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I'm trying to find a way to apply monthly percentage changes to forecast pricing. I set my problem up in excel to make it a bit more clear. I'm using SQL Server 2017.

We'll say all months before 9/1/18 are historical and 9/1/18 and beyond are forecasts. I need to calculate the forecast price (shaded in yellow on the sample data) using...

Forecast Price = (Previous Row Forecast Price * Pct Change) + Previous Row Forecast Price

Just to be clear, the yellow shaded prices do not exist in my data yet. That is what I am trying to have my query calculate. Since this is monthly percentage change, each row depends on the row before and goes beyond a single ROW_NUMBER/PARTITION solution because we have to use the previous calculated price. Clearly what is an easy sequential calculation in excel is a bit more difficult here. Any idea how to create forecasted price column in SQL?

You need to use a recursive CTE. That is one of the easier ways to look at the value of a calculated value from previous row:

DECLARE @t TABLE(Date DATE, ID VARCHAR(10), Price DECIMAL(10, 2), PctChange DECIMAL(10, 2));
INSERT INTO @t VALUES
('2018-01-01', 'ABC', 100, NULL),
('2018-01-02', 'ABC', 150, 50.00),
('2018-01-03', 'ABC', 130, -13.33),
('2018-01-04', 'ABC', 120, -07.69),
('2018-01-05', 'ABC', 110, -08.33),
('2018-01-06', 'ABC', 120, 9.09),
('2018-01-07', 'ABC', 120, 0.00),
('2018-01-08', 'ABC', 100, -16.67),
('2018-01-09', 'ABC', NULL, -07.21),
('2018-01-10', 'ABC', NULL, 1.31),
('2018-01-11', 'ABC', NULL, 6.38),
('2018-01-12', 'ABC', NULL, -30.00),
('2019-01-01', 'ABC', NULL, 14.29),
('2019-01-02', 'ABC', NULL, 5.27);

WITH ncte AS (
 -- number the rows sequentially without gaps
 SELECT *, ROW_NUMBER() OVER (PARTITION BY ID ORDER BY Date) AS rn
 FROM @t
), rcte AS (
 -- find first row in each group
 SELECT *, Price AS ForecastedPrice
 FROM ncte AS base
 WHERE rn = 1
 UNION ALL
 -- find next row for each group from prev rows
 SELECT curr.*, CAST(prev.ForecastedPrice * (1 + curr.PctChange / 100) AS DECIMAL(10, 2))
 FROM ncte AS curr
 INNER JOIN rcte AS prev ON curr.ID = prev.ID AND curr.rn = prev.rn + 1
)
SELECT *
FROM rcte
ORDER BY ID, rn

Result:

| Date | ID | Price | PctChange | rn | ForecastedPrice |
|------------|-----|--------|-----------|----|-----------------|
| 2018-01-01 | ABC | 100.00 | NULL | 1 | 100.00 |
| 2018-01-02 | ABC | 150.00 | 50.00 | 2 | 150.00 |
| 2018-01-03 | ABC | 130.00 | -13.33 | 3 | 130.01 |
| 2018-01-04 | ABC | 120.00 | -7.69 | 4 | 120.01 |
| 2018-01-05 | ABC | 110.00 | -8.33 | 5 | 110.01 |
| 2018-01-06 | ABC | 120.00 | 9.09 | 6 | 120.01 |
| 2018-01-07 | ABC | 120.00 | 0.00 | 7 | 120.01 |
| 2018-01-08 | ABC | 100.00 | -16.67 | 8 | 100.00 |
| 2018-01-09 | ABC | NULL | -7.21 | 9 | 92.79 |
| 2018-01-10 | ABC | NULL | 1.31 | 10 | 94.01 |
| 2018-01-11 | ABC | NULL | 6.38 | 11 | 100.01 |
| 2018-01-12 | ABC | NULL | -30.00 | 12 | 70.01 |
| 2019-01-01 | ABC | NULL | 14.29 | 13 | 80.01 |
| 2019-01-02 | ABC | NULL | 5.27 | 14 | 84.23 |

Demo on DB Fiddle

In SQL Server you can access values of previous / next rows by using the windowing functions LAG and LEAD. You need to define the order of the rows by specifying it in the OVER clause. You may need to wrap the select query, that returns prev/next values in a derived table or CTE, and then select from it and calculate your forecasts.

with cte as (SELECT [Date], Price, LAG(Price, 1) over(order by [Date]) as PrevPrice from TABLE)
select [Date], Price, Price - PrevPrice as PriceChange from cte