Apr 132019
Free Book Chapter
Drei Jahrzehnte nach der Wiedervereinigung unterscheiden sich Lebensumstände, Erfahrungen, Einstellungen, Wertorientierungen und politische Verhaltensweisen von Ostdeutschen und Westdeutschen immer noch deutlich. Im Wahlverhalten zeigt sich dies unter anderem darin, dass in den neuen Ländern Nichtwähler- und Wechselwähleranteile höher sind als im Westen. Auch bei der Wahlentscheidung gibt es fast schon klischeehafte Unterschiede: im Westen schneiden die “Bonner Parteien” besser ab, im Osten die Linkspartei und seit 2014 auch die AfD.

Als ich den Pre-Print zur (vermeintlichen) Stellung der AfD als ostdeutsche Regionalpartei online gestellt habe, ist mir aufgefallen, dass der Vorgängerbeitrag zum Wahlverhalten bei der Bundestagswahl 2013 in Ost-West-Perspektive im digitalen Nirwana gelandet war. Das ist nun korrigiert.

 bundestag photo

Photo by LoboStudioHamburg

Mar 202017
With just six months to go until the 2017 Bundestag election, this is perhaps the ideal time to reflect on the rather remarkable 2013 election. Perhaps there is also a very fine line between Political Science and Contemporary History, and the German electoral studies community has a particular gift to step exactly on that line without ever quite crossing over? Either way, German Politics (the journal) published a fine Special Issue on the 2013 election in Germany. The articles focus on a number of highly specific research questions: Ben Christian employs the Rolling Cross Section-component of the GLES to study how voters learn to identify what would be the “correct” electoral choice for them over the course of the campaign. Martin Elff and Sigrid Roßteutscher show that the link between dealignment and party decline (of the SPD in particular) is more nuanced than previously thought. Marc Debus demonstrates that – female Chancellor or not – gender had little effect on voting for the Christian Democrats in recent Bundestag elections.

Katsunori Seki and Guy D. Whitten pit various economic voting models against each other. Robert Rohrschneider and Stephen Whitefield show that as far as mainstream parties are concernend, “Europe” is still largely a non-issue in German Politics, even in these troubled times. Sascha Huber looks at motivations for coalition voting to explain the decline of the FDP in the last weeks preceding the 2013 election. Rüdiger Schmitt-Beck dissects the AfD’s 2013/2014 electorate into two groups: euro-sceptics and xenophobes. Heiko Giebler and Bernhard Weßels demonstrate that good campaigns made voters remember local candidates. Finally, yours truly casts another long and dirty look at partisan dealignment, which has almost come to a halt in Germany. And since German Politics is a somewhat arcane journal, you may want to have a look at the nearly identical author’s (pre-publication) version.

Nov 032014
Even the Washington Post has woken up to the fact that 25 years after the uprising in the GDR, Germany stubbornly remains divided economically, politically, and socially. In the great scheme of things, this may matter less than you might think: In Western Europe alone, the UK, Spain, Belgium, or Switzerland – countries that have been around as nation states for much longer than Germany’s current iteration – are similarly diverse.

But it keeps the German Politics crowd busy enough. I’m currently working on a piece that looks at the latest federal election in comparative (east vs west) perspective – something that I have done previously for the 1998, 2002, and 2009 elections (2005 was someone else’s turn). The biggest difference is of course in the results of the Left party, which, compared to the West German districts, is about four times as successful in the East (this figure is down from a 20:1 rate in the 1990s). But here is another Bundestagswahl fun fact: The Liberals – not longer represented in the federal parliament for the first time since 1949, because their national result was just below the five per cent electoral threshold – barely scraped beyond this threshold in the old West, where they garnered 5.1 per cent of the valid votes. Based on the western results, the former Christian Democrats/Liberal coalition could have continued. Once more, the Easterners brought about political change.lonesome politician


Nov 262013

Coalition Talks: Not Quite as Speedy as Papal Election (and less fun)

It’s crunch time in Berlin: A mere two months after the election, both the SPD and Merkel’s CDU have announced that they want to resolve all remaining issues during yet another sleepless night of haggling (the Bavarian CSU is more reluctant). Tomorrow, they want to present the ‘coalition treaty’, a 170+ page agenda for the next four years. If the SPD’s restless rank-and-file will approve of this document is (quite literally) a question for another day.

