Structural breaks

When the rules change

A structural break is a point in time after which the parameters of the process, its mean, variance or regression coefficients, are different. The cause may be a crisis, a policy change, a new market structure such as decimalisation or a change in the index a stock belongs to. Parameters estimated across a break are an average of two different worlds and describe neither, so forecasts built on them are biased. Breaks also fool other tools. A one-time shift in the mean looks like a unit root, so the ADF test fails to reject (Perron). A shift in the variance level looks like extreme persistence in a GARCH model.

If the break date is known, the Chow test compares the sum of squared residuals of one regression on all the data, $\mathrm{SSR}_p$, with those of separate regressions before and after the date, $\mathrm{SSR}_1+\mathrm{SSR}_2$. A large drop is evidence of a break, and the statistic follows an $F$ distribution with $k$ and $n-2k$ degrees of freedom, where $k$ is the number of parameters. The date is usually unknown. Then you try every candidate date, keep the one that minimises the combined squared residuals, and use the largest $F$ (the sup-$F$, or Quandt-Andrews, test). Since the date was chosen to maximise the statistic, ordinary $F$ critical values are far too low and special ones apply. The edges are trimmed, often by 15%, because a break near the ends leaves too few observations on one side. Several breaks can be found with the Bai-Perron method. The CUSUM test, which accumulates standardised forecast errors, flags the moment errors stop looking like zero-mean noise and is natural for monitoring a live model.

• Break: the mean, variance or coefficients change at some date; a full-sample fit averages two regimes.
• Chow for a known date; sup-$F$ for an unknown date, with special critical values and trimmed edges.
• CUSUM monitors forecast errors in real time; a mean shift can pass for a unit root.

Lab · what's graded

• A Chow statistic15 pts
• The drop in squared errors10 pts
• Averaging across a break15 pts
• Trimming the edges10 pts
• A shift that looks like a unit root15 pts
• Match the vocabulary15 pts
• Handling a break10 pts
• Estimating across a break10 pts