Additional calculus of M-estimation examples using geex

Example 4: Instrumental variables

Example 4 calculates an instumental variable estimator. The variables (\(Y_3, W_1, Z_1\)) are observed according to:

\[ Y_{3i} = \alpha + \beta X_{1i} + \sigma_{\epsilon}\epsilon_{1, i} \]

\[ W_{1i} =\beta X_{1i} + \sigma_{U}\epsilon_{2, i} \]

and

\[ Z_{1i} = \gamma + \delta X_{1i} + \sigma_{\tau}\epsilon_{3, i}. \]

\(Y_3\) is a linear function of a latent variable \(X_1\), whose coefficient \(\beta\) is of interest. \(W_1\) is a measurement of the unobserved \(X_1\), and \(Z_1\) is an instrumental variable for \(X_1\). The instrumental variable estimator is \(\hat{\beta}_{IV}\) is the ratio of least squares regression slopes of \(Y_3\) on \(Z_1\) and \(Y_3\) and \(W_1\). The \(\psi\) function below includes this estimator as \(\hat{\theta}_4 = \hat{\beta}_{IV}\). In the example data, 100 observations are generated where \(X_1 \sim \Gamma(\text{shape} = 5, \text{rate} = 1)\), \(\alpha = 2\), \(\beta = 3\), \(\gamma = 2\), \(\delta = 1.5\), \(\sigma_{\epsilon} = \sigma_{\tau} = 1\), \(\sigma_U = .25\), and \(\epsilon_{\cdot, i} \sim N(0,1)\).

\[ \psi(Y_{3i}, Z_{1i}, W_{1i}, \mathbf{\theta}) = \begin{pmatrix} \theta_1 - Z_{1i} \\ \theta_2 - W_{1i} \\ (Y_{3i} - \theta_3 W_{1i})(\theta_2 - W_{1i}) \\ (Y_{3i} - \theta_4 W_{1i})(\theta_1 - Z_{1i}) \\ \end{pmatrix} \]

SB4_estFUN <- function(data){
  Z1 <- data$Z1; W1 <- data$W1; Y3 <- data$Y3
  function(theta){
      c(theta[1] - Z1,
        theta[2] - W1,
        (Y3 - (theta[3] * W1)) * (theta[2] - W1),
        (Y3 - (theta[4] * W1)) * (theta[1] - Z1)
    )
  }
}

estimates <- m_estimate(
  estFUN = SB4_estFUN, 
  data  = geexex, 
  root_control = setup_root_control(start = c(1, 1, 1, 1)))

The R packages AER and ivpack can compute the IV estimator and sandwich variance estimators respectively, which is done below for comparison.

ivfit <- AER::ivreg(Y3 ~ W1 | Z1, data = geexex)
iv_se <- ivpack::cluster.robust.se(ivfit, clusterid = 1:nrow(geexex))

coef(ivfit)[2] 
coef(estimates)[4]
iv_se[2, 'Std. Error'] 
sqrt(vcov(estimates)[4, 4])

Example 5: Hodges-Lehmann estimator

This example shows the link between the influence curves and the Hodges-Lehman estimator.

\[ \psi(Y_{2i}, \theta) = \begin{pmatrix} IC_{\hat{\theta}_{HL}}(Y_2; \theta_0) - (\theta_1 - \theta_0) \\ \end{pmatrix} \]

The functions used in SB5_estFUN are defined below:

F0 <- function(y, theta0, distrFUN = pnorm){
  distrFUN(y - theta0, mean = 0)
}

f0 <- function(y, densFUN){
  densFUN(y, mean = 0)
}

integrand <- function(y, densFUN = dnorm){
  f0(y, densFUN = densFUN)^2
}

IC_denom <- integrate(integrand, lower = -Inf, upper = Inf)$value

SB5_estFUN <- function(data){
  Yi <- data[['Y2']]
  function(theta){

     (1/IC_denom) * (F0(Yi, theta[1]) - 0.5)
  }
}

estimates <- m_estimate(
  estFUN = SB5_estFUN, 
  data  = geexex,
  root_control = setup_root_control(start = 2))

The hc.loc of the ICSNP package computes the Hodges-Lehman estimator and SB gave closed form estimators for the variance.

theta_cls <- ICSNP::hl.loc(geexex$Y2)
Sigma_cls <- 1/(12 * IC_denom^2) / nrow(geexex)

## $geex
## $geex$parameters
## [1] 2.026376
## 
## $geex$vcov
##            [,1]
## [1,] 0.01040586
## 
## 
## $cls
## $cls$parameters
## [1] 2.024129
## 
## $cls$vcov
## [1] 0.01047198

Example 6: Huber center of symmetry estimator

This example illustrates estimation with nonsmooth \(\psi\) function for computing the Huber (1964) estimator of the center of symmetric distributions.

\[ \psi_k(Y_{1i}, \theta) = \begin{cases} (Y_{1i} - \theta) & \text{when } |(Y_{1i} - \theta)| \leq k \\ k * sgn(Y_{1i} - \theta) & \text{when } |(Y_{1i} - \theta)| > k\\ \end{cases} \]

