Unit Roots and Economic Recovery

This may seem like an esoteric subject, but the concept of “unit roots” has implications for practitioners who debate time series forecasts.

Consider the following two models of economic growth:

y_t = y_0 e^{\theta t}\varepsilon_t
y_t = y_{t-1} e^{\theta}\varepsilon_t
Where y_t is GDP at time t, \theta is a long-term exponential growth rate, and \varepsilon_t is a stochastic driver with mean 1.

The first model multiplies an initial value by an exponential growth trend and a random variable. For every y_t, the \varepsilon_t term will make y_t be above or below the long-term trend. However, these deviations are temporary and will have no impact on y_{t+1}.

In the second model, the most recent observation is multiplied by the growth rate and a new error term. Like in the previous model, y_t is affected by growth and random shocks. However, rather than \varepsilon_t generating a temporary deviation from a trend, \varepsilon_t generates a deviation from the trended prior observation. Because y_{t-1} is dependent on \varepsilon_{t-1}, y_t depends on all prior shocks. In this model, shocks have a permanent impact on future y.

The first model is said to be trend-stationary because the expected value and variance is identical throughout for all t after adjusting for the trend. The second model is harder to forecast. It will resemble a random walk with an upward drift.

The above graph from wikipedia shows what a trend-stationary and unit-root time series might look like after a negative shock. The blue line is trend-stationary and will return to the long-term trend. The green line has a unit root and will continue to drift upwards at the previous growth rate.

Surprisingly, there isn’t a consensus among economists over whether economic growth has a unit root. For example, in March 2009, Paul Krugman and Greg Mankiw had a public disagreement over whether GDP has a unit root. Mankiw believed the damage caused by the financial crisis would be persistent. Krugman entitled his response “roots of evil.” Although Krugman never took Mankiw’s subsequent offer to wager, Mankiw likely would have won the bet. The sustained recession appears to be the predictions of the Unit Root model playing out.

In the less-glamorous world of litigation support and business valuation, I’ve seen the following graph used to argue that area businesses were due a high growth trend as the local economy would soon be “reverting to the trend” but for the business interruption.

This is a valid point if, as the expert was implicitly assuming, the time series is trend-stationary. However, if Corporate Income Tax Revenue has a unit root, then there is no long-term trend to which the economy can revert.

The most common way to test whether a time series is trend-stationary is to use a Dickey-Fuller test. In my opinion, applying this test should be one of the first things an aspiring analyst does before working on a time series. Specifying the wrong model when a unit-root relationship is present can lead to spurious results and bad forecasts.