Panel GMM Estimation

• Panel Generalized Method of Moments (GMM) estimation is a technique for estimating parameters in dynamic binary panel data models using unconditional moment conditions.
• It constructs linearly independent moment functions based on time triples to overcome incidental parameter issues, ensuring root-n consistency and asymptotic normality.
• The method is implemented via a two-step GMM procedure with careful selection of moments and weighting matrices, making it applicable in economics and social sciences.

Panel Generalized Method of Moments (GMM) estimation is a methodological framework for estimating parameters in panel data models under conditions that allow for individual-specific heterogeneity, dynamic dependence, and potential endogeneity. In the context of dynamic binary response models with fixed effects, notably panel logit AR(1) and AR(p) models, recent results rigorously establish the identification, construction, dimension, and statistical properties of moment conditions required for root-n consistent and asymptotically normal GMM estimation. This approach has been developed through the work of Kitazawa, Honoré, and Weidner, with substantial extensions and formal dimension results provided by Kruiniger ("Further results on the estimation of dynamic panel logit models with fixed effects" (Kruiniger, 2020)).

1. Model and Data Structure

Panel GMM estimation is applied to data comprising $n$ individuals observed over $T$ periods, where the observations for each individual $i$ and time $t$ are binary outcomes $y_{it}\in\{0,1\}$, strictly exogenous covariates $x_{it}\in\mathbb{R}^K$, and unobserved individual fixed effects $\alpha_i$. The AR(1) panel logit model specifies

$$P(y_{it}=1\,|\,y_{i,t-1},\,x_{it},\,\alpha_i) = \Lambda(x_{it}'\beta + \gamma\,y_{i,t-1} + \alpha_i)$$

where $\Lambda(u) = e^u / (1+e^u)$, and $\theta = (\beta',\gamma)'$ collect the common parameters. For AR(2) extensions: $$P(y_{it}=1\,|\,y_{i,t-1},\,y_{i,t-2},\,x_{it},\,\alpha_i) = \Lambda(x_{it}'\beta + \gamma_1\,y_{i,t-1} + \gamma_2\,y_{i,t-2} + \alpha_i)$$ Strict exogeneity of covariates ($x_{it}$ independent of $\alpha_i$ for all $t$) is assumed. Initial conditions $y_{i0}$ (and $y_{i,-1}$ for AR(2)) are conditioned upon in both model specification and the derivation of moment functions.

2. Construction and Dimension of Valid Moment Functions

The central innovation in panel GMM estimation for dynamic logit models is the identification and explicit construction of linearly independent unconditional moment functions. For the AR(1) model with $T\geq 3$, the dimension of the space of valid moment functions is $2^T - 2T$, corresponding to a $2^T - 2T$-dimensional subspace of the $2^T$-dimensional space of functions $\mathbb{R}^{\{0,1\}^{T}}$. These "Kitazawa-Honoré-Weidner" moments are constructed as follows:

For each strictly increasing triple $(t, s, r)$ with $1 \le t < s < r \le T$, define

$$z_{i,t,s}(\theta) = (x_{i,t} - x_{i,s})'\beta + \gamma(y_{i,t-1} - y_{i,s-1})$$

Then, the moments are: $$m^{(a)}_i(t,s,r;\theta) = \begin{cases} \exp[z_{i,t,r}(\theta) + z_{i,r,s}(\theta)] - \exp[z_{i,t,s}(\theta)], & (y_{it},y_{is},y_{ir})=(0,1,0)\ \exp[z_{i,t,r}(\theta)] - \exp[z_{i,r,s}(\theta)], & (0,1,1)\ 0, & \text{otherwise} \end{cases}$$

$$m^{(b)}_i(t,s,r;\theta) = \begin{cases} \exp[z_{i,r,t}(\theta) + z_{i,s,t}(\theta)] - \exp[z_{i,s,r}(\theta)], & (y_{it},y_{is},y_{ir})=(0,0,1)\ \exp[z_{i,r,t}(\theta)] - \exp[z_{i,s,t}(\theta)], & (1,0,0)\ 0, & \text{otherwise} \end{cases}$$

These satisfy $$E[m^{(a)}_i(t,s,r;\theta_0)] = 0, E[m^{(b)}_i(t,s,r;\theta_0)] = 0$$ at the true value $\theta_0$. The set of all indicator-weighted linear combinations of these, with "pre-weights" depending on $(y_{i1},...,y_{i,t-1})$, span the full space of valid unconditional moments. Kruiniger provides a matrix proof demonstrating the rank and independence of these moments via a $2^T \times 2^T$ matrix $P$ whose rank is $2T$.

For the AR(2) model, dimension results are:

• With strictly exogenous covariates present and $T\in\{3,4,5\}$, the number of linearly independent moments is $2^T - 4(T-1)$.
• If covariates are omitted ($\beta = 0$), the count is $2^T - (3T-2)$.