The SPD kicked-off the talks with the announcement that they would no longer unconditionally rule out coalitions with the Left at the federal level (“but hey, no pressure”). It took the CDU some time to respond to this, but they did so with a vengeance: In Hesse, which held a land election on the day of the Federal election, the CDU has now entered coalition talks with the Greens. If these talk succeed, it would be the first CDU/Green coalition in a large non-city state (a coalition in Hamburg broke down relatively quickly, and so did a CDU/FDP/Green coalition in tiny Saarland).

Kanzleramt in Berlin
Werner Kunz / Foter.com / CC BY-NC-SA

First and foremost, this is a remarkable development in itself: Hesse has possibly one of the most polarised subnational party systems. Just a few years ago, then CDU leader and long-term minister president Roland Koch quite happily campaigned on the fact that the leaders of the SPD and the Greens had foreign-sounding names. But after the inconclusive election and four full rounds of sounding exercises involving all parties, a CDU/Green coalition is not longer unthinkable.

The Real Result: Less Segmentation?

But the Hessian regrouping also has a long-term, two-level strategic element (although all parties deny it): If the FDP does not rebound, the CDU needs an alternative. And much by the same token, the Greens don’t want to tie themselves too closely to the ailing SPD.

Black-green coalitions have been the stuff of political war games and academic debates for the last two decades. Now, they could become a reality. I’m not sure if the September election and this long, protracted negotiations in Berlin will indeed bring about a viable agreement between Christian Democrats and the SPD. But at any rate, they seem to have quickly reduced segmentation in the German party system.

Sep 252013

What’s the Matter with Germany?

At least in Germany, people begin to realise that Merkel may have painted herself into a corner by winning so gloriously (told you so first thing on Monday). While her Christian Democrats are by far the largest party group in the new Bundestag, she needs a coalition partner, but nobody wants to play.

Kanzleramt in Berlin
Werner Kunz / Foter / CC BY-NC-SA

The Social Democrats are not exactly keen to form a coalition with Merkel. They came out of the last Grand Coalition (2005-2009) very badly damaged and have hardly recovered from that electoral blow. And a new CDU/SPD coalition would be not so ‘grand’ any more. In 2005, the SPD and the Christian Democrats were not so far apart in terms of votes won. This time, there is a 16 point gap between the two.

The important NRW state party, which represents roughly a quarter of the party’s total membership,  is positioning itself against a Not-So-Grand Coalition. The party’s left-wing opposes on principal grounds, and because they feel that this would further strengthen the Left party. The party leadership is officially stalling just a bit, saying the situation is open. Talks will be ‘ergebnisoffen’ – non-directive. And they are seriously telling the Greens that it’s a dirty job, so perhaps they should do it.

A CDU/Green coalition on the other hand, while not impossible, is unlikely. The idea was all the rage a couple of years ago, but it did not work well in the Länder. Moreover, the party has moved to the left during the campaign (something they are regretting now) and returned to a more polarised view of the political world. The party leadership has resigned over the slightly disappointing result. Some of them will return, but it is not yet clear what the balance of power within the party will look like, and any new leadership will find it extremely difficult to sell a coalition with Merkel to the rank-and-file, who have a de facto veto.

This may very well be a thinly disguised attempt to drive the price of a black-red (or black-green) coalition upwards. In 1998 and 2002, it took roughly a month to form a government. In 2005, SPD and CDU needed 55 days. In 2009, the FDP/CDU/CSU government was sworn in about 40 days after the election. But what will happen if both the Greens and the SPD refuse to play (it is still silly season, but nobody’s talking CDU/Left. Yet)?

Is a Minority Government Possible? Will There Be New Elections?

In Germany, there is always a procedure, and in this case, it is spelled out in the constitution, whose framers were obsessed with stability (for very good reasons). The new parliament will be convened on October 22, 30 days after the election. That is the constitutional maximum. With this first plenary meeting (parliamentary parties haven been holding business meetings since Monday), Merkel’s second term as Chancellor will end, and so will the tenure of her ministers. The Federal President will however ask her ‘to continue to manage Germany’s affairs’ until a successor is appointed, and she is obliged to heed that request. So are the ministers. There is no ‘gap’: We’ll always have someone who tells us what to do.

