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Time-Varying Skewness and Real Business Cycles.

A growing literature in macroeconomics and finance has found important economic effects of variations in risk, in particular shocks to the volatility of key macroeconomic variables (such as total factor productivity). However, much less is known about the importance of shocks to the skewness of macroeconomic variables. (1)

In this paper, we seek to quantify the economic effects of skewness shocks. To this end, we augment a small open economy real business cycle model with a novel feature: discrete regime changes in the higher-order moments of exogenous shocks, modeled as shocks to total factor productivity (TFP). We assume that in each period the economy can be in one of two possible Markov states: an unrest state or a quiet state. The unrest state is assumed to be associated with a substantial increase in volatility and negative skewness of shocks. This assumption is motivated by our empirical findings about the moments of business cycles of many countries that experience political unrest (see the discussion of our calibration below). Hence, unrest is effectively a shock to the second-order and third-order moments of the distribution of economic shocks.

To solve the model, we develop a third-order perturbation method to approximate the endogenous reactions to shocks to the second-order and third-order moments of TFP. Existing methods to solve and simulate models (including global approximations to policy functions as in Judd [1996] or Richter et al. [2014] or perturbation methods as in Andreasen et al. [2017]) rely on Monte Carlo simulations to calculate the dynamics of third-order moments of endogenous quantities such as output and consumption. However, Monte Carlo simulations are problematic for the computation of higher-order moments such as skewness because these higher-order moments are more sensitive to simulation error. (2) To overcome this problem, we build upon the method of Andreasen (2017) to calculate generalized impulse response functions (GIRF) of third-order approximations of third-order moments of endogenous variables. Our solution method exploits computational symbolic algebraic manipulation to calculate the third-order moments without Monte Carlo simulations. This technical innovation is nontrivial, since it requires solving for the dynamics of over 20,000 polynomials, in the presence of a Markov-switching state, that are up to ninth order in the state variables. Furthermore, our approach is readily applicable to other DSGE models, especially those for which the dynamics of higher-order moments of endogenous variables are of interest.

Calibration: To calibrate the model, we document and exploit the substantial changes in higher-order moments of aggregate economic variables during periods of mass political unrest. Unrest episodes, which are well-documented by the political science literature (Chenoweth and Lewis 2013), are helpful in identifying higher-order moment shocks for several reasons. First, we find that these episodes are associated with substantial increases in the volatility and negative skewness of growth rates of output, consumption, and investment. For instance, on average, a year during an unrest episode is associated with a more than 50 percent increase in the volatility and a more than three times increase in the negative skewness of output growth. The changes in higher-order moments of aggregate variables (output growth, consumption growth, and investment growth) associated with an episode of unrest can be estimated with reasonable precision, since the database provides a relatively large number of country-year observations (with eighty-four unrest episodes between 1960 and 2006, each lasting more than five years on average).

Second, since the model assumes that shocks are common knowledge, we ideally want to identify shocks using events that are easily observed for all agents, at home or abroad. Mass unrest episodes are appropriate for this end, as they are major events, and agents in the economy as well as investors abroad do not need to be econometricians to learn that a campaign of mass political unrest is underway. Hence, the onset of an unrest episode is likely to have a direct effect on economic agents' perceptions of risk. Furthermore, since the impulse response exercises assume unanticipated shocks, we ideally want to use events that are ex-ante difficult to predict. Unrest episodes are again appropriate for this end, as it has been well-documented that mass unrest is largely unanticipated because it requires unpredictable shocks that enable a large number of nonstate actors to overcome informational and coordination problems. (3)

Results: Our model shows that the increase in volatility and especially negative skewness when the economy enters an episode of unrest has quantitatively substantial impacts on economic activities. In the baseline calibration, the observed changes in volatility and negative skewness can explain 21 percent of the observed drop in average output growth, 45 percent of the drop in average consumption growth, and 51 percent of the drop in average investment growth during unrest episodes. More importantly, the increase in negative skewness accounts for about half of these drops in growth.

Intuitively, when shocks become more negatively skewed, risk-averse agents know that realizations on the left tail of the distribution of shocks have become more likely. The increase causes agents to shift their portfolios to safer assets abroad and accumulate stocks of these safer assets, leading to capital outflow and drops in domestic investment and output. The consequences of this increased mass on the left tail are heightened under Epstein-Zin preferences. A Taylor expansion of the household's Bellman equation reveals that Epstein-Zin preferences punish and reward, respectively, the second and third central moments of the future value function. To a second-order approximation, Epstein-Zin preferences penalize the second central moment, i.e., variance. To a third-order approximation, the preferences gain an additional term that rewards the third central moment, which is the product of skewness and variance raised to the power of 3/2. Therefore, the quantitative effects of time-varying variance is amplified by time-varying negative skewness.

We demonstrate the quantitative significance of skewness by comparing the losses in economic activities during unrest under a second-order approximation to the same losses under a third-order approximation. The second-order approximation can account for only half of the economic losses that the third-order approximation can. Therefore, negative skewness is revealed as an important component of risk.

Related literature. Our paper is related to several strands of the literature on higher-order moments of business cycles. First, there is a growing body of research that emphasizes the importance of the time-varying volatilities of economic variables (e.g., Justiniano and Primiceri 2008; Caldara et al. 2012; Arellano et al. 2012; Christiano et al. 2014; and Gilchrist et al. 2014. The study that is the closest to ours in quantifying the impact of time-varying higher-order moments is Fernandez-Villaverde et al. (2011). They consider a stochastic volatility process for the real interest rate and explore the impacts of interest rate volatility shocks to economic activities. The primary difference between our paper and this literature is that while they focus only on shocks to second moments, we focus on shocks to both second-order and third-order moments.

Second, there is a related body of macro-finance research that stresses the importance of skewness (e.g., Ranciere et al. 2008; Barberis and Huang 2008; Guvenen et al. 2014; Salgado et al. 2015; Feunou et al. 2015; and Colacito et al. 2015). Our analysis is most related and complementary to that of Colacito et al. (2015), who show the importance of time-varying skewness in a macro-finance model with Epstein-Zin preferences. The major difference is that while they focus on the effects of skewness on financial variables (implied equity Sharpe ratios and equity risk premia), we focus on the effects on real economic variables (the growth rates of output, consumption, and investment). Also, while they focus on the United States, we focus on emerging and developing economies. Finally, while they calibrate the model by looking at analysts' forecasts for the U.S. economy, we look at the changes in higher-order moments of real economic variables during unrest episodes.

Finally, our paper is also related to a body of literature that emphasizes the importance of rare disasters in explaining macroeconomic phenomena (e.g., Barro 2006; Gourio 2012; Andreasen 2012; Gabaix 2012). A key insight from this literature is that variations in the probability of rare disasters, modeled as events on the far left tail of the distribution of shocks, can have first-order macroeconomic effects, as they influence the precautionary behaviors of risk-averse agents. Our paper points out that time-varying negative skewness has similar effects. This is because an increase in negative skewness implies a higher probability of states with very low consumption. However, our estimation approach is different and complementary to existing approaches in this literature. Since rare disasters occur infrequently in data, the literature usually does not estimate the time variation in the probability of disasters from data, (4) or it employs calibrations to proxies such as time-varying volatility of equity returns (e.g., Gourio et al. 2013). In contrast, we exploit the uncertainty associated with episodes of unrest to estimate the time variation in the skewness of economic shocks when the economies enter and exit unrest. (5)

Our paper is organized as follows. Section 1 describes our data sources and documents several stylized facts on business cycles during unrest episodes. Section 2 introduces unrest to a standard small open economy model and calculates how much of the stylized facts can be explained by changes in the distribution of shocks. Section 3 concludes.


Data Sources and Definitions

For economics and other data, we use annual panel macroeconomic data from 154 countries listed in the World Bank's World Development Indicators (WDI) database over the interval 1960-2006. This includes three time series: real output, real investment, and real consumption. We also use WDI data on the Gini coefficient and Alesina et al.'s (2003) data on ethnic, linguistic, and religious fractionalization as control variables

For mass unrest episodes, we use the Nonviolent and Violent Campaigns and Outcomes (NAVCO) dataset, version 2.0 (Chenoweth and Lewis 2013). NAVCO 2.0 provides a "consensus population" of all known continuous and large (having at least 1,000 observed participants) organized unrest campaigns between 1945 and 2006 (6) that satisfy a series of conditions, as detailed in Appendix C. Each episode has an onset year and an end year. The onset year is defined as the first year with a series of coordinated, contentious collective actions with at least 1,000 observed participants. The episode is recorded as over when peak participation drops below 1,000. Overall, the NAVCO dataset provides 157 episodes of nonviolent and violent mass political unrest around the world between 1945 and 2006. Of these, there are eighty-four episodes in the years between 1960 and 2006, the period for which we have both unrest and economic data. Over this period, the average duration of an episode is 5.99 years.

Examples include many pro-democracy movements of civil unrest in Latin America, the Philippines's People Power Revolution (1983-87), Indonesia's civil unrest against Suharto (1997-98), and Mozambique's RENAMO resistance movement (1979-1992); for a complete listing of these episodes, see Appendix A.1. As an illustration, Figure 1 plots the time series of the growth rate in aggregate economic variables for the Philippines around the People's Power Revolution.

Stylized Facts on Business Cycles During Unrest

We now investigate the relationship between unrest and macroeconomic activities. The goal of this section is to arrive at a set of moments that will be used as calibration targets for the structural model of the following section. We focus on the contemporaneous association between unrest in a given country-year and the growth rates of output, consumption, and investment. We follow others in the macroeconomic literature (e.g., Fernandez-Villaverde et al. 2011) and do not explicitly model why the higher-order moments change, nor do we attempt to make any causal claims about the contemporaneous causal impacts of unrest on output or vice versa.

We calculate the growth rates of output, consumption, and investment by the first difference in logs of the variable at constant 2005 USD and then remove a country-specific average growth rate from each series. That is, if the real output for country i in year t is [], then we calculate the raw growth rate as [DELTA][] [equivalent to] 100(ln [] - ln []). Then we take out the country's average growth rate to yield a demeaned output growth rate of [g.sub.Y,it] [equivalent to] [DELTA][] - 1/[T.sub.i][[SIGMA].sub.t][DELTA][Y.sub.i]. A similar method is applied to demean consumption and investment growth. We demean to isolate fluctuations at the business cycle frequency and to control for differences in country-specific average growth rates.

We then contrast the distributions of growth rates during unrest ([]|[] = 1) against moments during quiet times of no unrest ([]|[] = 0) in Figure 2. The left column of Figure 2 displays smoothed kernel estimates of the empirical probability density functions for the growth rates of output, consumption, and investment, and the right column displays the corresponding empirical cumulative distribution functions. The probability density functions are estimated by Epanechnikov kernels with a bandwidth of 2 percentage points for output and consumption, and 4 percentage points for investment. The figures suggest that the distributions of the growth rates are more negatively skewed during unrest episodes.

To have numerical comparisons, Table 1 displays the means, standard deviations, skewnesses, and kurtoses of (country-demeaned) output growth, consumption growth, and investment growth during and outside of unrest episodes. All confidence intervals are bootstrapped with 500 replications and are reported at the 95 percent level. The first two columns report the estimated moments. The third column reports the difference in the estimated moments, along with the p-value for a test of the null hypothesis that there is no difference between the corresponding moments. The fourth column reports the ratio of the estimated standard deviations, along with the p-value for the Levene test of the equality of variances. The fifth column reports the p-value for the Kolmogorov-Smirnov test of whether the two distributions of shocks (under unrest and no unrest) are the same.

