Let's see what Gross says when he actually gets down to CPI numbers.
First, the 1983 change to owner equivalent rent:
... a dubious assumption belied by the experience of the past 10 years during which the average cost of homes has appreciated at 3x the annual pace of the substituted owners’ equivalent rent (OER), and which would have raised the total CPI by approximately 1% annually if the switch had not been made.
Then he talks about the 1990's "Boskin report" changes:
In 2004, I claimed that these revised methodologies were understating CPI by perhaps 1% annually
That's it. So in 2004 he has a total "error" in the CPI of 2%.
To get the first 1% "error", he has to use something other than "owner equivalent rent". If the BLS were simply using house transaction prices in the CPI, then they would be reporting falling prices today and people would be screaming that BLS is cooking the numbers. OER may not be perfect, but any other system, used consistently, isn't going to satisfy the "inflation is horrendous" crowd.
To get the second 1% "error", he needs to say that if people spend about the same amount on a new computer today as they did 12 years ago, the price of computers hasn't gone down. The BLS says that we're buying better computers, therefore they say the price has gone down.
I think the BLS has the right theory on both of these issues. I won't claim that they've got the details perfect. (For example, it's very hard to get "owner equivalent rent" statistics on higher priced homes. Similarly, it's hard to say exactly how much computers have improved.)
Gross also says:
Others have actually tracked the CPI that “would have been” based on the good old fashioned way of calculation. The results are not pretty, but are undisclosed here because I cannot verify them.
He mentions Kevin Phillips, who wrote the Harper's article that's been posted here. As we know, Phillips seemed to get his numbers from John Williams. Williams says that the CPI has been low by 7%, a much bigger number than the 2% Gross is proposing. Probably, Gross "cannot verify" Williams' numbers because Williams won't disclose his calculations.
So when you get down to numbers that Gross is willing to back up, they are modest, and he's not necessarily correct.
Note that I'm not saying there are no clouds on the horizon. Eventually, the international trade issues have to balance out. I'm quite sure that consumer prices today don't reflect $130 barrel oil. If oil simply stays where it is, and the Fed creates enough money to keep nominal wages from falling, we'll see higher consumer prices just from oil working its way through the economy. But those are future events, the CPI is always backward-looking.
P.S. Note that Gross also says:
This isn’t a conspiracy blog and there are too many statisticians and analysts at the Bureau of Labor Statistics (BLS) and Treasury with rapid turnover to even think of it.