A Random Walk Among the Undead

The efficient market model was pronounced dead over a year ago at the World Economic Forum in Davos. But, Jeff Anderson-Lee, a commentator on Paul Krugman's blog, commented that "the 'efficient-market hypothesis' ... seems harder to kill than the undead."

What's going on here; why won't the theory die? In this case, I have to agree with Robert Shiller, the theory won't die because it's partly true! If you've been following my attempt to forecast the S&P 500, you know that I've produce a number of plausible forecasts for the market future from optimistic to pessimistic. One explanation is that I don't know what I'm doing, another is that the future is unknowable and a third is the efficient market hypothesis (EMH).

What is the efficient market hypothesis? That's a little difficult to present clearly because the concept has become overloaded with multiple meanings. The simplest form of the hypothesis is the random walk hypothesis, that is, stock prices move according to a random walk. The random walk hypothesis requires that the largest characteristic root of a state-space model including S&P 500 prices as the only output variable should be close to unity. For a monthly model I estimate running from January 1950 to January 2010, the largest characteristic root is 0.9986935 or, with rounding, unity.

A forecast with this model (displayed above) shows that our best prediction for the future of S&P 500 prices is the current price, as called for by the random walk hypothesis. Actually, the forecast above was not made with a pure random walk model but rather a random walk with drift model. The pure random walk equation is X(t) = X(t-1) + e(t-1) where e(t-1) is random, uncorrelated error. The random walk with drift is X(t) = a + X(t-1) + e(t-1) where a is the drift term. Supposedly, the drift term invalidates the random walk hypothesis, but even with drift the market is not very predictable.

Interestingly, plotting just X(t) = a + X(t-1) provides a very basic bubble predictor. From 1950 to 1995, a buy-and-hold investment strategy (advocated by EMH proponents) made some sense. No matter when you purchased the stock that tracked the S&P 500, you made money on any sale. After 1995, things became a lot more risky. Stocks purchased in 2000 and sold in 2010 generated huge loses. Buy-and-hold after 1995, in retrospect, wouldn't have made much sense as an investment strategy.

If EMH was restricted to random walk or random walk with drift models, it is a good basic starting point for understanding investment strategies and market bubbles. However, "efficient" should not be equated with "optimal." A casino is essentially a random walk with drift for the house. Where the EMH embraces visions of perfection, it obviously (after the dot-com bubble and the subprime mortgage bubble) goes too far. In future posts I'll struggle with what optimization would mean in the context of the stock market and struggle even harder with the question of whether, whatever the stock market is, it benefits the US economy.

Double Bubble, Toil and Trouble

The NY Times today posed the question "Fed Missed This Bubble. Will It See a New One?" Both Fed chairmen Alan Greenspan and Ben Bernanke famously missed the development of the dot-com and the Great Recession bubbles (even conservative News.max accepts this analysis). So, how hard is it really to identify bubbles?

In an earlier post I showed how feather forecasting from a state-space model estimated up to 1990, showed that the market was over-priced during both the dot-com and the subprime mortgage bubbles. Implicitly, the feather forecast points to a sustainable level for the market but doesn't really display the level explicitly. To do that we need to remove the cyclical components and the month-to-month shocks from the model and run a counterfactual simulation for the stock market. The simulation is displayed above. Rather than reaching almost 1600 in the peak of the dot-com bubble, the simulation suggests that the market should have been at about 600. And, 1000 would have been a better level when subprime mortgage crisis broke.

Why is it so difficult for the US Federal Reserve to identify bubbles? Part of the problem is that one of the major cyclical state variable is the US economy is driven by Fed policy. Another cyclical state variable involves more fundamental economic factors such as housing, corporate profits, oil prices and unemployment. Whether the Fed is counter-cyclical or pro-cyclical will have to be a topic for another posting but clearly the Fed is in the middle of market cycles and has trouble looking outside the box.

And, compared to my pessimistic forecast for the S&P 500, the forecast eliminating the cyclical state variables for the US economy is optimistic. In the figure above, the S&P 500 is back to pre-crisis levels by 2015. The usual disclaimers apply even more strongly to this counterfactual forecast.

At best, financial reform might reduce but not totally eliminate bubbles in the US economy. Since late 20th Century bubbles seem to last about a half decade, the pattern suggests a bubble investing strategy that would differ from the standard diversified portfolio approach--a topic for a future posting.

