Twitter: @pakremp


New: What is the probability that your vote will decide the election? (with Andrew Gelman). R code available here.

New: Updating the Forecast on Election Night with R. R code available here.


Last update: Tuesday, November 8, 3:24am ET.


This is a Stan implementation of Drew Linzer’s dynamic Bayesian election forecasting model, with some tweaks to incorporate national poll data, pollster house effects, correlated priors on state-by-state election results and correlated polling errors.

For more details on the original model:

Linzer, D. 2013. “Dynamic Bayesian Forecasting of Presidential Elections in the States.” Journal of the American Statistical Association. 108(501): 124-134. (link)

The Stan and R files are available here.


1455 polls available since April 01, 2016 (including 1129 state polls and 326 national polls).


Electoral College

Note: the model does not account for the specific electoral vote allocation rules in place in Maine and Nebraska.

National Vote

This graph shows Hillary Clinton’s share of the Clinton and Trump national vote, derived from the weighted average of latent state-by-state vote intentions (using the same state weights as in the 2012 presidential election, adjusted for state adult population growth between 2011 and 2015). In the model (described below), national vote intentions are defined as:

\[\pi^{clinton}[t, US] = \sum_{s \in S} \omega_s \cdot \textrm{logit}^{-1} (\mu_a[t] + \mu_b[t, s])\]

The thick line represents the median of posterior distribution of national vote intentions; the light blue area shows the 90% credible interval. The thin blue lines represent 100 draws from the posterior distribution. The fundamentals-based prior is shown with the dotted black line.

Each national poll (raw numbers, unadjusted for pollster house effects) is represented as a dot (darker dots indicate narrower margins of error). On average, Hillary Clinton’s national poll numbers seem to be running slightly below the level that would be consistent with the latent state-by-state vote intentions.

State Vote

The following graphs show vote intention by state (with 100 draws from the posterior distribution represented as thin blue lines):

\[\pi^{clinton}[t,s] = \textrm{logit}^{-1} (\mu_a[t] + \mu_b[t, s])\]

States are sorted by predicted Clinton score on election day.

Current Vote Intentions and Forecast By State

State-by-State Probabilities

Map

Pollster House Effects

Most pro-Clinton polls:

Poll Origin Median P95 P05
Saint Leo University 2.4 1.1 3.7
Public Religion Research Institute 1.7 0.6 2.8
RABA Research 1.6 0.4 2.8
AP 1.5 0.3 2.8
Michigan State University 1.4 -0.3 3.4
GQR 1.3 0.3 2.3
ICITIZEN 1.3 0.0 2.6
UNH 1.2 -0.1 2.4
WNEU 1.2 -0.4 3.0
University of Delaware 1.0 -0.5 2.6

Most pro-Trump polls:

Poll Origin Median P95 P05
Rasmussen -2.3 -2.8 -1.7
IBD -1.7 -2.7 -0.7
Remington Research Group -1.6 -2.4 -0.8
PPIC -1.5 -3.0 -0.1
Clout Research -1.4 -3.2 0.3
Emerson College Polling Society -1.4 -2.8 0.1
Hampton University -1.3 -2.9 0.1
InsideSources -1.3 -3.1 0.5
Dan Jones -1.2 -2.8 0.2
Dixie Strategies -1.2 -2.7 0.2

Discrepancy between national polls and weighted average of state polls

Data

The runmodel.R R script downloads state and national polls from the HuffPost Pollster website as .csv files before processing the data.

The model ignores third-party candidates and undecided voters. I restrict each poll’s sample to respondents declaring vote intentions for Clinton or Trump, so that \(N = N^{clinton} + N^{trump}\). (This is problematic for Utah).

When multiple polls are available by the same pollster, at the same date, and for the same state, I pick polls of likely voters rather than registered voters, and polls for which \(N^{clinton} + N^{trump}\) is the smallest (assuming that these are poll questions in which respondents are given the option to choose a third-party candidate, rather than questions in which respondents are only asked to choose between the two leading candidates).

Polls by the same pollster and of the same state with partially overlapping dates are dropped so that only non-overlapping polls are retained, starting from the most recent poll.

To account for the fact that polls can be conducted over several days, I set the poll date to the midpoint between the day the poll started and the day it ended.

