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Re-thinking Equilibrium Presidential Approval: Markov-Switching Error-Correction Simon Jackmany July 18, 1995 1 Long-Run Equilibria in Presidential Approval If a week is a long time in politics, then surely some forty odd years is a very long time. From the mid-1950s to 1992 the United States had eight dierent presidents, has been in and out of wars, recessions, economic booms, and political crises of one form and another. Technological developments have radically altered the way political information is generated and communicated, to say nothing of the content of that information over time. Given these changes in the political and economic context in which presidents operate, how likely is it that a single statistical regime will adequately capture the relationship between presidential approval and its determinants for the whole time series? Put dierently, is there any good reason to expect that the eects of the variables structuring presidential approval will be constant over the entire time series, relative to one another? One can well imagine circumstances in which ination might receive a heftier weight than other circumstances in determining presidential approval? Or unemployment, business expectations, or even the Prepared for the 12th Annual Political Methodology Summer Conference, Indiana University, Bloomington, Indiana, July 27-30, 1995. I thank Larry Bartels and Rob McCulloch for useful comments and discussion. Errors and omissions remain my own responsibility. y Department of Political Science, University of Chicago, 5828 S. University Ave, Chicago IL 60637. ([email protected]) 1 extent to which presidential approval exhibits inertia? Are rally events disruptions to a longrun equilibrium relationship between approval and economic conditions, or do rally events prompt a switch to a dierent statistical regime? Are returns to equilibria symmetric? That is, does over-approval return to equilibrium levels as quickly as under-approval? In short, how likely is it that the same vector of parameters will adequately tap the relationship between approval and its determinants over presidents of dierent partisan orientations, through war, peace, booms and recessions, and generational turnover in the electorate? 2 Markov-Switching Time Series Models Here I investigate these possibilities with what I call a \Markov switching error correction model". The mathematics of Markov chains provide a way of modelling changes in discrete states, and so the application of Markov chains to regime shifts is relatively straightforward. Markov chains are a simple way of modelling a discrete-valued random variable as a time series, where here the discrete-valued random variable is a latent indicator of the state the substantive process is in. At any time point t, t = 1; 2; : : : ; T , the state variable s takes on a integer value labelling the states (i.e., 8 t; s 2 f1; : : : ; N g, where N is the number of states. The Markov switching model I employ here considers just two states, though theoretically more are possible. Formally, a two-state Markov-switching model of a time series, y , is t t 1 y t 8 > < t X0 + Y 0 + u => : X0 + Y 0 + u t 1 t 1 1t t 2 t 2 2t if s = 1 if s = 2 (1) t t where X is a vector of exogenous variables (possibly including a unit vector), Y is a vector t t I restrict my attention to two states since this seems fairly standard in the econometric and statistics literature on these models, and increasing the number of states raises tough questions of interpretation and parsimony. Standard likelihood tests break down here since if the true process is characterized by N , 1 states, since the parameters of a N state model are unidentied under the null hypothesis. Hamilton (1994, 698) elaborates. See also Hansen (1992). In addition, my experience is that the programming and computational burdens increase apparently exponentially with the number of states. 1 2 of lags of y , fu g are sequences of random iid Gaussian disturbances with mean zero and variances < 1 and fs g is a Markov chain with two states f1,2g such that t it 2 t i P (s t = 2js , = 1) = p t 1 1 and P (s t = 1js , = 2) = p : t 1 2 Note that the two series of disturbances are independent of one another, and of the Markov process. Given the denition of the transition probabilities above, a more complete statement of the Markov process is 2 3 2 6 P (s = 1) 7 = 6 1 , p1 4 5 4 p2 t P (s t = 2) 32 3 7 6 P (s ,1 = 1) 7 : 54 5 t 1,p p1 2 P (s ,1 = 2) (2) t The rst term on the right-hand side of (2) is referred to as a transition matrix, which is dened in terms of the transition probabilities p and p (note that the columns of the transition matrix sum to one, consistent with the rules of probability). For simplicity, these transition probabilities are assumed constant over the sample (xed transition probabilities, or FTP) though time-varying transition probabilities (TVTP) have been recently considered in the literature. In the two-state, FTP case I consider here, p and p completely characterize the Markov chain, and are parameters to be estimated in addition to each state's structural parameters, = (0 ; 0 ); i = 1; 2. Estimation of this model occurs via iterative procedures described in the Appendix (EM, or a Gibbs sampler). In addition to estimating parameters these procedures also yield estimates of the time-varying mixing probabilities { the quantities on the left-hand side of (2) { which are simply optimal estimates of the probability that the process is in a particular state at a given point in time. If, as one would hope, the estimates of the states reect meaningful distinctions in the data, then these probabilities in turn convey substantive information 1 2 2 1 i i 2 i Filardo (1994) models transition probabilities as a function of exogenous variables, Durland and McCurdy (1994) model transition probabilities as duration-dependent, and Ghysels (1994) considers periodicity and seasonality in the Markov process. 2 3 about the serie's determinants at various time points. In an ideal world, an analyst would create interactions or transformations on the right-hand side of a regression equation to capture subtle uctuations, non-linearities and interdependencies in the structural component of a time-series. Here, the unobserved mixing probabilities optimally weight the estimated states, and so might be thought of as proxying for an \omnibus" variable, unobserved by the analyst, but shaping the way independent variables aect the dependent variable under study. 