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A STATE-SPACE APPROACH TO
ECONOMIC POPULARITY FUNCTIONS
Mark Pickup
University of Oxford
June 2006
Prepared for presentation at the 23rd Annual Summer Meeting of the
Society for Political Methodology, July 20-22 2006
Abstract:
Economic popularity functions are central to the debate over whether voters use evaluations
of the economy in their decision to support their government or not. This is of particular
importance to the key democratic principle of electoral accountability that parties in power
should and are held accountable for the outcomes of their actions and policies through the
electoral process. Given the evidence from many nations that the economy is an issue of
importance to the electorate, which they believe the government has control over, the
inconsistent findings with regards to the impact of the economy on party popularity has made
conclusive evaluations of the principle of electoral accountability elusive. This study
demonstrates that the difficulty lies in a series of methodological flaws found in current
approaches to developing popularity functions. Most analysts using public opinion timeseries data have not applied the necessary methods to take into account the problems which
such data can pose – problems such as complex error structures, shifting and compound nonstationary dynamics and noisy data. Accordingly, this study explicates a state-space Bayesian
approach that addresses these methodological issues. In doing so, it outlines a technique that
may be applied to a wide range of public opinion dynamic modelling issues.
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1.0 Introduction
Economic popularity functions are central to the debate over whether voters use evaluations
of the economy in their decision to support their government or not. This is of particular
importance to the key democratic principle of electoral accountability that parties in power
should and are held accountable for the outcomes of their actions and policies through the
electoral process. Given the evidence from many nations that the economy is an issue of
importance to the electorate, which they believe the government has control over, the
inconsistent findings with regards to the impact of the economy on party popularity has made
conclusive evaluations of the principle of electoral accountability elusive.1 This study
demonstrates that the difficulty lies in a series of methodological flaws found in current
approaches to developing popularity functions. Accordingly, this study explicates an
alternative approach that addresses these methodological issues. In doing so, it outlines a
technique that may be applied to a wide range of public opinion modelling issues.
Popularity functions use public opinion time-series data – either government or party
support over time.2 The statistical tools utilised to examine the dynamics of government and
party support have been problematic and, at times, inappropriate. Most analysts using such
data have not applied the necessary methods to take into account the problems which such
public opinion time-series processes can pose – problems such as complex error structures,
shifting and compound non-stationary dynamics and noisy data. In this study I examine the
many potential difficulties of working with aggregate, public opinion time-series data and
present a state-space Bayesian approach which alleviates many of these challenges providing
a framework for examining the impact of macro-economic conditions on public opinion.
I develop a different path of analysis for students of government and party popularity
from that traditionally taken, suggesting that popularity must be understood in terms of its
component dynamics and that each of these components is of substantive interest. Many of
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them represent the political context in which economic conditions are translated into
government or party support. These contextual dynamics include shifts in baseline support
produced by political events, a cycling in popularity which corresponds with the timing of
elections, trending in popularity, unequal variances in popularity over time, and varying
dynamics during distinct time periods. Other dynamics, such as error variances correlating
with time and conditional heteroscedasticity, are a product of the method of measuring
popularity – the public opinion poll. Many of these dynamics are non-stationary and represent
a statistical challenge when modelling the impact of economic conditions on government and
party support.
I demonstrate how to correct for the problems presented by these dynamics by
employing state-space models, estimated using Bayesian analysis to separate and explicitly
model the multitude of dynamic components – stationary and non-stationary – comprised by
public opinion towards governments and parties over time. This allows for the estimation of
economic popularity, properly accounting for statistical problems such as nonstationarity and
controlling for the political context in which economic conditions are translated into
government and party support. The state-space approach also provides a sophisticated method
by which to cope with noisy public opinion data containing measurement error variances that
correlate with time. At the same time, it is compatible with conventional approaches to
determining the appropriate lag structure of the measures of economic conditions and the
error structure of the stationary component of public opinion.
This study is outlined as follows. In part 2.0, I review current approaches to
developing economic popularity functions, identifying their shortcomings. As an example of
the challenges posed by the dynamics within government popularity time-series data, in part
3.0 I identify these dynamics within Canadian federal government popularity, between 1957
and 2000. I also introduce the state-space model and outline how it can be used to work with
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and around such problems. In part 4.0, I examine the outcome of this modelling technique
applied to Canadian government popularity. I demonstrate that the state-space model
effectively solves the problems identified in this study. To illustrate how this method can be
used to determine the impact of economic conditions on government popularity, in part 5.0, I
apply the technique to modelling the economic popularity function for the Canadian federal
Progressive Conservative government between 1984 and 1993.
2.0 Background: Economic Popularity Functions
While most economic popularity studies find that economic conditions are important in some
way, the range of variation between these studies in terms of the results and the different
statistical methodologies applied is greater than for traditional voting studies (Lewis-Beck
and Stegmaier, 2000). The earliest US research on popularity functions was done by J.
Mueller (1970). However, Lewis-Beck and Stegmaier identify the earliest published
popularly function ever as being C.A.E. Goodhart and R.J. Bhansali's British case in 1970.
Goodhart and Bhansali (1970) examine monthly measures of British government popularity
between 1947 and 1968. They find that levels of unemployment and the rate of inflation
influence the government's political popularity. They also find that the strength of the impact
of these economic conditions had increased over the time period under study. Further, they
find that government popularity may follow what they call a “natural path” between
elections. Such a path includes honeymoon effects, trending downwards after the honeymoon
and trending upwards leading into an election. Overall, this suggests that popularity follows
an inter-election cycle. This cycle is part of the political context in which economic
conditions are translated into government support. Goodhart and Bhansali attempt to control
for this inter-election “natural path” through the application of dummy and index variables.
It is commendable that Goodhart and Bhansali so early on recognised the need to
control for the political context in which economic popularity operates, even if the methods
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by which this was done were fairly crude. The reason it is so important is that the dynamics
in government popularity produced by this political context creates the problem of
nonstationarity within the popularity data. Nonstationarity is a very challenging statistical
process and a problem which is discussed further in the next section but in short, if a model
which assumes the time-series is a stationary process is used to model nonstationary
popularity data, the model is seriously misspecified. Therefore, any estimated results from
these models are unreliable, unstable and highly susceptible to spurious correlation. Very
peculiar results can be obtained when nonstationary data is modelled as stationary. To model
popularity, the data must be transformed so that it is stationary or a model that accounts for
the nonstationarity in the data must be used.
Given these considerations, it is surprising that many studies since Goodhart and
Bhansali are no more sophisticated in tackling this problem. Most studies will typically use
crude time-indices to control for trending before and/or after elections. Beyond this, many
very recent studies hardly address the issue of nonstationarity produced by political
contextual forces at all. This failure renders the statistical techniques used in the studies to
model popularity inappropriate and even the most recent work of highly respected analysts
exhibits this serious oversight.3 An example which represents much of what is done well in
the field and yet still exhibits this oversight is David Sanders’ examination of the impact of
objective macro-economic conditions on British government monthly popularity between
1974 and 1997 (Sanders, 2000). Sanders includes only event dummies to control for political
context. Further, while he uses the differenced form of unemployment and inflation to correct
for nonstationarity in these economic variables, the issue of potential nonstationarity in the
popularity data is not seriously considered.4 Political contextual factors which exhibit as
trending, cycling and shifts in baseline support are left completely unaddressed. Sanders'
modelling techniques are technically inappropriate and fail to fully take into account the
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dynamics of the political context in which economic conditions are translated into public
opinion and the consequent nonstationarity. This puts his findings into doubt.
Sanders’ study also demonstrates a further shortcoming of much of the popularity
research. Economic variables are entered into his model at lags of 0, 1 and 2 months. This is
done without any particular justification. It is reasonable to suspect that there will be some
lag in economic effects. Some time will be required for information regarding the state of the
economy to reach the electorate and to be processed into an opinion about the government.
What the appropriate lag is though is not obvious. There is no more reason to believe that it
will be one or two months than that it will be three or four months or even up to a year.