Model

The model is in the file state and national polls.stan. It has a backward component, which aggregates poll history to derive unobserved latent vote intentions; and a forward component, which predicts how these unobserved latent vote intentions will evolve until election day. The backward and forward components are linked through priors about vote intention evolution: in each state, latent vote intentions follow a reverse random walk in which vote intentions “start” on election day \(T\) and evolve in random steps (correlated across states) as we go back in time. The starting point of the reverse random walk is the final state of vote intentions, which is assigned a reasonable prior, based on the Time-for-change, fundamentals-based electoral prediction model. The model reconciles the history of state and national polls with prior beliefs about final election results and about how vote intentions evolve.

Backward Component: Poll Aggregation

For each poll \(i\), the number of respondents declaring they intended to vote for Hillary Clinton \(N^{clinton}_i\) is drawn from a binomial distribution:

\[ N^{clinton}_i \sim \textrm{Binomial}(N_i, \pi^{clinton}_i) \]

where \(N_i\) is poll sample size, and \(\pi^{clinton}_i\) is share of the Clinton vote for this poll.

The model treats national and state polls differently.

State polls

If poll \(i\) is a state poll, I use a day/state/pollster multilevel model:

\[\textrm{logit} (\pi^{clinton}_i) = \mu_a[t_i] + \mu_b[t_i, s_i] + \mu_c[p_i] + u_i + e[s_i]\]

What this model does is simply to decompose the log-odds of reported vote intentions towards Hillary Clinton \(\pi^{clinton}_i\) into a national component, shared across all states (\(\mu_a\)), a state-specific component (\(\mu_b\)), a pollster house effect (\(\mu_c\)), a poll-specific measurement noise term (\(u\)), and a polling error term (\(e\)) shared across all polls of the state (the higher \(e\), the more polls overestimate Hillary Clinton’s true score).

On the day of the last available poll \(t_{last}\), the national component \(\mu_a[t_{last}]\) is set to zero, so that the predicted share of the Clinton vote in state \(s\) (net of pollster house effects and measurement noise) after that date and until election day \(T\) is:

\[\pi^{clinton}_{ts} = \textrm{logit}^{-1} (\mu_b[t, s])\]

To reduce the number of parameters, the model only takes weekly values for \(\mu_b\), so that:

\[\mu_b[t, s] = \mu_b^{weekly}[w_t, s]\]

where \(w_t\) is the week of day \(t\).

National polls

If poll \(i\) is a national poll, I use the same multilevel approach (with random intercepts for pollster house effects \(\mu_c\)) but I add a little tweak: the share of the Clinton vote in a national poll should also reflect the weighted average of state-by-state scores at the time of the poll. I model the share of vote intentions in national polls in the following way:

\[\textrm{logit} (\pi^{clinton}_i) = \textrm{logit}\left( \sum_{s \in \{1 \dots S\}} \omega_s \cdot \textrm{logit}^{-1} (\mu_a[t_i] + \mu_b^{weekly}[w_{t_i}, s] + e[s]) \right) + \alpha + \mu_c[p_i] + u_i\]

where \(\omega_s\) represents the share of state \(s\) in the total votes of the set of polled states \(1 \dots S\) (based on 2012 turnout numbers adjusted for adult population growth in each state between 2011 and 2015). The \(\alpha\) parameter corrects for possible discrepancies between national polls and the weighted average of state polls. Possible sources of discrepancies may include:

  • the fact that when polls are not available for all states, polled states can be on average more blue or more red than the country as a whole (not a problem since the first 50-state Washington Post/SurveyMonkey poll in early September);
  • changes in state weights since 2012;
  • any possible (time-invariant) bias in national polls relative to state polls.

The idea is that while national poll levels may be off and generally not very indicative of the state of the race, national poll changes may contain valuable information to update \(\mu_a\) and (to a lesser extent) \(\mu_b\) parameters.

How vote intentions evolve

In order to smooth out vote intentions by state and obtain latent vote intentions at dates in which no polls were conducted, I use 2 reverse random walk priors for \(\mu_a\) and \(\mu_b^{weekly}\) from \(t_{last}\) to April 1:

\[\mu_b^{weekly}[w_t-1, s] \sim \textrm{Normal}(\mu_b^{weekly}[w_t, s], \sigma_b \cdot \sqrt{7})\]

\[\mu_a[t-1] \sim \textrm{Normal}(\mu_a[t], \sigma_a)\]

Both \(\sigma_a\) and \(\sigma_b\) are given uniform priors between 0 and 0.05.

