is summarized in the point estimates and associated
uncertainty of last period’s state estimates: βt−1 . The smoother, by running in reverse
from time T back to time 1, updates again each period’s estimate of βt conditional on all
information in the entire model. This smooths out some variation in the time-varying state
estimates by using information from the full time series.
Our final effect coefficient estimates are these smoothed time-varying coefficients. The
recursive estimation procedure is identified conditional on a starting value for β, β0 , and
estimates of Q, σv2 , and the starting value of the covariance matrix of innovations, Σ0 . These
are estimated via maximum likelihood. The process begins with initial values for each of
these latter parameters, and then the βt s are estimated using the Kalman filter. After this,
maximum likelihood is used to re-estimate the other parameters conditional on the estimates
of βt . This process is repeated until all the parameter estimates stabilize.
In the next section, we outline a series of empirical settings in which we apply this
methodology to test for the presence of transformative policy changes.
Testing for transformative policy change
The first setting in which we apply our state space model is to test for dynamic relationships
between inputs to the policy process and policy outputs is in replicating the tentative positive
results reported by True (2002). His test examined U.S. defense spending over several decades
and found that the relationship between prior spending, international tensions and wars, and
12
Soviet spending showed evidence of changing over time.
We collected similar data for the period 1948-2006. Unfortunately, lagged Soviet defense
spending is only available between 1965 and 2006 (with Russian defense spending substituted
following the collapse of the Soviet Union), which leaves too short a time series to apply our
preferred modeling technique. Nevertheless, we continue with analyzing the rest of the model
and will return to this problem with a version of the state space model augmented for dealing
with missing data.
The response variable is the percentage annual change in U.S. defense spending. Key
explanatory variables include a lag of the national unemployment rate, the constant U.S.
dollar level of the government’s international support spending (i.e. international aid), an
indicator variable for presidential election years in which an incumbent is competing, and
indicator variables for periods of war or heightened international tensions. This final set of
variables was recoded into a single indicator for our analysis. Since the state space model
can detect changes in the effect of the single indicator over time, and we have no theoretical
reason to distinguish among periods of war or tension, we have collapsed them into a single
indicator. The variable takes on a value of 1 during periods of war or tension, including:
the Korean War, the Vietnam War, the Reagan buildup, the first Gulf war, and from 2001
onward following the attacks of September 11th and the wars in Afghanistan and Iraq.
Finally, we also include the lagged constant U.S. dollar value of defense spending as a
control. In our first model, we include all of these variables. In a second specification, we
remove some of the variables with insignificant effects and add in an indicator for public
opinion on defense and a measure of the partisan ideology of the U.S. House of Representatives. Our public opinion measure comes from Gallup’s annual estimate of the proportion of
the U.S. population stating that defense is an important problem. House ideology is measured by averaging the DW-NOMINATE scores of all members of Congress during a given
session.1 Positive values on this measure are associated with conservative political ideology,
1
Data is found on the website: http://www.voteview.com/. For details on DW-NOMINATE, see Poole
& Rosenthal (1997)
13
and negative values are associated with liberal ideology. For this latter specification, data is
only available from 1959-2006. Summary statistics for all variables in both models can be
found in appendix A.
Our additional variables were chosen to broaden the range of explanatory factors captured
by the model. Whereas the occurrence of a war or international tension constitutes a feature
of the environment facing policy makers that likely influences defense spending, the first
model does not capture public opinion and only barely addresses partisan politics. However,
public attention and partisan politics are two other theoretically important factors that likely
influence policy change. Therefore, we capture public attention with the most important
problem measure. Although the election year indicator does capture some aspects of politics,
it is not explicitly partisan. Therefore, we add in an estimate of the ideology of the House
of Representatives. Transformative change in any of these variables is likely to reflect that
policy makers are learning, policy frames are changing, or that core beliefs about policy have
evolved.
Results
Dynamic coefficient estimates from our first model specification are plotted in figure 1.
There is some initial evidence here that inputs to the policy process do not have constant
effects over time. Periods of war and international tension are not consistently related
to increases in defense spending, for example. Furthermore, election years appear to be
associated either with greater or lesser spending. The bubble-shaped confidence intervals on
these two coefficients occur during periods in which the variable takes on a value of 0.
The rest of the estimates from the model, however, shows little evidence of transformative
changes. The lagged level of spending is consistently negative, such that higher levels of
spending last year are associated with negative adjustments to this year’s budget. The effects
of both international support spending and lagged unemployment are basically constant and
14
statistically indistinguishable from zero.
Intercept
Coefficient
0.4
Coefficient
War/Tension
1.0
0.2
0.5
0.0
−0.5
0.0
1950
1960
1970
1980
1990
2000
1950
1960
Lag Defense Spending
1970
1980
1990
2000
Lag Unemployment
0.02
1e−06
Coefficient
Coefficient
5e−07
0e+00
−5e−07
0.01
0.00
−1e−06
−0.01
1950
1960
1970
1980
1990
2000
1950
1960
International Support
1980
1990
2000
1990
2000
Election Year
0.001
0.2
0.000
Coefficient
Coefficient
1970
−0.001
0.0
−0.2
−0.002
−0.4
−0.003
1950
1960
1970
1980
1990
2000
1950
1960
1970
1980
Figure 1: Estimated time-varying coefficients from model one
Note: Effect coefficients are plotted over time. The solid line indicates the point estimate for the coefficient
each period, while gray areas are 95% confidence intervals. The dashed horizontal line indicates zero. Where
the gray area overlaps the dashed line, estimates are not statistically significant by conventional standards.
