Monday, April 27, 2015

Unit roots in English and Pictures

After my unit roots redux post, a few people have asked for a nontechnical explanation of what this is all about.

Suppose there is an unexpected movement in any of the data we look at -- inflation, unemployment, GDP, prices, etc.  Now, how does this "shock" affect our best estimate of where this variable will be in the future? The graph shows three possibilities.

Friday, April 24, 2015

Unit roots, redux

Arnold Kling's askblog and Roger Farmer have a little exchange on GDP and unit roots. My two cents here.

I did a lot of work on this topic a long time ago, in How Big is the Random Walk in GNP?  (the first one)  Permanent and Transitory Components of GNP and Stock Prices” (The last, and I think best one) "Multivariate estimates" with Argia Sbordone, and "A critique of the application of unit root tests", particularly appropriate to Roger's battery of tests.

The conclusions, which I still think hold up today:

Log GDP has both random walk and stationary components. Consumption is a pretty good indicator of the random walk component. This is also what the standard stochastic growth model predicts: a random walk technology shock induces a random walk component in output but there are transitory dynamics around that value.

Wednesday, April 22, 2015

The right to herd

Just when you thought financial regulation couldn't get more expansive and incoherent, our Justice Department comes in to defend morons' right to herd.

As explained in the Wall Street Journal at least, Mr. Navinder Singh Sarao is now under arrest, fighting extradition to the US, and his business ruined, for "spoofing" during the flash crash.

What is that? The Journal's beautiful graph at left explains.

The obvious question: Who are these traders who respond to spoofing orders by placing their own orders? Why is it a crucial goal of law and public policy to prevent Mr. Sarao from plucking their pockets? Is "herding trader" or "momentum trader" or "badly programmed high-speed trading program" or just simple "moron in the market" now a protected minority?

Why is Mr. Sarao being prosecuted and not all the people who wrote badly programmed algorithms that were so easily spoofed? If this caused the flash crash (how, not explained in the article) are they not equally at fault?

I don't mean by this a defense of the crazy stuff going on in high speed trading. As explained here, I think one second batch auctions are a much better market structure.  But the whole high speed trading thing is largely a response to SEC regulations in the first place, the order routing regulation, discrete tick size regulation, and strict time precedence regulation. A fact which will probably not enter at Mr. Sarao's trial (he doesn't seem to have billions for a settlement) and will give him little comfort in jail.

And maybe, just maybe, there is something more coherent here than the Journal lets on. I'll keep reading hoping to find it and welcome comments who can.

A larger thought. We still really want to rely on regulators to spot all the problems of finance and keep us safe from more crashes?

Update: Craig Pirrong excellent commentary here via a good FT alphaville post. Great quote:
The complaint alleges that Sarao employed the layering strategy about 250 days, meaning that he caused 250 out of the last one flash crashes. [my emphasis] I can see the defense strategy. When the government expert is on the stand, the defense will go through every day. “You claim Sarao used layering on this day, correct?” “Yes.” “There was no Flash Crash on that day, was there?” “No.” Repeating this 250 times will make the causal connection between his trading and Flash Clash seem very problematic, at best.
Update 2: Reading various commentaries that I can't find to cite any more, I realize that "front running" more than "herding" is the protected class. You "spoof" by putting in a bunch of orders just outside the current spread. The algorithms that respond to that think this behavior means some big orders coming, so try to front run those by buying. They cross the spread to take the small order you put on the other side. Or so the story goes. In any case, viewed as spoofers vs. front-runners it's harder still to have sympathy for the latter.

Update 3: Good Bloomberg View coverage from Matt Levine  and John Arnold, the source of the above front-running observation.

Monday, April 20, 2015

Consumption-based model and value premium

The consumption based model is not as bad as you think. (This is a problem set for my online PhD class, and I thought the result would be interesting to blog readers.)

I use 4th quarter to 4th quarter nondurable + services consumption, and corresponding annual returns on 10 portfolios sorted on book to market and the three Fama-French factors. (Ken French's website)
The graph is average excess returns plotted against the covariance of excess returns with consumption growth. (The graph is a distillation of Jagannathan and Wang's paper, who get any credit for this observation.  The lines are OLS cross-sectional regressions with and without a free intercept.)

