Quantum error-correcting codes have a distance parameter, conveying the minimum number of single spin errors that could cause error correction to fail. However, the success thresholds of the finite per-qubit error rate that have been proven for the likes of the toric code require them to work well beyond this limit. We argue that, without the assumption of being below the distance limit, the su...

Vector autoregressions are widely used in macroeconomic forecasting since they became known in the 1970s. Extensions including vector error correction models, co-integration and dynamic factor models are all rooted in the framework of vector autoregression. The three important extensions are demonstrated to have formal equivalence between each other. Above all, they all emphasize the importance...

Monetary policy-makers face a difficult task when evaluating the current state of the economy and deciding what actions are needed to achieve their objectives, such as keeping inflation within a given range. Because long and variable lags exist between a monetary policy action and its effects on economic variables, policy-makers need a way to assess whether their actions are having, or indeed w...