Chapter 11 — Statistical Inference for Multiple Regression

A regression gives you a single coefficient, but the range of plausible values is where significance actually lives. Does the CI clear zero?

A 95% confidence interval b_j ± t_{n−k, 0.025} × se(b_j) gives a range of plausible values for β_j. If the interval excludes zero, the coefficient is statistically significant at the 5% level. Narrower intervals mean more precise estimation, which improves with larger samples and lower multicollinearity. Wider intervals buy more confidence at the cost of precision.

Confidence level 95%

Variable

Coefficient

—

t-critical

—

CI Lower

—

CI Upper

—

Contains 0?

—

Try this

Pick size at 95%. The CI is [$36.44, $100.30] and excludes zero. Size is significant at the 5% level — the effect of size on price is unmistakable even in a multicollinear model.
Switch to bedrooms at 95%, then push to 90%. The CI still includes zero at both levels. Bedrooms cannot clear the bar even when you lower it — multicollinearity has swallowed any signal.
Set 99% on the size variable. The CI widens but still excludes zero. Robust significance — the effect is real at any conventional confidence level.

Take-away: A CI that excludes zero is the quickest significance test — and its width tells you how confident "significant" actually is. Read §11.3 in the chapter →

Not just "is it significant?" — test any claim about the size coefficient: is it exactly $68, less than $90, more than $30? Pick a null, get an answer.

The most common hypothesis test examines H₀: β_j = 0 — whether variable x_j has any partial effect on y. The t-statistic t = (b_j − β₀) / se(b_j) measures how many standard errors the estimate is from the hypothesized value. Reject H₀ when |t| > t_critical, or equivalently when the p-value < α. Generalizing to any β₀ tests any substantive claim about the coefficient.

Hypothesized value β₀ $0/sqft

b_size

—

t-statistic

—

p-value

—

Try this

Set β₀ = 0 (the standard significance test). t = 4.44, p < 0.001, REJECT. Size has a non-zero partial effect on price — and the triangle is deep in the tail.
Drag β₀ toward 68 (the estimate itself). t → 0 and p → 1.0. We can never reject that β equals its own sample estimate — an obvious sanity check.
Find the β₀ where p first crosses 0.05. That value is exactly the lower (~$36) or upper (~$100) CI endpoint. Tests and CIs are two views of the same object.

Take-away: A t-test against any β₀ is just a CI read from the side — the values you'd reject are precisely those outside the interval. Read §11.4 in the chapter →

Does adding more predictors always sharpen your estimate? No — when predictors overlap, extra variables buy noise and inflate standard errors without moving R² much.

The standard error se(b_j) = s_e / √Σ x̃_ji² reveals what makes estimates precise. Four levers: (1) a well-fitting model (small s_e), (2) large n, (3) high variation in x_j after controlling for others, and (4) low multicollinearity. When regressors are highly correlated, Σ x̃_ji² shrinks — and standard errors explode.

Model

b_size

—

SE(size)

—

t(size)

—

R²

—

Adj R²

—

Try this

Start with Size only. SE = $11.17, R² = 0.618. A clean, precise bivariate estimate — baseline for the experiment.
Add Bedrooms. SE jumps to $13.30 but R² stays at 0.618. Bedrooms overlaps with size (r = 0.52); it adds noise, not explanatory power.
Switch to the Full 6-variable model. SE grows to $15.39 (38% larger than the bivariate!); R² rises 3.3 percentage points to 0.651. A trivial fit gain for a serious precision cost — more variables ≠ more information.

Take-away: Every correlated regressor you add inflates the SE of the ones you already care about — add variables deliberately, with precision-for-parsimony in mind. Read §11.2 in the chapter →

Five individually-insignificant coefficients — could they be jointly significant anyway? t-tests can't answer; only the F-test can.

Individual t-tests cannot test multiple restrictions simultaneously because they ignore correlations between coefficient estimates. The F-test evaluates joint significance using the F(q, n−k) distribution, where q is the number of restrictions. It compares how much worse the restricted model fits the data than the unrestricted one (KC 11.5). Under homoskedasticity, F = [(RSS_r − RSS_u)/q] / [RSS_u/(n−k)] (KC 11.6). Large F = restrictions inconsistent with the data.

