Citing shadowstats is not really a good sign in any argument. Even if they might have a point, the whole site is not about that point.
The changes in CPI calculation (continuous consumer basket adjustment, etc.) are well documented, well known in econometrics, and is considered a sane thing. (After all you can't really equivocate a TV from the 50s and a TV now.)
And while it's always possible to make better adjustments, shadowstats does not argue for this, it just argues against a strawman conspiracy.
https://moneymaven.io/economonitor/emerging-markets/deconstr...
"For example, a can of tomato sauce that cost $.25 at Piggly Wiggly in 1982 cost $.79 at my local market in early 2015. Starting from the 1982 price, the CPI predicts that it should cost $.61 in 2015 while ShadowStats predicts that it should cost $2.64. Starting from the 2015 price and working backwards, the CPI predicts that it should have cost $.32 in 1982 while ShadowStats predicts that is should have cost $.08. Based on these calculations, we see that the CPI underestimates inflation, as measured by the Tomato Sauce Index: The ratio of the 2015 predicted price of $.61 to the 2015 actual price, $.79, is .77, an underestimate of 23 percent. The ratio of the ShadowStats prediction to the actual price is 3.32, an overstatement of 223 percent. For tuna, both indexes overestimate inflation, the CPI by 34 percent and ShadowStats by 478 percent, and so on."
also: http://blog.jparsons.net/2011/03/shadow-stats-debunked-part-...
And to address the "they miss it part". Well, probably most people don't buy stocks, and most people don't buy private equity limited partnership chunks, so ... CPI-U does not measure "asset bubbles".