# Random Walk With Drift

With limited data, forecasting rules or algorithms have to be relatively simple.

The simplest rule, of course, is the “no change” forecast, which turns out to be optimal for a simple random walk.

“Simple trending” is a slightly more complex algorithm. For a time series xt the forecast for the next period t+1, when the current period is period t, is xt + (xt – xt-1). In other words, the forecast follows from adding the difference between the current and previous period’s values to the value of the time series in the current period.

This forecast rule is suited to a random walk with drift

This is the situation where the value of the time series in period t equals the value of that time series in the preceding period, plus two terms – a drift term μ and the random error term εt.

xt = xt-1 + μ +εt

Generalizations involve letting the drift vary and embedding the result in white noise. Just as a random walk with noise is linked with simple exponential smoothing, so also random walks are amenable to two parameter or Holt exponential smoothing, resulting in forecasts which usually depart from the above simple trending rule. Nevertheless, simple trending can provide a quick approximation and itself can be the optimal forecast, when the drift term is determined by a random walk.

Macroeconomic Time Series and the Random Walk With Drift

One reason why this is important is that many macroeconomic time series appear to be essentially random walks with drift. This is controversial, but is persistently confirmed by various data since first advanced in a classic paper, one of whose authors is now President and CEO of the Philadelphia Federal Reserve Bank. The debate, which has wound on since the early 1980’s (summarized here), is couched in terms of whether macroeconomic time series have “unit roots.”

The stakes are large.

If, say, US Gross Domestic Product (GDP) is one form or other of a random walk with drift, there is no guarantee that total output will snap back to a “long run trend line” after a recession. GDP is subject to permanent effects from random shocks, and exhibits no tendency to return to a long-run deterministic path.

It’s not hard to find a graphic representation of what this would look like. Consider, for example, a chart of US nominal GDP for period 2001:Q1 to 2011:Q4.

In this period, US Gross Domestic Product expands from just below \$10 trillion to above \$15 trillion, with a major wobble over the Recession of 2008-2009. GDP “peaked” at \$14.4 trillion the 2nd Quarter of 2008, and only re-attained that level of output in the 2nd quarter of 2010. So while the growth rate could be argued to be “mean-reverting” - inasmuch as the slope of GDP growth appears to have reasserted itself after 2010 -there is no evidence yet output is returning to a “long run trend line.”

Forecasts of a Random Walk With Drift Can Have Infinite Variance

Another reason this talk about random walks with drift is important relates directly to forecasting.

Time series which manifest a random walk with drift play havoc with forecasting, since the variability of the forecast error increases without bound with longer forecast horizons. Beyond a certain point, there is a perfect fog about the future value of a variable whose data generating process is a random walk with drift.

This seems compatible with the errors in the macroeconomic forecasts documented in a variety of studies. Staff at the Philadelphia Federal Reserve Bank, for example, show that errors of the Survey of Professional Forecasters (SPF) GDP growth forecasts increase systematically with the forecast horizon. And, as we have seen in an earlier entry, the SPF do not seem to do any better than anyone else, when it comes to forecasting a recession.