3. GMM Estimation and Asymptotic Properties

A set of $L \ge \text{dim}(\theta)$ valid moments $m_i(\theta)$ is stacked to form the sample-averaged moment vector: $$g_n(\theta) = \frac{1}{n}\sum_{i=1}^n m_i(y_{i,1:T}, x_{i,1:T}; \theta) \in \mathbb{R}^L$$ The two-step GMM estimator is: $$\hat{\theta} = \arg\min_{\theta \in \Theta} g_n(\theta)' W_n g_n(\theta)$$ with $W_n$ a positive-definite weighting matrix. Standard procedure sets $W_n = I$, computes a preliminary $\hat{\theta}^{(1)}$, then estimates

$$\widehat{S} = \frac{1}{n}\sum_{i=1}^n m_i(\hat{\theta}^{(1)}) m_i(\hat{\theta}^{(1)})'$$

setting $W_n = \widehat{S}^{-1}$ for a second-stage estimate $\hat{\theta}^{(2)}$. Under regularity (smoothness of $m_i$ in $\theta$, identification criterion $$\text{rank } E[\partial_\theta m_i(\theta_0)] = \text{dim} \theta$$, $W_n \rightarrow S^{-1}$), asymptotic normality holds: $$\sqrt{n}(\hat{\theta} - \theta_0) \overset{d}{\longrightarrow} N(0, (G'WG)^{-1})$$ where $G = E[\nabla_\theta m_i(\theta_0)]$.

Because the dimension of valid moments is typically large, practitioners select a basis of linearly independent moments (e.g., one per distinct triple $(t,s,r)$, with appropriate indicator pre-weights) that provides full rank $G = \text{dim}(\theta)$ and optimal identification.

4. Implementation Workflow

The construction and estimation routines can be implemented directly in R, MATLAB, Python, or similar platforms. The stepwise procedure is:

1. Input Data: $\{y_{it}, x_{it}\}_{i=1,...,n; t=0,...,T}$
2. Select Index Triples: Choose $\mathcal{T} = \{(t,s,r): 1 \le t < s < r \le T\}$ and a basis of indicator "pre-weights" $w_{t,s}^{(\nu)}(y_{i1},...,y_{i,t-1})$ for $\nu = 1,...,N_{t,s}$.
3. Moment Construction:
   • For each $i$ and $(t,s,r)\in\mathcal{T}$, compute $z_{i,t,s}(\theta)$, $m^{(a)}_i$, and $m^{(b)}_i$ as above.
   • Linearly combine via $$m_i^{(\nu,t,s,r)}(\theta) = w_{t,s}^{(\nu)}(y_{i,1:t-1}) m^{(a)}_i(t,s,r;\theta)$$ (and similarly for $m^{(b)}_i$).
   • Stack all moments into an $L \times 1$ vector $m_i(\theta)$.
4. Moment Averaging: Form $g_n(\theta) = n^{-1} \sum_i m_i(\theta)$.
5. Minimization: Minimize $g_n(\theta)' W_n g_n(\theta)$ with initial $W_n = I$ to obtain $\hat{\theta}^{(1)}$.
6. Update Weighting: Compute $$\widehat{S} = n^{-1} \sum_i m_i(\hat{\theta}^{(1)}) m_i(\hat{\theta}^{(1)})'$$, set $W_n = \widehat{S}^{-1}$.
7. Final Estimation: Re-minimize to obtain efficient $\hat{\theta}$; compute estimated variance $\widehat{V} = (\hat{G}' W_n \hat{G})^{-1}$ with $$\hat{G} = n^{-1}\sum_i \partial_\theta m_i(\hat{\theta})$$.

All formulas and steps arise directly from Kruiniger's results: for AR(1), the triple-moment construction yields $2^T - 2T$ moments; for AR(2) with covariates, $2^T - 4(T-1)$ moments; for AR(2) without covariates, $2^T - (3T-2)$ moments for $T \le 5$. The estimator achieves root-n consistency and asymptotic normality when a sufficient number of differentiable, independent moments is used.

5. Identification, Selection, and Practical Considerations

Panel GMM estimation is predicated on constructing a basis of moment functions that is both valid (satisfying $E[m_i(\theta_0)] = 0$ at the true parameters) and linearly independent. Direct calculation of the dimension ensures no redundancy and avoids biases from over-specification. Moment functions are designed to be unconditional with respect to $\alpha_i$, leveraging strict exogeneity and avoiding the incidental parameters problem inherent in non-linear panel models. Selection of which moments to include from the high-dimensional candidate set should prioritize non-redundancy, finite-sample computational tractability, and statistical identification (rank condition).

Computational requirements are substantial when $T$ is large, as the number of moments and combinations increases exponentially. Optimizers should be robust to non-convexity and numerical instability that may arise in high-dimensional settings.

6. Relevance and Extensions

The panel GMM framework with explicit moment construction advances the estimation of dynamic discrete choice panel models with fixed effects, particularly by resolving longstanding concerns over incidental parameters and lack of identification at root-n rates. The formulas and dimension results for AR(1) and AR(2) models provide practitioners with the tools to implement robust, theoretically justified estimators in various contexts, including economics, epidemiology, and social science applications with short panels and binary outcomes. Extensions to other dynamic panel models, higher-order dynamics, and covariate structures are feasible within this blueprint, subject to the derivation of corresponding valid moment functions.

In summary, the panel GMM approach is rigorously grounded, fully explicit, and proven to deliver root-n consistent, asymptotically normal estimates for dynamic binary response panel models with fixed effects, given appropriate moment construction, identification checks, and computational implementation (Kruiniger, 2020).