Level-Based Estimation of Dynamic Panel Models

A.3.1 Proof of Lemma Lemma 1

From eqs. (7)–(8), θ_∘ is a solution of E[gi(θ)]=0h×1. We show that θ_∘ is indeed the unique solution. Let θ~=(α~,γ~,β~′,σ~μ2,τ~′) satisfy E[gi(θ~)]=0h×1.

We prove first that (γ~,β~′)=(γ∘,β∘′). Let Zi(t,l) and Z~i(t,l) denote the (t, l)-coefficient of Z_i and Z~i, respectively. Define the mappings J:{1,…,h}→{1,…,T−1} and J~:{1,…,h~}→{1,…,T−2} such that

Essentially, J(l) (or J~(l)) provides the number of the row that contains the nonzero element of column l of Z_i (or Z~i). Note that both J(l) and J~(l) are well-defined as each column of Z_i and Z~i contains only one nonzero coefficient. Now, for a given l=1,…,h~, define further (L1(l),L2(l))∈{1,…,h}2 such that

Observe that both L1(l) and L2(l) are well-defined as Z~i is a submatrix of Z_i and also each row of Z_i and Z~i does not contain nonzero repeated elements. Moreover, we must have L1(l)<L2(l) by construction of Z_i. Then, let DAB be nonstochastic h~×h matrix whose components are given by

For instance, when T = 3 and k = 1, we have

By construction, DAB must satisfy

as well as DABΨ(θ)=0h~×1 (see eqs. (5)–(7)). We refer to Ahn and Schmidt (1995, Appendix A.2) for further discussion and examples about these results. Observe that E[DABgi(θ~)]=DABE[gi(θ~)]=0h~×1, so eq. (15) yields Arellano and Bond’s (1991) system of linear equations:

Since E(Z~iΔxi) has full rank (Assumption 3), there is a unique solution to this system of (linear) equations and, as a result, we must have (γ~,β~′)=(γ∘,β∘′).

Regarding the other parameters, using the first equation of the system E[gi(θ~)]=0h×1, we obtain

and therefore α~=E(yi2)−γ~E(yi1)−E(xi2′)β~=E(yi2)−γ∘E(yi1)−E(xi2′)β∘=α∘. Using also the T-th equation of E[gi(θ~)]=0, it follows that

Proceeding in a similar manner, we can prove that τ~tx=τt∘x for every t = 1, …, T. Finally, exploiting the (T + 2)th equation of E[gi(θ~)]=0h×1 related to expression (3), we obtain

and therefore

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A.3.2 Proof of Theorem Theorem 1

1. By Theorem 2.6 in Newey and McFadden (1994), to establish consistency it suffices to check that the following conditions are satisfied:^[2]

   1. Ω˙→PΩ and Ω is positive definite;

   2. E[g_i(θ)] = 0 if and only if θ = θ_∘;

   3. θ_∘ ∈ interior(Θ) for some compact set Θ;

   4. g_i(⋅) is a continuously differentiable on interior(Θ) with probability one;

   5. E[supθ∈Θ‖gi(θ)‖∞2] is finite.

   Conditions (i) and (ii) follows immediately from Assumption 4 and Lemma 1, respectively. For condition (iii), we can take any compact set of R×(−1,1)×Rk×R×RkT+1 containing θ_∘. Note that the functions g_i(⋅) and Ψ(⋅) are well-defined on R×(−1,1)×Rk×R×RkT+1. Condition (iv) holds because g_i(⋅) is continuously differentiable on R×(−1,1)×Rk×R×RkT+1 for any realization of [(yi1,yi′)′,(xi1,xi)′]. In particular, closed-forms expression for the Jacobian matrix of Ψ(θ) are provided in Appendix A.2. Condition (v) holds for any compact set Θ because (yi1,yi′,xi1′,…,xiT′) has finite fourth moment.

2. By Theorem 3.4 in Newey and McFadden (1994), in addition to conditions (i)-(v), to establish asymptotic normality it suffices to verify:

   1. G′Ω−1G is nonsingular.

   2. E[supθ∈Θ‖▽θgi(θ)‖∞2] is finite.

   Condition (vi) follows immediately from Assumption 5, while (vii) follows by construction of g_i(θ) (see eq. (8)) and the characterization provided in Appendix A.2.

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A.3.3 Proof of Lemma Lemma 2

By Theorem 4.5 in Newey and McFadden (1994), conditions (i)–(vii) are sufficient to establish the consistency of the asymptotic variance estimator.

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A.3.4 Proof of Theorem Theorem 2

Denote ΩD=ΩD(γ∘,β∘′)=E[giD(θ∘)giD(θ∘)′] and consider the unfeasible estimator

The reason for considering such an estimator is that, by expression (12), it has the same asymptotic variance as θ^; see Hall (2005, Sec. 3.7). After partitioning

θ~ can be computed by solving the following linear system of equations:

where ▽(γ,β′)g¯1D(γ,β′)=(1/N)∑i=1N▽(γ,β′)gi,1D(γ,β′),

and the remaining terms are defined in a similar manner. From the inverse formula for symmetric partitioned matrices (Theil 1983, eq. 3.2) and since gi,1D(γ,β′) does not depends on θ∖γβ, i.e. ▽θ∖γβg¯1D(γ,β′)=0(h~′+T−2)×(kT+3), our unfeasible estimator (γ~,β~′) can be obtained by solving the (linear) system of equations

or, equivalently, by solving the following optimization problem:

From these expressions, it follows that we can ignore the presence of θ∖γβ, as well as Ω12D and Ω22D, when computing (γ~,β~′).

Following the arguments in the Proof of Theorem 1.2, it can be shown that the asymptotic variance of (γ~,β~′) is given by [G1D′(Ω11D)−1G1D′]−1, where

Since (γ~,β~′) and (γ^,β^′) are asymptotically equivalent, it follows immediately that Σγβ=[G1D′(Ω11D)−1G1D′]−1. Partition

note that G1,2D=E(Δx~i), and write

This inverse is obtained by applying eq. (3.2) of Theil (1983). We have that Υ is positive definite because ΩD=DΩD′ is positive definite (Assumption 4) and so is its inverse. Finally, it follows that

and therefore

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