Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
17,645 characters · 2 sections · 0 citation commands
Further results on the estimation ofdynamic panel logit models with fixed effects.
JEL\ classification: C12, C13, C23.
Keywords: dynamic panel logit models, exogenous regressors, fixed effects.
\setcounter{page}{0} \thispagestyle{empty}
\baselineskip=20pt
Kitazawa (2013, 2016) showed that the common parameters in the panel logit AR(1) model with strictly exogenous covariates and fixed effects are estimable at the root-n rate using the Generalized Method of Moments. Honor \'{e} and Weidner (2020) extended his results in various directions: they found additional moment conditions for the logit AR(1) model and also considered estimation of logit AR(p) models with $p>1$. In this note we prove a conjecture in their paper and show that for given values of the initial condition, the covariates and the common parameters $2^{T}-2T$ of their moment functions for the logit AR(1) model are linearly independent and span the set of valid moment functions, which is a $2^{T}-2T\,$ -dimensional linear subspace of the $2^{T}$-dimensional vector space of real valued functions over the outcomes $y\in \{0,1\}^{T}$. We also prove that when $p=2$ and $T\in \{3,4,5\},$ there are, respectively, $2^{T}-4(T-1)$ and $2^{T}-(3T-2)$ linearly independent moment functions for the panel logit AR(2) models with and without covariates.
\setcounter{page}{0} \thispagestyle{empty}
We adopt the notation of Honor\'{e} and Weidner (2020). In p.16 of their paper they define for triples of time periods $t,s,r\in \{1,2,...,T\}$ with $ t<s<r$ the moment functions $m_{y_{0}}^{(a/b)(t,s,r)}(y,x,\beta ,\gamma ).$ Let $z_{t,s}(y_{0},y,x,\beta ,\gamma )=(x_{t}-x_{s})^{\prime }\beta +\gamma (y_{t-1}-y_{s-1}).$ Then
In p.17 of their paper they conjecture that for $\gamma _{0}\neq 0$ (and arbitrary $y_{0}$, $x$ and $\beta _{0};$ index $i$ is omitted) any moment function $m_{y_{0}}(y,x,\beta ,\gamma )=\overline{w} (y_{1},...,y_{t-1})m_{y_{0}}^{(a/b)(t,s,r)}(y,x,\beta ,\gamma )$ for the panel logit AR(1) model with strictly exogenous regressors and $T\geq 3,$ where $\overline{w}_{y_{1},...,y_{t-1}}(y_{1},...,y_{t-1}):\{0,1\}^{t-1} \rightarrow \mathbb{R} $, can be written as
with weights $w_{y_{0}}^{(a/b)}(t,s,y_{1},...,y_{t-1},x,\beta ,\gamma )\in \mathbb{R} $ that are uniquely determined by the function $m_{y_{0}}(.,x,\beta ,\gamma ).$
We will prove this conjecture by showing for given values of $y_{0},$ $x,$ $ \beta $ and $\gamma $ (i) that the set of valid moment functions is a linear subspace of the $2^{T}$-dimensional vector space of real valued functions over the outcomes $y\in \{0,1\}^{T}$ that has a dimension of at most $ 2^{T}-2T,$ and (ii) that the $2^{T}-2T$ moment functions of the form $ w_{y_{1},...,y_{t-1}}(y_{1},...,y_{t-1})m_{y_{0}}^{(a/b)(t,s,T)}(y,x,\beta ,\gamma ),$ where $w_{y_{1},...,y_{t-1}}(y_{1},...,y_{t-1}):\{0,1\}^{t-1} \rightarrow \{0,1\}$ are $2^{t-1}$ linearly independent indicator functions and $1\leq t<s<T,$ are linearly independent and span this subspace.
