Oct. 24: In Polls, Romney\'s Momentum Seems to Have Stopped

The term “momentum” is used very often in political coverage - but reporters and analysts seldom pause to consider what it means.

Let me tell you what I think it ought to mean: that a body in motion tends to stay in motion. That is, it ought to imply that a candidate is gaining ground in the race - and, furthermore, that he is likely to continue to gain ground.

As a thesis or prediction about how polls behave, this notion is a bit dubious, especially in general elections. In races for the United States Senate, for instance, my research suggests that a candidate who gains ground in the polls in one month (say, from August to September) is no more likely to do so during the next one (from September to October). If anything, the candidate who gains ground in the polls in one month may be more likely to lose ground the next time around.

(Where might there be clearer evidence for momentum, as I've defined it? In primaries, especially when their are multipl e candidates in the race and voters are behaving tactically in choosing among them. But there is little evidence of it in general elections.)

The way the term “momentum” is applied in practice by the news media, however, it usually refers only to the first part of the clause - meaning simply that a candidate has been gaining ground in the polls, whether or not he might continue to do so. (I've used this phrasing plenty of times myself, so I have no real basis to complain about it.)

But there are other times when the notion of momentum is behind the curve - as it probably now is if applied to Mitt Romney's polling.

Mr. Romney clearly gained ground in the polls in the week or two after the Denver debate, putting himself in a much stronger overall position in the race. However, it seems that he is no longer doing so.

Take Wednesday's national tracking polls, for instance. (There are now eight of them published each day.) Mr. Romney gained ground in just one of the polls, an online poll conducted for Reuters by the polling organization Ipsos. He lost ground in five others, with President Obama improving his standing instead in those surveys. On average, Mr. Obama gained about one point between the eight polls.

This is the closest that we've come in a week or so to one candidate clearly having “won” the day in the tracking polls - and it was Mr. Obama.

The trend could also be spurious. If the race is steady, it's not that hard for one candidate to gain ground in five of six polls (excluding the two that showed no movement on Wednesday) just based on chance alone.

What isn't very likely, however, is for one candidate to lose ground in five of six polls if the race is still moving toward him. In other words, we can debate whether Mr. Obama has a pinch of momentum or whether the race is instead flat, but it's improbable that Mr. Romney would have a day like this if he still had momentum.

The FiveThirtyEight model looks at a broader array of polls - including state polls - in order to gauge the overall trend in the race.

Our “now-cast” also finds a slightly favorable trend for Mr. Obama over the course of the past 10 days or so. Mr. Romney's position peaked in the “now-cast” on Friday, Oct. 12, at which point it estimated a virtual tie in the popular vote (Mr. Obama was the projected “winner” by 0.3 percentage points). As of Wednesday, however, Mr. Obama was 1.4 percentage points ahead in the “now-cast”, meaning that he may have regained about 1 percentage point of the 4 points or so that he lost after Denver. Mr. Obama's chances of winning the Electoral College were up in the FiveThirtyEight forecast to 71 percent on Wednesday from 68.1 percent on Tuesday.

It's not yet clear how much of this, if any, has to do with the final presidential debate in Florida this Monday, which instant polls regarded Mr. Obama as having won. Instead, it's been more of a slow and unsteady trajectory for him, with Mr. Obama often taking two steps forward but then one step back. It's also not out of the question that the apparent trend just represents statistical noise.

At the same time, there is more reason to take a potential change in the polls seriously if it is precipitated by a news event like the debate. The tracking polls that were released on Wednesday contained only one full day of interviews that postdated the Florida debate. If the debate moved the needle toward Mr. Obama, it should become more apparent in the coming days.

The battleground state polls that came in on Wednesday were generally very close to o ur model's current projections. For instance, there were three Ohio polls published on Wednesday; one showed a tied race there, while the other two showed Mr. Obama ahead by margins of two and five points.That's pretty much what you'd expect to see out of a trio of Ohio polls if Mr Obama's lead there were about two points, which is where our model now has it.

Some of the polls, especially the Time Magazine poll which had Mr. Obama five points ahead in Ohio, seemed to set off a lot of discussion on Twitter, as though people were surprised that Mr. Obama still held the lead there.

But these polls are really nothing new. Since the Denver debate, Mr. Obama has held the lead in 16 Ohio polls against 6 for Mr. Romney. In Nevada, Mr. Obama has had the lead in11 poll, to Mr. Romney's 1. Mr. Obama has led in all polls of Wisconsin since the Denver debate, and he has had five poll leads in Iowa to one for Mr. Romney.

Part of the confusion (and part of the reason beh ind the perception that Mr. Romney is still gaining ground in the race) may be because of the headlines that accompany polls.

We're still getting some polls trickling in where the most recent comparison is to a poll conducted before the Denver debate. We should expect Mr. Romney to gain ground relative to a poll conducted before Denver. (Mr. Romney may have lost a point or so off his bounce, but he has clearly not lost all of it). But it isn't news when he does; Mr. Romney's Denver gains had long ago become apparent, and priced into the various polling averages and forecast models.

The question, rather, is whether Mr. Romney is gaining ground relative to the post-Denver polls - or if, as Wednesday's polls seemed to imply, the race instead may have ticked back slightly toward Mr. Obama.

Oct. 23: The Virtues and Vices of Election Prediction Markets

Tuesday featured what, on the surface, would seem to be a fairly neutral day of polling. Of the 11 national polls published on Tuesday, five showed President Obama ahead, four gave a lead to Mitt Romney and two had a tied race. On average among the polls, Mr. Romney led by 0.3 percentage points, the same as his average margin in the previous edition of the same polls.

The volume of battleground state polling was light on Tuesday, and what polls there were require a few cautions. For instance, an Old Dominion University poll of Virginia put Mr. Obama seven points ahead there â€" but the poll had an extremely long field period, having been started in September, with about two-thirds of i ts interviews coming before the first presidential debate in Denver. It thus receives little weight in the forecast.

