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Copulas explained

Let’s say you want to measure the relationship between multiple variables. One of the easiest ways to do this is with a linear regression (e.g., ordinary…



Let’s say you want to measure the relationship between multiple variables. One of the easiest ways to do this is with a linear regression (e.g., ordinary least squares). However, this methodology assumes that the relationship between all variables is linear. One could also use generalized linear models (GLM) in which variables are transformed, but again the relationship between the outcome and the transformed variable is–you guessed it–linear. What if you wanted to model the following relationship:

In this data, both variables are normally distributed with mean of 0 and standard deviation of 1. Additionally, the relationship is largely co-monotonic (i.e., as the x variable increases so does the y). Yet the correlation is not constant; the variables are closely correlated for small values, but weakly correlated for large values.

Does this relationship actually exist in the real world? Certainly so. In financial markets, returns for two different stocks may be weakly positive related when stocks or going up; however, during a financial crash (e.g., COVID, dot-com bubble, mortgage crisis), all stocks go down and thus the correlation would be very strong. Thus, having the dependence of different variables vary by the values of a given variable is highly useful.

How could you model this type of dependence? A great series of videos by Kiran Karra explains how one can use copulas to estimate these more complex relationships. Largely, copulas are built using Sklar’s theorem.

Sklar’s theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables.

Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae separately.

Each variable of interest is transformed into a variable with uniform distribution ranging from 0 to 1. In the Karra videos, the variables of interest are x and y and the uniform distributions are u and v. With Sklar’s theorem, you can transform these uniform distributions into any distribution of interest using an inverse cumulative density function (that are the functions F-inverse and G-inverse respectively.

In essence, the 0 to 1 variables (u,v) serve to rank the values (i.e., percentiles). So if u=0.1, this gives the 10th percentile value; if u=0.25, this gives the 25th percentile value. What the inverse CDF functions do is say, if you say u=0.25, the inverse CDF function will give you the expected value for x at the 25th percentile. In short, while the math seems complicated, we’re really just able to use the marginal distributions based on 0,1 ranked values. More information on the math behind copulas is below.

The next question is, how do we estimate copulas with data? There are two key steps for doing this. First, one needs to determine which copula to use, and second one must find the parameter of the copula which best fits the data. Copulas in essence aim to find the underlying depends structure–where dependence is based on ranks–and the marginal distributions of the individual variables.

To do this, you first transform the variables of interest into ranks (basically, changing x,y into u,v in the example above). Below is a simple example where continuous variables are transformed into rank variables. To crease the u,v variables, one simply divides by the maximum rank + 1 to insure values are strictly between 0 and 1.

Once we have the rank, we can estimate the relationship using Kendall’s Tau (aka Kendall’s rank correlation coefficient). Why would we want to use Kendall’s Tau rather than a regular correlation? The reason is, Kendall’s Tau measure the relationship between ranks. Thus, Kendall’s Tau is identical for the original and ranked data (or conversely, identical for any inverse CDF used for the marginals conditional on a relationship between u and v). Conversely, the Pearson correlation may vary between the original and ranked data.

Then one can pick a copulas form. Common copulas include the Gaussian, Clayton, Gumbel and Frank copulas.

The example above was for two variables but one strength of copulas is that can be used with multiple variables. Calculating joint probability distributions for a large number of variables is often complicated. Thus, one approach to getting to statistical inference with multiple variables is to use vine copulas. Vine copulas rely on chains (or vines) or conditional marginal distributions. In short, one estimat

For instance, in the 3 variable example below, one estimates the joint distributions of variable 1 and variable 3; the joint distribution of variable 2 and variable 3 and then one can estimate the distribution of variable 1 conditional on variable 3 with variable 2 conditional on variable 3. While this seems complex, in essence, we are doing a series of pairwise joint distributions rather than trying to estimate joint distributions based on 3 (or more) variables simultaneously.

The video below describes vine copulas and how they can be used for estimating relationships for more than 2 variables using copulas.

For more detail, I recommend watching the whole series of videos.

