# Julia Manual GMM

**URL:** <https://discourse.julialang.org/t/julia-manual-gmm/10487>\
**Category:** New to Julia\
**Tags:** question\
**Created:** [April 23, 2018, 4:00am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487 "2018-04-23T04:00:16Z")\
**Posts on this page:** 19\
**Page:** 1

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**Author:** ![James\_Stewart](https://avatars.discourse-cdn.com/v4/letter/j/9de053/32.png) [@James\_Stewart](https://discourse.julialang.org/u/James_Stewart)\
**Post date:** [April 23, 2018, 4:00am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/1 "2018-04-23T04:00:16Z")

</div>

Hey  
I am having a bit of trouble understanding how to manually do GMM and in particular how to do GMM in Julia.  
Here is the Stata code + data:

> use [http://www.stata-press.com/data/r13/docvisits](http://www.stata-press.com/data/r13/docvisits)  
> gmm (docvis - exp({xb:private chronic female income}+{b0})), instruments(private chronic female age black hispanic) twostep

As I understand it, I need to first find the residuals and then find estimates for theta but am not sure on how to do either. I am supposed to first assume i.i.d errors and then assume heteroskedasticity but no autocorrelation in the errors.

Is it something of the form:  
inv(n)_Z’(y-X_theta))'_W_(inv(n)_Z’(y-X_theta))

Where W is the weighting matrix for the heteroskedastic case and W=I for i.i.d  
Any help would be appreciated,  
Thanks

EDIT: Here is what I have so far, not sure how correct it is:

using Optim  
using CSV

data=CSV.read(“data Ass\_2.csv”, datarow=2)

docvis=Array(data[:docvis])  
y=docvis  
#x and z variables  
private=Array(data[:private])  
chronic=Array(data[:chronic])  
female=Array(data[:female])  
income=Array(data[:income])  
age=Array(data[:age])  
black=Array(data[:black])  
hispanic=Array(data[:hispanic])  
β0=ones(4412)  
#collection of instruments  
Z=[private chronic female age black hispanic]

#collection of regressors

X=[β0 private chronic female income]

“”"  
L\>K so X=Z inv(Z’Z)Z’X  
“”"  
X̂=(Z\*inv(Z’\*Z)\*Z’\*X)  
W=eye(6)  
θ=(inv(X̂’\*X̂)\*X̂’_y)  
 #The solution for Optimal β  
“”"  
β=inv(X’ZW_Z’\*X)_X’ZW_Z’_y  
eX=exp(X)  
βe=inv(eX’ZW_Z’\*eX)_eX’ZW_Z’_y  
“”"  
u=y-X̂_θ  
σ= u.^2

f(σ)=y-X̂\*θ  
optimize(f, [0, 0, 0, 0, 0])

---

<div class="post-metadata">

**Author:** ![anriseth](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/anriseth/32/3106_2.png) [@anriseth](https://discourse.julialang.org/u/anriseth)\
**Post date:** [April 23, 2018, 11:35am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/2 "2018-04-23T11:35:10Z")

</div>

You can include code on Discourse using code blocks,

```julia
using Optim
using CSV

data=CSV.read(“data Ass_2.csv”, datarow=2)

docvis=Array(data[:docvis])
y=docvis
#x and z variables
private=Array(data[:private])
chronic=Array(data[:chronic])
female=Array(data[:female])
income=Array(data[:income])
age=Array(data[:age])
black=Array(data[:black])
hispanic=Array(data[:hispanic])
β0=ones(4412)
#collection of instruments
Z=[private chronic female age black hispanic]

#collection of regressors

X=[β0 private chronic female income]

"""
 L>K so X=Z inv(Z’Z)Z’X
"""
X̂=(Z*inv(Z’*Z)*Z’*X)
W=eye(6)
θ=(inv(X̂’*X̂)*X̂’y)
#The solution for Optimal β
"""
β=inv(X’ZWZ’*X)X’ZWZ’y
eX=exp(X)
βe=inv(eX’ZWZ’*eX)eX’ZWZ’y
"""
u=y-X̂θ
σ= u.^2

f(σ)=y-X̂*θ
optimize(f, [0, 0, 0, 0, 0])