Table 1 shows that a period of unrest is associated with significant losses in growth. The per-year loss in output growth (relative to periods without unrest) is 1.92 percent, statistically significant at the 1 percent level. This estimated per-year loss is nontrivial, especially given that unrest is persistent once started. The estimated cumulative loss is relatively substantial at 11.50 percent of the base (pre-onset) year's output. (7) The annual loss in consumption growth is 1.22 percent, which is smaller than that of output growth. At the same time, investment growth losses are larger than output growth, at 3.96 percent. In cumulative terms, consumption and investment losses amount to 7.31 percent and 23.71 percent, respectively. Note that this ordering of the loss in investment, output, and consumption is consistent with the permanent income hypothesis, which predicts that investment is more sensitive to shocks than output, which is in turn more sensitive than consumption.

Furthermore, Table 1 shows that the standard deviations of the growth rates of output, consumption, and investment substantially increase during unrest episodes. The fourth column of Table 1 displays the ratio of standard deviations. We can see that the standard deviation of output growth is 53 percent larger in unrest, and the standard deviations of consumption and investment growth are 17 percent and 35 percent larger, respectively. The column also reports the p-values of Levene's test of equality of variances between various forms of unrest against the baseline of no unrest. The p-values show that all of these increases are highly statistically significant: well below 0.01 for all three.

Table 1 also shows that both output and consumption growth becomes more negatively skewed during unrest. The difference in the skewness between unrest and no unrest is -1.57 for output growth and -3.36 for consumption growth. The bootstrapped p-value for the hypothesis that the difference in skewness is equal to zero is 0.15 for output growth and 0.10 for consumption growth. While it is generally difficult to estimate higher-order moments of relatively infrequent events with great confidence, we believe that these differences in skewness are economically significant. The greater variance and larger left tail of many distributions are also visually discernible in Figure 2. (8) This discernible mass on the left tail corresponds to a continuous range from moderately to extremely bad outcomes. The difference between a period of unrest and a period with no unrest then is not the increased probability of a single disaster but an increase in the probability of a whole range of bad outcomes.

Finally, as the fifth column of Table 1 shows, under the Kolmogorov-Smirnov test, we can reject the hypothesis that the two distributions of shocks (under unrest and under no unrest) are the same, as the associated p-value is zero for each series (output growth, consumption growth, or investment growth).

We summarize our results in the following stylized fact: Fact: Episodes of mass political unrest are associated with statistically and economically significant economic costs: the distributions of output, investment, and consumption growth during unrest have lower means and higher variances than the distributions in periods of no unrest. In addition, the distributions of output and consumption growth are more negatively skewed during unrest.

One potential mechanism that could explain the increased volatility and negative skewness in economic activities is that unrest is associated with substantial increases in the probability of institutional disruptions. In Appendix A.3, we document that the probabilities of large political and government changes, including major changes in polity and coups, substantially increase during unrest episodes. Large political changes are often associated with significant changes in legal and economic institutions, such as the protection of property and investment, which are key determinants of investment and growth (Acemoglu and Robinson 2005; and Acemoglu et al. 2014). Therefore, unrest episodes can increase the probability and severity of economic disasters.



How much of observed declines in average output, consumption, and investment growth during unrest, as reported in the previous section, can be attributed to volatility and skewness shocks? To answer this question, we augment a standard small open economy with a regime-switching process for the volatility and skewness of TFP. We calibrate the regime-switching process to moments that were estimated from data in the previous section.

Consider a canonical small open economy model with a representative household. Domestic firms competitively produce a numeraire good [Y.sub.t] using capital [K.sub.t-1] and labor [H.sub.t], subject to TFP [[zeta].sub.t]:

[Y.sub.t] = [[zeta].sub.t][K.sup.[alpha].sub.t-1][([H.sub.t]).sup.1-[alpha]].

These firms take factor prices [R.sub.t] and [W.sub.t] as given. Their first-order conditions on their optimal choices of capital and labor equate these factor prices with the corresponding marginal products in production:

[W.sub.t] = (1 - [alpha])[zeta.sub.t][K.sup.[alpha].sub.t-1][H.sup.-[alpha].sub.t].

[R.sub.t] = [alpha][[zeta].sub.t][K.sup.[alpha].sub.t-1][H.sup.1-[alpha].sub.t].

Unrest shock. We introduce a regime-switching process. Let [u.sub.t] be an exogenous two-state Markov process, with [u.sub.t] = 1 representing the country being in unrest in period t and [u.sub.t] = 0 representing no unrest, or a quiet time, in period t. Transitional probabilities are calibrated to match the probability of unrest onset and the persistence of unrest observed in data.

To model how unrest affects economic activities in the most tractable way, we assume that unrest affects the TFP process. Intuitively, as unrest episodes are associated with significant economic and political instability, they will affect the productivity of many economic sectors by, for instance, affecting the efficiency of resource allocation (Acemoglu et al. 2014). Such effects can be captured in a reduced form by a wedge to TFP, as in Chari et al. (2007).

Remark. Recall that our goal is to analyze the extent to which the shocks to higher-order moments of aggregate macroeconomic variables that we observe during unrest can explain the observed average losses in output, consumption, and investment growth. To conduct this analysis in the simplest and clearest possible way, we assume that unrest is a shock only to higher-order moments of the TFP process and not to the first moment. Obviously, this is a simplifying assumption and will likely lead to underestimations of the economic impacts of unrest. The model can be extended to allow for the possibility that unrest affects the first moment as well, but this will complicate the analysis. We will show that, even without an immediate associated fall in average productivity, a higher-order moment shock is enough to generate large changes in macroeconomic aggregates in line with the data.

Speciically, assume that TFP [[zeta].sub.t] consists of a growth component [([g.sup.t]).sup.1-[alpha]] and a level component [A.sub.t]:

[[zeta].sub.t] = [([g.sup.t]).sup.1-[alpha]][A.sub.t],

where, for numerical simplicity, we have assumed that growth rate is a constant g. However, level component [A.sub.t] follows an autoregressive process with autoregressive parameter [rho] and i.i.d. shocks [[epsilon].sub.t]:

ln [A.sub.t] = [rho] ln [A.sub.t1] + [[epsilon].sub.t].

The stochastic process for [[epsilon].sub.t] depends on whether the economy is currently experiencing unrest. While in unrest ([u.sub.t] = 1), shock [[epsilon].sub.t] is distributed Normal Inverse Gaussian with mean 0, standard deviation [[sigma].sub.u], skewness [s.sub.u], and kurtosis [k.sub.u]. While not in unrest ([u.sub.t] = 0), shock [[epsilon].sub.t] is distributed Normal Inverse Gaussian with mean 0, standard deviation [[sigma].sub.q], skewness [s.sub.q], and kurtosis [k.sub.q]. The Normal Inverse Gaussian distribution has been used in the finance literature to model skewed distributions with fat tails (e.g. Barndorf-Nielsen 1997; Andersson 2001; and Mencia and Sentana 2012). The fact that the mean of [[epsilon].sub.t] is the same whether [u.sub.t] = 0 or [u.sub.t] = 1 reflects the assumption that unrest only affects higher-order moments of TFP. (9)

Preferences: As is now standard in the macro-finance literature (e.g., Gourio 2012; and Colacito and Croce 2013), we assume the representative household has recursive preferences as in Epstein and Zin (1989). These preferences allow us to distinguish between the intertemporal elasticity of substitution and risk aversion (captured by [zeta] and [gamma] below). Moreover, these preferences nest the standard expected utility with constant relative risk aversion (CRRA) as a special case.

Let [C.sub.t] denote household consumption in period t, and let [[??].sub.t] [equivalent to] [C.sub.t] - [[theta][omega].sup.-1][g.sub.t-1][H.sup.w.sub.t] denote labor-adjusted consumption, where [theta] and [omega] are preference parameters. Then, we follow the sign convention of Rudebusch and Swanson (2012) and define the representative household's preferences as:

[mathematical expression not reproducible] (1)

This convention ensures that the value function and the instantaneous payoff have the same sign.

Households supply capital and labor to the domestic firms, consume domestic goods, invest subject to an adjustment cost in capital, and trade noncontingent bonds in the international credit market:

[mathematical expression not reproducible]

subject to:

[mathematical expression not reproducible]

We assume that the interest rate households borrow at is a function of the aggregate stock of debt [D.sub.t]:

[mathematical expression not reproducible]

where [r.sub.t] is the interest rate, r* is a constant representing the world's risk-free interest rate, and d and [psi] are exogenous constants. This debt-elastic interest rate is a standard assumption to ensure that the equilibrium is stationary (e.g., Schmitt-Grohe and Uribe 2003).

Finally, a recursive equilibrium is defined as a set of policy functions for [C.sub.t], [V.sub.t], [K.sub.t], [D.sub.t], [Y.sub.t], [r.sub.t], [I.sub.t], [H.sub.t], [W.sub.t], and [R.sub.t] as functions of [K.sub.t-1], [D.sub.t-1], [A.sub.t], and [u.sub.t] such that all agent expectations are rational and the optimality conditions, constraints, and laws of motion described above hold.

Solution Method

One way to derive moments of output, consumption, and investment growth from the model is to simulate a very long time series in which the country transitions into and out of unrest with the same probabilities as in the data. But since unrest is rare, we would need an extraordinarily long simulated time series to reduce the Monte Carlo noise around our estimates of those higher-order moments. Instead, we adapt the pruning method from Andreasen et al. (2017) to get closed-form solutions for the paths of conditional moments of endogenous variables, the GIRF. We first describe how we calculate a GIRF and then how we use the GIRF to compare the model against the data. All details on the computational strategy, from approximation to pruning and the GIRF, are given in the Appendix.

We define the GIRF as follows. Let [y.sub.t] denote the log-deviation of output [Y.sub.t] from its steady-state value. Then [DELTA][y.sub.t] is the growth rate of output [Y.sub.t]. Let [X.sub.t] denote a vector of the first three powers of the growth rates of output, consumption, and investment:

[X.sub.t] [equivalent to] ([DELTA][y.sub.t], [DELTA][i.sub.t], [DELTA][c.sub.t], [([DELTA][y.sub.t]).sup.2], [([DELTA][i.sub.t]).sup.2], [([DELTA][c.sub.t]).sup.2], [([DELTA][y.sub.t]).sup.3], [([DELTA][i.sub.t]).sup.3], [([DELTA][c.sub.t]).sup.3]).

The GIRF is the evolution over time of the difference of conditional expectations of [X.sub.t] between two conditioning sets, differing with respect to two given time series of realizations of unrest, u = {[u.sub.t], -[varies] < t < [varies]} and [mathematical expression not reproducible]:

[mathematical expression not reproducible]

The first path, u, represents a country that starts with no unrest and then enters into unrest at t = 1 and stays there. That is, [u.sub.t] = 0 [for all]l [less than or equal to] 0 and [u.sub.t] = 1 [for all]l [greater than or equal to] 1. The second counterfactual path, [mathematical expression not reproducible], is one where the country never enters unrest: [mathematical expression not reproducible].

Remark. The GIRF is useful for our purposes for several reasons. First, we want to calculate the moments that would be uncovered from a simulation. The conditional expectations in the GIRF allow us to consider the effects of shocks over the course of the GIRF. This is important, since under a nonlinear approximation to the policy function, the presence of shocks will cause the ergodic moments of all variables to differ from those in the absence of shocks. Second, since [X.sub.t] contains powers and products of endogenous variables, we can find paths not just for conditional means, but also for conditional variances and skewnesses of the endogenous variables of interest given the paths for the components of [X.sub.t]. Moreover, the GIRF allows us to avoid measurement error, which is a problem for estimating higher-order moments of simulated series from a finite simulation length. While Andreasen et al. (2017) rely on SMM for higher-order moments, we use the computer algebra software Mathematica to calculate GIRFs for these moments symbolically, term by term, and avoid Monte Carlo error.

