Setup: I have a model that produces a vector of aggregate outcomes, $\theta$, based on parameters, $\beta$. The relationship $\theta=\Theta(\beta)$ is stochastic and analytically intractable, but I can simulate draws of $\theta$ given $\beta$. I denote the mean of $\theta$ obtained from a large number of simulations as $\theta_{sim}(\beta)$.

Estimation of $\beta$ with Indirect Inference: I estimate $\beta$ using indirect inference, which chooses $\hat{\beta}$ such that simulated $\theta$ are as close as possible to the $\theta_{data}$ observed in the real data. I.e. $$\hat\beta_{ii}=\arg\min_\beta (\theta_{sim}(\beta)-\theta_{data})W(\theta_{sim}(\beta)-\theta_{data})$$

For the weighing matrix $W$, I use the variance-covariance matrix of $\theta$s from a sample of independent cases (I do not use the other cases for the rest of the estimation, because I want to estimate the $\beta$s for one particular case).

While related, this is not GMM because the $\theta$ are not an average of some statistic/moment across many observations.

Question: I need to test a linear restriction. E.g. whether $\beta=0$. I have obtained estimates for the constrained and the unconstrained case. However. How can I do this?

In GMM I would construct a $\chi^2$-test by subtracting the value of the objective function in the unconstrained case from the value of the objective function in the constrained case, following Wooldridge (2002, p 200). These type of tests rely on the CLT, based on the moments being (i) averages over (ii) independent observations. Both points are not necessarily fulfilled in indirect inference.


Background: Some details that may help to understand. A simplified version of the model I am estimating starts with a matrix of i.i.d. Normal draws $\varepsilon_{(20x20)}$ with mean $\beta$ and variance $\sigma^2$. I do not observe $\varepsilon$ but instead I observe a function $G(\varepsilon)$ of it. This function's value is binary matrix $t_{20x20}$. The function has no closed form expression, but can be evaluated through simulations. Importantly, unlike $\varepsilon$, the elements of $t$ are not indepente anymore!

With indirect inference I estimate $\sigma$ and $\beta$. That is, using some statistic, $\theta$ (e.g., the rank of $t$ or the number of ones in it) I search for the values for $\sigma$ and $\beta$, which---across many simulations---on average result in the same $\theta$ as I observe in the data.

Since entries of $t$ are highly dependent, thus, I believe that most of the GMM results/tests do not directly apply.

Related stuff: This paper suggests a way to construct an indirect likelihood. However, it using requires an optimal $W$, whereby I am unsure whether my inverse VCV matrix fulfills this. Also, they seem to assume that I have data on (n) independent individuals.


You don't have independent obsevations, so the factor $\sqrt{n}$ with which one usually scales the likelihood ratio/other test statistics would make your test oversized. I know this is not much, but it seems plausible to argue that, if a likelihood ratio-type test, which do not scale by this factor (i.e., set n=1) rejects the null, you're safe in the sense that this test should be overly conservative

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    $\begingroup$ That's not a solution, but at least it's a starting point. Do you have a reference that could be usefull here? $\endgroup$ – sheß Dec 19 '19 at 16:34

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