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Yen Doesn’t Buy The BOJ Narrative Just Yet, But It Will

Yen Doesn’t Buy The BOJ Narrative Just Yet, But It Will

By Ven Ram, Bloomberg markets live reporter and strategist

There seems to be growing…



Yen Doesn't Buy The BOJ Narrative Just Yet, But It Will

By Ven Ram, Bloomberg markets live reporter and strategist

There seems to be growing conviction that Japan will exit negative rates in possibly just a couple of months, though the currency markets are underpricing that prospect.

Earlier Tuesday, data for January inflation showed not only faster-than-forecast headline numbers but also a core-core reading above 3% for a 13th straight month. The prints may convince the Bank of Japan that the sustainable inflation that it has long sought is here. Little wonder that overnight indexed swaps, which were assigning some 60% chance of a 10-basis point move from the BOJ in April, now reckon the probability is more like 80%.

The yen, though, hasn’t come to the party at all. Since the start of the year, the currency has slumped more than 6% against the dollar. That decline isn’t what is indicated by fundamentals, with the yen’s weakness looking completely out of sync with what ought to have happened.

A major part of what has happened with the yen is actually a dollar story. Fed fund futures, which were pricing a little more than three interest-rate cuts from the US central bank by June, are now wondering if policymakers will even deliver a single reduction by then. That has pushed up nominal and inflation-adjusted rates, sending the dollar far higher than reckoned.
There is also seasonality at play. The yen has weakened in four of the past five first quarters, except when the pandemic first struck the markets in 2020 and spurred investors to scramble for havens.
The Japanese currency is exceptionally undervalued at current levels, with its real-effective exchange rate near the cheapest it has ever been in history. Once the BOJ exits negative rates, expectations are that it will raise rates further and buoy the yen, although Japan won’t see any tightening anywhere on the scale that we have seen in in the other major economies as colleague Mark Cranfield notes.
The tide will turn decisively in favor of the yen whenever it becomes abundantly clear that inflation in the US is slowing to a crawl — allowing the Fed to cut rates as outlined in its December dot plot.

Tyler Durden Tue, 02/27/2024 - 22:00

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Will New Gavin Newsom Recall Effort Backfire Again?

Will New Gavin Newsom Recall Effort Backfire Again?

Authored by Susan Crabtree viaRealClear Wire,

Conservative activists in California hope…



Will New Gavin Newsom Recall Effort Backfire Again?

Authored by Susan Crabtree viaRealClear Wire,

Conservative activists in California hope the seventh time is a charm when it comes to recalling Gavin Newsom. The governor, they say, is focused on serving as a top surrogate for President Biden and raising a national profile for his own presidential run while neglecting the state’s deep budget deficit, rising crime rate, persistent homelessness, and sky-high cost-of-living – factors driving an exodus of people and businesses to other states.

But their latest effort to oust Newsom after so many failed attempts isn’t stoking the same fears among Democrats as in the past, and even some Republicans are worried the partisan effort will blow up in their faces. It has the potential, some GOP operatives caution, to overshadow waning support for progressive policies in San Francisco and elsewhere while galvanizing Democrats and big donors behind the term-limited but politically ambitious governor.

Democrats argue that the latest recall effort is an obvious attempt to blunt presumed Newsom’s White House aspirations by showing that there’s a backlash against him resurfacing at home. But far from a big cloud hanging over his head, top California Democratic strategists are casting it as little more than an annoyance.

“I don’t think it even merits a cloud – maybe a little bit of fog or haze,” Steve Maviglio told RealClearPolitics Monday. Maviglio served as the press secretary for former California Gov. Gray Davis, a Democrat who, in 2003, became the state’s second top executive to be recalled.

Republicans are so outnumbered by Democrats in California that one of their only political weapons is a recall petition. “They can either howl at the moon, or they can file a recall petition – those are their two choices,” said Garry South, a longtime Democratic strategist who managed Davis’ successful campaigns in 1998 and 2002.