Proof: Recall that $Pr(Y_{i}=y_{i}|Y_{i0}=y_{i0},X_{i}=x_{i},A_{i}=\alpha _{i})\equiv $
We drop the index $i.$ A valid moment function $m_{y_{0}}(y,x,\beta ,\gamma ) $ satisfies
or equivalently
Let $T\geq 2$ and $\alpha _{1}<\alpha _{2}<...<\alpha _{2^{T}}.$ Define the $ 2^{T}\times 2^{T}$ matrix $\bar{P}$ with $\bar{P}_{g,h}=p_{y_{0}}(y,x,\beta _{0},\gamma _{0},\alpha _{g})$ for $g,h=1,2,...,2^{T}$ with $ h=1+2^{0}y_{1}+2^{1}y_{2}+...+2^{T-1}y_{T}.$ Next let $P_{g,t}=\exp (x_{t}^{\prime }\beta _{0}+\alpha _{g})$ and define the $2^{T}\times 2^{T}$ matrix $\breve{P}$ with $\breve{P}_{g,h}=P_{g,T}^{y_{T}}\prod \limits_{t=1}^{T-1}(P_{g,t}(1+P_{g,t+1})/(1+P_{g,t+1}e^{\gamma _{0}}))^{y_{t}}$ for $g,h=1,2,...,2^{T}$ with $ h=1+2^{0}y_{1}+2^{1}y_{2}+...+2^{T-1}y_{T}.$ Finally, let $\overline{D}= \overline{D}(x,\beta _{0},\gamma _{0},\alpha )$ and $\breve{D}=\breve{D} (\gamma _{0})$ be nonsingular diagonal matrices with $\overline{D} _{g,g}=((1+P_{g,1}e^{\gamma _{0}})/(1+P_{g,1}))^{y_{0}}\prod\limits_{t=1}^{T}(1+P_{g,t})$ and $\breve{D} _{h,h}=\prod\limits_{t=1}^{T}\exp (-\gamma _{0}y_{t-1}y_{t})$ for $ g,h=1,2,...,2^{T}$ with $h=1+2^{0}y_{1}+2^{1}y_{2}+...+2^{T-1}y_{T}.$ Then it is easily verified that $\breve{P}=\overline{D}\bar{P}\breve{D}.$ Hence $ rk(\breve{P})=rk(\bar{P}).$ We also define $y^{S}=\tsum \nolimits_{t=1}^{T}y_{t}$ for later use.
We now show (i). If the model does not contain covariates, i.e., if $\beta _{0}=0,$ then $\breve{P}$ does not depend on $x$ and there exist $2^{T}-rk( \breve{P})$ linearly independent moment functions, which will not depend on $ x$. Furthermore, the number of linearly independent moment functions available for the model without covariates is at least as large as the number of linearly independent moment functions available for the model that does include them, i.e., that allows $\beta _{0}\neq 0$. In the appendix we show that $rk(\breve{P})\geq 2T$ irrespective of whether $\beta _{0}=0$ or $ \beta _{0}\neq 0$, that is, we prove Lemma 1, which states that the $2T$ columns of $\breve{P}$ corresponding to vectors $y$ with either the first $k$ or the last $k$ elements equal to 1 and the remaining elements (if any) equal to 0 for $k=0,1,2,...T$ are linearly independent.\thinspace \footnote{ More generally, any $2T$ columns of $\breve{P}$ will be linearly independent if they correspond to the following $2T$ $y$-vectors: the two $y$-vectors that satisfy $y^{S}=0$ or $y^{S}=T$ and for each $k\in $ $\{1,2,...,T-1\}$ two $y$-vectors that satisfy $y^{S}=k$, one with $y_{T}=0$ and the other with $y_{T}=1.$} Recall that $rk(\breve{P})=rk(\bar{P}).$ It follows that claim (i) is correct. We now show (ii):
It is easily seen that for any $t_{1}$ and $s_{1}$ with $t_{1}<s_{1}<T$, the $2^{t_{1}}$ moment functions $ w_{y_{1},...,y_{t_{1}-1}}(y_{1},...,y_{t_{1}-1})m_{y_{0}}^{(a/b)(t_{1},s_{1},T)} $ are linearly independent because the $2^{t_{1}-1}$ indicator functions $ w_{y_{1},...,y_{t_{1}-1}}(y_{1},...,y_{t_{1}-1})$ are linearly independent and $m_{y_{0}}^{(a)(t_{1},s_{1},T)}$ and $m_{y_{0}}^{(b)(t_{1},s_{1},T)}$ are linearly independent. Furthermore, any nontrivial linear combination of the moment functions $ w_{y_{1},...,y_{t_{1}-1}}(y_{1},...,y_{t_{1}-1})m_{y_{0}}^{(a/b)(t_{1},s_{1},T)}(y,x,\beta ,\gamma ) $ with $t_{1}<s_{1}<T$ is linearly independent of $ w_{y_{1},...,y_{t-1}}(y_{1},...,y_{t-1})m_{y_{0}}^{(a/b)(t,s,T)}(y,x,\beta ,\gamma )$ with $t<s<T$ and $(t,s)\neq (t_{1},s_{1})$ because only the former depends on $\exp [\pm z_{t_{1},s_{1}}(y_{0},y,x,\beta ,\gamma )],$ where $z_{t_{1},s_{1}}(y_{0},y,x,\beta ,\gamma )=(x_{t_{1}}-x_{s_{1}})^{\prime }\beta +\gamma (y_{t_{1}-1}-y_{s_{1}-1})$. This is still true when $\beta =0.$ Hence the $2^{T}-2T$ functions $ w_{y_{1},...,y_{t-1}}(y_{1},...,y_{t-1})m_{y_{0}}^{(a/b)(t,s,T)}(y,x,\beta ,\gamma )$ are linearly independent. They are also valid moment functions, see Honor\'{e} and Weidner (2020). It follows that they span a $2^{T}-2T$ \thinspace -dimensional linear subspace of the $2^{T}$-dimensional vector space of real valued functions over the outcomes $y\in \{0,1\}^{T}$ that contains the valid moment functions.