The GIRF provides the conditional moments in the first year of an unrest episode, the second year, and so on. The moments from the data presented in the previous section are weighted averages over the years in observed unrest episodes because years that are closer to the beginning of an episode are more likely observed than years that are many years after the beginning of an episode. If p = Pr([U.sub.t]|[U.sub.t-1] = 1), then the probability of a given observed year of unrest being the nth year of unrest (n [greater than or equal to] 1) within its respective episode is (1-p)[p.sup.n-1]. Thus, to construct the single value for average value of X on unrest, we take a weighted average of a GIRF where a country enters into unrest and stays there but with smaller and smaller weight given to later periods of unrest. That is, we calculate [mathematical expression not reproducible].


First, we calibrate the model's basic parameters using standard values from the small open economy literature. These numbers are listed in the top panel of Table 2. (10) We allow the values for Epstein-Zin preference parameters to vary within the standard range of values of the literature, surveyed in Table 3. (11)

Second, we calibrate parameters for the unrest process and the higher-order moments of TFP innovation [[epsilon].sub.t] to estimated moments from our empirical analysis in Section 1. Care must be paid to the calibration of the higher-order moments of TFP, both in unrest and in quiet times. The parameters chosen in the model govern the exogenous TFP process, but they are chosen to match the moments of endogenous quantities. It is relatively straightforward (as one could even rely on closed-form solutions) to choose the volatility of a shock process given a desired volatility of an endogenous quantity, such as output growth under a log-linear approximation to equilibrium. However, it is much less straightforward to choose higher-order moments of a shock process to match higher-order moments of a nonlinear approximation of the law of motion for an endogenous variable. Therefore, the parameters [[sigma].sub.q], [[sigma].sub.u], [s.sub.q], and [s.sub.u] are chosen so that the ergodic standard deviation and skewness of output growth, and the average generalized impulse responses of the standard deviation and skewness of output growth, match those in the data. (12)


Model's performance relative to data. We compare the average loss in output, investment, and consumption growth from the GIRF [[infinity].summation over (t=1)](1 - p)[p.sup.t-1][GIRF.sup.2]([DELTA][y.sub.t]) to the corresponding observed average loss in growth as documented in Section 1. Table 4 reports the percentage of observed growth loss that can be explained by the calibrated model. The overall effect is an endogenous response of endogenous variables to an unrest shock that increases the volatility and negative skewness of TFP shocks, with an interplay of capital adjustment costs and preferences over the time resolution of risk. In each panel, we report the percentage obtained by using the first-, second-, and third-order approximations of the solution to the model. Note that by construction, the percentage explained using a first-order approximation is zero, as we assume that unrest does not affect the first moment of TFP shocks. The columns report the results with different preference parameters.

Table 4 shows that, under the baseline specification (the first column), the model explains 21 percent of the average output growth loss, 45 percent of the average consumption growth loss, and 51 percent of the average investment growth loss. This amounts to an output growth loss of 0.40 percent per year, a consumption growth loss of 0.55 percent per year, and an investment growth loss of 2.01 percent per year. In cumulative terms over the average episode duration, this is an output growth loss of 2.41 percent, a consumption growth loss of 3.28 percent, and an investment growth loss of 12.09 percent.

The second column of Table 4 shows that, not surprisingly, the model can explain more with a larger coefficient of risk aversion ([gamma] = 20 instead of [gamma] = 10). There, the fractions of growth losses explained increase to 62 percent for output, 128 percent for consumption and 148 percent for investment (thus this calibration "overexplains" the losses in consumption and investment). On the other hand, when we shut down Epstein-Zin preferences and use a lower coefficient of risk aversion (the third column), the fractions of growth losses explained decrease to 9 percent, 11 percent, and 23 percent for output, consumption, and investment, respectively.

It is not surprising that the model cannot fully explain the observed losses, since we assume that unrest only affects higher-order moments of TFP shocks, and not the first-order moment, thus abstracting away from factors such as reallocation of resources between sectors of the economy that may directly affect the average productivity. (13)

However, the table shows that shocks to the higher-order moments of TFP alone can still explain a substantial fraction of the observed losses, especially in investment. Even without Epstein-Zin preferences and with a relatively low risk-aversion index, the model can still explain around a fourth of the observed loss in investment growth. Intuitively, in the model, when risk increases (either through the second-order or third-order moment of TFP), agents in the country shift away from domestic capital and into the internationally traded asset. This mechanism explains the drop in investment.

Role of negative skewness. One of our main findings is that negative skewness shocks play quantitatively important roles in driving business cycles. To see this, in rows labeled "second order" in Table 4, we show the fractions of observed losses explained under each calibration, but using an approximation of the solution of the model only to the second order, and thus efectively shutting down the endogenous response to the shock to the skewness of TFP. As the baseline column shows, the reaction to skewness is substantial: the fractions of average losses explained in the third-order rows are roughly doubling those explained in the second-order rows. Diferences of comparable magnitudes are also found in the two other calibration columns.

Why does skewness matter? Intuitively, agents in our model dislike negative skewness. To see this, let [mathematical expression not reproducible], the aggregate of utility from consumption and labor to the household. By the definition of household preferences, [mathematical expression not reproducible]. Let [v.sub.t] [equivalent to] [V.sup.1-[zeta].sub.t] so that when [gamma] = [zeta] and thus Epstein-Zin preferences reduce to expected utility preferences, [v.sub.t] is the usual definition of the value function for the household: [mathematical expression not reproducible]. The third-order Taylor approximation for [v.sub.t] around [v.sub.t+1] = [micro] [equivalent to] [E.sub.t][[v.sub.t+1]] is:

[mathematical expression not reproducible] (2)

The first three terms of the continuation payoff are well-known in the literature on Epstein-Zin preferences (e.g., Colacito et al. 2013). The first term is current utility. The second is the same discounted continuation payoff that appears in non-Epstein-Zin expected utility preferences. The third term is a "correction" to expected utility that penalizes future variance of the value function as long as [gamma] > [zeta]. (14) The fourth term is novel to a third-order approximation. Under the same assumption that [gamma] > [zeta] and [zeta] < 1, this term rewards positive skewness of the future value function and penalizes negative skewness. As [gamma] increases, the penalties for both volatility and negative skewness increase.

The term [Skew.sub.t] [[v.sub.t+1]] [Var.sub.t] [[[v.sub.t+1]].sup.3/2] is equal to [E.sub.t][[([v.sub.t+1]-[micro]).sup.3]], the third central moment of the value function. It shows that, for a given amount of skewness, the size of the third central moment increases in the variance. This is why skewness and variance are complementary in giving rise to precautionary motives in equilibrium.

Expression (2) is another way to see how these higher-order moments relate to a disaster risk. A disaster is an outcome on the far left tail. If variance increases, extreme events on both tails become more likely. If in addition skewness becomes more negative, the events far out on the lower tail specifically become more likely. Though we do not calculate a fourth-order approximation to this model, one can easily show that the next term in the above expansion would penalize the fourth central moment of the value function. An increase in the fourth-order moment, like an increase in negative skewness for a given second-order moment, also makes outcomes on the tails more likely. Therefore, by taking a higher-order approximation to the value function and by considering shock distributions with fat and skewed tails, we can recover some of the effects of what has been explored in the rare disaster literature.

Comparison with other studies. How do the results in Table 4 compare with other studies in the literature on the macroeconomic effects of risk? It is well-known that increases in second-order moments lead to economic slowdowns, though the range of models in the literature is wide and none are exactly comparable with the model in this paper in terms of modeling assumptions or forcing processes. For example, while using a very different model (a closed economy with heterogeneous firms, subject to a transitory shock to the second-order moment of a composite of technology and demand, on the monthly frequency), Bloom (2009) obtains effects of risk that are of the same order of magnitude as here, i.e., doubling the standard deviation of the forcing process leads to a decline in the level of output by 2 percentage points within the first six months. For the canonical small open economy model considered here, Fernandez-Villaverde et al. (2011) find that a transitory one-standard-deviation shock to second-order moment of innovations to the global interest rate (the interest rate that households in the small open economy pay on their international debt) can lead to declines in output levels in Argentina of 1.16 percentage points below steady state after sixteen quarters, or an average output growth loss of 0.29 percentage points per year, which is about 73 percent of what our baseline model predicts. Just as in Gourio's (2012) experiment with a transitory increase in the disaster probability, in our model, investment experiences the most significant decline and output contracts by a few percentage points. However, in that model, a disaster also entails some destruction of capital, so it is difficult to directly compare the two sets of numerical results.

Welfare. Finally, we evaluate the welfare loss due to the shock to the distribution of TFP. The change in the value function [V.sub.t] experienced in the first period of an unrest episode corresponds to the welfare loss from facing the more negatively skewed distribution of TFP. The loss can be evaluated by considering the following counterfactual scenario: suppose that household consumption is dictated by a social planner who ensures that households enjoy labor-adjusted consumption [mathematical expression not reproducible] (the steady-state level of labor-adjusted consumption in the model) during each period the economy is not in unrest and [mathematical expression not reproducible], where [[DELTA].sub.C] < 1, during each period the economy is in unrest. Suppose additionally that unrest follows the same stochastic switching process as in the data and the model but there are no other sources of uncertainty to the households. The value function of the household in this scenario takes on two values: [mathematical expression not reproducible] while not in unrest, and [mathematical expression not reproducible], where [[DELTA].sub.V] < 1, while in unrest. The value function takes the following form:

[mathematical expression not reproducible] (3)

The log-linearization of the above:

[mathematical expression not reproducible] (4)

For a given [mathematical expression not reproducible], we can calculate the change in labor-adjusted consumption [mathematical expression not reproducible] that would give rise to a fall of [mathematical expression not reproducible] in the value function below its steady-state value for each period spent in unrest. We take [mathematical expression not reproducible] as calculated from our GIRF.

Our estimates imply a [mathematical expression not reproducible] equal to -6.1 percent. In other words, the welfare loss due to increased volatility and skewness during unrest is equal to the welfare loss if consumption were 6.1 percent lower than its steady-state value in each period of unrest. How does this number compare with those in other studies? Lucas (1987) shows that eliminating all business cycle fluctuations for a representative agent with expected-utility preferences corresponds to 0.1 percent to 0.5 percent of steady-state consumption. Dolmas (1998) finds that the same exercise under Epstein-Zin preferences yields 2 percent to 20 percent of steady-state consumption, depending on the degree of risk aversion.


We estimate shocks to the volatility and skewness of business cycles by exploiting the uncertainty associated with episodes of political unrest. A small open economy real business cycle model calibrated to the estimated moments from data shows that higher-order moment shocks, especially increased negative skewness, play important roles in explaining the observed average decline in economic activities. In short, the paper demonstrates the quantitative importance of time-varying skewness of shocks in the context of a small open economy real business cycle model. Our paper makes several contributions to different threads of the macroeconomic literature. In the context of real business cycle and DSGE models, the mapping from the higher-order moments of exogenous processes to moments of endogenous variables, such as the mapping studied in this paper, is relatively underexplored. While the literature has deployed a number of mechanisms (e.g., adjustment costs on investment, debt-elastic interest rates, habit in consumption, and interest rate smoothing; see Smets and Wouters 2007) to help log-linearized models better replicate the first-order and second-order moments of observed time series, it is less clear how these mechanisms affect the model's ability to match third-order moments as well. Our paper suggests it may be important to know more about the endogenous mechanisms that help or hinder matching higher-order moments of models, given that these moments could be important for the consequences of aggregate risk. Additionally, our method of accurately calculating the GIRF of third-order moments may help future researchers analyze the dynamics of higher-order moments of macroeconomic aggregates in DSGE models while avoiding Monte Carlo error.