Unfortunately, it is unhelpful to just enter all possible lags and see what falls out. Economic
variables contain a great deal of autocorrelation. Including unemployment with a two-month
lag (for example) will greatly affect whether the estimated effect of unemployment with a
one-month lag is statistically significant. Moreover, economic variables are correlated; so,
including unemployment with a two-month lag will also greatly affect whether the estimated
effect of inflation lagged one-month (again, for example) will be statistically significant. A
much more purposive technique is required to determine the lags at which economic
variables should enter into a popularity model.
There are a few studies which have begun to take the necessary steps towards
addressing these important methodological issues. Paul Whiteley, in "Inflation,
Unemployment and Government Popularity: Dynamic Models for the United States, Britain
and West Germany," examines the impact of economic conditions on monthly measures of
government popularity in the United States, Britain and West Germany. Whiteley (1984) uses
a process developed by Box and Jenkins to specify the popularity functions. Clarke, Stewart
and Zuk (1986) use the Box-Jenkins approach do the same in their examination of party
popularity in Britain between 1979 and 1983. They apply this approach to an ARIMA model
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of popularity. They also include a number of sophisticated political event variables
controlling for the effects of strikes, internal party disputes, leadership popularity, etc. The
Box-Jenkins approach allows these researchers to account for autocorrelation and trending in
the independent and dependent variables through the appropriate differencing of these
variables. It also allows them to determine the lag structure of the independent variables.
Helmut Norpoth (1985) also applies the box-Jenkins approach to presidential
popularity in the US from 1961 to 1980. Norpoth goes one further than Clarke et al by
explicitly exploring the appropriate structure for the error process in the ARIMA model,
settling on a first order moving average process. Norpoth notes that there is resistance to
using approaches such as that of Box-Jenkins because it is felt that the procedure for
eliminating the nonstationary in popularity, produced by political contextual forces, may
throw out real economic effects – that is, "the baby may be thrown out with the bathwater"
(Norpoth, 1985). He further notes that this is particularly problematic with data that is noisy,
such as popularity data produced by public opinion polls. Consequently, he finds it necessary
to aggregate his time-series to the quarterly rather than monthly level. Norpoth is correct to
identify the potential problems produced by the noise (sampling error) inherent in measures
of popularity. In fact, the problem is even larger than he identifies. As I demonstrate in this
study, the variance of the measurement error within popularity time-series can also trend,
cycle and generally be heteroscedastic. This is yet another source of nonstationarity.
In addition to being inadequately able to handle noisy data, the Box-Jenkins approach
is not always able to fully account for the nonstationarity produced by the cycling within
popularity as identified by Goodhart and Bhansali. The Box-Jenkins approach eliminates
cycling by differencing the data by the cycle length. The cycling is the product of the interelection cycle and in the US case, where elections are evenly spaced, the Box-Jenkins
approach is appropriate. In the case of Parliamentary governments with unevenly spaced
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elections, the Box-Jenkins approach is unable to cope with nonstationarity produced by the
cycling. When cycling is driven by the timing of elections, as it often is, it will not have a
constant frequency. In which case, there is no way to difference the data in order to eliminate
it and this nonstationary component will remain.
Even when it is possible to eliminate cycling or trending through differencing, it may
not be desirable. While this may be a statistically acceptable procedure, it does not allow us
to examine these components.5 In public opinion series, these components are not only
statistical challenges, they are also substantively interesting. Trending and cycling are very
real dynamics within public opinion, with important political origins and consequences –
simply eliminating them forfeits the possibility of understanding them. Therefore, the BoxJenkins approach is inadequate if we wish to fully understand the dynamics of public opinion.
What is required is a method that explicitly models and accounts for non-stationary
dynamics which cannot or should not simply be difference away, while controlling for noisy
data with complexly heteroscedastic measurement errors. This study develops a state-space
Bayesian approach to modelling popularity to do just this. This approach is also used to
address each of the other methodological problems identified in this section.
The name "state-space" originates with the initial use of these models to represent the
state of a physical system (position, velocity, momentum, etc) in spatial form. When applied
to time-series public opinion data, state-space models are distinguishable from other
modelling approaches in that observations are regarded as made up of distinct elements, such
as trend, cycling, autoregression, equilibrium shift, measurement error and disturbance
components, each of which is modelled separately. This technique of modelling public
opinion is highly flexible and is capable of handling a much wider range of problems than its
primary alternative, the Box-Jenkins ARIMA approach (Durbin, 2004). James Durbin (2004)
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also notes that unlike the state space approach, the Box-Jenkins approach does not necessarily
lead to a unique model. Very different models can appear to fit the data equally well.
It is the state-space approach that is the basis of the Kalman filter. The Kalman filter
is recommended by Donald Green, Alan Gerber, and Suzanna De Boef (1999) to reduce
sampling error in public opinion time-series such as popularity and by Nathaniel Beck (1990)
to do the same in modelling US presidential popularity.6 This study goes far beyond the
Kalman filter. Not only does it use the state-space approach to reduce sampling error, it
employs this approach to account for error variances correlating with time and to control for
nonstationary dynamics within popularity produced by political contextual forces by
explicitly modelling them. The use of the state-space approach, as it is developed here, is
uniquely able to properly account for both unevenly spaced inter-election cycling within
popularity and complex measurement error. Having developed a technique that fully accounts
for nonstationarity, aspects of the Box-Jenkins technique are modified so that they can be
applied to determining the proper lag structure of economic variables within the state-space
framework. No other approach developed to date is able to handle each and every one of
these methodological challenges.
3.0 Government and Party Popularity in Canada, 1957 to 2000
Figure 1 is a plot of the Canadian government popularity.7 This aggregate popularity timeseries covers the period from 1957 to early 2000. It was constructed from individual level
survey data collected by the Canadian Institute of Public Opinion (CIPO or Gallup Canada).
The CIPO surveys are the most consistent measure of Canadian government popularity
publicly available. For more than 50 years Gallup has been asking Canadians: "If a federal
election were held today, which party's candidate do you think you would favour?"
Government popularity is defined as the percentage that indicate they would vote for the
party in government. (For the purposes of this study, the calculation is made excluding those
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that indicate they do not know for whom they would vote.) Since the early 1970s, Gallup has
run surveys including vote-intention questions monthly. Before this time, survey data is
available on a less consistent but still highly regular basis.8
3.1 Difficulties with Modelling Government Popularity
When examining time-series data (such as government popularity) it is necessary to account
for statistical challenges, such as nonstationarity in the series. A time-series
process ( y1 , y2 ,..., yT ) is said to be covariance stationary if:9
1) E ( y1 ) = E ( y2 ) = ... = E ( yt ) = µ ;
2) Var ( y1 ) = Var ( y2 ) = ... = Var ( yt ) = σ 2 ;
3) Cov ( y t , y t −1 ) = Cov ( yt , y t −τ ) = γ τ ;
where µ and σ 2 are the mean and variance of ( y1 , y2 ,..., yT ) . γ τ = Cov ( y t , y t −τ ) is called the
autocovariance at lag τ . Just as the autocovariances only depend on the lag, so do the
autocorrelations ρ (τ ) =
γτ γτ
= . Stationarity implies stability in the variance and covariance
σ 2 γ0
structure of the data. This, in turn, implies stability in the estimated relationships between
timeseries (e.g., government popularity and economic conditions). If timeseries data is
modelled assuming stationarity and it is violated then the inferences regarding relationships
between timeseries will be incorrect.
The first condition for stationarity will be violated if the mean of the time-series is
correlated with time – that is, the series trends up or down. The second condition will be
violated if the variances are correlated with time. This may occur if the underlying variance
of the process and/or the variance in the measurement process itself systematically changes
over time. The third condition will be violated if the autocorrelations are correlated with time.
This will occur in data that contains cycles.
Examining figure 1, The Canadian government popularity time-series is, at times,
clearly subject to each of these violations. Richard Johnston observes that there appears to be
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three distinct time periods since World War II, in which the dynamics of government
popularity are unique (1999). The earliest period extends back to the end of the War and ends
during the mid-seventies (shortly after the 1974 election). The second period continues from
the mid-seventies until the 1993 election and the most recent period picks up from there.
