Frisch-Waugh-Lovell Theorem
The Frisch-Waugh-Lovell (FWL) theorem says: the coefficient on \(X_1\) in \(Y = \beta_1 X_1 + \beta_2 X_2 + \varepsilon\) is identical to the slope from regressing the residualized \(Y\) on the residualized \(X_1\) — after partialling out \(X_2\) from both.
Drag the sliders to see it hold for any DGP.
#| standalone: true
#| viewerHeight: 900
library(shiny)
ui <- fluidPage(
tags$head(tags$style(HTML("
.eq-box {
background: #f0f4f8; border-radius: 6px; padding: 14px;
margin-bottom: 14px; font-size: 14px; line-height: 1.9;
}
.eq-box b { color: #2c3e50; }
.match { color: #27ae60; font-weight: bold; }
.coef { color: #e74c3c; font-weight: bold; }
"))),
sidebarLayout(
sidebarPanel(
width = 4,
sliderInput("n", "Sample size:",
min = 50, max = 500, value = 200, step = 50),
sliderInput("b1", HTML("True β<sub>1</sub>:"),
min = -3, max = 3, value = 1.5, step = 0.1),
sliderInput("b2", HTML("True β<sub>2</sub>:"),
min = -3, max = 3, value = -1, step = 0.1),
sliderInput("rho", HTML("Corr(X<sub>1</sub>, X<sub>2</sub>):"),
min = -0.9, max = 0.9, value = 0.6, step = 0.1),
sliderInput("sigma", HTML("Error SD (σ):"),
min = 0.5, max = 5, value = 1, step = 0.5),
actionButton("resim", "New draw", class = "btn-primary", width = "100%"),
uiOutput("results_box")
),
mainPanel(
width = 8,
plotOutput("plot_full", height = "350px"),
fluidRow(
column(6, plotOutput("plot_partial", height = "350px")),
column(6, plotOutput("plot_fwl", height = "350px"))
),
uiOutput("step_text")
)
)
)
server <- function(input, output, session) {
dat <- reactive({
input$resim
n <- input$n
b1 <- input$b1
b2 <- input$b2
rho <- input$rho
sigma <- input$sigma
# Generate correlated X1, X2
z1 <- rnorm(n)
z2 <- rnorm(n)
x1 <- z1
x2 <- rho * z1 + sqrt(1 - rho^2) * z2
eps <- rnorm(n, sd = sigma)
y <- b1 * x1 + b2 * x2 + eps
# Full OLS
full_fit <- lm(y ~ x1 + x2)
# FWL steps
ey <- resid(lm(y ~ x2)) # residualise Y on X2
ex <- resid(lm(x1 ~ x2)) # residualise X1 on X2
fwl_fit <- lm(ey ~ ex)
list(x1 = x1, x2 = x2, y = y,
ey = ey, ex = ex,
full_fit = full_fit, fwl_fit = fwl_fit,
b1 = b1, b2 = b2)
})
# --- Plot 1: Y vs X1 (naive scatter) ---
output$plot_full <- renderPlot({
d <- dat()
par(mar = c(5, 5, 4, 2))
plot(d$x1, d$y, pch = 16, col = "#3498db80", cex = 0.8,
xlab = expression(X[1]), ylab = "Y",
main = expression("Y vs " * X[1] * " (raw)"))
abline(lm(d$y ~ d$x1), col = "#e74c3c", lwd = 2.5)
naive_b <- round(coef(lm(d$y ~ d$x1))[2], 4)
legend("topleft", bty = "n", cex = 0.9,
legend = paste("Naive slope =", naive_b))
})
# --- Plot 2: residualised X1 (partial out X2) ---
output$plot_partial <- renderPlot({
d <- dat()
par(mar = c(5, 5, 4, 2))
plot(d$x1, d$ex, pch = 16, col = "#9b59b680", cex = 0.8,
xlab = expression(X[1]),
ylab = expression(e[X[1]]),
main = expression("Residualise " * X[1] * " on " * X[2]))
abline(h = 0, lty = 2, col = "gray50")
abline(lm(d$ex ~ d$x1), col = "#8e44ad", lwd = 2)
legend("topleft", bty = "n", cex = 0.85,
legend = expression("Variation in " * X[1] * " independent of " * X[2]))
})
# --- Plot 3: FWL regression ---
output$plot_fwl <- renderPlot({
d <- dat()
par(mar = c(5, 5, 4, 2))
plot(d$ex, d$ey, pch = 16, col = "#2ecc7180", cex = 0.8,
xlab = expression(e[X[1]]),
ylab = expression(e[Y]),
main = "FWL: Residual Y vs Residual X1")
abline(d$fwl_fit, col = "#e74c3c", lwd = 2.5)
fwl_b <- round(coef(d$fwl_fit)[2], 4)
legend("topleft", bty = "n", cex = 0.9,
legend = paste("FWL slope =", fwl_b))
})
# --- Results comparison ---
output$results_box <- renderUI({
d <- dat()
full_b1 <- round(coef(d$full_fit)["x1"], 4)
fwl_b1 <- round(coef(d$fwl_fit)[2], 4)
naive_b <- round(coef(lm(d$y ~ d$x1))[2], 4)
tags$div(class = "eq-box", style = "margin-top: 16px;",
HTML(paste0(
"<b>True β<sub>1</sub>:</b> ", d$b1, "<br>",
"<b>Full OLS β<sub>1</sub>:</b> <span class='coef'>", full_b1, "</span><br>",
"<b>FWL β<sub>1</sub>:</b> <span class='coef'>", fwl_b1, "</span><br>",
"<span class='match'>✓ They match!</span><br><br>",
"<b>Naive slope:</b> ", naive_b, "<br>",
"<small>(biased by omitting X<sub>2</sub>)</small>"
))
)
})
# --- Step explanation ---
output$step_text <- renderUI({
tags$div(class = "eq-box", style = "margin-top: 8px;",
HTML(paste0(
"<b>Steps:</b> ",
"(1) Regress Y on X<sub>2</sub> → residuals <i>e<sub>Y</sub></i> | ",
"(2) Regress X<sub>1</sub> on X<sub>2</sub> → residuals <i>e<sub>X₁</sub></i> | ",
"(3) Regress <i>e<sub>Y</sub></i> on <i>e<sub>X₁</sub></i> → ",
"slope = β<sub>1</sub> from full regression"
))
)
})
}
shinyApp(ui, server)Did you know?
- Ragnar Frisch and Jan Tinbergen won the very first Nobel Prize in Economics in 1969. Frisch coined the terms “econometrics,” “microeconomics,” and “macroeconomics.” The FWL theorem appeared in Frisch & Waugh (1933).
- Michael Lovell extended the result in 1963, showing it applies to any partitioned regression — not just the two-variable case. That’s why it’s FWL, not just FW.
- FWL is the theoretical foundation behind “partialling out” and “controlling for” variables. Every time you add a control to a regression, you’re implicitly doing the residualization that FWL describes.
- In machine learning, the same idea appears as “residualization” in double/debiased ML (Chernozhukov et al., 2018) — one of the most important recent developments in causal ML.