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If we look at the US M0 money supply after the global financial crisis (GFC):
we see that it doubled from 2010 to 2015.
If we look at the Consumer Price Index in the same period:
it rose by 8%.
So over the 5 year period from 2010 to 2015 the M0 had a geometric average of 15% and the CPI a geometric average of 1.6%.
According to the Quantity Equation the US therefore had a yearly real growth of more than 13% in this period?
Which sounds way to much.
What was the annual real growth rate in the US from 2010 to 2015?
I presume you mean growth rate of output. You can see the quarterly real growth rate in the US from 2010 to 2015 below (source Fred) with average quarterly growth rate being 0.6%:
According to the Quantity Equation the US therefore had a yearly real growth of more than 13% in this period?
No, that is definitely not implied by the quantitative theory of money (QTM) - I assume that is what you mean by the 'quantity equation'. First, you are using wrong data for that (at least for money supply). Second you are not using all data required for QTM is given by:
$$MV=PY$$
With, $M$ being total money supply (usually proxied by $M2$), $V$ velocity of money, $P$ price level, $Y$ real output. Taking logs and differentiating the expression wrt time we get:
$$\frac{\dot{M}}{M} + \frac{\dot{V}}{V} = \frac{\dot{P}}{P} + \frac{\dot{Y}}{Y} $$
The above says that the growth rate in $M$ + growth rate in $V$ should be equal to growth rate in $P$ plus growth rate in $Y$.
Now the total growth rate of $Y$ between 2010 and 2015 of $Y$ was $\approx 11.7 \%$ (see Fred data), the total growth rate of $P$ in between 2010 and 2015 was $\approx 9.4 \%$ (see Fred data), next the total growth rate of $M$, in the same time period, measured by $M2$ was approximately $39.6 \%$ (see Fred data), lastly the total growth in velocity $V$ over the same period was approximately $-14.9 \%$ (again see Fred data).
Consequently, plugin these into QTM we would get that:
$$39.5 \%+ (-14.3\%) - 9.4\% = \frac{\dot{Y}}{Y} $$
Hence QTM would predict that the growth rate of $Y$ would be:
$$ 15.8 \% \approx \frac{\dot{Y}}{Y} $$
This is still off by approximately $4.1\%$ given that the actual real GDP growth was $\approx 11.7\%$, however note QTM is extremely simplistic model of money market equilibrium.
More realistic description of money market equilibrium would be given by (see Woodford Interest & Prices pp 295):
$$M^s_t/P_t = L(Y_t,\Delta_t: \xi)$$
where $L$ is the money demand that is a function of real output and the interest rate differential between nonmonetary and monetary assets $\Delta \equiv \frac{i_t-i^m_t}{1+i_t}$ conditional on vector of shocks $\xi$. You can view this as the more nuanced version of the equilibrium relationship given by QTM since here $MV=PY \implies M/P =Y/V$ and we replace crude $Y/V$ with actual money demand that depends on output and interest rates. In this more nuanced model there does not need to be 1:1 correspondence between LHS and RHS since $L$ will be some function (e.g. it could be $\alpha+ \beta Y - \gamma \frac{i_t-i^m_t}{1+i_t}$, where $\alpha$ and $\beta$ and $\gamma$ would have to be empirically fitted).