SB6_estFUN <- function(data, k = 1.5){
  Y1 <- data$Y1
  function(theta){
    x <- Y1 - theta[1]
    if(abs(x) <= k) x else sign(x) * k
  }
}

estimates <- m_estimate(
  estFUN = SB6_estFUN, 
  data  = geexex,
  root_control = setup_root_control(start = 3))

The point estimate from geex is compared to the estimate obtained from the huber function in the MASS package. The variance estimate is compared to an estimator suggested by SB: \(A_m = \sum_i [-\psi'(Y_i - \hat{\theta})]/m\) and \(B_m = \sum_i \psi_k^2(Y_i - \hat{\theta})/m\), where \(\psi_k'\) is the derivative of \(\psi\) where is exists.

theta_cls <- MASS::huber(geexex$Y1, k = 1.5, tol = 1e-10)$mu

psi_k <- function(x, k = 1.5){
  if(abs(x) <= k) x else sign(x) * k
}

A <- mean(unlist(lapply(geexex$Y1, function(y){
  x <- y - theta_cls
  -numDeriv::grad(psi_k, x = x)
})))

B <- mean(unlist(lapply(geexex$Y1, function(y){
  x <- y - theta_cls
  psi_k(x = x)^2
})))

## closed form covariance
Sigma_cls <- matrix(1/A * B * 1/A / nrow(geexex))

results <- list(geex = list(parameters = coef(estimates), vcov = vcov(estimates)), 
                cls = list(parameters = theta_cls, vcov = Sigma_cls))
results

## $geex
## $geex$parameters
## [1] 4.82061
## 
## $geex$vcov
##            [,1]
## [1,] 0.08356179
## 
## 
## $cls
## $cls$parameters
## [1] 4.999386
## 
## $cls$vcov
##           [,1]
## [1,] 0.0928935

Example 7: Sample quantiles (approximation of \(\psi\))

Approximation of \(\psi\) with geex is EXPERIMENTAL.

Example 7 illustrates calculation of sample quantiles using M-estimation and approximation of the \(\psi\) function.

\[ \psi(Y_{1i}, \theta) = \begin{pmatrix} 0.5 - I(Y_{1i} \leq \theta_1) \\ \end{pmatrix} \]

SB7_estFUN <- function(data){
  Y1 <- data$Y1
  function(theta){
    0.5  - (Y1 <= theta[1])
  }
}

Note that \(\psi\) is not differentiable; however, geex includes the ability to approximate the \(\psi\) function via an approx_control object. The FUN in an approx_control must be a function that takes in the \(\psi\) function, modifies it, and returns a function of theta. For this example, I approximate \(\psi\) with a spline function. The eval_theta argument is used to modify the basis of the spline.

spline_approx <- function(psi, eval_theta){
  y <- Vectorize(psi)(eval_theta)
  f <- splinefun(x = eval_theta, y = y)
  function(theta) f(theta)
}

estimates <- m_estimate(
  estFUN = SB7_estFUN, 
  data  = geexex, 
  root_control = setup_root_control(start = 4.7),
  approx_control  = setup_approx_control(FUN = spline_approx,
                                      eval_theta = seq(3, 6, by = .05)))

A comparison of the variance is not obvious, so no comparison is made.

## $geex
## $geex$parameters
## [1] 4.7
## 
## $geex$vcov
##           [,1]
## [1,] 0.1773569
## 
## 
## $cls
## $cls$parameters
## [1] 4.708489
## 
## $cls$vcov
## [1] NA

Example 8: Robust regression

Example 8 demonstrates robust regression for estimating \(\beta\) from 100 observations generated from \(Y_4 = 0.1 + 0.1 X_{1i} + 0.5 X_{2i} + \epsilon_i\), where \(\epsilon_i \sim N(0, 1)\), \(X_1\) is defined as above, and the first half of the observation have \(X_{2i} = 1\) and the others have \(X_{2i} = 0\).

\[ \psi_k(Y_{4i}, \theta) = \begin{pmatrix} \psi_k(Y_{4i} - \mathbf{x}_i^T \beta) \mathbf{x}_i \end{pmatrix} \]

psi_k <- function(x, k = 1.345){
  if(abs(x) <= k) x else sign(x) * k
}

SB8_estFUN <- function(data){
  Yi <- data$Y4
  xi <- model.matrix(Y4 ~ X1 + X2, data = data)
  function(theta){
    r <- Yi - xi %*% theta
    c(psi_k(r) %*% xi)
  }
}

estimates <- m_estimate(
  estFUN = SB8_estFUN, 
  data  = geexex,
  root_control = setup_root_control(start = c(0, 0, 0)))

m <- MASS::rlm(Y4 ~ X1 + X2, data = geexex, method = 'M')
theta_cls <- coef(m)
Sigma_cls <- vcov(m)

results <- list(geex = list(parameters = coef(estimates), vcov = vcov(estimates)), 
                cls = list(parameters = theta_cls, vcov = Sigma_cls))
results