2.1 Labelling States And here lies a danger with this approach. In allowing statistical estimation procedures to tease out dierences between (unobserved) regimes, and allowing the Markov process to evolve independent of known dynamics in the data, one is faced with a problem of ex post interpretation. What do the states mean? What does it mean for the process to be in state \1" with high probability? An analogy here may be useful; I see this problem as akin to producing ex post substantively meaningful interpretations of the dimensions produced by exploratory factor analysis. As I show below, carefully interpreting each state's parameter estimates allows one to make some conclusions about what the states mean; alternatively, one might attempt to impose some structure on the Markov process directly via explanatory variables, though such work is still in its infancy, and doesn't pin down the issue of stateidentication. Another approach to this problem is motivated by Bayesian ideas: typically the researcher will have some hunches about what distinguishes the states, and this prior information ought to be incorporated in any modelling exercise. Examples of prior information might include beliefs that variability in one state is greater than others (i.e., have the belief that > be reected in a prior, see Albert and Chib 1993) or that the eect of one parameter is greater than in another (e.g., specify a prior that > , see McCulloch and Tsay 1994). Given the diculties of a formal, classical test of the number of states, this Bayesian approach seems attractive. Sensitivity analysis casts further light on the problem. By easing the 1 1;k 4 2;k 2 priors on the parameters that distinguish the states a researcher can monitor the extent to which the states become less distinct (McCulloch and Tsay 1994, 524). Increasingly diuse priors might lead to estimates of the structural parameters that are constant across states or estimates of state probabilities that hover close to .5 (in the two state setup). In either case it would seem that at least on the basis of the data alone are insucient in say, by the parameter estimates 3 A single-state error-correction model of presidential approval I begin by presenting estimates of a single-regime, single-equation error correction model of presidential approval series in Table 1. Data sources are listed in the Appendix. Since my focus here is on the Markov-switching model, I will pause only briey to explain the error-correction set-up. A longer elaboration appears in Jackman (1995). Where one has strong beliefs over the direction of causality in an error-correction context (in particular, that re-equilibration can be conned to movement in just one variable) the typical two-stage estimation process can be collapsed into a single regression (e.g., Beck 1992, 243). Formally, the equilibrium relationship can be written as y t =! +! x + ; t (3) = y , !^ , !^ x (4) 0 1 t with ^ t 0 t 1 t as the estimate of disequilibrium component of y to be used as an additional regressor in the second-stage regression in dierences. If the analyst can be condent that x is (weakly) exogenous to y then re-equilibration occurs via changes in y , and one can proceed directly to the estimation of the regression in dierences. That is, the analyst could estimate the \second" stage of the error-correction t t t t 5 Table 1: Single-Equation Error Correction Regression Analysis of Presidential Approval Coecient Variable Estimate Std Error ,:82 4:78 7:35 1:03 4:60 2:36 ,3:16 1:83 ,:18 :04 ,:11 :19 ,:68 :36 :046 :028 :12 :04 4:24 :66 :90 Turning Points Correctly Predicted (%) 70:7 DW 1:96 n.b., dependent variable is APP . Seven unreported indicator variables \dummy out" rst observation of each presidency. N=139. 0 Intercept 1 Rally 2 Gulf War 3 Vietnam APP ,1 , !1 INFL ,1 , !2 Unemp ,1 , !3 S& P 500 ,1 , !4 BEXP ,1 r2 (dierences) r2 (levels) t t t t t t Structural Parameters, Equilibrium Relationship, Bootstrapped Sampling Distributions, Summary Statistics Parameter Variable Point Estimate Median 5{95 percentiles pr > 0 ! INFL , ,:60 ,:58 [-2.41, 1.35] :71 ! Unemp , ,3:85 ,3:85 [-7.41, -.56] :03 ! S& P 500 , :26 :24 [-.002, .62] :95 ! BEXP , :68 :68 [.30, 1.22] :997 1 1 t 2 t 1 3 4 t t 1 1 6 system, y = + z + ^ , + ; 0 t 1 t t 1 (5) t in one-step with y = + z + y , , ! x , + ; t 1 0 t 1 t 1 t 1 t (6) where z are sources of short-term uctuation in y . I denote the intercept in the \one-step" equation as to distinguish it from ; the former collapses the intercepts from both the levels and dierences equations, while the two-step procedure yields unique estimates of each intercept (see Beck 1992, n7). Also note that in this compact form the coecient on the lagged exogenous \levels" variable, x , , is , ! ; i.e., the equilibrium structural parameters are not recovered directly from this \reduced form" error-correction model. One approach is to estimate equation (6) via non-linear least squares; however, since is estimated separately when using a least-squares routine, I recover an estimate of ! by dividing the OLS estimate of , ! by -1 times the OLS estimate of , ,^, and bootstrap to derive the sampling distribution of this quotient of two regression estimates. Since I have condensed the usual two-step procedure into one equation some explanation of the model is warranted. I hypothesize that equilibrium levels of presidential approval are driven by the real economy (ination, unemployment, and changes in the stock market), plus expectations about business conditions (the BEXP variable). Short-term changes in approval are a function of lagged out-of-equilibrium approval, Rally events, an additional indicator for the Gulf War, and the Vietnam battle deaths variable. All variables are described more fully in the Appendix. My classication of the candidate regressors into the \equilibrium" or \short-term" types closely follows the approach used by other analysts modelling presidential approval in an error-correction framework (e.g., Ostrom and Smith 1992). In the second part of Table 1 I report the results of a re-sampling procedure used to recover the structural coecients from the equilibrium relationship, ! ; : : : ; ! . These estimates by and large repeat the ndings of MacKuen, Erikson, and Stimson (1992). The equilibrium relationship between approval and the economic variables is fairly robust, though the business expectations variable washes out the eects of ination; in the t t 0 0 t 1 1 1 1 1 7 4 aggregate, it seem that the prospective information contained in expectations about the economic future is more consequential for presidential approval than a seemingly concrete indicator like the current rate of ination. The error correction mechanism is statistically signicant but does not speedily return approval to levels we would expect on the basis of prevailing economic conditions and economic expectations; only 18% of \out-of-equilibrium" approval is corrected in each quarter, implying that (^ + 1) = :82 20% of out-ofequilibrium approval persists some two years ahead. The contemporaneous eects of \irrational" expectations, rally events and other determinants of approval are but a modest portion of the cumulative long-run eects of these variables; for any given contemporaneous eect, Z , the cumulative eect is Z=(,^) where ^ is the estimate of the lagged approval in equation (6). The estimated dynamics here suggest that approval is fairly \sticky", with the contemporaneous eects accounting for only 18% of the total long-run eect of an input to the presidential approval error-correction system. 