The best-looking result of the day for Mr. Romney was probably an American Research Group poll of New Hampshire, which had him up two points there â€" although its prior poll of the state had Mr. Romney up four points, so this isn't necessarily a sign of a change in his favor.

Similarly, a SurveyUSA poll of Ohio had Mr. Obama three points ahead there â€" a decent result for him, but the same as in its previous poll of the state.

Mr. Obama's chances of winning the Electoral College were down slightly in the FiveThirtyEight forecast, to 68.1 percent from 70.3 per cent on Monday. Some of the decline was because of the sharp decrease in the stock market on Tuesday, which still has some influence on the FiveThirtyEight economic index. Some of the rest is because among the various national polls, the one that the model rates most highly is the national tracking poll run by ABC News and The Washington Post, and it showed a slight decline for Mr. Obama.

At the political betting market Intrade, however, traders seemed to have a much stronger opinion about the day's news. Mr. Romney's stock, which can be read as a forecast of the chance that he will win the Electoral College, rose to 45.4 percent from 39.7 over the course of the day.

The probabilistic forecasts issued by FiveThirtyEight have been quite close to Intrade and those at other trading and betting markets over the course of the election. The spread as of Tuesday night â€" with Intrade implying that Mr. Obama has a 55 percent chance of winnin g the election, and FiveThirtyEight a 68 percent chance â€" is much wider than usual.

However, Intrade is also something of an outlier relative to other prediction markets. Another prediction market, Betfair, gave Mr. Obama about a 63 percent chance of winning as of early Wednesday morning. The consensus of sports books, similarly, implied that he has about a 65 percent chance. These values are not significantly different from the FiveThirtyEight forecast.

The general question of whether FiveThirtyEight ought to be better than prediction and betting markets is an interesting one. I am far from an efficient-market hypothesis purist, but markets are tough to beat in most circumstances. Furthermore, the FiveThirtyEight forecasts are public information, and bettors can use our forecasts along with those of our competitors to calibrate their estimates of the outcomes.

A 2009 paper by David Rothschild found that FiveThirtyEight's forecasts were somewhat better th an Intrade's over the course of the 2008 presidential election cycle. Mr. Rothschild wrote, however, that Intrade's forecasts would have outperformed FiveThirtyEight's had they been corrected to adjust for the market's tendency to overrate the chances of the trailing candidate (what is known as a favorite-long shot bias). Of course, if that bias still applies in Intrade's pricing, it would imply that the market is overrating Mr. Romney's chances now as he is the underdog in the race.

The more pertinent question for the time being may be why Intrade differs from the other betting markets, and from the sports books.

It is unusual for differences like these to persist for very long in well-functioning markets, because they would potentially enable a trader to make a risk-free profit by selling Mr. Romney's stock short at Intrade, where it is expensive, and then betting on him at one of the sports books or at another prediction market, where it is relatively cheap. The fact that these relatively wide spreads exist between the different markets suggest that they are relatively illiquid and that they entail high transaction costs, which may limit their efficiency.

In fact, these differences between Intrade and the other prediction markets have persisted throughout much of the past two months. Intrade has consistently shown slightly more favorable odds for Mr. Romney than other prediction and gambling sites.

According to Brandon Adams, a professional poker player and a teaching fellow at Harvard, Intrade's prices are not necessarily considered more reliable by professional gamblers. Mr. Adams says that Betfair and the sports book Pinnacle, both of which put Mr. Obama's odds at about 63 percent as of early Wednesday morning, feature more sophisticated market participants.

However, Intrade is cited far more frequently by the American news media. That at least opens up the possibility that someone could place a wager on Mr. Romney (or Mr. Obama) at Intrade in order to influence the news media's perceptions about which candidate has the momentum.

At several points on Tuesday and early Wednesday, what appeared to be a single trader bought a large number of Mr. Romney's shares at Intrade, at one point boosting Mr. Romney's chances to about 49 percent from 41 percent over the span of a few minutes. The betting patterns echo similar ones in the pricing of John McCain and Mr. Obama's stock at Intrade late in the 2008 cycle.

It is not necessarily safe to conclude that represents an effort to manipulate the market, or public perceptions about Mr. Romney's chances. Instead, it could just be someone who thinks Mr. Romney is a good bet relative to the price being offered.

A sophisticated bettor, however, would probably have sought out somewhere else to place his bets on Mr. Romney, since even before the recent market action, they offered a more favorable price than Intrade did.

So me people are also inclined to conclude that the “rogue trader” may have inside information about the state of the race. Personally, I think this is unlikely. The overall amount of the bet early on Tuesday equaled about $18,000. That is a large bet in everyday terms, but not by the standards of a professional gambler or investor who felt he had a sure thing.

A truly sophisticated trader, for instance, could make bets in the actual stock market, buying up certain stocks in industries that might be expected to perform well under a Romney presidency (for instance, those of oil or coal companies) while shorting those that would probably benefit from Mr. Obama's remaining in office (for example, certain health care stocks or green-technology companies). Although it would require some skill to build such portfolios, the potential payoff would be much greater â€" measured in the millions rather than the thousands of dollars.

My conclusion, then, is that the recent b ets in Intrade are unlikely to represent the behavior of a sophisticated trader. Beyond that, it is hard to infer much about the motivation behind them.

Markets of any kind are potentially subject to herding behavior. Even if the “rogue trader” does not have superior information about the campaign, other traders might alter their behavior assuming that he did.

Or traders could be betting on speculation â€" buying up Mr. Romney's stock, for example, in the hope of selling it at a higher price later, even if that price diverged from their estimate of the “fundamentals” in the race.

Coming up with a probabilistic assessment of the chances of each candidate's winning the Electoral College is intrinsically a very challenging task. ”Gut-feel” probability estimates are not likely to be much good. It is the sort of problem, in my view, that benefits from a more rigorous approach, i.e. building a model. Yet, models are constrained by the lack of historic al data and the infrequency of presidential elections â€" only since about 1980 has state-by-state polling of the election been common.

