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I'm struggling with the naming of different elements in GMM. There doesn't seem to be consistency in the literature. What do we even call the moment integrand $g$? I take a stab below with help from Wikipedia.

Detail

The wiki on GMM describes the set-up of GMM using

$$m(\theta_0) \equiv \operatorname{E}[\,g(Y_t,\theta_0)\,]$$ $$\hat{m}(\theta) \equiv \frac{1}{T}\sum_{t=1}^T g(Y_t,\theta).$$

I believe they imply the names of these things are:

  • $m(\theta)$: a population/theoretical moment function
  • $\operatorname{E}[\,g(Y_t,\theta_0)\,]$: a population/theoretical generalized moment
  • $\operatorname{E}[\,g(Y_t,\theta_0)\,]=c$: a population/theoretical generalized moment condition
  • $\operatorname{E}[\,g(Y_t,\theta_0)\,]=0$: a population/theoretical generalized moment condition for GMM.
  • $\hat{m}(\theta)$: a sample generalized moment function
  • $\frac{1}{T}\sum_{t=1}^T g(Y_t,\theta)$: a sample generalized moment
  • $g(Y_t,\theta)$???

Do we just call $g(Y_t,\theta)$ a moment integrand function? That seems clunky, especially if we want to refer to these functions on their own, outside of the GMM model-estimation context. What about model statistic? I like that it's more general. But strictly speaking, these aren't statistics since they're functions of parameters. So what about model statistic generator or model statistic function?

In most materials on GMM, no one even gives this thing a name. Those that do usually call it something or other which conflicts with the above terminology, e.g., moment function. Maybe the above terminology needs to change? Can someone suggestion a consistent naming scheme here?

UPDATE: For my particular use-case, I don't use GMM, but my set-up is very similar. The $g$ actually take the form $\ell(f(Y_t,\theta),A_t)$ where $\ell$ is a loss like squared error, $f(Y_t,\theta)$ is some model statistic function (My terminology. I say function to emphasize that a statistic is technically a transformation of raw sample data and is not a function of model parameters. This is. So when evaluated at specific model parameters $\theta$, it produces a model statistic), and $A_t$ are some targets or actuals. Since $\ell$ is a loss, $\operatorname{E}[\ell(\cdot)]>0$. There is no hope of making $\ell$ an orthogonality condition function in this form. So, in my more specific setting, I'm inclined to call the $g=\ell$ losses, $f$ model statistic functions, and $\hat{m}(\theta)$ cost functions. If someone answers with a better solution, I'll take that as the answer. Otherwise, I'm going to take this solution of mine to be the answer.

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  • $\begingroup$ do you really need a name for all these parts? $\endgroup$ – Aksakal May 22 at 14:54
  • $\begingroup$ @Aksakal the $g$ is most important to me. I think the others are probably fine. But I include them here because even these are not consistent across the literature. $\endgroup$ – lowndrul May 22 at 15:06
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If you really need to call it a name, I suggest "orthogonality condition function" for $g(Y_t,\theta)$. That's how it's referred to in Hansen's paper: "Large Sample Properties of Generalized Method of Moments Estimators", see Assumption 2.3

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  • $\begingroup$ Ok, but what would you call a $g(Y_t,\theta)$ such that $\operatorname{E}[g(Y_t,\theta)]\neq 0$. Doesn't seem right having its name depend on the particular condition it's used for in GMM, no? $\endgroup$ – lowndrul May 22 at 15:23
  • $\begingroup$ @lowndrul, it's a condition, it may or may not hold, that shouldn't grant the name change $\endgroup$ – Aksakal May 22 at 15:26
  • $\begingroup$ For my use case, I'm not actually using GMM but, rather, minimizing a weighted sum of these functions by taking as given data $Y_t$ and choosing $\theta$. These functions are distances between $Y_t$ and $\theta$ like squared-error. The set-up is very similar to GMM. But since $g>0$, there's no hope of satisfying any orthogonality conditions in the given form. $\endgroup$ – lowndrul Jun 3 at 20:37

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