Fama-MacBeth Regression in R
The Data Hall
2026.5
1 Introduction
The Fama-MacBeth (1973) regression is a two-pass procedure widely used in empirical asset pricing to test factor models (e.g., CAPM, Fama-French). It addresses cross-sectional correlation of residuals in panel data.
There are three key steps:
Step 1 – Time-Series Regressions (N regressions): For each asset \(i\), regress its excess returns on factor returns over the full time series to estimate factor loadings (\(\hat{\beta}_i\)).
Step 2 – Cross-Sectional Regressions (T regressions): For each time period \(t\), regress the cross-section of asset returns on the estimated betas to get period-specific risk premia (\(\hat{\lambda}_t\)).
Step 3 – Time-Series Average: Average the \(T\) cross-sectional estimates to obtain the final risk premium estimates, and use their time-series standard deviation for inference.
2 Load Required Packages
# Install if not already installed
packages <- c("tidyverse", "broom", "knitr", "kableExtra", "ggplot2", "lmtest", "sandwich")
installed <- packages %in% rownames(installed.packages())
if (any(!installed)) install.packages(packages[!installed])
library(tidyverse)
library(broom)
library(knitr)
library(kableExtra)
library(ggplot2)
library(lmtest)
library(sandwich)3 Data Simulation
Since we are replicating the methodology, we simulate a panel dataset of 25 stocks over 60 months with two factors: Market (Mkt) and SMB (Small Minus Big), consistent with the Fama-French framework.
set.seed(42)
N <- 25 # Number of stocks
T <- 60 # Number of time periods (months)
# ── True parameters ──────────────────────────────────────────────────────────
true_lambda_mkt <- 0.06 # True market risk premium (annualised ~6%)
true_lambda_smb <- 0.03 # True SMB risk premium
# ── Simulate factor returns ───────────────────────────────────────────────────
factor_data <- tibble(
t = 1:T,
mkt = rnorm(T, mean = 0.005, sd = 0.04), # Monthly market excess return
smb = rnorm(T, mean = 0.002, sd = 0.03) # Monthly SMB return
)
# ── Simulate stock-level betas ────────────────────────────────────────────────
stock_betas <- tibble(
stock = paste0("Stock_", sprintf("%02d", 1:N)),
beta_mkt = runif(N, 0.5, 1.8),
beta_smb = runif(N, -0.5, 1.2)
)
# ── Simulate stock returns using factor model + idiosyncratic noise ───────────
panel_data <- expand_grid(stock = stock_betas$stock, t = 1:T) %>%
left_join(stock_betas, by = "stock") %>%
left_join(factor_data, by = "t") %>%
mutate(
alpha = rnorm(n(), mean = 0.001, sd = 0.002),
eps = rnorm(n(), mean = 0, sd = 0.03),
ret = alpha + beta_mkt * mkt + beta_smb * smb + eps
)
# Preview
head(panel_data, 10) %>%
kable(caption = "Panel Data (first 10 rows)", digits = 4) %>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
full_width = FALSE)| stock | t | beta_mkt | beta_smb | mkt | smb | alpha | eps | ret |
|---|---|---|---|---|---|---|---|---|
| Stock_01 | 1 | 0.5879 | -0.2701 | 0.0598 | -0.0090 | 0.0032 | 0.0064 | 0.0473 |
| Stock_01 | 2 | 0.5879 | -0.2701 | -0.0176 | 0.0076 | 0.0000 | 0.0510 | 0.0387 |
| Stock_01 | 3 | 0.5879 | -0.2701 | 0.0195 | 0.0195 | 0.0001 | -0.0002 | 0.0061 |
| Stock_01 | 4 | 0.5879 | -0.2701 | 0.0303 | 0.0440 | 0.0024 | -0.0245 | -0.0162 |
| Stock_01 | 5 | 0.5879 | -0.2701 | 0.0212 | -0.0198 | -0.0011 | -0.0015 | 0.0152 |
| Stock_01 | 6 | 0.5879 | -0.2701 | 0.0008 | 0.0411 | 0.0009 | 0.0024 | -0.0073 |
| Stock_01 | 7 | 0.5879 | -0.2701 | 0.0655 | 0.0121 | -0.0021 | -0.0468 | -0.0137 |
| Stock_01 | 8 | 0.5879 | -0.2701 | 0.0012 | 0.0332 | 0.0033 | -0.0051 | -0.0101 |
| Stock_01 | 9 | 0.5879 | -0.2701 | 0.0857 | 0.0296 | 0.0005 | -0.0126 | 0.0303 |
| Stock_01 | 10 | 0.5879 | -0.2701 | 0.0025 | 0.0236 | 0.0001 | 0.0147 | 0.0099 |
4 Step 1: Time-Series Regressions (N Regressions)
For each stock \(i\), we estimate:
\[r_{it} = \alpha_i + \beta_{i,\text{mkt}} \cdot \text{Mkt}_t + \beta_{i,\text{smb}} \cdot \text{SMB}_t + \varepsilon_{it}\]
This gives us the estimated factor loadings \(\hat{\beta}_{i,\text{mkt}}\) and \(\hat{\beta}_{i,\text{smb}}\).
