Download Presidential Elections and the Stock Market:

Presidential Elections and the Stock Market:
Comparing Markov-Switching and (FIE)GARCH Models of Stock Volatility1
David Leblang
Associate Professor
Dept. of Political Science
University of Colorado
[email protected]
Bumba Mukherjee
Assistant Professor
Dept. of Political Science
Florida State University
[email protected]
Abstract: Existing theoretical research on electoral politics and financial markets predict that
when investors expect left parties –Democrats (US), Labor (UK)—to win elections market
volatility increases. In addition, current econometric research on stock market volatility
suggests that Markov-Switching models provide more accurate volatility forecasts and fit
stock price volatility data better than linear or non-linear GARCH (Generalized
Autoregressive Conditional Heteroscedasticity) models. We take issue with both of these
claims. We construct a formal model which predicts that if traders anticipate that the
Democratic candidate will win the Presidential election stock market volatility decreases. Using
two data sets from the 2000 Presidential election we test our claim by estimating several
GARCH, Exponential-GARCH (EGARCH), Fractionally Integrated Exponential-GARCH
(FIEGARCH) and Markov-Switching models. We also conduct extensive out-of-sample
forecasting tests to evaluate these competing statistical models. Results from the out-ofsample forecasts show—in contrast to prevailing claims—that GARCH and EGARCH
models provide substantially more accurate forecasts than the Markov-Switching models.
Estimates from all the competing statistical models support the predictions from our formal
model.
1 Prepared for presentation at the 2003 Political Methodology Summer Meetings, Minneapolis, MN. We are
grateful to Charles Franklin, Christopher Wlezien, and Andre Gibson of the Chicago Mercantile Exchange for
providing data and to Jude Hays, William Bernhard and Brian Gains for helpful discussions regarding the
calculation of election night probabilities.
1. Introduction
There is an extensive empirical literature focusing on the relationship between politics
and financial markets. Scholars studying currency, stock and bond markets have examined the
role that electoral systems, elections, partisanship, and political uncertainty play in shaping
both the value and volatility of financial assets (e.g., Freeman, Hays and Stix 2000; Martin and
Moore 2003; Leblang and Bernhard 2000a, 2000b; Blomberg and Hess 1997, Lobo and Tufte
1998; Alesina, Roubini and Cohen 1997; Cohen 1993; Gartner and Wellershoff 1995; Herron
2000; Herron et al 1999; Goodhart and Alter 2003; McGillivray 2000, 2002; Roberts 1994;
Gemill 1992, 1995). This literature shares two broad similarities. First, from a methodological
perspective, scholars working in this research area typically use similar econometric tools -inclusive of ARMA, GARCH (Generalized Autoregressive Conditional Heteroscedasticity),
EGARCH2 or Markov-Switching3 models—to test theoretical predictions4 on samples of highfrequency stock price and currency volatility data.5 Some of these scholars favor the use of
Markov-Switching models claiming that Markov-Switching models are more accurate and
provide better forecasts than a variety of linear and nonlinear GARCH models (Turner, Stratz
and Nelson 1989; Kim, Morley and Nelson 2002; Van Norden and Schaller 1993; Sola and
Timmerman 1994; Simonato 1992). This claim is surprising since, in reality, there is almost no
2 Applications of the GARCH and/or (E)GARCH models to test hypotheses on currency or stock price
volatility include Leblang and Bernhard (2000a, 2000b), Bollerslev et al (1992), Ramchand and Susmel (1998),
Poon and Taylor (1992). An excellent literature review that surveys applications of the family of GARCH
models in the financial economics literature can be found in Fan (2002).
3 Application of the Markov-Switching Model (Hamilton 1989; 1994) in macroeconomics, financial economics
and political economy is far too vast to cite in a single footnote; however, for a literature review that provides a
detailed bibliography of various applications of Markov-Switching models, see McCulloch and Tsay (1994).
Among political economists, Freeman, Hays and Stix (1999, 2000) and Blomberg and Hess (1997) have used
Markov-Switching Models to estimate the impact of political variables on exchange-rate volatility.
4 GARCH and Markov-Switching models are not the only econometric techniques used by scholars. For
instance, Herron (2000) and Herron et al (1999) use nonlinear least squares, while Foster and Vishwanathan
(1995) use GMM estimation to examine the impact of economic and/or political variables on stock market
volatility. However, given the widespread use of GARCH and Markov switching models, we concentrate our
analysis on comparing the forecasting performance of these two popular estimation techniques.
5 Financial time series data exhibit excess kurtosis, autoregressive conditional heteroscedasticity (ARCH),
nonlinearity and nonstationarity.
1
work that seriously evaluates the statistical merits of these two competing sets of models –
GARCH versus Markov-Switching.6 Second, from a substantive viewpoint, the predominant
theoretical prediction and empirical finding in the literature is that if traders or investors
anticipate electoral victory by a “left” party –Democrats in the US or Labor in Britain—then
the prices of various financial assets—exchange rates, stock indices, bond prices—will become
increasingly volatile (Herron 2000; Cohen 1993; Alesina, Roubini and Cohen 1997; Freeman,
Hays and Stix 2000; Gemmill and Saflekos 2000).
In this paper we take issue with these methodological and substantive claims. We use
GARCH, EGARCH, FIEGARCH and Markov-Switching models to analyze the impact of
electoral information, uncertainty and partisanship on stock price volatility during the 2000
U.S. Presidential Election. We find, using two different data sets, that GARCH type models
outperform Markov-Switching models in a number of important ways. Results from the outof-sample forecasting tests – which included the RMSE and MAE statistics as well as realized
volatility regressions – show that the GARCH and especially the EGARCH models more
accurately forecast volatility than Markov-Switching models.
In addition to this methodological contribution our paper provides an important
substantive addition to the burgeoning literature on democratic politics and financial markets.
In particular, we construct a formal model of speculative trading where stock traders and the
market–maker rationally respond to political information about potential electoral outcomes.
Our model predicts that higher ex ante uncertainty amongst traders about the electoral
outcome increases stock price volatility. More importantly, the model also predicts that if stock
To the best of our knowledge, only Pagan and Schwert (1990) have compared the forecasting performance of
GARCH, EGARCH and Markov-Switching models by using pre-war US stock returns data. There are a
number of key differences between our paper and their work. First, our Markov-Switching model is estimated
with time-varying transition probabilities whereas their model contains constant transition probabilities.
Second, our specifications contain numerous political and electoral variables to capture the behavior of stock
market returns. Finally, they do not compare forecasts between FIEGARCH and Markov-Switching models
that we do here.
6
2
traders expect a Democrat to win the Presidential election, then stock market volatility decreases.
This theoretical prediction stands in stark contrast to existing findings in the literature which
show that bond yields and stock price volatility increases when investors anticipate leftoriented parties to win elections (Herron 2000; Gemmill and Saflekos 2000; Alesina, Roubini
and Cohen 1997; Freeman, Hays and Stix 1999; Cohen 1993).
This paper is organized as follows. We present the formal model in the next section.
In section 3 we describe the data, variables and results from GARCH, EGARCH,
FIEGARCH and Markov-Switching models. We report and compare the out-of-sample
forecasts from all the estimated models in section 4 before concluding the paper.
2. The Model
2.1 Players, Sequence of Moves
To examine how uncertainty about the outcome of a Presidential election, the
partisan identification of the Presidential candidates and political information from a public
source affects stock price volatility, we construct a model of speculative trading. There are
two players in our model: (i) a group of strategic traders7, I, where i ∈ I denotes a single
trader and (ii) a market-maker who sets prices based on the strategic behavior of traders.
The traders in our model are primarily interested in trading an asset with a terminal
value, v, which is known to them. Before trading, all players observe public signals s ∈ S that
provide political information on the potential outcome of the presidential election and the
policies that will be adopted by the party that wins the elections. More substantively, then, s
includes aspects such as opinion polls, news events or campaign promises by candidates and is
7 The traders in our model trade stocks for short-run personal financial gain and as service to clients. Following
models of speculative trading by Kyle (1984, 1985), Admati and Pfleiderer (1988), we assume that traders and
the market maker are risk-neutral to capture the linear relationship between risk and stock returns.
3
correlated with v. The political information8 learnt from observing s influences the strategic
behavior of traders in our model, which – we show later—affects the price, p, of the traded
stock set by the market-maker and the variance9 in the traded stock’s price. We denote the
mean price of the traded stock as p 0 and the variance as v − p 0 .10
After observing the public signal s, each trader submits to the market maker the order
xi ( I , s, v) + qi . The component xi ( I , s, v) indicates the segment of the order that is placed by
a trader given the political information provided by the signal s, the number of traders (I) and
his (or her) expectations about the terminal value, v, of the traded stock. qi indicates the
amount of nondiscretionary liquidity trading11 done by each trader. The market maker sets
prices based on the information arising from the signal s and net order flows12
I
n = ∑ xi ( I , s, v) + q that satisfy the zero profit condition.13
i =1
The sequence of moves in the game is as follows: First, political information that
stems from the public signal s is observed by all players. Following observation of s, traders
submit orders to the market maker who then determines the price knowing s and the net
order flow. After the market-maker sets the price, trade occurs and the terminal value of the
traded stock is revealed. We now describe below the players’ utility functions.
Unlike existing models, we do not analyze the case of endogenous information acquisition where traders incur
search costs to acquire information. This is because electoral information exogenously stems from public
sources and players don’t incur search costs to acquire electoral information, which is publicly observable.
9 We use the terms “variance” or “variability” to mean “volatility” in the traded stock’s price (or vice-versa).
10 p also denotes the full information price of the traded stock.
0
8
Following Kyle (1984, 1986) and Foster & Vishwanathan’s (1995) models, we let q i denote the stocks traded
by a trader for purely portfolio-balancing reasons. This portfolio-balancing is done to diversify risk or to serve
the interests of institutional clients, rather than for maximizing his short-run personal profit. Hence, the
argument q i is not optimized in each trader’s maximization problem, but is assumed to be correlated to s.
11
12
Note that q =
I =1
∑q
i
i
13
We formally define the zero-profit condition later.
4
2.2 Utility Functions
Scholars have noted that there always exists variance in the traded stock’s price and the
amount of liquidity trading by traders. 14 This is obvious given that stock prices and liquidity
trading are highly sensitive to variability in trading behavior. To formalize this variance, we
denote the unconditional variance of the stock’s terminal value, v, as σ v2 and the unconditional
variance of q as σ q2 . Recent scholarship in the financial economics literature15 has also
emphasized that public signals that provide economic or political information exhibit variance
largely because such signals stem from heterogeneous sources. We thus denote variance in the
signal as σ s2 and we let s 0 denote the expected value (mean) of s.16
The behavior of traders in our model is influenced by their beliefs over the
parameters s,v and q. Instead of using the normal distribution to characterize beliefs,17 we
follow Owen and Rabinovitch (1987) and Foster and Vishwanathan (1995) by assuming that
s,v and q – with unconditional 18expected values s 0 , p 0 and q0 -- are jointly distributed
according to a multivariate elliptically contoured (hereafter ECC)19 distribution,
 s 

s 
 v  ~ ECC   p0 , ∑, f (.) 
 q 0 

q
 
 0 

(1)
For this, see Kyle (1984, 1985, 1986) and Gallant, Rossi and Tauchen (1992).
For this, see Kyle (1985) and Harris and Raviv (1993).
16 Substantively, s denotes political information about the election result that is expected by traders.
0
17 Kyle (1984, 1985) and Admati and Pfleiderer (1988) use the normal distribution to characterize beliefs.
18 This occurs when the expected value of the parameters are not dependent on s.
19 Typical examples of multivariate elliptically contoured distributions are the multivariate t- distribution as well
as distributions that belong to the compound normal class. We use a multivariate elliptically ECC distribution
to represent traders’ beliefs over the parameters v,p,s and q because it has the property that a higher absolute
difference between the realized public signal s and its mean s o leads to more variability in the traded stock’s
price. This property is vital for the theoretical analysis in this paper. In Appendix B, we provide formal
definitions from Cambanis, Huang and Simons (1981) and Johnson (1987) to characterize some technical
properties of the multivariate ECC distribution, which are used to prove propositions from our model.
14
15
5
In (1), f(.) is the function that characterizes the density of the distribution20. ∑ is the
unconditional variance-covariance matrix21, which is defined as:
σ s 2

Σ =  σ sv

 0
σ sv
σ v2
0
0 

0 

σ q 2 
(2)
Equation (1) merely defines the multivariate ECC distribution. The distribution in (1) is,
however, inadequate because we need to assume in our model that traders’ beliefs about the
variance in the stock’s price, v − p 0 and liquidity trading is conditional on the signal s.22
To do so, we assume that the traders’ beliefs over v − p0 and σ q2 given s are represented by
v − p | s
the conditional distribution  2 0  .23 More formally, the conditional distribution of v − p 0
σ q | s 
and σ q2 given s is jointly distributed according to the multivariate ECC distribution,
(
)

 σ sv / σ s 2 (s − s0 )  ( s − s0 ) 2   Θ 0 
 v − p0 | s 


, h



,
(.)
f
 σ 2 | s  ~ ECC 
2
  σ 2   0 σ q 

q
 q

s
 

0

 