8 8 4 Two states of presidential approval I turn now to a two-state Markov-switching error-correction model. Each state's parameter estimates are reported in Table 2, along with estimates of the transition probabilities. In each state an error-correction mechanism is supported by the data; the parameter estimate on the lagged level of presidential approval is unambiguously within the negative unit interval in both cases. However, in State 1, the error correction eect is almost three times as strong as in State 2. In State 1 over 30% of the previous quarter's out-of-equilibrium presidential approval is dissipated, while only 11% is corrected by the State 2 error-correction mechanism. In State 1 then, the past carries far less weight than contemporaneous inuences in shaping presidential approval. State 2 though is characterized by a high degree of inertia in presidential approval; out-of-equilibrium presidential approval lingers for a much greater degree than in State 1. Figure 1 displays the dierence between State 1 and State 2 in these patterns of per8 Table 2: Markov-Switching Error-Correction Model of Presidential Approval Coecient Variable State 1 State 2 Intercept ,13:91 10:03 (5:14) (4:22) Rally 8:87 7:08 (1:07) (:93) Gulf War ,5:55 6:31 (3:49) (1:87) Vietnam ,4:74 ,2:30 (2:16) (1:53) APP , ,:30 ,:10 (:05) (:04) , ! INFL , :30 ,:41 (:20) (:17) , ! Unemp , ,2:04 :10 (:43) (:30) , ! S&P 500 , ,:11 :12 (:03) (:02) , ! BEXP , :36 ,:04 (:05) (:04) 2:89 2:89 Transition Probabilities :65 :33 n.b., dependent variable is APP . Seven unreported indicator variables \dummy out" rst observation of each presidency. constant across states (identifying constraint). Transition probabilities are probability of changing to other state at t + 1, conditional on being in given state at t. Standard errors in parentheses. N=139. Structural Parameters, Equilibrium Relationships, Bootstrapped Sampling Distributions, Summary Statistics Parameter Variable Point Est. Median 5-95 % prop > 0 State 1: ! INFL , :97 :97 [-.11, 2.34] :93 ! Unemp , ,6:70 ,6:70 [-9.28,-4.62] 0 ! S&P 500 , ,:34 ,:34 [-.60,-.16] :001 ! BEXP , 1:18 1:18 [.89, 1.61] 1 State 2: ! INFL , ,3:96 ,3:95 [-9.16,-1.39] :008 ! Unemp , :93 :93 [-3.49, 9.03] :63 ! S&P 500 , 1:14 1:14 [.63, 2.69] :998 ! BEXP , ,:41 ,:40 [-1.66, .17] :13 0 1 2 3 1 t 1 1 t 2 1 t 3 1 t 4 t 1 t 1 1 t 2 t 1 3 4 1 1 t 1 1 t 1 t 2 t 1 3 4 t t 1 9 100 Figure 1: Dierences in Presidential Approval's Durability, State 1 vs State 2. 0 20 Shock (%) 40 60 80 State 1 State 2 0 5 10 15 Time (Quarters) sistence, for an identical, nominal shock. After twelve quarters about 25% of the shock is retained in the State 2 regime. However, in State 1, after just four quarters the eect of the nominal shock has already dropped to roughly this 25% level. New information exerts a considerably more powerful inuence in State 1 than in State 2, the latter exhibiting considerable persistence in presidential approval. Figure 2 displays the actual period-specic error-correction eects, bounded by the \pure" State 1 error-correction eect (-.31) and the \pure" State 2 eect (-.10); estimated period-specic eects between these \pure" types reects the estimated probability of mixing between the two states. The dierence across the two states in the coecients for lagged business expectations (BEXP) is also noteworthy and large. In State 1 business expectations are an important determinant of presidential approval; the bottom panel of Table 2 reports estimates of the structural parameters of the equilibrium part of the error-correction model, for both states. Business expectations in State 1 pick up a large positive coecient. Each point of the Survey of Consumer's 200-point BEXP scale translates into roughly 1.1 points of presidential 10 Figure 2: Estimated Period-Specic Error-Correction Eects Eisenhower JFK LBJ Nixon Ford Carter Reagan Bush -0.10 -0.15 -0.20 -0.25 -0.30 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 approval. Contrast State 2. Both reduced form and structural parameters for BEXP are indistinguishable from zero in State 1. These two dierences alone represent quite a departure from standard stories about business expectations and presidential approval. The relationship between presidential approval, its past, and economic expectations is fairly subtle; the evidence here indicates that the eects of lagged approval and business expectations on current approval is not constant, but is stronger at some times than at others. And it is only in State 1 that I nd a strong, signicant relationship between economic expectations and presidential approval. In State 2 there is no relationship between economic expectations and presidential approval. The process characterized by the State 2 parameter estimates might be termed a \steady-state" or \business-as-usual" type of presidential approval, characterized by high 11 1.00 0.75 0.50 0.25 0.0 1958 1960 Eisenhower 1956 LBJ 1966 1968 1970 Nixon 1972 1974 Ford 1976 1980 Carter 1978 1982 1986 Reagan 1984 Figure 3: Estimated Probability of State 1. 1964 JFK 1962 12 1988 1992 Bush 1990 inertia (a low rate of error-correction), and responsiveness to the usual \objective" economic culprits (with the exception of unemployment) and rally events. And moreover, State 2 is the more prevalent state in these data (see Figure 3). In Figure 4 I graph the predicted values of change in presidential approval implied by State 1, State 2, and the overall model (the latter quantity being the weighted sum of the state-specic predicted values). By and large, State 1 is associated with falls in presidential approval; about seventy percent of the data points in the rst panel of Figure 4 are below zero. Given the faster error-correction mechanism implied by the State 1 estimates, it seems that State 1 brings presidential approval back down to levels we would be more likely to expect on the basis of prevailing economic conditions, with economic expectations also contributing signicantly. In contrast, State 2's contribution to the mixture-model span a larger range Probability of being in State 1 13 \ t 1956 -10 0 10 20 1956 -5 0 5 10 15 20 25 -15 1956 -10 -5 0 5 1958 Eisenhower 1958 Eisenhower 1958 Eisenhower 1960 1960 1960 1962 JFK 1962 JFK 1962 JFK 1964 1964 1964 1966 LBJ 1966 LBJ 1966 LBJ 1968 1968 1968 1974 1976 1972 Nixon 1974 1976 Ford 1978 1970 1972 Nixon 1974 1976 Ford 1978 1980 Carter 1980 Carter 1980 Carter 1978 State 2, Predicted Effects 1972 Ford Complete Model, Predicted Effects 1970 1970 Nixon State 1, Predicted Effects 1982 1982 1982 1986 1986 1984 1986 Reagan 1984 Reagan 1984 Reagan 1988 1988 1988 1992 1992 1990 1992 Bush 1990 Bush 1990 Bush Figure 4: Predicted Change in Presidential Approval (APP ), Markov-Switching Error-Correction Model Change in Presidential Approval than State 1's contributions, in part a reection of the slower error-correction mechanism in State 2. And since the overall process tends to gravitate towards State 2 (recall the transition probabilities reported in Table 2), it seems that State 1 seems to operate as a short, sharp corrective to out-of-equilibria approval around rally events. Prominent examples of State 1 period-specic corrections in my data include the Watergate revelations (73:2), associated with a decline of 15 points in approval of Nixon; an estimated 6.5 point rebound in Ford's approval after his pardon of Nixon (75:2); poor economic conditions under Carter, leading to period-specic declines of 6.5 points of approval in both 79:1 and 79:2; uctuations in Carter's approval through the Iranian hostage crisis (a estimated 4 point boost to approval in 79:4, but a more-than-osetting 5 point decline in 80:2); the Iran-Contra revelations under Reagan (86:4, but with the correction specic to State 1 coming in 87:1, estimated to be on the order of -11 points of approval); and three, large, negative impacts on Bush' approval rating in 90:4, 91:4, and 92:1, estimated to be -11, -7, and -11 points, respectively. Large State 2 estimated contributions include a 6.5 point increase in Nixon's approval in 72;2, associated with the Moscow U.S.