Johnston describes the first period as exhibiting no special inter-election rhythm. In the
second, he notes a consistent cycle. Each election is followed by a honeymoon period in
which popularity increases. Subsequently, popularity drops below the level of the
government's election return and bottoms out. Popularity then begins to recover as the
government enters the next election. Underlying these cycles is a long downward trend. In the
third period (after 1993), this downward trend ceases and government popularity surges up
beyond the 50 percent level. According to Johnston, this level of popularity is largely
sustained for the entire period except during election campaigns when popularity temporarily
spikes downwards to produce a vote return within the forties (Johnston, 1999). Post-1993
also exhibits cycling, although it appears to differ in frequency and magnitude from that
before it.
Such trending and cyclicity and changing dynamics over time are violations of all
three conditions of weak stationarity. One way to further explore these potential violations of
weak stationarity is to examine the autocorrelation (AC) and partial autocorrelation (PAC)
functions for government popularity. The autocorrelation ρ (τ ) of a time-series process at lag
τ is a measure of correlation between time-series observations τ units apart and the partial
autocorrelation is a measure of correlation between time-series observations τ units apart
after the correlation at intermediate lags have been controlled or ‘partialed out’(McCleary,
Hay, Meidinger, and McDowall, 1980). In order to accommodate the possibility of distinctive
government popularity periods, the autocorrelation functions for each period are examined
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separately (figures 2a to 2f). Note that a 1975/1979 separation between the first and second
periods is used. This necessity is explained further below.
Both the autocorrelation functions for the 1957-1975 (figure 2a) and 1979-1993
(figure 2c) periods indicate some autoregressive process at work but they decay far too
slowly to be simple AR(1) processes. The partial autocorrelation functions for these first two
periods (figures 2b & 2d) suggest government popularity cannot be modelled as a first-order
autoregressive process. If this was the case, the partial autocorrelation would simply show a
single large value at lag 1 – both PACs have substantial values at lags two and three.
The slow decline of the autocorrelation functions is almost certainly the product of
trending. It may also be the product of cycling. However, observations regarding cyclicity are
inconclusive. The autocorrelation and partial autocorrelation functions do not show classical
indications of fixed period cycling (this would exhibit itself as autocorrelation function peaks
at regular intervals) but they would not be expected to because an inter-election government
popularity cycle would have an uneven frequency – one based on the inconsistent timing of
elections. It is not clear what the partial autocorrelation for such a process would look like.
The slow decay within the autocorrelation functions also suggests the possibility that
while part of the dynamics of government popularity is stationary, part of it is likely
integrated. An integrated public opinion process is one in which all shocks carry over from
one period to next. The value of the time-series at any time t is equal to the sum of all
previous shocks in addition to the current shock. Substantively, an integrated process within
government popularity may represent permanent shifts in baseline support for the
government (i.e., a change in equilibrium). In a stationary process a shock to the time-series
will decay over the following periods. The speed at which the shock decays depends upon the
degree of memory in public opinion. Because all shocks decay, the value of the time-series
tends towards a constant equilibrium value. A stationary process within government
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popularity may represent the temporary and decaying impact of events such as changes in
economic conditions. A time-series process which contains both integrated and stationary
components is called fractionally integrated (Wlezien, 2000). The primary difficulty of
fractional integration is that it suggests that the popularity equilibrium is shifting.
The autocorrelation and partial autocorrelation functions (figures 2e & 2f) for the
1993-2000 period contains evidence of an autoregressive process of orders 1 and 4, and
evidence of a seventh order moving average process. However caution is advised, as the
potential presence of cycling and trending makes the use of AC and PAC functions in the
determination of the lag structure of government popularity inappropriate. Testing for such
cycling and trending can be done by regressing the popularity timeseries on cycling and
trending variables.
The cycling variables are described further in the next section but because the cycling
is of an uneven frequency, two terms are required to describe it. It is the joint significance of
both components that determines the significance of the cycle. Separate trending variables are
used for Liberal and Conservative governments. Newey-West regressions are required due to
the heteroscedasticity and serial correlation within the error component. The results presented
in table 1 are compelling. The Newey-West regressions indicate statistically significant
trending in the first two periods and statistically significant cycling in the second two periods.
The residuals from the regression of popularity on cycling and trending terms form a
popularity series with these nonstationary components partialled out. When the AC and PAC
functions of the new popularity series are examined (figure 3a to 3f) much clearer
autoregressive processes are evident in the first two periods. It is still clear, though, that the
autoregressive process contains lags greater than one. In particular, lags of two or three are
suggested. The AC and PAC functions for the third period changes little and there continues
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to be evidence of an autoregressive process of orders 1 and 4, and of a seventh order moving
average process.
It is less straightforward to determine the potential for uneven variances over time and
the violation of the second condition of weak stationarity. From a conceptual point of view,
however, the violation of this condition is very plausible. Given the length of the time-series
(about 43 years), it is risky to assume that the dynamics of government popularity, and
therefore the variance of the error term in whatever model we use to represent those
dynamics, will remain constant.10 It is for this very reason that each of the three periods
identified above are considered separately.
There is the further problem that the size of the measurement error component (and
therefore the variances) of our model's error term may be correlated with time. Since 1974,
Gallup has regularly used sample sizes of just over 1000 respondents. Before that time, many
of the Gallup poll results used much smaller sample sizes (although, sometimes much larger).
Moreover, the fifties and sixties component of the time-series contains a number of missing
values at the monthly level of measurement. This means more values in an analysis must be
interpolated. These interpolated values will, of course, contain larger errors than those which
were directly measured.11 These circumstances could possibly produce greater variances in
the earlier part of the time-series, compared to the later. A trend which runs counter to this
but which may also produce complications is the increasing number of respondents since the
early 1990s (except during election months) that indicate they do not know for whom they
would vote. The increasing number of “don't know” respondents may produce greater
variances in the latter part of the time-series, compared to the middle.
Figure 3 graphs the number of decided voters interviewed each month. In addition to
the increased frequency with which public opinion is measured over time and the general
decline in the number of valid respondents indicating a vote preference since the early 1990s,
14
the figure also reveals how spikes in measurement accuracy occur around elections. This is
produced by the combination of two phenomena. Leading into an election, there is an
increase in the number of polls, while at the same time the number of undecided voters drops
significantly. It is generally known that non-response rates are greater between elections than
closer to them (Penniman, 1981). These non-response rates can be substantial, ranging from
25 to 30 percent between elections and dropping down to around 10 to 20 percent during
elections. This has the potential to produce a cycling in the measurement accuracy and
possibly even contribute to cycling in the popularity series, violating both the second and
third conditions of stationarity. This may also cause error variances to not only correlate with
time but for them to be autoregressive.
The potential for the problem of variances correlating and cycling with time can be
explored by partialing out the trending and cycling components of government popularity by
OLS and testing for heteroskedasticity conditional on time (not the individual government
trends) and the inter election cycle. The residuals from this regression can also be tested for
autoregressive conditional heteroskedasticity (ARCH). Estimated Breusch-Pagan LM
statistics for heteroscedasticity conditional on time and the inter election cycle are presented
in table 1, along with the LM statistics for ARCH effects of order up to and including 3.
These are presented separately for each of the three periods.
The Breusch-Pagan LM statistics indicate heteroscedasticity conditional on time is
statistically significant for the first two periods and heteroscedasticity conditional on the inter
election cycle is statistically significant for the second two periods. This suggests the
variances in government popularity are correlated over time in some complex way. The LM
statistics for autoregressive heteroscedasticity are significant at lags of 1 through 3.12 This is
clear evidence of autoregressive heteroscedasticity in government popularity.13
15
Overall, the evidence suggests great potential for the violation of the covariance
stationarity assumption in time-series models of Canadian popularity data. The sources of
many of these violations have been previously overlooked, although the problem of
nonstationarity produced by inter-election cycles and trending at least has not gone unnoticed
and various solutions have been employed. As mentioned, the most advanced solution has
been the Box-Jenkins approach, which in many circumstances is inadequate in dealing with
inter-election cycling.