(3)
where Θ = σ v2 − (σ sv2 / σ s2 ) and h : ℜ → ℜ + .
The use of the multivariate ECC distribution unfortunately increases the technical
complexity of our model. Yet this distribution is extremely useful because of two reasons.
First, it helps us to analyze how traders’ uncertainty about which candidate will win the
In Appendix B, we formally characterize the unconditional distribution for the parameters, s,v and q. The
unconditional distribution is also a multivariate elliptically contoured distribution. We show that the mean,
variance and the function f(.) fully characterizes the multivariate elliptically contoured distribution in this case.
21 The term “unconditional” variance-covariance matrix implies the variance-covariance matrix of the
parameters v,q and s, when they are not conditional on s
22 Each trader is aware that all players adjust their behavior after observing s and that this affects the variance in
the traded stock’s price. Hence, ex ante, traders will have beliefs about how s may influence the variance in the
stock price. We also denote the variance of the traded stock’s price given n and s as (v − p0 ( n, s) | n, s) . For
proofs of propositions from our model, we examine variance in the stock price given s (or given n and s).
23 This conditional distribution captures how observation of political information about the potential electoral
outcome stemming from the signal s affects trader’s beliefs over the variance v − p 0 and q.
20
6
Presidency and the arrival of new political information about electoral results affects the
variability in the traded stock’s price. Second, it also allows us to analyze how traders’ ex ante
expectation of whether the Democratic or Republican candidate will win the election – i.e. the
candidates’ partisan identification-- affects stock market volatility.
We now turn to describe the players’ utility functions. Let p (n,s) be the price that the
market maker sets knowing the net order flow, n, and the realized political information from
the signal s. Following standard models of speculative trading (Admati and Pfleiderer 1988;
Foster and Vishwanathan 1995), the market maker in our model uses a linear pricing rule,
p(n, s) = b(s) + α (s)n
(4)
where b > 0 is some positive constant and α is the weight24 that the market maker places on
adjusting prices (upward or downward) as a function of net order flows given the realized
public signal s.25 The market maker’s objective is to set p(n,s) such that it satisfies the
expected zero profit condition, i.e.
p(n, s ) = E[v | n, s ]
(5)
where E[v | n, s ] is the expected terminal value of the traded stock conditional on n and s.
Given p(n,s), each trader maximizes,
arg max E [(v − p (n, s )) xi ( I , s, v) + qi | v, s ]
(6)
xi (.)
where E is the expectation operator. Substituting (4) into (6) yields the following utility
function for each trader,
α > 0 but bounded from below and above. That is, α ∈ [α ,α ] , α ∈ ℜ + .
∂p(n, s )
∂α (.)
25 Observe that in (4),
> 0 and
> 0 which indicates a monotonic relationship between n
∂n
dα
and α as well as p and α . We exploit this monotonic relationship to demonstrate in the proofs that the higher
24
We assume that
(lower) the variance in net order flows, the greater (lesser) the weight the market maker places toward adjusting
stock prices and hence the higher (lower) the variance in the traded stock price.
7
arg max E [(v − b( s ) − α ( s )n) xi ( I , s, v) + qi | v, s ]
(7)
xi (.)
We now present equilibrium results and comparative statics from our model.
2.3 Equilibrium Results and Comparative Statics
The solution concept that we use is Nash equilibrium. With respect to our model,
this means that each trader takes the strategies of all other traders as well as the terms of
trade –summarized by the market maker’s price-setting strategy—as given while choosing his
or her best response. Solving for the Nash equilibrium leads to the following result,
Lemma 1: The Nash equilibrium in the single period model is:
p(n, s ) = α n + E[v | s ]
xi ( I , s, v) = γ (v − b( s ))
α =
σq I Θ
s (1 + I )
and γ =
σ qI 2
s I Θ
(8)
(9)
Proof: See Appendix A.
th
Lemma 1 formally characterizes the order that the i trader places in a Nash equilibrium, the
Nash asset price and α that is set by the market maker. It also characterizes the parameter γ
that denotes the “intensity” with which each trader trades the stock in equilibrium, i.e. the
th
degree of the i trader’s trading (buying and/or selling) activity in equilibrium.26 We present
some comparative statics from the equilibrium solution below. These are used later to explain
the causal logic underlying various theoretical predictions from our model,
Corollary (to Lemma 1): (i) ∂α ∂I > 0 and ∂γ ∂I > 0 . (ii) α and γ is monotonic in s.
Proof: See Appendix A
In Kyle (1985,1986) and Foster and Vishwanathan’s (1995) models, γ is a proxy for the amount of stocks
that traders trade in equilibrium. Hence, the greater the intensity of trading, the greater the amount of trading
(or vice-versa). Note that the function γ (s ) in (8) increases when the difference between v and b(s) increases.
26
8
∂γ ∂I > 0 indicates that as the number of traders who trade increases, the intensity
of trading activity by each trader also increases. This is not surprising since the entry of
additional traders will drive down marginal trading profits, therein giving incentives to each
trader to increase his level of trading. ∂α ∂I > 0 implies that the weight that the market
maker places on adjusting prices increases when the number of traders that engage in
speculative trading increases. This is intuitive because an increase in the number of traders
leads to ∂γ ∂I > 0 , which directly affects the degree of net order flows by traders. This, in
turn, forces the market maker to adjust prices more often.
The comparative static result that is more important is the finding that both the
intensity of trading activity, γ , and the weight that the market-maker places on adjusting
prices, α , are monotonic in s. This implies that the trading behavior of traders and the pricesetting behavior of the market maker are directly affected by the nature27 of political
information that arrives via s. We show below that the monotonicity of γ and α in s plays a
critical role in explaining how uncertainty about Presidential election results and expectations
of which candidate will win the Presidency affects variability in the stock price. Second,
given that the market-maker and the traders’ behavior is affected by the signals s, it is likely
that the conditional variance in the traded stock’s price v − p0 | s , will also be influenced by
the political information that is learnt from observing s. The following result, Lemma 2,
formally proves the aforementioned claim,
Lemma 2: The variance in the traded stock’s price Var (v − p 0 | s ) is monotonic in s.
By “nature” of the political information/signal s, we mean information about whether the Democratic or the
Republican candidate will win the election.
27
9
Proof: See Appendix A.
The causal logic behind the result in Lemma 2 is as follows. Specifically, we find that
under certain conditions with respect to s –which we specify in more detail later—the
variance in the net order flows by traders increase, while under other conditions, it decreases.
Now when the variance in n increases, the weight that the market-maker places on adjusting
prices upward and downward increases.28 This serves to increase the variance in the traded
stock’s price.29 Conversely, if the variance in n decreases, the market-maker is less likely to
adjust the stock price and this engenders a decline in stock price volatility.30
The brief discussion of the result in Lemma 2 gives rise to the following question:
Under what conditions does political information from the signal s increase or decrease
variability in the traded stock’s price? We show that if the arrival of “new” political
information via s increases uncertainty about which candidate will win the Presidency, then the
variance in the traded stock’s price also increases. We also demonstrate that the direction of
political information from s matters; specifically, stock market volatility is affected by signals
that reveal whether the Democratic or the Republican candidate will win the Presidency.
We begin by analyzing how the degree of electoral uncertainty about the outcome of a
Presidential election can affect the variance in the traded stock’s price. Specifically, in our
model, uncertainty among traders occurs when new information from the updated signals, s A ,
about which Presidential candidate deviates from their expectation about the likely winner of
the Presidential election –this expectation being formed by the signal s0 . More formally,
This follows from the monotonicty of α with respect to n.
Since the Nash equilibrium asset price in (8) is monotonic in α and n, an increase in the variance of either of
these two parameters leads to a higher variance in the traded stock’s price.
30 In the proof of Lemma 2, we also prove that Var (v − p | s ) increases when the variance in n increases.
0
28
29
10
uncertainty about the electoral outcome among traders occur when s A deviates from it’s mean
s0 . Hence, the degree of electoral uncertainty is defined as the absolute difference s A − s0 .
This definition of electoral uncertainty can be illustrated by the following example.
Suppose that, on average, opinion polls predict that the Democratic candidate is likely to win
the elections, this is the information that stems from s0 . Assume further that new political
information from s A indicates that recent opinion polls favor the Republican candidate and
that the election race may be a close finish. The significant deviation between s A and s0 in this
case is likely to increase uncertainty among traders about the outcome of the Presidential
election. In contrast, if s A and s0 converge – i.e. if s A shows that the Democratic candidate is
still likely to win the election --then uncertainty about the electoral outcome will be low. Given
the conceptualization of electoral uncertainty in our model, we state the following result:
Proposition 1: The variability of the stock’s price Var (v − p 0 | (| s A − s 0 ) |) increases when electoral
uncertainty about which candidate will win the presidency increases; i.e. when the difference s A − s0 increases.
Proof: See Appendix A.
Three reasons explain the result in Proposition 1. First, when new political
information from s A is substantially different from what traders’ expect, we find that the
variance in net order flows by traders increases.31 The proof of Lemma 2 (Appendix A), shows
that this contributes to a higher variance in the traded stock’s price. Second, the proof of
proposition 1 shows that an increase in s A − s0 leads to an upward revision of beliefs about
the variance in the stock price among traders. This has a feedback effect that contributes to a
higher conditional variance in the stock’s price. Third, we claim that when uncertainty about
the election result increases, then traders find it difficult to assess, ex ante, how the electoral
31
We prove this claim in the proof of Proposition 1 in Appendix A.
11
outcome will affect the stock price ex post. We find that in equilibrium, traders rationally
hedge against this increased uncertainty by temporarily selling the traded asset and maintaining
larger cash balances. This, in turn, leads to a sharp downward price movement.
Note, however, that after the stock’s price declines to some threshold, the demand
for buying the stock rapidly increases therein inducing the market maker to revise the price
upwards.32 The upward revision in prices can be mean-reverting, that is, converge back to
p 0 , or, depending on the degree of order flows, it can rise above p 0 . In short, when
political uncertainty about the outcome of presidential elections increases, we will observe
relatively sharp dips and spikes in the price movement, which essentially implies increased
variability in the traded stock’s price.33
Our claim that greater electoral or (more broadly) political uncertainty leads to a
higher variability in stock prices is not surprising.34 But does the partisan identification of the
two Presidential candidates also affect stock market volatility? If so, how? As an answer to
these questions, we make the counter-intuitive claim that anticipation of a Democratic
victory reduces variability in the traded asset’s price, while expectations of a Republican
victory increases variability in the stock price.35
More specifically, we argue that traders generally believe that the post-election
economic policies that an incoming Democratic President will implement are not likely to
deviate from his party’s pre-electoral policy announcements that are likely to be “left-ofcenter” oriented. Our model demonstrates that this belief serves to lower the variability in
32 Traders on the NYSE market floor typically buy stocks in large amounts when stock prices fall and there are
arbitrage opportunities in the immediate future; see, Gallant, Rossi and Tauchen (1992).
33 We prove in a more detailed version of our model that increasing uncertainty leads to higher trading volume.
34 McGillivray (2002) shows that under coalition governments, uncertainty with respect to implementation of
policies by the governing coalition leads to higher stock price volatility. In a different context Freeman, Hays
and Stix (2000) show that more uncertainty is correlated with higher exchange rate volatility.
35 We use the term “counter-intuitive” since prevailing theoretical and empirical results by Herron (2000), for
example, indicate that investor’s expectation of electoral victory by left parties – Labor (UK) or Democrats
(US)—and not conservative parties increases stock price volatility.
12
the stock price when the market expects a Democrat to win the Presidency. Conversely, we
argue that traders typically perceive ex ante that an incoming Republican President will
“move further to the right” and implement relatively more extreme anti-inflationary, antiwelfare and tax-reduction policies than those proposed by him and his party before the
elections. We show below that the market’s anticipation of further deviation to the right by
an incoming Republican President engenders higher variability in the stock’s price.
To see this more formally, let s D be the pre-electoral signals (for e.g. campaign promises)
provided by the Democratic candidate with respect to the economic policies that he intends to
implement in office. From a substantive viewpoint, we assume that s D reveals information that
the Democratic candidate will implement in office policies that lower unemployment, but leads
to higher inflation, welfare and taxes. This assumption follows from existing results in rational
partisan literature (Alesina & Rosenthal 1995) and the general reputation of Democrats being
liberal. Now let ŝ D represent the post-election policy signal that traders expect the Democratic
candidate to announce (and implement) after he wins the elections. The extent to which ŝ D is
expected to deviate from s D is formalized as | sˆ D − s D | where | sˆ D − s D | ≠ ∅ . 36
Likewise, we denote s R as the pre-electoral policy signals/policy promises that are
made by the Republican candidate. Based on the reputation of Republicans as conservative, we
assume that the signal s R reveals information that the Republican candidate will implement in
office conservative policies that help to reduce taxes and inflation. We let ŝ R represent the
post-election policy signal that the market expects the Republican candidate to announce if he
wins the elections. The degree to which ŝ R is expected to deviate from s R is | sˆ R − s R | ≠ ∅ .
36
Observe without loss of generality that the sign of (| sˆD ( R ) − sD ( R ) | ) will be the same as | sˆ D ( R ) − s D ( R ) | .
2
13
Given our earlier discussion, we presume that | sˆ D − s D | < | sˆ R − s R | and that
lim sˆD → sD | sˆ D − s D | < ε . This formalizes our idea that traders expect less deviation between
the pre-electoral policy signals and post-electoral policy announcements by the winning
Democratic candidate. We now state the following result from our model:
Proposition 2: When traders anticipate the Republican candidate to win the Presidency, then the
∂ E[Var(v − p | (| sˆR − s R |))
variance/volatility in the stock price increases. This is because
> 0.
∂ (| sˆR − s R | 2 )
Proof: See Appendix A.
The causal intuition that explains the result in the above proposition is as follows. We claim
that traders believe that after elections, a victorious Republican President will announce
