Comparing GMM results

General Usage
1

I am working on implementing a linear GMM estimator in Julia. As a first step, I compare 2 ways of getting parameter estimates: numerically minimizing the loss function

function LossFn(θ, y, X, Z)
    W = diagm(0 => ones(size(Z, 2)))
    g = [(y[i] - X[i, :]'*θ)*Z[i] for i=1:size(Z, 2)]
    loss = g'*W*g
    return loss
end
gmm   = optimize(θ1 -> LossFn(θ1, y, x, z), ones(6))

and using analytical solution:

inv(X'*Z*W*Z'*X)*X'*Z*W*Z'*y

I was surprised to find that the results are different.

2

Different how? Like obviously or slightly different? If they compare as equivalent with isapprox then they are sufficiently the same for this purpose.

1 Like
3

Obiviously different, making me wonder where my code is wrong lol

4

I think the error is size(Z, 2).

5

Yea is Z just a vector here or is it a matrix? Because you call size(Z,2) which indicates that it does have a second dimension but you are indexing it with only Z[i] which suggests it does not.

1 Like
6

Z is a N-by-L matrix of instrument, in which each row is an observation and each column is an exogenous variable.

function LossFn(θ, y, X, Z)
    W = diagm(0 => ones(size(Z, 2)))
    g = [(y[i] - X[i, :]'*θ)*Z[i, :] for i=1:size(Z, 1)]
    gbar = mean(g, dims = 1)[1]
    loss = gbar'*W*gbar
    return loss
end

I think this should get it right. Here g[i] is the value of moment funtion for observation i:
g_i = (y_i - x_i'\theta)z_i
and gbar gives sample moment
\bar{g} = \frac{1}{N}\sum_{i}^N g_i.

7

Likely that you want .* when constructing g

8

y is N-by-1 and X is N-by-K. The model is overidentified so we have K<L. Therefore

(y[i] - X[i, :]'*θ)*Z[i, :]

is a scaler muptiplying a 1-by-L vector.

I think this should be right, but the results are still quite different. Don’t know where I made another stupid mistake again.

9

Still haven’t figured out why the previous function was wrong, but found that the following matrix-based function works correctly:

function LossFn(θ, y, X, Z)
    W = diagm(0 => ones(size(Z, 2)))
    g = z'*(y - x*θ)
    loss = g'*W*g
    return loss
end