    # st: xtabond2

 From Michčle Sennhauser To Subject st: xtabond2 Date Mon, 23 Jan 2006 16:10:03 +0100

```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,
Michčle

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