# Run one OLS regression per stock
ts_results <- panel_data %>%
group_by(stock) %>%
do(tidy(lm(ret ~ mkt + smb, data = .))) %>%
ungroup()
# Extract betas (slope coefficients only)
estimated_betas <- ts_results %>%
filter(term %in% c("mkt", "smb")) %>%
select(stock, term, estimate) %>%
pivot_wider(names_from = term, values_from = estimate) %>%
rename(beta_mkt_hat = mkt, beta_smb_hat = smb)
estimated_betas %>%
kable(caption = "Step 1: Estimated Factor Loadings per Stock", digits = 4) %>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
full_width = FALSE) %>%
scroll_box(height = "400px")| stock | beta_mkt_hat | beta_smb_hat |
|---|---|---|
| Stock_01 | 0.4929 | -0.2392 |
| Stock_02 | 1.2404 | 1.0472 |
| Stock_03 | 0.5443 | -0.1766 |
| Stock_04 | 0.6518 | 1.2262 |
| Stock_05 | 1.1945 | 1.0912 |
| Stock_06 | 1.1874 | -0.2539 |
| Stock_07 | 0.7203 | 0.7942 |
| Stock_08 | 0.7117 | 0.2026 |
| Stock_09 | 1.2150 | -0.4295 |
| Stock_10 | 1.6167 | 0.9617 |
| Stock_11 | 0.9409 | 0.0444 |
| Stock_12 | 0.7856 | 0.0674 |
| Stock_13 | 0.7912 | 0.0591 |
| Stock_14 | 1.1561 | 0.3142 |
| Stock_15 | 0.4728 | -0.0379 |
| Stock_16 | 1.4387 | 0.3456 |
| Stock_17 | 0.7244 | -0.4609 |
| Stock_18 | 1.1138 | 0.1458 |
| Stock_19 | 1.1585 | 1.0218 |
| Stock_20 | 1.0756 | 0.0949 |
| Stock_21 | 1.4055 | 0.0456 |
| Stock_22 | 0.7792 | 0.1952 |
| Stock_23 | 0.8831 | -0.3394 |
| Stock_24 | 1.8404 | 0.0153 |
| Stock_25 | 1.1748 | -0.5045 |
4.1 Visualise Estimated Betas
estimated_betas_long <- estimated_betas %>%
pivot_longer(cols = c(beta_mkt_hat, beta_smb_hat),
names_to = "factor", values_to = "beta") %>%
mutate(factor = recode(factor,
beta_mkt_hat = "Market Beta",
beta_smb_hat = "SMB Beta"))
ggplot(estimated_betas_long, aes(x = reorder(stock, beta), y = beta, fill = factor)) +
geom_col(show.legend = FALSE) +
facet_wrap(~factor, scales = "free_x") +
coord_flip() +
labs(title = "Estimated Factor Loadings (Step 1)",
x = "Stock", y = "Beta Estimate") +
theme_minimal(base_size = 11) +
scale_fill_manual(values = c("Market Beta" = "#2c7bb6", "SMB Beta" = "#d7191c"))
5 Step 2: Cross-Sectional Regressions (T Regressions)
For each time period \(t\), we regress the cross-section of stock returns on the estimated betas:
\[r_{it} = \lambda_{0t} + \lambda_{\text{mkt},t} \cdot \hat{\beta}_{i,\text{mkt}} + \lambda_{\text{smb},t} \cdot \hat{\beta}_{i,\text{smb}} + \alpha_{it}\]
This yields T estimates of the risk premia \(\hat{\lambda}_t\).