```

---

<div class="post-metadata">

**Author:** ![anriseth](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/anriseth/32/3106_2.png) [@anriseth](https://discourse.julialang.org/u/anriseth)\
**Post date:** [April 23, 2018, 11:44am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/3 "2018-04-23T11:44:16Z")

</div>

You define a function `f(σ)=y-X̂*θ` that does not depend on the input variable `σ`. So your function is constant. I think you may benefit from a better understanding of how to define methods and functions in Julia, see the resources at [Get started with Julia](https://julialang.org/learning/)

Optim works well to minimise least squares problems, but you can also consider using [LsqFit.jl](https://github.com/JuliaNLSolvers/LsqFit.jl)

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**Author:** ![mcreel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mcreel/32/30088_2.png) [@mcreel](https://discourse.julialang.org/u/mcreel)\
**Post date:** [April 23, 2018, 1:14pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/4 "2018-04-23T13:14:58Z")

</div>

I have a function which will do this, pretty easily, at [https://github.com/mcreel/Econometrics.jl/blob/master/src/GMM/gmmresults.jl](https://github.com/mcreel/Econometrics.jl/blob/master/src/GMM/gmmresults.jl)

It gives results as follows:

```julia
julia> gmmresults();
************************************************************
GMM example, two step
GMM Estimation Results BFGS convergence: Normal
Observations: 100
Hansen-Sargan statistic: 1.84211
Hansen-Sargan p-value: 0.1747

                estimate st. err t-stat p-value
           1 -0.13380 0.10507 -1.27341 0.20588
           2 1.01924 0.15808 6.44768 0.00000

************************************************************
 ************************************************************
GMM example, CUE
GMM Estimation Results BFGS convergence: Normal
Observations: 100
Hansen-Sargan statistic: 1.56835
Hansen-Sargan p-value: 0.21045

                estimate st. err t-stat p-value
           1 -0.10846 0.10572 -1.02592 0.30745
           2 0.98518 0.14669 6.71591 0.00000

************************************************************

julia> 

```

For your problem, you would just need to define the appropriate moments function. An example which uses IV to estimate a linear model similar to yours is here: [https://github.com/mcreel/Econometrics/blob/master/Examples/GMM/MeasurementErrorIV.jl](https://github.com/mcreel/Econometrics/blob/master/Examples/GMM/MeasurementErrorIV.jl)

By the way, the gmmresults() function is general purpose, for possibly nonlinear in the parameters moment conditions. As such, it’s using Optim.jl, which is not necessary in the case of linear in parameters moment conditions. But, it works.

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<div class="post-metadata">

**Author:** ![James\_Stewart](https://avatars.discourse-cdn.com/v4/letter/j/9de053/32.png) [@James\_Stewart](https://discourse.julialang.org/u/James_Stewart)\
**Post date:** [April 23, 2018, 8:51pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/5 "2018-04-23T20:51:03Z")

</div>

From the Stata Code I provided earlier, am I understanding correctly that the moment conditions would be something like:

> [y-theta[1]exp(X[1]) y-theta[2]exp(X[2]) … y-theta[5]exp(X[5])]

Where there are 5 moment conditions?

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<div class="post-metadata">

**Author:** ![James\_Stewart](https://avatars.discourse-cdn.com/v4/letter/j/9de053/32.png) [@James\_Stewart](https://discourse.julialang.org/u/James_Stewart)\
**Post date:** [April 24, 2018, 2:30am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/6 "2018-04-24T02:30:58Z")

</div>

Just found out that we are required to use

> Optim  
> Optimize

Any thoughts on this?

> β0=ones(n)  
> #collection of instruments  
> Z=[private chronic female age black hispanic]
> 
> X=[β0 private chronic female income]  
> W1=eye(6)
> 
> Q(θ)=(inv(n)_(Z)_(y-X_θ’))'W1(inv(n)_(Z)_(y-X_θ’))
> 
> optimize(Q)

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<div class="post-metadata">

**Author:** ![Yifan\_Liu](https://avatars.discourse-cdn.com/v4/letter/y/4da419/32.png) [@Yifan\_Liu](https://discourse.julialang.org/u/Yifan_Liu)\
**Post date:** [February 15, 2020, 7:17am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/7 "2020-02-15T07:17:56Z")

</div>

Any julia GMM package now?