An alternative to the Box-Jenkins approach to controlling for trending and cycling, is
employing an arbitrary election cycle count variable and a length in office trending variable.14
The difficulty with including an election count variable is that it is arbitrary. This is evident
in the wide variety of count variables employed in different studies.15 Samuel Kernell argues
that such variables do nothing but measure time and are inappropriately used in such
models.16 Moreover, these counts usually only take into account the election period and do
not account for the potential cyclicity of the data during times far from an election. This can
be misleading (Pickup, 2005).
Neither of these methods is able to control for the complexity found in the
measurement error variances. Nor do they account for the problem that popularity is likely a
fractionally integrated process. So, what is the solution? In order to truly understand the
dynamics of public opinion, it is necessary to model explicitly the various components of
which it comprises. In addition to the cycling and trending components, it is necessary to
model fully integrated shifts in baseline support (e.g., those produced by political events) and
stationary shocks/deviations (e.g., those produced by changing economic conditions) with
decaying effects. At the same time, this has to be done in a way that accounts for the
heteroscedasticity in the error component of the popularity time-series and the changing
dynamics of popularity over time.
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3.2 The State-Space Approach to Modelling Government Popularity Dynamics
I suggest an inductive approach, providing a great deal of flexibility in modelling different
types of effects. It is a structural approach in that it builds models which explicitly include
components representing the various dynamics of public opinion. The basis of the
methodological option proposed is to express the time-series model in state-space form and
include separate shocks, cycling, trending, baseline shifts and measurement error
components. The basic model that can incorporate each of these components is:
The Government Popularity State-Space Model
GOVVOTEt = α t + Βt + cyct + ν t
Observation Equation
(including measurement error)
α t = ρα t −1 + γ 1LIBt + γ 2 PCt + ε tα
Stationary Component
(including drift)
Βt = β1LIBt + β 2 PCt + τ 1libtrendt + τ 2 pctrendt + ε tΒ
Baseline Component
(including long-term trending)
cyct = Θ1 sin(λθ ) + Θ 2 cos(λθ ) + ε tcyc
Cycling Component
where
•
t = 1,..., T at monthly increments;
•
ε tα ~ 0,σ ε2α , ε tΒ ~ 0,σ ε2 , ε tcyc ~ 0,σ ε2
(
t
)
(
Β
t
)
(
cyc
t
), ν ~ (0,σ ) and COV (ε
2
t
νt
x
t
,ν t ) = 0 , where
x = α, Β, and cyc
•
σν t is the standard deviation of the estimated sampling error calculated as
p t (1 − p t ) / N t , where p is the proportion of valid respondents supporting the
t
government at time = t and Nt is the sample size. The sample size is calculated as the
number of decided voters polled in each survey. If more than one poll was performed in
any given month, the individual responses were combined and overall aggregate
popularity values were calculated. The sample sizes in these cases would be the total
number of decided voters obtained from combining the polls.17
17
•
λ is the frequency (1/wavelength) of the popularity cycle and is defined by the length of
the inter-election period. (Note: fixing the wavelength to the inter-election period means
that it varies from one election to the next.)
•
ρ , γ 1 , γ 2 , β1 , β 2 , τ 1 , τ 2 , Θ1 , and Θ 2 are parameters to be estimated.
In the state-space model (also known as an unobserved component model), empirical
(observed) values of party popularity (GOVVOTE) are considered the sum of (unobserved)
structural elements Βt , cyct , and α and measurement error ν t . The first three elements
contain error terms and are therefore stochastic. GOVVOTEt is the time series that we observe
and therefore the equation describing it (first line of the government popularity state-space
model) is called the observation equation. The sum of the three structural components
(second, third and fourth lines of the government popularity state-space model) represents the
state of the system, or more precisely the state of government popularity. These components
cannot be directly observed but in order to understand the dynamics of popularity we need to
infer their behaviour.
We can estimate the precision of each observation within the GOVVOTEt time series.
This precision is the inverse of the variance of the measurement error term. Given the
precision of each observation and given the hypothesised structure of the structural
components, Markov Chain Monte Carlo (MCMC) methods can be used to estimate the
parameters of the structural components, such that they maximise the likelihood of each
observation being made. Because the model contains a memory component, the estimation
also maximises the likelihood of each observation being made given all other observations
that were made – that is, the estimation of the state of popularity at a particular time t takes
into account the estimated state of popularity at all other time points, giving greatest weight
to those time points temporally closest to t. This is the Bayesian aspect of the estimation and
it can be done because we know that government popularity in a given month is not
18
independent of government popularity in the months immediately preceding it or following.
This is particularly useful for those months in which surveys contain few valid respondents or
no survey at all was conducted and popularity has to be interpolated (Jackman, 2000).18
Each of the structural components represent different dynamics. The cyct component
explicitly accounts for any inter-election cycling that may exist within the government
popularity series. Estimated parameters Θ1 and Θ 2 can be used to calculate the cycle
amplitude =
Θ12 +Θ22
Θ
1
and phase = COS −1 ( amplitude
) for the inter-election cycle. The component α
captures the effects of variables (presently unmeasured) not related to baseline shifts or
cycling. Once included, these effects would produce deviations (shocks) in popularity from
its baseline (i.e., equilibrium). The ρ term represents and controls for the first order
autoregression [AR (1)] within government popularity and determines the length of time an
event which produces a deviation from baseline (and is modelled through the α component)
continues to have an effect on popularity. The fading impact of these events describes a
stationary process. Further autoregressive and/or moving average components can be
included if appropriate. Ultimately, it is through the α component that economic conditions
are entered into the state-space model. Economic effects are modelled this way because they
are expected to have long-term effects which are cumulative but not permanent. In other
words, the electorate responds to the state of the economy over the long-term but a party is
not punished or rewarded for the state of the economy in any given month forever after.
Βt is a measure of baseline support for the party, including long-term trending but
excluding cycling. It is the equilibrium value for party popularity. Political variables
producing shifts in baseline support can be entered through this term. In its present form it
includes two such political variables. Their effects are measured by β1 and β 2 . The first
indicates a Liberal government and the second a Conservatives government. The magnitude
19
of these separate constants reflects the underlying support with which the Liberals or
Conservatives begin their term in government.
Once included, the effect of political events modelled by the Βt component can be
designed to represent a temporary or permanent shift in equilibrium. They are modelled as an
integrated process. Such dynamics would represent an immediate shift in popularity in
response to the occurrence of some political event. This shift remains forever after or until
some other occurrence. In other words, the impact of the event becomes fully integrated. An
occurrence can include the end of an event, in which case the impact of the event lasts only
for the duration of the event. For example, this is generally the type of effect that political
events such as national crises are expected to have on popularity. Popularity responds
immediately to the crisis and that response is sustained for the duration of the event but once
it is over, popularity almost immediately returns to previous levels.
The proposed method of separately modelling integrated and stationary processes is
an alternative solution to the problem of fractional integration within the popularity timeseries from those suggested by Janet Box-Steffensmeier and Andrew Tomlinson (2000).19
The impact of two different forms of trending can be calculated by the γ and τ terms.
τ 1 and τ 2 measure long-term trending in the government popularity series, depending upon
whether the Liberals or PCs are in power. The trend is modelled simply as a linear increase or
decrease in the popularity equilibrium. The trend would generally be expected to be negative.
Its structure is based on the idea that a government is formed through a coalition of interests.
The longer a party is in power, the harder it becomes to hold this coalition together.
Parameters γ 1 and γ 2 measure memory-based, short-term trending – also known as
drift. It is theoretically based on the notion that a party may gain or lose popularity by virtue
of the fact that it is in government. On one hand, the party in government may be unable to
avoid decisions that are inherently unpopular. On the other hand, it may have the resources to
20
implement popular programs, which the other parties do not have. This trending component
assumes that the impact of these actions produces a shift in popularity only so long as they
are remembered. This memory is measured by ρ . Unlike the long-term trending, short-term
trending is not ever-increasing or decreasing. Eventually, the shift in popularity produced by
recent actions will be offset by the diminishing impact of past actions, as they are forgotten.