policies that curb inflation, but will also cut taxes more aggressively than what he promised
prior to elections. Consequently, traders expect better economic prospects and higher stock
returns in the near future under a Republican administration. This initially leads to “rational
exuberance” in the market where demand for stocks rises rapidly which, in turn, engenders an
increase in the expected price of the traded stock, as proved below,
Lemma 3: Let | sˆ R − s R | = λ , where λ ≠ ∅ . From the implicit function theorem,
Proof: See Appendix A.
dp(.)
> 0.
dλ
Now when the expected stock price increases owing to increased demand, traders
have rational incentives to sell the stock in order to take advantage of a short-run arbitrage
opportunity.37 We prove in the appendix (see Lemma 4) that such selling behavior engenders
a mean reversion in the stock’s price. After the stock price reverts to it’s mean, traders have
incentives to invest in the stock again owing to a cheaper buying price and expectations of
higher returns under a Republican administration. This consequently leads to an increase in
Since traders on the NYSE market floor trade largely for purposes of short-term financial gain, they
rationally respond to an upward increase in a stock price by selling it to acquire short-term profits.
37
14
the traded stock’s price. Thus, rapid switching between buying and selling behavior by
traders within a short-time period generates increased volatility in the stock price when the
market expects a Republican victory.
In contrast, our model predicts that variability in the stock’s price decreases when
traders anticipate that a Democrat will win the Presidency. More formally,
Proposition 3: When traders expect the Democratic candidate to win the Presidency, then variability in
the stock price decreases because lim sˆD → sD Var (v − p 0 | (| sˆ D − s D |)) < 0
Proof: See Appendix A.
Two reasons explain the above result. First, as mentioned earlier, traders expect that
policies implemented by the incoming Democratic President are less likely to deviate from
his pre-electoral policy announcements. This implies that the degree of ex ante uncertainty
among traders about the kind of economic policies that the Democratic candidate will follow
ex post – after the Democratic candidate wins the elections – is low. The proof of
Proposition 3 demonstrates that such low uncertainty reduces variability in the stock’s price.
Second, we argue that reputation matters in that traders perceive ex ante that an
incoming Democratic President will follow “left-oriented” measures – inflationary policies and
relatively higher taxes—that are detrimental to the stock market.38 In Appendix A, we state and
formally prove two lemmas from our model -- Lemmas 5 and 6 – which demonstrate that ex
ante perceptions of Democrats being “bad” for the stock market decreases the incentives and
intensity with which traders trade stocks. This serves to decrease the variance in net order
flows and the weight that the market maker places toward adjusting the stock price (Lemma 6).
Both these factors lead to lower volatility.
In sum, our model provides the following four testable hypotheses:
Santa-Clara and Volkanov (2002) show that the average expected returns in the stock market are 1.8% higher
under a Republican administration than under a Democratic government, thus justifying our assumption.
38
15
Hypothesis 1: Information arrival about electoral outcomes affects stock price volatility.
Hypothesis 2: Increased uncertainty about the electoral result increases volatility.
Hypothesis 3: If traders expect the Democratic candidate to win, then volatility decreases.
Hypothesis 4: If traders expect the Republican candidate to win, then volatility increases.
3. Empirical tests
3.1 Sample, Data and Variables
We test the hypotheses listed above on two distinct samples related to the 2000
Presidential election. The first sample is comprised of daily observations during the 2000
Presidential campaign —end of day returns for the S&P 500 and aggregated daily national
polling results. The second sample examines how actors trading S&P futures during the night
of November 7, 2000 – the night of the election – respond to information regarding the
likelihood of a candidate winning the Electoral College. These samples are discussed in turn.
The 2000 Presidential Campaign:
We examine the response of stock market returns to the arrival of political
information using a sample of daily observations from January 6, 2000 – November 6, 2000.
Data limitations prevented us from extending the sample backwards and November 6 was
the last day that polling information was available. We use returns (log changes in daily
closing prices multiplied by 100) on the Standard and Poor’s 500 index as our dependent
variable.39 To measure political information that captures expectations of a Gore (i.e.
Democrat) victory, we utilize polling data that indicates, for each day, Gore’s share of the
two major-party vote. These data, collected and used by Wlezien (2001) and Wlezien and
Erikson (2001), are based on an aggregation of 295 separate national polls conducted during
the 2000 presidential campaign. Missing values were filled in using linear interpolation.40
39 The S&P 500 index includes 80% industrials, 3% utilities, 1% transportation and 15% financial companies.
Their market value is roughly 80% of the value of all equities traded on the NYSE; see www.standardpoors.com
40 See Wlezien (2001) for a detailed discussion of this variable
16
We also include additional variables to control for other unmeasured influences on
stock market volatility. These include two dummy variables capturing “closing days effects;”
that is, effects on market activity that result from weekends or holidays. Closing day effects
variables measure the number of days BEFORE day t that the market was closed and the
number of days AFTER the day t that the market will be closed.41 It is expected that these
variables will have a positive and statistically significant effect on stock market volatility.
Finally, we include a variable measuring (the log of) trading volume because studies find that
including trading volume substantially accounts for observed volatility in stock market returns
(Gallant et al, 1992). Since volume data is not available for the S&P500 indexes, we use total
daily volume traded on the New York Stock Exchange (hereafter, NYSE) as a proxy. These
variables, their sources and measurement, are summarized in Table 1 in Appendix B.
--Insert Table 1 about here--
Election Night: November 7, 2000
A second laboratory within which to examine the effect of political information on
stock market volatility was created the evening of November 7, 2000. As the evening
progressed network and cable news outlets (as well as the major wire services) “called” the
electoral outcome of each state. These calls constitute the arrival into the market of political
information, information that affects the strategic decisions of traders.
The NYSE is open for trading between 9:30am-4:30pm Eastern Standard Time; it
closes, therefore, prior to the reporting of election results. After hours traders can trade
options and futures contracts through the GLOBEX electronic trading system. GLOBEX,
developed by Reuters and the Chicago Mercantile Exchange, is an automated system that
In their analysis of currency markets, Beine, Laurent and Lecourt (1999) find this specification more
parsimonious than the inclusion of a set individual business day dummy variables.
41
17
provides information about trades (bid & ask prices), routs orders, and executes trades. Using
the GLOBEX system, individuals can trade a variety of futures, options and interest rates.
The GLOBEX system reports information on the price and volume for every individual
transaction during the trading session. We use this “tick” data to track the movements of
futures prices for the S&P 500 Index.42 These tick data were aggregated to provide the average
price and total volume of trades for each minute during the trading session. To avoid overlap
with the NYSE, and because a five minute lag is used the sample period for the overnight data
set is 4:35pm on November 7th through 8:59am on November 8th.
We measure the arrival of political information by constructing a variable that
estimates the probability that Gore will win a majority of electors in the electoral college and,
thus, will become the 43rd president. This measure is based on state level polls for each of the
50 states and exploits the fact that these polls contain a degree of sampling uncertainty. As
each state is called over the evening of November 7th and into the morning of November 8th,
traders update their priors regarding the likelihood of a Gore victory.
The prior for each state is calculated using the final state level poll available. Table 2
(Appendix B) reports information regarding the sample size of the poll (sample size), the
percentage of respondents responding with a preference for Gore (Gore %) and for Bush
(Bush %) and the share of the two-party vote for Gore (Gore/(Bush+Gore))43. This
information is used to test the null hypothesis that, in the population, Gore’s share of the
two party vote is greater than or equal to 0.50001 against the alternative hypothesis that
Gore’s share is less than 0.50001. The p-values for rejection of the null are also listed in
42 A future is a legally binding agreement to buy or sell the cash value of the asset at a specific future date. In
the case of the futures used here the maturity date was November 15, 2000.
43 We are grateful to Charles Franklin and Chris Wlezien for sharing these data; see Franklin (2001) and
Wlezien (2001).
18
table two. Higher p-values indicate the probability of making an error by rejecting the null
that Gore will win the state.
The mapping of the share of the two-party vote for Gore and the p-value for rejecting
the null results in a S-shaped relationship. For example, Gore’s share of the two-party vote in
Massachusetts was 63.4%. The p-value for rejection of the null that Gore would get at least
50% and win the state is 1.00, indicating that it is certain that an error will be made if that state’s
electoral votes are given to Bush. Likewise, the p-value for rejection of the null for Texas is
0.000 meaning that there is zero chance out of a thousand that Gore will win that state.
Since these polls contain sampling uncertainty there is a probability that an error will
be made by rejecting the null hypothesis. The second step in variable construction is to
exploit this sampling uncertainty. This is done by randomly drawing from a uniform [0,1]
distribution and creating a variable Q with observations for each state. Denoting the p-value
for rejection of the null hypothesis P, if Q is less than P then Gore wins state i and gets all of
state i’s electoral votes. This is done for each state. If Gore wins sufficient states to give
him more than 270 electoral votes then he wins the election. Third, the process in step two
is repeated 1,000 times and the proportion of Gore victories is recorded. This measure--the
probability that Gore wins the Electoral College—is graphed in figure 1 (see Appendix B).
At the beginning of the evening, at 3:45pm, the probability of Gore winning the Electoral
College was .378; that is, he won 378 out of the 1,000 elections.
Finally, this probability is updated over the course of the election as each state was
called by CNN. As a state is called the probability of winning the state in question goes to
either zero or one, depending on whether the state is called for Bush or for Gore, and steps
two and three are repeated. Continuing this procedure until 6:21am on November 8th, when
Wisconsin was called, results in a variable that measures the probability for each minute, that
19
Gore will win the Electoral College. As can be seen in figure 1, this probability increases
dramatically at 7:52 when Florida is called for Gore and then declines at 8:55pm when CNN
takes Florida from Gore’s win column. Likewise, the probability that Gore wins the
Electoral College took a nosedive when Florida was called for Bush at 1:18am and then
increased when Florida was once again labeled a “toss-up.”
Similar to the daily data set, we control for the total volume traded during each
minute. We also control for the anticipated time interval between the current and previous
trade. O’Hara (1995) argues that “if market participants can learn from watching the timing
of trades, then the adjustment of prices to information will also depend on time” (p.169).
Engle (1996) empirically implements this idea and argues that the expected duration between
trades should have a statistically significant effect on the mean and price changes.
3.2 Statistical Models: GARCH, EGARCH, FIEGARCH
Because we are interested in the effect of political information on stock market
volatility, we utilize the Generalized Autoregressive Conditional Heteroscedasticity Model
introduced by Engle (1982) and extended by Bollerslev (1986). A GARCH model is
comprised of two equations: one for the conditional mean and the other for the conditional
variance. In the GARCH (1,1) specification, the conditional mean can be written as:
∆Pt = λ + ε t , ε t ~ N (0, σ t2 )
(10)
where ∆Pt is the change in closing price of the stock market index observed at time t, λ is a
constant and ε t is an error term that is normally distributed44 with mean zero and variance
σ 2t . Note that the mean is specified as following a random walk with a drift; no exogenous
variables are thought to influence the mean change in price.
We can also use the generalized exponential, student-t or double exponential distribution if desired. But we
did not need to do so since we found that the residuals from the estimated models are conditionally normal
44
20
The unique feature of GARCH models is that we can specify how the conditional
variance evolves ( σ 2t ) over time in response to both past values and to exogenous shocks.
The conditional variance for the standard GARCH (p, q) model is:
q
p
i =1
i =1
σ 2t = ω + ∑ α i ε 2t −i + ∑ β iσ 2t −i
(11)
Using the lag or backshift operator45, equation (11) can be rewritten as:
σ 2t = ω + α ( L)ε 2t + β ( L)σ 2t
(12)
In most cases there is one ARCH and one GARCH term. With exogenous variables affecting
the conditional variance, the GARCH(1,1) can thus be written as:
σ 2t = ω + α 1ε 2t −1 + β 1σ 2t −1 + δ i I i ,t
(13)
The variance σ 2t , called the conditional variance, is the one-period ahead forecast
variance based on all information available at time t-1. The conditional variance is a function
of four terms: the constant (ω) the ARCH term ( ε 2t −1 ), the GARCH term ( σ 2t −1 ), and a set of
exogenous variables (Ii,t). GARCH models are often used to analyze financial time series
because it is assumed that economic agents form expectations about this period’s variance
based the long term mean of the variance ( ω ), the forecasted variance from the prior period
( σ 2t −1 ), and new information about volatility gleaned in the prior period ( ε 2t −1 ). GARCH
models can account for the large clustering of errors observed in financial time series where
large deviations in the conditional variance are often followed by other large deviations.
While the standard GARCH model is useful it has two limitations. First, volatility in
tick data tends to persist over long periods of time. Second, it is likely that positive and
negative shocks from prior periods may have differential effects on price volatility at the
45
The backshift operator can be represented as:
α ( L) = α 1 L +...+α q Lq
21
and
β ( L) = β 1 L +...+ β p Lp
current point in time. To deal with these problems, Bollerslev and Mikkelsen (1996) have
developed the following Fractionally Integrated Exponential GARCH (FIEGARCH) model,
which we present in more detail in Appendix B:
ln(σ 2t ) = ω + δ i I i ,t + φ ( L) −1 (1 − L) − d [1 + α ( L)]g ( zt −1 )
(14)
In (14), [1 − β ( L)] = φ ( L)(1 − L) d and g ( zt ) = θzt + γ [| zt |− E | zt |] . Ignoring the g(zt-1) term for
a moment, equation (14) says that (the log of) volatility is a function of the constant (ω), a set
of exogenous variables (δIxi,t) measured at time t, the ARCH term (α), the GARCH term (β)
and the fractional integration parameter (d). As in standard ARIMA models, d measures the
speed at which shocks to the dependent variable (in this case, the variance) die out over time.
If d equals zero then shocks have no memory and equation (14) collapses to the standard