In these circumstances, traders may be looking for an anchor to calibrate their assessment of the probabilities. It is certainly possible that FiveThirtyEight itself serves something of this function.

The FiveThirtyEight model is sometimes perceived as being incredibly bold for having Mr. Obama as a two-to-one favorite despite what is certainly very close polling, but the few other models that seek to frame the election in probabilistic terms tend to give Mr. Obama an even clearer advantage, putting his odds at between 80 and 95 percent.

I would be happy to engage those other forecasters in a (good-natured) argument about why I think their models are considerably too confident about Mr. Obama's chances. But if the FiveThirtyEight model is thought of as “Obama-leaning” relative to the conventional wisdom, it is “Romney -leaning” relative to the other models that use a similar approach based on state polling data.

For betting markets like Intrade, it would be useful to have a dozen FiveThirtyEight-like models in the public sphere. The consensus of models or forecasts is very often better than even the best-performing members of the group. (This is the sense in which I remarked on Twitter on Tuesday that I thought FiveThirtyEight was “overrated.”)

But I also think that efforts to look at the polls in a systematic way are likely to be superior to much of what passes for polling analysis, which looks at them on an ad-hoc basis. Trying to decode the meaning of an individual poll is not a terribly useful exercise when 20 or 30 of them are released every day.

Instead, debates about overall, macro-level strategies for processing the polls and other information about the election â€" for instance, how one might reconcile state polls against the national ones â€" are liable t o be much more informative.

They could be especially informative to betting markets, serving as a better anchor against which hard-to-quantify elements about the election could be weighed. Betting markets certainly have some advantages over model-based approaches, like being able to react to news events before their effects show up in the polls.

But these advantages may be outweighed if most participants in the market are prone toward herding, speculation and similar behaviors â€" and if the markets are not liquid enough, and do not offer high enough stakes, to be attractive to highly sophisticated traders.

So is FiveThirtyEight better than Intrade? I don't know about that as a general proposition.

But I'd be happy to bet on FiveThirtyEight relative to the price that Intrade offered on Mr. Obama on Tuesday, which was not characteristic of rational market behavior.

Why Arizona Isn\'t a Battleground State (and Why It May Be Soon)

We continue our Presidential Geography series, a one-by-one examination of each state's political landscape and how it's changing. Here is Arizona, the Grand Canyon State. FiveThirtyEight spoke with Bruce Merrill, David Berman and Joseph Garcia of Arizona State University's Morrison Institute for Public Policy. Mr. Merrill and Mr. Berman are senior research fellows, and Mr. Garcia is the director of the institute's Latino Public Policy Center. FiveThirtyEight also spoke with Chip Scutari and Dave Cieslak, co-founders of the political consulting firm Scutari & Cieslak Public Relations.

While the Republican primaries were still unfolding, the Obama campaign suggested it could compete in Arizona. That looks to have been a bit optimistic: polls show Mitt Romney leading comfortably.

But the idea that Arizona would be a battleground in 2012 made some sense. President Obama made substantial gains in the Southwest in 2008, carrying three of Arizona's neighbors - Nevada , Colorado and New Mexico - that had voted Republican in 2004. Arizona was the exception, but Mr. Obama had a good excuse: there was an Arizonan on the ballot, Senator John McCain.

In 2012 that would not be an issue; Mr. Romney has a lot of “home” states (he was governor of Massachusetts, grew up in Michigan and has a vacation home in New Hampshire), but Arizona isn't one of them.

In addition, Democrats have some advantages in the Grand Canyon State. It is almost one-third Hispanic, the fourth-highest share in the nation, and polls show Latinos favoring Mr. Obama over Mr. Romney by about two to one. Arizona is also the ninth-most-urban state by population, with a majority of its residents clustered in and around Phoenix.

Nonetheless, Arizona simply isn't that near to the tipping point. In a presidential election that is at all close nationally - and the 2012 race is certainly close - Arizona is not a truly competitive state, yet. (More on that yet in a moment.)

Hispanics may comprise about 30 percent of the state, but they make up 25 percent of the state's voting-age population, because they skew much younger than non-Hispanic whites. In addition, roughly one-third of Arizona Latinos are undocumented and cannot vote. Even among Hispanics who can vote, turnout has lagged behind that of non-Hispanic whites.

In percentage terms, Hispanics account for roughly half as much of the Arizona electorate, 16 percent, as they do the state's overall population, 30 percent.

And even though the state's Latino voters have helped move Arizona toward the political left, the state started far to the political right. Republicans carried Arizona in every presidential race from 1952 to 1992, the only state to stay Republican over that period. Arizona was home to Barry Goldwater, one of the most conservative Republicans to run for president since 1952. Mr. Goldwater exemplified the political ethos of the state: conservative, with a strong libertarian streak.

Arizona is a Republican-leaning state largely because Maricopa County is a Republican-leaning county. With about 60 percent of the electorate, Maricopa County's political preferences tend to carry the day.

While Phoenix itself skews Democratic, most of the vast suburbs do not. “The East Valley - including Scottsdale, Chandler, Gilbert a nd Mesa - tend to be quite Republican,” Mr. Berman said. The East Valley also has a large Mormon community. (Arizona is the fifth-most Mormon state, but members of the Church of Jesus Christ of Latter-day Saints make up just 5 percent of the state over all).

The Republican Party also dominates in western and southeastern Arizona. Mohave and Yavapai Counties, in the northwest, are older, more rural, less diverse and more socially conservative than the Phoenix suburbs.

Maricopa County is so big that even though it leans Republican, Democrats get the bulk of their votes there, too. But the main Democratic-leaning county in Arizona is Pima County, which is more than one-third Hispanic and includes Tucson, where the University of Arizona is situated

Politics in Arizona's less populated counties tends to be determined by a mix of demographics and religion. Democrats normally carry Coconino County, which includes Flagstaff and is about a quarter Native American and 15 percent Hispanic; Apache County, which is about 72 percent American Indian; and Santa Cruz County, which is almost 83 percent Hispanic.

