4% growth

I wrote last week on the simple factual question of whether and how often the US has experienced 4% real GDP growth in the past.

The deeper question, is that growth possible again? I answered yes, it's surely possible as a matter of economics. 

A few have asked me "why do so many of your colleagues disagree?" It's a question I hate. It's hard enough to understand the economy, I don't pretend to understand how others respond to media inquiries. And I don't like the invitation to squabble in public. 

It has taken me some time to reflect on it, though, and I think I have a useful answer. I think we actually agree.

As I read through the many economists' quotes in the media, I don't think there is in fact substantial disagreement on the economic question -- is it economically possible for the U.S. to grow at 4% for a decade or more? Their caution is political. They don't think that any of the announced candidates (at least with a prayer of being elected) will advocate, let alone get enacted, a set of policies sufficiently radical to raise growth that much. 

This is a sensible position. When I answer the question, is 4% growth for a decade economically possible, my answer is whether the most extreme pro-growth policies would yield at least that result. A  short list:

• The tax code is thoroughly reformed to do nothing but raise revenue with minimal distortion -- a uniform consumption tax and no income, corporate, estate etc. taxes, or deductions.
• A dramatic regulatory reform. For example 
• Simple equity-financed banking in place of Dodd-Frank. 
• Private health-status insurance (with, if needed, on-budget voucher subsidies) in place of Obamacare. 
• An end to the mess of energy subsidies and interference. No more fuel economy standards, HOV lanes, Tesla tax credits, windmill subsidies, and so on and so on. (If you want to control carbon, a uniform carbon tax and nothing else.) 
• Many agencies cease to exist. 
• No more endless waits for regulatory decisions. 
• No more witch hunts for multibillion dollar settlements.
• Thorough overhaul of social programs to remove disincentives. Most help comes via on-budget vouchers.
• No more agricultural subsidies.
• No more subsidies, period. Fannie and Freddie closed down.  
• Unilateral free trade. 
• Essentially open immigration -- anyone can work.  
• Much labor law rolled back. Uber drivers can be contractors, thank you. Most occupational licenses removed -- anyone can work.  
• Drug legalization.
• School vouchers. 
• And so on. Essentially, every single action and policy is re-oriented toward growth. 

This program removes a lot of level inefficiencies. 10% increase in level over 10 years is 1% more growth per year. Labor force participation increases. The labor force itself grows. We get a spurt of productivity growth just from greater efficiency without needing big investments. And then innovation and new businesses, investment, technology kicks in.

There would be a lot of lawyer, accountant, lobbyist, compliance officer, and regulator unemployment. Well, Uber needs drivers.

Politically, this is free-market libertarian nirvana.

I think my fellow economists might agree that 4% growth for a decade is possible with such a program. In fact, it we can likely get to 4% with much less than all of these policies. They might complain about inequality or other objectives.  But most of all, they might say it's unlikely that the new President and Congress will enact anything like such a program.

That's a very reasonable view. I also agree that typical proposals -- a  small reduction in corporate rates, a twiddle here a tweak there, the typical small promise to improve regulation -- will not have one tenth the needed effect.

But, dear colleagues, they asked us about economic possibility, not our guess about political probability. Let's answer the question they asked us.  It would be better to say: "Sure, 4% growth is economically possible. But I don't think any politician will advocate the policies necessary to produce it."  If we were to say that more often, rather than give up at the outset, we just might get such policies and politicians.

You never know what's "politically feasible." In 1955, civil rights was "politically infeasible." In 2005 gay marriage was "politically infeasible." Politics sticks in the mud for 100 years and then changes faster than we imagine.

I think there actually are quite a few politicians who would do some of the radical things that need to be done. They need to hear from us that it could work, as a matter of economics, and let them handle the politics.

We will soon see a first test: Can any candidate show up in Iowa, and say "Ladies and Gentlemen, government subsidized corn ethanol is a rotten idea." Then, can they say something vaguely coherent on immigration and trade.  The campaign season is young. Let's not prejudge them. 