We turn now to a revised model specification. Since the relatively short time series
and numerous parameters being estimated places high demands on our limited amount of
data, we can improve our estimates if we can eliminate unimportant information from the
model. In our second model specification, plotted in figure 2, we replace unemployment and
international support spending with public opinion and Congressional ideology and rerun
the model on the time period from 1959-2006.
The results in this second model specification are estimated with greater certainty and
further sharpen up some of the trends glimpsed in the first. Here, we see again that lagged
15
Intercept
House Ideology
0.4
Coefficient
Coefficient
2
0.3
0.2
0
0.1
−2
0.0
1960
1970
1980
1990
2000
1960
1970
Lag Defense Spending
1980
1990
2000
Public Opinion
0e+00
Coefficient
Coefficient
0.6
−5e−07
0.4
0.2
−1e−06
0.0
1960
1970
1980
1990
2000
1960
1970
War/Tension
1980
1990
2000
Election Year
Coefficient
Coefficient
0.04
0.2
0.0
−0.2
0.02
0.00
−0.02
1960
1970
1980
1990
2000
1960
1970
1980
1990
2000
Figure 2: Estimated time-varying coefficients from model two
Note: Effect coefficients are plotted over time. The solid line indicates the point estimate for the coefficient
each period, while gray areas are 95% confidence intervals. The dashed horizontal line indicates zero. Where
the gray area overlaps the dashed line, estimates are not statistically significant by conventional standards.
spending is associated with a fall in the percentage change in defense spending and the
constant positive intercepts suggests a generally positive trend over time. This model’s
time series commences after the negative effect estimated for election years in model one.
However, in this model, the election year effect has essentially disappeared as the coefficient
is constant never reaches statistical significance.2
Periods of war and international tension are even more clearly inconsistently related
to changes in defense spending. Whereas the Vietnam era clearly saw higher spending
associated with the war, by the Gulf War the estimated effect had reversed and in the
2
As a check on the within sample predictive accuracy of the two models, appendix B contains plots of
predicted values from each model against the actual values of the response variable.
16
period of the Afghanistan and second Iraq wars there is no clear association between the
crises and changes in defense spending. Instead, from around the Reagan era onward, strong
public sentiment that defense is an important problem is positively associated with spending
changes. This positive relationship is consistent with previous eras, but the relationship
between public opinion and defense spending is barely statistically insignificant prior to this.
Finally, House ideology shows dramatic transformations. Left-right ideology in the House has
a positive association with changes in defense spending until the Reagan era, at which point
is becomes indistinguishable from zero. In the early 1990s this association reverses briefly
and becomes negative, and then it moves toward positive again though it is statistically
insignificant from the mid-90s onward.
The coefficient on House ideology reaches its lowest point in 1989, at which time a one
standard deviation increase in ideology (i.e. a shift toward being more conservative) would be
associated with a predicted 11.5% decrease in defense spending. This is in contrast to 1959,
when the coefficient was at its highest point and a one standard deviation increase in ideology
would have been associated with a predicted 16.2% increase in defense spending. The shifts
in the impact of war and international tension are almost as pronounced. In 1966, after the
major escalation of U.S. involvement in Vietnam following the Gulf of Tonkin Resolution,
the war was associated with an estimated 7% increase in defense spending. However, in 1991
during the first Gulf War, that conflict was associated with an estimated almost 5% drop in
defense spending. It is worth noting that these effects are not only statistically significant
in that they differ from the null of zero, but these shifts are also statistically significantly
different from one another. That is, the shifts we observe in the coefficients are significant
and substantively large changes.
17
Conclusion
We take these initial results as compelling evidence in support of the most important implication of our strongest theories of policy change: relationships are dynamic in the policymaking process. Testing the exact nature and mechanisms of these transformative changes is
beyond the scope of this paper, but the econometric evidence is strongly suggestive that the
key inputs to defense spending changes we have analyzed do not have constant relationships
to policy output. Rather, their impacts on policy change over time.
This opens up multiple possibilities for future theoretical and empirical research. Chiefly,
there is a dearth of theoretical expectations regarding the timing and nature of transformative policy changes. With the empirical tools to test these expectations in hand, deriving
such theoretical expectations is a promising way forward. Furthermore, the different causal
mechanisms underlying the general theoretical expectation of transformational policy change
provide a launching point for generating new hypotheses that can be tested using the framework we advocate.
The planned next steps for this paper are to run comparable analyses across a large
set of budgetary categories and to implement versions of the current analysis augmented
for handling missing data in order to expand the time frames we can analyze. The modeling framework we have advocated can also generalize to handle non-normally distributed
response variables, allowing us to examine policy outputs other than budget changes and
other plausibly normally distributed phenomena.