Their posterior marginal distributions are shown below. The median day-to-day total standard deviation of vote intentions is about 0.4%. The model seems to find that most of the changes in latent vote intentions are attributable to national swings rather than state-specific swings (national swings account on average for about 91% of the total day-to-day variance).

Forward Component: Vote Intention Forecast

Final outcome

I use a multivariate normal distribution for the prior of the final outcome. Its mean is based on the Time-for-Change model – which predicts that Hillary Clinton should receive 48.6% of the national vote (based on Q2 GDP figures, the current President’s approval rating and number of terms). The prior expects state final scores to remain on average centered around \(48.6\% + \delta_s\), where \(\delta_s\) is the excess Obama performance relative to the national vote in 2012 in state \(s\).

\[\mu_b[T, 1 \dots S] \sim \textrm{Multivariate Normal}(\textrm{logit} (0.486 + \delta_{1 \dots S}), \mathbf{\Sigma})\]

For the covariance matrix \(\mathbf{\Sigma}\), I set the variance to 0.05 and the covariance to 0.025 for all states and pairs of states – which corresponds to a correlation coefficient of 0.5 across states.

  • This prior is relatively imprecise as to the expected final scores in any given state; for example, in a state like Virginia, which Obama won by 52% in 2012 (a score identical to his national score), Hillary Clinton is expected to get 48.6% of the vote, with a 95% certainty that her score will not fall below 38% or exceed 59%.

  • State scores are also expected to be correlated with each other. For example, according to the prior (before looking at polling data), there is only a 3.4% chance that Hillary Clinton will perform worse in Virginia than in Texas. If the priors were independent, this unlikely event could happen with a 10% probability.

The covariance matrix implies that the correlation between the 2012 state scores and 2016 state priors is expected to be about 0.94 (as opposed to 0.89 if covariances were set to zero). The simulated distribution of correlations between state priors and 2012 scores is in line with observed correlations of state scores with previous election results since 1988 [http://election.princeton.edu/2016/06/02/the-realignment-myth/].

To put it differently, the model does not have a very precise prior about final scores, but it does assume that most of this uncertainty is attributable to national-level swings in vote intentions.

How vote intentions evolve

From election day to the date of the latest available poll \(t_{last}\), vote intentions by state “start” at \(\mu_b[T,s]\) and follow a random walk with correlated steps across states:

\[\mu_b^{weekly}[w_t-1, 1 \dots S] \sim \textrm{Multivariate Normal}(\mu_b^{weekly}[w_t, 1 \dots S], \mathbf{\Sigma_b^{walk}})\]

I set \(\mathbf{\Sigma_b^{walk}}\) so that all variances equal \(0.015^2 \times 7\) and all covariances equal 0.00118 (\(\rho =\) 0.75). This implies a 0.4% standard deviation in daily vote intentions changes in a state where Hillary Clinton’s score is close to 50%. To put it differently, the prior is 95% confident that Hillary Clinton’s score in any given state where she is currently polling around 50% should not move up or down by more than 0% over the remaining 0 days until the election.

Poll house effects

Each pollster \(p\) can be biased towards Clinton or Trump:

\[\mu_c[p] \sim \textrm{Normal}(0, \sigma_c)\]

\[\sigma_c \sim \textrm{Uniform}(0, 0.1)\]

Discrepancy between national polls and the average of state polls

I give the \(\alpha\) parameter a prior centered around the observed distance of polled state voters from the national vote in 2012 (this was useful until early September, when lots of solid red states had still not been polled and the average polled state voter was more pro-Clinton than the average US voter.):

\[\bar{\delta_S} = \sum_{s \in \{1 \dots S\}} \omega_s \cdot \pi^{obama'12}_s - \pi^{obama'12}\]

\[\alpha \sim \textrm{Normal}(\textrm{logit} (\bar{\delta_S}), 0.2)\]

Measurement noise

The measurement noise term \(u_i\) is normally distributed around zero, with standard error \(\sigma_u^{national}\) for national polls, and \(\sigma_u^{state}\) for state polls. I give both standard errors a uniform distribution between 0 and 0.10.

