3
$\begingroup$

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.

$\endgroup$
3
+300
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.