The president will then go to parliament and propose a successor. But ‘then’ is relative. Interestingly, the constitution which is usually very precise, does not stipulate a time-frame. Leading commentators say the time-frame must be ‘appropriate’. Four weeks are ok. Six weeks would be ok, too, I think. But how much longer?

In theory, the president could come up with any proposal, but in practice, his suggestion has always been based on a viable coalition agreement between the parties, since his proposal must be confirmed by more than one half of its members (as opposed to more than half the members happening to be in the chamber at that point). The constitution is extremely wary of unstable majorities, let alone minority government.

Should the president’s candidate not be elected (this has never happened in the past), parliament has 14 days to make up their minds. During this period, they can elect anyone who manages to get the votes of more than half of the members without the president having any say.

Failing this, parliament will have one last vote on the Chancellor. Under this rule, the person receiving the most votes wins. If, by some happy coincidence, the number of these votes exceeds the number of half the members of parliament, the new Chancellor is sworn in and appointed. If it is less, the president has a choice: Within seven days, he can either appoint the Chancellor to lead a minority government or trigger new elections. Again, the choice is his in theory, but in practice, he would consult with the parties.

So where does that leave us?

Should both the Greens and the SPD refuse to join a Merkel government, they could still form a coalition with the Left or negotiate a toleration arrangement. But that seems unlikely, since the political costs would be very high, and it is not clear that all leftist MPs would vote for him. A small group of SPD and/or Green MPs could vote for Merkel to get on with it, without the parties entering a formal coalition, or one of the two parties could officially decide to tolerate her. As long as that would give her a majority in the inaugural vote, the president would have to appoint her. Or the parties could agree on having another election, with unknown consequences (AfD and/or FDP entering parliament, losses for all but the Left?).

Given these prospects, a CDU/SPD coalition will probably look like the lesser evil once everybody has calmed down a bit. But that might take some time.

Sep 232013
It’s been a bit of a nailbiter yesterday, and every single pundit in the country must be rubbing their bloodshot eyes. So it’s obviously not a brilliant idea to blog about it just now. But there seems to be a largish elephant in the room (not related to sleep deprivation) that nobody seems to have noticed so far.

A Historical Result

lonesome politician
FDP: Going Nowheremicagoto / Foter / CC BY-NC

Without doubt, this is a very exciting result that warrants a lot of superlatives or near-superlatives. Merkel’s Christian Democrats have bounced back from their second-worst result since 1949 to heights they have not seen since the highly unusual 1990 (re-unification) election. At 41.5 per cent, they came awfully close to an outright majority, something they have not achieved since 1957 (although then they had a much bigger share of the vote ).

The Social Democrats, on the other hand, have hardly recovered from their devastating 2009 result. 25.7 per cent is still the second-worst result since the war. But the combined vote share of the two major parties – often described as ‘former major parties’ by pundits – has gone up for the first time since 2002.

Both the Greens (at some stage projected to garner 15 per cent) and the Left have lost more than 20 per cent of their support compared to their 2009 results, and for the first time since 1990, the number of parties in parliament has gone down. And that is of course because the FDP has gone from 14.6 per cent (their best result ever) to 4.8 per cent (their worst result ever) and is not represented in parliament for the first time since 1949.

To put this in perspective, let me remind you that during the 64 years, the FDP was not holding government positions only from 1956 to 1961, from 1966 to 1969, and from 1998 to 2009. In other words, they were in government for roughly 70 per cent of the time, usually holding key positions (Foreign Affairs, Economy, Justice) and punching far above their electoral weight. For most German Politics aficionados, it will take some time to get used to the idea of them not having a national presence. Moreover, their result, combined with the relatively strong showing of the AfD means that the number of wasted votes must be near its all time high, with proportionality going out of the window.

But there is something else.

The Coalition Could Have Had a Viable Majority in Parliament

In the past, the FDP has survived (and some times thrived) on a diet of tactical considerations. Their loyal supporters are few and far between, but often, supporters of the CDU would give them with their list votes to bring about a centre-right majority. Most of the time, the CDU would not openly encourage this behaviour but would also refrain from discouraging it. Sometimes, the two parties even came up with joint position papers for future governments, signalling that they were not exactly a pre-electoral alliance but very much part of the same camp.