\[\sigma_u^{national} \sim \textrm{Uniform}(0, 0.1)\] \[\sigma_u^{state} \sim \textrm{Uniform}(0, 0.1)\]

Polling error

To account for the possibility that polls might be off on average, even after adjusting for pollster house effects, the model includes a polling error term shared by all polls of the same state \(e[s]\). For example, the presence of an unexpectedly large share of Trump voters (undetected by the polls) in a given state would translate into large positive \(e\) values for that state. This polling error will remain unknown until election day; however it can be included in the form of an unidentified random parameter in the likelihood of the model, that increases the uncertainty in the posterior distribution of \(\mu_a\) and \(\mu_b\).

Because I expect polling errors to be correlated across states, I use a multivariate normal distribution:

\[e \sim \textrm{Multivariate Normal}(0, \mathbf{\Sigma_e})\]

To construct \(\mathbf{\Sigma_e}\), I set the variance to \(0.04^2\) and the covariance to 0.00175; this corresponds to a standard deviation of about 1 percentage point for a state in which Clinton’s score is close to 50% (or a 95% certainty that polls are not off by more than 2 percentage points either way); and a 0.7 correlation of polling errors across states.


Recently added polls

Entry Date Source State % Clinton / (Clinton + Trump) % Trump / (Clinton + Trump) N (Clinton + Trump)
2016-11-08 SurveyMonkey AK 37.2 62.8 319
2016-11-08 SurveyMonkey AL 39.6 60.4 1029
2016-11-08 SurveyMonkey AR 37.8 62.2 837
2016-11-08 SurveyMonkey AZ 51.1 48.9 2296
2016-11-08 SurveyMonkey CA 67.8 32.2 2359
2016-11-08 SurveyMonkey CO 54.8 45.2 2333
2016-11-08 SurveyMonkey CT 60.2 39.8 1221
2016-11-08 SurveyMonkey DE 57.3 42.7 327
2016-11-08 SurveyMonkey FL 54.8 45.2 3806
2016-11-08 SurveyMonkey GA 50.0 50.0 2177
2016-11-08 SurveyMonkey HI 64.2 35.8 345
2016-11-08 SurveyMonkey IA 38.8 61.2 1514
2016-11-08 SurveyMonkey ID 35.1 64.9 471
2016-11-08 SurveyMonkey IL 60.7 39.3 1622
2016-11-08 SurveyMonkey IN 39.8 60.2 1496
2016-11-08 SurveyMonkey KS 43.0 57.0 1127
2016-11-08 SurveyMonkey KY 42.2 57.8 1184
2016-11-08 SurveyMonkey LA 43.3 56.7 884
2016-11-08 SurveyMonkey MA 67.8 32.2 1199
2016-11-08 SurveyMonkey MD 64.8 35.2 1070
2016-11-08 SurveyMonkey ME 51.8 48.2 662
2016-11-08 SurveyMonkey MI 50.0 50.0 2768
2016-11-08 SurveyMonkey MN 53.5 46.5 1111
2016-11-08 SurveyMonkey MO 45.5 54.5 1204
2016-11-08 SurveyMonkey MS 43.5 56.5 661
2016-11-08 SurveyMonkey MT 35.4 64.6 368
2016-11-08 SurveyMonkey NC 54.4 45.6 2813
2016-11-08 SurveyMonkey ND 35.6 64.4 272