## $geex
## $geex$parameters
## [1] -0.04050369  0.14530196  0.30181589
## 
## $geex$vcov
##             [,1]          [,2]          [,3]
## [1,]  0.05871932 -0.0101399730 -0.0133230841
## [2,] -0.01013997  0.0021854268  0.0003386202
## [3,] -0.01332308  0.0003386202  0.0447117633
## 
## 
## $cls
## $cls$parameters
## (Intercept)          X1          X2 
##  -0.0377206   0.1441181   0.2988842 
## 
## $cls$vcov
##             (Intercept)            X1            X2
## (Intercept)  0.07309914 -0.0103060747 -0.0241724792
## X1          -0.01030607  0.0020579145  0.0005364106
## X2          -0.02417248  0.0005364106  0.0431120686

Example 9: Generalized linear models

Example 9 illustrates estimation of a generalized linear model.

\[ \psi(Y_i, \theta) = \begin{pmatrix} D_i(\beta)\frac{Y_i - \mu_i(\beta)}{V_i(\beta) \tau} \end{pmatrix} \]

SB9_estFUN <- function(data){
  Y <- data$Y5
  X <- model.matrix(Y5 ~ X1 + X2, data = data, drop = FALSE)
  function(theta){
    lp <- X %*% theta
    mu <- plogis(lp)
    D  <- t(X) %*% dlogis(lp)
    V  <- mu * (1 - mu)
    D %*% solve(V) %*% (Y - mu)
  }
}

estimates <- m_estimate(
  estFUN = SB9_estFUN, 
  data  = geexex, 
  root_control = setup_root_control(start = c(.1, .1, .5)))

Compare point estimates to glm coefficients and covariance matrix to sandwich.

m9 <- glm(Y5 ~ X1 + X2, data = geexex, family = binomial(link = 'logit'))
theta_cls <- coef(m9)
Sigma_cls <- sandwich::sandwich(m9)

results <- list(geex = list(parameters = coef(estimates), vcov = vcov(estimates)),  
                cls = list(parameters = theta_cls, vcov = Sigma_cls))
results

## $geex
## $geex$parameters
## [1] -1.1256071  0.3410619 -0.1148368
## 
## $geex$vcov
##             [,1]         [,2]         [,3]
## [1,]  0.35202094 -0.058906883 -0.101528787
## [2,] -0.05890688  0.012842435  0.004357355
## [3,] -0.10152879  0.004357355  0.185455144
## 
## 
## $cls
## $cls$parameters
## (Intercept)          X1          X2 
##  -1.1256070   0.3410619  -0.1148368 
## 
## $cls$vcov
##             (Intercept)           X1           X2
## (Intercept)  0.35201039 -0.058903546 -0.101534539
## X1          -0.05890355  0.012841392  0.004358926
## X2          -0.10153454  0.004358926  0.185456314

Example 10: Testing equality of success probabilities

Example 10 illustrates testing equality of success probablities.

\[ \psi(Y_i, n_i, \theta) = \begin{pmatrix} \frac{(Y_i - n_i \theta_2)^2}{n_i \theta_2( 1 - \theta_2 )} - \theta_1 \\ Y_i - n_i \theta_2 \end{pmatrix} \]

SB10_estFUN <- function(data){
  Y <- data$ft_made
  n <- data$ft_attp
  function(theta){
    p <- theta[2]
    c(((Y - (n * p))^2)/(n * p * (1 - p))  - theta[1], 
      Y - n * p)
  }
}

estimates <- m_estimate(
  estFUN = SB10_estFUN, 
  data  = shaq, 
  units = 'game', 
  root_control = setup_root_control(start = c(.5, .5)))

V11 <- function(p) {
  k    <- nrow(shaq)
  sumn <- sum(shaq$ft_attp)
  sumn_inv <- sum(1/shaq$ft_attp)
  term2_n  <- 1 - (6 * p) + (6 * p^2)
  term2_d <- p * (1 - p) 
  term2  <- term2_n/term2_d
  term3  <- ((1 - (2 * p))^2) / ((sumn/k) * p * (1 - p))
  2 + (term2 * (1/k) * sumn_inv) - term3
}

p_tilde <- sum(shaq$ft_made)/sum(shaq$ft_attp)
V11_hat <- V11(p_tilde)/23

# Compare variance estimates
V11_hat

## [1] 0.0783097

vcov(estimates)[1, 1]

## [1] 0.1929791

# Note the differences in the p-values
pnorm(35.51/23, mean  = 1, sd = sqrt(V11_hat), lower.tail = FALSE)

## [1] 0.02596785

pnorm(coef(estimates)[1], 
      mean = 1, 
      sd = sqrt(vcov(estimates)[1, 1]),
      lower.tail = FALSE)

## [1] 0.1078138

This example shows that the empircal sandwich variance estimator may be different from other sandwich variance estimators that make assumptions about the structure of the \(A\) and \(B\) matrices.