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In their preprint, Rocha et al. suggest a new type of residual for linear regression models with heteroscedasticity. They call their new residual PCA residuals. I have tried to replicate some of their examples but failed. I would be very grateful if someone could give me a hint where I'm wrong.

Background

Briefly, they assume the usual linear model of the form in matrix notation $$Y=X\beta + \epsilon$$

where $Y = (Y_1\ldots, Y_n)^T$, $X = (X_{ij})$ is an $n\times p$ matrix, $\beta = (\beta_1,\ldots, \beta_p)^T$ and $\epsilon = (\epsilon_1, \ldots, \epsilon_n)^T$. Further, let $\Omega = \mathrm{Cov}(\epsilon) = \mathrm{diag}(\sigma_1^2, \ldots, \sigma_n^2)$ (with $0<\sigma_i^2<\infty$ for $i=1,\ldots, n$). They show that the residuals satisfy $$ \hat{\epsilon} = (I_n - H)y\sim N_n(0, (I_n - H)\Omega) $$ where $I_n$ is the identity matrix and $H$ the hat matrix with diagonal elements $h_i$. There exist a number of known heteroscedasticity-consistent estimators for $\Omega$ but I will focus on $HC_3$ here:

$$ \widehat{\Omega}_{3} = \mathrm{diag}(1/(1-h_i)^{2})\widehat{\Omega} $$

where $\widehat{\Omega}=\mathrm{diag}(\hat{\epsilon}^2_{1}, \ldots, \hat{\epsilon}^2_{n})$.

PCA residuals

To get the PCA residuals, perform a spectral decomposition (Eigendecomposition) of the form

$$ (I_n - H)\widehat{\Omega}_{3} = Q\widehat{\Lambda}Q^{-1} $$

where $Q$ is an orthogonal matrix of eigenvectors and $\Lambda = \mathrm{diag}(\overbrace{\lambda_1, \ldots, \lambda_{n-p}}^{n-p}, \underbrace{0, \ldots, 0}_{p})$. That means that the last $p$ eigenvalues are zero. Finally, the standardized PCA residuals are defined as

$$ R_{i} = \frac{Q\hat{\epsilon}}{\sqrt{\lambda_{i}}},\quad i = 1,\ldots, n-p. $$

One problem in my calculations below (I think) is that this covariance matrix $(I_n - H)\widehat{\Omega}_{3}$ is not positive and symmetric.

An example

The authors use the states.dta dataset to illustrate the use of their PCA residuals. I have tried to replicate their calculations but was not able to.

Consider the multiple linear regression model $$ \mathrm{csat}_i = \beta_0 + \beta_1\mathrm{percent}_i + \beta_2\mathrm{high}_i + \beta_3\mathrm{percent}_{i}^2 + \epsilon_i, \quad i = 1,\ldots, 51 $$

Here are my steps in R:

library(foreign)

dat <- read.dta("http://github.com/IQSS/workshops/blob/master/R/Rstatistics/dataSets/states.dta?raw=True")

mod <- lm(csat~percent + high + I(percent^2), data = dat)

Calculate the heteroscedasticity-consistent matrix $\widehat{\Omega}_{3}$:

res2 <- as.numeric(residuals(mod)^2)
omegahat_HC3 <- diag(res2/(1 - hatvalues(mod))^2)

Now perform the spectral decomposition (note that the decomposition results in some complex numbers but I don't know if that's a problem):

X <- model.matrix(mod)
Ident <- diag(nobs(mod))
Hatmatrix <- tcrossprod(svd(X)$u)

covmat <- (Ident - Hatmatrix)%*%omegahat_HC3

decomp <- eigen(covmat) # Spectral decomposition
Q <- decomp$vectors     # Eigenvectors
Lambda <- decomp$values # Eigenvalues

Finally, calculate the PCA residuals (note that I'm multiplying the residuals with the transpose of $Q$, because the results where nonsensical otherwise. But I'm unsure why that's the case):

res <- as.numeric(residuals(mod))

res_pca <- t(Q)%*%res # Unstandardized PCA residuals

n <- nobs(mod)
p <- length(coef(mod))

res_pca_std <- Re(res_pca[1:(n - p)]/sqrt(Lambda[1:(n - p)]))

The authors show the following index plot of the PCA residuals PCA_residual_plot

Here is the index plot of my calculations: PCA_residual_plot_replication

My PCA residuals are obviously not quite the same but I don't know here I was going wrong with my calculations.

Can anybody explain why my calculations are off and why?

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  • $\begingroup$ Hi: I only looked quickly but see if you get the hatmatrix straight from the output of the lm. you re calcing it using brute force using solve ( when it's already been calculated during the lm call ) which could be dangerous numerically. it's generally safer to use the QR decomp which I think is what the lm does. $\endgroup$ – mlofton Apr 20 at 14:31
  • $\begingroup$ Also, check this out because Doug is a serious expert with these issues so whatever he says is often really useful . stat.wisc.edu/courses/st849-bates/lectures/Orthogonal.pdf $\endgroup$ – mlofton Apr 20 at 14:38

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