The weight matrix (N-2X(0tθ)X(tθ) )-1 is proportional to the inverse of the (asymp-
totic) covariance matrix of N -1X (0tθ)∆tθ when ∆itθ is IID across i. The consis-
tency of βb Dx(tθ) relies on Assumptions (A), (B1), and (E).
Two modifications of βb Dx(tθ) exist: First, if var(∆itθ) varies with i, we can
increase the efficiency of (38) by replacing xi0(tθ) xi(tθ) by xi0(tθ) (∆ditθ)2xi(tθ) , which
gives an asymptotically optimal GMM estimator of the form (24). Second, instead
of using X (tθ) as IV matrix for ∆Xtθ, we may either, if K = 1, use Y (tθ), or, for
arbitrary K, (X(tθ) : Y (tθ)), provided that also (C1) is satisfied.
Equation in levels, IV’s in differences. Using ∆X(t) as IV matrix for Xt
(for notational simplicity we omit the ‘dot’ subscript on X.t and y∙t), we get
the following estimator of β , specific to period t levels, utilizing all admissible x
difference IV’s,
(39)
-1
βbLx(t) = Xt0(∆X(t)) (∆X(t))0(∆X(t)) (∆X(t))0Xt
-1
× Xt0(∆X(t)) (∆X(t))0(∆X(t)) (∆X(t))0yt
-1
hPixi0t(∆xi(t))ihPi(∆xi(t))0(∆xi(t))i-1hPi(∆xi(t))0xiti
× hPixi0t(∆xi(t))ihPi(∆xi(t))0(∆xi(t))i-1 hPi(∆xi(t))0yiti
It exists if (∆X(t))0(∆X(t)) has rank (T - 2)K, which requires N ≥ (T - 2)K.
This estimator examplifies (23), utilizes the OC E[(∆xi(t))0eit] = 0(T-2)K, 1 — which
follows from (34) - and minimizes the quadratic form
[N-1(∆X(t))0t]0[N-2(∆X(t))0(∆X(t))]-1[N-1(∆X(t))0t].
The weight matrix [N -2 (∆X (t) ) 0 (∆X (t) )]-1 is proportional to the inverse of the
(asymptotic) covariance matrix of N-1(∆X(t)) 0t when it is IID across i. The
consistency of βb Lx(t) relies on Assumptions (A), (B1), (D1), (D2), and (E).
Three modifications of βb Lx(t) exist: First, if var(it) varies with i, we can increase
the efficiency of (39) by replacing (∆xi(t))0(∆xi(t))by(∆xi(t))0(bit)2(∆xi(t)), which
gives an asymptotically optimal GMM estimator of the form (24). Second, instead
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