EGARCH model. However, if d equals 1 then (14) becomes the Integrated EGARCH model.
Bollerslev and Mikkelsen (1996) find that the FIEGARCH model fits tick data well.
Turning our attention to the g(zt-1) term, this part of the equation captures the idea
that volatility responds differently to “good news” than to “bad news.” Nelson (1991) noted
that “to accommodate the asymmetric relation between stock returns and volatility
changes…the value of g(zt) must be a function of both the magnitude and the sign of zt.”
This is accomplished with the θ and γ terms in g(zt-1) where θ captures the sign and γ
captures the magnitude of past errors. Substantively this term means that the negative errors
in the prior period will have a larger effect on the conditional variance than positive shocks.
If θ and γ in g(zt-1) equal zero then equation (14) becomes the FIGARCH (p,d,q) model
3.3 Results from the GARCH, EGARCH and FIEGARCH models
We estimate GARCH models on the daily and overnight samples. The results of these
models are contained in Tables 3 and 4. Cell entries in both tables are maximum likelihood
22
parameter estimates with Bollerslev-Wooldridge semi-robust standard errors in parentheses.
Beginning with the daily sample in Table 3, we note that the Ljung-Box Q statistic, indicating
no residual serial correlation, suggests that the differenced S&P price series follows a random
walk. The squared Ljung-Box statistic is also statistically insignificant; this suggests that there is
no remaining ARCH in the residuals. The Jarque-Bera statistic also prevents us from rejecting
the null hypothesis of normally distributed residuals.
--Insert Table 3 about here-Turning our attention to the specification in column one in Table 3, the ARCH term
(α) is not statistically significant while the GARCH term (β) is significant at the .05 level. This
means that while random errors from the prior period ( ε 2t −1 ) does not significantly affect the
conditional variance at time t, the conditional variance from time t-1 does. We also note that
the sum of the ARCH and GARCH terms ( αˆ + βˆ ) is significantly less than one indicating a
non-integrated GARCH process. The coefficients on the variables included in the conditional
variance are—for the most part—consistent with our expectations. Stock market volatility is
greater the days after traders return from vacation. Daily stock Price volatility is also
significantly (in both substantive and statistical terms) higher as a result of increased trading.
In column one, we test the hypothesis that expectation of Gore’s victory decreases
stock market volatility. Column one uses the percentage of people polled expressing a
preference for Al Gore to test hypothesis three.46 The coefficient on the GORE variable is
negative and statistically significant, indicating that a higher likelihood that Gore will win the
popular vote for president—and subsequently become the President—decreases the
volatility of the S&P 500 index.
While we report the results of contemporaneous information arrival—that is, measures at time t—there is no
substantive or statistical difference if we use lagged (t-1) measures of information arrival.
46
23
Hypothesis two suggests that electoral uncertainty drives stock market volatility as
uncertain information will bias traders’ forecasts. In column two we operationalize the idea
of uncertainty by calculating a measure of entropy E=1-4(p-.5)2 where p is Gore’s share of
the two-party vote. The entropy measure is greatest when p is closest to .5; the intuition
being that there is little uncertainty about an outcome when the probability of an electoral
victory is .10 or .90 and great uncertainty about an outcome when the probability is equal to
.5 (see Freeman, Hays and Stix 2000). The entropy measure is substituted for the GORE
variable in column two. Note initially that the entropy measure has very little variance: it has
a mean of 99.57, a minimum of 97.39 and a maximum of 99.9, reflecting the fact that Gore’s
share of the two party vote in opinion polls hovered around 50% for most of the 2000
campaign. Interestingly, when incorporated in the GARCH model the entropy variable has
a negative, as opposed to its hypothesized positive, effect on stock market volatility. This is
not only contrary to expectations but to prior (e.g., Freeman, Hays and Stix, 2000) research
as well as to the results we report later using the overnight sample.
Testing hypothesis one—that the arrival of political information affects stock market
volatility—is difficult to operationalize using polling data. We conceive of information arrival
in terms of changes in the percentage of the population reporting a preference for Gore. As
such, we use the change in GORE to measure this concept. As indicated in column three, this
variable is negative not statistically significant. This suggests that information arrival does not
play a role in stock market volatility. This result should be interpreted with caution, however,
as it is far from clear that we have measured the concept correctly.
In results not reported here we checked the robustness of these findings by using a
different dependent variable and by also including a number of other political and economic
variables in the conditional variance equation. For the three models reported in table 3 we
24
obtain very similar results using the return on the Dow Jones Industrial Average rather then
on the S&P 500. We also experimented with including political variables such as the dates of
the republican and democratic conventions and the dates when republican and democratic
challengers dropped out of the race. We also included a dummy variable for the period after
the democratic convention interacted with GORE. Economic variables included dates of
Federal Open Market Committee meetings, dates of interest rate changes and continuous
variables measuring both the level of and change in the three-month Treasury bill rate. In no
case did inclusion of any of these variables significantly (in both a statistical and substantive
sense) alter the results reported in table 3.
A second laboratory within which to examine the effect of political information on
stock market volatility was created the evening of November 7, 2000. As the evening
progressed network and cable news outlets (as well as the major wire services) “called” the
electoral outcome of each state. These calls constitute the arrival into the market of political
information, information that affects the strategic behavior of traders.
Table 4 includes a variety of econometric specifications to capture the price dynamics
of S&P futures the night of November 7, 2000. In initial specification of the GARCH model
residual diagnostics revealed remaining serial correlation so an AR(1) and MA(1) term were
included. This is consistent with Bollerslev and Mikkelsen (1996) and the findings
summarized in Dacorogna et al (2001) that volatility in high frequency financial data is
persistent. In addition, because the Jarque-Bera test consistently rejects the null hypothesis of
normality we utilize Bollerslev-Wooldridge semi-robust standard errors. The models in Table
4 also include two control variables. Following Engle (1996) we include a variable measuring
the expected duration between trades in both the mean and variance equation. We also
include a variable measuring the quantity traded each minute.
25
--Insert Table 4 about here-Column one is the basic GARCH specification that includes the lagged (by five
minutes) variable measuring the probability that Gore will win the electoral college.47 While
the coefficient on this variable is negative and statistically significant—providing support for
both hypotheses three and four—the Ljung-Box test reveals remaining ARCH in the residuals.
We re-estimate the model in column one using an EGARCH specification under the
assumption that accounting for the asymmetry nature of shocks to volatility will render the
residuals white noise. This intuition is born out in column two: again the variable measuring
the probability that Gore will win the Electoral College is negative and statistically significant
and the diagnostics reveal no remaining ARCH. Solving one problem, however, leads to
another as we now see that the sum of ARCH and GARCH terms ( αˆ + βˆ ) is not significantly
different from one indicating the existence of an integrated GARCH process.
The solution, as presented in columns three-five, is to estimate a FIEGARCH model.
The coefficients for these models lend support for our four hypotheses. The FIEGARCH
model using the probability that Gore will win the Electoral College is well behaved and passes
all diagnostic tests. The coefficient on GORE is negative and statistically significant providing
support for hypothesis three. Political uncertainty—operationalized as entropy—exerts a
positive and statistically significant effect on stock market volatility. Finally, the arrival of
political information—measured by the calling of “tossup” states—increases volatility.
We obtain the same results if we substitute the return on NASDAQ futures for S&P
futures. (Unfortunately futures for the Dow Jones Industrial Average did not exist in 2000).
The results are also unchanged if we include a set of dummy variables reflecting the times
A five-minute lag was chosen since that is the average amount of time that it takes for a trade to be executed
by the GLOBEX system. Changing the lag from between one and ten minutes did not alter our results.
47
26
when Florida was called for and subsequently taken away from both Gore and Bush. These
findings are compelling from an experimental point of view: using a nightly sample minimizes
the likelihood that exogenous factors such as earnings reports, interest rate expectations, or
other events influenced the behavior of these asset prices.
3.4 Estimating a Markov Switching Model
We also test the hypotheses from our theoretical model by estimating a Markov
Regime-Switching model with time-varying transition probabilities (Freeman, Hays and Stix
2000; Diebold, Lee and Weinbach 1994).48 In our Markov-Switching model, we assume that
the stock price series is governed by a two-state, first order Markov-Switching process. 49
Each state is characterized by a high (or low) variance and mean that corresponds to a
separate regime. The series that we observe is thus a “mixture” of these two regimes50 where
this mixture is determined by a probabilistic transition between the two states. More
formally, we estimate an autoregressive specification where the mean and variance is subject
to switches between two states that evolves according to a first-order Markov process:
∆ pt = µ S t + φ (∆ pt −1 − µ S t −1 ) + ε t , ε t ~ N (0,σ S2t ) , St ∈ {0,1}
µ St = S t µ1 + (1 − S t ) µ 2
 2
2
2
σ St = S t σ 1 + (1 − S t )σ 2
(15)
A Markov-Switching model has useful statistical properties in that it can account for nonlinearities,
nonstationarity, clustering, serial correlation and fat-tailed distributions of stock price volatility (Hamilton 1989,
1994; Engel 1994; Durland and McCurdy 1994; Filardo 1994).
49 In the Markov-Switching model that we estimate here, the probability that stock market returns and volatility
is in either state (“regime”) 1 or 2 at time t is a function of the state that it was in t-1. The term “state”/ regime
1 denotes a situation of high variance and high mean, while state/regime 2 implies low variance and low mean.
50 Similar to Hamilton’s (1994: 685-696 ) Markov-Switching model, stock prices and volatility in our Markov
model is drawn from a mixture of two normal densities. We assume that high volatility corresponds to a high
mean and low volatility to a low mean. This follows from models in the theoretical finance literature, which
presumes that rational agents invest in highly volatile assets only if the conditional mean (or risk-premium) of
the returns of the volatile asset is high; see Merton (1973), Turner, Stratz and Nelson (1989).
48
27
In (15), µ1 denotes high mean and σ 12 high variance, while µ 2 represents low mean
and σ 22 low variance. φ denotes the AR coefficient. The states , St , are generated by a
realization of the first-order Markov process with transition probabilities,
Pr( St = 1 | St −1 = 1) = p11,t , Pr( St = 2 | St −1 = 1) = 1 − p11,t
Pr( St = 2 | St −1 = 2) = p22,t , Pr( St = 1 | St −1 = 2) = 1 − p22,t
(16)
p11,t denotes the probability of the state of high volatility, St = 1 , and p22,t indicates the
probability of the low volatility state St = 1 . To examine the effect of exogenous variables
on volatility, we allow the transition probabilities in (16) to depend on electoral uncertainty,
information arrival, partisanship and trading volume via the logistic specification,
p11,t =
p22,t =
exp(c1 + β1,k xt′−1,k )
1 + exp(c1 + β1,k xt′−1,k )
1 − p11,t = 1 −
exp(c2 + β 2,k xt′−1,k )
1 + exp(c2 + β 2,k xt′−1,k )
1 − p22,t = 1 −
exp(c1 + β1,k xt′−1,k )
1 + exp(c1 + β1,k xt −1 )
exp(c2 + β 2,k xt′−1,k )
1 + exp(c2 + β 2,k xt′−1,k )
(17)
xt′−1, k is the vector of relevant political variable(s) (k=1)51 that affect the transition probabilities,
while β1, k , β 2, k denotes the coefficient to be estimated and ci (i=1,2) is the constant.52 Notice
that if β1,k > 0 ( β1,k < 0 ), then dp11,t dxt′−1, k > 0 ( dp11,t dxt′−1, k < 0 ) ∀xt′−1, k ∈ ℜ+ . This implies
that for β1, k > 0 ( β1, k < 0 ) the probability of remaining in state 1 increases (decreases). If
β 2,k > 0 ( β 2,k < 0 ), then dp22, t dxt′−1, k > 0 ( dp22,t dxt′−1, k < 0 ) ∀xt′−1, k ∈ ℜ + . Hence,
for β 2,k > 0 ( β 2,k < 0 ), the probability of remaining in state 2 increases (decreases). We briefly
derive the log-likelihood of the Markov-Switching model in Appendix B.53
51
52
53
To accurately interpret coefficients, we include one independent variable separately in each Markov model.
Since xt′−1, k influences ∆Pt via p11 , p22 , it allows for nonlinear relationships between xt′−1, k and ∆Pt .
We estimate our Markov-Switching model via the EM Algorithm (Diebold, Lee and Weinbach 1994: 690-695).
28
We first discuss the results derived from estimating a Markov-Switching model on
the daily data set between 6th January to 7th November 2000. A battery of pretest results –
reported in Table 5 – establishes the existence of regime switching in stock market volatility
in this data set. Specifically, Wald tests rejects the null hypothesis of equality of means (7.97)
and variances (12.85) and the null of independent switching between states at the 1% level
(59.61).54 Both the Hansen (1992, 1996b) and the Garcia (1998) tests reject the null
hypothesis of no switching in the mean and variance at the 1% level.55
--Insert Table 5 about here-In model (1), Table 5, the coefficient of β1,1 for the entropy variable is positive and
significant, while β 2,1 is negative and significant. The coefficient of the high variance state
σ 12 is significantly positive and about seven times higher than the coefficient of the low
variance state σ 22 for the entropy variable. The estimated transition probability p11,t = 0.966
is significant and almost equal to 1. The results mentioned above indicate that increased
uncertainty over the Presidential electoral outcome significantly increases the probability that
stock prices will enter and remain in the state of high volatility.
The Markov-Switching results in model (2) confirm the predictions from the
theoretical model. The coefficient β 2,1 in this model is positive and highly significant, while
β1,1 is negative and significant. The coefficient of low variance σ 22 is roughly 17 times
higher than the coefficient of high variance σ 12 . Not surprisingly, we find that the estimated
transition probability p22,t = 0.974 is close to 1 and highly significant. Put together, these
“Independent switching between states” means that the current state is not dependent on the previous state.
Garcia (1998) and Hansen (1992, 1996b) provide likelihood-ratio tests to test the null of a single regime in
the mean and variance. Garcia derives the asymptotic null distribution of the LR statistic to obtain critical
values; in his test, the 1% critical value for rejecting the null is 17.67. Hansen (1992, 1996b) uses empiricalprocess theory to derive the upper bound of his standardized LR statistic.
54
55
29
results demonstrate that as the probability of a Democrat’s (i.e. Gore’s) victory increases,
stock price volatility declines and the persistence of the low volatility state increases.
In model (3), the coefficients of both β 2,1 and σ 22 are positive, but insignificant. This
result indicates that information arrival does not significantly affect stock price volatility. In
