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Is it possible to have an integrated random walk process by linearly combining finite number of random walks?

Is it possible to have a random walk process by linearly combining I(0) processes?

Thanks

Cagdas Ozgenc
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1 Answers1

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Linear combinations of individually stationary processes will always be stationary. In this context we talk about variance-covariance stationarity, that is, the mean $(\mu)$ and autocovariance $(\gamma)$ will be independent of time, $E(Y_t)=\mu$ for all $t$, $E(Y_t-\mu)(Y_{t-j}-\mu)=\gamma_{j}$ for all $t$ and $j$. Try to make a linear combination of two stationary processes and estimate the mean and autocovariance of this combination, then you'll find that they are independent of time and hence stationary.

fredrikhs
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  • What about combining I(1)s? That would also be an I(1)? By the way I am not assuming independence between the processes. – Cagdas Ozgenc Aug 24 '13 at 12:17
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    Combinations of I(1) processes can be both I(1) and I(0). If the a linear combination of two I(1) processes are I(0) the two processes are considered to cointegrated and there is a long term relationship between them. If there is no relationship between them, the combination of the two I(1) processes would be I(1). – fredrikhs Aug 24 '13 at 12:28
  • So the bottom line is you can only decrease the order of integration (if cointegrated) but not increase. Thanks – Cagdas Ozgenc Aug 24 '13 at 12:31
  • I guess you can put it that way. :-) – fredrikhs Aug 24 '13 at 12:32
  • Is it possible to linearly combine I(1) processes and result in a fractionally integrated process, long memory process for example? – Cagdas Ozgenc Aug 24 '13 at 12:36
  • Sorry, I'm not able to answer this. – fredrikhs Aug 24 '13 at 12:39