{ Soviet summit, and a 5 point boost in 73;1, associated with Nixon's re-inauguration and the end of draft for the Vietnam War; the Watergate revelations, associated with estimated 5 point declines in Nixon's approval in both 73:3 and 73:4; Ford's pardon of Nixon (roughly -5 points in 75:1); 14 the rally in support for Carter during the early stages of the Iran hostage crisis (9 points in 79:4, 5 points in 80:1, but a 7 points decline in 80:2); a rally in approval of Reagan after the killing of U.S. Marines in Beirut and the invasion of Grenada (83:4, 6 points); a 5 point fall in approval coinciding with the Iran-Contra revelations (86;4); and a mammoth 25 point boost to George Bush's approval in 91:1 at the time of the Gulf War. Figure 5 is a slightly dierent interpretation of these data: a scatter-plot of the mixture model's predicted change in presidential approval by the probability of State 1, along with a least squares regression t (and 95% pointwise condence bounds) and a loess t. This graphical representation of the data repeats the story of Figure 4 quite starkly. The more \pure" instances of State-1-type-approval are associated with the more precipitous predicted falls in presidential approval. The labelling of the more extreme observations also reveals the volatility of the approval-generating mechanism under George Bush's tenure: 90:2, 90:4, 91:1, 91:4, and 92:1 are among the extremes in both the probability of either state and the predicted change in presidential approval. I am condent that the dierences between States 1 and 2 here are statistically signicant. The single-state error-correction model of presidential approval has a log-likelihood -389.61, an estimated standard error (^) of the regression of 4.24, and an r of .66. The mixture model I present here has a log-likelihood of -368.10, an estimated standard error of 2.89 (which is held constant across the states; see the Appendix for more details), and an r of .78. Minus twice the dierence in the log-likelihoods is the test statistic for a likelihood ratio test; put simply, the likelihood ratio test here tests whether the dierence in the loglikelihoods is signicant given the extra degrees of freedom consumed in estimating the extra parameters of the mixture model. Twice as many regression parameters are estimated in the mixture model than in the single-state model, and two transition probabilities are also estimated, for a total of 11 extra parameters. Accordingly, the likelihood ratio test statistic 2 2 15 \ Figure 5: Estimated Change in Presidential Approval (APP ) and Probability of State 1 (P [S = 1]). t t 74:4 91:1 10 75:2 0 55:3 79:1 -10 16 Predicted Change in Presidential Approval 20 81:1 79:291:4 90:4 87:1 92:1 90:2 80:2 73:2 0.0 0.2 0.4 0.6 Probability of State 1 0.8 1.0 here is ,2 (,389:61 + 368:10) 43:02 which asymptotically follows a distribution with 18 degrees of freedom (16 structural parameters per state { including the seven dummy variables for change of presidencies { plus the two transition parameters). In this instance the null hypothesis can be rejected fairly overwhelmingly; the critical p = :05 value for a distribution with 18 degrees of freedom is 28.87. The improved t of the Markov-mixing error correction model is not a clever statistical sleight of hand, or simply a consequence of estimating more than twice as many parameters as the single-state ECM. A model which allows switching between two statistical regimes quite clearly is a better characterization of the dynamics of presidential approval than the single-state model. My ndings suggest that past ndings about the relationship between economic expectations and presidential approval require revising. In particular, economic expectations matter in structuring presidential approval, but only when State 1 mixing probabilities are non-zero. Whenever State 2 dominates, my estimates are that economic expectations have no eect on presidential approval. Economic expectations carry large weight only in State 1, and a standard single-regime time-series analysis of presidential approval that nds large eects for economic expectations might be picking up on the eects of economic expectations in the relatively few cases where State 1 mixing probabilities are close to one, in much the same way as outlying observations can cause a linear regression to yield a signicant relationship between two variables. A standard single-equation analysis in eect aggregates across States 1 and 2, nding an fairly impressive overall eect for economic expectations. The switching-regime model tells a more realistic story. Economic expectations are part of the mix of considerations which comes into play to help drive approval back towards more \realistic" levels after the shock of a rally event. Economic expectations are not a constant determinant of presidential approval, but exert powerful eects on presidential approval from time to time, helping to speed the return of presidential approval to levels more in line with economic circumstances after the intrusion of a rally event. The variability in the eect of economic expectations on presidential approval is for all practical purposes equivalent to the mixing probabilities plotted over time in Figure 3. Since economic expectations count for 2 2 17 nought in State 2, at any given point in the time series the eect of economic expectations on presidential approval is the equal to the the coecient on economic expectations in State 1 times the probability that the process is in State 1. The dierences across states in other independent variables are of interest as well: 3 Ination diminishes presidential approval in State 2 (t ,2:5, reduced form coecient, and the structural parameter for the equilibrium relationship in levels is -3.96 and highly signicant). But there is also some weak evidence to suggest that ination modestly enhances approval in State 1 (t 1:46 for the reduced form coecient, and the structural parameter is small but signicantly positive). Unemployment has a statistically signicant eect on presidential approval only in State 1, and in the anticipated negative direction. This eect is large relative to the ination results; a one-time additional percentage point of unemployment in State 1 leads to roughly a one-time two point decrease in presidential approval, while the eect in the equilibrium relationship is much larger: a sustained increase in unemployment would lead to a new equilibrium level of presidential approval 7 points lower than the prior equilibrium level (see the estimates of the structural parameters in the bottom panel of Table 2). The variable tapping changes in the stock-market, S&P 500, has a small, negative eect in State 1, but has a reasonably impressive structural parameter in State 2 (approximately 1.14, and clearly non-zero). The Gulf War variable is at rst glance an odd case, since it picks up a statistically signicant positive coecient in State 2 (t 3:4), but a marginally signicant negative coecient in State 1 (t ,1:6). Two caveats are in order here. First, the Gulf War eect can not be interpreted without also considering the eect of the Rally variable. The predicted eects plotted over time in Figure 4 are generated in this fashion: i.e., form y^ by multiplying the observed data on X by the estimated coecients for each state, but condition on the probability of being in either state by multiplying y^ by that probability. 