model (4), the coefficients of β1,1 and σ 12 for the trading volume variable are both positive
and significant. This confirms existing empirical results that indicate a positive correlation
between trading volume and higher stock price volatility (Gallant, Rossi and Tauchen 1992).
The coefficient of the parameter, φ , is also highly significant in all four columns in Table 5.
Finally, the Ljung-Box Q-Statistics for lags 1 and 3 for each model shows that the MarkovSwitching model eliminates much of the serial correlation in the residuals till the third lag.
To check for robustness, we also estimated each model in Table 5 on a smaller sub-sample,
i.e. an “in-sample”, of 139 observations56 (results not reported here). There was no
difference in the results obtained for the sub-sample for each model.
The estimates of the Markov-Switching model on stock volatility data from the night
of November 7th, 2000-- see Table 6-- produces no surprises. Wald tests reject the null
hypothesis of equality of means (6.83), variances (12.46) and independent switching between
states (39.21) at the 1% level. The Hansen (1992, 1996b) and Garcia (1998) tests also reject the
null hypothesis of no switching in the mean and variance at the 1% level.
--Insert Table 6 about here-The positive and significant coefficients of β1,1 and σ 12 for the entropy variable in model (5),
Table 6, shows that there was certainly a significant correlation between increased uncertainty
over which candidate will win the election and higher volatility during the night of November
56
The rest of the sample is used for “out-of-sample” forecast tests, which are described in the next section.
30
7th 2000. In contrast, the positively significant coefficients of β 2,1 > 0 , σ 22 > 0 for the Gore
variable in this data set (see model (6)) suggests that when expectations of a Gore (a
democrat’s) victory increased on the night of the November 7th, 2000, stock market volatility
decreased. In fact, the coefficient of low variance, σ 22 , is 16 times higher than σ 12 , therein
indicating that expectations of a Gore victory was strongly correlated with lower volatility.57
The significant coefficients of β1 > 0 and σ 12 > 0 for the Bush variable in model (6)
demonstrate that traders’ expectation of a Bush victory on the night of November 7th, 2000 was
correlated with significantly higher volatility. The coefficients for the information arrival variable
in this data set provide ambiguous results. In particular, β1,1 , β 2,1 and σ 12 , σ 22 for this variable
are insignificant (see model (7)). This result is different from the FIEGARCH model where the
coefficient for information arrival is significant. As before, the coefficients of β1,1 and σ 12 for
the trading volume variable are both positive and significant. The AR parameter, φ , is highly
significant in all the five models in Table 6. The Ljung-Box Q-Statistics for lags 1 and 3
demonstrates that each Markov-Switching model eliminates much of the serial correlation in the
residuals till the third lag. We also estimated each model in Table 6 on a smaller sub-sample, i.e.
an “in-sample” of 385 observations58 for this data set (results not reported here). There was no
difference in the results obtained for the sub-sample for each model.
4. Comparing Volatility Forecasts
In this section, we examine which estimator –GARCH, EGARCH, FIEGARCH or
the Markov-Switching model—provides more accurate volatility forecasts to judge the
relative performance of these models. We first compare the Akaike Information Criterion
57
58
The estimated transition probability p22 = 0.989 in model ( 5) confirms this result.
The remainder of the sample is used for “out-of-sample” forecast tests, as described in the next section.
31
(AIC) and Bayesian Information Criterion (BIC) statistics of all the models estimated for the
daily data set. For the daily data set, the AIC statistic of each of the three estimated GARCH
models is 1.24, while the BIC statistics are 1.36, 1.37 and 1.38 respectively. These AIC and
BIC values are much lower than the lowest obtained AIC (382.25) and BIC (417.78) statistic
from the estimated Markov-Switching models for the daily data set (see the last two rows in
Table 5). Hence, in terms of information criteria, the GARCH models outperform all the
Markov-Switching models in this case.
Another way of comparing the GARCH and the Markov-Switching models is
through out-of-sample forecast errors. Out-of-sample tests are effective since they control
for the possibility of over-fitting and hence provide a useful framework for evaluating the
merits of competing models. We used 80 observations from 08/17/2000 to 11/06/2000 in
the daily data set for the out-of-sample forecast evaluation in the daily data set.59 Similarly,
we used 350 observations from the last 6 hours in the overnight data set for the out-ofsample forecast evaluation.60 Two well-known criteria were used to evaluate the forecast
errors from the models across both the data sets. These are the RMSE (Root Mean Square
Error) and the MAE (Mean Absolute Error).61 Panel A, Table 7 reports the RMSE and
MAE statistics for the GARCH, EGARCH and FIEGARCH models from the daily and
overnight data sets. Panel A, Table 8 reports the RMSE and MAE statistics for all the
Markov-Switching models in the daily and overnight data sets.
--Insert Tables 7 and 8 about here--
We estimated all the GARCH and Markov-Switching models on this smaller sub-sample of 80 observations
(results not reported but available on request). After doing so, we evaluated the out-of-sample-error forecasts.
60 We estimated all the GARCH, EGARCH, FIEGARCH and Markov-Switching models on this smaller subsample of 359 observations (results not reported). We then evaluated the out-of-sample error forecasts.
61 The (root) mean squared error provides a quadratic loss function, which disproportionately weighs large
forecast errors more heavily relative to mean absolute error. As a result, the RMSE may be particularly useful in
forecasting situations when large forecast errors are disproportionately more serious than small errors.
59
32
Observe that for the daily data set, the RMSE statistic of all the GARCH models is
0.134 and the MAE statistic is 0.011. These values are substantially lower than the lowest
RMSE (1.227) and MAE (0.319) statistics from the Markov-Switching models for this data
set. The RMSE value for the GARCH, EGARCH and FIEGARCH models based on 350
observations from the overnight data set is 0.001, while the MAE statistic is either 0.0003 or
0.0005 for these models. Once again, these values are significantly lower than the lowest
RMSE (1.184) and MAE (0.329) statistics obtained from the Markov-Switching models for
this data set. Put together, the RMSE and MAE statistics discussed above unambiguously
demonstrate that all the GARCH, EGARCH and FIEGARCH models provide more
accurate forecasts and fit the data better than any of the Markov-Switching models.
To check and compare the out-of-sample volatility forecasts, we also estimated the
following ex post volatility regression62 for each GARCH and Markov-Switching model,
σ t2 = α + βσˆ t2−1 + ut
(18)
In (18), the measure of ex post (i.e. realized) volatility, σ t2 , is square of log change of daily
closing prices (x 100) of S& P 500 index for the daily data set and square of log change of
prices (x 100) of S&P 500 futures for the overnight data set. The term σˆ t2−1 denotes the
forecasted conditional variance derived from the estimated out-of-sample GARCH and
Markov-Switching models.63 The procedure that we adopted to estimate the above
volatility/variance regression is as follows. We first estimated each of the GARCH (Table 3),
GARCH, EGARCH, FIEGARCH (Table 4) and Markov-Switching models (Tables 5 and 6)
on the out-of-sample observations from the daily and overnight data set. We then derived the
62
This regression is also known as the Mincer-Zarnowitz (1969) regression.
63
In appendix B, we describe how we estimated
σˆ t2−1
for our Markov-Switching model.
33
forecasted conditional variance for each of these estimated models. The volatility regression in
(18) was estimated for each model by using their forecasted conditional variance.
After estimating (18) for each model, we checked if αˆ = 0 and βˆ = 1 (in each case)
because the aforementioned hypothesized values indicate that the relevant model provides
perfect forecasts. Note that if the estimated for a model is βˆ > (<) 0 , then that model
underestimates (overestimates) the true realized volatility in the data (Pagan and Schwert
1990: 283). Results from estimating (18) for each GARCH model for the daily data set is
reported in Panel B, Table 7. The estimated coefficient, α̂ , for all the GARCH models are
well above 0. The estimated β̂ coefficient of the three GARCH models (-0.07, -0.068, 0.13)
are insignificant and below 1. This indicates that the GARCH models do not provide
accurate volatility forecasts for this data set.
The estimated α̂ and β̂ for each Markov-Switching model for the daily data set –
see Panel B, Table 8 -- fare much worse. The intercept estimate α̂ for all these models are
much below 0, while the insignificant slope coefficient estimate β̂ of the Markov-Switching
models are much higher than 1. This demonstrates that there exists a substantial bias in the
forecasts from the Markov-Switching models and that these models are underestimating the
degree of volatility in the daily data set. Moreover, it suggests that the GARCH models
provide relatively better forecasts than the Markov-Switching models.
The estimated volatility regressions for the GARCH, EGARCH and FIEGARCH
models from the overnight data set –see Panel B, Table 7– provide mixed results. The
estimated α̂ of all the GARCH, EGARCH and FIEGARCH models are approximately
equal to zero. It is also significant for the EGARCH (p(gore)) and FIEGARCH (entropy)
models. The estimates of β̂ for the FIEGARCH models are disappointing since they are
34
insignificant and well below 1. However, the estimated β̂ for the EGARCH model is slightly
encouraging because it is significant and marginally different from 1.
The estimate of α̂ for all the Markov-Switching models in the overnight data set is
again well below their hypothesized value of 0 (see Panel B, Table 8). Likewise, the slope
coefficient estimate β̂ of the Markov-Switching models are substantially higher than 1. In
sum the volatility regression results for all models in both data sets is disappointing. Yet,
combined with the RMSE and MAE statistics, the volatility regression results show that:
(i)
The GARCH and especially the EGARCH model outperforms the MarkovSwitching models with respect to out-of-sample forecasts in both the daily and
overnight data set and,
(ii)
The EGARCH model provides the most accurate volatility forecast compared to
the FIEGARCH and Markov-Switching models.
Our out-of-sample error and volatility forecast results are surprising. They challenge the
claim that Markov-Switching models provide more accurate forecasts of stock price volatility
than various linear and non-linear GARCH models (Kim, Morley and Nelson 2002; Van
Norden and Schaller 1993; Sola and Timmerman 1994).64 The analysis presented in this
section thus raises a key question: Why do Markov-Switching models provide poor volatility
forecasts in our study? We provide brief answers to this question in the conclusion.
5. Conclusion
This paper makes two main contributions. First, unlike the existing literature, we
argued and empirically demonstrated here that anticipation of a Democratic victory decreases
stock price volatility. Second, in sharp contrast to methodological claims in the literature, our
out-of-sample forecasts show that the GARCH and EGARCH models provide more accurate
64 Our out-of-sample forecasts are similar to Akgiray 1989 and Franses and Van Djick, 1996 who found that
GARCH models fit stock market data better than random-walk models. Our analysis differs from their works
in that we compare the out-of-sample forecasts between various GARCH and Markov-Switching models,
which is not done by the aforementioned scholars.
35
forecasts of stock volatility than the Markov-Switching models. We posit three plausible
reasons below to explain why the Markov-Switching models provided the worst forecasts.
First, the Markov-Switching models that we have estimated here –and which is also
commonly used in the empirical literature—cannot account for the presence of ARCH and
GARCH effects within each regime. This is problematic since we know from the estimates of
all the GARCH models, we know that the ARCH and GARCH terms significantly affect the
conditional variance and that this, in turn, plays an important role in determining the degree
of realized volatility. Thus, by not taking into account the effects of ARCH and GARCH on
the conditional variance, the Markov-Switching models are underestimating the degree of
realized volatility in both the daily and overnight data set.65
Second, on more technical grounds, it is well known that the Markov-Switching
model places an upper bound on the conditional variance that is too small (Pagan and
Schwert, 1990: 283). As a result, volatility estimates from the Markov-switching models are
typically too low, which weakens their ability to predict realized volatility. This problem is
evident in our analysis where the estimates of β̂ from the volatility regressions of the
Markov-Switching models show that these models are seriously underestimating the degree
of volatility in both two data sets. Third, in comparison to the Markov-Switching model, the
GARCH, EGARCH and FIEGARCH models estimated are not only better at capturing the
persistence of volatility, 66 but can also account for the differential effects of positive and
negative shocks on stock price volatility. This is crucial since volatility in daily and especially
tick (overnight) data tends to persist over long periods of time. The persistence of volatility,
65 Instead of estimating a “plain-vanilla” Markov-Switching model, it might have been more appropriate to
estimate Markov-Switching-GARCH models (Dueker 1997, Klaassen 2002).
66 Note that, by construction, the switch between the states in our Markov-switching model is determined by a
first-order Markov process. Combined with the absence of ARCH and GARCH terms within regimes, this
impairs the ability of Markov-switching models to account for temporal persistence of volatility.
36
in all likelihood, affects the conditional variance and hence the degree of realized volatility.
Further, it is possible that differential effects of shocks that stem from both “good” and
“bad” news about the expected electoral outcome exert a powerful influence on realized
volatility. The inability of Markov-Switching models to account for volatility persistence and
the differential effects of shocks could be a key cause for it’s poor forecasting performance.
Although we found that Markov-Switching models seriously underestimate the
degree of stock market volatility, we need to do more extensive out-of-sample forecasting
tests and use an experimental design by sampling the data at different intervals to confirm
our methodological claims. We also need to estimate all the statistical models used here on
different data sets; i.e. on data of stock price movements from other Presidential election
years and from other advanced industrial democracies such as Britain, for example. If the
obtained parameter estimates and out-of-sample forecasts from different data sets are similar
to those reported there, then the methodological and theoretical claims that we have posited
in this paper will be truly generalizable.
37
Appendix A
Proof of Lemma 1: Since the market maker uses the linear pricing rule, p (n, s ) = b( s ) + α ( s )n the
ith trader maximizes,
E [(v − b( s ) − α ( s )n) xi ( I , s, v) + qi | v, s ]
= [v − b( s ) − α ( s )∑ xi ( I , s, v)] xi ( I , s, v) + qi − α ( s )[ xi ( I , s, v)]2
where n =
∑x
(1)
j ≠i
j
( I , s, v)] + q . The first-order condition with respect to xi ( I , s, v) is,
j
[v − b( s ) − α ( s )∑ xi ( I , s, v) − 2α ( s)[ xi ( I , s, v)] = 0
(2)
j ≠i
and the second-order condition is − 2α ( s ) < 0 or α ( s ) > 0 .From (2), we obtain after some algebra
1
[v − b( s ) − α ( s ) ∑ x j ( I , s , v )]
α ( s)
j
1
x ( I , s, v ) =
[v − b( s)]
( I + 1)α ( s)
xi ( I , s , v ) =
(3)
(4)
1
. The market-maker treats the linear strategy of traders as given and sets
( I + 1)α ( s)
p (n, s) = E [v | s, n] . Since elliptically contoured distributions have linear conditional expectations
Hence γ ( s ) =
(see property (3) in Appendix B), we obtain
E [v − E[v | s ] | s, n] =


