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Analysis of Interactive Fixed Effects Dynamic\ Linear Panel Regression with Measurement Error
This paper studies a simple dynamic linear panel regression model with interactive fixed effects in which the variable of interest, say $ Y_{it}^{\ast },$ contains measurement error:
Here $Y_{it}$ is the observed variable and ${\Greekmath 0111} _{it}$ represents measurement error. The term ${\Greekmath 0115} _{i}^{0}f_{t}^{0}$ describes unobserved interactive fixed effects.\footnote{ In this paper, we consider a single factor, that is, the dimensions of $ f_{t} $ and ${\Greekmath 0115} _{i}$ are equal to one. The extension to the multiple factor case is straightforward, but omitted due to space limitation.}$^{ \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}}$\footnote{ When interpreting ${\Greekmath 0115} _{i}^{0}$ as individual specific fixed effects, the term $f_{t}^{0}$ represents the (time-varying) linear projection coefficient of $Y_{it}^{\ast }$ on ${\Greekmath 0115} _{i}^{0}$ (holding $ Y_{it-1}^{\ast }$ constant). This allows the effect of the unobserved individual characteristic ${\Greekmath 0115} _{i}^{0}$ on $Y_{it}^{\ast }$ to be time-varying. Alternatively, one can interpret $f_{t}^{0}$ as a common time specific shock (a common factor) and ${\Greekmath 0115} _{i}^{0}$ then describes reaction to the common shock (a factor loading).} The goal of the paper is to estimate ${\Greekmath 010B} _{0}$ when both the number of individuals $N$ and the number of time periods $T$ are large.\footnote{ We consider large $N,T$ approximations to characterize the bias due to the incidental parameters ${\Greekmath 0115} _{i}^{0}f_{t}^{0}$, see e.g. Bai (2009) and Hahn and Kuersteiner (2004).}
The dynamics of the observed variable $Y_{it}$ can be written as
where $U_{it}={\Greekmath 010F} _{it}+{\Greekmath 0111} _{it}-{\Greekmath 010B} _{0}{\Greekmath 0111} _{it-1}.$ There are two noticeable features in equations $\left( \ref{model.unobserved}\right) $ and $\left( \ref{model.observed}\right) $ compared to the widely studied dynamic panel regression model. First, the individual effects take an interactive form instead of the time invariant form. Secondly, the variable of interest $Y_{it}^{\ast }$ is not observed but measured with error. To our knowledge, combining these two features in dynamic linear panel regression models has not been studied in the large $N,T$ panel literature.
We expect two hurdles in estimating ${\Greekmath 010B} _{0}.$ One is the presence of the interactive fixed effects ${\Greekmath 0115} _{i}^{0}f_{t}^{0}$ which might cause a so-called incidental parameter problem in both the cross section and the time dimension. The second one is that the composite error $U_{it}$ in the observed variable equation $\left( \ref{model.observed}\right) $ is correlated with the lagged dependent variable $Y_{it-1}$ and we may therefore need to use instrumental variables (IVs).
The main contribution of the paper is to find a valid estimation method that overcomes these two problems. The proposed estimator is a nested two-step estimator based on least squares minimization in the first step and distance minimization for some of the first step parameter estimates in the second step\footnote{ An alternative approach would be to use the common correlated effect methods suggested by Harding and Lamarche (2011). Both approaches have their own merits and weaknesses. Comparing these different methods is not our interest in this paper.}. Following Moon, Shum and Weidner (2012) (hereafter MSW), we call this method the LS-MD estimation method. This approach was used in estimating endogenous quantile regression models by Chernozhukov and Hansen (2006, 2008) and in estimating the random coefficient logit demand model by MSW.
