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Describe the difficulties that arise when modelling time series that contain unit roots.

FRM® Part I · Quantitative Analysis · Non-Stationary Time Series · FRM1.QA.11.d

The idea

Unit roots make ordinary regression analysis unreliable. If two independent series each contain a unit root, regressing one on the other typically gives a high R², a large t-statistic and a very low Durbin-Watson statistic, even though there is no relationship. This is a spurious regression. The problem gets worse, not better, as the sample grows, because the t-statistic does not follow its usual distribution. The estimated autoregressive coefficient is also biased downward in finite samples and has a non-standard distribution, so ordinary critical values over-reject. Shocks have permanent effects and there is no mean reversion. The forecast error variance grows without bound as the horizon lengthens. Typical remedies are to difference the series or, if the series share a common stochastic trend, to model their cointegration. The usual mistake is to trust a high R² and a significant t-statistic in levels.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which combination is the classic warning sign of a spurious regression between two series in levels?

  1. A low R² together with a Durbin-Watson statistic near 2Why not: This is a poor but otherwise unremarkable fit with uncorrelated residuals.
  2. A high R² together with a Durbin-Watson statistic near 2Why not: This indicates a good fit with uncorrelated residuals, which is not the warning sign.
  3. A high R² together with a very low Durbin-Watson statistic - correct
  4. A low R² together with a very low Durbin-Watson statisticWhy not: The spurious pattern is characterised by a high R², not a low one.

Highly persistent residuals push the Durbin-Watson statistic far below 2 while the trending data give a high R². A rule of thumb is that R² above DW signals trouble.

A harder one

Two independent series that each follow a random walk are regressed on each other in levels using 200 observations. The result is R² 0.64, a slope t-statistic of 18 and a Durbin-Watson statistic of 0.09. What is the best interpretation and response?

  1. The relationship is genuine, because the t-statistic far exceeds 1.96 and the sample is largeWhy not: Trusts the t-statistic although it is not valid when both series have unit roots.
  2. The relationship is spurious and collecting more data will remove the problemWhy not: The problem is not a small-sample issue; with more data the spurious t-statistic gets larger.
  3. The relationship is spurious and taking logs of both series and re-estimating in levels will fix itWhy not: Taking logs does not remove a unit root; the stochastic trend remains.
  4. The relationship is probably spurious; re-estimate in first differences or test for cointegration - correct

The very low Durbin-Watson statistic and high R² are the signature of regressing unit-root series on each other. Differencing removes the stochastic trends; if the series cointegrate, an error-correction model is appropriate.

The hardest tier

A shock of size 3 hits a series at time t. Compared with the effect on the forecast 10 periods later, which pair is correct for a random walk and for a stationary AR(1) with coefficient 0.8?

  1. Random walk: 0.00; AR(1): 0.32Why not: Assumes that shocks to a random walk die out, which is the stationary behaviour; they are permanent.
  2. Random walk: 3.00; AR(1): 0.32 - correct
  3. Random walk: 3.00; AR(1): 2.40Why not: Applies the persistence coefficient once rather than raising it to the power of the horizon.
  4. Random walk: 3.00; AR(1): 0.11Why not: Forgets to multiply the decay factor by the size of the shock.

For the random walk the effect stays at 3. For the AR(1) it decays as phi^h: 3 x 0.8^10 = 0.32. Persistent shocks are one reason unit roots complicate forecasting.

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