2016-11-08 SurveyMonkey NE 39.8 60.2 869
2016-11-08 SurveyMonkey NH 57.0 43.0 599
2016-11-08 SurveyMonkey NJ 57.3 42.7 1226
2016-11-08 SurveyMonkey NM 47.4 52.6 584
2016-11-08 SurveyMonkey NV 52.3 47.7 1062
2016-11-08 SurveyMonkey NY 64.0 36.0 1965
2016-11-08 SurveyMonkey OH 48.9 51.1 2517
2016-11-08 SurveyMonkey OK 35.6 64.4 1144
2016-11-08 SurveyMonkey OR 58.6 41.4 1388
2016-11-08 SurveyMonkey PA 50.0 50.0 2504
2016-11-08 SurveyMonkey RI 55.8 44.2 357
2016-11-08 SurveyMonkey SC 46.2 53.8 1545
2016-11-08 SurveyMonkey SD 40.7 59.3 395
2016-11-08 SurveyMonkey TN 42.9 57.1 1304
2016-11-08 SurveyMonkey TX 47.2 52.8 2678
2016-11-08 SurveyMonkey UT 56.7 43.3 991
2016-11-08 SurveyMonkey VA 54.4 45.6 1965
2016-11-08 SurveyMonkey VT 66.3 33.7 390
2016-11-08 SurveyMonkey WA 61.2 38.8 1233
2016-11-08 SurveyMonkey WI 51.1 48.9 1976
2016-11-08 SurveyMonkey WV 35.3 64.7 401
2016-11-07 ABC 52.2 47.8 1998
2016-11-07 Angus Reid 52.2 47.8 1024
2016-11-07 Bloomberg 51.8 48.2 679
2016-11-07 CBS 52.3 47.7 1226
2016-11-07 FOX 52.2 47.8 1191
2016-11-07 Franklin Pierce 52.2 47.8 928
2016-11-07 IBD 48.8 51.2 862
2016-11-07 Lucid 52.9 47.1 791
2016-11-07 Monmouth University 53.2 46.8 703
2016-11-07 Rasmussen 51.1 48.9 1320
2016-11-07 UPI 51.6 48.4 1544
2016-11-07 YouGov 52.3 47.7 3155
2016-11-07 UPI AK 43.6 56.4 290
2016-11-07 Ipsos AL 41.3 58.7 538
2016-11-07 UPI AL 38.9 61.1 324
2016-11-07 Ipsos AR 41.8 58.2 469
2016-11-07 UPI AR 39.4 60.6 304
2016-11-07 Data Orbital AZ 48.4 51.6 500
2016-11-07 Ipsos AZ 47.2 52.8 763
2016-11-07 UPI AZ 46.2 53.8 329
2016-11-07 Ipsos CA 64.8 35.2 1268
2016-11-07 UPI CA 61.1 38.9 482
2016-11-07 Ipsos CO 53.3 46.7 626
2016-11-07 Remington Research Group CO 50.6 49.4 1546
2016-11-07 UPI CO 53.2 46.8 322
2016-11-07 Ipsos CT 55.6 44.4 464
2016-11-07 UPI CT 57.9 42.1 311
2016-11-07 UPI DE 56.2 43.8 295
2016-11-07 Ipsos FL 50.5 49.5 1210
2016-11-07 Opinion Savvy FL 51.1 48.9 802
2016-11-07 Quinnipiac FL 50.5 49.5 804
2016-11-07 Remington Research Group FL 48.4 51.6 2187
2016-11-07 UPI FL 50.0 50.0 412
2016-11-07 CBS GA 46.7 53.3 915
2016-11-07 Ipsos GA 46.7 53.3 622
2016-11-07 Landmark GA 48.4 51.6 1140
2016-11-07 UPI GA 46.8 53.2 344
2016-11-07 UPI HI 67.7 32.3 296
2016-11-07 Ipsos IA 50.0 50.0 524
2016-11-07 UPI IA 49.5 50.5 325
2016-11-07 Ipsos ID 36.1 63.9 270
2016-11-07 UPI ID 35.5 64.5 299
2016-11-07 Ipsos IL 56.7 43.3 703
2016-11-07 UPI IL 57.4 42.6 379
2016-11-07 Ipsos IN 41.3 58.7 498
2016-11-07 UPI IN 45.7 54.3 318
2016-11-07 Ipsos KS 41.3 58.7 477
2016-11-07 UPI KS 41.9 58.1 305
2016-11-07 Ipsos KY 39.6 60.4 625
2016-11-07 UPI KY 37.5 62.5 315