A.1 Details of NAVCO Unrest Data

NAVCO provides detailed information on 250 nonviolent and violent mass political campaigns between 1945 and 2006. These campaigns constitute a "consensus population" of all known cases satisfying the following conditions. Each episode is a series of observable (i.e., tactics used are overt and documented), continuous (distinguishing from one-of events or revolts) mass tactics or events that mobilize nonstate actors in pursuit of a political objective. The NAVCO dataset also provides, among other information, the country, the main participating groups, the documented objective of the movement in each year of the campaign, the presence of violence in each year of the campaign, and the degree to which the movement was successful at achieving the documented objective. We focus on episodes whose objectives belong to one of the following categories:

(0) Regime change indicates a goal of "overthrowing the state or substantially altering state institutions to the point that it would cause a de facto shift in the regime's hold on power."

(1) Significant institution reform indicates a goal of "changing fundamental political structures to alleviate injustices or grant additional rights."

(2) Policy change indicates a goal of "changes in government policy that fall short of changes in the fundamental political structures, including changes in a state's foreign policy."

For a complete listing of NAVCO unrest episodes, see the Online Appendix C.

A.2 Estimates of Onset and Continuation Probabilities

We investigate how likely unrest is to start and how persistent it is once it starts. We establish that unrest is rare but persistent. These facts are important for understanding the economic consequences of higher-order shocks to business cycles.

Let a dummy variable [] take the value of one during episodes of unrest and zero during years with no unrest, where i denotes a country and t denotes a year. We estimate both the probability of unrest onset (i.e., the probability of unrest conditional on no unrest the previous year) and the probability of unrest continuation (i.e., the probability of unrest conditional on there being unrest in the previous year). To assess whether the probability of unrest is a function of other observable characteristics of a country, we estimate two probit models, one for onset and one for continuation. Each probit predicts [] = 1 as a function of a constant and a vector [] of control variables, including lagged real GDP growth minus the country-specific average growth rate [mathematical expression not reproducible], religious, ethnic, and linguistic fractionalization (all on a scale of 0 to 1), and income inequality (measured with the Gini coefficient). To control for region-specific factors that might influence the overall probability of a given country experiencing unrest, we include a term [[gamma].sub.Region(i)] as a region-fixed effect. (15) We do not include country-fixed effects because this would effectively exclude any country from our sample that has never experienced unrest. Instead, we want to include all countries in our sample to exploit not just variation within countries but between them as well. The fact that many countries never experience unrest is informative to estimating the probability of onset. The two probit regressions are:

Pr([]|[] = 0) = [PHI] ([[gamma].sub.Z0][] + [[gamma].sub.0] + [[gamma].sub.0,Region(i)]) (onset) (5)

Pr([]|[] = 1) = [PHI] ([[gamma].sub.Z1][] + [[gamma].sub.1] + [[gamma].sub.1,Region(i)]) (continuation) (6)

where [PHI] is the cumulative distribution function of the standard normal distribution.

Our baseline estimations, reported in Table 5, indicate that the onset of unrest is rare: the estimated onset probability is 1.4 percent per year. However, once it starts, unrest tends to last for several years: the estimated continuation probability in Table 6 is 83.3 percent per year. This continuation probability implies that the average duration of unrest episodes is 5.99 (= 1/1-0.833).

In summary, we find that the onset of unrest is rare. But once started, unrest is persistent, leading to relatively lengthy episodes.

A.3 Political Risks Associated with Unrest

We document that the probability of large political changes increases significantly in each year of unrest. To the extent that any large political change entails at least a temporary disruption of the economy, an increase in the probability of disruptive events might help make sense of the increase in the left tail of the distributions of output and consumption growth documented in the next section. We estimate a series of probit regressions to predict a set of political disruptions: (1) coups, (2) positive changes in the Polity index, (3) negative changes in the Polity index, (4) large positive changes in the Polity index (greater than five points), and (5) large negative changes in the Polity index (greater than five points). (16) Each probit regression is specified as in equation (6), as a function of a constant, an indicator for current unrest, the difference between lagged real GDP growth and a country-specific average real GDP growth, and the interaction between current unrest and lagged real GDP growth. Let [] be an indicator for one of the political disruptions. We estimate:

Pr([]) = [PHI] ([[gamma].sub.U][] + [[gamma].sub.Z][] + [[gamma].sub.zu][][] + [[gamma].sub.0]) (7)

There are a few differences between this specification and the specification of unrest onset and continuation in equation (6). First, we estimate one probit for each political disruption []. Second, in equation (6), we estimate the probits conditional on the presence of lagged unrest and the absence of lagged unrest separately. Here, we estimate one probit including both unrest and its interactions with the controls in one step. We do this to test hypotheses that the probability of each political disruption is significantly different in the presence and absence of unrest. Third, for simplicity, we include in the vector of controls [] just one control: the difference between lagged output growth and country-specific average output growth. We find that unrest is associated with increases in the probability of all kinds of political changes.


B.1 Derivation of the Household Problem

First, we pose the problem in recursive form

[mathematical expression not reproducible]

The associated first-order conditions and envelope condition are:

[mathematical expression not reproducible].

These lead to:

[mathematical expression not reproducible]

B.2 Full Set of Equilibrium Conditions

The equilibrium conditions are (with additional variables introduced for convenience):

[mathematical expression not reproducible] (8)

[mathematical expression not reproducible] (9)

[mathematical expression not reproducible] (10)

[W.sub.t] = [theta][Z.sub.t-1][H.sup.[[omega]-1.sub.t]] (11)

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

[Y.sub.t] = [A.sub.t][K.sup.[alpha].sub.t-1][([Z.sub.t][H.sub.t]).sup.1-[alpha]] (14)

[W.sub.t] = (1-[alpha])[A.sub.t][K.sup.[alpha].sub.t-1][Z.sup.1-[alpha].sub.t][H.sup.[alpha].sub.t] (15)

[R.sub.t] = [alpha][A.sub.t][K.sup.[alpha]-1.sub.t-1][Z.sup.1-[alpha].sub.t][H.sup.1-[alpha].sub.t] (16)

[Y.sub.t] + [D.sub.t]/1+[r.sub.t] = [D.sub.t-1] + [C.sub.t] + [l.sub.t] + [phi]/2[([K.sub.l]/[K.sub.t-1]-[g.sub.t]).sup.2] [K.sub.t-1] (17)

[I.sub.t] = [K.sub.t] - (1 - [delta])[K.sub.t-1] (18)

[mathematical expression not reproducible] (19)

The equilibrium conditions, scaled ([c.sub.t] = [C.sub.t]/[Z.sub.t-1], [mathematical expression not reproducible], [h.sub.t] = [H.sub.t], [w.sub.t] = [W.sub.t]/[Z.sub.t-1], [v.sub.t] = [V.sub.t]/[Z.sub.t-1], [mathematical expression not reproducible], [y.sub.t] = [Y.sub.t]/[Z.sub.t-1], [k.sub.t] = [K.sub.t]/[Z.sub.t], [a.sub.t] = [A.sub.t]) and simplified:

[mathematical expression not reproducible] (20)

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

[w.sub.t] = [theta][h.sup.w-1.sub.t] (23)

[[kappa].sub.t] = [k.sub.t]/[k.sub.t-1]-1 (24)

[mathematical expression not reproducible] (25)

[mathematical expression not reproducible] (26)

[y.sub.t] = [a.sub.t][k.sup.[alpha].sub.t-1][([g.sub.t][h.sub.t]).sup.1-[alpha]] (27)

[w.sub.t][h.sub.t] = (1 - [alpha])[y.sub.t] (28)

[R.sub.t][k.sub.t-1] = [alpha][y.sub.t] (29)

[mathematical expression not reproducible] (30)

[i.sub.t] = [k.sub.t][g.sub.t] - (1 - [delta])[k.sub.t-1] (31)

[r.sub.t] = r* + [psi]([e.sup.([d.sub.t]-[bar.d])] - 1) (32)

log([a.sub.t+1]) = [rho]log([a.sub.t]) + [eta]([u.sub.t][[sigma].sub.u] + (1 - [u.sub.t])[[sigma].sub.q])[c.sub.t+1] (33)

log([g.sub.t]) = log([g.sub.q]) + [eta][u.sub.t] log([g.sub.u]) (34)

[[epsilon].sub.t+1] ~ i.i.d.N(0, 1) (35)

[u.sub.t+1] ~ Markov, 0 or 1 with constant transition matrix. (36)

Steady state at [eta] = 0:

[mathematical expression not reproducible] (37) [mathematical expression not reproducible] (38) [mathematical expression not reproducible] (39)

w = 0[h.sup.w-1] (40)

1 = [beta][g.sup.-[zeta]](1 + r) (41)

r = R - [delta] (42)

y = a[k.sup.[alpha]][(gh).sup.1-[alpha]] (43)

wh = (1 - [alpha])y (44)

Rk = [alpha]y (45)

[mathematical expression not reproducible] (46)

i = k(g - 1 + [delta]) (47)

r = r* (48)

a = 1 (49)

[kappa] = 0 (50)

g = [g.sub.q]. (51)

B.3 Notes on Solution Method and GIRFs

To approximate the solution to equilibrium of our model, we use a higher-order perturbation method with the pruning algorithm of Andreasen et al. (2017). Because we calculate GIRFs for higher-order moments of endogenous variables, deriving analytic representations for the GIRFs, as Andreasen et al. (2017) do, would be extremely algebraically tedious. Instead, we rely on the computer algebra software Mathematica to compute these higher-order moments. This section describes our computational strategy.

The equilibrium conditions can be stated in the following form:

0 = [E.sub.t][F([y.sub.t+1], [y.sub.t], [x.sub.t+1], [x.sub.t], [u.sub.t+1], [u.sub.t])]. (52)

The vector of equations F includes all optimality conditions, constraints, and the law of motion for the exogenous process. The vector [y.sub.t] is the vector of control variables: [mathematical expression not reproducible]. The perturbation parameter, [eta], is 1 in the model of interest but set to 0 at the point of approximation. The vector [x.sub.t] is the vector of continuous states, including the perturbation parameter (17): [log([k.sub.t-1]), [d.sub.t-1], log([a.sub.t]), [eta]]. [u.sub.t] is the indicator for unrest, which can only take the values 0 and 1.