The total drift that would occur in a party's popularity due to these actions if they were to
remain in government indefinitely is calculated as γ /(1 − ρ ) . This value is the equilibrium
level for the short-term trend – the total increase or decrease in popularity produced by the
short-term trend.
The inclusion of a short-term trend beginning at the beginning of a governments term
in office is an explicit recognition that government popularity may not begin in equilibrium
the day (or month) after the election. This is an assumption that is regularly made in
timeseries modelling which could easily be violated by the nature of government popularity.
As with the violation of other important assumptions, the state-space approach rectifies it by
explicitly modelling the dynamic that produces the violation.
The sum of Βt , cyct and α (the state of government popularity) represents “filtered”
values of government popularity, in that they exclude ν t , the “noise” produced by survey
measurement error (Harvey, 1993). Including the measurement error term and estimating its
standard deviation based on the number of valid respondents each month allows us to
explicitly account for the variations in measurement accuracy produced by fluctuating sample
sizes – that is, increased accuracy over time with rising numbers of polls, decreased accuracy
over time with increasing numbers of undecided voters, spikes in accuracy near and during
election months and cycling accuracy between elections.
In order to account for the potential of three distinct government popularity periods,
the time-series model is estimated separately for each proposed period. As previously noted,
21
the model for the second period has to start in 1979. This is a consequence of including the
short-term trending term (drift). It requires that the time-series begin the month after an
election and the first election to follow the one held in 1974 occurred in 1979. With monthly
measures of popularity, the first period contains 222 time points, the second period contains
172 time points and the third period contains 84 time points. The next section discusses the
results of this modelling exercise.
4.0 Government Popularity Dynamics
Table 2 contains median values of the Bayesian-estimated distributions of the parameters for
the state-space model of government popularity.20 Parameter estimations were made using
Winbugs.21 Winbugs performs Bayesian analysis of statistical models using MCMC
methods.22 The order of the AR and MA terms to include in the α component was
determined by examining the AC and PAC functions in figure 3. Only those terms required to
remove serial correlation in the α component error term were retained (more on this below).
The first-order autoregressive factor in the first period is about 0.84, suggesting a
large amount of public opinion memory in the government popularity process from one
month to the next. In a stationary process, the AR(1) is greater than -1 and less than 1. If it is
between 0 and 1, then values closer to 1 indicate longer memory than values closer to 0. The
third-order autoregressive factor is statistically insignificant and only 0.008. In the second
period, the first-order autoregressive factor is 0.91 and the third-order term is insignificant at
0.04. The first-order autoregressive factor in the third period is much less than that of the
second period at 0.47. The fourth-order autoregressive factor is insignificant at 0.16.
During the first period (1957-1975), there is a statistically significant cycle but its
amplitude is only 1.4 percent. This suggests that Johnston's (1999) observation that before
1974 government popularity did not follow any inter-election dynamic is not completely
22
accurate. However, one can hardly be blamed for failing to visually catch such a small
fluctuation.
During the second period (1979-1993), a clear cycle is evident. The amplitude is 5
percentage points, meaning that government popularity fluctuated by 10 percent throughout
the cycle. The time shift (phase) of the cycle indicates that when elections occurred,
popularity was 9 percent of the cycle away from reaching its maximum. This means that in a
full five-year inter-election period, an election occurs almost 5 months before popularity
reaches its maximum in the cycle. It is incorrect to assume that if an election occurs before
popularity reaches a maximum in its cycle, the government would have received a greater
proportion of the vote if the election was delayed. The rise in popularity subsequent to an
election is probably a function of the election having occurred (a honeymoon). What is being
described here is simply the position of elections along the popularity cycle, which may occur
for a number of reasons.
The findings regarding the second period cycle are consistent with Johnston's
observations (1999). Following an election, government popularity trends upwards through a
traditional honeymoon period. Popularity then proceeds to decline to its nadir before
beginning to make a recovery. Soon after popularity has risen to average or just above
average levels for the government, an election is called.23 Although the amplitude is smaller
and elections are held earlier in the cycle, the government popularity cycle in the third period
(1993-2000) is similar to that of the second period. The amplitude of the cycle is significant
at 2.7 percentage points. The time shift suggests that elections occur when popularity is
around 35 percent of the cycle away from reaching its maximum – that is, elections occur
over 21 months before popularity reaches its maximum in a full five-year inter-election time
period. (Of course, the inter-election period never reached a full five years between 1993 and
2000.)
23
In the first period, the short-term trend for the PCs is statistically insignificant, while
the long-term trend is negative. This is consistent with the long-term decline in popularity of
the Diefenbaker government. The long-term and short-term trends for Liberal governments
are not significant. It should be noted that as expected, the long-term trend for both Liberal
and PC governments in all periods is estimated to be downward or zero.
The difference between the Liberal and PC constant values ( β1 and β 2 ) is an
indicator of the difference between the underlying popularity of Liberal and PC governments
before trending occurs. In the first period, the difference is over 23 percentage points in
favour of PC governments. This reflects the difference between the large majority initially
put together by Diefenbaker and the initial minority government of Lester Pearson.
The long-term decline in popularity during the second period is only significant for
the Liberal government. This is a surprising result given the very large decline in popularity
apparently experienced by the Mulroney PC government. Although, it may be explained by
the fact that only one term was included to account for trending during both Clark and
Mulroney's PC governments. The difference in popularity dynamics between the two
governments is fairly large and the insignificance of the common trend term suggests that
separate trends should be estimated for each government. As will be seen in the next section,
estimating a separate trend for the Mulroney government does produce a negative and
significant results as would be expected. The short-term trend for both Liberal and PC
governments is insignificant.
The estimated difference between Liberal and Conservative underlying popularity in
the second period reflects a similar problem as the estimation of trending. The low value for
the PCs compared to the Liberals is largely a product of the low popularity of the Clark
minority government. This obscures the high popularity of the Conservatives in the rest of the
period. This too is rectified in the next section. Distinctive features of the third period are the
24
particularly high constant for the Liberals (67 percent) and the insignificance of either short
or long-term trending. This is consistent with the sustained popularity of the Chrétien
government during this time.
Having modelled the various components of government popularity, we can now plot
the predicted government popularity based on the deterministic parts of the government
models (figure 5) – that is, the cycling, trending, and baseline components. Many of the
largest movement in party popularity are predicted by these components. However, there is
clearly still a great deal of residual movement to be explained. Part of this movement will, of
course, be measurement error but a great deal of it can be attributed to economic conditions
and political events not explicitly included in the model so far.
An approximation of the degree of movement that may be explained by such factors
can be obtained by comparing the residual movement not explained by the cycling, trending
and baseline components and that which cannot be attributed to measurement error to the
total variance of the original popularity time-series. These values are presented in Table 3.
The percentage of the total variance that remains to be explained is 18.8, 27.6 and 11.8 for
the first, second and third periods respectively. It is within this remaining movement that we
expect to find the impact of economic conditions.
Specifically, it is within the residual movement of the government popularity α
component that we expect to find the impact of economic conditions. Therefore, it is
necessary to examine the estimated α component to determine if any nonstationarity remains.
Remember the α component is assumed to be stationary and this assumption needs to be
confirmed before economic effects are modelled through it.
That the residuals of the α component are a white noise process can be confirmed by
applying the Q-test to them. The results of these tests are presented in table 2. The null
hypothesis is that the residuals are white noise. We are unable to reject the null for the models
25
from any of the three periods. Residual variances can also be examined by regressing the
square of the residuals on time and inter-election cycling. The results are also presented in
table 2. The results suggest that the variances are not correlated with time and do not cycle
with the inter-election period. This indicates that explicitly modelling the cyclical and
trending components of the time-series and explicitly accounting for variations in
measurement accuracy removes the problem of error variances correlating or cycling with
time from the α component. Based on this evidence and the Q-test, α is a stationary process,
meaning that the nonstationarity in the popularity time-series, produced by political
contextual variables and complex measurement error processes, has been removed and so it is
appropriate to model the impact of economic conditions through it.