2016-11-07 Ipsos LA 40.9 59.1 557
2016-11-07 UPI LA 42.1 57.9 313
2016-11-07 Ipsos MA 59.6 40.4 456
2016-11-07 UPI MA 61.5 38.5 317
2016-11-07 Ipsos MD 62.2 37.8 648
2016-11-07 UPI MD 59.6 40.4 309
2016-11-07 Ipsos ME 54.5 45.5 234
2016-11-07 UPI ME 56.8 43.2 293
2016-11-07 Ipsos MI 50.5 49.5 554
2016-11-07 UPI MI 52.1 47.9 314
2016-11-07 Ipsos MN 55.0 45.0 610
2016-11-07 UPI MN 52.1 47.9 317
2016-11-07 Ipsos MO 45.5 54.5 736
2016-11-07 UPI MO 47.3 52.7 353
2016-11-07 Ipsos MS 40.4 59.6 345
2016-11-07 UPI MS 43.8 56.2 317
2016-11-07 UPI MT 43.2 56.8 322
2016-11-07 Ipsos NC 49.5 50.5 604
2016-11-07 NYT Upshot NC 50.0 50.0 704
2016-11-07 Quinnipiac NC 51.1 48.9 800
2016-11-07 Remington Research Group NC 48.4 51.6 2414
2016-11-07 UPI NC 49.5 50.5 339
2016-11-07 UPI ND 41.5 58.5 289
2016-11-07 Ipsos NE 37.6 62.4 298
2016-11-07 UPI NE 40.4 59.6 320
2016-11-07 Ipsos NH 51.2 48.8 272
2016-11-07 UNH NH 56.3 43.7 602
2016-11-07 UPI NH 53.3 46.7 285
2016-11-07 Ipsos NJ 55.8 44.2 566
2016-11-07 UPI NJ 57.3 42.7 334
2016-11-07 Albuquerque Journal NM 52.9 47.1 428
2016-11-07 Ipsos NM 51.9 48.1 217
2016-11-07 UPI NM 55.3 44.7 280
2016-11-07 Ipsos NV 51.7 48.3 497
2016-11-07 Remington Research Group NV 49.5 50.5 1632
2016-11-07 UPI NV 51.6 48.4 307
2016-11-07 Ipsos NY 60.7 39.3 1162
2016-11-07 UPI NY 60.8 39.2 444
2016-11-07 Ipsos OH 50.5 49.5 721
2016-11-07 Remington Research Group OH 49.4 50.6 2276
2016-11-07 TargetSmart OH 48.2 51.8 991
2016-11-07 UPI OH 49.5 50.5 328
2016-11-07 Ipsos OK 33.0 67.0 604
2016-11-07 UPI OK 35.1 64.9 313
2016-11-07 Ipsos OR 56.0 44.0 534
2016-11-07 UPI OR 54.3 45.7 300
2016-11-07 CBS PA 51.1 48.9 819
2016-11-07 Ipsos PA 51.6 48.4 888
2016-11-07 Remington Research Group PA 50.5 49.5 2442
2016-11-07 UPI PA 50.5 49.5 370
2016-11-07 UPI RI 61.5 38.5 298
2016-11-07 Ipsos SC 47.3 52.7 574
2016-11-07 UPI SC 44.2 55.8 321
2016-11-07 UPI SD 42.1 57.9 296
2016-11-07 Ipsos TN 41.9 58.1 708
2016-11-07 UPI TN 41.5 58.5 319
2016-11-07 Ipsos TX 44.3 55.7 1067
2016-11-07 UPI TX 42.6 57.4 444
2016-11-07 CBS UT 36.5 63.5 480
2016-11-07 Ipsos UT 45.2 54.8 388
2016-11-07 UPI UT 39.4 60.6 227
2016-11-07 Christopher Newport VA 53.3 46.7 1074
2016-11-07 Ipsos VA 52.9 47.1 553
2016-11-07 Remington Research Group VA 51.1 48.9 2768
2016-11-07 UPI VA 53.2 46.8 323
2016-11-07 UPI VT 65.3 34.7 290
2016-11-07 Ipsos WA 55.3 44.7 734
2016-11-07 UPI WA 58.9 41.1 312
2016-11-07 Ipsos WI 53.5 46.5 724
2016-11-07 Remington Research Group WI 54.4 45.6 2448
2016-11-07 UPI WI 54.3 45.7 350
2016-11-07 Ipsos WV 37.0 63.0 388
2016-11-07 UPI WV 37.8 62.2 278
2016-11-07 UPI WY 29.5 70.5 304