The solution to this model is a set of policy functions of the following form, where [mathematical expression not reproducible] and the two shocks [[epsilon].sup.u.sub.t+1] and [[epsilon].sup.q.sub.t+1] follow two i.i.d. Normal Inverse Gaussian processes, described in the text:

[y.sub.t] = g([x.sub.t], [u.sub.t]) (53)

[x.sub.t+1] = h([x.sub.t], [u.sub.t]) + [eta]S([u.sub.t+1])[[epsilon].sub.t+1]. (54)

More specifically, for the state vector,

[mathematical expression not reproducible] (55)

At the point of approximation, the system is at a nonstochastic steady state in [x.sub.t] and [y.sub.t]: [x.sub.t] = [] = [log([]), [], 0, 0] and [y.sub.t] = []. Since the unrest and no-unrest states are completely symmetric at [eta] = 0 by construction, the process [u.sub.t] is irrelevant for the steady states of [x.sub.t] and [y.sub.t]. Therefore, the following is true for all values of [u.sub.t+1] and [u.sub.t]:

0 = F([], [], [], [], [u.sub.t+1], [u.sub.t]).(56)

We use a standard third-order perturbation method (e.g., Judd 1996) to construct Taylor series approximations to h(x, 0), h(x, 1), g(x, 0), and g(x, 1). Those Taylor series approximations yield the coefficients [mathematical expression not reproducible], and [mathematical expression not reproducible], conformably reshaped:

h(x, 0) [approximately equal to] [h.sub.0x]x + 1/2[H.sub.0xx](x [cross product] x) + 1/6[H.sub.0xxx](x [cross product] x [cross product] x).(57)

This implies that the law of motion for [x.sub.t] and [y.sub.t] can be approximated to third order as:

[mathematical expression not reproducible] (58)

[mathematical expression not reproducible] (59)

However, it is well-known (e.g., in Kim et al. 2008; and Den Haan and De Wind 2012) that third-order approximations like the above can have undesirable statistical properties, such as explosive simulated paths and spurious steady states. Andreasen et al. (2017) extend Kim et al. (2008) and use a pruning algorithm to eliminate these undesirable properties. The second-order pruning algorithm separates simulated components of [x.sub.t] and [y.sub.t] into first-order components [x.sup.f.sub.t] and [y.sup.f.sub.t], second-order components [x.sup.s.sub.t] and [y.sup.s.sub.t], and third-order components [x.sup.r.sub.t] and [y.sup.r.sub.t]. The simulated quantities of interest are [x.sup.f.sub.t] + [x.sup.s.sub.t] + [x.sup.r.sub.t] and [y.sup.f.sub.t] + [y.sup.s.sub.t] + [y.sup.r.sub.t], and the components evolve linearly.

Let [mathematical expression not reproducible]. The constant vector [[0, 0, 0, 1].sup.t] reflects the fact that the law of motion for the perturbation parameter is simply [eta] = 1.

Following the approach in Andreasen et al. (2017), we have:

[mathematical expression not reproducible] (60)

[mathematical expression not reproducible] (61)

[mathematical expression not reproducible] (62)

[mathematical expression not reproducible] (63)

[mathematical expression not reproducible] (64)

[mathematical expression not reproducible] (65)

At this point, we deviate from the notation in Andreasen et al. (2017). Let

[mathematical expression not reproducible]

Expanding the above, we find that

[mathematical expression not reproducible]

Remember that [C.sub.t+1] is a function of [[epsilon].sub.t+1].

[mathematical expression not reproducible]


[mathematical expression not reproducible] (66)

[mathematical expression not reproducible] (67)

[mathematical expression not reproducible] (68)

[mathematical expression not reproducible]

Similarly, for controls [y.sub.t], we have [mathematical expression not reproducible], where [mathematical expression not reproducible].

In this paper. we are interested in the growth rates of the controls.

[mathematical expression not reproducible]

To calculate the average change in the first three moments of output, investment, and consumption growth during unrest, we use the concept of GIRF from Andreasen et al. (2017) and Koop et al. (1996). In particular, we calculate the unconditional moments of all endogenous variables for two fixed paths for the unrest process. The first path is for a country that starts with no unrest and then enters into unrest at t =1 and stays there. That is, [u.sub.t] = 0 [for all]t [less than or equal to] 0 and [u.sub.t] = 1 [for all]t [greater than or equal to] 1. Denote this path for [u.sub.t] as u. The second counterfactual path is one where the country never enters unrest: [u.sub.t] = 0 [for all]t. Denote this path for [u.sub.t] as [mathematical expression not reproducible]. Andreasen et al. (2017) condition on an initial value of the state vector [z.sub.0]. We instead focus on an unconditional expectation over the entire range of t to be able to arrive at a single path of moments for our exercise. The generalized IRF for the state variables [z.sub.t] is the difference, at each point in time t, of the unconditional mean of [z.sub.t] along the path u and the unconditional mean of [z.sub.t] along the path [mathematical expression not reproducible]:

[mathematical expression not reproducible] (69)

Andreasen et al. (2017) derive separate expressions for the evolution over time of the variances of controls. We take a different approach, which we find to be simpler, especially in dealing with third-order moments. We expand the set of objects we find a GIRF of from [DELTA][y.sub.t] to [mathematical expression not reproducible], so that we can compute one GIRF for all the moments of interest in one pass. For example, vec(Var([DELTA][y.sub.t])) = E[([DELTA][y.sub.t]) [cross product] ([DELTA][y.sub.t])] - E[[DELTA][y.sub.t]] [cross product] E[[DELTA][y.sub.t]], and the skewness of [DELTA][y.sub.t] is similarly a function of E[([DELTA][y.sub.t]) [cross product] ([DELTA][y.sub.t]) [cross product] ([DELTA][y.sub.t])]. Using the expression [mathematical expression not reproducible] and expanding the Kronecker products in [X.sub.t], we have matrices [mathematical expression not reproducible] and [mathematical expression not reproducible] such that

[mathematical expression not reproducible] (70)

where [mathematical expression not reproducible]

To calculate the law of motion for [Z.sub.t], we expand the Kronecker products in the definition of [Z.sub.t] using the law of motion [mathematical expression not reproducible] and arrive at the law of motion [mathematical expression not reproducible] for matrices [mathematical expression not reproducible] and [mathematical expression not reproducible].

For any [Z.sub.0], the independence of [e.sub.t+1] and [Z.sub.t] implies for states and controls (noting that [mathematical expression not reproducible] and similarly for [mathematical expression not reproducible] and [mathematical expression not reproducible]):

[mathematical expression not reproducible] (71)

[mathematical expression not reproducible] (72)

Let [Z.sub.0] be the fixed point of the law of motion for E[[Z.sub.t+1][Z.sub.0], u], conditional on t < 0, in other words, conditional on no unrest at time t or t + 1.

[mathematical expression not reproducible] (73)

From the elements of [Z.sub.0], we can calculate the ergodic means, variances, covariances, skewnesses, and sundry third moments of all elements of [z.sub.t] conditional on no unrest. This is the starting point of our GIRF. For t = 1, 2, 3..., use the laws of motion for [X.sub.t] and [Z.sub.t] to iterate forward, conditional on both the path u and the counterfactual path [mathematical expression not reproducible], and use those paths to calculate the GIRF for [X.sub.t]:

[mathematical expression not reproducible] (74)

This notation is very condensed. For example, [mathematical expression not reproducible] is a very large matrix, with very large polynomials containing terms with order as high as [[epsilon].sup.9.sub.t+1]. There are a large number of terms in every element of these matrices, even if expressed as Kronecker products; that is why we rely on the symbolic manipulation of Mathematica to expand these polynomials. Even after exploiting the very high degree of symmetry and redundant terms in [Z.sub.t], there are over 20,000 unique elements in that vector. Mathematica can handle these calculations very quickly, calculating the GIRFs for both states and controls in under two minutes.



Acemoglu, Daron, Tarek A. Hassan, and Ahmed Tahoun. 2014. "The Power of the Street: Evidence from Egypt's Arab Spring." Working Paper 20665. Cambridge, Mass.: National Bureau of Economic Research. (November).

Acemoglu, Daron, and James A. Robinson. 2005. Economic Origins of Dictatorship and Democracy. Cambridge: Cambridge University Press.

Alesina, Alberto, Arnaud Devleeschauwer, William Easterly, Sergio Kurlat, and Romain Wacziarg. 2003. "Fractionalization." Journal of Economic Growth 8 (June):155-94.

Andersson, Jonas. 2001. "On the Normal Inverse Gaussian Stochastic Volatility Model." Journal of Business & Economic Statistics 19 (January): 44-54.

Andreasen, Martin M. 2012. "On the Effects of Rare Disasters and Uncertainty Shocks for Risk Premia in Non-Linear DSGE Models." Review of Economic Dynamics 15 (July): 295-316.

Andreasen, Martin M., Jesus Fernandez-Villaverde, and Juan F. Rubio-Ramirez. 2017. "The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications." Review of Economic Studies 85 (January): 1-49.

Arellano, Cristina, Yan Bai, Patrick J. Kehoe. 2012. "Financial Frictions and Fluctuations in Volatility." Federal Reserve Bank of Minneapolis Research Department Staff Report 466 (Revised November 2018).

Barberis, Nicholas, and Ming Huang. 2008. "Stocks as Lotteries: The Implications of Probability Weighting for Security Prices." American Economic Review 98 (December): 2066-100.

Barndorf-Nielsen, Ole E. 1997. "Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling." Scandinavian Journal of Statistics 24 (March): 1-13.

Barro, Robert J. 2006. "Rare Disasters and Asset Markets in the Twentieth Century." Quarterly Journal of Economics 121 (August) :823-66.

Bloom, Nicholas. 2009. "The Impact of Uncertainty Shocks." Econometrica 77 (May): 623-85.

Caldara, Dario, Jesus Fernandez-Villaverde, Juan F. Rubio-Ramirez, and Wen Yao. 2012. "Computing DSGE Models with Recursive Preferences and Stochastic Volatility." Review of Economic Dynamics 15 (April): 188-206.

Chari, V. V., Patrick J. Kehoe, and Ellen R. McGrattan. 2007. "Business Cycle Accounting." Econometrica 75 (May): 781-836.

Chenoweth, Erica, and Orion A. Lewis. 2013. "Unpacking Nonviolent Campaigns: Introducing the NAVCO 2.0 Dataset." Journal of Peace Research 50 (May): 415-23.

Chenoweth, Eerica, and Maria J. Stephan. 2011. Why Civil Resistance Works: The Strategic Logic of Nonviolent Conflict. New York: Columbia University Press.

Christiano, Lawrence J., Roberto Motto, and Massimo Rostagno. 2014. "Risk Shocks." American Economic Review 104 (January): 27-65.

Colacito, Riccardo, Max Croce, Steven Ho, and Philip Howard. 2013. "BKK the EZ Way: An International Production Economy with Recursive Preferences." Working Paper (April).

Colacito, Riccardo, and Mariano M. Croce. 2013. "International Asset Pricing with Recursive Preferences." Journal of Finance 68 (December): 2651-686.

Colacito, Riccardo, Eric Ghysels, Jinghan Meng, and Wasin Siwasarit. 2015. "Skewness in Expected Macro Fundamentals and the Predictability of Equity Returns: Evidence and Theory." Working Paper (August).

Den Haan, Wouter J., and Joris De Wind. 2012. "Nonlinear and Stable Perturbation-Based Approximations." Journal of Economic Dynamics and Control 36 (October): 1477-497.

Dolmas, Jim. (1998). "Risk Preferences and the Welfare Cost of Business Cycles." Review of Economic Dynamics 1 (July): 646-76.

Edmond, Chris. 2013. "Information Manipulation, Coordination, and Regime Change." Review of Economic Studies 80 (October): 1422-458.

Epstein, Larry G., and Stanley E. Zin. 1989. "Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework." Econometrica 57 (July): 937-69.

Fernandez-Villaverde, Jesus, Pablo Guerron-Quintana, Juan F. Rubio-Ramirez, and Martin Uribe. 2011. "Risk Matters: The Real Effects of Volatility Shocks." American Economic Review 101 (October): 2530-561.

Feunou, Bruno, Mohammed R. Jahan-Parvar, and Cedric Okou. 2015. "Downside Variance Risk Premium." Bank of Canada Working Paper 2015-36 (October).

Gabaix, Xavier. 2012. "Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance." Quarterly Journal of Economics 127 (May): 645-700.

Garcia-Cicco, Javier, Roberto Pancrazi, and Martin Uribe. 2010. "Real Business Cycles in Emerging Countries?" American Economic Review 100 (December): 2510-531.

Gilchrist, Simon, Jae W. Sim, and Egon Zakrajsek. 2014. "Uncertainty, Financial Frictions, and Investment Dynamics." Working Paper 20038. Cambridge, Mass.: National Bureau of Economic Research. (April).