The Dot-com and Subprime Mortgage Bubbles: An Example of State-Space Forecasting Techniques

In a prior post, I described state space models and why I think they are useful for time series forecasting. I'm using state-space models to generate forecasts for 2010 and plan to look at how well the models did in 2011. However, I can pretend that I was trying this same exercise in 1990 by estimating the model only on historical data up to 1990, using the 1990 model to forecast forward to the present and see how well the model would forecast two notorious crises that standard economic models seem unable to have predicted, the dot-com bubble from 1998-2000 and the subprime mortgage crisis from 2007-2010. For this exercise, I'll just look at the stock market, particularly the S&P 500 index using data from Yahoo! Finance.

First, as a point of comparison, I estimated a state-space model for the entire period. The model uses the state of the US economy as an input variable and the volumes and price data for the S&P 500 as the output variables. The fit of the model for S&P 500 volumes and prices in the figure above (dotted red line) is excellent.Using a model estimated only up to 1990, the price fit is still good but the volume forecast (upper panel above) starts to deteriorate after 2005. Notice for both series that the model tends to miss the turning points--a common feature of all forecasting models (the models don't know that the bubble has burst until after it has happened).

Although models appear unable to capture turning points, they are able to identify developing bubbles using feather forecasting. A feather forecast uses the model to predict a number of periods into the future (thirty-six in the figure above for monthly S&P 500 data), starting from each period in the sample. Up to 1995, the model forecasts basic linear growth for S&P 500 prices. In 1995, however, the feather forecast identifies the developing dot-com bubble as having started in 1995 rather than the conventional dating of 1998. From 1995 to well into 2001, the feather forecast is predicting a collapse in prices. Indeed, from 1995 to 2007 (the last point where a 36 month feather forecast can be made) the feather forecast is warning of a collapse in prices. A conservative investing strategy would have been not to ignore the developing bubble but realize that one would have to be vigilant and liquid with any new investments made after 1995.

This then brings us back to a future forecast made with the model estimated up to 1990. The model is essentially predicting a sell off for 2010 (increasing volumes and decreasing prices). It will be interesting to see what actually happens over the next year. The model cannot predict monthly rallies and sell-offs or the turning points for recovery, but studying the feather forecast suggests that the S&P 500 cannot realistically reach 1000 for a few more years into the future. In any event, the usual disclaimers apply to this prediction.

Forecasting Disclaimer

At the beginning of 2010, I'm publishing a collection of economic and environmental forecasts based on state-space time series models. Given that economists were unable to forecast the financial crisis of 2007-2010, I should probably explain why I think my approach is better.

First, I'm not sure my approach is better. That's part of the purpose for doing the forecasts in early 2010. At the start of 2011 (or any point in between), I can look back at the forecasts, see what went wrong (if anything) and try to figure out the reasons for any failures.

The state-space approaches are attractive because they address a number of known problems with existing forecasting models. State-space models forecast based on the "unobserved" state of the system being studied. The state variables are the minimum collection of independent, orthogonal variables that connect the model's input variables to the output variables. Typically, economic forecasting models have a rigid set of structural equations that are used to predict each output variable. Real human systems, however, are too flexible and too emergent to be modeled with a rigid set of equations.

A related attractive feature of state-space models is that there are two sources of error variation, one for the state-vectors and another for the output variables. Typically, economic forecasting models have one source of error for each output equation. And, typically the error distributions are modeled using a bell-shaped normal distribution. Nouriel Roubini, for one, has strongly criticized economic forecasting models both for their unrealistic structure, estimation techniques and assumptions about error terms. Rather than assuming that the functional form of the error term is known, state-space models can easily use a non-parametric bootstrapping technique to estimate confidence intervals around forecasts and the Kalman filter to compute the system state while forecasting (after an approximate system state is used to estimate the model).

Another interesting aspect of state-space models is the use of computer simulation to understand the forecasts. For example, one can estimate a state-space model for a based period (say 1950-1990 for the US economy) and then use simulation techniques to calculate the unobserved system state (using the Kalman filter) and the time path of the output variables to any point in the future. I'll present an example of the analysis in a later post.

For all these reasons, I find state-space models attractive for forecasting. For some of the answers to how well the models actually perform, we'll have to wait until 2011. However, we can also analyze two notorious historical episodes (the dot-com bubble and the financial crisis of 2007-2010) using a model estimated on US data up to 1990. Could these two crises have been predicted with state-space models? That will be the subject of my next post.