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Suppose

`x_(t)`

is stationary with zero mean. Let

`\epsi _(t)=x_(t)-\sum_(i=1)^(h-1) a_(i)x_(t-i)`

and

`\delta _(t-h)=x_(t-h)-`

`\sum_(j=1)^(h-1) b_(j)x_(t-j)`

be the two residuals where

`a_(1),dots,a_(h-1)`

and

`b_(1),dots,b_(h-1)`

are chosen so that they minimize the mean-squared errors

`E[\epsi _(t)^(2)]`

and

`E[\delta _(t-h)^(2)]`

. The PACF at lag

`h`

was defined as the cross-correlation between

`E[\epsi _(t)]`

and

`E[\delta _(t-h)]`

; that is,

`\phi _(h,h)=(E[\epsi _(t)\delta _(t-h)])/(\sqrt(E[\epsi lon_(t)^(2)]E[\delta _(t-h)^(2)]))`

Let

`R_(h)`

be the

`h\times h`

matrix with elements

`\rho (i-j)`

for

`i,j=1,dots,h`

, and let

`\rho _(h)=`

`(\rho (1),\rho (2),dots,\rho (h))`

be the vector of lagged autocorrelations,

`\rho (h)=corr(x_(t+h),x_(t))`

. Let

`tilde(\rho )_(h)=(\rho (h),\rho (h-1),dots,\rho (1))`

be the reversed vector. In addition, let

`x_(t)^(h)`

denote the BLP of

`x_(t)`

given

`{x_(t-1),dots,x_(t-h)}`

, i.e.,

`x_(t)^(h)=\alpha _(h,1)x_(t-1)+dots+\alpha _(h,h)x_(t-h)`

. Show that

`\phi _(h,h)=(\rho (h)-tilde(\rho )_(h-1)^(')R_(h-1)^(-1)\rho _(h))/(1-tilde(\rho )_(h-1)^(')R_(h-1)^(-1)tilde(\rho )_(h-1))=\alpha _(h,h)`