Convergence checks

With 4 chains and 2000 iterations (the first 1000 iterations of each chain are discarded), the model runs in less than 15 minutes on my 4-core Intel i7 MacBookPro.

##  [1] "Inference for Stan model: state and national polls."                          
##  [2] "4 chains, each with iter=2000; warmup=1000; thin=1; "                         
##  [3] "post-warmup draws per chain=1000, total post-warmup draws=4000."              
##  [4] ""                                                                             
##  [5] "                   mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat"
##  [6] "alpha             -0.01       0 0.01 -0.03 -0.02 -0.01 -0.01  0.00  2401 1.00"
##  [7] "sigma_c            0.05       0 0.01  0.04  0.04  0.05  0.05  0.06   966 1.00"
##  [8] "sigma_u_state      0.06       0 0.00  0.05  0.06  0.06  0.06  0.07  1114 1.00"
##  [9] "sigma_u_national   0.02       0 0.01  0.01  0.01  0.02  0.02  0.03   678 1.01"
## [10] "sigma_walk_a_past  0.02       0 0.00  0.01  0.01  0.02  0.02  0.02  1806 1.00"
## [11] "sigma_walk_b_past  0.01       0 0.00  0.00  0.00  0.01  0.01  0.01   808 1.01"
## [12] "mu_b[33,2]        -0.26       0 0.07 -0.40 -0.31 -0.26 -0.21 -0.13  3390 1.00"
## [13] "mu_b[33,3]        -0.43       0 0.07 -0.56 -0.48 -0.43 -0.39 -0.30  3389 1.00"
## [14] "mu_b[33,4]        -0.41       0 0.07 -0.54 -0.46 -0.41 -0.37 -0.29  3060 1.00"
## [15] "mu_b[33,5]        -0.08       0 0.06 -0.20 -0.12 -0.08 -0.04  0.04  3090 1.00"
## [16] "mu_b[33,6]         0.59       0 0.06  0.47  0.54  0.59  0.63  0.70  2904 1.00"
## [17] "mu_b[33,7]         0.10       0 0.06 -0.01  0.06  0.10  0.14  0.21  2811 1.00"
## [18] "mu_b[33,8]         0.29       0 0.07  0.16  0.24  0.29  0.33  0.42  2551 1.00"
## [19] "mu_b[33,9]         0.33       0 0.07  0.19  0.29  0.33  0.38  0.46  2519 1.00"
## [20] "mu_b[33,10]        0.03       0 0.06 -0.08 -0.01  0.03  0.07  0.15  2639 1.00"
## [21] "mu_b[33,11]       -0.09       0 0.06 -0.20 -0.13 -0.09 -0.05  0.02  2690 1.00"
## [22] "mu_b[33,12]        0.61       0 0.07  0.47  0.56  0.61  0.66  0.76  3246 1.00"
## [23] "mu_b[33,13]       -0.11       0 0.06 -0.23 -0.15 -0.11 -0.07  0.02  2523 1.00"
## [24] "mu_b[33,14]       -0.58       0 0.07 -0.71 -0.62 -0.58 -0.53 -0.44  3021 1.00"
## [25] "mu_b[33,15]        0.37       0 0.06  0.25  0.33  0.37  0.41  0.49  3025 1.00"
## [26] "mu_b[33,16]       -0.31       0 0.06 -0.43 -0.35 -0.31 -0.27 -0.19  2560 1.00"
## [27] "mu_b[33,17]       -0.31       0 0.06 -0.44 -0.36 -0.31 -0.27 -0.19  2769 1.00"
## [28] "mu_b[33,18]       -0.48       0 0.07 -0.61 -0.53 -0.48 -0.44 -0.35  3138 1.00"
## [29] "mu_b[33,19]       -0.34       0 0.06 -0.46 -0.38 -0.34 -0.29 -0.22  2601 1.00"
## [30] "mu_b[33,20]        0.59       0 0.06  0.46  0.54  0.59  0.63  0.71  2665 1.00"
## [31] "mu_b[33,21]        0.63       0 0.06  0.51  0.59  0.63  0.67  0.76  2779 1.00"
## [32] "mu_b[33,22]        0.17       0 0.06  0.05  