# Solved – an integrated time series

In this question a commenter says that "differencing a series that is not integrated is certainty problematic from the statistical perspective". What is an integrated time series, and why is differencing a series that is not integrated problematic?

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Consider the first difference \$Delta u_t\$ of a linear process (a fairly general way of stating that something does not have a unit root) \$u_t=sum_{j=0}^inftypsi_jepsilon_{t-j}\$ with \$psi_0=1\$ and \$sum_{j=0}^infty|psi_j|<infty\$, i.e. \$\$ Delta u_t=sum_{j=0}^inftypsi_jepsilon_{t-j}-sum_{j=0}^inftypsi_jepsilon_{t-j-1} \$\$ The long-run variance of \$Delta u_t\$ is zero, so that a stationary process should not be differenced "too" often, as the estimated long-run variance for example enters the denominator of t-ratios, and having a population quantity that is zero should not be in a denominator.

We find the \$MA\$ coefficient sequence of \$Delta u_t\$, call it \$d(L)\$. We then show that \$d(1)^2=0\$.

Write \$\$ Delta u_t=epsilon_t+sum_{j=1}^infty(psi_j-psi_{j-1})epsilon_{t-j}equivsum_{j=0}^infty d_jepsilon_{t-j} \$\$ with \$d_0=psi_0=1\$ and \$d_j=psi_j-psi_{j-1}\$. Hence \$sum_{j=0}^infty d_j=1+psi_1-psi_{0}+psi_2-psi_{1}+psi_3-psi_{2}+ldots=0\$.

The long-run variance can be written as \$J=sigma^2(sum_{j=0}^infty d_j)^2\$. Hence, \$J=0\$.

This is because, in general, the long-run variance of an \$MA(infty)\$ process \$Y_t=mu+sum_{j=0}^inftypsi_jepsilon_{t-j}\$ can be written as \$\$ J=sigma^2biggl(sum_{j=0}^inftypsi_jbiggr)^2 \$\$ Take \$sigma^2=1\$ w.l.o.g. Writing out the right-hand side gives begin{eqnarray*} biggl(sum_{j=0}^inftypsi_jbiggr)^2&=&psi_0psi_0+psi_0psi_1+psi_0psi_2+psi_0psi_3+ldots\ &&+quadpsi_1psi_0+psi_1psi_1+psi_1psi_2+psi_1psi_3+ldots\ &&+quadpsi_2psi_0+psi_2psi_1+psi_2psi_2+psi_2psi_3+ldots\ &&+quadpsi_3psi_0+psi_3psi_1+psi_3psi_2+psi_3psi_3+ldots\ &=&ldots\ &=&sum_{j=0}^inftypsi_j^2+2sum_{j=0}^inftypsi_jpsi_{j+1}+2sum_{j=0}^inftypsi_jpsi_{j+2}+2sum_{j=0}^inftypsi_jpsi_{j+3}+ldots\ &=&gamma_0+2gamma_1+2gamma_2+2gamma_3+ldots\ &=&J end{eqnarray*} where the second-to-last line uses expressions for autocovariances of \$MA(infty)\$-processes.

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