6
$\begingroup$

For OLS, $\hat{\beta} = (X'X)^{-1}X'y$, and $\text{var}(\hat{\beta}) = (X'X)^{-1} X' \sigma^2 I X (X'X)^{-1}$. I can reproduce these "by hand".

For WLS, with heteroskedastic errors and weights in diagonal $W$, $\hat{\beta} = (X'WX)^{-1}X'Wy$, and

$\text{var}(\hat{\beta}) = (X'WX)^{-1} \hspace{2 pt} X'W \hspace{2 pt} \text{diag}(\sigma^2_i) \hspace{2 pt} WX \hspace{2 pt} (X'WX)^{-1}$. I can also reproduce these "by hand".

Sandwich standard errors act on the variance estimates by substitututing estimates for $\sigma^2_i$. For example for HC0 (Zeiles 2004 JSS) the squared residuals are used. I can also reproduce these "by hand" both for OLS and WLS (see code below).

However, summary(lm_wls) produces an SE estimate of 0.25081. I cannot reproduce this estimate "by hand". See below at the "### <-- HERE" arrow for the spot. What gets plugged in for "diag(sig^2_i)" there?

###########################################################
### DATA GENERATION #######################################
###########################################################
set.seed(1234)
# Generate a covariate
x <- rnorm(100)
# Generate the propensity score
ps <- (1 + exp(-(-.5 + .8*x)))^-1
# Generate the exposure (i.e., treatment) variable
z <- rbinom(n = 100, size = 1, prob = ps)
# Generate the outcome
y <- 1.1*x - .6*z + rnorm(100, sd = .5)

### Estimate the average treatment effect by OLS
###########################################################
### ORDINARY LEAST SQUARES REGRESSION #####################
###########################################################
### OLS via lm()
lm_ols <- lm(formula = y ~ x + z)
summary(lm_ols)

### Verify OLS 'by hand'
X <- cbind(1, x, z)
(betas <- solve(t(X) %*% X) %*% t(X) %*% y)
(resid_var <- sqrt(sum(lm_ols$residuals^2)/(100 - 3)))
(var_betas <- solve(t(X) %*% X) %*% 
              (t(X) %*% diag(resid_var^2, 100, 100) %*% X)  
              %*% solve(t(X) %*% X))
sqrt(diag(var_betas)) # SEs -- Compare to summary(lm_ols)

### Sandwich SEs based on squared residuals 
### (see p.4 of Zeiles 2004 JSS)
library(sandwich)
(vcovHC(x = lm_ols, type = "HC0", sandwich = TRUE))
sqrt(diag(vcovHC(x = lm_ols, type = "HC0", sandwich = TRUE)))

### Verify Sandwich SEs for the OLS model
var_betas <- solve(t(X) %*% X) %*% 
             t(X) %*% diag(lm_ols$residuals^2) %*% X %*% 
             solve(t(X) %*% X)
sqrt(diag(var_betas)) # SEs -- Compare to sandwich SEs

### Estimate the average treatment effect by propensity
### score weighting
###########################################################
### INVERSE PROPENSITY SCORE WEIGHTING ####################
###########################################################
### Estimate propensity scores
glm1 <- glm(z ~ x, family = "binomial")
ps_est <- predict(glm1, type = "response")
### Create inverse ps weights
wts <- (z/ps_est) + (1-z)/(1-ps_est)
### Estimate average treatment effect via WLS
lm_wls <- lm(formula = y ~ z, weights = wts)
summary(lm_wls)

### Verify WLS
X <- cbind(1, z)
W <- diag(wts)
(betas2 <- solve(t(X) %*% W %*% X) %*% t(X) %*% W %*% y)
(resid_var2 <- sqrt(sum(wts*(lm_wls$residuals^2))/(100 - 2)))
(var_betas2 <- solve(t(X) %*% W %*% X) %*% 
           (t(X) %*% W %*% "diag(sig^2_i)" %*% t(W) %*% X) %*%  ### <-- HERE
           solve(t(X) %*% W %*% X))
sqrt(diag(var_betas2)) # SEs

### Sandwich SEs with sandwich package
library(sandwich)
vcovHC(x = lm_wls, type = "HC0", sandwich = TRUE)
sqrt(diag(vcovHC(x = lm_wls, type = "HC0", sandwich = TRUE)))

### Verify Sandwich SEs for the WLS model
(var_betas3 <- solve(t(X) %*% W %*% X) %*% 
           (t(X) %*% W %*% diag(lm_wls$resid^2) %*% 
           t(W) %*% X) %*% solve(t(X) %*% W %*% X))
sqrt(diag(var_betas3)) # SEs -- Compare to sandwich SEs 
$\endgroup$

1 Answer 1

7
$\begingroup$

Essentially you already computed everything you need. The missing piece is just that the sig_i should be the residual standard error divided by the corresponding square root of the weight. In OLS this isn't necessary because all weights are 1.

sig_i <- resid_var2 / sqrt(wts)
var_betas2 <- solve(t(X) %*% W %*% X) %*% (t(X) %*% W %*% diag(sig_i^2) %*% t(W) %*% X) %*% solve(t(X) %*% W %*% X)

And then you get:

sqrt(diag(var_betas2))
##                   z 
## 0.1760843 0.2508150 

which matches the output of summary() and vcov():

sqrt(diag(vcov(lm_wls)))
## (Intercept)           z 
##   0.1760843   0.2508150 

Even more familiar might be the equations as $\hat \sigma^2 (X^\top W X)^{-1}$ where the terms from the full sandwich (= bread * meat * bread) have already been simplified to just the bread:

var_betas2a <- resid_var2^2 * solve(t(X) %*% W %*% X)
sqrt(diag(var_betas2a))
##                   z 
## 0.1760843 0.2508150 
$\endgroup$
2
  • 1
    $\begingroup$ This is exactly what I was looking for. Can you recommend a regression textbook that works out the matrix algebra for WLS? And thanks for your work on sandwich and other useful packages. $\endgroup$
    – bsbk
    Jun 13, 2017 at 20:38
  • 2
    $\begingroup$ @bsbk Thanks, glad if the packages are useful. As for WLS: This should be covered in all standard econometrics textbooks, e.g., the 2003 edition of Greene's Econometric Analysis covers it in Chapter 11.5. $\endgroup$ Jun 13, 2017 at 21:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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