Gourio, Francois. 2012. "Disaster Risk and Business Cycles." American Economic Review 102 (October): 2734-766.

Gourio, Francois, Michael Siemer, and Adrien Verdelhan. 2013. "International Risk Cycles." Journal of International Economics 89 (March): 471-84.

Guvenen, Fatih, Serdar Ozkan, and Jae Song. 2014. "The Nature of Countercyclical Income Risk." Journal of Political Economy 122 (June): 621-60.

Judd, Kenneth L. 1996. "Approximation, Perturbation, and Projection Methods in Economic Analysis." In Handbook of Computational Economics Vol. 1, edited by H.M. Amman, D.A. Kendrick, and J. Rust. Amsterdam: North Holland, 509-85.

Justiniano, Alejandro, and Giorgio E. Primiceri. 2008. "The Time-Varying Volatility of Macroeconomic Fluctuations." American Economic Review 98 (June): 604-41.

Kim, Jinill, Sunghyun Kim, Ernst Schaumburg, and Christopher A. Sims. 2008. "Calculating and Using Second-Order Accurate Solutions of Discrete Time Dynamic Equilibrium Models." Journal of Economic Dynamics and Control 32 (November): 3397-414.

Koop, Gary, M. Hashem Pesaran, and Simon M. Potter. 1996. "Impulse Response Analysis in Nonlinear Multivariate Models." Journal of Econometrics 74 (September): 119-47.

Kuran, Timur. 1989. "Sparks and Prairie Fires: A Theory of Unanticipated Political Revolution." Public Choice 61 (April): 41-74.

Lucas, Robert E. 1987. Models of Business Cycles, Volume 26. Oxford: Basil Blackwell.

Marshall, Monty G., and Keith Jaggers. 2002. "Polity IV Project: Political Regime Characteristics and Transitions, 1800-2002."

Marshall, Monty G., and Donna Ramsey Marshall. 2011. "Coup D'etat Events, 1946-2012 Codebook." Center for Systemic Peace.

Mencia, Javier, and Enrique Sentana. 2012. "Distributional Tests in Multivariate Dynamic Models with Normal and Student-t Innovations." Review of Economics and Statistics 94 (February): 133-52.

Nakamura, Emi, Jon Steinsson, Robert Barro, and Jose Ursua. 2013. "Crises and Recoveries in an Empirical Model of Consumption Disasters." American Economic Journal: Macroeconomics 5 (July): 35-74.

Ranciere, Romain, Aaron Tornell, and Frank Westermann. 2008. "Systemic Crises and Growth." Quarterly Journal of Economics 123 (February): 359-406.

Richter, Alexander W., Nathaniel A. Throckmorton, and Todd B. Walker. 2014. "Accuracy, Speed and Robustness of Policy Function Iteration." Computational Economics 44 (December): 445-76.

Rudebusch, Glenn D., and Eric T. Swanson. 2012. "The Bond Premium in a DSGE Model with Long-Run Real and Nominal Risks." American Economic Journal: Macroeconomics 4 (January): 105-43.

Salgado, Sergio, Fatih Guvenen, and Nicholas Bloom. 2015. "Skewed Business Cycles." Working Paper (June).

Schmitt-Grohe, Stephanie, and Martin Uribe. 2003. "Closing Small Open Economy Models." Journal of International Economics 61 (October): 163-85.

Smets, Frank, and Rafael Wouters. 2007. "Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach." American Economic Review 97 (June): 586-606.

Vissing-Jorgensen, Annette, and Orazio P. Attanasio. 2003. "Stock-Market Participation, Intertemporal Substitution, and Risk-Aversion." American Economic Review 93 (May): 383-91.

Lance Kent and Toan Phan

Amazon, (work completed while the author was affiliated with the College of William & Mary); Federal Reserve Bank of Richmond, This paper supersedes an earlier working paper entitled "Business Cycles and Revolutions." The authors are grateful for helpful discussions with Daron Acemoglu, Filipe Campante, Matthias Doepke, B.R. Gabriel, Elias Papaioannou, and Romain Wacziarg. The authors would also like to thank seminar participants at the Atlanta Fed, Midwest Macro, the National University of Singapore, the 2014 ASSA Meetings, Duke University, the 2014 ISNIE meetings, the 2014 ESEM meetings in Toulouse, Queen's University Belfast, Utrecht University, and Maastricht University for their suggestions and feedback. The views expressed herein are those of the authors and not those of the Federal Reserve Bank of Richmond or the Federal Reserve System.

(1) See the related literature section for a discussion of existing studies.

(2) The calculation of skewness and other higher-order moments is sensitive to the tails of the distribution of interest. Since realizations on the tails are rare, many Monte Carlo draws are needed to ensure that the tails are sufficiently sampled. Therefore, for a given simulation length, the influence of Monte Carlo simulation error is going to be much more pernicious for higher-order moments, such as skewness, than for lower-order moments, such as the mean.

(3) See, e.g., Kuran (1989), Chenoweth and Stephan (2011), and Edmond (2013). We also verify this in our probit analysis to predict the onset of unrest in Appendix A.2.

(4) E.g., Nakamura et al. (2013) allow disasters to be correlated across countries but suppose that the probability of a given country entering into a disaster "on its own" is fixed over time.

(5) It is important to note that we do not identify unrest episodes themselves as disasters.

(6) More recent versions of the dataset include more recent years.

(7) If p is the continuation probability and x is the annual loss, then the cumulative loss is estimated to be [mathematical expression not reproducible].

(8) While there is also a visibly larger left tail for the distribution of investment growth, the bootstrapped difference in the skewness in investment growth between unrest and no unrest is not significantly different from zero, with a p-value of 0.83. This is because there are a few observations of investment growth that are very large in absolute value on both sides of the distribution (consistent with sharp falls in investment and subsequent rebounds), and the bootstrapped estimate of the difference in skewness is sensitive to these outliers.

(9) The unrest shock in each period t affects the distribution of the TFP in period t, and because TFP is autocorrelated, the unrest shock will affect the distribution of future TFP terms too. This is different from a "news shock" that does not affect current TFP, only future TFP.

(10) The sensitivity parameter of interest rate to debt is simply set to a small value to avoid a unit root, as in Garcia-Cicco et al. (2010) and Schmitt-Grohe and Uribe (2003).

(11) In Vissing-Jorgensen and Attanasio (2003), the estimated risk-aversion parameter can take a wide range of values, as large as thirty. To be conservative, we only set the maximum risk-aversion to be twenty.

(12) We do not attempt to match kurtoses exactly, since we approximate equilibrium only to third order. We choose the kurtosis of the TFP processes high enough to permit existence of the Normal Inverse Gaussian distribution for the calibrated second-order and third-order moments. The calibrated kurtosis of TFP is approximately equal to that of the empirical distribution of output growth.

(13) For example, if sectors of the economy differ not only with respect to average productivity, but also exposure to political uncertainty under unrest, we might see a reallocation of capital to relatively inefficient sectors, driving up the share of output growth loss explained. Recent work by Acemoglu et al. (2014) provides evidence that, during the Egyptian experience of the Arab Spring, firms that had closer ties to the threatened regime suffered greater losses on the Egyptian stock market than firms that did not. Exploring the macroeconomic significance of this and other micro risks associated with political unrest would be complementary to our analysis and is outside of the scope of this paper.

(14) Remember, we are using a calibration where [zeta] < 1, so [micro] > 0 and [micro](1-[zeta]) > 0.

(15) The regions, as classified by the World Bank, are: East Asia and Pacific, Europe and Central Asia, Latin America and Caribbean, Middle East and North Africa, South Asia, and sub-Saharan Africa.

(16) Data for coups come from Marshall and Marshall (2011) and data for Polity come from Marshall and Jaggers (2002).

(17) We include the perturbation parameter in the definition of the state vector to simplify notation. Andreasen et al. (2017) include a brief discussion of this notation in the extensive appendix to their paper.
Table 1 Empirical Moments In and Out of Unrest

                  No unrest      Unrest         Difference  Ratio
                  (c.i.)         (c.i.)         [p-value]   [p-value]

Output growth
Mean                0.17          -1.75          -1.92
                   (0.03,0.32)   (-2.46,-1.04)   [0.00]
Standard Dev.       5.62           8.63                      1.53
                   (5.30,5.95)    (7.05,10.20)              [0.00]
Skewness           -0.69          -2.26          -1.57
                  (-1.68,0.30)   (-4.16,-0.35)   [0.15]
Kurtosis           23.61          23.38          -0.23
                  (15.95,31.27)  (13.86,32.91)   [0.97]
Consumption grth
Mean                0.12          -1.10          -1.22
                  (-0.11,0.36)   (-1.90,-0.31)   [0.00]
Standard Dev.       8.34           9.80                      1.17
                   (7.69,8.99)    (7.30,12.29)              [0.00]
Skewness            0.40          -2.96          -3.36
                  (-1.38,2.17)   (-6.52,0.60)    [0.10]
Kurtosis           33.46          39.88           6.42
                  (17.34,49.57)   (6.66,73.10)   [0.73]
Investment grth
Mean                0.40          -3.56          -3.96
                  (-0.19,1.00)   (-6.00,-1.12)   [0.00]
Standard Dev.      20.29          27.45                      1.35
                  (18.69,21.89)  (22.93,31.96)              [0.00]
Skewness           -0.94          -0.71           0.24
                  (-2.52,0.64)   (-2.14,0.73)    [0.83]
Kurtosis           31.78          15.11         -16.66
                  (17.77,45.80)  (11.21,19.00)   [0.03]

                  K-S test

Output growth     [0.00]
Standard Dev.
Consumption grth  [0.00]
Standard Dev.
Investment grth   [0.00]
Standard Dev.

Notes: Empirical moments in and out of unrest with bootstrapped 95
percent confidence intervals (in brackets) and p-values (in square
brackets) on hypothesis tests that there is no difference between the
two distributions. The first two columns report the estimated moments.
The third column reports the difference in the estimated moments, along
with the p-value for a test of the null hypothesis that there is no
difference between the corresponding moments. The fourth column reports
the ratio of the estimated standard deviations, along with the p-value
for the Levene test of the equality of variances. The fifth column
reports the p-value for the Kolmogorov-Smirnov test of whether the two
distributions of shocks (under unrest and no unrest) are the same.

Table 2 Calibrated Parameters


                 From literature
[alpha]          Capital share in production
[beta]           Discount factor
[delta]          Depreciation
[phi]            Adjustment costs to capital
g                Trend growth rate
[theta]          Disutility from labor
[omega]          Disutility from labor
d                Steady-state debt level
[psi]            Interest rate sensitivity to debt
[zeta]           Inverse intertemporal elasticity of substitution
[gamma]          Risk aversion
                 Estimates from data
[p.sub.onset]    Probability of unrest onset
[p.sub.cont.]    Probability of unrest continuation
                 Chosen to match target
[[sigma].sub.q]  Std. of TFP shock [[epsilon].sub.t] in quiet times
[s.sub.q]        Skewness of TFP shock [[epsilon].sub.t] in quiet times
[k.sub.q]        Kurtosis of TFP shock [[epsilon].sub.t] in quiet times
[[sigma].sub.u]  Std. of TFP shock [[epsilon].sub.t] during unrest
[s.sub.u]        Skewness of TFP shock [[epsilon].sub.t] during unrest
[k.sub.u]        Kurtosis of TFP shock [[epsilon].sub.t] during unrest

                 Value        Source/Target

                 From literature
[alpha]            0.32       Garcia-Cicco et al. (2010)
[beta]             0.922      -
[delta]            0.126      -
[phi]              3.3        -
g                  1.005      -
[theta]            0.224      -
[omega]            1.6        -
d                  0.007      -
[psi]            [10.sup.-5]  -
[zeta]             0.9 to 5   Table 3
[gamma]            5 to 20    Table 3
                 Estimates from data
[p.sub.onset]      0.014      Appendix A.2
[p.sub.cont.]      0.833      Appendix A.2
                 Chosen to match target
[[sigma].sub.q]    2.75       Table 1
[s.sub.q]         -1.10       Table 1
[k.sub.q]         22          (*)
[[sigma].sub.u]    4.66       Table 1
[s.sub.u]         -2.83       Table 1
[k.sub.u]         22          (*)

Notes: (*) means chosen sufficiently high to permit existence of Normal
Inverse Gaussian distribution.

