Mean-Reversion in Growth Rates and Convergence

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Brad DeLong posted about the recent paper by Pritchett and Summers (PS) on “Asiaphoria” and mean-reversion in growth rates. PS found several things:

  • Growth rates are not persistent. The growth rate over the last 10 years has very little information about the growth rate over the next 10 years. Growth rates “regress to the mean” as PS say.
  • Growth in developing countries tends to take place in bursts of growth and bursts of stagnation. This is different from rich countries where growth variation tends to consist of mild variation around a trend rate.
  • There is no reason to believe that rapidly growing economies today (China and India) will necessarily continue to grow rapidly.

Brad’s response is to take their evidence as a fundamental challenge to the standard Solow model explanation for why growth rates differ.

Lant Pritchett and Larry Summers are now trying to blow this up: to say that just as the neoclassical aggregate production function is a very bad guide to understanding the business cycle, as the generation-old failure of RBC models tells us, so the neoclassical aggregate production function and the Solow growth model built on top of it is a bad guide to issues of growth and development as well.

This is an overreaction. The mean-reversion and “bursts” that PS find are perfectly consistent with a Solow model including shocks.

Let’s start with the finding that regressing decadal growth rates on prior-decadal growth rates gives you a coefficient of something like 0.2-0.3. PS call this mean-reversion. I think it’s an artifact of convergence. Let’s imagine an economy that is following the Solow model precisely. It is very poor in 1960, and growth from 1960-1970 is about 10% per year. By 1970 it is much better off, and so growth from 1970-1980 slows to 5% per year. By 1980 this has gotten the country to steady state, so from 1980-1990 it grows at 2% per year. From 1990-2000 it is still at steady state, so grows at 2% a year again.

Now regress decadal growth rates (5,2,2) on prior-decade growth rates (10,5,2). What do you get? A line with a slope of about 0.397. Why? Because growth rates slow down as you approach steady state. Play with the numbers a little and you can make the slope 0.3 if you want to. The point is that convergence will generate just such a pattern in growth rates.

What about the unpredictability of growth rates? PS find that the correlation of growth rates across periods is very low. This is more problematic for convergence, on the face of it. If convergence is true, then growth rates across decades should be tightly correlated. In other words, even if the slope of the toy regression I ran above is less than one, the R-squared should be large.

In my toy example, the country systematically converges to 2% growth, and the R-squared of my little regression is 0.86. PS find much smaller R-squares in their work. The conclusion is that growth rates in the next decade are very unpredictable. So does this mean that convergence and the Solow model are wrong? No. The reason is that once you allow for any kind of meaningful shocks to GDP per capita, the short-run growth rates get very noisy, and you lose track of the convergence. It doesn’t mean it isn’t there, it just is hard to see.

Let me give you a clearer demonstration of what I mean. I’m going to build an economy that strictly obeys convergence, with the growth rate related to the difference between actual GDP per capita and trend GDP per capita.

More formally, let

\displaystyle  y_{t+1} = (1+g)\left[\lambda y^{\ast}_t + (1-\lambda)y_t \right] + \epsilon_{t+1} \ \ \ \ \ (1)

where {g} is the long-run growth rate of potential GDP, {y^{\ast}_t} is potential GDP in year {t}, {y_t} is actual GDP in year {t}, and {\epsilon_{t+1}} are random shocks to GDP in year {t+1}. This formula mechanically captures convergence to trend GDP per capita, but with the additional wrinkle of shocks occurring in any given period that push you either further away or closer to trend. {\lambda} is the convergence parameter, which I said in some recent posts was about 0.02, meaning that 2% of the gap between actual and trend GDP per capita closes every period.

I simulated this over 100 periods, with {g=0.02}, {\lambda=0.02}, {y^{\ast}_0 = 20} and {y_0 = 5}. The country starts well below potential. I then let there be a shock to {y} every period, drawn from a normal with mean 0, variance 0.25. Here are the results of one run of that simulation.