Graham County is reliably Republican. It is 31 percent Hispanic but also 26 percent Mormon. Navajo County has been more marginally Republican. It is 43 percent American Indian but 20 percent Mormon.

The Bellwether

Maricopa County accounts for so much of Arizona's electorate that it is almost guaranteed to be a good bellwether for the statewide vote. It was just one percentage point more Republican-leaning than the state in 2008 and was just two percentage points more Democratic-leaning in 2004 and 2000.

The Bottom Line

Mr. Romney is a 98 percent favorite in Arizona, according to the current FiveThirtyEight forecast. But the days when Republicans can count on carrying the Grand Canyon State may be numbered.

More and more voters are registering as independents. The number of unaffiliated voters has surpassed the number of registered Democrats and is soon expected to overtake the number of Republicans, Mr. Berman said. The ideological middle of Arizona's electorate may also gain more sway if voters pass Proposition 121, an initiative on the November ballot that would replace the current party primaries with one primary open to all voters. In the open primary system, the top two finishers advance to the general election regardless of party.

But the most powerful potential factor pushing Arizona to the political center remains the state's Latino population and the prospect that Latino turnout rates will rise. Even with depressed turnout, Hispanics doubled as a share of the electorate to 16 percent in the 2008 presidential election from 8 percent in 1992.

As in Texas, it's long been predicted that this is the year Arizona Latinos will really go to the polls.

“Everyone's been waiting for Latinos to flex their political muscle,” Mr. Garcia said.

Mr. Merrill added, “I've been here 40 years, and it hasn't happened so far.”

The November elections may not draw a flood of new Latino voters, but there are reasons to expect an uptick. “The Hispanic population is much more mobilized and engaged, in part because of the anti-illegal immigration campaign that was for a couple years the thing that got the Republicans going here,” Mr. Merrill said.

Latino voters have also been energized by the push to pass the Dream Act for undocumented young people and Mr. Obama's decision to stop the deportation of young undocumented immigrants under certain conditions, Mr. Garcia said.

Signs of an energized Hispanic electorate are apparen t in the Democratic Senate candidate Richard H. Carmona‘s surprisingly strong campaign against Representative Jeff Flake, who was thought to have the seat of Senator Jon Kyl, who is retiring, locked up. Arizona hasn't had a Democratic senator in almost 18 years.

Mr. Carmona, who is Puerto Rican, is currently a modest underdog in the FiveThirtyEight forecast; Mr. Flake is a 69 percent favorite. But “the Carmona-Flake race is an interesting example of Arizona starting to moderate a little bit,” Mr. Cieslak said. “It's going to come down to the wire.”

If the Republican Party does not improve its standing among Hispanics, it seems almost inevitable that Arizona will become substantially more competitive. Hispanics will make up one in four registered voters by 2030, according to an analysis by the Morrison Institute for Public Policy.

“I think this is the last election you could win without the Latino vote,” Mr. Garcia said.

Oct. 22: Ohio Has 50-50 Chance of Deciding Election

The supposition that Ohio is the most important state in presidential elections is not always true. The decisive state in the 2000 election, of course, was Florida. (Al Gore also could have won by carrying New Hampshire or Missouri, both of which were slightly closer than Ohio that year.)

In 2008, Barack Obama won by a clear national margin, so the Electoral College did not really come into play. But had he suffered a sudden decline at the end of the race, Colorado or Iowa - not Ohio - would have been the decisive state (or what we call the tipping-point state in FiveThirtyEight parlance).

But this year, all the clichés about Ohio are true. In our most recent simulations, Ohio has provided the decisive vote in the Electoral College about 50 percent of the time.

We will return to the Buckeye State in a moment. Let me first give you a whirlwind tour of the three dozen polls released on Monday. I am going to be somewhat more ci rcumspect than usual because what we will mostly be watching for the next few days is whether Monday night's debate had any effect on the race - something that polls before the debate won't tell us.

Monday's Polls

How close is the race according to national polls?

There were a remarkable number of them, 15, released on Monday. Pretty much every pollster with a pulse weighed in. (Unnecessarily meticulous note: I am excluding one case of a poll that was published on Monday but which conducted its interviewing back in September.)

Seven of the polls showed President Obama ahead, six had Mitt Romney ahead and two showed the race tied.

If you average the 15 polls, they show a tie, exactly. Neither candidate had an edge, even in the second decimal place.

There is a debatable case that Mr. Obama's national polls are a pinch stronger than a week or so ago, when there was a period of a few days when they seemed to suggest that Mr. Obama was trailing by perhaps one percentage point.

Our method uses both state polls and national polls to calibrate its estimate of the national popular vote; the state polls, in our view, have suggested a slightly stronger outcome for Mr. Obama all year than that implied by the national polls alone.

We do show just a tiny bit of improvement for Mr. Obama, however. In our “now-cast” - our estimate of what would happen in an election held today - Mr. Obama's advantage in the popular vote is now taken to be 1.2 percentage points, up from a low of 0.3 point on Oct. 12.

Mr. Obama also seems to be holding onto thin leads in the polling averages in the states that are most essential to his path forward in the Elec toral College.

Two polls of Wisconsin published on Monday, for example, gave him leads of 3 and 5 percentage points.

There were four polls published of Pennsylvania on Monday, showing Mr. Obama ahead by margins varying from 3 percentage points to 10. Our forecast model comes in toward the lower end of that range, showing a lead there of about 4.5 points for Mr. Obama right now.

A quick word about Pennsylvania: the FiveThirtyEight forecast model has liked it for a long while as a high-upside play for Mr. Romney, since winning it would devastate Mr. Obama's electoral map and since Mr. Romney has relatively few other opportunities to play offense. Pennsylvania is almost certainly a more plausible win than Minnesota or Michigan, for instance.

But as Monday's polls ought to attest, it is also a high-risk play - Mr. Romney has only about a 7 percent chance of winning Pennsylvania, in our estimation. Any state that Mr. Romney is trailing in by 4 or 5 percent age points at a time when he is tied in national polls is not essential to the electoral calculus. It might be a worthwhile luxury expenditure given Mr. Romney's flush cash situation, but it is probably not more than that.