But this year (following the FDP’s defeat in Bavaria only a week before the General election), the CDU sent out a clear, high-profile “everyone for themselves” message to their voters. I can see three reasons for that. First, recent electoral reforms designed to make the system more proportional mean that the CDU would not benefit from a by-product of tactical CDU/FDP voting, the so-called ‘surplus seats’. Second, the ‘loan vote’ strategy has recently backfired in Lower Saxony, leaving a weakened CDU on the opposition benches. Third, the CDU may well have anticipated a Grand Coalition after Bavaria, and in that case, bolstering the FDP would not have made sense.

But this was probably a self-fulfilling prophecy. Though it looked very close yesterday night, Merkel did not win an outright majority. Christian Democrats and FDP together, on the other hand, are stronger than the three left parties combined: 46.3 vs 42.7 per cent. That would have been enough for Merkel to continue the centre-right coalition (her preference), with the added benefit of having a much more docile, dependent partner.

Negotiating a coalition with the Social Democrats will be tough. The party is licking its wounds and is highly reluctant to enter such an arrangement after the 2009 disaster that followed their last co-operation with the Christian Democrats. A CDU/Green coalition, while arithmetically feasible, seems highly unlikely at the moment, so the SPD will try to extract a large premium from the Christian Democrats for going into government with them. In the end, coalition talks could fail, and Germany could go to the polls again.

Without doubt, this result is a great triumph for Merkel. But I think the CDU leadership may have outwitted themselves, and the stern, slightly grumpy expression Merkel wore as she left the celebrations seems to confirm it.

Sep 202013
We’re now officially in the final throes of a very bad case of election fever. Or at least the wonks are. The latest and probably last crop of major polls is in. Moreover, last Sunday brought us the Bavarian Land (i.e. State) election, which was obviously seen not as an election in its own right but rather as a Bellwether for the big one. This is nonsense.

What’s the Matter with Bavaria?

Last Sunday, the Bavarian Christian Democrats received a cool 47.7 per cent of the valid votes. That is enough to give them a majority of the seats in parliament. The Liberals, with whom they governed from 2008, are no longer required and, to add insult to actually injury, actually no longer represented in parliament, as they got a not-so-cool 3.3 per cent.

Just a week before the national election, journalists, politicians and pundits are desperate for every shred of insight into what’s to come, and so the idea of a Grand (CDU/SPD) Coalition gained a lot of currency very quickly. Predictably, the FDP launched a new plea for “loan votes” from CDU supporters. Slightly less predictably, the CDU declared that they had no votes to spare, and that their supporters should vote straight.

But the Bavarian election is not very informative. Bavarians made a judgement on a different government, voted for different parliament, under different rules and facing a different party system. Merkel’s CDU does not even exist in Bavaria, which has its own Christian Democratic party, the Bavaria-centric CSU. Both parties do not even form an electoral alliance for the Bundestag election. They have merely chosen to co-operate in parliament for the last 17 times (and will do so again). In a very real sense, the current coalition consists of three parties, and the CSU has often been the most quarrelsome of them.

The CSU is the party of Bavarian nationalism, and single-party rule by the CSU is more or less the norm. The CSU has been part of every post-war government with the exception of the 1954-57 gang-of-four coalition (an abomination), and has governed alone from 1947-50 and then again from 1966-2008. A liberal presence in the Bavarian parliament, on the other hand, is not the norm. The party was out from 1982 to 1990, and then from 1994 to 2008. The Greens have been doing better than the Liberals in every election since 1982, and that is saying something in Bavaria. And yet, there is a whiff of change in the air.

The Final (?) Polls Are Coming In

In Germany, the two major public TV channels are the main sponsors for frequent large-scale polling. In the past, they stopped publishing results ten days before the election, but this is not a legal requirement: The law only bans the publication of findings from exit polls before the polling stations close on Sunday at six. Publishing results based on interviews conducted just before voters entered the polling station would be technically legal.

This year, broadcaster ZDF broke with tradition and published their final poll yesterday. Moreover, findings from three other major polls were published today, bringing the total of new polls since my last post to five. Two of them are straddling the allegedly ominous Bavarian election, while three of them were conducted this week. What do they tell us?

The Final Estimates


This close to the election, the model is becoming a lot more confident, i.e. credibility intervals for Sunday are now pretty narrow. The upward trend for the SPD (Fingergate notwithstanding) and the corresponding downward trend for the CDU that became visible last week are now more pronounced, but this should be seen in proportion: Predictions for both parties are within a point or two where they were seven weeks ago.

