---
title: "Linear regression and smooth GMM"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Linear regression and smooth GMM}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 6,
                      fig.height = 5)
```

This vignette (synthetic data) shows the two estimating-equation
interfaces.  Both compute influence contributions and pass them to the
same core construction, `aersn()`.

## Linear regression

For OLS with regressors $z_t$ and residuals $\hat u_t$, the influence
contribution of observation $t$ to the coefficient vector is
$\hat Q^{-1} z_t \hat u_t$ with $\hat Q = n^{-1}\sum_t z_t z_t^\top$.  The
intercept and any control coefficients are estimated jointly; their effect
on the contributions of the coefficients of interest is retained.

```{r}
library(aersn)
set.seed(21)
n <- 300
x1 <- as.numeric(arima.sim(list(ar = 0.6), n))
x2 <- as.numeric(arima.sim(list(ar = 0.3), n))
u <- as.numeric(arima.sim(list(ar = 0.5), n))
y <- 1 + 0.8 * x1 - 0.4 * x2 + u
fit <- aersn_lm(y ~ x1 + x2, target = c("x1", "x2"))
fit
ref <- aersn_reference(fit, draws = 4000, seed = 1)
aersn_test(fit, null = c(0.8, -0.4), reference = ref)
confint(fit, reference = ref)
plot(fit, reference = ref, null = c(0.8, -0.4))
```

A fitted `lm` object can be passed instead of a formula.  Weighted least
squares and aliased coefficients are not supported.

## Smooth GMM

`aersn_gmm()` takes the moment contributions $m_t(\hat\beta_n)$, the average
Jacobian $\hat D = n^{-1}\sum_t \partial m_t(\hat\beta_n)/\partial\beta^\top$,
the weighting matrix $\hat W$, and the estimate.  It forms
$\hat M = (\hat D^\top\hat W\hat D)^{-1}\hat D^\top\hat W$ and the
contributions $\hat\psi_{n,t} = -\hat{\dot h}_n\hat M m_t(\hat\beta_n)$ for the
target $h(\beta)$.  The function warns when the first-order condition
$\hat D^\top\hat W\bar m_n$ is not close to zero, which indicates that the
supplied estimate does not solve the GMM problem.

Two-stage least squares with two instruments for one endogenous regressor:

```{r}
set.seed(22)
z1 <- rnorm(n); z2 <- rnorm(n); v <- rnorm(n)
x <- z1 + 0.5 * z2 + v
y <- 1 + x + 0.6 * v + rnorm(n)
Z <- cbind(1, z1, z2); X <- cbind(1, x)
W <- solve(crossprod(Z) / n)
Dm <- -crossprod(Z, X) / n
gZy <- crossprod(Z, y) / n
beta <- -solve(t(Dm) %*% W %*% Dm, t(Dm) %*% W %*% gZy)   # 2SLS
moments <- Z * as.numeric(y - X %*% beta)
gmm <- aersn_gmm(moments, Dm, beta, weight = W, target = 2, names = "slope")
gmm
aersn_test(gmm, null = 1, draws = 20000, seed = 1)
```

## Conditions

The sufficient conditions are those of the manuscript's Assumption
"Sufficient conditions for smooth GMM": a stationary ergodic moment
process, $\sqrt n$-consistency, full column rank of the population Jacobian,
consistent Jacobian and weighting matrix, smooth moments, and a joint
functional central limit theorem for the moments and their Jacobian.
Under these conditions the path based on the estimated contributions is
uniformly equivalent to the path based on the true influence function, so
the hull-based inference is valid.  Non-smooth moments (quantile
regression) and nonstandard rates are outside this scope.