To provide an example of the utility of the state-space approach, the next section
builds a popularity function for the Progressive Conservative government in Canada between
1984 and 1993. This was a period of high volatility for government popularity and relatively
large movements in economic conditions, particularly economic growth and inflation (figure
6). Johnston (1999), using an error correction model, argues that government popularity
during this period benefited from both inflation and economic growth. In contradiction, if one
were to apply the Box-Jenkins approach or the approach used by Sanders (2000), the
conclusion would be that neither economic growth nor inflation had any impact on Canadian
government popularity during this period.24 This makes this period an interesting one to
demonstrate the state-space approach.
5.0 Modelling Progressive Conservative Government Popularity, 1984-1993
Based on the government popularity model presented in section 3.0, the following is the
Progressive Conservative government popularity state-space equation which models the
impact of economic conditions through the α component.
26
Progressive Conservative Government Popularity State-Space Equation, 1984-1993
PCVOTE t = α t + Β t + cyct + ν t
α t = ρα t −1 + γ 1 + δ 1 INFt −k + δ 2 GDPt −k + δ 3GDPt −k INFt − k + ε tα
Β t = β1 + τ 2 pctrend t + ε tΒ
cyct = Θ1 sin(λθ ) t + Θ 2 cos(λθ ) t + ε tcyc
where
•
ρ , δ 1 , δ 2 , δ 3 , γ 1 , are parameters to be estimated.
•
β1 ,τ 1 , Θ1 and Θ 2 are estimated in the state-space model excluding economic variables
and are fixed at these estimated values.
•
INF is the year-over-year change in the consumer price index.
•
GDP is year-over-year percentage change in real personal income per capita.
The cyct and Βt components have already been described.
The component α acts as the dependent variable in the equation which estimates the
impact of economic conditions on popularity. The ρ term within the α equation represents
the first order autoregression within government popularity. AC and PAC functions for
Progressive Conservative government popularity during this period, with trending and
cycling partialled out, suggest that only an AR(1) term is required. The Q-test for white noise
applied to the estimated α residuals (45.04, P = 0.269) confirm this.25
The model of the impact of economic conditions includes an interaction term between
changes in real per capita income and inflation. This is done for theoretical reasons. It is quite
possible that in the minds of the electorate, the gains or losses in popularity produced by
changes in real income are mediated by the degree of concurrent inflation. For example,
when inflation is high the popularity gains to be made by a party in government from
economic growth may not be as great as when inflation is more moderate.26
27
The above model of government popularity also includes economic variables lagged
by k. This may include a particular economic variable being included with more than one lag.
In order to determine the lag structure of the economic variables, a modified version of the
Box-Jenkins approach to this issue is proposed.
The first step in the Box-Jenkins approach is to determine the univariate model (or
transfer function) for all time series – in this case, for the party popularity and economic
variables. If nonstationarity is evident then the series are differenced appropriately. This
strategy is appropriate for the economic variables but cannot be applied to the party
popularity time series. As noted before, if nonstationarity is in part produced by cycling with
an inconsistent frequency (based on the timing of elections), differencing the series will not
correct the problem. Fortunately, the state-space approach allows us to extract the stationary
α t component of party popularity in which we expect to find the effects of economic
conditions. It is this component that we utilise rather than the differenced popularity series
usually used in the Box-Jenkins approach.
Once the transfer function is determined for each stationary economic series, it can be
inverted and applied to both the economic series and the α t component of the government
popularity series estimated excluding economic variables. The resulting series are referred to
as prewhitened. The cross-correlation function between each prewhitened series (economic
and popularity) is calculated in order to identify potential lags at which each economic series
should enter the popularity model.
Having identified the appropriate lag structure, the model is estimated and the
resulting residuals are examined. If the residuals are not white noise, an appropriate model for
the noise component may be identified. If this is necessary, the model is reestimated and the
residuals are examined again. Once the appropriate models have been identified, the new
28
model residuals can be cross-correlated with the prewhitened economic variables to identify
any lagged relationships which might have been missed.
Applying this procedure to PC government popularity between 1984 and 1993, it is
determined that government popularity is cross-correlated with GDP, inflation and the
interaction between GDP and inflation at lag 1.27 Accordingly a Progressive Conservative
government economic popularity model was estimated (table 4). It included inflation, GDP
and their interaction, lagged 1 month. GDP, inflation and its interaction with GDP prove to be
statistically significant. Based on these results, the government economic popularity model is:
Progressive Conservative Government Popularity State-Space Equation, 1984-1993
PCVOTE t = α t + Β t + cyct + ν t
α t = 0.684α t − 1 − 0.021 − 0.004GDPt −5 INFt −1 − 0.007 INFt −1 + 0.019GDPt −1 + ε tα
Β t = 0.427 − 0.002 pctrend t + ε tΒ
cyct = −0.0075 sin(λθ ) t + 0.0208 cos(λθ ) t + ε tcyc
Examining this model of government popularity, both the long-term and short-term trending
in popularity is negative and statistically significant. The coefficient on the long-term trend
component suggests a loss of a quarter of a percentage point per month. The cycling
component is also statistically significant, with an amplitude of approximately 10 percentage
points. These results suggest strong trending and inter-election cycling. The phase of the
cycle is 0.04, suggesting elections occurred about 2.4 months before the peak in PC
government popularity.
As for the economic popularity function built through the α t component, it is evident
that during the 1984-1993 period, the Conservative government was rewarded for growth in
GDP and penalised for inflation (one month prior). At the beginning of its term in office in
November of 1984, growth in GDP was 4.86 percent and inflation was 2.6 percent. In the
29
following month, the government benefited from the economy by 2.4 percentage points. Six
years later, in March of 1991, growth in GDP was -4.6 percent and inflation was 5.8 percent.
The Conservative government was consequently punished by 2.1 percentage points.
The interaction between GDP and inflation acted to strongly moderate the negative
impact of inflation when economic growth was high. In October of 1987 when inflation was
3.4 percent but growth in GDP was a healthy 4.9 percent, the penalty to the Conservatives
was offset. In fact, overall, they were rewarded approximately a quarter of a percentage point.
Growth in GDP was able to (just) trump inflation. Unfortunately for the Conservatives,
economic growth rarely hit such high levels. Average growth in GDP was 1.1 percent and
average inflation was 3.3 percent. In a given month, these economic conditions would
produce a 1.7 percent decline in popularity.
In addition to the impact of economic conditions in an individual month, the presence
of memory within public opinion makes it important to also considered the cumulative
impact of economic conditions. This is the monthly contribution of economic conditions if
they are held at some constant level. If economic conditions are held at a constant level, such
as average inflation and GDP levels, their contribution to popularity quickly reaches some
equilibrium. This equilibrium depends upon the degree of memory and the level of economic
conditions. The fact that the AR(1) term is positive suggests that the cumulative impact will
be larger than that in a given month. The cumulative impact of these average GDP growth
and inflation levels in any given month is minus 5.3 percentage points. This suggests that PC
popularity on average 5.3 percentage points lower than it would have been if it was not held
accountable by the voting public for economic conditions during the period.
The economic effects on government popularity found so far excludes the interelection cycle. It is a separate question as to whether cycling in popularity has an economic
source. To get at this, it is necessary to determine if inter-election cycling exist within
30
economic conditions. Such cycling in GDP growth and inflation can be modelled using the
same methods used to model the inter-election cycling in government popularity. In doing so,
a statistically significant inter-election cycle is found in both economic variables. Figure 7 is
a plot of the deterministic part of the two economic cycles along with a plot of the
deterministic part of the government popularity cycle.
Using this approach, we cannot make statistical conclusions but the results are highly
suggestive. The peaks in government popularity occur within months of the peak in GDP.
The inflation cycle is offset such that during these periods inflation remains relatively low.
The nadir in popularity subsequently occurs when GDP bottoms out and inflation remains
relatively high. This is consistent with the economic effects found within the stationary
component of government popularity. The causal story behind this pattern is the subject for
another paper, although the timing of the cycles relative to the timing of elections does hint
strongly at some political business cycle explanation. It is important to note that no other
standard method of developing economic popularity functions is in a position to explore the
potential of such economic effects. Other methods either inappropriately ignore or
unhelpfully eliminate them.
