Fireworks provide a few minutes of joy, but the economic and environmental costs last much longer

Rawpixel.com/ShutterstockEvery Bonfire Night, people pay for a box of fireworks so their friends and family can enjoy the colourful display. Within a few minutes, it’s all over. But the price on the box doesn’t reflect the hidden costs. The smoke inhaled by neighbours, the animals disturbed or the public services called out when things go wrong.

Deep-π*: a neural network estimate of trend inflation

Most empirical models of trend inflation treat it as a latent random walk driven by unobserved shocks, so they cannot say which economic conditions move it. Deep- π*makes the innovations to the trend a possibly nonlinear function of a large number of macroeconomic indicators, estimated with neural networks, so that Shapley values can be used to attribute every movement of the trend to the indicators behind it. Two complementary loss functions define a current trend, which fits inflation today, and a forward-looking trend, which best predicts inflation over the next year.

Deep-π*: a neural network estimate of trend inflation

Most empirical models of trend inflation treat it as a latent random walk driven by unobserved shocks, so they cannot say which economic conditions move it. Deep- π*makes the innovations to the trend a possibly nonlinear function of a large number of macroeconomic indicators, estimated with neural networks, so that Shapley values can be used to attribute every movement of the trend to the indicators behind it. Two complementary loss functions define a current trend, which fits inflation today, and a forward-looking trend, which best predicts inflation over the next year.

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