Some Republican political players across the state are also disheartened by the tired feel of the repeated long-shot effort. “I would much rather focus on the legislative and [federal and local] races where we could win, but they seem to want flashy politicking, not the hard work of retail politics,” one conservative activist told RCP.

The California Republican Party provided $125,000 for the 2021 recall effort but sidestepped questions on Monday about whether it would support it this year.

“We are eight days out from our March 5 primary and several weeks into pre-election voting,” California GOP Chairwoman Jessica Millan Patterson said in a statement. “While Gavin Newsom has been an absolute disaster for our state – from accruing a record $73 billion budget deficit to hosting the nation’s largest homeless population to flatlining the quality of our schools and allowing criminals to thrive – the CAGOP’s attention is on turning out the vote in the primary election and supporting our endorsed candidates who can fix our broken state.”

Newsom wasted no time connecting the recall to Trump, confidently predicting it would go down in flames.

“Trump Republicans are launching another wasteful recall campaign to distract us from the existential fight for democracy and reproductive freedom,” he tweeted Monday. “We will defeat them.”

Newsom triumphed over a 2021 recall that made it on the ballot and was organized by the same conservative activists. The first-term governor overwhelmingly defeated the effort to oust him after a jittery summer in which polls predicted the race as a dead heat. After Newsom was fortified by nearly $75 million in unlimited campaign donations from his committees and allies, he sailed to victory over Republican talk-radio host Larry Elder.

The deep blue state’s Democratic voters showed up for Newsom in droves, with 61.9% voting to keep Newsom and only 38.1% voting to remove him. It was essentially the same margin he won by in his first campaign for governor against businessman John Cox in 2018. After the failed recall, Newsom’s support dipped only slightly. He came back in 2022 to win a second term by 59.2% to 40.8% against state Senator Brian Dahle.

Newsom beat back the 2021 recall so strongly it only strengthened his reelection and bolstered his political ambitions, South argued. President Biden and Vice President Kamala Harris stumped with Newsom in the final days before the election. Former President Barack Obama cut a television ad, deeming the election a “matter of life and death” – the difference between “protecting kids from COVID or putting them at risk, helping Californians recover or taking them backward.”

Newsom hadn’t benefited from that level of Democratic star power since former President Bill Clinton flew in for a last-minute rally when polls showed his 2003 race for San Francisco mayor against a Green Party candidate much tighter than expected.

During his 2021 recall victory remarks, Newsom twice said he was “humbled” by the experience. But he also viewed the landslide as a validation of his strict COVID policies and efforts to lean into the country’s culture wars and hinted at his long-held White House ambitions.

“We are enjoying an overwhelming ‘no’ vote tonight here in the state of California,” Newsom told a crowd of cheering supporters gathered in Sacramento to view the recall returns. “But ‘no’ was not the only thing that was expressed tonight. I want to focus on what we said ‘yes’ to as a state. We said ‘yes’ to science, ‘yes’ to vaccines, we said ‘yes’ to ending the pandemic.”

“We said ‘yes’ to diversity, we said ‘yes’ to inclusion, we said ‘yes’ to pluralism,” he declared. “We said ‘yes’ to those things that we hold dear as Californians, and I would argue, as Americans.”

Recall organizers, led by Rescue California, acknowledge that Newsom was emboldened by beating the recall, but insist that it should have motivated him to focus his attention  on fixing California’s problems instead of skewering red states and serving as a star surrogate for Biden. Half the state’s voters seem to agree.

Last fall, when Newsom was amplifying his national profile, his standing among California voters hit an all-time low, with 49% disapproving of his job as governor, a UC Berkeley Institute of Governmental Studies/Los Angeles Times poll found. His approval rating in the late October poll fell to 44%, an 11-point slide from February when 55% of voters approved his performance.

“He’s using California taxpayer money to fly around the country and the world to support a national political agenda for president when he’s not even qualified, at this point, to run the state of California as the deficit numbers approach $100 billion,” Ann Dunsmore, Rescue California’s campaign director, told RCP.

