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This question is regarding “Estimating cross-section common stochastic trends in nonstationary panel data”, Bai (2004). On p. 152 he writes: The model: $$X_{it}=\lambda_{i0}F_{t}+\lambda_{i2}F_{t-2}+e_{it},\,Eq.\,\left(1\right) $$can be rewritten as: $$X_{it}=\left(\lambda_{i0}+\lambda_{i2}\right)F_{t}-\lambda_{i2}\left(\Delta F_{t-1}+\Delta F_{t-2}\right)+e_{it},\,Eq.\,\left(2\right) $$where $X_{it} $ and $F_{t}$ are scalars (for simplification) and $F_{t}$ is a random walk, $F_{t}=F_{t-1}+u_{t}$.

If I multiply $Eq.\,\left(2\right)$ out I get:

$$X_{it}=\lambda_{i0}F_{t}+\lambda_{i2}F_{t-2}+\lambda_{i2}\left(F_{t}-F_{t-1}-F_{t-2}+F_{t-3}\right)+e_{it}$$

So unless the term $\lambda_{i2}\left(F_{t}-F_{t-1}-F_{t-2}+F_{t-3}\right)=0 $ the statement above will not hold. My problem is that I cannot get this term to equal zero, even after using the definition for $F_{t}=F_{t-1}+u_{t}$

Before this he considers a more general model but when I can't rewrite this simpler one there is no point in looking at the more general one. Any hints on how to solve this would be appreciated.

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  • $\begingroup$ I'll try a simple argument: the formulas (16) and (17) in the paper cannot match unless $\gamma'_{ip}=0$ because otherwise (17) involves $F_{t-p-1}$ (coming from $\Delta F_{t-p}$) while (16) does not involve $F_{t-p-1}$. Does that make sense? $\endgroup$
    – Richard Hardy
    Commented Oct 27, 2015 at 20:26
  • $\begingroup$ Usually when going from levels to first differences, the number of lags changes by one. E.g. given a VECM$(p)$ process, the corresponding process in levels is VAR$(p+1)$. So perhaps (17) should have one lag less than it has (the last lag is included by mistake)? $\endgroup$
    – Richard Hardy
    Commented Oct 27, 2015 at 20:37
  • $\begingroup$ @RichardHardy: Yes I know how to rewrite a VAR to a VECM and I was thinking about that as well at one point but didn't think there would such a mistake in the paper. It was after all published in the Journal of Econometrics. I was thinking that there was something else I had missed somehow. Thanks for the comment. $\endgroup$
    – Plissken
    Commented Oct 28, 2015 at 11:47

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I think (17) should be $$ X_{it}=\gamma_{i0}'F_t-\gamma_{i1}'\Delta F_{t}-\cdots-\gamma_{ip}'\Delta F_{t-p+1}+e_{it}. $$ By expanding this $$ X_{it}=\gamma_{i0}'F_t-\gamma_{i1}'(F_{t}-F_{t-1})-\cdots-\gamma_{ip}'(F_{t-p+1}-F_{t-p})+e_{it}\\ =(\gamma_{i0}'-\gamma_{i1}')F_t+(\gamma_{i1}'-\gamma_{i2}')F_{t-1}+\cdots+(\gamma_{ip-1}'-\gamma_{ip}')F_{t-p+1}+\gamma_{ip}'F_{t-p}+e_{it} $$ which with the definitions $\gamma_{ik}=\lambda_{ik}+\cdots+\lambda_{ip}$ means that $(\gamma_{ik-1}-\gamma_{ik})=\lambda_{ik}$ giving you (16): $$ X_{it}=\lambda_{i0}'F_t+\lambda_{i1}'F_{t-1}+\cdots+\lambda_{ip}'F_{t-p}+e_{it}. $$ The same error seems to have been reproduced in the text, which isn't surprising since he probably just used (17) to obtain it in the first place.

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  • $\begingroup$ That's pretty much what I thought, too. $\endgroup$
    – Richard Hardy
    Commented Oct 27, 2015 at 20:59
  • $\begingroup$ Great! If there is a mistake in the paper as both you and @RichardHardy comment on then everything works out great. Thank for the help! $\endgroup$
    – Plissken
    Commented Oct 28, 2015 at 11:50

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