Friday, April 17, 2015

Macro Handbook 2

Last week I attended the first half of the conference on the Handbook of Macroeconomics Volume 2, organized by John Taylor and Harald Uhlig, held at Hoover. The conference program and most of the papers are here.  The second half will be in Chicago April 23-25, program here

Overall, this Handbook is shaping up as a very useful resource.  Really good summary and review papers are a natural way in to long literatures. Bad summary and review papers are long and boring. The conference produced the first kind. Most of the papers are rough first drafts, so make a note to come back when they're finished. A few highlights (with apologies to authors I've left out; I can't review them all here.)

Thursday, April 16, 2015

Banking at the IRS

A while ago in two blog posts here and here I suggested many ways other than currency to get a zero interest rate if the government tries to lower rates below zero. Buy gift cards, subway cards, stamps;  prepay bills, rent, mortgage and especially taxes -- the IRS will happily take your money now and you can credit it against future tax payments; have your bank make out a big certified check in your name, and sit on it, don't cash incoming checks. Start a company that takes money and invests in all these things (as well as currency).

Chris and Miles Kimball have an interesting essay exploring these ideas "However low interest rates might go, the IRS will never act like a bank." Their central point: sure that's how things work now. But with substantial negative interest rates, all of these contracts can change. It's technically possible in each case for people and businesses to charge pre-payment penalties amounting to a negative nominal rate.

Reply: Sure, in principle. Nominal claims can all be dated, and positive or negative interest charged between all dates.

But this did not happen in the US and does not happen in other countries for positive inflation and high nominal rates,  despite symmetric incentives, and at rates much higher than the contemplated 3-5% or so negative rates.  Yes,  with large nominal rates there is pressure to pay faster,  inventory cash-management to reduce people's holdings of depreciating nominal claims, but this pervasive indexation of nominal payments did not break out. The IRS did not offer interest for early payment.

More deeply, what they're describing is a tiny step away from perfect price indexing. If all nominal payments are perfectly indexed to the nominal interest rate, accrued daily, then it's a tiny change to index all prices themselves to the CPI, accrued daily. If "how much you owe me," say to rent a house, is legally, contractually, and mechanically determined as a value times e^rt, and changes day by day, then e^(pi t) is just as easy.

So, price stickiness itself would (should!) disappear under this scenario.

Price stickiness has always been a bit of a puzzle for economists. As the Kimballs speculate how easy it is to index payments to negative interest rates, so economists speculate how easy it is to index payments to inflation. Yet it seems not to happen.

So this point of view strikes me as a bit of a catch-22 for its advocates, who generally are of the frame of mind that prices and nominal contracts are sticky and that’s why negative nominal rates are a good idea to "stimulate demand" in the first place.  If we can have negative nominal rates and change all these legal and contractual zero-rate promises to allow it, then prices won't be sticky any more!   Conversely, I should be cheering, as it amounts to a broad push to unstick prices. That has long seemed to me the natural policy response to the view that sticky prices are the root of all our troubles. It would allow negative rates, but eliminate their need as well.

Alas, the world seems remarkably resistant to time-indexing all payments.

Wednesday, April 15, 2015

Gdefault needs not Grexit

The little grumpy cartoon usually represents me pounding my coffee down in agreement as the WSJ exposes some idiocy. Last week, alas, I spilled my grumpy coffee in disagreement with a little part of its otherwise excellent  "The case for letting Greece go."
Thursday marks another deadline in Greece’s struggle to avoid default, as a €450 million payment to the International Monetary Fund comes due. Athens says it will meet this obligation, but sooner or later Prime Minister Alexis Tsipras and his government will miss a payment to someone if it doesn’t agree with creditors on a new bailout. An exit from the euro would then be a real possibility.
Please can we stop passing along this canard -- that Greece defaulting on some of its bonds means that Greece must must change currencies. Greece no more needs to leave the euro zone than it needs to leave the meter zone and recalibrate all its rulers, or than it needs to leave the UTC+2 zone and reset all its clocks to Athens time. When large companies default, they do not need to leave the dollar zone. When cities and even US states default they do not need to leave the dollar zone. A common currency means that sovereigns default just like large financial companies. (Yes, a bit of humor in the last one.)