3 t t t 18 In 90:4 both Rally and Gulf War take on the value 1; in 91:1 they both take on the value 2. The eect of the Gulf War rally is thus partitioned into two components; the part due to a general \rally" eect, and an additional component picking up to the extent to which the Gulf War is an extraordinary rally event. Second, the eects reported in Table 2 have to be multiplied by the period-specic probabilities of being in either state in order so as to derive the actual eect for a specic period. The massive boost in approval for Bush associated with Gulf War comes in 91:1, when the process appears to be in State 2, thereby ignoring the negative coecient on this rally in State 1. The \rally" eects of the Gulf War at time t can be expressed formally as y^ = p(s = 1) y^ + p(s = 2) y^ ; h i h i 0 0 ^ ^ ^ ^ = P [s = 1] X + 1 , P [s = 1] X ; t t1 t t t2 t t 1 t t 2 where P^ [s = 1] is the estimated probability that the system is in State 1 (and since there are only two states here, one minus this quantity is the probability that the system is in State 2), and ^ denotes a vector of the relevant parameter estimates from State 1 (and similarly for State 2), and X is a vector of the relevant variables (Rally and Gulf War) measured at time t. Substituting the relevant parameters estimates from Table 2, state probabilities, and values of Rally and Gulf War for 90:4 and 91:1, I obtain the following predicted values for the Gulf War rally: t 1 t 90:4 = 1) 1 , P^ (s = 1) Rally Gulf War y^ y^ 1.00 0.00 1 1 3.32 0.00 91:1 0.00 t P^ (s t t t 1.00 2 t 2 st =1 st =2 y^ t 3.32 0.00 26.78 26.78 As this table makes clear, since presidential approval switches from \pure" State 1 in 90:4 to \pure" State 2 in 91:1, the calculations here are somewhat simplied. Combining the \generalized" rally eect and the incremental boost due to the Gulf War with the estimated state-probabilities produces a small \rally" eect in 90:4 (on the order of three points in approval). In 91:1 this eect is enormous { over eight times as large { worth roughly 27 points of approval to Bush. 19 5 Volatility between states and approval of Bush Recalling Figure 3, it is apparent that State 1 is not at all prevalent these data; presidential approval enters State 1 with probabilities equal to one in only four instances. Of these four instances of \pure" State 1 presidential approval, two occur in the Bush presidency. But, these cases aside, the dynamics of approval for George Bush more closely resemble those captured by State 2. Table 3: Summary Statistics, Probability of State 1, by Presidents President Mean Median Std Dev N Eisenhower :29 :22 :27 14 JFK :36 :35 :19 12 LBJ :28 :25 :22 17 Nixon :37 :36 :22 23 Ford :34 :33 :22 9 Carter :38 :33 :26 16 Reagan :34 :34 :27 32 Bush :40 :31 :40 16 Table 3 summarizes the smoothed state probabilities by presidencies, and this information is also represented graphically in Figure 6. The median probability of approval being in State 1 is just .25 under George Bush; the second lowest median tendency towards State 1 I observe in the 8 presidencies I analyze. But at the same time the smoothed state probabilities exhibit their greatest variability under George Bush; while the median probability of \State-1-typeapproval" is comparatively low under George Bush, the mean probability is the highest among the 8 presidencies (indicating a quite skewed distribution of state probabilities under Bush), and the standard deviation for the Bush-specic state probabilities is almost twice as large as the next largest presidency-specic standard deviation. Finally, it is also during Bush's tenure that I nd 50% (three out of six) of the few instances where the system enters State 1 with probabilities greater than .95. This volatility in the state probabilities (observable in the timeseries plotted Figure 3) stems in no small measure from the volatility in presidential approval itself (see Figure 7). The mixture-model here provides a way of addressing the volatility in the approval series by providing estimates of the state probabilities, which appear quite volatile 20 0.0 0.2 0.4 0.6 0.8 1.0 Figure 6: Estimated Probabilities of State 1, by Presidents Eisenhower JFK LBJ Nixon Ford Carter Reagan Bush themselves under Bush's tenure. The rapidly varying state probabilities observed under the Bush presidency in turn imply rapidly varying eects of the right-hand side variables on presidential approval. In short, these data strongly suggest that there is something unique about the translation of economic perceptions into political sentiments during Bush's tenure. The low median tendency towards \State-1-type" presidential approval observed under George Bush, coupled with high volatility in the state probabilities in turn suggests that the mix of considerations brought to bear in assessing George Bush was itself volatile. Recall that lagged values of the business expectations variable and unemployment carry more weight in State 1 than in in State 2, whereas approval is much \stickier" in State 2 and responsive to ination and recent changes in the stock market. But the exceptional features here are the abrupt transitions in 21 and out of the two states under George Bush. Doesn't this characterization strikes something of a chord when we recall the presidency of George Bush? The pattern of volatility I report above bears a resemblance to the changing perceptions of Bush as president: a sound, economic manager inheriting the reins of power from a popular two-term president, stumbling in the summer 1990 budget negotiations with Congress as the economy weakened, triumphant commander-in-chief and leader of a new world order six months later, and then seemingly helpless to halt economic recession and a growing sense of national malaise that lingered long into 1992, if not up until the election itself. Looking closely at the smoothed state probabilities, approval for George Bush enters \pure" State 1 on three occasions; 90:4 (p 1.00), 91:4 (p .97), and 92:1 (p 1.00). All three quarters involve some of the three most precipitous declines in Bush's approval: -10.67 points in 90:4, -11.33 points in 91:4, and -15.67 points in 92:1. In these same three quarters the Survey of Consumers business conditions expectations measure, BEXP, was 75, 100 and 107.7, respectively, below or close its median under Bush of 105.3, and even further below the median of BEXP for the whole time series of 111.3. The other variable that carries large weight in State 1, unemployment, was rising through this period, further depressing approval for Bush. 4 In fact, this last reading on BEXP in 92:1, the highest of the three here, asks respondents to imagine the business conditions of 93:1 relative to 92:1, and may possibly reect an optimism as to the state of the economy under a dierent President. I am reluctant to push this point too hard, since while I found a cycle in BEXP corresponding to the presidential electoral calendar, I have no evidence that this cycle is associated with (aggregated) beliefs about the probability of the incumbent president being re-elected. Nonetheless, the monthly data on BEXP from early 1992 reveal a denite bounce from low readings in late 1991. Between November and December 1992, after the election, BEXP jumps 20 points (116 to 136), the single largest positive increase under Bush's presidency save for the 33 points increase between February and March 1991 after the conclusion of the Gulf War (98 points to 131). 