Iγ ( s )Θ | s
I
n−
( E[v | s ] − b( s )

2
2
σ | s + I γ ( s) Θ | s  ( I + 1)α ( s )

2
q
This implies that,
α ( s) =
Iγ ( s )Θ | s
σ | s + I 2γ ( s ) 2 Θ | s
(5)
(6)
2
q


I
(7)
b( s ) = E[v | s ] − α ( s ) 
( E[v | s ] − b( s )
 ( I + 1)α ( s )

This yields b( s ) = E[v | s ] . Rearranging (6), we get Iγ ( s)Θ | s = α ( s)[σ q2 | s + I 2γ ( s) 2 Θ | s ] . From
γ ( s) and the conditional variance matrix in equation (3) in the text, we obtain,


I2
I
Θ + σ q2  =
Θ
2
2
 ( I + 1) α ( s )
 ( I + 1)α ( s )
α ( s) 
α (s) = α =
where | Θ | > 0 and
σq I
Θ and γ ( s ) = γ =
(1 + I ) s
I 2σ q
s I Θ
(8)
(9)
Θ > 0 . Q.E.D
Proof of Corollary to Lemma 1: Part (i) Let s′ ∈ S , s′′ ∈ S and s ∈ S , such that s ′′ ≥ s ′ ≥ s .
Suppose further that | s′ − s |≠ ∅ and | s"− s |≠ ∅ . Since S is bounded, it follows that,
α (| s ′ − s |) =
σq I
(1 + I ) | s ′ − s |
Θ≥
σq I
(1 + I ) | s ′′ − s |
38
Θ = α (| s ′′ − s |) and
γ (| s′ − s |) =
I 2σ q
| s′ − s | I Θ
Then α (| s ′ − s |) =
γ (| s′ − s |) =
≥
I 2σ q
| s′′ − s | I Θ
σq I
Θ≤
(1 + I ) | s ′ − s |
I 2σ q
| s′ − s | I Θ
≤
= γ (| s′′ − s |) . Now suppose that s ′′ ≤ s ′ ≤ s .
I 2σ q
| s′′ − s | I Θ
I 2σ q
| s′′ − s | I Θ
= γ (| s′′ − s |) and
= γ (| s′′ − s |) . ∴ α , γ are monotonic in s.
Part (ii) Differentiating α with respect to I, we obtain,
(
)
∂α σ q s Θ 1 / 2 I + I / 2 I − I
=
>0
∂I
( s + Is) 2
where
Θ > 0 . Differentiating γ with respect to I, we obtain,
(10)
(
)
∂γ σ q I Θ 3I / 2 I
=
> 0 . Q.E.D
∂I
IΘs 2
Proof of Lemma 2: We prove that Var (v − p 0 | s ) is monotonic in s given n and s. Let the signal s A
denote the arrival of new political information where s A ≠ s 0 . Define the variance of n as σ n =| n − n 0 |
where n0 is the mean level of net order flows. From Property 5 of ECC distributions (see Appendix B),
 (| n − n |2 ) (| s − s |2 ) 
σ2
(11)
Var (v − p0 | s ) = g  2 2 0 2 + A 2 0  2 2 q 2 Θ
 I γ Θ +σq
 I γ Θ +σq
σ
s
A


From (11), observe that Var (v − p0 | s ) monotonically increases when | s A − s 0 | increases. Also
observe that Var (v − p 0 | s ) monotonically increases when σ n increases. Q.E.D.
Proof of Proposition 1: We use a multivariate t distribution (which is also an ECC distribution) with
k degrees of freedom and x =3 variables to prove that volatility increases under higher uncertainty,
i.e. when | s A − s 0 | increases. The conditional density for the multivariate t distribution is
d ( s , v, q ) =
Γ[1 / 2(k + x)]
(det ∑) −1 / 2
Γ[1 / 2k ]((k − 2)π ) x / 2
−(k +v) / 2

 s A − s0 


1

( s A − s0 v − p0 q ) ∑ −1  v − p0 
× 1 +
k −2
 q 



(12)
Given (12), the conditional variance between v and the unconditional expected price is,
2
 (| s − s | 2 ) 


Θ = k − 2 1 k + (| s A − s 0 | ) k  Θ
(13)
Var[v − p 0 || s A − s 0 |] = h A 2 0
2


−
−
k
k
k
1
2
σ
σ
sA
sA




Observe in (13) that an increase in the deviation of s A from it’s mean s0 will lead to an increase in the
conditional variance of the traded stock’s price. In footnote 28, we claimed that σ n increases when
I
| s A − s 0 | increases. Note that n = ∑ xi ( I , (| s A − s0 |), v] + q(| s A − s0 |) . Since xi (.) and q are
i =1
monotonic in | s A − s 0 | , an increase in | s A − s 0 | increases the variance of xi (.) , q and n. Q.E.D
39
We state and prove in the following Lemma (which we term as Chu’s Lemma), Chu’s (1973) first and
second characterization of the expected absolute value of the random variable a that is drawn from an
elliptically contoured distribution. This characterization is used to prove propositions 2 and 3
Chu’s Lemma: Let the random variable “a” be drawn from an elliptically contoured description with mean 0. Then


σa
 z dW ( z )

2π ∞
 z 2 dW ( z )1 2 

 ∫0
∞
E[| a |] is a monotonic function of its variance σ and E[| a |] = ∫ 2
2
a
0
where z represents the random variable that determines the variance of the independent random variable “a”. From
Chu’s (1973) second characterization (see Property 5, Appendix B), we obtain E[| a |] =
where
2
2 π
E[ Z ]
σa
(E[Z]1 2
Z represents the random variable that multiplies the unit normal in Chu’s (1973) second characterization.
Proof: From Royden (1968: 270) and Chu’s (1973) characterization of ECC distributions, we get,
+∞
∞
∞
∞
−∞
−∞
0
−∞ 0
E[| a |] = ∫ | a | l (a )da = ∫ | a | ∫ j (a, z )dW ( z )da ≡ ∫
∫
∞
| a | j (a, z )da dW ( z )
(14)
where dW(z) is a finite measure and | a | j (a, z ) is a positive function. Equation (14) implies that,


σa
 z dW ( z )

2π ∞
 z 2 dW ( z )1 2 

 ∫0
∞
E[| a |] = ∫ 2
0
(15)
In (15), one can observe that E[| a |] is monotonic in σ a . The second characterization of E[| a |]
when it belongs to the compound class of normal distributions (taken from Chu (1973) is,
E[| a |] =
2
2 π
E[ Z ]
σa
(E[Z]1 2
(16)
Proof of Proposition 2: We prove this proposition by assuming that | sˆR − sR | increases and by using
Chu’s Lemma. Since variance in the stock price is influenced by ŝR --which is a random variable – we let
Var[v − p | (| sˆ R − s R |) = a = U × e and E (Var[v − p | (| sˆ R − s R |)) = E[U | (| sˆ R − s R |) . From
Chu’s(1973) characterization of ECC distributions in Property 5 and the proof of Chu’s lemma:
∞
E[U | (| sˆ R − s R
∫ Zr (Z ) g (| sˆ
|) =
∫ r (Z ) g (| sˆ
0
R
∞
R
0
− s R |) / Z )dz
− s R |) / Z )dz
(17)
where r(.) is the density function of z and g the normal density function of (| sˆ R − s R |) / Z . Now
observe that
∂E[U | (| sˆR − sR |)
∂E[U | (| sˆR − sR |)
> 0 when
> 0 . From (17), note that
∂ (| sˆR − sR |)
∂ (| sˆR − sR |) 2
∞
∞
∂E[U | (| sˆR − sR |)
= ∫ Zr ( Z ) g (| sˆR − sR |) / Z )dz × ∫ 1 / Zr ( Z ) g (| sˆR − sR |) / Z )dz
2
0
0
∂ (| sˆR − sR |)
∞
+ ∫ r ( Z ) g (| sˆR − sR |) / Z )dz
0
Equation (18) is strictly positive which implies that
∂E[U | (| sˆR − sR |)
> 0 Q.E.D.
∂ (| sˆR − sR |) 2
40
(18)
Proof of Lemma 3: From Lemma 1, α =
σq I
(1 + I )(| sˆ R − s R |)
Θ , p(n, s ) = αn + E [v | s] . Hence,
∂p(.) ∂I ∂p(.) ∂n
∂α
< 0 , λ = (| sˆ R − s R |) . For α and p(n,s), we obtain the Jacobian, J = 
.
∂α ∂n 
∂λ
 ∂α ∂I
Note that ∂α ∂I = 0, ∂α ∂n > 0, ∂p (.) ∂I = 0 and ∂p (.) ∂n > 0. Hence |J|< 0. From the
∂p(.) ∂λ ∂p(.) ∂n
dp(.)
dp(.)
implicit function theorem
= −
| J | which ⇒
> 0. Q.E.D