Tuesday, April 14, 2015

Blanchard on Countours of Policy

Olivier Blanchard, (IMF research director) has a thoughtful blog post, Contours of Macroeconomic Policy in the Future. In part it's background for the IMF's upcoming conference with the charming title Rethinking Macro Policy III: Progress or Confusion?” (You can guess my choice.)

Olivier cleanly poses some questions which in his view are likely to be the focus of policy-world debate for the next few years.  Looking for policy-oriented thesis topics? It's a one-stop shop.

Whether these should be the questions is another matter. (Mostly no, in my view.)

As a blogger, I can't resist a few pithy answers. But please note, I'm mostly having fun, and the questions and essay are much more serious.

Thursday, April 2, 2015

The sources of stock market fluctuations

How much do dividend-growth vs. discount-rate shocks account for stock price variations?

An under-appreciated point occurred to me while preparing for my Coursera class and to comment on Daniel Greewald, Martin Lettau and Sydney Ludvigsson's nice paper "Origin of Stock Market Fluctuations" at the last NBER EFG meeting

The answer is, it depends the horizon and the measure. 100% of the variance of price dividend ratios corresponds to expected return (discount rate) shocks, and none to dividend growth (cash flow) shocks.  50% of the variance of one-year returns corresponds to cashflow shocks. And 100% of long-run price variation corresponds to from cashflow shocks, not expected return shocks. These facts all coexist

I think there is some confusion on the point. If nothing else, this makes for a good problem set question.

The last point is easiest to see just with a plot. Prices and dividends are cointegrated. Prices correspond to dividends and expected returns. Dividends have a unit root, but expected returns are stationary. Over the long run prices will not deviate far from dividends. So 100% of long-enough run price variation must come from dividend variation, not expected returns.
Ok, a little more carefully, with equations.

A quick review: 

The most basic VAR for asset returns is \[ \Delta d_{t+1} = b_d \times dp_{t}+\varepsilon_{t+1}^{d} \] \[ dp_{t+1} = \phi \times dp_{t} +\varepsilon_{t+1}^{dp} \] Using only dividend yields dp, dividend growth is basically unforecastable \( b_d \approx 0\) and \( \phi\approx0.94 \) and the shocks are conveniently uncorrelated. The behavior of returns follows from the identity, that you need more dividends or a higher price to get a return,  \[ r_{t+1}\approx-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}% \] (This is the Campbell-Shiller return approximation, with \(\rho \approx 0.96\).) Thus, the implied regression of returns on dividend yields, \[ r_{t+1} = b_r \times dp_{t}+\varepsilon_{t+1}^{r} \] has \(b_r = (1-\rho\phi)+0 = 1-0.96\times0.94 = 0.1\) and a shock negatively correlated with dividend yield shocks and positively correlated with dividend growth shocks.

The impulse response function for this VAR naturally suggests "cashflow" (dividend) and "expected return" shocks, (d/p). (Sorry for recycling old points, but not everyone may know this.)

Three propositions:
  • The variance of p/d is 100% risk premiums, 0% cashflow shocks
Iterate forward the return identity, to get \[ dp_{t} =\sum_{t=1}^{\infty}\rho^{j-1}r_{t+j}-\sum_{t=1}^{\infty}\rho ^{j-1}\Delta d_{t+j} \] multiply by \(dp_t\) and take expectations (all variables are demeaned) \[\sigma^{2}\left( \log\frac{P_{t}}{D_{t}}\right) =\sigma^{2}\left( dp_{t}\right) =\sum_{t=1}^{\infty}\rho^{j-1}cov(dp_{t},r_{t+j})-\sum _{t=1}^{\infty}\rho^{j-1}cov(dp_{t},\Delta d_{t+j}), \] But \(b_d \approx 0 \), so the dividend growth terms are all zero, and 100% of the variance of price-dividend ratios corresponds to time-varying expected returns. (I know this will bore people familiar with it and befuddle those who are not. "Discount rates" has a bit more leisurely review and citations)