6.0 Conclusion
This study endeavors to make a substantial methodological contribution to the modelling of
economic popularity functions using difficult time-series popularity data. It proposes and
demonstrates the utility of a unique structural approach to accounting for the great number of
sources of nonstationarity – cycling, trending, shifts in equilibrium and complex patterns of
measurement error – within such data. The basis of this structural approach is the use of the
state-space form of time series modelling. In this approach, the observed values of popularity
are regarded as being made up of distinct unobserved components and measurement error.
Many of the unobserved components represent the political context in which
31
economic popularity operates. These dynamics are each modelled separately. This allows the
analyst to explore the dynamics of the political context in which economic conditions are
translated into government popularity. Another important nonstationary dynamic of measured
popularity is measurement error. By explicitly modelling this error, the state-space model
accounts for error variances which correlate and cycle with time. The unobserved
components also include a stationary process. This stationary component is extracted from
the nonstationary dynamics and is also modelled separately. Consequently, it is appropriate to
use modelling techniques that assume stationarity to model the impact of economic
conditions through it. Through this approach, the statistical problems posed by trending,
inter-election cycling and measurement error variances cycling and correlating with time are
addressed. This approach also provides one of the most sophisticated methods to date to
account for a shifting equilibrium due to integrated processes.
As demonstrated, it is possible to determine the effects of economic conditions on
government popularity between 1984 and 1993 using the methods developed in this study.
Growth in GDP was beneficial and high inflation was detrimental. There is little question that
the economy mattered to government popularity and that the Conservative government
between 1984 and 1993 was held accountable for it. This is in contrast to Johnson's perverse
finding that inflation helped the government during this period and the complete lack of
findings using other methods.
The example of the state-space approach provided also reveals the very real potential
of a political business cycle story connecting inter-electoral cycling in government popularity
and inter-electoral cycling in economic conditions. This is a finding that is missed by other
approaches and further demonstrates the utility of the Bayesian state-space approach to
economic popularity functions and its advantages over other approaches.
32
The difficulties of developing economic popularity functions outlined here are not
limited to government popularity timeseries. Many public opinion timeseries exhibit very
similar patterns. A final strength of the state-space approach worth identifying is its
flexibility. By breaking public opinion dynamics down into its constituent parts, the forces
within public opinion and effects of forces acting on public opinion are revealed. The statespace approach, as it has been outlined, has the potential of being a valuable resource to a
great number of longitudinal public opinion analyses.
33
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34
1
For a recent discussion of the importance of economic conditions on government support see Duch and
Stevenson, forthcoming .
2
In the US case, the popularity function is actually developed for that of the President. While there are
important measurement differences between Presidential and government popularity, the modelling techniques
and issues are identical. Therefore, in this study the term government popularity will be assumed to include
presidential popularity.
3
Prominent examples include Mueller (1970), Kernell (1978), and Beck (1991).
4
Sanders differenced the economic variables on the basis that the series were unit root processes.
5
It should be noted that differencing a timeseries often introduces complex dynamics in the error structure
which then must be accounted for.
6
(Green, Donald, Alan Gerber, and Suzanna De Boef. 1999. Tracking Opinion over Time: A Sampling Method
for Reducing Sampling Error. Public Opinion Quarterly 63:178-192.) and (Beck, Nathaniel. 1990. Estimating
Dynamic Models Using Kalman Filtering. Political Analysis 1.)
7
Figure 1 uses Kalman filtered data, making the earlier period of the time-series easier to interpret by including
interpolated values for the months in which no poll was reported. Harvey, A. C. 1993. Time Series Models. 2nd
ed. Cambridge, Mass.: MIT Press.
8
Many (although not all) economic popularity studies use and focus on government popularity, rather than party
popularity. This is done because the electoral accountability debate is ultimately about support for the governing
party. There are strong arguments for modelling the popularity of the Liberal and Conservative parties
separately, regardless of whether they are in government or opposition. This is called party popularity and is
defined as the percentage of poll respondents that indicate they would vote for a particular party. However, this
paper builds a model of government popularity so that the result can be compared with those from previous
studies.
9
These are actually the conditions for weak stationarity, which is sufficient for our purposes.
10
Consequently, some analysts suggest only modelling short periods of time (Clarke and Zuk, 1987).
11
Aggregating the data to a quarterly level does not solve this problem. Quarterly measurements made later in
the time-series will still be more accurate and have smaller variances than those earlier in the time-series.
12
This does not necessarily mean that the residual variances are autoregressive up to an order of 3 or beyond.
Strong first order autoregression will produce statistically significant coefficients at higher lags.
13
Of course, the heteroscedasticity and autoregressive heteroscedasticity could be caused by an error process
other than that produced by measurement error.
14
Lewis-Beck, Michael. 1988. Economics and Elections. Ann Arbor: University of Michigan Press.
15
Ibid.
16
Kernell, Samuel. 1978. Explaining Presidential Popularity. How Ad Hoc Theorizing, Misplaced Emphasis,
and Insufficient Care in Measuring One's Variables Refuted Common Sense and Led Conventional Wisdom
Down the Path of Anomalies. The American Political Science Review 72 (2):506-522.
17
Including the separate measurement error term (ν t ) is closely related to Nathaniel Beck's use of the Kalman
filter to estimate presidential popularity (Beck, 1990).
Missing values are handled by using interpolation and then assigning large standard errors to the interpolated
value. Within the Bayesian framework, this allows the surrounding time points which have more certain
measurements to be used to fill in for the missing data.
19
If fully integrated processes contained in the popularity series are not explicitly modelled in the Βt
18
component, they are likely to be captured by the residuals of the
α
component. This is because while the Βt
component can be used to model fully integrated political dynamics, this is done through the way in which the
political variables are constructed. The component itself is not fully integrated. It contains no memory term
whatsoever. Therefore, the α component may remain somewhat fractionally integrated.
20
Note: popularity dependent variable is entered into the model as a proportion rather than a percentage.
21
All models were estimated using two chains with varying initial values.
22
Details on the guidelines used to determine significance of Bayesian estimated parameters can be obtained
from the author at [email protected].
23
The election called by Prime Minister Turner in 1984 appears to be an exception, in that the month in which
the election was called Turner's popularity had just dipped below its average. However, the poll results just
previous to the month in which the election was called were above-average and it was likely this information
that Turner had available when making his decision.
24
The author would be happy to provide the results from either of these methods applied to this period.
35
The equation describing the α component is similar to the partial adjustment model recommended by
Nathaniel Beck (1991) and used by Johnston (1999).
26
It is important to note that this interaction does not represent a real economic interaction between changes in
real GDP and inflation. The interaction is hypothesised to be purely within the opinions of the electorate. If this
were an economic rather than a public opinion model, the structure would certainly be different.
27
The crosscorrelation table actually reveals no significant results but the largest crosscorrelations for all three
variables (GDP, inflation and their interaction) occur at a lag of 1. The economic variables were therefore
entered accordingly.