Remark 1: The analysis above is also valid when there are no covariates, i.e., $\beta _{0}=0$.
Remark 2: When $\beta _{0}\neq 0,$ then $\breve{P}$ depends on $x$ and part (i) of the proof implies that $rk(\breve{P})\geq 2T.$ However, part (ii) of the proof shows that there exist at least $2^{T}-2T$ linearly independent moment functions, which in turn implies that $rk(\breve{P})\leq 2T$. We conclude that $rk(\breve{P})=2T.$ When $\beta _{0}=0,$ the proof of the conjecture still implies that $rk(\breve{P})=2T$.
Remark 3: It follows from the result under (i) that there are no valid moment functions when $T=2.$ In other words, GMM estimation of the panel logit AR(1) model with fixed effects and possibly strictly exogenous covariates is not possible for $T=2.$ Our proof is more general than that of Honor\'{e} and Weidner (2020) for this claim because we also cover the case where the values of $\alpha $ can only be finite. In their proof, Honor\'{e} and Weidner (2020) chose two of the four different values of $\alpha $ equal to $\pm \infty ,$ which leads to probabilities that are equal to 1 for the events where all elements of $y$ are either zero or one. This unnecessarily restricts the moment functions a priori. In contrast, we also allow all the probabilities of observing a $y$-vector with only zeros or only ones to be less than 1.
Remark 4: The analysis above can also be extended to panel logit AR($p$) models with fixed effects and $p>1$.
Remark 5: The analysis above can also be used for the static panel logit model, i.e., when $\gamma _{0}=0.$ In that case $\breve{P} _{g,h}=\prod\limits_{t=1}^{T}P_{g,t}^{y_{t}}.$ When also $\beta _{0}=0,$ $ \breve{P}$ is equal to a matrix with columns from a Vandermonde matrix of rank $T+1.$ It follows that when $\gamma _{0}=0,$ the set of valid moment functions is a linear subspace of the $2^{T}$-dimensional vector space of real valued functions over the outcomes $y\in \{0,1\}^{T}$ that has at most dimension $2^{T}-(T+1)$ and in particular that when $T=2,$ there exists at most one valid moment condition.
When $p=2,$ we have
where $y_{i}^{(0)}=(y_{i0},$ $y_{i,-1})^{\prime }$ and $\gamma _{0}=(\gamma _{1,0},\gamma _{2,0})^{\prime }.$ We drop the index $i.$ Let us redefine $ \bar{P}$ as a $2^{T}\times 2^{T}$ matrix with $\bar{P} _{g,h}=p_{y^{(0)}}(y,x,\beta _{0},\gamma _{0},\alpha _{g})$ for $ g,h=1,2,...,2^{T}$ with $h=1+2^{0}y_{1}+2^{1}y_{2}+...+2^{T-1}y_{T},$ and let us redefine $\breve{P}$ as a $2^{T}\times 2^{T}$ matrix with $\breve{P} _{g,h}=P_{g,T}^{y_{T}}\prod\limits_{t=2}^{T-1}\left( P_{g,t-1}(\frac{ (1+P_{g,t})(1+P_{g,t+1})}{(1+P_{g,t}e^{\gamma _{1}})(1+P_{g,t+1}e^{\gamma _{2}})})^{1-y_{t-2}}(\frac{1+P_{g,t}}{1+P_{g,t}e^{\gamma _{1}+}{}^{\gamma _{2}}})^{y_{t-2}}\right) ^{y_{t-1}}\times \newline \left( P_{g,T-1}(\frac{1+P_{g,T}}{1+P_{g,T}e^{\gamma _{1}}})^{1-y_{T-2}}( \frac{1+P_{g,T}}{1+P_{g,T}e^{\gamma _{1}+}{}^{\gamma _{2}}} )^{y_{T-2}}\right) ^{y_{T-1}}$ for $g,h=1,2,...,2^{T}$ with $ h=1+2^{0}y_{1}+2^{1}y_{2}+...+2^{T-1}y_{T}.$ Note that with these new definitions of $\bar{P}$ and $\breve{P},$ we still have $\breve{P}=\overline{ D}\bar{P}\breve{D}$ for some nonsingular diagonal matrices $\overline{D}= \overline{D}(x,\beta _{0},\gamma _{0},\alpha )$ and $\breve{D}=\breve{D} (\gamma _{0})$.