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<div class="post-metadata">

**Author:** ![mcreel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mcreel/32/30088_2.png) [@mcreel](https://discourse.julialang.org/u/mcreel)\
**Post date:** [February 15, 2020, 4:22pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/8 "2020-02-15T16:22:30Z")

</div>

I’d say GMM is something that doesn’t really need a package. The estimator is defined by two things, the moment conditions, and the weight matrix (or just the moment conditions, if the continuously updating version is used). The following code will compute the ordinary two-step GMM estimator and everything needed to compute the estimated covariance matrix, given a function defining the moment conditions and the weight matrix (fminunc is just a convenience frontend to Optim). There’s not much there to need a dedicated package. I’d say it would be more appropriate to place GMM into a more general econometrics package.

```julia
# ordinary GMM with provided weight matrix
# weight should be gXg positive definite
function gmm(moments, theta, weight)
    # average moments
    m = theta -> vec(mean(moments(theta),dims=1)) # 1Xg
    # moment contributions
    momentcontrib = theta -> moments(theta) # nXg
    # GMM criterion
    obj = theta -> ((m(theta))'weight*m(theta))
    # do minimization
    thetahat, objvalue, converged = fminunc(obj, theta)
    # derivative of average moments
    D = (Calculus.jacobian(m, vec(thetahat), :central))' 
    # moment contributions at estimate
    ms = momentcontrib(thetahat)
    return thetahat, objvalue, D, ms, converged
end

```

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<div class="post-metadata">

**Author:** ![Albert\_Zevelev](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/albert_zevelev/32/11844_2.png) [@Albert\_Zevelev](https://discourse.julialang.org/u/Albert_Zevelev)\
**Post date:** [February 15, 2020, 4:57pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/9 "2020-02-15T16:57:19Z")

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@Yifan_Liu  
This is the best one I found so far:

> **[GitHub - schrimpf/GMMInference.jl: Methods for inferences for GMM models.](https://github.com/schrimpf/GMMInference.jl)**
>
> Methods for inferences for GMM models. Contribute to schrimpf/GMMInference.jl development by creating an account on GitHub.

Also see:

> **[GitHub - floswald/SMM.jl: Simulated Method of Moments for Julia](https://github.com/floswald/SMM.jl)**
>
> Simulated Method of Moments for Julia. Contribute to floswald/SMM.jl development by creating an account on GitHub.

By @floswald

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<div class="post-metadata">

**Author:** ![Paul\_Soderlind](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/paul_soderlind/32/1753_2.png) [@Paul\_Soderlind](https://discourse.julialang.org/u/Paul_Soderlind)\
**Post date:** [February 15, 2020, 9:29pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/10 "2020-02-15T21:29:07Z")

</div>

I believe the example by @mcreel is very good. In case you want more of something close that, you may consider my lecture notes/notebook at [GMM forth and back](https://github.com/PaulSoderlind/EmpiricalFinancePhD/blob/master/GMM.ipynb)

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<div class="post-metadata">

**Author:** ![Albert\_Zevelev](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/albert_zevelev/32/11844_2.png) [@Albert\_Zevelev](https://discourse.julialang.org/u/Albert_Zevelev)\
**Post date:** [February 15, 2020, 10:33pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/11 "2020-02-15T22:33:42Z")

</div>

These are all the Julia codes I’m aware of which can do GMM (none is a registered pkg):  
@Paul_Schrimpf: [GitHub - schrimpf/GMMInference.jl: Methods for inferences for GMM models.](https://github.com/schrimpf/GMMInference.jl)

@mcreel: [Econometrics/src/GMM at main · mcreel/Econometrics · GitHub](https://github.com/mcreel/Econometrics/tree/master/src/GMM)

