# How to Test Linear Hypotheses about Parameters in Simulation-Based Indirect Inference

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.

Appendix

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