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## Naive approach to finding best L1 fit to a vector time series

The problem is given $v(t)$ a vector time series find a matrix $A$ that minimizes the function

$E(A) = \sum_{t=0}^{T-1} \| v_{t+1} - A * v_t \|_1$

The straightforward solution of this without any worry about computation time is

obj<-function(A){ sum(abs(v-A%*%lag(v,k=1))) }

A0<-matrix(0,nrow=n,ncol=n)

fit<-optim( A0, obj)

print(fit$par) print(fit$value)

This simple model is unlikely to describe the volatility process obviously but it’s useful to understand ergodicity.  Abstractly, ergodic theory studies iterates of a map $T:X\rightarrow X$.  Terry Tao’s ergodic theory notes are here.