∂α ∂n 
dλ
dλ
 ∂α ∂λ
Lemma 4: When | sˆ R − s R | increases, then p(n,s) can revert to its mean p0 .
Proof of Lemma 4: Note that if p (n, s ) = E[v | s ] , then p (n, s ) = p0 . From equation (9) in text
and monotonicity of γ and α in s, it follows that γ and α decreases when | sˆ R − s R | increases.
Suppose lim | sˆR − sR | j → ∞ . Then lim α j → 0 which ⇒ for lim , p (n, s) = E[v | s ] = p0 Q.E.D
j →∞
j →∞
j →∞
Proof of Proposition 3: Define s′D ∈ S such that | s′D − sD | < | sˆD − sD | . From (18), this implies
∂E[U | (| s′D − sD |)
∂E[U | (| sˆD − sD |)
<
. If we continue this process and define s′D′ ∈ S
∂ (| s′D − sD |)
∂ (| sˆD − sD |2 )
∂E[U | (| s′D′ − sD |)
∂E[U | (| s′D − sD |)
<
. Taking the
such that | s′D′ − sD | < | s′D − sD | , then
2
∂ (| s′D′ − sD | )
∂ (| s′D − sD |2 )
limit lim sˆ D → s D , we find that lim sˆ D → s D Var[v − p | (| sˆ D − s D |)] strictly decreases. Q.E.D
that
Lemma 5: Suppose lim | sˆ D − s D |→ 0 . Then γ (| sˆD − sD |) strictly decreases as lim | sˆ D − s D |→ 0 .
sˆD → s D
sˆD → s D
Proof of Lemma 5: From Lemma 1, γ ( s ) = γ =
I σq
2
. Substitute | sˆ D − s D |= s .
s I Θ
∂γ (.)
Differentiating with respect to | sˆ D − s D | leads to
< 0 . Suppose sˆ D < s D or
∂ (| sˆD − sD |)
∂γ (.)
sˆ D > s D . Since
< 0 , γ (.) strictly decreases as lim | sˆ D − s D |→ 0 . Q.E.D
sˆD → s D
∂ (| sˆD − sD |)
Lemma 6: n and α (| sˆ D − s D |) strictly decreases in | sˆ D − s D | as lim | sˆ D − s D |→ 0
sˆD → s D
Proof of Lemma 6: α ( s ) = α =
σq I
(1 + I ) s
Θ from Lemma 1. Substitute | sˆ D − s D |= s .
∂α (.)
< 0 , which implies that α (.)
∂ (| sˆD − sD |)
strictly decreases as lim | sˆ D − s D |→ 0 . When α (| sˆD − sD |) strictly decreases, then xi (.)
Differentiating with respect to | sˆ D − s D | shows that
sˆD → s D
decreases for each i and, in the aggregate, n decreases. Q.E.D
41
Appendix B
The FIEGARCH (1,d,1) Model:
To motivate the discussion, rewrite the expression for the conditional variance in equation (13) by
dropping the exogenous variables, adding ε 2t to both sides, and moving σ 2t to the right hand side:
ε 2t = ω + (α 1 + β 1 )ε 2t −1 + ν t − β 1ν t −1
where ν t = ε − σ
2
t
2
t .
2
t .
(B1)
Note that the GARCH (1,1) representation in (B1)can be thought of as an
ARMA model for ε It follows that the GARCH (1,1) model is covariance stationary iff α 1 + β 1 < 1 .
In high frequency data, α 1 + β 1 = 1 , which is an Integrated GARCH (IGARCH) model since
α 1 + β 1 = 1 implies a unit root for ε 2t in (B1). As in an ARMA model, the existence of a unit root
means that shocks to the conditional variance die out very slowly (Bollerslev et al. 1992:15). This is in
contrast with the expectation that volatility is mean reverting. An IGARCH model can be rewritten as:
φ ( L)(1 − L)ε 2t = ω + ν t − β 1ν t −1
(B2)
To deal with long-memory in volatility, Baillie, Bollerslev and Mikkelsen (1996) introduce the Fractionally
Integrated GARCH or FIGARCH (p,d,q) model. Denoting the fractional integration parameter by d and
adding d to the first difference operator, Baillie et al (1996) derive the FIGARCH model:
φ ( L)(1 − L) d ε t2 = ω + ν t − β 1ν t −1
(B3)
As in standard ARIMA type models, the fractional differencing parameter, d, indicates the speed at
which shocks to ε 2t die out over time. Baillie et al (1996) and Bollerslev and Mikkelsen (1996) find that
the FIGARCH model fits high frequency data quite well.
The second problem with the standard GARCH model is that it assumes that both positive
and negative innovations/error ( ε 2t ) have the same effect on the conditional variance. This is
problematic since a positive (negative) shock or innovation may have a larger effect on the conditional
variance than a negative (positive) one. This phenomena is quite common in the analysis of stock (or
currency) returns when investors engage in herding behavior-- a negative shock leads larger volatility
than a positive shock. To deal with asymmetric effects of shocks, Nelson (1991) developed an
Exponential GARCH (EGARCH) which from equation (12) can be written as follows:
ln(σ 2t ) = ω + αzt −1 + γ 1 (| zt −1 |− E (| zt −1 |)) + β 1 ln(σ 2t −1 )
(B4)
where zt represents standardized innovations (εt/σt), and E is the expectations operator In B4, the
conditional variance is a function of four terms: the constant, the GARCH term ( σ t2−1 ) and two
ARCH terms--an asymmetric component (zt-1) and a symmetric component (| zt −1 |− E (| zt −1 |)) .
Consider (| zt −1 |− E (| zt −1 |)) which measures deviations between realized and expected innovations and
can hence capture how unexpected innovations affect conditional volatility. γ is typically greater than
zero. Thus, if (|zt| > E|zt|), then future volatility will be higher than its average level. However, if
|zt| < E|zt|, then future volatility will be lower than average. αzt-1 provides for the asymmetric effect
of the standardized innovations. If γ>0, then a positive (negative) value for α implies that positive
(negative) shocks will have a larger effect on future volatility than negative (positive) shocks.
Bollerslev and Mikkelsen (1996) introduce the Fractionally Integrated Exponential GARCH
model (FIEGARCH). This approach combines the FIGARCH and EGARCH specifications.
Rewriting equation (B4) using the backshift operator:
ln(σ 2t ) = ω + [1 − β ( L)]−1[1 + α ( L)]g ( zt −1 )
(B5)
where g ( zt ) = θzt + γ [| zt |− E | zt |] . Factorizing the autoregressive polynomial
[1 − β ( L)] = φ ( L)(1 − L) d , Bollerslev and Mikkelsen (1996) derive the FIEGARCH model:
ln(σ 2t ) = ω + φ ( L) −1 (1 − L) − d [1 + α ( L)]g ( zt −1 )
42
(B6)
Derivation of Log-Likelihood for Markov-Switching Model:
We briefly describe the log-likelihood for the Markov-Switching model with an AR(1)
specification. For more details, readers can consult Diebold, Lee and Weinbach (1994: 285-289) and
Hamilton (1994: 690-695). Let θ1 = ( µ1 ,σ 12 ,φ )′ and θ 2 = ( µ 2 ,σ 22 ,φ )′ denote the parameter vectors
that require to be estimated. Define θ = (θ1 ,θ 2 ) and t = 1,2,…T. Let Pr( s1 = 1) = ρ denote the
probability that at t=1, the state is 1. Hence, Pr( s1 = 2) = 1 − ρ denotes the probability that at t=1,
the state is 2. Define the indicator function at t=1 for state 1 as I ( s1 = 1) and I ( s1 = 2) for state 2. To
avoid notational confusion, we let ∆pt = yt . Finally, note that p11, t = {st = 1 | st −1 = 1} ,
1 − p11,t = {st = 2 | st −1 = 1} , p22,t = {st = 2 | st −1 = 2} and 1 − p22,t = {st = 1 | st −1 = 2} , where
p11,t = exp( xt' −1β1 ) 1 + exp( xt' −1β1 ) . Using Diebold, Lee and Weinbach’s (1994: 285-289) terminology,
we can define the “complete-data likelihood” of the Markov Switching model in log form as:
f ( yT | yT −1 , sT ; θ) = I ( s1 = 1)[log ρ + log f ( y1 | s1 = 1;θ1 )] + I ( s1 = 0)[log(1 − ρ ) + log f ( y1 | s1 = 1;θ1 )]
T
+ ∑{I ( st = 1) log f ( yt | yt −1 , st = 1;θ1 ) + I ( st = 2) f ( yt | yt −1 , st = 1;θ1 )
t =2
+ I ( st = 1, st −1 = 1) log( p11,t ) + I ( st = 2, st −1 = 1) f ( yt | yt −1 , st = 1;θ1 ) log(1 − p11,t )
(B7)
+ I ( st = 2, st −1 = 2) log( p22,t ) + I ( st = 1, st −1 = 2) f ( yt | yt −1 , st = 1;θ1 ) log(1 − p22,t )
Since in practice, the complete data cannot be observed, we require the “incomplete-data” log
likelihood which can be obtained by summing over all possible state sequences:
 1
log f ( yT | yT −1 , xT , sT ; θ) = log ∑
 s1 = 0
1