# Merge estimated betas back into the panel
panel_with_betas <- panel_data %>%
left_join(estimated_betas, by = "stock")
# Run one OLS regression per time period
cs_results <- panel_with_betas %>%
group_by(t) %>%
do(tidy(lm(ret ~ beta_mkt_hat + beta_smb_hat, data = .))) %>%
ungroup()
# Extract lambdas
lambdas <- cs_results %>%
select(t, term, estimate) %>%
pivot_wider(names_from = term, values_from = estimate) %>%
rename(
lambda0 = `(Intercept)`,
lambda_mkt = beta_mkt_hat,
lambda_smb = beta_smb_hat
)
head(lambdas, 10) %>%
kable(caption = "Step 2: Cross-Sectional Risk Premia per Period (first 10)", digits = 5) %>%
kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
full_width = FALSE)| t | lambda0 | lambda_mkt | lambda_smb |
|---|---|---|---|
| 1 | 0.01631 | 0.03845 | -0.00519 |
| 2 | 0.01366 | -0.01663 | -0.01152 |
| 3 | 0.00268 | 0.01754 | 0.03018 |
| 4 | -0.00558 | 0.03368 | 0.03063 |
| 5 | 0.02625 | -0.00256 | -0.00634 |
| 6 | -0.01360 | 0.00272 | 0.03813 |
| 7 | 0.01163 | 0.04956 | 0.01215 |
| 8 | -0.00983 | -0.00198 | 0.05761 |
| 9 | 0.01899 | 0.07479 | 0.01745 |
| 10 | -0.00223 | 0.00426 | 0.02850 |
5.1 Visualise Risk Premia Over Time
lambdas_long <- lambdas %>%
pivot_longer(cols = c(lambda_mkt, lambda_smb),
names_to = "factor", values_to = "lambda") %>%
mutate(factor = recode(factor,
lambda_mkt = "Market Risk Premium",
lambda_smb = "SMB Risk Premium"))
ggplot(lambdas_long, aes(x = t, y = lambda, colour = factor)) +
geom_line(linewidth = 0.8) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "grey40") +
facet_wrap(~factor, ncol = 1, scales = "free_y") +
labs(title = "Step 2: Time-Varying Risk Premia from Cross-Sectional Regressions",
x = "Time Period (Month)", y = expression(hat(lambda)[t])) +
theme_minimal(base_size = 11) +
theme(legend.position = "none") +
scale_colour_manual(values = c("Market Risk Premium" = "#2c7bb6",
"SMB Risk Premium" = "#d7191c"))
6 Step 3: Time-Series Average of Risk Premia
The final Fama-MacBeth estimates are the time-series means of the \(T\) cross-sectional estimates. Standard errors are based on the time-series standard deviation (Newey-West adjusted for serial correlation).
