Let me first summarize the statistical caliber of the three. M0 is cash in circulation, which is currency; M1 = currency + travelers’ checks + demand deposits + other check deposits; M2 = currency + travelers’ checks + demand deposits + other check deposits.
+ Small Time Deposits + Savings and Money Market Deposit Accounts + Money Market Fund Shares (Retail).
In short, compared with M0, M1 has added some assets that are not currency but are highly liquid, such as demand deposits. Compared with M1, M2 has added some assets that are less liquid than M1.
Your question actually addresses a difficulty in formulating monetary policy. We cannot know with certainty which indicator can truly measure currency. In research, we generally observe whether the trends of these indicators are consistent. If they are consistent, it will be easy to say.
However, in fact, the growth rates of M1 and M2 often diverge.
If the growth rate of M1 is greater than M2, it means that the growth rate of corporate demand deposits is greater than the growth rate of time deposits, businesses and residents have active transactions, micro entities have strong profitability, and economic prosperity increases.
If the growth rate of M1 is smaller than M2, it means that enterprises and residents choose to deposit funds in banks in the form of regular deposits, micro-individual profitability declines, investment opportunities are limited in the future, excess funds begin to precipitate from the real economy, and economic performance declines.
Our research has found that the trend of the difference between my country's M1 increment and M2 increment has a relatively strong correlation with the trend of the stock index.