18
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A
Appendix: Summary statistics
Year
% Change in defense spending
Lagged defense spending
International support spending
War/Tension
Lagged unemployment
Election year
Min
Max
Mean Std Dev
Obs
1948
2006
- 59.00
-0.28
0.88
0.03
0.15 59.00
82334.00 510603.00 311106.83 93948.39 59.00
62.00 17711.00
4441.44 3455.43 59.00
0.00
1.00
0.39
0.49 59.00
0.00
9.71
5.53
1.65 59.00
0.00
1.00
0.17
0.38 59.00
Table A.1: Summary Statistics for Model 1
Year
% Change in defense spending
Lagged defense spending
War/Tension
House Ideology
% Defense most important prob.
Election year
Min
Max
Mean Std Dev
Obs
1959
2006
- 48.00
-0.12
0.17
0.02
0.07 48.00
238744.00 510603.00 339319.00 70873.23 48.00
0.00
1.00
0.40
0.49 48.00
-0.13
0.15
-0.02
0.08 48.00
0.00
0.47
0.15
0.14 48.00
0.00
1.00
0.17
0.38 48.00
Table A.2: Summary Statistics for Model 2
22
B
Appendix: Within-sample predictions from models
The following plots show the predictive accuracy of the two models. Both figures plot two
lines. In solid black, we see the actual values of the response variable: percentage annual
change in defense spending. The dashed line is the model’s prediction for each period, with
its 95% confidence interval in gray. The predictive accuracy of model 2 is stronger, with the
black line depicting actual change in defense spending being within the confidence interval
for the prediction most of the time.
Predicted and actual spending changes
1.0
0.5
0.0
−0.5
1950
1960
1970
1980
1990
2000
Figure A.1: Model 1: Predicted and actual values of change in defense spending
Note: The solid black line is the actual value of the response variable. The dashed line is the model’s
prediction of the response for each period, with its 95% confidence interval in gray.
23
Predicted and actual spending changes
0.2
0.1
0.0
−0.1
1960
1970
1980
1990
2000
Figure A.2: Model 2: Predicted and actual values of change in defense spending
Note: The solid black line is the actual value of the response variable. The dashed line is the model’s
prediction of the response for each period, with its 95% confidence interval in gray.
24
C
Appendix: Model details
The model exposition we adopt is that outlined in Shumway & Stoffer (2010). Their work
provides a general version of the specific approach we utilize. The estimation of the timevarying covariates is recursive, starting at time one and progressing to time t. This entire
recursive estimation is repeated with each of the maximum likelihood estimates of the additional parameters of the model, beginning with their initially chosen starting values and
ending with the stable final estimates. The process proceeds as follows:
1. Select initial values for the parameters: β0 , Q, σv2 , and the covariance matrix of innovations (or prediction errors) Σ0 .
2. Run the Kalman filter to obtain values for the innovations (prediction errors) from the
model, v, and their covariance.
3. Use the estimates obtained from the Kalman filter to estimate β0 , Q, σv2 , and Σ0 using
maximum likelihood.
4. Repeat step 2 using the estimates from step 3 in place of the starting values selected
in step 1.
5. Repeat step 4 until the estimates of β0 , Q, σv2 , and Σ0 or the likelihood stabilizes.
The recursion for the Kalman filter proceeds as follows. At time 0, the following two
steps are unique and take the place of steps one and two in the next list:
1. Calculate an expectation of β1 , conditional on β0 . We assume that β follows a random
walk, therefore our initial expectation about next period is simply our current estimate:
βt|t−1 = βt−1|t−1 . Likewise: β1|0 = β0
2. Calculate an expectation of the covariance of innovations to β1 , conditional on Σ0 and
Q. We refer to this as P : P1|0 = Σ0 + Q.
From time 1 through time t, the following steps are taken:
1. βt|t−1 = βt−1|t−1
2. Pt|t−1 = Pt−1|t−1 + Q
3. Calculate the predicted value of y conditional on expectations from time t − 1: yt|t−1 =
Xt βt|t−1
4. Calculate the prediction error in time t: ηt|t−1 = yt − yt|t−1
5. Calculate the Kalman gain for period t, i.e. the proportion of uncertainty in each
parameter in βt attributable to uncertainty regarding the parameter relative to the
full uncertainty in the model:
Kt =
Pt|t−1 Xt0
Xt Pt|t−1 Xt0 + σv2
25
6. Update estimate of effect coefficients, βt|t , based on prediction error and Kalman gain:
βt|t = βt|t−1 + Kt ηt|t−1
The value of βt|t is our estimate of βt .
7. Update expectation of the covariance of parameters, Pt|t , based on Xt and σv2 :
Pt|t = Pt|t−1 −
Pt|t−1 Xt0
Xt Pt|t−1
Xt Pt|t−1 Xt0 + σv2
This can also be expressed as:
Pt|t = [I − Kt Xt ]Pt|t−1
26