4 22 6 Modelling Approval, more generally The estimates I report here suggest a recasting of some common understandings about presidential approval and its determinants, and perhaps especially economic expectations. The Markov-switching model suggests fairly strongly that a single-state model is an inadequate approximation of the process generating presidential approval, at least over the long, quarterly time series I analyze here. Put quite blandly, \dierent things carry dierent weights at dierent times." Specically, economic expectations and unemployment are not a \constant" in the mix of considerations brought to bear in assessing a president. The role of economic expectations and unemployment is actually more limited than conventional single-state analyses suggest. Economic expectations and unemployment appear to help reequilibrate presidential approval after rally events, as part of a State 1 \corrective" or even \counter-shock". Recall that the set of parameter estimates specic to State 1 give heavy weight to unemployment levels and economic expectations in the error-correction component of the model. Furthermore, the transition probabilities show that State 1 occurs infrequently and is not at all durable; as a result, State 2 dominates the mixture of state probabilities. State 1 then appears as something of a \short, sharp, shock" to presidential approval, hastening the return to normalcy after a rally event has driven approval to levels higher than we would expect given economic conditions. This short, sharp, (counter) shock is characterized by the aggregate public paying more attention to sources of possible economic insecurity than is typical: unemployment and economic expectations. In the more normal State 2 these variables count for nought in shaping equilibrium levels of approval; approval appears to stand in an equilibrium relationship with ination and changes in the stock market, and reverts only slowly to this equilibrium after a rally event. Cast in this light, State 1 seems somewhat peculiar: economic considerations that appear to matter little for most of the time abruptly become germane in assessing a president, and often, hot on the heels of a rally event. These data appear to suggest a bitter and perhaps even ironic political reality for presidents. Short term boosts in presidential approval quickly dissipate in response 23 to the aggregate public's attention turning to economic insecurities that do not appear to drive assessments of presidents before or even during the rally. Unemployment and economic expectations quickly claw approval back down into equilibrium, even though unemployment and economic expectations do not form part of a major component of the typical (State 2) equilibrium. In short, while approval may rally from time to time, presidents can apparently rest assured that the public will quickly nd something to grumble about. One interpretation of these results is that after being distracted from economic concerns by a rally event, the aggregate public's attention turns to quite weighty economic matters|the state of the job market and the medium-term economic future|before settling back into a more \normal" focus on ination and short-term changes in the stock market. Pushing interpretations like this too far here is somewhat risky. Embedded in the model I present here is the assumption that the Markov process governing the transitions from state to state is independent of the other dynamics in the model, namely, the error correction mechanism linking presidential approval to economic conditions. Nothing here necessarily links occurrences of State 1 to out-of-equilibrium presidential approval, although, ex post it does appear that this is a reasonable characterization of the switching process. In future elaborations I aim to model the transition probabilities as time-varying, with a response to dis-equilibrating rally events and other shocks an explicit part of the switching process. Other extensions might be to compare how this switching model does compares with an ARCH setup, given what appears to be increasing volatility in approval under Bush's tenure. A Appendix A.1 Survey Data The Survey of Consumers is a periodic survey of the consumer attitudes and expectations, conducted by the Survey Research Center of the University of Michigan. The Survey was initiated in 1946, and provides a roughly quarterly series of economic expectations and 24 retrospections since the mid 1950s. Since 1978 the Survey has been administered monthly. The measure of business expectations I use (BEXP) is derived from responses to the following item in the Survey of Consumers: And how about a year from now, do you expect that in the country as a whole, business conditions will be better, or worse than they are at present, or just about the same? A.2 Aggregation The dierence between the percentage of respondents replying \better" and \worse" (a \balance statistic") is added to 100, to yield an aggregate score with theoretical bounds of 0 and 200. I average the monthly relative scores to from the post{1978 surveys to quarters, so as to maintain comparability with the pre-1978 data. The method of aggregation employed here is not as ad hoc as it might rst seem. The key quantity is the dierence between the proportions of respondents reporting \higher" or \better" expectations and respondents reporting \lower" or \worse" expectations; adding one hundred to the result is an arbitrary scaling factor introduced for convenience. This technique has been in use since at least the early 1950s (Pesaran 1987, 212) and was given an explicit econometric foundation by Theil (1952). A.3 Sources for Economic and Political Variables All economic data are from CITIBASE, and the Rally measures and Vietnam measures I employ follow fairly standard usages in the literature. The Rally measure is simply an indicator, scored 1 for the following rallies: the Geneva Summit (55:3), the Cuban missile crisis (62:4), the Moscow summit (72:2), the Paris peace talks (73:1), the Mayaguez incident (75:2), the Camp David accords (78:4), the assassination attempt (81:2), the Grenada invasion (83:4), and the Gulf War (90:4, 91:1). Negative events (scored -1) include Nixon's pardon (74:4) and the Iran{Contra controversy (86:4). Quarters 73:2{73:4 are coded -1 (Watergate), and the Iran{hostages crisis is coded 2 for 79:4, 1 for 80:1, and -1 in 80:2 (see 25 Figure 7: Gallup Presidential Approval, by quarters. Eisenhower JFK LBJ Nixon Ford Carter Reagan Bush 80 70 60 50 40 30 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 MacKuen, Erikson, and Stimson 1992, 609). Vietnam is measured as tens of thousands of U.S. battleeld deaths, by quarter. The Gallup presidential approval series (see Figure 7) is originally from Edwards (1990), aggregated to quarters or months by simple averaging, and is part of the approval.asc data set Neal Beck contributed to the maxlik archive, available from Statlib (anonymous ftp to lib.stat.cmu.edu) in the general sub-directory. Pre-1953 Gallup approval numbers come from Gallup (1980). A.4 Switching Regime Models in Time Series The discussion here closely follows that in Hamilton (1994, ch22). Models with regime shift are not uncommon in econometrics, but what is distinctive here is that the regime shifts 26 are not discrete. Econometric models with regime change typically involve shifts from one regime to others, and often the shifts are once-and-for-all, and one might use a Chow test or some recursive estimation procedure to nd where a structural break takes place. The Markov switching model I consider here is also a \mixing" model, in the sense that at any given time, all regimes (or \states") have non-zero contributions to the structural component of the model. At a given time point the dependent variable is a modelled as mixture of the various regimes, and the sum over the weights of the regimes at any time point is one. Accordingly, the Markov switching model is a varying parameter model of sorts, where although the structural parameters associated with any