Mr. Romney got a stronger poll in Iowa on Monday, where a survey from Rasmussen Reports showed a tied race there. That is down from a two-percentage-point lead for Mr. Obama in Rasmussen's prior poll of the state.

In Ohio, polls split the uprights between our forecast there, which projects Mr. Obama ahead by about two percentage points. A Quinnipiac University poll (conducted in conjunction with CBS News) had Mr. Obama five percentage points ahead, but a Suffolk University poll had a tied race.

There was also a poll published by Pulse Opinion Research, the parent of Rasmussen Reports, in Ohio on Monday, and that one had Mr. Romney up by one percentage point. This is a slightly unusual case, however, as we classify Rasmussen Reports and Pu lse Opinion Research polls together for purposes of the model since it is essentially the same poll conducted under different brand names. Rasmussen Reports itself published an Ohio poll late last week that showed Mr. Obama up by one percentage point instead. The Rasmussen-branded poll is actually the more recent of the two (despite having been published earlier) and so receives more weight in our forecast.

These details would not be worth writing about in any other state - but Ohio is Ohio, and whoever wins it is extremely likely to win the election.

Ohio, Ohio, Ohio

We are now running about 40,000 Electoral College simulations each day. In the simulations that we ran on Monday, the candidate who won Ohio won the election roughly 38,000 times, or in about 95 percent of the cases. (Mr. Romney won in about 1,400 simulations despite losing Ohio, while Mr. Obama did so roughly 550 times.)

Whether you call Ohio a “must-win” is a matter of semantics, b ut its essential role in the Electoral College should not be hard to grasp.

Were he to lose Ohio, Mr. Romney would have a number of undesirable, although not impossible, options.

The most favorable path, in the view of the model, would be for Mr. Romney to carry both Iowa and Nevada.

Of the two states, Iowa is the easier get. The polls there show a split between ties and leads for Mr. Obama, as opposed to Nevada, where they are mainly split between smaller leads for Mr. Obama and larger ones. In addition, in Nevada, Democrats have a significant voter-registration advantage and are building a large lead in early voting; the polls there have also tended to underestimate Democratic performance in recent years.

But Mr. Romney does not get to pick and choose if he loses Ohio; he would need to win both Iowa and Nevada under this plan. Furthermore, he would need to win New Hampshire to avert a 269-269 tie, where the polls have been inconsistent at best, but seem to show Mr. Obama slightly ahead, on average. On top of all that, he would need to win both Colorado and Virginia. The race is so close in both states that the model has fluctuated between showing them as blue states and red states with almost every new poll that comes in.

The major alternative would be for Mr. Romney to win Wisconsin, which would allow him to lose both Iowa and Nevada (although not Colorado or Virginia). But Republicans have had a number of seemingly favorable periods in Wisconsin - after Democrats' failed attempt to recall the state's governor, Scott Walker, after Mr. Romney named Representative Paul D. Ryan of Wisconsin as his running mate, and during Mr. Romney's overall surge in the polls recently - and have never quite pulled ahead in the average of polls there. Wisconsin is certainly a big factor in the election, but may also be the sort of state where Mr. Romney has a pretty easy path to 48 or 49 percent of the vote, and a hard one to 50 p ercent.

Or Mr. Romney could carry Pennsylvania, but this is less likely still, especially as Pennsylvania is highly demographically similar to Ohio but slightly more Democratic-leaning. In our simulations on Monday, Mr. Romney lost Pennsylvania 99.5 percent of the time when he also lost Ohio.

But if Ohio is almost a must-win for Mr. Romney, the same case could be argued for Mr. Obama. Were he to lose Ohio, Mr. Obama would then need to carry either Virginia or Colorado - along with holding Wisconsin, Iowa and Nevada.

In other words, Mr. Obama would need to carry at least one of the states where he is now tied in the forecast rather than leading. Thus, Mr. Obama's narrow lead in Ohio accounts for the bulk of his overall advantage in the forecast right now. Were Ohio decreed to Mr. Romney by fiat, Mr. Obama's chances of winning would decline to 57 percent from 70 percent in the forecast.

Alternatively, Mr. Obama could carry either Florida or North Caroli na, but as in the case of Mr. Romney and Pennsylvania, these permutations are just not very likely. In the simulations on Monday, Mr. Obama won Florida just 0.4 percent of the time that he lost Ohio, and North Carolina only 0.2 percent of the time when he did so.

Unlikely does not equal impossible, but Ohio is central enough in the electoral math that it now seems to matter as much as the other 49 states put together.

I am not sure whether I should be congratulating you or consoling you if you happen to be reading this in Toledo.

Obama Unlikely to Get Big Debate Bounce, but a Small One Could Matter

Instant-reaction polls following Monday night's debate in Boca Raton, Fla., judged President Obama to be the winner.

A CBS News poll of undecided voters who watched the debate found 53 percent giving it to Mr. Obama, 23 percent to Mitt Romney and 24 percent declaring it a tie. Mr. Obama's margin of victory in the poll was slightly wider than Mr. Romney's following the first presidential debate in Denver, which a similar CBS News poll gave to Mr. Romney at 46 percent to 22 percent.

Other polls, conducted among a broader group of voters rather than just undecided ones, suggested a smaller margin for the president.

A Public Policy Polling survey of voters in 11 swing states who watched the debate found them giving it to Mr. Obama, 53 percent to 42 percent.

A CNN poll of registered voters who watched the debate put Mr. Obama ahead, 48 percent to 40 percent. That was similar to Mr. Obama's 46-39 margin in a CNN poll of the second debate, and much smaller than Mr. Romney's 67-25 advantage in the first one.

An online poll by Google Consumer surveys had Mr. Obama winning, 45.1 percent to 35.3 percent. His roughly 10-point margin in the poll is smaller than in a Google poll after the second debate, which gave it to Mr. Obama by 17 points, or Mr. Romney's after the first, which he won by 22 points.

