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Autoregressive Models

"Arguing in My Spare Time," No. 3.20

by Arnold Kling

August 24, 2000

Several aimst readers have recommended to me "The Age of Spiritual Machines," by Ray Kurzweil. Earlier this month, I finally got around to reading it.

Kurzweil asks us to think about computers as if they were a new species of intelligence introduced onto the planet. We tend not to think of them that way, because they are still quite stupid. His claim, however, is that they are making and will continue to make progress at such a rate that computers' share of total intelligence on earth will increase over the next 100 years from less than 1 percent to over 99 percent.

Most of Kurzweil's book is devoted to tracing through the implications of that shift. He argues that it will raise issues in many fields, including medicine, sex, politics, philosophy, and economics. On the latter, he argues for a "law of accelerating returns," meaning that in terms of one of my earlier essays that he is a believer in the positive second derivative.

I am particularly interested in the thinking behind Kurzweil's forecast that computers will constitute a significant share of earthly intelligence in the near future. It is a fascinating exercise in forecasting a nonlinear economic process.

Economic forecasting is notoriously difficult. We do not do terribly well forecasting six months ahead, and as the term of the forecast lengthens, we get progressively less accurate.

I was in graduate school during the energy crisis. At the time, a group of "systems theorists" used a complex nonlinear model to forecast a sort of "doomsday scenario" in which depletion of the earth's natural resources led to economic catastrophe. Economists did not believe the story, but we could not categorically prove that the systems theorists were wrong.

We did not have a precise alternative forecast to give. Instead, our unease with the "doomsday scenario" reflected our general discomfort with making precise long-term forecasts of nonlinear processes.

My unease with Kurzweil's book is similar. In a sense, he is asking us to buy into the following: a nonlinear process that has increased computer intelligence from, say, 0.000001 percent of total earthly intelligence in 1960 to, say, 0.002 percent of total earthly intelligence in 2000, will continue to increase at a predictable rate until it has reached, say, 99 percent of all earthly intelligence in 2099.

Kurzweil is relying heavily on an autoregressive model. An autoregressive model is a model in which you use the statistical properties of the past behavior of a variable (its derivatives with respect to time, in some sense) to predict its behavior in the future.

For example, in my essay on the second derivative, I predicted that Yahoo's page views eventually will level off at 1.5 to 2.0 billion per day. I arrived at this figure by using a crude autoregressive model. I went back to each quarterly release that Yahoo had issued over the past several years and took the figure for page views. I then took the first and second derivatives of these pages views with respect to time, and based my forecast on those derivatives.

My simple autoregressive model ignored what economists would call the "structural factors" underlying the empirical behavior of page views. That is, it takes no account of differences in Internet adoption rate across countries, the rate of introduction of new services on Yahoo, or other underlying causal factors. Implicitly, the autoregressive model assumes that these causal factors will continue to interact as they have in the past. If this is the case, then the model will have value in forecasting the future. If not, then the model is likely to break down.

Kurzweil places heavy reliance on Moore's Law, which is an autoregressive model. The law, first articulated by Gordon Moore of Intel, states that the surface area of a transistor will fall by 50 percent approximately every 12 to 24 months.

As a first approximation, Moore's Law means that the cost of computing power will fall at a rate of 50 percent every two years. This might be referred to as Moore's Corollary. However, most writers tend to use Moore's Law and Moore's Corollary interchangeably.

Kurzweil's forecast requires Moore's Corollary to continue to hold long after Moore's Law has expired. That is, he projects that the cost of computing power will continue to come down at the same rate, even after the ability to shrink chip densities has reached its physical limit. He cites several technological strategies for continuing to enhance computer power beyond that point. However, he gives no reason for those technologies to continue to obey Moore's Law or what I call Moore's Corollary.

In economics, it is considered dangerous to rely on an autoregressive model in a time of structural change. A change in computer technology from chips to some other method certainly would qualify as such a change. That is why many economists prefer structural models, that specify the forces driving a system, rather than an autoregressive model, which is a black box.

Within structural models, economists like to see prices, along with supply and demand responses. One of the main objections that economists had to the "doomsday scenario" models was that they lacked these basic economic elements.

A structural model of the price of computing power would incorporate supply and demand. The industry joke, "Andy figures out how to create more computing power and Bill figures out how to use it," is a model of the supply and demand for processing speed.

The key point is that both demand and supply have to be shifting in order for Moore's Law to continue to hold. Once Bill loses the ability to entice people to buy bloatware that requires faster computers, Andy is going to find that the return on investment for taking chips to the next level is no longer there. At that point, computer power will stop following Moore's Law. We may be close to that point now, if we have not already reached it.

Today, much of the increase in computing intelligence is taking place outside of the Andy/Bill realm, on the Internet. That is a structural change. It does not necessarily mean that computing intelligence is growing more slowly than Moore's Law. It may in fact be growing faster. It will be difficult to tell, in part because traditional measures such as chip density or instructions per second (MIPS) are becoming less relevant as indicators of progress.

For Kurzweil's forecast to hold, the demand and supply for machine-based intelligence both have to grow rapidly. I think that my skepticism concerns the demand side. I am not sure that we will demand technological change at ever-increasing speed.

Can human beings absorb the pace of change necessary to continue to increase the demand for machine intelligence? "The Industry Standard" magazine includes in its reviews of new technology gadgets an "expected useful life," meaning the amount of time before the gadget is likely to become obsolete. In many cases, their estimate is less than a year.

Somehow, it feels to me like we're approaching a limit in the process of innovation. Call it adoption drag or cultural friction. Maybe it is not as big a deal as it appears. Maybe there are other offsetting factors. But my guess is that Kurzweil's "law of accelerating returns scenario" is not much more reliable as a forecast than the "doomsday scenario" was 25 years ago.