Make consistent normalizing constant in Algorithm

[vvb610]

I believe the test statistic needs to be changed for asymptotics and seems most natural to send the number of markets to infinity?

[chrisconlon]

No we ned the constant to be "N" = \sum_t dim(\mathcal{J}_t) since we're summing over all products and markets.

[vvb610]

I thought more about this and I believe one can get away with sending the number of markets $T \to \infty$ (for the sake of asymptotics) in the following way if one wishes to have a different number of goods in each market (the previous edits only work if each market has the same number of goods). [EDIT: I think that what I wrote in the commit would actually also work for doing the asymptotics with a variable number of goods in each market.]

Define the population moment

$$h(\omega_{t}^m, z_{t}) = \mathbb{E}_t \left[\frac{1}{\mathcal{J}_t}\sum_{j=1}^{\mathcal{J}_t} \omega_{jt}^m \cdot \hat{g}(z_{jt})\right]$$

The empirical sample moment is

$$\hat{h}(\hat{\omega}^m, z) = \frac{1}{T}\sum_{t=1}^T\left(\frac{1}{\mathcal{J}_t}\sum_{j=1}^{\mathcal{J}_t} \hat\omega_{jt}^m \cdot \hat{g}(z_{jt})\right)$$

The GMM objective is

$$Q(\eta^m) = (\hat{h}(\omega^m, z))^2$$

The DGP is that one observed a market $t$ with all $\mathcal{J}_t$ goods (with $J_t$ bounded) from a process that simultaneously samples the entire market $iid$ from some distribution, which makes it clear that the expectation is over $t$ in the population moment.

It's a bit odd for me to think about the data-generating process being over [product $\times$ markets], but I'd be curious if there's a DGP that supports that and what the expectation is over. The reason that I think that taking $JT \to \infty$ is potentially odd is that for a given market, each of the products have dependent data (eg., the shares must sum to $1$, etc.) so that I don't think one can think that each product $\times$ market is an $iid$ sample from some distribution. [EDIT: I suppose one can index the observed product $\times$ markets by $i \in {1, ..., N}$ where $N := \sum_t^T J_t$ and define $X_i := (p_i, \xi_i, x_i, s_i, z_i)$ and assume that the data-generating process simultaneously produces $X_1, ..., X_N$. Some of the data-points $X_i$ are dependent (since they come from the same market), but then we apply some "finite-block-dependent" asymptotic theory -- this seems a bit less satisfying.]