When would you keep an error-correction term rather than simply differencing two cointegrated series?

## The choice Once two series are known to be non-stationary, you have two respectable ways to model them together: 1. **Difference both and model the changes.** Simple, robust, and correct in the sense that inference is valid. 2. **Keep the levels information via an error-correction term.** The model becomes changes-plus-gap: short-run dynamics from lagged differences, plus a term proportional to how far the pair sat from its long-run relation last period. The Granger representation theorem makes this precise: if two I(1) series are cointegrated, an error-correction representation exists, and conversely. So option 2 is not an optional embellishment — it is the correctly specified model whenever cointegration truly holds. A pure-differences model is then **misspecified by omission**: the correction term belongs in the equation and has been left out. ## What each choice costs **Differencing costs you the level.** After differencing, the model can no longer see that one series is far above its usual relation to the other. At a one-step horizon this rarely matters much — the next change is dominated by short-run dynamics either way, and a differences model is often within noise of the error-correction model on one-step accuracy. As the horizon lengthens, the gap matters more: the error-correction model's forecasts are anchored by the equilibrium, while the differences model's forecasts fan out with no anchor at all. Anything defined on the *spread* — a pairs trade, a hedge ratio, a convergence bet, a capacity-versus-demand balance — is entirely about the level relation and is unmodellable in pure differences. **Error correction costs you robustness.** The cointegrating coefficient is estimated from history. If the relationship ends — a merger, a technology substitution, a regulatory change, a shift in one firm's business model — the correction term does not go quiet. It reports an ever-growing disequilibrium and pushes the forecast harder in the direction of a convergence that will never come. A differences model, ignorant of the level, simply carries on. That asymmetry matters when real money or real capacity sits behind the forecast: the failure mode of the richer model is confident and unbounded, the failure mode of the simpler model is merely uninformative. A statistical point in error correction's favour: the cointegrating coefficient is *super-consistent*, converging much faster than an ordinary regression coefficient, so it is usually estimated sharply even in moderate samples. The uncertainty that hurts is not sampling noise in the coefficient — it is whether the relationship still exists. ## How I decide - **Horizon.** Short horizon, high frequency, one-step decisions: difference. Multi-period forecasts, planning cycles, anything that must not fan out without an anchor: error correction. - **Is the decision about the spread?** If yes, differencing has removed the decision variable, and the question answers itself. - **Is there a reason for the equilibrium to exist?** Shared input costs, an arbitrage relation, a physical constraint, the same customers. A cointegration test that passes with no such story is a screening artefact, and its correction term is a liability. - **Does it survive out of sample and across subsamples?** Fit on early history, check the adjustment on later data, and look for a break. A relationship that holds only in the full sample has usually already ended. - **Who monitors it?** Adopting error correction is a commitment to watch the spread and to have a rule for retiring the model. Without an owner and a trigger, the sophistication is a hazard rather than an asset. ## The organisational angle The deeper judgment is about who consumes the forecast. A richer model that is right most of the time and catastrophically wrong when a regime ends may be worse for a business than a plainer model that is mildly wrong all the time — because the plainer model's errors are the kind people plan around, while the richer model's errors arrive as a surprise at the worst moment. Making that call explicitly, rather than defaulting to whichever model scored better on backtest accuracy, is what the question is really probing. State the horizon, state the failure mode you can tolerate, and choose accordingly.