Growth is just too important to give up on so easily. Sclerotic growth is the economic issue of our time. Economists should be cheering any policy agenda focused on growth.  If you think the policies needed to give us growth are hard, and out of the current political mainstream, that's ever more reason to keep reminding people that growth is possible and needs big changes, not to confuse "it's unlikely they'll do it" with "it's economically impossible."

Update: Response to Noah Smith's comment on this post  here 

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.

A linear trend in GDP is only visible ex-post, like a "bull" or "bear" market.  It's not "wrong" to detrend GDP, but it is wrong to forecast that GDP will return to the linear trend or to take too seriously correlations of linearly detrended series, as Arnold mentions.  Treating macro series as cointegrated with one common trend is a better idea.

Log stock prices have random walk and stationary components. Dividends are a pretty good indicator of the random walk component. (Most recently, here.)

Arnold asks  "In stock market returns, econometricians have been able to identify long-term mean reversion even though the short run is a random walk. Can something similar be done with GDP data?" Answer: Yes, and  Permanent and Transitory Components is it.

Both Arnold and Roger claim that unemployment has a unit root. Guys, you must be kidding. Actually, this makes a great test case for my point in "A critique", that it is a bad idea it is to blindly run unit root tests and then impose that structure.

A unit root means a random walk component. A random walk will eventually pass any upper and lower limit. Look at it. That's as stationary a series as you're going to find in economics. ("Look at the plot" and "think about the units" are the Cochrane unit root tests.)

Yes, unemployment like other stationary ratios in macro (consumption/GDP, hours/day, etc.)  have important and frequently overlooked low-frequency movements. But they are far from random walks, and they like unemployment have a very large transitory component at business cycle frequencies. When unemployment is above 8%, it is a good bet that it will decline over the next 5 years.

If you apply unit root tests to an hour of second by second temperature data from 9 to 10 AM you will think it has both a linear trend and a unit root. Millisecond data will not help you to detect climate change.  That's why unit root tests are a problem. You have to think, and consider the span of data you have and the frequency of mean reversion that makes economic sense in your data.

The tests are about infinite horizon behavior which you can never tell with finite horizons. However, they can alert you to low-frequency movement in your data, which can make ordinary distribution theory a bad guide. So can looking at a plot.

As far as I can tell, "Potential GDP" is equivalent to a two sided filter. It looks great ex-post. None of this is inconsistent with Arnold's view that standard calculations of potential GDP gaps do little to forecast GDP growth, especially in real time.

Jumps and diffusions

I learned an interesting continuous time trick recently. The context is a note, "The fragile benefits of endowment destruction" that I wrote with John Campbell, about how to extend our habit model to jumps in consumption. The point here is more interesting than that particular context.

Suppose one time series \(x\), which follows a diffusion, drives another \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the second equation describes how habits \(y\) respond to consumption \(x\). The same kind of structure might describe how invested wealth \(y\) responds to asset prices \(x\), or how option prices \(y\) respond to stock prices \(x\).

Now, suppose we want to extend the model to handle jumps in \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What do we do about the second equation? \(y_t\) now can jump too. On the right hand side of the second equation, should we use the left limit, the right limit, or something in between?

The usual answer is to use the left limit. We generalize the model to jumps this way: \[dx_t = \sigma dz_t+ dJ_t \] \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] where \(y_{t_{-}}\) denotes the left limit.

That approach has some weird properties however. Suppose \(y_{t_-}=1\), and \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose there are two jumps of size 1/2, one at time \(t\) and one at time \(t+\varepsilon\). Now \(y\) jumps up to 1.5 after the first jump, and then jumps another \(1.5 \times 0.5 = 0.75\), ending up at \(y_{t+\varepsilon} =2.25\). Two half jumps have a different response than one full jump.