...
f ( yT | yT −1 , xT , sT ; θ) 
∑
∑
s2 =0
sT = 0

It is intractable to maximize the above log-likelihood with respect to θ . We use the same EM
1
(B8)
algorithm as used by Diebold, Lee and Weinbach (1994: 287-288) and Hamilton (1994:pp#) for
maximization of the incomplete-data likelihood. Interested readers can refer to the pages in Diebold
et al (1994) where the EM algorithm has been described.
Calculation of Conditional Variance, σˆ t2 , From Markov-Switching Model:
We show how to compute σˆ t2 from our Markov-Switching model; obtaining σˆ t2−1 is trivial
given σˆ t2 . Let µ S t = α1St + α 2 and σ S2t = ω1St + ω 2 ; recall ε t ~ N (0,σ S2 ) . Suppose that the stock
t
price index was in regime 1 at t-1. From Pagan and Schwert (1990: 275), σˆ t2 is derived from
[ E{σ ( St ) | St −1 = 1}]2 + var{σ ( St ) | St −1 = 1} + E{[ µ ( St ) − E ( µ ( St ))]2 | St −1 = 1} which yields,
[ω 2 + ω1 p11 ]2 + ω12 p11 (1 − p11 ) + α12 p11 (1 − p11 )
(B9)
Suppose that the stock price index was in regime 2 at t-1. From Pagan and Schwert (1990: 275), σˆ t2 is
derived from [ E{σ ( St ) | St −1 = 2}]2 + var{σ ( St ) | St −1 = 2} + E{[ µ ( St ) − E ( µ ( St ))]2 | St −1 = 2} , i.e. from,
[ω 2 + ω1 (1 − p22 )]2 + ω12 p22 (1 − p22 ) + α12 p22 (1 − p22 )
Multiplying (B9) and (B10) by the estimates of the conditional probabilities of being in each regime
given data through t-1 gives the estimate of the conditional variance σˆ t2 , which is then lagged to
obtain σˆ t2−1 .
43
(B10)
Table 1: Data, Variables and Sources
VARIABLE
Daily Sample
Return on S&P 500 Index
Volume Traded
Before
After
Gore
Overnight Sample
Return on S&P 500 Futures
Volume Traded
Duration
Gore
Information Arrival
MEASURE
SOURCE
(d(log(sp)))*100 – daily closing
price
log(volume)
Number of non-trading days prior
to day t
Number of non-trading days after
day t
Share of Gore’s 2-party vote based
on aggregation of polls
finance.yahoo.com
(d(log(sp)))*100 – average price
per minute
log(volume) – total volume traded
per minute.
Predicted duration between trades
based on hazard model.
Probability that Gore will win
Electoral College
Time when CNN called PA, OH,
MI, IL, WI, MO, WA, LA
finance.yahoo.com
Wlezien(2001); Franklin (2001)
Chicago Mercantile Exchange
GLOBEX database
Chicago Mercantile Exchange
GLOBEX database
Simulation based on times called
on air by CNN
CNN
Summary Statistics From the Daily Sample
Variable
Dlsp
Before
After
Ldv
Rpctgore
Entropy
drpctgore
Mean
0.0092563
0.3926941
0.3926941
20.73713
47.79178
99.57449
0.0183875
Std. Dev
1.339722
0.8356405
0.8356405
0.1593171
2.4058
0.4410992
1.963907
Min
-6.00451
0
0
19.92897
41.92393
97.39109
-5.957844
Max
4.654578
3
3
21.13898
55.29616
99.9999
6.304455
N
219
219
219
219
219
219
219
Summary Statistics From the Overnight Sample
Variable
dlsp
sq
st_dur
l5gore
l5entropy
ltossup
Mean
0.0001917
3.767513
0.558605
0.4392081
0.8901627
0.0050761
Std. Dev
0.0305541
8.12765
1.291843
0.154233
0.2278682
0.0711021
Min
-0.2961127
0
-6.94119
0.021
0.082236
0
44
Max
0.1990449
110
2.204567
0.657
0.999984
1
N
985
985
985
985
985
985
Table 2: State Probabilities, Polls and p-values (Overnight Sample)
State
Alaska
Alabama
Arkansas
Arizona
California
Colorado
Connecticut
Delaware
Florida
Georgia
Hawaii
Iowa
Idaho
Illinois
Indiana
Kansas
Kentucky
Louisiana
Massachusetts
Maryland
Maine
Michigan
Minnesota
Missouri
Mississippi
Montana
North Carolina
North Dakota
Nebraska
N. Hampshire
New Jersey
New Mexico
Nevada
New York
Ohio
Oklahoma
Oregon
Pennsylvania
Rhode Island
South Carolina
South Dakota
Tennessee
Texas
Utah
Virginia
Vermont
Washington
Wisconsin
West Virginia
Wyoming
Sample
Size
400
625
286
423
600
400
447
625
600
512
261
603
633
600
600
600
625
660
401
627
400
600
1015
600
625
628
625
586
1007
801
843
425
625
700
600
625
600
600
370
625
300
500
625
914
625
400
500
400
536
412
Gore %
26
38
44
39
45
38
48
42
48
37
50
44
30
50
30
32
41
38
52
52
47
51
47
46
41
37
41
35
31
39
41
45
43
54
43
39
45
50
47
38
33
46
30
27
41
52
50
39
39
32
Bush %
47
55
47
49
44
47
32
46
46
53
31
43
56
42
53
55
51
46
30
38
36
44
37
46
52
49
48
47
56
45
36
45
47
37
50
54
44
42
29
53
51
51
64
59
49
36
42
44
41
57
Gore/
(Bush+Gore)
0.356
0.409
0.484
0.443
0.506
0.447
0.600
0.477
0.511
0.411
0.617
0.506
0.349
0.543
0.361
0.368
0.446
0.452
0.634
0.578
0.566
0.537
0.560
0.500
0.441
0.430
0.461
0.427
0.356
0.464
0.532
0.500
0.478
0.593
0.462
0.419
0.506
0.543
0.618
0.418
0.393
0.474
0.319
0.314
0.456
0.591
0.543
0.470
0.488
0.360
P-Value
0.000
0.000
0.287
0.010
0.607
0.017
1.000
0.127
0.697
0.000
1.000
0.609
0.000
0.983
0.000
0.000
0.003
0.007
1.000
1.000
0.996
0.964
1.000
0.498
0.002
0.000
0.024
0.000
0.000
0.021
0.970
0.498
0.132
1.000
0.032
0.000
0.607
0.983
1.000
0.000
0.000
0.124
0.000
0.000
0.013
1.000
0.974
0.113
0.280
0.000
Electoral
Votes
3
9
6
8
54
8
8
3
25
13
4
7
4
22
12
6
8
9
12
10
4
18
10
11
7
3
14
3
5
4
15
5
4
33
21
8
7
23
4
8
3
11
32
5
13
3
11
11
5
3
*New Mexico and Oregon were not called before markets closed on November 8, 2000.
45
Time Called by
CNN (est)
12:00am (B)
8:00pm (B)
12:12am (B)
11:51pm (B)
11:00pm (G)
11:41pm (B)
8:00pm (G)
8:00pm (G)
see below
7:59pm (B)
11:00pm (G)
5:00am (G)
10:00pm (B)
8:00pm (G)
6:00pm (B)
8:00pm (B)
6:00pm (B)
9:21pm (B)
8:00pm (G)
8:00pm (G)
10:10pm (G)
9:24pm (G)
10:25pm (G)
10:47pm (B)
8:00pm (B)
10:00pm (B)
8:14pm (B)
9:00pm (B)
9:00pm (B)
12:07am (B)
8:00pm (G)
not called*
1:31am (B)
9:00pm (G)
9:19pm (B)
8:00pm (B)
not called*
9:24pm (G)
9:00pm (G)
7:00pm (B)
9:00pm (B)
11:03pm (B)
8:00pm (B)
10:00pm (B)
7:33pm (B)
7:00pm (G)
12:08am (G)
6:21am (G)
10:46pm (B)
9:00pm (B)
Table 2 (Continued): Florida
Time
7:52pm
8:55pm
1:18am
2:58am
Event
Called for Gore
Toss-up (taken away from Gore)
Called for Bush
Toss-up (taken away from Bush)
Table 3: Daily GARCH Models (N=218 in all models)
Mean
Intercept
Variance
Intercept
ARCH
GARCH
Before
After
log(Volume )
Gore
(1)
(2)
(3)
-0.009
(0.074)
0.005
(0.075)
-0.001
(0.072)
-60.267*
(0.378)
0.138
(0.084)
0.628*
(0.144)
-82.08*
(0.285)
0.134
(0.127)
0.726*
(0.207)
-53.82*
(1.81)
0.136
(0.139)
0.704*
(0.254)
-0.656
(0.739)
0.572*
(0.228)
-1.09
(1.78)
0.545
(0.308)
-0.313
(0.818)
0.822
(0.551)
3.14*
(0.003)
-0.134*
(0.007)
4.195*
(0.015)
2.49*
(0.009)
Entropy
-0.07*
(0.003)
d(Gore)
-0.175
(0.154)
Diagnostics
p-value
p-value
p-value
LB(10)
0.1742
0.1742
0.1742
LB2(10)
0.1050
0.0978
0.0965
Jarque-Bera
0.6201
0.6209
0.6207
AIC
1.24
1.24
1.24
BIC
1.36
1.38
1.37
Notes: Cell entries are maximum likelihood estimates with semi-robust standard errors in parentheses.
46
Table 4: Overnight GARCH Models (N=985 in all models)
(1)
(2)
-0.001*
(0.0003)
AR(1)
MA(1)
Mean
Intercept
Duration
Variance
Intercept
ARCH
GARCH
(3)
(4)
(5)
-0.0004
(0.0004)
-0.0004
(0.0004)
-0.0004
(0.0005)
-0.0004
(0.0005)
0.025
(0.26)
0.413*
(0.159)
0.279*
(0.150)
0.274*
(0.130)
0.268*
(0.133)
0.066
(0.248)
-0.239
(0.175)
-0151
(0.167)
-0.131
(0.167)
-0.132
(0.150)
-0.0027*
(0.0010)
-0.0005
(0.001)
-0.0005
(0.001)
-0.0006
(0.002)
-0.0005
(0.003)
0.0015*
(0.00003)
0.158*
(0.026)
-1.64*
(0.117)
0.30*
(0.03)
-3.09*
(0.344)
0.459*
(0.045)
-3.67*
(0.373)
0.468*
(0.046)
-4.603*
(0.487)
0.443*
(0.045)
0.686*
(0.008)
0.801*
(0.014)
0.025
(0.068)
0.049
(0.048)
0.060
(0.037)
0.063
(0.048)
0.089*
(0.040)
0.120
(0.048)
0.081*
(0.038)
0.120*
(0.015)
0.018*
(0.002)
-0.406*
(0.075)
0.121*
(0.018)
0.332*
(0.034)
0.080*
(0.006)
-1.11*
(0.218)
0.124*
(0.018)
0.350*
(0.034)
0.080*
(0.007)
0.071*
(0.018)
0.399*
(0.036)
0.085*
(0.007)
EGARCH
fraction (d)
Duration
Volume
P[Goret-5]
0.0001
(0.001)
0.001*
(0.0001)
-0.0024*
(0.0004)
Entropyt-5
0.232*
(0.100)
Info Arrivalt-5
1.544*
(0.700)
Diagnostics
LB(12)
0.5176
0.5043
0.5269
0.4764
0.4682
2
LB (12)
0.0725
0.1151
0.1345
0.1254
0.1201
Jarque-Bera
0.0000
0.0000
0.0000
0.0000
0.0000
AIC
-5257
-4735
-4784
-4769
-4784
BIC
-5208
-4681
-4725
-4710
-4725
Notes: Cell entries are maximum likelihood estimates with semi-robust standard errors in parentheses.
47
Table 5: Markov-Switching Estimates for Daily Sample (N=218)
Parameters
µ1
µ2
β 1,1
c1
β 2,1
c2
σ 12
σ 22
φ
p11,t
p 22,t
(1)
Entropy
1.113
(0.647)
0.702
(0.422)
2.254
(0.660)
1.710
(0.512)
-0.471
(0.289)
1.587
(0.978)
(2)
Gore
0.392
(0.237)
1.436
(0.203)
-0.415
(0.308)
0.895
(0.556)
3.142
(0.395)
1.280
(0.764)
(3)
Information
0.413
(0.289)
1.031
(0.244)
-0.397
(0.325)
0.593
(0.417)
2.119
(0.324)
0.977
(0.743)
(4)
Volume
-0.141
(0.078)
0.375
(0.082)
0.698
(0.110)
0.190
(0.146)
0.423
(0.265)
0.147
(0.136)
2.360
(0.315)
0.325
(0.643)
0.334
(0.036)
0.966
(0.083)
0.684
(0.194)
0.183
(0.225)
3.134
(0.512)
0.359
(0.048)
0.712
(0.186)
0.974
(0.181)
0.237
(0.149)
1.874
(0.419)
0.293
(0.051)
0.664
(0.297)
0.955
(0.214)
0.598
(0.151)
0.147
(0.102)
0.264
(0.067)
0.950
(0.214)
0.708
(0.129)
-245.342
-230.246
-211.348
-316.022
Wald Tests
H 0 : µ1 = µ 2
7.97 **
H0 :σ = σ
12.85**
H 0 : p 22 = 1 − p11 59.61**
2
1
2
2
LRT Tests:
Garcia
85.14**
Hansen
5.98**
Log Likelihood
Ljung-box Q-statistics
LB-1
LB-3
0.141 (0.718)
0.925 (0.428)
0.157 (0.697) 0.125 (0.724)
0.976 (0.411) 0.912 (0.433)
AIC
342.45
394.73
382.25
BIC
430.22
422.15
417.78
Notes: Standard errors reported in parentheses. * 5% level, ** 1% level.
48
0.158 (0.530)
0.934 (0.522)
376.23
439.06
Table 6: Markov-Switching Estimates for Overnight Sample (N=985)
Parameters
µ1
µ2
β 1,1
c1
β 2,1
c2
σ 12
σ 22
φ
p11
p 22
(5)
Entropy
1.462
(0.852)
0.341
(0.536)
(6)
Gore
0.282
(0.165)
1.012
(0.314)
(7)
Bush
0.564
(0.307)
0.196
(0.288)
(8)
Information
0.555
(0.321)
0.482
(0.346)
(9)
Volume
-0.156
(0.081)
0.450
(0.092)
1.941
(0.299)
1.027
(0.673)
-0.335
(0.299)
0.583
(0.337)
2.424
(0.307)
0.632
(0.422)
0.276
(0.185)
0.373
(0.191)
0.677
(0.122)
0.208
(0.315)
-0.558
(0.318)
0.664
(0.397)
3.529
(0.411)
1.119
(0.689)
0.714
(0.397)
0.878
(0.522)
0.459
(0.312)
0.248
(0.196)
0.222
(0.139)
0.135
(0.172)
1.862
(0.218)
0.265
(0.410)
0.185
(0.228)
2.977
(0.432)
4.056
(0.493)
0.386
(0.219)
0.458
(0.293)
0.677
(0.442)
0.662
(0.138)
0.148
(0.087)
0.455
(0.027)
0.988
(0.125)
0.701
(0.163)
0.413
(0.038)
0.771
(0.159)
0.989
(0.146)
0.298
(0.034)
0.991
(0.127)
0.698
(0.141)
0.340
(0.052)
0.797
(0.403)
0.824
(0.397)
0.236
(0.061)
0.932
(0.145)
0.783
(0.191)
-179.062
-314.311
0.138 (0.755)
1.011 (0.205)
-148.57
-211.02
0.215 (0.731)
1.226 (0.512)
-136.35
-221.63
Wald Tests
H 0 : µ1 = µ 2
6.83**
H0 :σ = σ
12.46**
H 0 : p 22 = 1 − p11 39.21**
2
1
LRT Tests:
Garcia
Hansen
2
2
85.27**
7.25**
Log Likelihood
-127.686
-165.801
-138.452
Ljung-box Q-statistics
LB-1
0.104 (0.836) 0.129 (0.784) 0.157 (0.695)
LB-3
0.819 (0.514) 0.929 (0.371) 1.233 (0.299)
AIC
-106.68
-109.65
-114.22
BIC
-214.51
-203.26
-259.37
Notes: Standard errors reported in parentheses. * 5% level, ** 1% level
49
Table 7 : Error and Volatility Forecasts from all GARCH models
GARCH
(RpctGore)
Daily Sample
GARCH
GARCH
(Entropy) (Information
Arrival)
Overnight Sample
GARCH EGARCH FIEGARCH FIEGARCH
(P(Gore)) (P(Gore))
(P(Gore))
(P(Gore))
FIEGARCH
(Information
Arrival)
Panel A.
Error Forecasts:
RMSE
0.134
0.134
0.134
0.001
0.001
0.001
0.001
0.001
MAE
0.011
0.011
0.011
0.0003
0.0003
0.0005
0.0003
0.0005
α̂
1.32
(0.58)
1.33
(0.57)
0.94
(0.51)
0.0001
(0.0001)
0.0002
(0.0001)
0.0001
(0.0001)
0.0004
(0.0001)
0.0001
(0.0001)
βˆσˆ t2−1
-0.07
(0.28)
-0.068
(0.26)
0.13
(0.23)
0.26
(0.087)
0.328
(0.101)
-0.0001
(0.0001)
-0.0001
(0.0002)
-0.0001
(0.0001)
R2
0.0009
0.0009
0.0037
0.0261
0.0252
0.023
0.024
0.022
Panel B.
Volatility regression
Notes: Standard Errors in Parentheses.
50
Table 8: Error and Volatility Forecasts from all Markov Switching models
Markov
(RpctGore)
Daily Sample
Markov
Markov
(Entropy) (Information
Arrival)
Markov
(Volume)
Overnight Sample
Markov Markov
Markov
Markov
(P(Gore)) (Bush)
(Entropy)
(Information
Arrival)
Markov
(Volume)
Panel A.
Error Forecasts:
RMSE
1.227
1.273
1.351
1.462
1.184
1.878
1.367
1.421
1.193
MAE
0.385
0.379
0.371
0.319
0.329
0.401
0.388
0.390
0.365
α̂
-0.406
(0.284)
-0.418
(0.288)
-0.402
(0.291)
-0.395
(0.274)
-0.327
(0.290)
-0.319
(0.279)
-0.334
(0.293)
-0.309
(0.282)
-0.328
(0.275)
βˆσˆ t2−1
3.993
(1.048)
4.112
(0.997)
4.079
(1.023)
3.988
(1.103)
5.643
(1.482)
5.187
(1.311)
5.889
(1.263)
5.074
(1.107)
5.631
(1.505)
R2
0.0003
0.0002
0.0005
0.0003
0.0009
0.0011
0.0012
0.0010
0.0011
Panel B.
Volatility
regression
Notes: Standard Errors in Parentheses.
51
Figure 1
52
Time (EST)
32.13
31.47
31.21
30.55
30.29
30.03
2:58am: FL
from Bush
29.37
29.11
28.45
1:18am:
FL for Bush
28.19
27.53
6pm: IN & KY
for Bush
7:33pm: VA
for Bush
27.27
27.01
26.35
26.09
25.43
11pm:CA
for Gore
25.17
24.51
24.25
23.59
23.33
23.07
8:55pm: FL
from Gore
22.41
22.15
21.49
21.23
0.6
20.57
20.31
20.05
19.39
19.13
18.47
18.21
0.5
17.55
0.2
17.29
0.3
17.03
16.37
16.11
15.45
Pr(Gore Victory)
0.7
6:21am: WI
for Gore
7:52: FL
for Gore
0.4
5am: IA
for Gore
0.1
0
Appendix B (Continued)
Properties of Elliptically Contoured Distribution (Used for Proofs in Appendix A.)
For proofs of the lemmas and propositions in Appendix A, we used the following definitions
and properties of the Elliptically Contoured (ECC) Distribution. These are taken from Cambanis,
Huang and Simons (1981), Johnson (1987), Foster & Vishwanathan (1995) and Chu (1973).
Property 1: A random variable x is said to have an elliptically contoured distribution if its characteristic
function is of the form,
E [a it/x ] = a
it/µ
× f (t' ∑ -1 t )
where µ is the mean vector of the random variable x and f (.) satisfies the necessary conditions f
(0)=1, | f (t ) |< 1 ∀t ≠ 0, and f (t) is continuous.
Property 2: Assume that the random variable x has ∑ -1 = I . The elements of x are mutually
independent if and only if x ~ N (0, I ) . The Normal Distribution is the only elliptically contoured
distribution that has the independence property.
Property 3: If x ~ ( µ , , ∑, f ) and we partition x' = ( x1′, x′2 ) and therefore,
∑
∑ =  11
 ∑ 21
∑12 

∑ 22 
From the above matrix, we obtain,
E [ x1 | x 2 ] = ∑12 ∑ −221 x 2
If E [ x1 | x2 ] = 0 , then x1 , x2 are semi-independent. Elliptically contoured distributions are the only
distributions with linear conditional expectations.
Property 4: From Property 4, it holds that Var [ x1 | x2 ] is independent of x2 if and only if
x ~ N (0, Ω) .
Property 5: If the random variable a is drawn from an elliptically contoured multivariate distribution
with mean zero, then from Chu’s (1973) lemma it follows that,
∞
l(a) = ∫0 u(a,z )dW(z)
where dW (.) is a weighting function on [0, ∞) that can assume negative values and
∫
∞
0
dW ( z ) = 1 . u(.,z ) is the normal density with mean zero and variance z 2Ω . If dW (z ) ∈ ℜ +
then it is itself a density and l(a) is in the compound normal class. Following Chu (1973) and Foster
and Vishwanathan (1995) a second characterization of the random variable a when it is drawn from
an elliptically contoured multivariate distribution is a =
variable independent of the normal random variable e.
53
Z × e where Z is a positive random
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