one regime are constant, the relative contributions of the regimes vary over time. The contribution of a given independent variable varies over time, since the contribution of an independent variable in the overall model is equal to the sum over the regimes of the relevant parameter estimate times the time-specic probability of being in each of the states. Further, the mixing weights (state probabilities) evolve via a Markov process, which I specify as independent of the dynamics linking past observations of the dependent variable to current values. This model is known in econometrics as \the Hamilton model" after Hamilton's (1989, 1990) use of this type of model to analyze U.S. GDP growth (e.g., Hansen 1992, Lam 1990). But the model has been known to statisticians for some time as a \hidden Markov" model (Pesaran and Potter 1992) or a \doubly stochastic" model (Tjostheim 1986) and can also be thought of as special type of random coecient model (Tjostheim 1994; Nicholls and Quinn 1982), sometimes referred to as a \suddenly changing autoregressive model" (SCAR) (Tyssedal and Tjostheim 1988; Karlsen and Tjostheim 1990; Granger and Terasvirta 1993, 18{9, 144{5). The more common deterministic multiple regime setup can be considered a special case of the Markov-switching model if one (or more) of the states in the Markov process is an absorbing state (i.e., once the chain got to one or more of the states it remains in that subset of states with probability one). 5 I am grateful to Nick Polson for enlightening me as to the history of this class of model, and referring me towards the Tjostheim articles in particular. 5 27 A.4.1 Markov Chains Formally, a Markov chain is just a simple mathematical characterization of how a discrete variable like s changes over time. If s takes on integer values f1; 2; : : : ; N g then a fairly simple specication of its dynamics might be to assume that the probability that s equals some particular value j depends only on s , : t t t t P (s t 1 = j js , = i; s , = k; : : :) = P (s = j js , = i) = p : t 1 2 t t t 1 (A1) ij In words, we say that s is an N -state Markov chain and p is the \probability that the chain is in state j given that it was in state i". p is a transition probability. Transition probabilities sum to one at any given time: t ij ij p 1 +p 2 +:::+p i i iN = 1: (A2) As I show in the text, transition probabilities are usually gathered in a transition matrix: P 2 6 6 6 = 666 6 4 p11 p21 p12 p22 p1 p2 3 ::: p 1 7 7 ::: p 2 7 7: 7 . : : : .. 7 7 5 N ... ... N (A3) N ::: p N NN In my application this reduces to a 2 by 2 matrix, since I have only a 2-state model. A.4.2 Mixing Models The form of the mixing model I use here is a special case of the following generic case presented by Hamilton (1994, 690). Let y be an endogenous variable, and X be a k vector of observations on exogenous variables. Let Y = (y ; y , ; : : : ; y ; X ; X , ; X ) be a vector containing all observations up to time t. If switching between regimes follows the simple Markov process described above, then the conditional density of y is t t t t t 1 1 t t 1 1 t f (y js t t = j; X ; Y , ; ); t 28 t 1 (A4) where is a vector of parameters characterizing this conditional density. With N dierent states in the process, there will be N dierent densities (i.e., j = 1; 2; : : : ; N in (A4). These densities can be collected in an N by 1 vector . For my error-correction setup, there are the unknown regression parameters to be estimated for each state ( ; ; : : : ; ; ! ; ! ; : : : ; ! ), error-correction parameters ( ; ; : : : ; ), as well as state-specic error variances ( ; ; : : : ; ). For state j then, t 1 1 2 2 1 N 2 N 2 1 N 2 2 2 N y = Z + y , , ! X , + ; t t j j t 1 j j t 1 (A5) tj which can be written more compactly (without loss of generality) as y = t j X + ; (A6) t t with N (0; ); 8 t. Given the assumption of Normality, 2 tj j 2 f (y js = 1; X ; 6 =4 3 2 6 1 ; 1 ) 7 5=6 4 3 p21 exp ,(y ,2 X ) 7 (A7) , X ) 7 5; , ( y 1 p2 exp f (y js = 2; X ; ; ) 2 for the two-state model I employ. Note that the conditional densities depend on only the current state, and not on what state the process happened to be in at t , 1. Hamilton (1994, 691) considers relaxations of this, as does Lam (1990); it is also possible to impose some structure on the Markov process via independent variables (e.g., Filardo 1994), but here I prefer to keep the specication relatively simple, keeping the structural dynamics of the model conned to an error-correction framework, and having the mixing be between two regimes, each co-integrated. The unconditional density of y is just a weighted sum of the state-specic densities in , where the weights are simply the probabilities that the process is in each of the states, at a particular time point. These probabilities are unobserved by the analyst in any interesting, non-deterministic process, but here evolve according to a simple Markov process. The full log-likelihood for the model is thus t t t t t t t 2 2 t 2 1 1 t t 2 t 2 2 2 2 1 2 t t ln L() = XX T N p(s t t=1 = j ) f (yjs = j; X ; ; ); t j =1 29 t 2 t j j (A8) P where = ( ; )0 = ( ; ; : : : ; ; ; ; : : : ; )0. Note again that p(s = j ) = 1; 8 t. Since this is a mixture model it is necessary to constrain the across the N mixtures: this is because the log-likelihood function in (A8) has no global maximum, since a singularity arises whenever one of the N mixtures is imputed to be equal to its mean with no variance. Note that this also occurs if the algorithm attempts to t one data point as a separate regime (Filardo 1994, 301). At this singularity the log-likelihood is innite. This problem is inherent in all mixture models and my experience is that unconstrained optimization algorithms will typically ounder on this problem. Setting all the variance parameters equal across the states (i.e., dropping the j subscript on in (A8)) is a common solution to this problem when working with mixture distributions. See Jackman (1994, 352) and the references there. Preliminary work with alternative estimation strategies permitting a relaxation of this constraint (e.g., an unreported implementation of a Gibbs sampler ) suggests that for these data the equality constraint on the regime-specic variances is reasonable. 2 1 2 N 2 1 2 2 2 N j =1 N t j 6 2 j 7 A.4.3 Inferences about Mixing Probabilities Where do the p(s ) = j in (A8) come from? This is where the Markov process comes into play. Let P (s = j jY ; ) be the best guess about s based on the information in the sample data up through time t and the parameters . These conditional probabilities, for j = 1; 2; : : : ; N can be stacked in a vector ^ j . The \one-step-ahead" forecast of this vector is written ^ j , which is just the probability that the process will be in state j at time t + 1, given sample information and the parameters up through time t. Optimal inferences and forecasts result from a ltering algorithm that iterates on the following pair of equations: ^ ^ j = 0 ^j , ; (A9) 1 ( j , ) t t t t t t t+1 t t t 1 t t t t t 1 t The researcher typically waits as one of the variance parameters shrinks towards zero and the log-likelihood starts to increase rapidly, after having apparently settled down at a local maximum. The EM algorithm employed here gives only linear convergence but has the same diculty, just more slowly and less dramatically. 7 See McCulloch and Tsay (1993, 1994) and Albert and Chib (1993) for examples. 