Dunsmore also points out that Newsom will have little choice but to raise taxes to reduce the deficit because the state Constitution prohibits the government from declaring bankruptcy without voter approval. Right now, Newsom is backing a proposition calling for $6 billion in new spending to curb homelessness after the state has spent more than $20 billion on the issue during his time in office while the problem grew worse.

In early January, Newsom and Democrats who run the state legislature also ushered in new health insurance payments for all illegal immigrants. Newsom this week continued taking the fight to conservative states, unveiling a six-figure ad campaign and online petition effort in several Republican-controlled states that he said are trying to ban out-of-state travel for abortions and related medications. Newsom paid for the ads with a national political action committee he launched last year with $10 million from leftover state campaign funds.

Crime and cost of living are driving more and more businesses from the state, Dunsmore said, “and now Newsom’s coming out with an abortion ad?”

Right now, people are trying to make ends meet, and it’s getting worse, not better,” she said. “Our tax base is gone. Just how out of touch is he?”

Recall petitions can be launched easily in California, but they face formidable hurdles. This one would require signatures equal to 12% of the turnout in the last election – roughly 1.31 million verified signatures.

Dunsmore won’t say how much Rescue California, which successfully led the effort to recall Davis in 2003, plans to spend on the effort. The 2021 recall required 1.5 million signatures, which it exceeded by 126,000. It cost organizers roughly $8 million, a fraction of the $78 million Newsom amassed to fight it.

This year, however, Dunsmore says they don’t plan to use costly paid signature-gatherers posted outside shopping centers. Instead, she said her organization and its partners already have the infrastructure in place – supporters’ physical addresses and emails – from the last effort.

Garry South, however, cautions that the 2021 recall only went forward after an unprecedented four-month extension for signature-gathering. A judge appointed by GOP Gov. Arnold Schwarzenegger approved the rare extension to compensate for COVID lockdowns when signature-gathering was far more difficult.

“That’s never happened before in a recall election in California,” South said, adding that there’s “no way” recall organizers will amass the required signatures with an all-volunteer effort.

The 2003 recall effort against Gov. Gray Davis got a boost when wealthy GOP Rep. Darrell Issa poured $2 million of his own money into signature-gathering efforts because he was aiming to become governor. But Schwarzenegger, a blockbuster Hollywood actor, stepped into the race and won it, thwarting Issa’s political ambitions.

Even Davis’ low poll numbers didn’t stop him from winning election in 2002. On Election Day in 2002, South recalls that Davis’ approval rating was only 22%, and he still won with 47.4% of the vote to the GOP candidate Bill Simon’s 42.4%.

“Newsom beat the recall with nearly 62% of the vote and then was reelected to a second term with 59%,” South said. “Those are not the metrics of someone who is going to be recalled.”

Susan Crabtree is RealClearPolitics' national political correspondent.

Tyler Durden Tue, 02/27/2024 - 22:20

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Stock Bull Market Might Just Be Getting Started, But…

Stock Bull Market Might Just Be Getting Started, But…

Authored by Simon White, Bloomberg macro strategist,

The rally in equities might…



Stock Bull Market Might Just Be Getting Started, But...

Authored by Simon White, Bloomberg macro strategist,

The rally in equities might have much further to go, based on the positive outlook for liquidity.

It might not seem like it after a seemingly relentless advance and fevered speculation, but the new bull market is comparatively mild versus the postwar past.

Yet that could change. Excess liquidity - the difference between real money growth and economic growth - shows that the stock rally could have much further to go, turning a so-far historically below-par bull market into one that’s above the past average.

There are many reasons why this might not transpire...

As a natural cynic, I’m more comfortable when the outlook is pessimistic (no room for disappointment) versus when it is optimistic (plenty of opportunity to end up with egg on your face when things do go wrong after all). But sometimes the data just isn’t there to support a downbeat view.

That’s the case today. One of the best medium-term drivers of stock returns is excess liquidity. It’s an intuitive measure: when money, which is created by banks and central banks, is growing faster in real terms than GDP, liquidity is left which is “excess” to the needs of the real economy, and which thus tends to find its way into risk assets.