25
36
TABLES AND FIGURES
37
Table 1: Tests of Cycling, Trending, Heteroscedasticity and Autoregressive Conditional Heteroscedasticity
1957-1975
1979-1993
Lag
Coefficient
Newey-West
Coefficient
Newey-West
Standard Errors
Standard Errors
Cycle1
0.86
0.010
1.02
5.88
Cycle2
0.36
0.008
1.32
2.31
Liberal trend
0.03
-0.07
0.07
-0.20
PC trend
0.02
0.02
-0.04
-0.14
Constant
1.57
1.67
50.31
40.66
N
222
—
213
—
F
P-value
F
P-value
Cycle F-test
0.37
0.689
16.89
0.000
Breusch-Pagan LM test for heteroscedasticity conditional on:
Chi2
P-value
Chi2
P-value
0.046
0.000
Time
3.97
18.61
1.42
0.000
0.000
Cycle
25.21
LM test for autoregressive conditional heteroskedasticity, H0: no ARCH effects
Lag
Chi2
P-value
Chi2
P-value
1
0.0000
0.0000
66.6
144.3
2
0.0000
0.0000
83.6
143.6
3
0.0000
0.0000
83.6
143.6
Coefficient
-1.14
2.08
-0.02
—
55.67
84
F
6.32
1993-2000
Newey-West
Standard Errors
0.79
0.63
0.02
—
0.92
—
P-value
0.000
Chi2
2.20
7.89
P-value
0.138
0.019
Chi2
9.3
10.7
11.5
P-value
0.0023
0.0049
0.0093
38
Table 2: Estimated Parameters of Government Popularity State-Space Model
1957-1975
Standard
Deviation
Median
AR(1)
α ARMA
0.1696
0.8057
AR(3)
0.1431
0.0085
Cycle
Progressive
Conservatives
Residuals
1993-2000
Standard
Deviation
Median
0.2031
0.4676
—
—
AR(4)
—
—
0.1252
—
0.0403
—
0.1619
0.1638
MA(7)
—
—
—
—
0.4863
-0.2181
0.0075
0.0136
0.0165
0.0507
0.0113
0.0271
—
0.1849
—
0.0862
—
0.3471
Trend – Short
Trend – Long
Constant
0.0301
0.0003
0.1015
0.0286
-0.0002
0. 0078
-0.0065
0. 5092
0.0476
0.0012
0.0873
-0.0408
0.0000
0.3098
0.0066
0.0015
0.0676
Trend – Short
0.0119
0.0155
0.0086
0.0046
—
—
Trend – Long
0.0010
-0.0046
0.0025
0.0019
—
—
Constant
0.0286
0.5369
0.0244
0.3327
—
—
Test Statistic
26.444
-1.43
0.49
P-value
0.951
0.154
0.6159
Test Statistic
23.112
-1.63
2.59
P-value
0.985
0.105
0.0778
Test Statistic
37.518
0.51
2.05
P-value
0.583
0.613
0.1356
Amplitude
Phase
Liberals
1979-1993
Standard
Deviation
Median
0.1300
0. 9139
White noise Q-Test
Trending t-test
Cycling F-test
0.6651
Notes:
1) Phase is calculated post-Bayesian estimation and so no SE for its distribution is provided but the statistical significance of the parameters used in the
calculation indicate the significance of phase.
2) The popularity dependent variable is entered into the model as a proportion rather than a percentage.
3) Bolded values are determined to be statistically significant based on Bayesian estimated distribution of parameters.
39
Table 3: Government Popularity Variance, Explained and Unexplained
1957-1975
1979-1993
1993-2000
Total Variance
Variance Explained by Deterministic Components
Variance Attributable to Measurement Error
Unexplained Variance
39.6
123.1
16.3
19.7
94.4
3.8
1.1
1.1
0.7
18.8
27.6
11.8
Popularity is expressed as percentage for the purposes of this table
40
Table 4: Progressive Conservative Economic Popularity, 1984-1993
Box-Jenkins
SE
Median
0.1014
Memory AR(1)
0.6839
0.02063 0.09604
Amplitude
—
Phase
0.040
0.04041 0.4268
Constant
7.86E-04 -0.00246
Trend – Long
0.0114
Trend – Short
-0.0207
0.0014
InflationxGDP(t-1)
-0.0041
0.0030
Inflation (t-1)
-0.0067
0.0066
GDP (t-1)
0.0185
Q-Test P-value
Residuals
41
Figure 1: Government Popularity
Government Popularity 1957-2000 with Elections Demarcated
0.70
0.65
0.60
Government Popularity
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002
Year
42
Figure 2: Government Popularity Autocorrelation Functions
Figure 2a.
Figure 2b.
Partial Autocorrelation Function: Goverment Popularity
1957-1974
1957-1974
-0.20
-0.50
Autocorrelations of govvote
0.00
0.50
Partial autocorrelations of govvote
0.00 0.20 0.40 0.60 0.80
1.00
Autocorrelation Function: Goverment Popularity
0
5
10
15
20
25
Lag
0
5
10
15
20
25
Lag
Bartlett's formula for MA(q ) 95% confidence bands
95% Confidence ba nds [se = 1/sq rt(n)]
Figure 2c.
Figure 2d.
1979-1993
-0.50
Autocorrelations of govvote
0.00
0.50
Partial autocorrelations of govvote
0.00
0.50
1.00
Partial Autocorrelation Function: Goverment Popularity
1979-1993
1.00
Autocorrelation Function: Goverment Popularity
0
5
10
15
20
0
25
5
10
15
20
25
Lag
Lag
95% Confidence ba nds [se = 1/sq rt(n)]
Bartle tt's formula for MA(q ) 95% confidence bands
Figure 2e.
Figure 2f.
Partial Autocorrelation Function: Goverment Popularity
1993-2000
1993-2000
0
5
10
15
Lag
Bartlett's formula for MA(q) 95% confidence bands
20
25
-0.40
-0.40
Partial autocorrelations of govvote
-0.20
0.00
0.20
0.40
Autocorrelations of govvote
-0.20
0.00
0.20
0.40
Autocorrelation Function: Goverment Popularity
0
5
10
15
20
25
Lag
95% Confidence bands [se = 1/sqrt(n)]
43
Figure 3: Decycled/Detrended Government Popularity Autocorrelation Functions
Figure 5a.
Figure 5b.
1957-1974
-0.50
Autocorrelations of detrendedgov
0.00
0.50
Partial autocorrelations of detrendedgov
-0.20 0.00 0.20 0.40
0.60 0.80
Partial Autocorrelation Function: Goverment Popularity
1957-1974
1.00
Autocorrelation Function: Goverment Popularity
0
5
10
15
20
25
Lag
0
Bartlett's formula for MA(q) 95% confidence bands
5
10
15
20
25
Lag
95% Confidence ba nds [se = 1/sqrt(n)]
Figure 5c.
Figure 5d.
1979-1993
-0.50
Autocorrelations of detrendedgov
0.00
0.50
Partial autocorrelations of detrendedgov
-0.20
0.00
0.20
0.40
0.60
0.80
Partial Autocorrelation Function: Goverment Popularity
1979-1993
1.00
Autocorrelation Function: Goverment Popularity
0
5
10
15
20
0
25
5
10
15
20
25
Lag
Lag
95% Confidence ba nds [se = 1/sq rt(n)]
Bartle tt's form ula for MA(q ) 95% confidence bands
Figure 5e.
Figure 5f.
1993-2000
-0.40
Partial autocorrelations of detrendedgov
-0.40
-0.20
0.00
0.20
0.40
Partial Autocorrelation Function: Goverment Popularity
1993-2000
Autocorrelations of detrendedgov
-0.20
0.00
0.20
0.40
Autocorrelation Function: Goverment Popularity
0
5
10
15
20
25
0
5
10
Lag
Bartlett's formula for MA(q) 95% confidence bands
15
20
25
Lag
95% Confidence bands [se = 1/sqrt(n)]
44
Figure 4: Number of Valid Decided Voters Interviewed Each Month
Number of Decided Voters Interviewed Each Month
7000
6000
5000
N
4000
3000
2000
1000
0
1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002
Year (tick mark identifies beginning of year)
45
Figure 5: Predicted Government Popularity from Deterministic Parts of State-Space Popularity Model
Predicted Government Popularity
Government Popularity
0.6
0.4
0.2
0.0
1956
1960
Measured
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
Year
Predicted
95% Confidence
46
Figure 6: Government Popularity, Economic Growth and Inflation, 1984-1993
0.6
8
Inflation
GDP Growth
PC Popularity
7
0.5
5
4
0.4
3
2
1
0.3
0
-1
0.2
-2
-3
PC Governement Popularity
Inflation and GDP Growth
6
0.1
-4
-5
-6
1984
0.0
1986
1988
1990
1992
1994
Year
47
Figure 7: Government, Inflation and GDP Growth Cycles, 1984-1993
0.3
0.2
0.0
-0.3
1984
1985
1986
1987
1988
1989
1993 ELECTION
-0.2
1988 ELECTION
-0.1
1984ELECTION
Cycle
0.1
1990
1991
1992
1993
Government Cycle
GDP Cycle
Inflation Cycle
1994
Year
48