The properties of the quasi-maximum likelihood estimator (QMLE), which minimizes the sum of squared residuals, for large $N$, $T$ linear panel regressions with interactive fixed effects were discussed in Bai (2009), and Moon and Weidner (2010). However, this estimation method cannot be used to estimate model $\left( \ref{model.observed}\right) $ since the regressor $ Y_{it-1}$ is endogenous w.r.t. the error $U_{it}$ through the lagged measurement error ${\Greekmath 0111} _{it-1}.$ In this case, we may use instrumental variables. Since $U_{it}$ has an $MA(1)$ type serial dependence structure, we have $E\left( U_{it}Y_{it-1-s}\right) =0$ for all $s\geq 1.$ This suggests to choose $Z_{it}=\left( Z_{1,it},...,Z_{L,it}\right) ^{\prime }=\left( Y_{it-2},...,Y_{it-1-L}\right) ^{\prime }$ for the IVs of the endogenous regressor $Y_{it-1}$. The question, then, is how to use the instrumental variables $Z_{it}$ to estimate ${\Greekmath 010B} _{0}$ in the presences of interactive fixed effects ${\Greekmath 0115} _{i}^{0}f_{t}^{0}$ when both $N$ and $ T $ are large.
The estimation method we consider in this paper is a two-step least-squares minimum distance (LS-MD) estimation. This was recently proposed by MSW for estimating the BLP demand model. A similar multi-step estimation idea was also used in Chernozhukov and Hansen (2006, 2008) in estimating endogenous quantile regressions with IVs.
The LS-MD estimation consists of the following two steps: Step 1: For given $ {\Greekmath 010B} ,$ we solve the least squares problem augmented by the instrumental variables $Z_{it},$ that is, we run the OLS regression of $Y_{it}-{\Greekmath 010B} Y_{it-1}$ on $Z_{it}$ with interactive fixed effects ${\Greekmath 0115} _{i}f_{t}$ and solve
where ${\Greekmath 010D} =\left( {\Greekmath 010D} _{1},...,{\Greekmath 010D} _{L}\right) ^{\prime }$, $ {\Greekmath 0115} =\left( {\Greekmath 0115} _{1},...,{\Greekmath 0115} _{N}\right) ^{\prime }$ and $ f=\left( f_{1},...,f_{T}\right) ^{\prime }.$ Step 2: For some positive definite weight matrix $W_{NT}^{{\Greekmath 010D} }$, we estimate ${\Greekmath 010B} $ by minimizing the length of $\hat{{\Greekmath 010D}}\left( {\Greekmath 010B} \right) $ as
The idea of the LS-MD method is that since $Z_{it}$ is excluded in the regression equation $\left( \ref{model.observed}\right) $ the coefficient of $Z_{it}$ should be zero when ${\Greekmath 010B} ={\Greekmath 010B} _{0}$. When there is no interactive fixed effect one can show that the LS-MD estimator is equivalent to the conventional 2SLS estimator for an appropriate weight matrix $ W_{NT}^{{\Greekmath 010D} }$.
The iid assumptions of ${\Greekmath 010F} _{it}$ and ${\Greekmath 0111} _{it}$ are made for simplicity of the analysis. Later, an extension to a non-iid case will be discussed. Assumption (ref)(i) also assumes that the measurement error ${\Greekmath 0111} _{it}$ is classical in the sense that ${\Greekmath 0111} _{it}$ has zero mean and is uncorrelated with $Y_{it}^{\ast }.$ Later we discuss how to extend\ our method to some special cases of non-classical measurement error. Assumption (ref)(ii) assumes that the factors are strong, which is standard in the factor analysis literature. Assumption (ref)(vi) assumes that ${\Greekmath 010B} _{0}\neq 0$, otherwise the IVs become irrelevant.
