Research 1: We test the invariance-of-bethypothesis from Kyle and Obizhaeva (2016) for the Tokyo Stock Exchange (TSE).The pooled regression coefficients of the logarithm of the number of trades onthe logarithm of trading activities range from 0.665 to 0.669, close to thetheoretical value of two-thirds predicted by the invariance-of-bet hypothesis.European markets data also confirms the two-thirds relations implied by marketinvariance. Our results suggest that using alternative transaction data reducesthe measurement errors in variables such as the number of trades and the tradesizes, explaining why the two-thirds relation might not hold using more recentU.S. data.Research 2:We compare empirical measures of intraday market depth for U.S. stocks from 2004--2023 based on quoted bid-ask spreads and size with theoretically implied measures of market depth based on market microstructure invariance. Quoted dollar depth is measured as the implied slope of a linear market impact function. Invariance-implied dollar depth is proportional to the two-thirds power of the ratio of dollar volume to returns variance. Consistent with previous research, both of these measures of market depth show show high correlation and a pattern of increasing depth throughout the day. The increase in volume at the end of the day leads to increases in  theoretically implied depth which are economically significantly larger than increases in empirical quoted depth.  This result is consistent with the hypothesis that a significant fraction of end-of-day volume does not contribute to market liquidity. This finding is consistent with the hypothesis that institutional investors cross orders at the end of the day in a manner which reduces trading costs for themselves but lessens liquidity provided to other traders.