There is, obviously, some disagreement on the magnitude of Mr. Obama's advantage - the polls surveyed different types of voters and applied different methods to do so.

But averaging the results from the CBS News, CNN and Google polls, which conducted surveys after all three presidential debates along with the one between the vice-presidential candidates, puts Mr. Obama's margin at 16 points.

That compares favorably to Mr. Obama's average 10-point margin after the second debate, and Vice President Joseph R. Biden's 6-point margin against Representative Paul Ryan, but is smaller than Mr. Romney's average 29-point win in Denver.

The first presidential debate produced roughly a 4-point bounce in head-to-head polls toward Mr. Romney, while the second presidential debate brought no appreciable bounce toward Mr. Obama.

It is tempting to split the difference, and assume that Mr. Obama might get a 1- or 2-point bounce in the polls, but there are some mitigating factors. The pace of the debate was slow, and it was competing against high-profile baseball and football games, which may have kept viewership down.

Voters have more information about the candidates than they did before the first debate, meanwhile, which means that their additional impressions of the candidates could make less d ifference at the margin. Historically, the bounces following the third presidential debate, in the years when one was held, were smaller than after the first two.

Finally, the subject of the debate, foreign policy, is not as important to most voters as economic policy, although some voters may have judged the candidates on style regardless of the substance of the conversation.

Still, with the contest being so tight, any potential gain for Mr. Obama could matter.

Mr. Obama was roughly a 70 percent Electoral College favorite in the FiveThirtyEight forecast in advance of the debate, largely because he has remained slightly ahead in polls of the most important swing states.

If Mr. Obama's head-to-head polling were 2 points higher right now, he would be a considerably clearer favorite in the forecast, about 85 percent. A 1-point bounce would bring him to 80 percent, and even a half-point bounce would advance his position to being a 75 percent favorite in t he forecast.

Still, Mr. Obama should not take even that for granted. There have been some past debates when the instant-reaction polls judged one candidate to be the winner, but the head-to-head polls eventually moved in the opposite direction.

Referendum or Choice, Which Candidate Will Show Fighting Spirit?

The conventional wisdom about this presidential election, which this blog often endorsed, is that it represented a competition between two competing paradigms.

Mitt Romney, this theory held, would seek to make the election a referendum on President Obama's performance - hoping that voters' dissatisfaction with the direction of the country in general, and the economy in particular, would lead them to vote him out of office, almost regardless of what the alternative was.

Mr. Obama, it was thought, would seek to frame the election as a choice - defeating Mr. Romney on the basis of his own superior favorability ratings and, perhaps, his strong oratorical and campaigning skills.

Like any theory, this one was an oversimplification: every election is both a referendum and a choice to some degree. But the election mostly played to form in the spring and summer.

Then came the two defining events of the campaign - and both went against type.

At their convention in Charlotte, N.C., Democrats - especially former President Bill Clinton and Vice President Joseph R. Biden Jr. - gave a vigorous defense of Mr. Obama's performance in office. Although Democrats also took opportunities to tweak Mr. Romney, the gain Mr. Obama made in head-to-head polls after the convention seemed most to be driven by increased voter satisfaction with his performance.

Mr. Obama's approval ratings improved, rising to about 49 percent, where they remain today - probably just enough to make an incumbent the favorite given a strong turnout. Voter perceptions about the economy, including the consumer confidence numbers, also improved.

Then came the first presidential debate, in Denver. Mr. Romney, for the first time, had the chance to share a stage with the president - and based on the postdebate polls, he defeated Mr. Obama soundly.

While Mr. Romney critiqued Mr. Obama's performance in office at the de bate, he also articulated his own vision for the country - in a clearer way, probably, than he had at his own convention in Tampa, Fla.

Mr. Romney's favorability ratings shot up after the debate - to the point that they are now net-positive in most surveys and often the equal of Mr. Obama's. And Mr. Romney has gained in the head-to-head polls to make the presidential race very close, even as Mr. Obama's approval ratings and favorability numbers held steady.

In Charlotte, in other words, Mr. Obama demonstrated that perhaps he could win a referendum election after all. And in Denver, Mr. Romney showed that he would have a fighting chance even if voters saw the election as a clash of personalities.

So what does this mean for the final debate, on Monday night in Boca Raton, Fla.?

To the extent that foreign policy gives the commander in chief a home-court advantage, the timbre of the debate is probably more up to Mr. Obama. He could tout his own foreign po licy accomplishments, which voters tend to rate more highly than his handling of economic affairs. Or he could seek to draw sharper contrasts with Mr. Romney, perhaps hoping to convince voters that Mr. Romney is unready for office.

But so far this year, in both primary and general election debates, the best defense has often been a good offense. Mr. Obama made the mistake in Denver of seeming too willing to rest on his laurels, and what was then a four- or five-point lead in head-to-head polls. What might have seemed like presidential composure in the briefing room came across as aloofness and even disinterestedness in the debate hall.

Even if Mr. Obama thinks he can win a referendum election - and he possibly can - he will need to engage his opponent and show some fighting spirit.

Mr. Romney's calibration of these moods may be a little different, because his worst moments in the debates have tended to be characterized by his being hot-tempered. Still, Mr. Romney seems to trail Mr. Obama just slightly in the most important swing states, meaning that Mr. Romney will not necessarily benefit from playing the debate to a draw.

Oct. 21: Uncertainty Clouds Polling, but Obama Remains Electoral College Favorite

The bad news for President Obama: it's been almost a week since the second presidential debate, in Hempstead, N.Y., one that instant-reaction polls said was a narrow victory for him. But there is little sign that this has translated into a bounce for Mr. Obama in his head-to-head polls against Mitt Romney. Instead, the presidential race may have settled into a period of relative stability.

There is bad news for Mr. Romney as well, however. The “new normal” of the presidential campaign is considerably more favorable for him than the environment before the first debate, in Denver. However, it is one in which he still seems to be trailing, by perhaps 2 percentage points, in the states that are most vital in the Electoral College.

