Revisiting the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg" display="inline" id="d1e9793"><mml:mi mathvariant="bold">∪</mml:mi></mml:math>-shaped patterns in volatility and price impacts: Novel results using trade-time es

When measured using trade-time aggregation, intraday patterns in trading activity remain ∪ -shaped, but estimates of volatility and Kyle’s lambda fall sharply from open to close. ∪ -shaped patterns in volatility and Kyle’s lambda found using commonly-used calendar-time aggregation reflect over-aggregation biases when trading activity is high as near the open and close. Indicative of imperfectly-competitive liquidity provision, trade-time aggregation also reveals that in active markets, expected trade imbalances are positively priced and unexpected trade imbalances are more strongly priced when they share the sign of expected imbalances, while in less active markets expected trade imbalances are negatively priced. • When properly measured using trade-time aggregation, intraday patterns in trading activity are ∪ -shaped, but estimates of volatility and Kyle’s lambda fall sharply from open to close. • ∪ -shaped patterns in volatility and Kyle’s lambda found using commonly- used calendar-time aggregation reflect over-aggregation biases when trad- ing activity is high as near open and close. • Contrary to competitive liquidity provision, in active markets returns per- sist, expected trade imbalances are positively priced and unexpected trade imbalances are more strongly priced when they have the same sign as ex- pected imbalances. • In further contrast to competitive liquidity provision, in less active markets prices revert and expected trade imbalances are negatively priced.