Things look a bit different for the smaller parties. The Greens are now projected to garner just under 10 per cent. In early August, the model was giving them about 13 per cent (with a very wide credibility interval). Looking at the curves, one can be pretty certain that there has been real movement: They have lost about one quarter of the support they had in mid-August. Conversely, the lot of the Left has improved. They are now projected to end up with about eight per cent, up from roughly six per cent that were predicted in August.

The most interesting case is of course the FDP. Back in August, the model thought they would scrape by with about five point something per cent. The probability that they would enter parliament at all was estimated at roughly 70 per cent. Seven weeks on, the estimate for their vote share has hardly changed, but the credibility interval is much more narrow. The model is now 95 per cent certain that they will garner at least five per cent. Reports of their death seem slightly exaggerated. Or my model could be totally wrong.


The probability of a majority for the current coalition is now estimated at 82 per cent. Whether the CDU likes it or not, voters will not ask for permission to vote tactically. If they do, the probability goes up to 85 per cent. These figures are down by seven/nine points from last week, reflecting the (modest) decline of the CDU and the slightly better performance of the SPD and the Left. But the probability of a Red-Green government is still zero per cent.

The probability of a (politically infeasible) Red-Red-Green majority is 15/18 per cent (lower if CDU supporters help the FDP). In line with previous post, I assume that there is a one in ten chance that the SPD might go back on their word, and a nine in ten chance that they would form a Grand Coalition with the CDU. This puts the Merkel-O-Meter (TM) at 98 per cent (irrespective of tactical voting).

Finally, the AfD is still making an awful lot of noise. While their party leader has recently lost a libel suit against a leading pollster, their supporters in the social media tend to say things that would make you want to avoid them in dark alleys. So far, one single poll has put them at exactly five per cent, and that was based on an online access panel. Everyone else has them at four per cent or less. If 40 per cent of the current support for “other” parties is for them, their probability of getting into parliament is zero. If their share of the “other” vote was more like 60 per cent, they would be almost certainly in.

The Outlook

While the pundits’ and politicians seem to brace themselves for a Grand Coalition and the death bell is ringing for the FDP, the model says that a new mandate for the current coalition is still by far the most likely outcome. That feels distinctly odd.

German politicians have a mantra that goes “Polls are not actually election results”, and pollster routinely claim that they are not making predictions. But colleague Bernhard Weßels over at the WZB has an excellent blog about the fit between polls and election results (in German). Usually they are very close. But there are some embarrassing exceptions. So stay tuned for post-election analysis, subterfuge and model tweaking.

Sep 192013
Thank you for more than 28,000 clicks on that photo!  Dome of the Reichstag building - La cúpula del Reichstag - Reichstagskuppel Berlin
alles-schlumpf / Foter / CC BY-NC-SA

Why Would I Want to Pool the Polls?

Pre-election polls are noisy for a number of reasons. First, there is sampling error. For n=1000, the confidence interval for a party whose true support is 40 per cent ranges from 37 to 43 per cent, which is more than most people would think. And this assuming simple random sampling. For multi-stage sampling, you could end up with one to two extra points at each end. Then there are house effects: Pollsters dress up their raw figures and different ways, use different sampling frames and slightly different modes and questions. And finally, political events and media coverage on the day of the poll will have effects, especially early on when many voters are undecided.

Combining results from different polls is one obvious strategy to deal with these problems: The combined sample size is bigger, and there is hope that the various sources of bias might offset each other. Hopefully.

Where Do the Data Come From?

The very useful site wahlrecht.de publishes margins from seven large German pollsters. Excluding INSA (they use an online-access panel), I check this site regularly and generate a data set from it that you can download here. To companies post (relatively) raw data, which I find preferable. What the others do to their figures, we cannot know.

How Does It Work?