Test type

F-statistic

—

df₁, df₂

—

p-value

—

Try this

Start with Overall F. F = 6.83, df = (6, 22), p < 0.001, REJECT. At least one variable in the full model is doing real work — the regression is jointly significant.
Switch to Subset F. F = 0.42, df = (5, 22), p = 0.83, FAIL. The five extra variables are jointly indistinguishable from zero — size alone carries the model.
Reconcile the two results. The overall F is significant; the subset F is not. Size contributes all the signal; bedrooms-through-monthsold contribute nothing. Parsimony wins on the evidence.

Take-away: F-tests reveal joint significance that t-tests miss — use them to justify dropping groups of correlated regressors. Read §11.5 in the chapter →

A side-by-side table answers one question in one glance: does adding variables actually improve the model, or does it just run up the complexity bill?

To test whether a subset of regressors belongs in the model, compare the restricted and unrestricted models with an F-test, and compare them on adjusted R², AIC, and BIC. If the F-statistic is small (large p-value), the extras don't meaningfully improve fit and the simpler model is preferred. Raw R² always rises with more regressors; adjusted R² and the information criteria reward parsimony and can fall as you add noise.

Variable	Model 1	Model 2	Model 3

Try this

Scan the R² row. It rises monotonically (0.618 → 0.651). Guaranteed mathematically — raw R² is a one-way ratchet.
Scan Adjusted R². Model 1 (0.603) wins; Model 3 (0.555) loses. The five extra regressors are penalised for weak explanatory power and adj R² flips the verdict.
Scan the F-statistic row. Model 1's F (43.58) dwarfs Model 3's (6.83). Fewer regressors, but each one earns its keep — a cleaner "joint significance" story.

Take-away: When R² and adjusted R² disagree, believe adjusted R² — parsimony is not a stylistic preference, it's a statistical one. Read §11.7 in the chapter →

If error variance changes with x, your textbook SEs lie. Robust SEs don't — but do they change any conclusions on this dataset?

Heteroskedasticity-robust (HC1) standard errors provide valid inference without assuming homoskedasticity. Coefficient estimates remain unchanged; only the standard errors — and thus t-statistics and p-values — may differ. For cross-sectional data, reporting robust SEs alongside standard SEs is modern best practice: it costs nothing when homoskedasticity holds and protects you when it fails.

SE type

Variable	Coef	SE	t	p

Try this

Compare SE(size) across Standard and Robust. $15.39 vs $15.36 — essentially identical. For size, the two worldviews agree completely.
Compare SE(bathrooms). Standard $15,721 vs robust $19,284 — a 23% jump. Heteroskedasticity in this small cross-section inflates bathrooms' uncertainty once you account for it.
Scan every row for a significance flip. None here. In 29-observation datasets the differences rarely decide significance — but in larger, more heteroskedastic data, robust SEs can flip conclusions.

Take-away: Always report robust (HC1) SEs alongside standard SEs — when they disagree, the robust ones are the honest reference. Read §11.7 in the chapter →

Code Summary

You've explored the key concepts interactively — now reproduce them in code. These self-contained blocks cover everything you practiced above. Pick your language, copy the code, and run it.

# =============================================================================
# CHAPTER 11 CHEAT SHEET: Statistical Inference for Multiple Regression
# =============================================================================

# --- Libraries ---
import numpy as np                              # numerical operations
import pandas as pd                             # data loading and manipulation
import matplotlib.pyplot as plt                 # creating plots and visualizations
import pyfixest as pf                           # OLS regression with R-style formulas
# !pip install pyfixest                         # uncomment if running in Google Colab
from scipy import stats                         # t and F distributions for inference

# =============================================================================
# STEP 1: Load data directly from a URL
# =============================================================================
# pd.read_stata() reads Stata .dta files; the dataset has 29 houses from Davis, CA
url = "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_HOUSE.DTA"
data_house = pd.read_stata(url)

print(f"Dataset: {data_house.shape[0]} observations, {data_house.shape[1]} variables")
print(data_house[['price', 'size', 'bedrooms', 'bathrooms', 'lotsize', 'age']].describe().round(2))

# =============================================================================
# STEP 2: Estimate the full multiple regression model
# =============================================================================
# Formula syntax: 'y ~ x1 + x2 + ...' includes an intercept automatically
# pf.feols() estimates and returns the fitted model in one call
fit_full = pf.feols('price ~ size + bedrooms + bathrooms + lotsize + age + monthsold',
                    data=data_house)

n = int(fit_full._N)
k = len(fit_full.coef())
df = n - k  # degrees of freedom for t and F tests

print(f"\nSize effect: ${fit_full.coef()['size']:,.2f} per sq ft (p = {fit_full.pvalue()['size']:.4f})")
print(f"R-squared: {fit_full._r2:.4f} ({fit_full._r2*100:.1f}% of variation explained)")
print(f"Degrees of freedom: n-k = {n}-{k} = {df}")