  •  The variance of returns is 50% due to risk premiums, 50% due to cashflows. 
\[ r_{t+1}=-\rho dp_{t+1}+dp_{t}+\Delta d_{t+1}% \] \[ \varepsilon_{t+1}^{r} =-\rho\varepsilon_{t+1}^{dp}+\varepsilon_{t+1}^{d} \] \[ \sigma^{2}\left( \varepsilon_{t+1}^{r}\right) =\rho^{2}\sigma^{2}\left( \varepsilon_{t+1}^{dp}\right) +\sigma^{2}\left( \varepsilon_{t+1}^{d}\right) \] The variance of the two shocks comes out very close to a 50/50 decomposition at an annual horizon. It's a lot more expected return at a daily horizon, and less at longer horizons. Here I use the fact that dividend growth and dividend yield shocks are basically uncorrelated.

Why are returns and p/d so different?  Current cash flow shocks affect returns. But a shock to dividends, when prices rise at the same time, does not affect the dividend price ratio. (This is the essence of the Campbell-Ammer return decomposition.)

The third proposition is less familiar:
  • The long-run variance of stock market values (and returns) is 100% due to cash flow shocks and none to expected return or discount rate shocks.
Here's why: \[ \Delta p_{t+1} =-dp_{t+1}+dp_{t}+\Delta d_{t+1} \] \[ p_{t+k}-p_{t} =-dp_{t+k}+dp_{t}+\sum_{j=1}^{k}\Delta d_{t+j} \] so as k gets big, \[ {var} (p_{t+k}-p_{t}) \rightarrow 2 {var}(dp_t) + k {var}(\Delta d_{t}) \] The first term approaches a constant, but the second term keeps growing. As above the central fact is that P and D are cointegrated while expected returns are stationary.

This is related to a point made by Fama and French in their Equity Premium paper. Long run average returns are driven by long run dividend growth  plus the average value of the dividend yield. The difference in valuation -- higher prices for given set of dividends -- can affect returns in a sample, as higher prices for a given set of dividends boost returns. But that mechanism can't last. (Avdis and Wachter have a nice recent paper formalizing this point.)  It's related to a similar point made often by Bob Shiller: Long run investors should buy stocks for the dividends.

A little more generality as this is the new bit.

\[ p_{t+k}-p_t = dp_{t+k}-dp_t + \sum_{j=1}^{k}\Delta d _{t+j} \] \[ p_{t+k}-p_t = (\phi^{k}-1)dp_t + \sum_{j=1}^{k}\phi^{k-j} \varepsilon^{dp}_{t+j} +  \sum_{j=1}^{k} \varepsilon^d _{t+j} \] \[ var(p_{t+k}-p_t) = \frac{(1-\phi^{k})^2}{1-\phi^2} \sigma^2(\varepsilon^{dp}) + \frac{(1-\phi^{2k})}{1-\phi^2}  \sigma^2(\varepsilon^{dp}) +  k\sigma^2(\varepsilon^d) \] \[var(p_{t+k}-p_t) = 2\frac{(1-\phi^{k})}{1-\phi^2} var(\varepsilon^{dp}_{t+1})  + k var(\varepsilon^d_{t+j})\] So you can see the last bit takes over. It doesn't take over as fast as you might think. Here's a graph using sample values,

At a one year horizon, it's just about 50/50. The dividend shocks eventually take over, at rate 1/k. But at 50 years, it's still about 80/20.

Exercise for the interested reader/finance professor looking for problem set questions: Do the same thing for long horizon returns, \( r_{t+1}+r_{t+2}+...+r_{t+k} \) using \(r_{t+1} = -\rho dp_{t+1} + dp_t + \Delta d_ {t+1} \) It's not so pretty, but you can get a closed form expression here too, and again dividend shocks take over in the long run.

Be forewarned, the long run return has all sorts of pathological properties. But nobody holds assets forever, without eating some of the dividends.

Disclaimer: Notice I have tried to say "associated with" or "correspond to" and not "caused by" here! This is just about facts. The facts have just as easy a "behavioral" interpretation about fads and bubbles in prices as they do a "rationalist" interpretation. Exercise 2: Write the "behavioralist" and then "rationalist" introduction / interpretation of these facts. Hint: they reverse cause and effect about prices and expected returns, and whether people in the market have rational expectations about expected returns.