The most straightforward idea in poll-pooling is calculating a moving (and possibly weighted). A more principled approach is model-based. My model borrows heavily from Simon Jackman’s (2005) paper and from Chris Hanretty’s application of a similar model to Italy, but differs in some respects. First, I treat the polls as draws from a multinomial distribution to account for Germany’s moderate multi-partyism. The parameters of this distribution depend on the relative strength of latent support for each party. Modelling the results as multinomial implies the constraint that the estimated shares must some to unity, which is useful. Second, like Jackman and Hanretty, I assume that latent support for each party follows a random walk (today’s support is yesterdays support plus a random quantity), but I allow for a drift: a linear trend in latent support over the course of the campaign. Third, I assign each poll to a week, because there are relatively few polls, and field-times are relatively long. Put differently, I assume that public opinion moves from week to week (but not from day to day).

The model estimates latent party support since January 2013 and makes predictions for the outcome of the election. The code (R & Bugs) is here.

Does It Work?

Honestly, I have no idea. This is work in progress, so take the findings with a pinch of salt.

What Could Possibly Go Wrong?

Everything? Anything? The most obviously dubious assumption of the model is that the polls are unbiased on average. Latent linear trends are a close second.

Sep 152013

State of Play: From Momentum to Peer’s Finger

The German media have been particularly excitable this week. They kicked off with floating the idea that there was momentum in the polls (“SPD up by almost two points!”) and concluded with covering the infamous magazine cover portraying Peer Steinbrück flaunting his finger. Moreover, the idea that the eurosceptic AfD might enter parliament has gained some traction. Of course, there is zero new evidence, but some days ago a leading pollster said that he could not rule out a result north of five per cent (the threshold).

Given that they have been polling between three and four per cent and considering the margin of error, that is a very sensible statement. By this morning, it had morphed into “Pollsters Expect that AfD will force Grand Coalition”. My totally scientifically objective hybrid human-computer qualitative analysis of the press (i.e. me browsing the internet) shows that the more pro-business papers are pushing that story – presumably because the AfD’s leader, a professor of macro economics, has the ear of a few editors.

The party does have, however, an unusually strong social media presence. If that is indicative of anything is anyone’s guess.

The Polls

The most recent iteration includes three new polls from week 36 (the week after the debate) as well as two from this week, and they are as noisy as ever. Ratings for the CDU are particularly variable (between 39 and 44 per cent), which is to be expected since the sampling error is bigger for parties that are close to 50 per cent. But according to model, this is mostly noise, and the party is still solidly somewhere in the area of 40 per cent (or slightly more).


Estimates/Predictions for the Major German Parties (2013). Click for larger image

The SPD’s support has indeed risen a bit over the last couple of weeks (all polls before finger-gate). But in the great scheme of things, that does not seem to matter much: The party is still stuck well under 30 per cent. Looking on the bright side, they will probably do better than the abysmal 23 per cent they got in 2009.

The decline of the Greens seems to be levelling out. Their recent drop in popularity not withstanding, the model predicts that the party will come third at about ten percent.

Support for the Left is basically constant and stable, while the FDP continues to inch upward. According to the model, Merkel’s preferred coalition partner will almost certainly make it into parliament.


Estimates/Predictions for the Smaller German Parties (2013). Click for larger image

The estimated probability of a new mandate for the current coalition is now 89 per cent. If tactical voting is taken into account, that number goes up to 94 per cent. The probability of a red-green majority is constant at virtually zero per cent. The SPD has repeatedly ruled out that they would accept any arrangement involving the Left. In line with previous posts, I assume that there is a 10 per cent chance that they nonetheless consider the unspeakable. Plugging this into the calculation, the probability of Merkel winning a third term is estimated at 98.9/99.4 per cent (with/without tactical voting for the FDP).

The Outlook

Over the next six days, the CDU’s main problem will be the complacency of their voters: With the race virtually run, they might simply be too lazy to turn out to vote. The SPD, on the other hand, seems to be making very small noises that imply that they might be interested in entering another Not-So-Grand Coalition. And the FDP will make a desperate appeal to their supporters (whom?) while trying to convince at least some CDU voters that yellow-blue is the new black.

Much will depend on the outcome of today’s state election in Bavaria: While Bavaria is basically representative for, well, Bavaria, politicians, journalists and voters will inevitably take it as a bellwether for the big one. Let’s wait and see then, shall we?

Sep 142013
I finally got around to clean/update the code for my model that pools the 2013 German pre-election polls, estimates latent party support and comes up with the most marvellous predictions (TM). You may download it (complete with the most recent version of the polling data) from my dataverse. If you spot a bug or some other problem, please drop me a line.

I will be back soon with the latest polls.