Table 3 Epstein-Zin Parameter Calibrations in the Literature

                                       [zeta]  [gamma]

Fernandez-Villaverde et al. (2011)      5       5
Colacito et al. (2013)                  0.9    10
Vissing-Jorgensen and Attanasio (2003)  0.9    20

Table 4 Numerical Results

                                 Numerical Results
                    Baseline     High risk av.      No EZ, low risk av.
                    [zeta]-0.9,  [zeta]-0.9,        [zeta]-5,
                    [gamma]-10   [gamma]-20         [gamma]-5

Output growth
First order          0             0                 0
Second order        11            21                 6
Third order         21            62                 9
Consumption growth
First order          0             0                 0
Second order        22           742                 7
Third order         45           128                11
Investment growth
First order          0             0                 0
Second order        27            51                16
Third order         51           148                23

Notes: Numerical results for the percentages of the empirically
observed average losses in the growth rates of output, consumption,
and investment that are explained by the model. The rows show the
percentage explained by using first-order, second-order, and
third-order approximations of the solution to the model.

Table 5 Estimated Onset Probability

Onset                                       Baseline      (2)

[mathematical expression not reproducible]                -0.003
Ethnic Frac
Language Frac
Religion Frac
Europe, Central Asia
Latin America, Caribbean
Middle East, North Africa
North America
South Asia
Sub-Saharan Africa
constant                                    -2.209 (***)  -2.147 (***)
                                            (0.03)        (0.04)
Pr([U.sub.i,t]|[U.sub.i,t-1] = 0)            0.014         0.016
                                            (0.00)        (0.00)
N                                            9272          5910

Onset                                       (3)           (4)

[mathematical expression not reproducible]  -0.004        -0.005
                                            (0.01)        (0.02)
Ethnic Frac                                  0.500 (**)
Language Frac                               -0.050
Religion Frac                               -0.227
Gini                                                      -0.014
Europe, Central Asia
Latin America, Caribbean
Middle East, North Africa
North America
South Asia
Sub-Saharan Africa
constant                                    -2.240 (***)  -1.386 (***)
                                            (0.11)        (0.46)
Pr([U.sub.i,t]|[U.sub.i,t-1] = 0)            0.013         0.083
                                            (0.00)        (0.07)
N                                            5357          599

Onset                                       (5)

[mathematical expression not reproducible]  -0.001
Ethnic Frac
Language Frac
Religion Frac
Europe, Central Asia                        -0.095
Latin America, Caribbean                     0.021
Middle East, North Africa                   -0.005
North America                                no obs.
South Asia                                   0.432 (**)
Sub-Saharan Africa                           0.125
constant                                    -2.182 (***)
Pr([U.sub.i,t]|[U.sub.i,t-1] = 0)            0.01
N                                            5771

Notes: Probit coefficient estimates to predict onset of unrest,
[], and derived probabilities. [DELTA][] denotes real
GDP growth (= 100 x (ln [Y.sub.t] - ln [Y.sub.t-1])). Standard errors
in parentheses. East Asia is the baseline region for the specification
with region FE. (*): p < 0:10. (**): p < 0:05. (***): p < 0:01.

Table 6 Estimated Continuation Probability

Continuation                                 Baseline

[mathematical expression not reproducible]
Ethnic Frac'n
Language Frac'n
Religion Frac'n
Europe, Central Asia
Latin America, Caribbean
Middle East, North Africa
North America
South Asia
Sub-Saharan Africa
constant                                      0.967 (***)
Pr([U.sub.i,t]|[U.sub.i,t-1] = 1)             0.833
N                                           732

Continuation                                 (2)            (3)

[mathematical expression not reproducible]    0.011 (*)      0.009
                                             (0.01)         (0.01)
Ethnic Frac'n                                                0.423
Language Frac'n                                              0.406
Religion Frac'n                                             -1.022 (**)
Europe, Central Asia
Latin America, Caribbean
Middle East, North Africa
North America
South Asia
Sub-Saharan Africa
constant                                      0.995 (***)    0.974 (***)
                                             (0.06)         (0.18)
Pr([U.sub.i,t]|[U.sub.i,t-1] = 1)             0.840          0.835
                                             (0.02)         (0.04)
N                                           590            558

Continuation                                (4)       (5)

[mathematical expression not reproducible]  -0.016     0.010
                                            (0.03)    (0.01)
Ethnic Frac'n
Language Frac'n
Religion Frac'n
Gini                                         0.006
Europe, Central Asia                                  -0.800 (***)
Latin America, Caribbean                               0.317
Middle East, North Africa                             -0.044
North America                                         no obs.
South Asia                                            -0.199
Sub-Saharan Africa                                     0.007
constant                                     0.694     1.001 (***)
                                            (0.75)    (0.18)
Pr([U.sub.i,t]|[U.sub.i,t-1] = 1)            0.756     0.842
                                            (0.24)    (0.04)
N                                           74       590

Notes: Probit coefficient estimates to predict onset of unrest,
[], and derived probabilities. [DELTA][] denotes real
GDP growth (= 100 x (ln [Y.sub.t] - ln [Y.sub.t-1])). Standard errors
in parentheses. East Asia is the baseline region for the specification
with region FE. (*): p < 0:10. (**): p < 0:05. (***): p < 0:01.

Table 7 Estimated Probability of Political Events

                          []     [DELTA][] > 0

[U.sub.i,t]                0.494 (***)       0.802 (***)
                          (0.07)            (0.07)
Lagged output             -0.005            -0.017 (***)
                          (0.00)            (0.00)
[U.sub.i,t*]Lagged        -0.002            0.004
output growth[dagger]
                          (0.01)            (0.01)
constant                  -1.585 (***)      -1.696 (***)
N                       6500              6500

                          [DELTA][] < 0

[U.sub.i,t]                0.431 (***)
Lagged output             -0.005
[U.sub.i,t*]Lagged         0.002
output growth[dagger]
constant                  -1.937 (***)
N                       6500

                          [DELTA][] > 5

[U.sub.i,t]                0.849 (***)
Lagged output             -0.019 (**)
[U.sub.i,t*]Lagged         0.003
output growth[dagger]
constant                  -2.397 (***)
N                       6500

                          [DELTA][] < -5

[U.sub.i,t]                0.420 (**)
Lagged output             -0.007
[U.sub.i,t*]Lagged         0.003
output growth[dagger]
constant                  -2.437 (***)
N                       6500

Notes: Probit coefficient estimates to predict other political
upheavals as functions of current unrest and derived probabilities.
[dagger] relative to country-specific average output growth:
[mathematical expression not reproducible]. Probabilities evaluated at
lagged real output growth equal to country-specific average. Standard
errors in parentheses. (*): p < 0:10. (**): p < 0:05. (***): p < 0:01.

Table 8 List of Episodes of Mass Political Campaigns

Country                 Begin  End   Campaign
                        year   year

  1 Afghanistan         1978   1978  Afghans
  2 Afghanistan         1992   1996  Taliban/Anti-Government Forces
  3 Afghanistan         2001   2006  Taliban Resistance
  4 Albania             1989   1991  Albania Anti-Communist
  5 Algeria             1962   1963  Former Rebel Leaders
  6 Algeria             1992   2006  Islamic Salvation Front
  7 Angola              1975   2002  UNITA
  8 Argentina           1973   1977  ERP/Monteneros
  9 Argentina           1977   1983  Argentina pro-democracy movement
 10 Argentina           1987   1987  Argentiana coup plot
 11 Bangladesh          1987   1990  Bangladesh Anti-Ershad
 12 Belarus             1988   1991  Belarus Anti-Communist
 13 Belarus             2006   2006  Belarus Regime Opposition
 14 Benin               1989   1990  Benin Anti-Communist
 15 Bolivia             1952   1952  Bolivian Leftists
 16 Bolivia             1977   1982  Bolivian Anti-Junta
 17 Brazil              1984   1985  Diretas ja
 18 Bulgaria            1989   1989  Bulgaria Anti-Communist
 19 Burma               1988   2006  Karens
 20 Burma               1988   1990  Burma pro-democracy movement
 21 Burundi             1972   1973  First Hutu Rebellion
 22 Burundi             1988   1988  Second Hutu Rebellion
 23 Burundi             1991   1992  Tutsi supremacists
 24 Burundi             1993   2002  Third Hutu Rebellion
 25 Cambodia            1970   1975  Khmer Rouge
 26 Cambodia            1978   1979  Anti-Khmer Rouge
 27 Cambodia            1989   1997  Second Khmer Rouge
 28 Chad                1968   1990  Frolinat
 29 Chad                1994   1998  Chad rebels
 30 Chile               1973   1973  Pinochet-led rebels
 31 Chile               1983   1989  Anti-Pinochet Movement
 32 China               1956   1957  Hundred Flowers Movement
 33 China               1966   1968  Cultural Revolution Red Guards
 34 China               1976   1979  Democracy Movement
 35 China               1989   1989  Tiananmen
 36 Colombia            1946   1953  Liberals of 1949
 37 Colombia            1964   2006  Revolutionary Armed Forces of
                                     Colombia and National
                                     Liberation Army
 38 Costa Rica          1948   1948  National Union Party
 39 Croatia             1999   2000  Croatian Institutional Reform
 40 Cuba                1956   1959  Cuban Revolution
 41 Czechoslovakia      1989   1990  Velvet Revolution
 42 Djibouti            1991   1994  Afar insurgency
 43 Dominican Republic  1965   1965  Dominican leftists
 44 Egypt               2000   2005  Kifaya
 45 El Salvador         1977   1991  Salvadoran Civil Conflict
 46 Ethiopia            1981   1991  Tigrean People's Liberation Front
 47 France              1960   1962  Pro-French Nationalists
 48 Georgia             2003   2003  Rose Revolution
 49 Ghana               1949   1950  Convention People's Party movement
 50 Ghana               2000   2000  Anti-Rawlings
 51 Greece              1963   1963  Anti-Karamanlis
 52 Greece              1973   1974  Greece Anti-Military
 53 Guatemala           1954   1954  Conservative movement
 54 Guatemala           1961   1996  Marxist rebels (URNG)