# Full regression table (coefficients, std errors, t-stats, p-values, R\u00b2)
fit_full.summary()

# =============================================================================
# STEP 3: Confidence intervals — does the CI exclude zero?
# =============================================================================
# 95% CI: b_j \u00b1 t_critical \u00d7 se(b_j)
# If the interval excludes zero, the coefficient is statistically significant at 5%
conf_int = fit_full.confint()
print("\n95% Confidence Intervals:")
print(conf_int.round(2))

# Manual calculation for the size coefficient
coef_size = fit_full.coef()['size']
se_size   = fit_full.se()['size']
t_crit    = stats.t.ppf(0.975, df)  # 0.975 = upper tail for two-sided 95% CI

ci_lower = coef_size - t_crit * se_size
ci_upper = coef_size + t_crit * se_size
print(f"\nSize 95% CI: [${ci_lower:.2f}, ${ci_upper:.2f}]  (excludes zero \u2192 significant)")

# =============================================================================
# STEP 4: Hypothesis test on a single coefficient
# =============================================================================
# Test H\u2080: \u03b2_size = 50 vs H\u2081: \u03b2_size \u2260 50
# t = (b_j - hypothesized value) / se(b_j)
null_value = 50
t_stat = (coef_size - null_value) / se_size
p_value = 2 * (1 - stats.t.cdf(abs(t_stat), df))

print(f"\nTest H\u2080: \u03b2_size = {null_value}")
print(f"  t-statistic: {t_stat:.4f}")
print(f"  p-value: {p_value:.4f}")
print(f"  Decision: {'Reject' if p_value < 0.05 else 'Fail to reject'} H\u2080 at 5% level")

# Test of statistical significance: H\u2080: \u03b2_size = 0
t_stat_zero = coef_size / se_size
print(f"\nTest H\u2080: \u03b2_size = 0")
print(f"  t-statistic: {t_stat_zero:.4f}, p-value: {fit_full.pvalue()['size']:.6f}")
print(f"  Size IS statistically significant at 5% level")

# =============================================================================
# STEP 5: Joint F-test — are groups of coefficients jointly significant?
# =============================================================================
# Overall F-test: H\u2080: all slope coefficients = 0
overall_F = fit_full.wald_test()
print(f"\nOverall F-test: F = {overall_F.stat.values[0]:.4f}, p = {overall_F.pval.values[0]:.6e}")
print(f"  At least one variable matters \u2192 model is jointly significant")

# Subset F-test: are variables other than size jointly significant?
# H\u2080: \u03b2_bedrooms = \u03b2_bathrooms = \u03b2_lotsize = \u03b2_age = \u03b2_monthsold = 0
# Use R matrix: each row zeroes out one coefficient
vars_to_test = ['bedrooms', 'bathrooms', 'lotsize', 'age', 'monthsold']
all_vars = list(fit_full.coef().index)
R = np.zeros((len(vars_to_test), len(all_vars)))
for i, v in enumerate(vars_to_test):
    R[i, all_vars.index(v)] = 1
subset_F = fit_full.wald_test(R=R)
print(f"\nSubset F-test (drop 5 vars, keep size):")
print(f"  F = {subset_F.stat.values[0]:.4f}, p = {subset_F.pval.values[0]:.4f}")
print(f"  Extra variables are NOT jointly significant \u2192 simpler model preferred")

# =============================================================================
# STEP 6: Model comparison — restricted vs unrestricted
# =============================================================================
# Restricted model: only size as predictor
fit_restricted = pf.feols('price ~ size', data=data_house)

# Manual F-test: restricted vs unrestricted
rss_r = np.sum(fit_restricted._u_hat**2)
rss_u = np.sum(fit_full._u_hat**2)
q = len(fit_full.coef()) - len(fit_restricted.coef())  # number of restrictions
F_manual = ((rss_r - rss_u) / q) / (rss_u / (n - len(fit_full.coef())))
p_manual = 1 - stats.f.cdf(F_manual, q, n - len(fit_full.coef()))
print(f"\nF-test: Restricted (size only) vs Full model:")
print(f"  F = {F_manual:.4f}, p = {p_manual:.4f}")

# Compare fit statistics
print(f"\n{'Model':<25} {'R\u00b2':>8} {'Adj. R\u00b2':>9}")
print("-" * 44)
for name, m in [('Size only', fit_restricted), ('Full model', fit_full)]:
    print(f"{name:<25} {m._r2:>8.4f} {m._adj_r2:>9.4f}")