First, look at the 10-year growth rates over time. There is a downward trend if you look at it, but this is masked by a lot of noise in the growth rate. You have what look distinctly like two growth booms, about period 25 and period 50.

10-year Growth Rates

Second, look at the correlation of the average growth rate in one “decade” and the average growth rate in the prior “decade”. This is essentially what Pritchett and Summers do. I’ve also included the fitted regression line, so you can see the relationship. There is none. The coefficient on the prior-decade growth rate is 0.05, so pretty severe mean-reversion. The R-squared is something like 0.16. A high growth rate one decade does not indicate high growth the following decade, and the current decadal growth rate provides very little information on growth over the next decade.

Correlation of Growth Rates over time

But this model has mechanical convergence built into it, just with some extra noise dropped on top to make things interesting. And with sufficient noise, things are really interesting. If you looked at this plot, you’d start talking about growth accelerations and growth slowdowns. What happened in period 25 to boost growth? Did this economy democratize? Was there an opening to trade? And what about the bust around period 40? For a poor country, that is low growth. Was there a coup? We see plenty of “bursts” of growth and “bursts” of stagnation (or low growth) here. It’s a function of the noise I built in, not a failure of convergence.

By the way, take a look at the log of output per worker over time. This shows a bumpy but steady upward trend. The volatility of the growth rate doesn’t look as dramatic here.

Log output per worker

If I turned up the variance of the noise term, I’d be able to get even wilder swings in output, and wilder swings in growth rates. In a couple simulations I played with, you get a negative relationship of current growth rates to past growth rates – but in every case there was convergence going on.

Why are growth rates so un-forecastable, as PS find? Because of convergence, the noise doesn’t just cancel out over time. If a country gets a big negative shock today, then the growth rate is going to be low this year. But now the country is particularly far below trend GDP per capita, and so convergence kicks in and makes the growth rate larger than it normally would be. And because convergence works slowly, it will be larger than normal for several periods afterwards. There is a natural tendency for growth rates to be uncorrelated in the presence of shocks, but that is again partly because of convergence, not evidence of its absence. There are lots of reasons that the Solow model could be the wrong way to look at growth. But this isn’t one of them.

I think the issue here is that convergence gets “lost” behind all the noise in the data. Over long periods of time, convergence wins out. [“The arc of history is long, but it bends towards Robert Solow”? Too much?] Growth rates start relatively high and end up asymptoting towards the trend growth rate. But for any small window of time – say 10 years – noise in GDP per capita can swamp the convergence effects. In the growth literature we tend to look at differences of 5 or 10 years to “smooth out” fluctuations. That’s not sufficient if one wants to think about convergence, which operates over much longer time periods.

PS are absolutely right that we cannot simply extrapolate out China and India’s recent growth rates and assume they’ll continue indefinitely. We should, as growth economists, account for the gravitational pull that convergence puts on growth rates as time goes forward. But just like gravity, convergence is a relatively weak force on growth rates. It can be overcome in the short-run by any reasonably-sized shock to GDP per capita.

You don’t think “Oh my God, gravity is broken!” every time you see an airplane overhead. So don’t take abnormal growth rates or uncorrelated growth rates as evidence that convergence isn’t occurring.

Cyclones and Economic Growth

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I finally got a chance to read through a recent paper by Solomon Hsiang and Amir Jina on “The Causal Effect of Environmental Catastrophe on Long-Run Economic Growth: Evidence From 6,700 Cyclones”. The paper essentially does what it says on the tin – regresses the growth rate of GDP on lagged exposure to cyclones for a panel of countries over the period 1950-2008. By cyclone, the authors mean any hurricane, typhoon, cyclone, or tropical storm in this period.

Hsiang and Jina 2014 Table 1

One thing I like about this paper is that they do not bury the lede. Table 1 in the introduction gives you an instant grasp of the magnitude of what they find. They compare the cumulative effect of various disasters on GDP. A storm at the 90th percentile in strength (based on wind speeds and/or energy) reduces GDP by 7.4% after 20 years, similar in size to a banking or financial crisis. This is a big effect. As a point of reference, 20 years after World War II Germany’s GDP was already back on it’s pre-war trend.