Country                 Target

  1 Afghanistan         Afghan government
  2 Afghanistan         Afghan regime
  3 Afghanistan         Afghan government
  4 Albania             Communist regime
  5 Algeria             Ben Bella regime
  6 Algeria             Algerian government
  7 Angola              Angolan government
  8 Argentina           Argentina regime
  9 Argentina           Military junta
 10 Argentina           Attempted coup
 11 Bangladesh          Military rule
 12 Belarus             Communist regime
 13 Belarus             Belarus government
 14 Benin               Communist regime
 15 Bolivia             Military junta
 16 Bolivia             Military juntas
 17 Brazil              Military rule
 18 Bulgaria            Communist regime
 19 Burma               Burmese government
 20 Burma               Military junta
 21 Burundi             Tutsi influence in government
 22 Burundi             Tutsi influence in government
 23 Burundi             Buyoya regime
 24 Burundi             Power-sharing/Tutsi-dominated government
 25 Cambodia            Cambodian government
 26 Cambodia            Cambodian government
 27 Cambodia            Cambodian government
 28 Chad                Chadian government
 29 Chad                Chadian regime
 30 Chile               Allende regime
 31 Chile               Augusto Pinochet
 32 China               Communist regime
 33 China               Anti-Maoists
 34 China               Communist regime
 35 China               Communist regime
 36 Colombia            Conservative govt
 37 Colombia            Colombia govt and US influence
 38 Costa Rica          Calderon regime
 39 Croatia             Semi-presidential system
 40 Cuba                Batista regime
 41 Czechoslovakia      Communist regime
 42 Djibouti            Djibouti regime
 43 Dominican Republic  Loyalist regime
 44 Egypt               Mubarak regime
 45 El Salvador         El Salvador government
 46 Ethiopia            Ethiopian government
 47 France              French withdrawal from Algeria
 48 Georgia             Shevardnadze regime
 49 Ghana               British Rule
 50 Ghana               Rawlings govt
 51 Greece              Karamanlis regime
 52 Greece              Military rule
 53 Guatemala           Arbenz leftist regime
 54 Guatemala           government of Guatemala

Table 9 List of Episodes of Mass Political Campaigns

Country               Begin  End   Campaign
                      year   year

 55 Guyana            1990   1990  Anti-Burnham / Hoyte
 56 Haiti             1985   1985  Anti-Duvalier
 57 Hungary           1956   1956  Hungary Anti-Communist
 58 Hungary           1989   1989  Hungary pro-dem movement
 59 India             1967   1971  Naxalite rebellion
 60 Indonesia         1949   1962  Darul Islam
 61 Indonesia         1956   1960  Indonesian leftists/Anti Sukarno
 62 Indonesia         1997   1998  Anti-Suharto
 63 Iran              1977   1978  Iranian Revolution
 64 Iran              1981   1982  Iranian Mujahideen
 65 Iran              1982   1983  KDPI
 66 Iraq              1959   1959  Shammar Tribe and pro-Western
 67 Iraq              1991   1991  Shiite rebellion
 68 Ivory Coast       2002   2005  PMIC
 69 Kenya             1990   1991  Anti-Arap Moi
 70 Kyrgyzstan        1990   1991  Kyrgyzstan Democratic Movement
 71 Kyrgyzstan        2005   2005  Tulip Revolution
 72 Laos              1960   1975  Pathet Lao
 73 Lebanon           1958   1958  Anti-Shamun
 74 Lebanon           1975   1975  Lebanon leftists
 75 Lebanon           2005   2005  Cedar Revolution
 76 Liberia           1989   1990  Anti-Doe rebels
 77 Liberia           1992   1995  NPFL & ULIMO
 78 Liberia           1996   1996  National patriotic forces
 79 Liberia           2003   2003  LURD
 80 Madagascar        1991   1993  Active Forces
 81 Madagascar        2002   2002  Madagasar pro-democracy movement
 82 Malawi            1959   1959  Nyasaland African Congress
 83 Malawi            1992   1993  Anti-Banda
 84 Maldives          2003   2006  Anti-Gayoom
 85 Mali              1990   1992  Mali Anti-Military
 86 Mexico            1987   2000  Anti-PRI
 87 Mexico            2006   2006  Anti-Calderon
 88 Mongolia          1989   1990  Mongolian Anti-communist
 89 Mozambique        1979   1992  Renamo
 90 Nepal             1990   1990  The Stir
 91 Nepal             1996   2006  CPN-M/UPF
 92 Nicaragua         1978   1979  FSLN
 93 Nicaragua         1980   1990  Contras
 94 Niger             1991   1992  Niger Anti-Military
 95 Nigeria           1993   1998  Nigeria Anti-Military
 96 Oman              1964   1976  Popular Front for the Liberation
                                   of Oman and the Arab Gulf (PFLOAG)
 97 Pakistan          1968   1969  Anti-Khan
 98 Pakistan          1983   1983  Pakistan pro-dem movement
 99 Pakistan          1994   1995  Mohajir
100 Panama            1987   1989  Anti-Noriega
101 Papua New Guinea  1988   1988  Bougainville Revolt
102 Paraguay          1947   1947  Paraguay leftist rebellion
103 Peru              1980   1995  Sendero Luminoso (The Shining
                                   Path) Senderista Insurgency
104 Peru              1996   1997  Tupac Amaru Revolutionary Movement
                                   (MRTA) - Senderista Insurgency
105 Peru              2000   2000  Anti-Fujimori
106 Philippines       1946   1954  Hukbalahap Rebellion
107 Philippines       1972   2006  New People's Army
108 Philippines       1983   1986  People Power
109 Philippines       2001   2001  Second People Power Movement
110 Poland            1956   1956  Poznan Protests

Country               Target

 55 Guyana            Burnham/Hoyte autocratic regime
 56 Haiti             Jean Claude Duvalier
 57 Hungary           Communist regime
 58 Hungary           Communist regime
 59 India             Indian regime
 60 Indonesia         Indonesian government
 61 Indonesia         Sukarno regime
 62 Indonesia         Suharto rule
 63 Iran              Shah Reza Pahlavi
 64 Iran              Khomenei regime
 65 Iran              Iranian regime
 66 Iraq              Qassim regime
 67 Iraq              Hussein regime
 68 Ivory Coast       Incumbent regime
 69 Kenya             Daniel Arap Moi
 70 Kyrgyzstan        Communist regime
 71 Kyrgyzstan        Akayev regime
 72 Laos              Laotian government
 73 Lebanon           Shamun regime
 74 Lebanon           Lebanese government
 75 Lebanon           Syrian forces
 76 Liberia           Doe regime
 77 Liberia           Johnson regime
 78 Liberia           Liberian govt
 79 Liberia           Taylor regime
 80 Madagascar        Didier Radsiraka
 81 Madagascar        Radsiraka regime
 82 Malawi            British rule
 83 Malawi            Banda regime
 84 Maldives          Maumoon Abudul Gayoom's regime
 85 Mali              Military rule
 86 Mexico            Corrupt govt
 87 Mexico            Calderon regime
 88 Mongolia          Communist regime
 89 Mozambique        Mozambique government
 90 Nepal             Monarchy/Panchayat regime
 91 Nepal             Nepalese government
 92 Nicaragua         Nicaraguan regime
 93 Nicaragua         Sandinista regime
 94 Niger             Military rule
 95 Nigeria           Military rule
 96 Oman              Oman government
 97 Pakistan          Khan regime
 98 Pakistan          Zia al-Huq
 99 Pakistan          Pakistani government
100 Panama            Noriega regime
101 Papua New Guinea  Papuan regime
102 Paraguay          Morinigo regime
103 Peru              Peruvian government
104 Peru              Peruvian government
105 Peru              Fujimori govt
106 Philippines       Filipino government
107 Philippines       Filipino government
108 Philippines       Ferdinand Marcos
109 Philippines       Estrada regime
110 Poland            Communist regime

Table 10 List of Episodes of Mass Political Campaigns

Country           Begin  End   Campaign
                  year   year

111 Poland        1968   1968  Poland Anti-Communist I
112 Poland        1970   1970  Poland Anti-Communist II
113 Poland        1976   1976  Poland Warsaw worker uprising
114 Poland        1980   1989  Solidarity
115 Portugal      1973   1974  Carnation Revolution
116 Romania       1987   1989  Anti-Ceaucescu rebels
117 Russia        1990   1961  Russia pro-dem movement
118 Rwanda        1961   1964  Watusi
119 Rwanda        1990   1994  Tutsi rebels
120 Rwanda        1994   1994  Patriotic Front
121 Senegal       2000   2000  Anti-Diouf
122 Serbia        1996   2000  Anti-Milosevic
123 Sierra Leone  1991   1996  RUF
124 Slovenia      1989   1990  Slovenia Anti-Communist
125 Somalia       1982   1991  Somalia clan factions; SNM
127 South Africa  1952   1961  South Atrica First Defiance
128 South Africa  1984   1964  South Atrica Second Defiance
129 South Korea   1960   1960  South Korea Student Revolution
130 South Korea   1979   1980  South Korea Anti-Junta
131 South Korea   1987   1987  South Korea Anti-Military
132 Sri Lanka     1971   1971  JVP
133 Sri Lanka     1972   1672  LTTE
134 Sudan         1985   1985  Anti-Jaafar
135 Sudan         1985   2005  SPLA-Garang faction
136 Sudan         2003   2006  JEM/SLA
137 Syria         1980   1982  Muslim Brotherhood
138 Taiwan        1976   1985  Taiwan pro-democracy movement
139 Tajikistan    1992   1997  Popular Democratic Army (UTO)
140 Tanzania      1992   1992  Tanzania pro-democracy movement
141 Thailand      1966   1981  Thai communist rebels
142 Thailand      1973   1973  Thai student protests
143 Thailand      1992   1962  Thai pro-dem movement
144 Thailand      2005   2006  Anti-Thaksin
145 Uganda        1980   1986  National Resistance Army
146 Uganda        1986   2006  LRA
147 Ukraine       2001   2004  Orange Revolution
148 Uruguay       1963   1672  Tupamaros
149 Uruguay       1984   1985  Uruguay Anti-Military
150 Venezuela     1958   1958  Anti-Jimenez
151 Venezuela     1958   1963  Armed Forces for National
                               Liberation (FALN)
152 Yugoslavia    1968   1968  Yugoslavia student protests
153 Yugoslavia    1970   1971  Croatian nationalists
154 Zambia        1990   1991  Zambia Anti-Single Party
155 Zambia        2001   2001  Anti-Chiluba
156 Zimbabwe      1974   1979  Zimbabwe African People's Union
157 Zimbabwe      1982   1987  PF-ZAPU guerillas

Country           Target

111 Poland        Communist regime
112 Poland        Communist regime
113 Poland        Communist regime
114 Poland        Communist regime
115 Portugal      Military rule
116 Romania       Ceacescu regime
117 Russia        Anti-coup
118 Rwanda        Hutuy regime
119 Rwanda        Hutu regime
120 Rwanda        Hutu regime and genocide
121 Senegal       Diouf govt
122 Serbia        Milosevic regime
123 Sierra Leone  Republican government
124 Slovenia      Communist regime
125 Somalia       Siad Barre regime
127 South Africa  Apartheid
128 South Africa  Apartheid
129 South Korea   Rhee regime
130 South Korea   Military junta
131 South Korea   Military government
132 Sri Lanka     Sri Lankan government
133 Sri Lanka     Sri Lankan occupation
134 Sudan         Jaafar Nimiery
135 Sudan         Sudanese government
136 Sudan         Janjaweed militia
137 Syria         Syrian regime
138 Taiwan        Autocratic regime
139 Tajikistan    Rakhmanov regime
140 Tanzania      Mwinyi regime
141 Thailand      Thai government
142 Thailand      Military dictatorship
143 Thailand      Suchinda regime
144 Thailand      Thaksin regime
145 Uganda        Okello regime
146 Uganda        Museveni government
147 Ukraine       Kuchma regime
148 Uruguay       Uruguay government
149 Uruguay       Military rule
150 Venezuela     Jimenez dictatorship
151 Venezuela     Betancourt regime
152 Yugoslavia    Communist regime
153 Yugoslavia    Yugoslav government
154 Zambia        One-party rule
155 Zambia        Chiluba regime
156 Zimbabwe      Smith/Muzorena regime
157 Zimbabwe      Mugabe regime
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Author:Kent, Lance; Phan, Toan
Publication:Economic Quarterly
Date:Mar 22, 2019
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