0.13  0.17  0.21  0.30  2637 1.00"
## [33] "mu_b[33,23]        0.09       0 0.06 -0.03  0.05  0.09  0.13  0.21  2482 1.00"
## [34] "mu_b[33,24]        0.16       0 0.06  0.04  0.12  0.16  0.20  0.29  3071 1.00"
## [35] "mu_b[33,25]       -0.21       0 0.06 -0.33 -0.26 -0.22 -0.17 -0.09  2751 1.00"
## [36] "mu_b[33,26]       -0.23       0 0.07 -0.36 -0.28 -0.23 -0.19 -0.10  3336 1.00"
## [37] "mu_b[33,27]       -0.39       0 0.07 -0.53 -0.44 -0.39 -0.34 -0.25  3224 1.00"
## [38] "mu_b[33,28]        0.03       0 0.06 -0.09 -0.02  0.03  0.07  0.14  2565 1.00"
## [39] "mu_b[33,29]       -0.47       0 0.07 -0.62 -0.52 -0.47 -0.42 -0.33  3158 1.00"
## [40] "mu_b[33,30]       -0.43       0 0.07 -0.56 -0.48 -0.43 -0.38 -0.30  2778 1.00"
## [41] "mu_b[33,31]        0.10       0 0.06 -0.01  0.06  0.10  0.14  0.22  2962 1.00"
## [42] "mu_b[33,32]        0.32       0 0.06  0.20  0.27  0.32  0.36  0.44  2863 1.00"
## [43] "mu_b[33,33]        0.15       0 0.06  0.03  0.11  0.15  0.19  0.27  2797 1.00"
## [44] "mu_b[33,34]        0.02       0 0.06 -0.10 -0.02  0.02  0.06  0.14  2634 1.00"
## [45] "mu_b[33,35]        0.49       0 0.06  0.37  0.45  0.49  0.54  0.62  2690 1.00"
## [46] "mu_b[33,36]       -0.03       0 0.06 -0.15 -0.08 -0.03  0.01  0.08  2644 1.00"
## [47] "mu_b[33,37]       -0.60       0 0.07 -0.73 -0.65 -0.60 -0.56 -0.47  3293 1.00"
## [48] "mu_b[33,38]        0.24       0 0.06  0.12  0.20  0.24  0.28  0.37  3050 1.00"
## [49] "mu_b[33,39]        0.09       0 0.06 -0.02  0.05  0.09  0.13  0.21  2499 1.00"
## [50] "mu_b[33,40]        0.33       0 0.07  0.19  0.28  0.33  0.38  0.47  2861 1.00"
## [51] "mu_b[33,41]       -0.14       0 0.06 -0.27 -0.19 -0.14 -0.10 -0.02  2802 1.00"
## [52] "mu_b[33,42]       -0.40       0 0.07 -0.53 -0.44 -0.40 -0.35 -0.26  3348 1.00"
## [53] "mu_b[33,43]       -0.37       0 0.07 -0.49 -0.41 -0.37 -0.32 -0.24  2938 1.00"
## [54] "mu_b[33,44]       -0.21       0 0.06 -0.33 -0.25 -0.21 -0.17 -0.09  2533 1.00"
## [55] "mu_b[33,45]       -0.30       0 0.06 -0.42 -0.34 -0.30 -0.26 -0.18  2842 1.00"
## [56] "mu_b[33,46]        0.14       0 0.06  0.02  0.10  0.14  0.18  0.26  2848 1.00"
## [57] "mu_b[33,47]        0.73       0 0.07  0.58  0.68  0.73  0.78  0.87  2971 1.00"
## [58] "mu_b[33,48]        0.31       0 0.06  0.19  0.27  0.32  0.36  0.44  3109 1.00"
## [59] "mu_b[33,49]        0.11       0 0.06  0.00  0.07  0.11  0.15  0.23  2753 1.00"
## [60] "mu_b[33,50]       -0.58       0 0.07 -0.71 -0.63 -0.58 -0.54 -0.45  3174 1.00"
## [61] "mu_b[33,51]       -0.93       0 0.08 -1.07 -0.98 -0.93 -0.88 -0.78  3328 1.00"
## [62] ""                                                                             
## [63] "Samples were drawn using NUTS(diag_e) at Tue Nov  8 09:01:10 2016."           
## [64] "For each parameter, n_eff is a crude measure of effective sample size,"       
## [65] "and Rhat is the potential scale reduction factor on split chains (at "        
## [66] "convergence, Rhat=1)."