After beginning to rise in the first half of last year, and supporting the equity rally that began in March, excess liquidity has continued to rise. It is difficult for markets to sell off significantly when there is plenty of risk-asset-supporting liquidity sloshing around the system.

Fiscal and monetary policy are conspiring to keep excess liquidity climbing despite the cumulative impact of higher rates coursing through the economy.

First, what has been driving excess liquidity so far?

It has three main elements: inflation, economic growth and narrow money, with the latter responsible for most of the measure’s rise over the last year.

But that’s not the full picture.

Excess liquidity is a global measure, made up of the money and economic growth of countries in the G10, in dollar terms. That means a weaker dollar boosts non-US excess liquidity.

As the chart below shows, it’s the weaker dollar - down over 9% from its September 2022 highs - that has been the biggest driver of excess liquidity.

We can blame fiscal policy here. The US’s expansive deficit has been one of the most important longer-term negative influences on the dollar. There is little sign the deficit is about to improve by much, based on (no doubt conservative) Congressional Budget Office forecasts. Government finances are also unlikely to be straitened whoever the next president is, meaning the primary trend in the dollar (DXY) is likely to remain down.

We can also blame monetary policy for the dollar’s malaise and excess liquidity’s buoyancy. The latter looked like it was about to start turning lower last year, but was saved in the nick of time by the Federal Reserve’s pivot in December.

How? On a shorter-term basis (6-9 months), the dollar is led by the real yield curve. The US currency is driven at the margin by the real return of foreign investors in long-term US assets. In the latter months of 2023, the real yield curve had been steepening, as longer-term real yields were rising more than shorter ones.

Then the Fed came with its still unfathomable pivot. Shorter-term real yields fell, but their longer-term counterparts fell by more, and the curve re-flattened. What was a strong supportive sign for the dollar returned to being a weight on it – and thus a continued tailwind for excess liquidity.

It’s not just liquidity that could charge the bull market further. The absence of a US recession, which continues to look off the cards for the time being, also bolsters the case that equities should not soon face a steep selloff. Traditional recession indicators have been misleading in this pandemic-addled cycle, but it has become increasingly clear a downturn in the US is now less likely than not.

Furthermore, the rally might be on shakier legs if sentiment and technicals were overly bullish, but they are not yet historically stretched. The net number of stocks making new 52-week highs, the number trading above their 200-day moving average or their upper Bollinger band, and the advance-decline line are all high but have been higher. Moreover, sentiment is net bullish but not at extremes, while retail allocation to stocks is only at its 5-year average.

Leadership is narrow, with only a handful of stocks driving the advance, but there is little historically to show that this leads to sub-par returns. And when markets eclipse new highs, as the S&P did a few weeks ago, it acts as a psychological all-clear that we are indeed in a new bull market. Whether you agree that’s justified or not, the catch-up money that floods the market creates its own momentum.

No bull market comes without risks and this one is no different. The biggest is a recession. While, as mentioned above, that does not look likely in the near term, a sudden and unanticipated economic slump (either endogenous or due to an exogenous shock) would decimate returns. Also, a bull market that does not begin either during a recession or within 18 months of one is unusual, with only one postwar example (1966).

Equities experience their largest drawdowns in recessions, and given there is little ex ante to indicate one is coming in the current environment, it would likely be particularly devastating.

A blow-off top is another risk. Even then, despite the upset one would cause, it might not be enough to kick-start a new bear market. Inflation, too, will pose a risk to stocks, but to their real returns, unless price growth’s revival is particularly abrupt or steep (bull and bear markets are, sub-optimally, based off nominal returns). A persistent bear-steepening of the yield curve would be the sign the rally is at risk.

To misquote John Templeton, bull markets are born on pessimism, but they grow on liquidity. As long as excess liquidity is supported, the market is primed to keep grinding higher, regardless of how cynical you might be.

Tyler Durden Tue, 02/27/2024 - 14:40

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