In kolesarm/GMMSensitivity: Optimal Sensitivity Analysis in Generalized Method of Moments Models

library("knitr")
knitr::opts_knit$set(self.contained = FALSE)
knitr::opts_chunk$set(tidy = TRUE, collapse=TRUE, comment = "#>",
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Introduction

The package GMMSensitivity implements estimators and confidence interval for sensitivity analysis in moment condition models considered in @ak18sensitivity. In this vignette, we demonstrate the implementation of these confidence intervals using the demand model for automobiles from @blp95.

The package includes the dataset blp, which contains estimates of the @blp95 model, as implemented by @ags17. A description of the dataset can be obtained via the documentation in R, using ?blp. We give additional description here:

G : Matrix with 31 rows and 17 columns, estimate of derivative of the moment condition $G=\partial E[g(w_i,\theta)]/\partial \theta$, evaluated at initial estimate $\hat{\theta}_{\text{initial}}$. The initial estimate corresponds to a second-step GMM estimate, as computed by @ags17.

H : Vector of length 17, estimate of derivative of average markup $h(\theta)$, evaluated at $\hat{\theta}_{\text{initial}}$.

W : Weight matrix used to obtain the estimate $\hat{\theta}_{\text{initial}}$. It corresponds to an estimate of the inverse of the variance of the moment condition $\Sigma$.

g_init : Average moment condition $n^{-1}\sum_{i=1}^{n}g(w_i,\theta)$, evaluated at estimate $\hat{\theta}_{\text{initial}}$ of $\theta$ from @blp95.

h_init : Estimate of the average markup, $h(\hat{\theta}_{\text{initial}})$.

names : Two lists, one for names of the moment conditions, and one for elements of theta

ZZ : Gram matrix of the instruments, used to specify the set $\mathcal{C}$

Sig : Variance of moment condition, estimate of $\Sigma$, given by the sample variance of the moment condition evaluated at $\hat{\theta}_{\text{initial}}$.

sdZ : vector of standard deviations of the instruments

perturb : scaling parameter to give interpretable meaning to violations of moment conditions. For demand-side moments, it corresponds to an estimate of $\delta_{d}=0.01\overline{p}/(\overline{y}/\alpha)$, and for supply-side moments, it corresponds to an estimate of $\delta_{s}=-0.01\overline{p}/\overline{m c}$, where $\overline{p}$ is average price, $\overline{y}$ is average income, $\overline{m c}$ is average marginal cost, and $\alpha$ is a parameter in the utility function. With this scaling, if a given demand-side instrument enters the utility function with coefficient $\delta_{d}$, a consumer is willing to pay 1\% of the average car price for a unit increase in the instrument. If a given supply-side instrument enters the cost function with coefficient $\delta_s$, increasing the instrument by one unit decreases the marginal cost by 1\% of the average car price.

n : Sample size, number of model/years.

library("GMMSensitivity")

Usage

The package implements estimators, confidence intervals, and efficiency calculations for the model (in the notation of @ak18sensitivity) \begin{equation} g(\theta_{0})=c/\sqrt{n},\quad c\in \mathcal{C},\quad \mathcal{C}={B\gamma\colon \norm{\gamma}_{p}\leq M}. \end{equation}

Suppose that we want to allow all excluded instruments in the @blp95 application to be potentially invalid. Fix $B$ to a scaling matrix so that if the $j$th supply-side instrument is invalid with $\gamma_{s j}=1$, this means that changing the instrument by one standard deviation changes the marginal cost by $\gamma_{s j}$\% of the average car price, and if the $j$ the demand-side instrument is invalid with $\gamma_{d j}=1$, then the consumer willingness to pay for one standard deviation change in the instrument is $\gamma_{d j}\%$ of the average 1980 car price. Let $p=2$, and $M=\sqrt{# I}$, where $#I$ is the number of invalid instruments so that $\gamma=1$ is included in the set (this is the same scaling as described in @ak18sensitivity in the paper). Then the confidence interval can be constructed as follows:

## Construct estimate of initial sensitivity
blp$k_init <- -drop(blp$H %*% solve(crossprod(blp$G, blp$W %*% blp$G),
                                        crossprod(blp$G, blp$W)))
## list collecting initial estimates of H, G, Sigma, n, g(thetahat), initial
## sensitivity k, and initial estimate of average markup h(thetahat)
eo <- list(H=blp$H, G=blp$G, Sig=blp$Sig, n=blp$n, g_init=blp$g_init,
           k_init=blp$k_init, h_init= blp$h_init)

## Rows corresponding to invalid instruments
I <- vector(mode="logical", length=nrow(eo$G))
I[c(6:13, 20:31)] <- TRUE

## Matrix B, scaled as described in the paper
B0 <- blp$ZZ %*% diag(sqrt(blp$n)*abs(blp$perturb)/blp$sdZ)

## Value of M
M0 <- sqrt(sum(I))

## Select columns of B0 corresponding to invalid instruments
OptEstimator(eo, B0[, I], M=M0, p=2, alpha=0.05, opt.criterion="FLCI")

The efficiency $\kappa_{*}$ for this confidence interval can be computed using the EffBounds function (which can also be used to compute efficiency of one-sided confidence intervals):

EffBounds(eo, B0[, I], M=M0, p=2)$twosided

In contrast the CI based on the initial estimate is much wider:

OptEstimator(eo, B0[, I], M=M0, p=2, alpha=0.05, opt.criterion="Valid")

A specification test for whether the value $M=M0$ is too low, that is a test of the hypothesis $H_0\colon M\leq M0$, can be conducted using the Jtest function:

## Update eo so that Sig corresponds to the initial estimate of
## Sigma, so that thetahat_initial minimizes n*g(theta)Sig^{-1} g(theta),
## and the J statistic is given by the value of this minimum.
eoJ <- eo
eoJ$Sig <- solve(blp$W)
jt <- Jtest(eoJ, B0[, I], M=M0, p=2, alpha=0.05)

Here J is the $J$-statistic, p0 is the p-value of the usual $J$-test (that assumes $M=0$ under the null), pC is the $p$-value of the test, and Mmin is the smallest value of $M$ that is not rejected. Rescaling by $\sqrt{# I}$, we obtain the result in Table 1 in @ak18sensitivity:

jt$Mmin/M0

If were only concerned about the validity of the demand-side instrument "demand_firm_const" (number of cars produced by the same firm), then since this is the sixth instrument, an analogous analysis could be conducted as:

I <- vector(mode="logical", length=nrow(eo$G))
I[6] <- TRUE
OptEstimator(eo, B0[, I, drop=FALSE], M=1, p=2, alpha=0.05, opt.criterion="FLCI")
EffBounds(eo, B0[, I, drop=FALSE], M=1, p=2)$twosided
OptEstimator(eo, B0[, I, drop=FALSE], M=1, p=2, alpha=0.05, opt.criterion="Valid")
Jtest(eoJ, B0[, I, drop=FALSE], M=1, p=2, alpha=0.05)

References

kolesarm/GMMSensitivity documentation built on Sept. 17, 2020, 5:47 p.m.