\[\hat{\lambda} = \frac{1}{T} \sum_{t=1}^{T} \hat{\lambda}_t, \qquad \text{SE}(\hat{\lambda}) = \frac{s(\hat{\lambda}_t)}{\sqrt{T}}\]
T_obs <- nrow(lambdas)
fm_results <- lambdas %>%
summarise(
across(c(lambda0, lambda_mkt, lambda_smb),
list(
Mean = ~mean(.x, na.rm = TRUE),
SD = ~sd(.x, na.rm = TRUE),
SE = ~sd(.x, na.rm = TRUE) / sqrt(T_obs),
t_stat = ~mean(.x, na.rm = TRUE) / (sd(.x, na.rm = TRUE) / sqrt(T_obs)),
p_value = ~2 * pt(-abs(mean(.x, na.rm = TRUE) /
(sd(.x, na.rm = TRUE) / sqrt(T_obs))),
df = T_obs - 1)
),
.names = "{.col}__{.fn}"
)
) %>%
pivot_longer(everything(), names_to = c("Parameter", "Stat"), names_sep = "__") %>%
pivot_wider(names_from = Stat, values_from = value) %>%
mutate(
Significance = case_when(
p_value < 0.01 ~ "***",
p_value < 0.05 ~ "**",
p_value < 0.10 ~ "*",
TRUE ~ ""
),
Parameter = recode(Parameter,
lambda0 = "Intercept (λ₀)",
lambda_mkt = "Market Risk Premium (λ_mkt)",
lambda_smb = "SMB Risk Premium (λ_smb)")
)
fm_results %>%
mutate(across(where(is.numeric), ~round(.x, 5))) %>%
kable(caption = "Step 3: Fama-MacBeth Final Estimates",
col.names = c("Parameter", "Mean", "Std Dev", "Std Error",
"t-Statistic", "p-Value", "Sig.")) %>%
kable_styling(bootstrap_options = c("striped", "hover"),
full_width = FALSE) %>%
column_spec(2, bold = TRUE) %>%
footnote(general = "*** p<0.01, ** p<0.05, * p<0.10. Standard errors computed as SD/sqrt(T).")| Parameter | Mean | Std Dev | Std Error | t-Statistic | p-Value | Sig. |
|---|---|---|---|---|---|---|
| Intercept (λ₀) | 0.00282 | 0.01648 | 0.00213 | 1.32464 | 0.19040 | |
| Market Risk Premium (λ_mkt) | 0.00215 | 0.04832 | 0.00624 | 0.34400 | 0.73207 | |
| SMB Risk Premium (λ_smb) | 0.00386 | 0.02959 | 0.00382 | 1.00913 | 0.31703 | |
| Note: | ||||||
| *** p<0.01, ** p<0.05, * p<0.10. Standard errors computed as SD/sqrt(T). |
7 Summary: Fama-MacBeth Regression Workflow
summary_df <- tibble(
Step = c("Step 1", "Step 2", "Step 3"),
Description = c(
"Time-Series Regressions (N = 25 regressions, one per stock)",
"Cross-Sectional Regressions (T = 60 regressions, one per month)",
"Time-Series Average of cross-sectional coefficients"
),
Output = c(
"Estimated betas: β̂_i,mkt and β̂_i,smb",
"Period-specific risk premia: λ̂_mkt,t and λ̂_smb,t",
"Final risk premia estimates with t-statistics and p-values"
)
)
summary_df %>%
kable(caption = "Fama-MacBeth Regression – Three-Step Summary") %>%
kable_styling(bootstrap_options = c("striped", "hover"),
full_width = TRUE) %>%
column_spec(1, bold = TRUE, color = "white", background = "#2c7bb6")| Step | Description | Output |
|---|---|---|
| Step 1 | Time-Series Regressions (N = 25 regressions, one per stock) | Estimated betas: β̂_i,mkt and β̂_i,smb |
| Step 2 | Cross-Sectional Regressions (T = 60 regressions, one per month) | Period-specific risk premia: λ̂_mkt,t and λ̂_smb,t |
| Step 3 | Time-Series Average of cross-sectional coefficients | Final risk premia estimates with t-statistics and p-values |
8 Interpretation
- Market Risk Premium (\(\hat{\lambda}_{\text{mkt}}\)): The estimated compensation per unit of market beta. A positive and significant value supports the CAPM.
- SMB Risk Premium (\(\hat{\lambda}_{\text{smb}}\)): The estimated compensation for exposure to the size factor.
- The Fama-MacBeth standard errors naturally account for cross-sectional correlation in returns because each cross-sectional regression is run separately per period.
Note: For robustness in real applications, consider Newey-West (1987) adjusted standard errors to account for autocorrelation in the \(\hat{\lambda}_t\) series.
9 References
- Fama, E. F., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. Journal of Political Economy, 81(3), 607–636.
- Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3–56.
- Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708.