@Paul_Soderlind: [https://github.com/PaulSoderlind/EmpiricalFinancePhD/blob/master/GMM.ipynb](https://github.com/PaulSoderlind/EmpiricalFinancePhD/blob/master/GMM.ipynb)

magnus dahlquist: [GitHub - MagnusDahlquist/PhD403: PhD 403: Empirical Asset Pricing](https://github.com/MagnusDahlquist/PhD403)

@sglyon: [GitHub - sglyon/GMM.jl](https://github.com/sglyon/GMM.jl)

Chris Hedtrich: [https://github.com/chrished/NonlinearGMM.jl](https://github.com/chrished/NonlinearGMM.jl)

---

<div class="post-metadata">

**Author:** ![Yifan\_Liu](https://avatars.discourse-cdn.com/v4/letter/y/4da419/32.png) [@Yifan\_Liu](https://discourse.julialang.org/u/Yifan_Liu)\
**Post date:** [February 15, 2020, 11:23pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/12 "2020-02-15T23:23:16Z")

</div>

thanks for your reply. I wrote my GMM functions in a style similar to the ones Paul\_Soderlind mentioned in this thread.

The reason I am asking this is that I wan to see if there is another package that can export the results of a GMM function to LaTex tables.

---

<div class="post-metadata">

**Author:** ![mcreel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mcreel/32/30088_2.png) [@mcreel](https://discourse.julialang.org/u/mcreel)\
**Post date:** [February 16, 2020, 8:24am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/13 "2020-02-16T08:24:49Z")

</div>

The function gmmresults.jl, mentioned a while ago upthread, uses the gmm code I posted recently, above, to compute the estimator, then it prints out nicely formatted results. To do this, it uses the [PrettyTables.jl](https://github.com/ronisbr/PrettyTables.jl) package. This package allows export to LaTeX. gmmresults.jl doesn’t use this feature, but it should be quite easy to modify the code to add this option.

 ![image](https://global.discourse-cdn.com/julialang/original/3X/e/0/e059bbb43978a5af5aa941e010ed57c1ddf9b00a.png)

By the way, thanks to all for the links, there are lots of good resources out there!

---

<div class="post-metadata">

**Author:** ![Yifan\_Liu](https://avatars.discourse-cdn.com/v4/letter/y/4da419/32.png) [@Yifan\_Liu](https://discourse.julialang.org/u/Yifan_Liu)\
**Post date:** [February 16, 2020, 7:01pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/14 "2020-02-16T19:01:29Z")

</div>

Just curious, does this PrettyTables package have the potential of translating the results of any econometrics functions into LaTex tables?

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<div class="post-metadata">

**Author:** ![mcreel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mcreel/32/30088_2.png) [@mcreel](https://discourse.julialang.org/u/mcreel)\
**Post date:** [February 17, 2020, 7:42am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/15 "2020-02-17T07:42:10Z")

</div>

As far as I know, it will just nicely format tables, and has the option to export LaTeX. But the user must supply the basic table input.

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<div class="post-metadata">

**Author:** ![jbshannon](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jbshannon/32/38838_2.png) [@jbshannon](https://discourse.julialang.org/u/jbshannon)\
**Post date:** [May 7, 2021, 9:24pm UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/16 "2021-05-07T21:24:07Z")

</div>

I’m working on my own implementation of GMM for a project and thought I’d contribute to the many excellent implementations in this thread with a minimal example that makes good use of unicode characters to correspond to familiar mathematical notation. This example assumes you have already programmed a function `g(z,θ)` that returns an L \times T matrix of L moment contributions computed at T observations (i.e. the t-th column of the output of `g` is equal to g(z\_t,\theta).

```julia
using Optim, ForwardDiff
function gmm(g, z, θ₀, W)
    # Parameter estimates
    f(θ) = (ḡ = mean(g(z,θ), dims=2); dot(ḡ, W, ḡ)) # GMM criterion
    result = Optim.optimize(f, θ₀, LBFGS(); autodiff=:forward) # run optimization
    θ̂ = Optim.minimizer(result) # access result

    # Standard errors: (G'WG)⁻¹G'WΩ̂WG(G'WG)⁻¹ →ᵖ Var(√T̅(θ̂-θ))
    T = size(z,1)
    G = ForwardDiff.jacobian(θ -> mean(g(z,θ), dims=2), θ̂) # G →ᵖ E[∂g/∂θ′]
    D = inv(G'*W*G)
    Ω̂ = (ĝ = g(z,θ̂); ĝ*ĝ'/T) # Ω̂ = T⁻¹Σₜ[g(zₜ,θ̂)g(zₜ,θ̂)′] →ᵖ Ω
    V̂ = D*G'*W*Ω̂*W*G*D # (G'WG)⁻¹G'WΩ̂WG(G'WG)⁻¹
    σ̂ = sqrt.(diag(V̂/T))

    return θ̂, σ̂
end

