Dear stata users,
It would be great if you could write me if I understood system-GMM
xtabond2) correctly: I would be very (!) grateful.
My model that I want to estimate is:
y{i,t} = b0 + b1 * y{i ,t-1} + b2 * x1{i,t} + ... + bk * x{i,t} +
c{i}+u{i,t}
in expected values:
E(y{i,t}) = E(b0 + b1 * y{i ,t-1} + b2 * x1{i,t} + ... + bk * xk-1{i,t}
+c{i} + u{i,t})
E(u{i,t} = E( y{i,t} - b0 - b1* y{i, t-1} - b2 * x1{i,t} - ... - bk
*xk-1{i,t} -c)
The instrument variable for y{i,t-1} is (y{i,t-1} - y{i,t-2})=Z1
The instrumental variables for x1,...,xk-1 are x1 till xk-1 themselves
because they are exogen.
The orthogonality conditions for the level equation are:
m1 = E( Z1(y{i,t} - b0 - b1* y{i, t-1} - b2 * x1{i,t} - ... - bk
*xk-1{i,t} -c))
m2 = E( x1(y{i,t} - b0 - b1* y{i, t-1} - b2 * x1{i,t} - ... - bk
*xk-1{i,t} -c))
m3 = E( x2(y{i,t} - b0 - b1* y{i, t-1} - b2 * x1{i,t} - ... - bk
*xk-1{i,t} -c))
.
.
.
mk-1+2, if k-1 = numer of rhs. I need at least k+1 instruments because of
the indiviual effect c. (is this correct?)
Then xtabond2 uses also the equation in differences, which is:
(y{i,t} - y{i,t-1}) = a*(y{i,t-1} - y{i,t-2}) + (x{i,t} - x{i,t-1})*b
+(u{i,t} - u{i,t-1})
in expected values:
E(y{i,t} - y{i,t-1}) = E(a*(y{i,t-1} - y{i,t-2}) + (x{i,t} - x{i,t-1})*b
+(u{i,t} - u{i,t-1})) or
E ( u{i,t} - u{i,t-1}) = E( (y{i,t} - y{i,t-1}) - a*(y{i,t-1} -{i,t-2}) -
(x{i,t} - x{i,t-1})*b)
The instrument variable for (y{i,t-1} - y{i,t-2}) are y{i,t-2} or
y{i,t-3}=Z2
The instrument variables for (x{i,t} - x{i,t-1}) are x{i,t-1} or x{i,t-2}
The orthogonality conditions for the difference equation are:
m21 = E( Z2( (y{i,t} - y{i,t-1}) - a*(y{i,t-1} - y{i,t-2}) -
(x{i,t} -x{i,t-1})*b))
m22 = E(x1( (y{i,t} - y{i,t-1}) - a*(y{i,t-1} - y{i,t-2}) -
(x{i,t} -x{i,t-1})*b))
.
.
.
m2k-1+2
Then system GMM creates the vector M = [m1, m2, ..., mk-1+2, m21,
m22,....,m2k-1+2], which has to be as close to 0 as possible. To minimize
M, system-GMM creates m_ (m with a - over it = mean) = (1/n) * SUM [from i
= 1 to n] of (M).
Then it creates the vector:
q = mean(M) ' * A * mean(M), A = weighting matrix = asymptotic covariance
mattric of n^(1/2) * mean(M) or = lim n-> infinitiy var[n^(1/2)*mean(m)]
q has to be minimized by chosing bo till bk-1 and c.
Is it ok to discribe the method of system-GMM like that? Stata doesn't
explicitly give an estimation for the individual unobserved effect. That's
just integrated in the residuals, right?
stata also does a sargan test of overidentified restricitons. is it a
problem when the H0 is rejected? Does this mean that my instruments aren't
exogenous?
I thank you a lot for your help!
With my sincere thanks,
Michèle
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