6 30 ^ j t+1 t = P ^ j ; (A10) t t where is the N by 1 vector of conditional densities dened (A4), P is the N by N transition matrix dened in (A3), 1 is a N by 1 unit vector, and the symbol denotes element-by-element multiplication. Thus the denominator in the expression (A9) is just the weighted sum of the conditional densities in , where the weights are just the t , 1 forecasted probabilities of being in the state corresponding to a particular element of . ^ j itself is just a vector of normalized, weighted, conditional densities, where the weights are the forecasted state-probabilities calculated at t , 1 and the normalization is with respect to the sum of these quantities, ensuring that the probabilities sum to unity. Some insight into the justication for this algorithm comes by noting that since X is exogenous with respect to s , the j th element of ^ j , is equivalent to P (s = j jX ; Y , ; ). Also, recall that the j th element of is f (y js = j; X ; Y , ; ). Thus the j th element of the numerator in (A9) is the product of these two quantities, which in turn is the conditional joint density of y and s , since the conditioning arguments are identical in the separate marginal densities. Thus the \conditional" probability in (A9) is just the joint density divided by the \marginal" density of y (Hamilton 1994, 693). As in most Kalman-lter applications, there is also a smoothing step involved here, which is used to derive estimates of the elements of P , the matrix of transition parameters. In estimating these transition probabilities it is optimal to exploit all sample information through time T . Smoothing algorithms provide a way of calculating the ^ conditional on the sequence obtained by iterating on the equations in (A9) and (A10). For the simple rst-order Markov chain considered here, the smoothing algorithm recommended by Hamilton (1994, 694) is ^ j = ^ j P 0 [^ j ^ j ] (A11) t t t t t t t 8 t t t t t t 1 t t t t t 1 1 t t t t T t+1 T t t t+1 t Here exogeneity is taken to mean that there is no information in X about s beyond that contained in Y , . But in my application y y , which includes the lagged level of y , , but this does not seem to violate the exogeneity condition, since y , is in Y , in any event. Alternatively, the error-correction regression could be trivially re-parameterized as a regression in levels with a lagged dependent variable. 8 t t t t 1 t 1 t t 31 1 t 1 where the sign denotes element-by-element division, and ^ j is calculated (and stored) when iterating on (A9) and (A10). This smoothing algorithm starts with t = T , 1 and with ^ j from (A10) with t = T . With the full set of smoothed probabilities, ^ j ; t = T , 1; T , 2; : : : ; 1 one has a T by N matrix of probabilities with generic row-column element (t; j ) equal to P (s = j jY ; ^ ). With this matrix maximum likelihood estimates of the transition probabilities are simply t+1 t T T t T t T p^ ij = P T P (s = j; s ,1 = ijY ; ^ ) ; P (s ,1 = ijY ; ^ ) =2 P t=2 t t (A12) T T t t T which is simply the number of times state i is estimated to have been followed by state j , divided by the number of times the process was in state i (Hamilton 1994, 695), and where ^ is the maximum likelihood estimate of . A.4.4 Estimation With an arbitrary (but reasonable) vector of starting values for ^ j and P one can start the algorithm dened by (A9) and (A10). These starting guesses are only used in the rst iteration of the estimation algorithm and the consequences of wildly implausible choices of starting values seem rarely serious. Starting values are also required for the structural parameters I have gathered here in . For these I simply use the results of the single-state ECM, replicated across the N states (here N = 2); in this case one needs to be sure that some asymmetry between the states is captured in the choice of starting values for the transition parameters and/or ^ j otherwise the algorithm fails to improve on the (perfectly symmetric) starting values. With starting values of , say , the conditional densities dened in (A4) can be used to cycle forwards through the data on (A9) and (A10), and then back through the data using (A11) to obtain smoothed estimates of P (s = j jY ; ^ ). One then estimates each of the N regressions in (A6) via GLS, with weights for the tth observation in the j th regression equal to the square root of the smoothed estimate of probability P (s = j jY ; ^ ). This application of GLS yields an updated estimate of = ( ; ; : : : ; )0. An updated 10 10 (0) t t t t 1 32 2 N t 1.00 0.75 0.50 0.25 0.0 Prob of State 1 Prob of State 1 1.00 0.75 0.50 0.25 0.0 1.00 0.75 0.50 0.25 0.0 Prob of State 1 Prob of State 1 1.00 0.75 0.50 0.25 0.0 JFK JFK Eisenhower JFK Eisenhower Eisenhower JFK Eisenhower LBJ LBJ LBJ LBJ Ford Ford Carter Carter Ford Carter Ford Carter Iteration 12 - change in llh = 0.06312 Nixon Iteration 7 - change in llh = 4.415 Nixon Iteration 3 - change in llh = 0.02193 Nixon Iteration 1 Nixon Reagan Reagan Reagan Reagan Bush Bush Bush Bush Prob of State 1 JFK JFK Eisenhower JFK Eisenhower Eisenhower JFK Eisenhower LBJ LBJ LBJ LBJ Ford Carter Ford Carter Ford Carter Ford Carter Iteration 20 - change in llh = 0.004621 Nixon Iteration 9 - change in llh = 1.497 Nixon Iteration 5 - change in llh = 4.171 Nixon Iteration 2 - change in llh = 0.5121 Nixon Reagan Reagan Reagan Reagan Figure 8: Iterative History, State Probabilities, Markov-Switching Model of Presidential Approval 1.00 0.75 0.50 0.25 0.0 1.00 0.75 0.50 0.25 0.0 1.00 0.75 0.50 Prob of State 1 Prob of State 1 Prob of State 1 0.25 0.0 1.00 0.75 0.50 0.25 0.0 33 Bush Bush Bush Bush estimate of is obtained by simply taking (1=T ) of the combined sum of squares from the N separate regressions. Together, the updates of and form ^ . Updates of the transition probabilities are obtained by applying (A12) to the current round of smoothed probabilities. The smoothed estimate ^ j replaces the starting values used to start the iterations on (A9) and (A10) and another iteration commences. This estimation procedure is an application of the EM algorithm (Dempster, Laird, and Rubin, 1977). Convergence is linear but occurs reasonably quickly with modest computing power. In S+ on a HP 715/64 workstation I obtained convergence (dened by the loglikelihood increasing by less than 10, ) in roughly 10 minutes; since convergence of the EM algorithm is linear, I obtained convergence much more rapidly with less strenuous convergence criteria. Additional overhead was consumed monitoring convergence graphically: at each iteration I plotted the estimates of the state probabilities. These estimates at selected iterations are displayed in the panels in Figure 8. The algorithm appears to settle on estimates of the state probabilities quickly: for instance, its the distinctiveness of successive observations during the Bush presidency becomes apparent after only four or ve iterations. Like most optimization algorithms, a good deal of time is spent in the neighborhood of what is at least a local optimum. 2 2 1 (1) T 8 B References Albert, J. and S. Chib. 1993. Bayes Inference Via Gibbs Sampling of Autoregressive Time Series Subject to Markov Mean and Variance Shifts. Journal of Business and Economic Statistics. 11:1-15. Beck, Nathaniel. 1992. The Methodology of Cointegration. Political Analysis. 4:237{47. Dempster, A.P., N.M. Laird, and D.B. Rubin. 1977. Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Association. Series B. 39:1{38. Durland, J. Michael and Thomas H. McGurdy. 1994. Duration-Dependent Transitions in a Markov Model of U.S. GNP Growth. 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