The FiveThirtyEight forecast was essentially unchanged again on Sunday, with Mr. Obama retaining a 67.6 percent chance of winning the Electoral College, little different from his 67.9 percent odds on Friday and Saturday.

These estimates might seem to be incongruous with national polls that show a nearly tied race. But the FiveThirtyEight method is, principally, an Electoral College simulation, and therefore relies more heavily on state-by-state polls. Our estimates of the popular vote in the critical states are highly similar to those of other Web sites that use different methods to calculate them.

This is expressed in the chart below, which compares the current FiveThirtyEight “now-cast” in each battleground state (our snapshot of what would happen in an election held today) against those of three of our competitors, RealClearPolitics, HuffPost Pollster and Talking Points Memo.

There are only three states that are “called” differently by the different methods. In Colorado, RealClearPolitics gives a slight edge to Mr. Romney, while the other three sites have Mr. Obama just ahead. However, it would be easy to overestimate the importance of these differences. RealClearPolitics has Mr. Romney only 0.2 percentage points ahead there, while the most favorable result for Mr. Obama, from The Huffington Post, has him up by 1.3 percentage points.

In Virginia, RealClearPolitics and The Huffington Post show an exact tie, while FiveThirtyEight and Talking Points Memo give a slight advantage to Mr. Romney.

Finally, in New Hampshire, FiveThirtyEight and The Huffington Post have Mr. Obama ahead, while the other two sites have the race going for Mr. Romney.

Still, these states - Virginia, Colorado and New Hampshire - are not quite at the electoral tipping point. Instead, Mr. Obama could win the Electoral College by winning Ohio, Wisconsin, and either Iowa or Nevada.

All the methods have Mr. Obama just ahead in each of these states. In Iowa, his margin runs from 1.5 percentage points to 2.5 percentage points, according to the different methods, with FiveThirtyEight representing the lowest point in the consensus. In Nevada, it is between 2.1 points and 5.2 points, with FiveThirtyEight again reflecting the lowest end of the range.

FiveThirtyEight is somewhere in the middle of the pack in Ohio, where the different methods give Mr. Obama an advantage of between 1.4 and 2.6 points, and in Wisconsin, where the estimates run between a 2.8- and a 3.8-point advantage.

One can certainly debate how to translate Mr. Obama's apparent edge in each state into his overall probability of winning the Electoral College. It is a challenging problem from a mathematical standpoint, particularly given that the states cannot be expected to behave independently from one another. (That is, i f Mr. Obama underperforms his polls in one swing state like Ohio, he is also more likely to do so in similar ones like Iowa.) It is a problem that we thought about very carefully in designing the model this spring, and in advance of the 2008 election. But because there have been only about eight presidential elections in which there has been a rich amount of state-by-state polling data, any approach will inherently be speculative to some degree.

At the same time, off-the-cuff or “gut-feel” estimates of these probabilities are probably not a good substitute for an effort to determine them through a more rigorous, empirical and rule-driven means. Our intuitive estimates of probabilities are normally quite poor, especially in situations that involve a relatively large degree of mathematical complexity and in which we have relatively little hands-on experience.

To be sure, these problems also present difficulties for anyone seeking to design a forecast model.

But if you accept the premise that Mr. Obama is ahead by some (small) margin in the tipping-point states, something that all the different methods agree on, it then becomes a question of how much doubt you should have about that advantage given the intrinsic uncertainty in polling.

Saying that the race “could go either way” is an obviously correct statement - but also one devoid of insight.

We dare to pose a more difficult question, the one that a gambler or an investor might naturally ask: What are the odds?

We calculate Mr. Obama's odds as being about two chances out of three.

Not only will the underdog - Mr. Romney - win some of the time, but he should win some of the time if we have estimated the odds correctly. If the set of candidates you have listed as 67 percent favorites in fact win 95 percent of the time, or 100 percent of the time, you've done something wrong. Over the long run, such candidates should win two out of three times - no less and no more.

Of course, it takes a very long time to realize the long run in presidential elections, since there is only one of them every four years. To the extent that one is evaluating the accuracy of political forecast models - whether they calibrate the odds correctly - it is probably better to look at something like races for the Senate. In that case, there are roughly 35 races held every other year, as opposed to just one every four years. Although these races are not completely independent from one another (there have been years in which Democrats or Republicans overperformed their Senate polls across the board), they are substantially more informative on balance for measuring the effectiveness of a series of forecasts.

Having indulged your patience with this philosophical detour, let me turn your attention back to the presidential race.

Sunday's Polls

Four of the five national tracking polls published on Sunday showed a gain for Mr. Romn ey, but there was one clear exception: a poll for Investor's Business Daily, which showed Mr. Obama surging to a 5.7-percentage-point lead from 2.6 percentage points previously.

What to make of the Investor's Business Daily poll? Probably about the same thing that you make of the Gallup poll, which had Mr. Romney up by 7 points among likely voters, and had him gaining an additional point on Sunday.

You should be highly skeptical that these polls represent the true condition of the race: polls that look like outliers normally prove to be so. This is certainly the case with the Gallup poll, which has performed quite poorly in the past when it has diverged from the consensus of other surveys.

B ut the Investor's Business Daily poll, like the Gallup poll, also has a history of sharp swings to and fro (sometimes before conveniently settling in with the consensus at the very end of the race).

When I point out that polls like Gallup or Investor's Business Daily have a history of behaving in this way, my goal is not to browbeat them into changing their methods. Instead, as strange as this might sound, I would much rather that the pollsters paid no attention to FiveThirtyEight. When pollsters feel as though they are under pressure to conform to expectations about the race, they may herd toward the poll averages, which would reduce their independence and would therefore reduce the benefit of aggregating different polls together.

My intention, rather, is to provide a counterweight to what seem like misinformed views about the presidential race. Sometimes the outlying polls receive more attention from the news media, when they probably deserve less.

On Sun day afternoon, I looked up how often each tracking poll had been cited in the Lexis-Nexis news database over the course of the past week. (The criterion I used for this search was to look for instances in which the pollster's name appeared within 25 words of the term “tracking” and also either of the terms “poll” or “survey.”)