Before we present the next assumption, we introduce some further notation. We use $\left[ a_{it}\right] _{it}$ to denote an $N\times T$ matrix with elements $a_{it}.$ For a full column rank matrix $A,$ let $\mathbb{P} _{A}=A\left( A^{\prime }A\right) ^{-1}A^{\prime }$ and $\mathbb{M}_{A}=I- \mathbb{P}_{A}.$ We use notation $Y=\left[ Y_{it}\right] _{it},$ $Y_{-k}= \left[ Y_{it-k}\right] _{it},$ $Z=\left[ Z_{it}\right] _{it},$ $U=\left[ U_{it}\right] _{it},$ ${\Greekmath 010F} =\left[ {\Greekmath 010F} _{it}\right] _{it},$ ${\Greekmath 0111} = \left[ {\Greekmath 0111} _{it}\right] _{it},$ and ${\Greekmath 0111} _{-1}=\left[ {\Greekmath 0111} _{it-1}\right] _{it}.$ Define ${\Greekmath 0115} ^{0}=\left( {\Greekmath 0115} _{1}^{0},...,{\Greekmath 0115} _{N}^{0}\right) ^{\prime }$ and $f^{0}=\left( f_{1}^{0},...,f_{T}^{0}\right) ^{\prime }.$ We also define the $NT$-vectors $y_{-1}=vec\left( Y_{-1}\right) $ and $z=vec\left( Z\right) .$
Assumption (ref) is a relevance condition on the instruments. It demands that the explanatory power of the instruments $Z_{it}$ for the endogenous regressor $Y_{it-1}$, given by $\frac{1}{NT}y_{-1}^{\prime } \mathbb{P}_{z}y_{-1}$, is larger than the joint explanatory power for $ Y_{it-1}$ of the true factor loading ${\Greekmath 0115} ^{0}$ together with any other factor loading ${\Greekmath 0115} $, given by $\frac{1}{NT}y_{-1}^{\prime }\mathbb{P} _{I_{T}\otimes \tilde{{\Greekmath 0115}}}y_{-1}$. If there are no interactive fixed effects included in the model, then the assumption simplifies to the standard relevance condition $\frac{1}{NT}y_{-1}^{\prime }\mathbb{P} _{z}y_{-1}>0$, which is satisfied for ${\Greekmath 010B} ^{0}\neq 0$.
Suppose that Assumption (ref) holds, and consider the special case where $f_{t}^{0}$ has mean zero and is distributed independently over $ t $. Then, Assumption (ref) is equivalent to\footnote{ For the proof of this, we refer to the supplementary appendix which is available at http://www.cemmap.ac.uk/publications.php.}
Thus, by imposing an appropriate lower bound on $|{\Greekmath 010B} _{0}|,$ one can guarantee that the lagged values of $Y_{it}$ are sufficiently relevant instruments. The conclusion that an appropriate lower bound on $|{\Greekmath 010B} _{0}| $ is sufficient for the relevance assumption Assumption (ref) can be extended to cases where $f_{t}^{0}$ is correlated across $t$, but in general it is not possible to give such a convenient analytic expression as in $\left( \ref{suff_cond}\right) $ for the lower bound.\footnote{ A non-zero mean of $f_{t}^{0}$ can result in situations where Assumption (ref) is not satisfied for any value of ${\Greekmath 010B} _{0}$. The assumption that $f_{t}^{0}$ is mean zero would not be restrictive if we would include a conventional individual specific fixed effect in the model, in addition to the interactive fixed effect --- or equivalently (from an asymptotic perspective), one can demean $Y_{it}$ separately for each $i$ before estimating the model with only interactive effects.} Note that the lower bound in $\left( \ref{suff_cond}\right) $ goes to zero when $\Sigma _{{\Greekmath 0115} }\Sigma _{f}$ becomes small relative to ${\Greekmath 011B} _{{\Greekmath 010F} }^{2}$, i.e., the bound is not restrictive when the relative influence of the factors on $Y_{it}$ is small.
To present the limiting distribution of $\hat{{\Greekmath 010B}},$ we need to introduce some further notation. Define the $NT$-vectors $y_{-1}^{{\Greekmath 0115} f}$ and $ z_{l}^{{\Greekmath 0115} f}$ by $y_{-1}^{{\Greekmath 0115} f}=vec\left( \mathbb{M}_{{\Greekmath 0115} ^{0}}Y_{-1}\mathbb{M}_{f^{0}}\right)$, and $z_{l}^{{\Greekmath 0115} f}=vec\left( \mathbb{M}_{{\Greekmath 0115} ^{0}}Z_{l}\mathbb{M}_{f^{0}}\right)$, where $l=1,...,L.$ Let $u=vec\left( U\right) $ and $z^{{\Greekmath 0115} f}=\left( z_{1}^{{\Greekmath 0115} f},...,z_{L}^{{\Greekmath 0115} f}\right) .$
Define $G=\limfunc{plim}_{N,T \rightarrow \infty}\frac{1}{NT}y_{-1}^{{\Greekmath 0115} f\prime }z^{{\Greekmath 0115} f}=\frac{{\Greekmath 011B} _{{\Greekmath 010F} }^{2}}{1-{\Greekmath 010B} _{0}^{2}} \left( {\Greekmath 010B} _{0},{\Greekmath 010B}_0^2,...,{\Greekmath 010B} _{0}^{L}\right)^{\prime }$, and
Notice that under Assumption (ref), the limits $G$ and $W$ are well defined. Also, notice that under Assumption (ref), we have $GWG^{\prime }>0.$
Define
where
MSW showed that under Assumption (ref), as $N,T\rightarrow \infty $ with $\frac{N}{T}\rightarrow {\Greekmath 0114} ^{2},$ where $0<{\Greekmath 0114} <\infty , $ we can approximate
Notice that as $N,T\rightarrow \infty $ with $\frac{N}{T}\rightarrow {\Greekmath 0114} ^{2},$ where $0<{\Greekmath 0114} <\infty ,$ under Assumptions (ref) we can show that
where $b=\left( b_{1},...,b_{L}\right) ^{\prime }$, and
\newline Combining $\left( \ref{appr.alphahat}\right) $ and $\left( \ref {limit.normality}\right) ,$ we have the following theorem.