```

You can test it out by comparing it to OLS using this moment function:

```julia
function g(z,θ)
    y = @view z[:,1]
    X = @view z[:,2:end]
    ε̂ = y - X*θ
    transpose(X.*ε̂)
end

```

Understanding the mechanics of actually getting GMM to work was a bit of a hurdle for me coming from Matlab, so I hope this is helpful to others making the switch.

---

<div class="post-metadata">

**Author:** ![gabriel\_omondi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gabriel_omondi/32/29290_2.png) [@gabriel\_omondi](https://discourse.julialang.org/u/gabriel_omondi)\
**Post date:** [October 15, 2021, 7:00am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/17 "2021-10-15T07:00:20Z")

</div>

Hello. I am trying the same but my codes are not working. How to introduce the weight W=eye(6) to my model?

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<div class="post-metadata">

**Author:** ![gabriel\_omondi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gabriel_omondi/32/29290_2.png) [@gabriel\_omondi](https://discourse.julialang.org/u/gabriel_omondi)\
**Post date:** [October 15, 2021, 7:02am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/18 "2021-10-15T07:02:24Z")

</div>

The below is a code for my case  
using Pkg  
import Pkg;  
using DataFrames;  
using CSV;  
using Lathe;  
using GLM;  
using Statistics;  
using StatsPlots;  
using MLBase;  
df=DataFrame(CSV.File(“C:Users/GABBYMOSH/OneDrive/Desktop/Work/hsng.csv”)  
first(df,5);  
names(df)  
#Multiple linear Model  
fm=@formula(rent~hsngval+pcturban)  
linearRegressor=lm(fm,df)  
##GMM  
hsngval=Array(df[:hsngval]);  
pcturban=Array(df[:pcturban]);  
faminc=Array(df[:faminc]);  
reg2=Array(df[:reg2]);  
reg4=Array(df[:reg4]);  
#Instrument variables  
Z=[pcturban faminc reg2 reg4];  
B0=ones(50);  
X=[B0 hsngval pcturban];  
rent=Array(df[:rent]);  
y=rent;

Please help complete it.Processing: hsng.xlsx…

---

<div class="post-metadata">

**Author:** ![jlapeyre](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jlapeyre/32/4514_2.png) [@jlapeyre](https://discourse.julialang.org/u/jlapeyre)\
**Post date:** [October 16, 2021, 12:04am UTC](https://discourse.julialang.org/t/julia-manual-gmm/10487/19 "2021-10-16T00:04:36Z")

</div>

I can’t answer everything, but this might help a bit: Assuming `eye(6)` is supposed to return a 6x6 floating point identity matrix, then this will work

```julia
using LinearAlgebra: I
W = Matrix{Float64}(I, 6,6)

```

Also, the first item in the following topic tells you how to quote your code in a Discourse post so that it is formatted nicely.

> [@Please read: make it easier to help you](https://discourse.julialang.org/t/please-read-make-it-easier-to-help-you/14757):
>
> Welcome to the Julia Discourse! We are enthusiastic about helping Julia programmers, both beginner and experienced. This public service announcement (PSA) outlines best practices when asking for help. Following these points makes it easier for us to help you and more likely you’ll get a prompt, useful answer. Keywords are highlighted to make it easier to refer to specific points. Choose a descriptive title that captures the key part of your question, eg “plots with multiple axes” instead of …