The formula for $\breve{P}_{g,h}$ suggests that a second conjecture of Honor \'{e} and Weidner (2020), henceforth H&W, namely that the number of linearly independent moment functions for the general panel logit AR($p$) models with covariates is given by $l=2^{T}-(T-p+1)2^{p},$ is plausible: when $p$ increases by one, the number of possible values for a $p$-tuple $ (y_{t-p},y_{t-p+1},\ldots ,y_{t-1}),$ namely $2^{p},$ doubles, while the number of different sets of $p$ consecutive elements of $ \{y_{1},y_{2},...,y_{T-1}\}$ that appear in the products of powers in $ \breve{P}_{g,h}$ decreases by one (this number equals $T-2$ when $p=2$) and the factors in $\breve{P}_{g,h}$ whose power depends on either $y_{T}$ or $ y_{0}$ account for $2^{p}$ more possibilities, which explains the $(T-p+1)$ part of the formula. To prove H&W's second conjecture for $p>1,$ one can in principle follow a similar proof strategy as for the case where $p=1$. However, when $p>1,$ things are a bit more complicated. As H&W demonstrate, when $p>1,$ the number of linearly independent moment functions for the general panel logit AR($p$) model is smaller than the number of linearly independent moment functions for the panel logit AR($p$) model without covariates (i.e., with $\beta _{0}=0$). One can relatively easily establish the latter number for different values of $T$ by using a proof strategy similar to that for the case $p=1.$ The difference between the two numbers of moment functions is equal to the number of linearly independent "special" moment functions that are only valid for "special" versions of the model, e.g. the model with $\beta _{0}=0,$ but not for the general model. Thus by subtracting the number of these special moment functions from the total number of linearly independent moment functions for the model with $\beta _{0}=0,$ one obtains the number of linearly independent moment functions for the general model.
H&W claim that they have found all moment functions for the general model when $T\leq 5$. However, their claim is premature as they have not shown that there cannot be more than $l$ moment functions for the general model when $T\leq 5$.\thinspace \footnote{ H&W have found one moment function for the panel logit AR(2) model with $ \beta _{0}=0$ (given the value of $y^{(0)}$) when $T=3$, which is a "special" moment function that is only valid when $x_{2}=x_{3}.$ However, they have not shown that when $T=3$, there is only one moment function for this model.} We have shown this above for $p=1$ (and any $T$) and we will show this in the appendix for $p=2$ and $T\leq 5.$
For the panel logit AR(2) model without covariates (i.e., with $\beta _{0}=0$ ), one can show that $rk(\breve{P})=4(T-1)-(T-2)=3T-2,$ so that there are $ 2^{T}-(3T-2)$ linearly independent moment functions available for this model.\thinspace \footnote{ A proof strategy for the claim that $rk(\breve{P})=3T-2$ is discussed in the appendix.} One can easily obtain these by solving the system $\bar{P} _{[3T-2]}\bar{M}_{3T-2}=0,$ where $\bar{P}_{[3T-2]}=\bar{P} _{[3T-2]}(e^{\gamma _{1,0}},$ $e^{\gamma _{2,0}})$ is a $(3T-2)\times 2^{T}$ matrix that consists of (any) $3T-2$ rows of the matrix $\bar{P}$, each evaluated at/corresponding to different values for the $\alpha _{g},$ and $ \bar{M}_{3T-2}$ is a $2^{T}\times (2^{T}-(3T-2))$ matrix with $rk(\bar{M} _{3T-2})=2^{T}-(3T-2)$. The $2^{T}-(3T-2)$ columns of $\bar{M}_{3T-2}$ span the nullspace of $\bar{P},$ which is the space of valid moment functions for the panel logit AR(2) model without covariates.