Notice that the bias $b$ in the limit distribution is due to the incidental parameters ${\Greekmath 0115} _{i}^{0}f_{t}^{0}$ and the lagged dependent variables as IVs, which is similar to the bias in Moon and Weidner (2010). This bias can be consistently estimated and is correctable, for details we refer to Moon and Weidner (2010) and Moon, Shum, and Weidner (2011).
In this section we investigate the finite sample properties of the LS-MD estimator $\hat{{\Greekmath 010B}}$ through small scale Monte Carlo simulations. The data generating process is
where ${\Greekmath 010B} _{0}\in \left \{ 0.2,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }0.5,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }0.8\right \} ,$ $ \left \{ {\Greekmath 0115} _{i}\right \} ,\left \{ f_{t}\right \} ,\left \{ {\Greekmath 0111} _{it}\right \} \sim iid$ $N\left( 0,0.4\right) $ and $\left \{ {\Greekmath 010F} _{it}\right \} \sim iid$ $N\left( 0,1\right) .$ We consider various combinations of $N\in \left \{ 20,50,100\right \} $ and $T\in \left \{ 20,50,100\right \} .$ We use $Z_{it}=Y_{it-2}$ as an instrument. Notice that ${\Greekmath 010B} _{0}=0.2$ violates the sufficient identification $\left( \ref {suff_cond}\right) $.
The finite sample properties of $\hat{{\Greekmath 010B}}$, obtained in simulations with 1000 repetitions, are reported in Table 1. Except for the case of ${\Greekmath 010B} _{0}=0.2$ with small samples, the LS-MD estimator $\hat{ {\Greekmath 010B}}$ performs well in finite samples.\footnote{ We also investigated the finite sample properties of the bias corrected estimator and found that analytical bias correction simultaneously reduces the bias and the standard deviation of the estimator, except when both $ {\Greekmath 010B} ^{0}$ and $T$ are small. We omit the detailed results due to space limitation, and since the biases in Table 1 without bias correction are already quite small relative to the corresponding standard deviations.} When ${\Greekmath 010B} _{0}=0.2,$ the finite sample properties improve as either $N$ and $T$ increases.
Choice of Instrumental Variables: It is well known in the GMM literature that the choice of moment conditions --- the choice of the lag length $(L)$ in our setup --- is one of the important factors that affect the finite sample properties of the GMM estimator. Various moment condition selection procedures have been proposed in the literature. These include, for example, the minimization of the (higher order) approximated mean squared error (e.g., Donald and Newey (2001), Okui (2009), and Kuersteiner (2010)) or of the asymptotic coverage error (e.g., Okui (2009)). However, it is not straightforward to apply these procedures to the LS-MD estimator. First, the LS-MD estimator has a bias even in the first order approximation. Secondly, the key approximation techniques used in the literature (e.g., Nagar's expansion and the Edgeworth expansion) are not available in the exiting literature for the LS-MD estimator. Developing a procedure for selection of $L$ is therefore beyond the scope of this paper.
Extensions: Our LS-MD estimation can be used for more sophisticated cases. We briefly discuss how to extend our simple model.