Hsiang and Jina 2014 Figure 9

We might think that this is due to particularly slow convergence rates following cyclones. That is, the cyclone is a big shock that pushes the economy below steady state, and then it simply takes a long time for the economy to recover back to that steady state. But Hsiang and Jina’s figure 9 shows that this isn’t the kind of trajectory we see in places hit by cyclones. The full effect of the cyclone isn’t felt until nearly 15 years after. So the cyclones appear to have long-lasting effects pushing economies below their pre-storm trends. This implies some sort of change in behavior – lowering savings/investment rates, increasing depreciation rates, lowering human capital accumulation, limiting technology adoption – something that puts a persistent drag on the level of GDP.

Hsiang and Jina 2014 Figure 22

Making things worse is that countries are hit by multiple cyclones over time, and the negative impacts of one cyclone (as in their figure 9) is then accumulated with the negative impact of other cyclones to really push down GDP. They do some counter-factuals with their estimated effects to see what growth would have looked like across countries if there had been no cyclones at all from 1950-2008. Their figure 22 shows the distribution of growth rates in panel A with and without cyclones, and panel B shows the implied growth rate of world GDP with and without cyclones. There’s a sizable effect, with world GDP growth being about 1.4% per year higher without these storms.

For particular countries, the effects can be startlingly large. Take the Philippines, which has one of the highest exposures to tropical cyclones of any country in the world. In Hsiang and Jina’s counter-factual, GDP per capita would be higher by 2,000%, making the Philippines just about as rich as the U.S. Believable? Maybe not, but it gives you a sense of how much the negative impacts of these cyclones build upon each other through continued exposure. For places like Jamaica, Madagascar, or the Philippines, exposure to cyclones constitutes a persistent negative shock to GDP per capita that is difficult to overcome.

Time for some skepticism. In estimating these effects, Hsiang and Jina use 20-year lags of exposure to cyclones to estimate their effects, and hence are able to create figures like those in their figure 9 above. But their evidence does not rule out long-run convergence back to trend. If the shock of a cyclone is felt over about 15 years, and it then takes 30 years to return to trend, Hsiang and Jina will not be able to identify that. They’d only be capturing the initial negative shock, and not the recovery. This matters because we want to know whether the cyclones have (a) permanently lowered the standard of living or (b) act as temporary (but perhaps long-lived) reductions in standards of living. To put it into regular language, we want to know if the response to a cyclone is “Screw it, I’m not going to bother building a new house at all” or “Crap, it sure is going to take me a long time to rebuild my house”.

Hsiang and Jina do look at how exposure effects GDP for different sub-samples based on how repeated their exposure is to cyclones. For countries in the lowest two quintiles of exposure to cyclones, the implied negative effects are very large (I’m having a hard time interpreting the scale on their figure 19, so I’m not sure of the exact magnitude). For the three top quintiles, though, the effects of cyclones are much smaller in estimated size. The estimated effects are negative, and statistically indistinguishable from the effects in their pooled sample. However, the effects are also statistically indistinguishable from zero in most cases – except for the highest exposure countries.

This doesn’t quite settle the matter, though. Even though any individual storm may not cause any statistically significant drop in GDP per capita for high-exposure countries, this does not mean that they are unaffected by storm exposure. They may have adopted option (a) above – the “screw it” response – and so have a permanently lower trend for GDP per capita. The Hsiang and Jina paper cannot tell us anything about this, because they are only estimating the short-run effect of exposure to any particular storm, not the long-run adaptation to being exposed (which is differenced out and/or slurped up by the country-level time trend in their regressions).

Regardless, the paper is an interesting read, the latest in an increasing number of studies on economic growth that use detailed-level geographic/climate/weather data. Seeing the effect of the shocks of these cyclones out to 20 years in table 1 is a little startling, and gives you some appreciation for how geographic shocks remain as pertinent as economic ones to growth prospects.