How to count monthly retention user in bigquery?

I have raw data as below. With each line is the record of an transaction of user, and the month when they made the transaction

What I want is to calculate the number of user who made order in a month and the number of repeated user (RETENTION) from last month, then I can know how many % of user is repeated user.

The desired result should look like this

How can I do it in bigquery?

One way to do it is to do it is through a self-join with the same table and a 1-month delay. That way, we match user&month combinations with user&previous-month to see if it's a returning user. For example, using the 2M row public table bigquery-public-data.hacker_news.stories and a particular user:

Note that prev_month is null (we used LEFT OUTER JOIN) for 2014-02-01 as the user was not active during 2014-01-01. We are joining on author and lagged months with:

FROM authors AS a 
LEFT OUTER JOIN authors AS b
ON a.author = b.author
AND a.month = DATE_ADD(b.month, INTERVAL 1 MONTH)

Then we count a user as repeating if the previous month was not null:

COUNT(a.author) AS num_users,
COUNTIF(b.month IS NOT NULL) AS num_returning_users

Note that we do not use DISTINCT here as we already grouped by author and month combinations when defining orders as CTE. You might need to take this into account for other examples.

Full query:

WITH
  authors AS (
  SELECT
    author,
    DATE_TRUNC(DATE(time_ts), MONTH) AS month
  FROM
    `bigquery-public-data.hacker_news.stories`
  WHERE
    author IS NOT NULL
  GROUP BY 1,2)

SELECT
  *,
  ROUND(100*SAFE_DIVIDE(num_returning_users,
      num_users),2) AS retention
FROM (
  SELECT
    a.month,
    COUNT(a.author) AS num_users,
    COUNTIF(b.month IS NOT NULL) AS num_returning_users
  FROM
    authors AS a
  LEFT OUTER JOIN
    authors AS b
  ON
    a.author = b.author
    AND a.month = DATE_ADD(b.month, INTERVAL 1 MONTH)
  GROUP BY 1
  ORDER BY 1
  LIMIT 100)

and results snippet:

which are correct results, i.e. for 2007-03-01:

Performance is not too fancy but in this case we are selecting only the fields needed for the aggregated data so scanned data is low and execution time not too high (~5s).

An alternative is to use EXISTS() inside COUNTIF() instead of the join but it takes longer for me (~7s). Query