Suppose instead we extend the original model to jumps by taking the jump limit of a continuous process. Imagine that we observe realizations of \(\{dz_t\}\) that get closer and closer to a jump in \(dx_t\), and let's find what happens to \(y_t\). The general solution to the first set of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, in the limit \(\Delta \rightarrow 0\) that \(x_t\) takes a jump of size \(dJ_t\), the jump-limit of a continuous movement is \[ dy_{t} \equiv y_t -y_{t_-} = y_{t_-}(e^{dx_{t}}-1) = y_{t_-}\sigma dz_t + y_{t_-}e^{dJ_t}\] rather than \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] So, the left-limit method produced a response to a jump that was different from the response to a continuous process arbitrarily close to a jump. For example, the left-limit approach can produce a negative \(y_t\), but this method, like the diffusion process, cannot fall below zero. This method also produces a response to two half jumps that is the same as the response to a full jump.

As you can see, the difference is whether the state variable \(y_t\) gets to change during the jump. In the left-limit approach, the same \(y_{t_-}\) gets applied to the whole jump. In the continuous-limit version, \(y_t\) implicitly gets to move while the jump in \(x_t\) is moving.

A nonlinear function of a jump is a little novel, but there's nothing wrong with it, and it exists in the continuous time literature. We don't see it that often, because when you're only studying one series it's easier to just change the distribution of the jump process instead. This question occurs when you can see both series x and y and you want to model the relationship between them.

Which is right?

Which extension to jumps is correct? Both are mathematically correct. There is nothing wrong with writing down a model in which the response to a jump is different from the response to continuous movements arbitrarily close to jumps. The answer depends on the economic situation.

For example, consider models with bankruptcy constraints. Agents who can continuously adjust their investments may always avoid bankruptcy in a diffusion setting. If we extend such a model to jumps with the continuous limit approach, implicitly preserving the investor's ability to trade as fast as asset prices change even in the jump limit, we will preserve bankruptcy avoidance in face of a jump in prices. However, if we model portfolio adjustment to jumps with the left-limit generalization, agents may be forced in to bankruptcy for price jumps.

Sometimes, one introduces jumps precisely to model a situation in which prices can move faster than agents can adjust their portfolios, so agents may be forced to bankruptcy. Then the left-limit generalization is correct. But if one wants to extend a model to jumps for other reasons, while avoiding bankruptcy, negative consumption, negative marginal utility (consumption below zero or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, and so forth, then one should choose a generalization in which the jump gives the same result as the continuous limit.

Similarly, when extending option pricing models to jumps, one may want to model the jump in such a way that investors cannot adjust portfolios fast enough. Then the left-limit extension is appropriate, and investors must hold the jump risk. But one may wish to accommodate jumps in asset prices to better fit asset price dynamics while maintaining investor's ability to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence between jumps and the limiting diffusion.

A little more general treatment

A little more generally, suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We want to add \(dJ_t\) to the first equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If there is a jump \(dJ_t\), \(y\) moves by an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The limit of a continuous movement solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you see the crucial difference, whether the state variable gets to move "during" the jump. We can write this as a differential, by writing the solution to this last differential equation as \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] and then \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]

So, you don't have to extend the model to jumps with the left-limit approach, and you don't have to swallow the idea that a jump has a different response than an arbitrarily close continuous-sample-path movement. The last equation shows you how to modify the model to include jumps in a way that preserves the property that the jump has the same effect as its continuous limit.

The point

Why a blog post on this? I asked a few continuous-time gurus, and none of them had seen this issue before. If someone knows where this has all been worked out with proper is dotted and ts crossed, I would like to know and cite it properly. (I would think the literature on option pricing with jumps had done it, but I couldn't find a reference.) Or perhaps it hasn't been done and someone wants to do it. I'm not good enough at the technical aspects of continuous time to write this with the right precision and generality.

And it's a cool trick that may be useful to someone outside of the narrow context that we had for it.

Update: 

Perhaps the right application is stock prices and option prices. When stock prices jump, someone must have studied the case that option prices move by the same amount the Black-Scholes formula gives for the same size stock price movement. Does anyone have a citation to that case?