It turns out that the Gallup national tracking poll was cited in the news media more often than the other six national tracking polls combined. Although the national tracking polls show, on average, a race that is about tied, they would have conveyed the impression of a four-point lead for Mr. Romney if weighted based on how often they were cited in news accounts.

< p>The simplest and best defense against this is merely to take the average of the polls.

Even averaging methods can conceal a fair amount of intrinsic uncertainty, however.

The nine national polls included in the RealClearPolitics average as of Sunday evening, for instance, contained about 12,000 interviews between them. (Collectively, they showed Mr. Obama ahead by a trivial margin of 0.2 percentage points.)

The margin of sampling error on a 12,000-person sample is larger than you might think: about plus or minus 2 percentage points in measuring the difference between the two major candidates. So the national polls reflected in the RealClearPolitics average could in fact be consistent with anything from a two-point lead for Mr. Romney to one of about the same margin for Mr. Obama.

And that calculation is based on just one type of error, that which is unavoidable from taking a random subsample of a larger population.

In fact, there are many othe r types of ambiguities in polling. Whether polling firms include cellphones in their samples seems to make a difference of about 2 percentage points, for instance. (The firms that include cellphones normally show stronger results for Mr. Obama.)

The likely-voter models that polls employ vary substantially from one another, suggesting everything from a situation in which Mr. Romney is performing about one point better among likely voters than among the broader universe of registered voters to one in which he has as much as a six- or seven-point advantage instead.

Some demographic groups are harder to get on the phone than others, and the techniques used to correct for this vary in quality. Polls that are especially “bouncy” may use poorly designed demographic weighting algorithms.

These same issues also apply in polls conducted at the state level, of course.

There are three major defenses that the FiveThirtyEight model employs in recognition of the se problems.

First, some of the “noise” in polling is predictable, since some polling firms consistently show results that are more favorable for one or the other candidate. In the cases in which the pattern is most consistent, we correct for this by means of our house effects adjustment.

Second, we seek to use state polling data and national polling data in a holistic way. It might seem, if you are looking at the state polls and the national polls in a very literal-minded way, that Mr. Obama is quite likely to win the Electoral College despite losing the popular vote. We certainly do allow for some possibility of this, and it has tended to increase over the course of the year. (Our estimate is that the chances would be about 7 percent in an election held today, although there is also a 3 percent chance of the reverse occurring and Mr. Romney winning the Electoral College despite losing the popular vote.)

To regard the probability as being substantially higher than that - taking the RealClearPolitics state and national polling averages might imply a 20 percent probability of a split outcome, for instance - may be to overestimate the degree of precision in both state and national polling averages.

What appears to be a tied race in national polls, as I mentioned, could fairly easily be a two-point lead for Mr. Obama just based on statistical variance alone, in which case there would be no appreciable difference between the national polls and the polls in the tipping-point states. By the same calculation, of course, the national polls could also be consistent with a two-point lead for Mr. Romney.

But our finding is that both state and national polls provide useful information about the overall condition of the race. The state polls can be aggregated, and weighted based on each state's projected turnout, to provide an estimate of the national popular vote. In the past, this technique would have provided an estimate of the national popular vote that would sometimes have been more accurate than the one from national polls themselves.

Mr. Obama's swing state polls are consistent with the hypothesis that he holds some sort of Electoral College advantage, but they are also consistent with the hypothesis that the national polls have the race slightly wrong. If you start out with the premise that the national polls are unimpeachably precise, then you might not give much weight to this hypothesis. But they are not all that precise as a matter of either theory or practice.

The opposite hypothesis - that the state polls are overestimating Mr. Obama's standing - is also a valid one to consider. If you return to the first chart in this article, you'll find that the FiveThirtyEight “now-cast” provides the most optimistic estimate for Mr. Obama's standing in the national popular vote (it puts him ahead by 0.8 percentage points). But we are generally on the lower end of the consensus on Mr. Obama's standing in swing states like Ohio, Iowa and Nevada. This is because, when the model detects an overall discrepancy between state and national polls, it adjusts both of them in opposite directions to bring them closer to one another. If after that adjustment the model still finds evidence that one candidate is performing especially well in the most important states, it will then attribute the remainder of the difference to a potential popular vote and Electoral College split.

Following a similar theme, we also use both national and state polls to inform our view about the overall trend in the race. The state polls published over the past several days, including on Sunday, have shown some stronger and weaker numbers for the candidates, but they are generally consistent with a race that has not changed much since Mr. Romney made his gains after the Denver debate.

Be wary of analyses that suggest that one candidate is surging or declining by an espe cially large amount in one or two particular states in a way that diverges from the overall trend in the race. Many of these apparent changes merely reflect statistical noise. We are at the point, certainly, where if Ohio moves a little less than the national polls, or Florida moves a little more, that is potentially significant. But most people's first instinct is to attribute too much meaning to these changes.

That brings up the third line of defense in the FiveThirtyEight model: our estimate of the uncertainty in the calculation. These estimates are based on how well polling averages, and presidential forecasting models, have performed under real-world conditions, rather than under idealized assumptions that, for instance, the uncertainty in polls is limited to statistical sampling error alone, or that the vote in each state is independent of the next one.

Right now we're at a point where, for better or for worse, these methodological choices will make quite a lot of difference in your evaluation of the race. The best I can do is to explain why we made the choices that we did in designing the model, and how they are playing out given the information that is coming our way.

There remains an outside chance that the race will break clearly toward one or the other candidate, after the third debate on Monday or because of some intervening news event, but the odds are strong that we will wake up on Nov. 6 with a reasonable degree of doubt about the winner. For that matter, we may wake up on Nov. 7 still uncertain about who won.

Nonetheless, stipulating that the race is clearly very close is not an adequate substitute for placing any kinds of odds on it at all. And the central premise behind why we see Mr. Obama as the modest favorite is very simple: he seems to hold a slight advantage right now in enough states to carry 270 electoral votes.