Two part question really.
What is the formula for calculating DWar?
Also, as I understand it; it seems to rely on some stats that were historically considerably less stringently monitored than say home runs or hits. If that's the case, does the accuracy of the stat dip exponentially after a given number of years back - is there a known cut-off line where the numbers stop being trustworthy? Or is my assertion incorrect?
The basic formula for WAR is as follows:
WAR = (Batting Runs + Base Running Runs + Fielding Runs + Positional Adjustment + League Adjustment +Replacement Runs) / (Runs Per Win)
Different sources will have different ways of calculating these values.
Baseball Reference (ESPN's source) uses:
bWAR:
RS (Runs Scored) = Runs per Win + (mwRAA (a modified weighted runs above average) + Base Running Runs + Park Factor + Position Adjustment + Replacement Level)
RA (Runs Allowed) = Runs per Win - DRS (Defensive Runs Saved)
Win - Loss% (W-L%)= (RS^x)/(RS^x + RA^x)
where RS = (league Runs/Game /2) + Player Offensive Runs
RA = (league Runs/Game /2) - Player Defensive Runs Saved
x = (Runs Per Game involving Player)^.285
Runs Per Game involving Player = 53.6 * (League Runs Per Out + (Player Runs Batting + Player Runs Double Plays + Player Runs Base Running + Player Positional Adjustment - Player Fielding Runs)/(6 * Player Innings).
Wins Above Average (WAA) = (W-L% - .500) * Player Games Played
Wins Run Scoring Environment (WRSE) = Player Total Runs / ((2 * (league Runs/Game)^.715) - (2 * (Runs per Game involving Player)^.715))
Wins Above Replacement bWAR = WAA + Wins Run Scoring Environment
Fangraphs uses:
fWAR = (wRAA + UZR (Ultimate Zone Rating) + wSB (weighted Stolen Base runs) + UBR (Ultimate Base Running) + Positional Adjustment + League Adjustment + Replacement Level)/ Runs Per Win
Runs per win can be found here
To answer your first question. Extrapolating DWAR from fWAR is:
fDWAR = UZR / Runs Per Win
For an in depth analysis of how to calculate UZR see here
As you can probably see extrapolating DWAR from bWAR is slightly more difficult.
W-L% = ((League Runs/Game / 2)^(Runs/Game involving Player))/((League Runs/Game / 2)^(Runs/Game involving Player) + ((League Runs/Game / 2)- Player Defensive Runs)^(Runs/Game involving Player))
DWAA = (W-L% - .5) * Games Played
DWRSE = Player Defensive Runs / ((2 * (league Runs/Game)^.715) - (2 * (Runs per Game involving Player)^.715))
DWAR = DWAA + DWRSE
For an in depth analysis of how to calculate DRS see here
To answer your second question. The stats used for modern defensive wins are circa 2002 (baseball reference uses them beginning in 2003) before this year TZR is used in place of UZR or DRS. TZR is a system devised by Sean Smith with a brief description here, which assigns hits by batters to position players based on the batted ball type and location. These have been reconstructed to a degree from historical play by play sources but the data becomes significantly more derived if only batted ball type and location of outs are available. So if you want to keep all things equal it is less fair to compare DWAR for players before 2003 with current stats, the further you go back the closer you will get to a point in which all of the stats are based on play by play data of varying quality. To be fair box scores where of much greater quality in the early 20th century, compared with those from the 40's and on due to greater consumption of baseball over radio and more in depth written reviews.
I wouldn't say there is a point at which there is an exponential decay, especially because the brilliant folks who worked on these analysis have gone through painstaking steps to go back through history and try to correct their methods when glaring errors become apparent.
On a side note Caught stealing data can be found to be variable depending on the source of the data especially for baserunner. See this post.
What is the formula for calculating DWar?
To quote from Wikipedia:
There is no clearly established formula for WAR.
does the accuracy of the stat dip exponentially after a given number of years back
I see no reason the accuracy would dip exponentially. That's a phrase with a very specific mathematical meaning, and I don't see a mechanism here that would produce an exponential change in the accuracy.
is there a known cut-off line where the numbers stop being trustworthy?
Even if one existed, that would depend on exactly which (D)WAR formula you're using.