# =============================================================================
# STEP 7: Robust standard errors — valid under heteroskedasticity
# =============================================================================
# HC1 (White's correction) provides valid inference without constant variance
fit_robust = pf.feols('price ~ size + bedrooms + bathrooms + lotsize + age + monthsold',
                      data=data_house, vcov='HC1')

print("\nStandard vs Robust SEs:")
print(f"{'Variable':<14} {'Coef':>10} {'Std SE':>10} {'Robust SE':>10} {'p (robust)':>10}")
print("-" * 56)
for var in fit_full.coef().index:
    print(f"{var:<14} {fit_full.coef()[var]:>10.2f} "
          f"{fit_full.se()[var]:>10.2f} {fit_robust.se()[var]:>10.2f} "
          f"{fit_robust.pvalue()[var]:>10.4f}")

# =============================================================================
# STEP 8: Coefficient plot — visual summary of significance
# =============================================================================
# Error bars crossing zero = not significant at 5%
params_plot = fit_full.coef()[1:]  # exclude intercept
ci_plot = conf_int.iloc[1:, :]

fig, ax = plt.subplots(figsize=(10, 6))
y_pos = np.arange(len(params_plot))
ax.errorbar(params_plot.values, y_pos,
            xerr=[params_plot.values - ci_plot.iloc[:, 0].values,
                  ci_plot.iloc[:, 1].values - params_plot.values],
            fmt='o', markersize=8, capsize=5, capthick=2, linewidth=2)
ax.axvline(x=0, color='red', linestyle='--', linewidth=1.5, alpha=0.7, label='H\u2080: \u03b2 = 0')
ax.set_yticks(y_pos)
ax.set_yticklabels(params_plot.index)
ax.set_xlabel('Coefficient Value')
ax.set_title('Coefficient Estimates with 95% Confidence Intervals')
ax.legend()
ax.grid(True, alpha=0.3, axis='x')
plt.tight_layout()
plt.show()

Open empty Colab notebook →

* =============================================================================
* CHAPTER 11 CHEAT SHEET: Statistical Inference for Multiple Regression
* =============================================================================

* --- Setup ---
clear all                                // start with a clean workspace
set more off                             // do not pause output for long results

* =============================================================================
* STEP 1: Load data directly from a URL
* =============================================================================
* use loads a Stata .dta file; "clear" drops any data already in memory
use "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_HOUSE.DTA", clear

describe                                 // list all variables, types, and labels
display "Observations: " _N              // _N is Stata's built-in observation count
summarize price size bedrooms bathrooms lotsize age

* =============================================================================
* STEP 2: Estimate the full multiple regression model
* =============================================================================
* regress fits OLS: first variable is y, the rest are x's
* Stata includes an intercept automatically (use ", noconstant" to suppress)
regress price size bedrooms bathrooms lotsize age monthsold

// After regress, Stata stores results you can reference:
display "Size effect: " _b[size]         // coefficient on size
display "R-squared:   " e(r2)            // R-squared
display "Obs:         " e(N)             // number of observations
display "df_residual: " e(df_r)          // degrees of freedom (n - k)

* =============================================================================
* STEP 3: Confidence intervals — does the CI exclude zero?
* =============================================================================
* By default, regress output shows 95% CIs in the last two columns
* Use ", level(XX)" to change the confidence level (e.g., 90% or 99%)
regress price size bedrooms bathrooms lotsize age monthsold, level(95)

// Manual calculation for the size coefficient
display "Size 95% CI lower: " _b[size] - invttail(e(df_r), 0.025) * _se[size]
display "Size 95% CI upper: " _b[size] + invttail(e(df_r), 0.025) * _se[size]
// invttail(df, p) returns the critical value; 0.025 for two-sided 95% CI

// Change to 99% CI to see the interval widen
regress price size bedrooms bathrooms lotsize age monthsold, level(99)

* =============================================================================
* STEP 4: Hypothesis test on a single coefficient
* =============================================================================
* After regress, Stata's output already includes t-stats and p-values for
* the standard test H0: beta = 0.
regress price size bedrooms bathrooms lotsize age monthsold

// Test a non-zero null: H0: beta_size = 50 vs H1: beta_size != 50
// test performs a Wald test on the most recent estimation
test size = 50
// Stata reports F-statistic and p-value; for a single restriction, F = t^2

// lincom computes the estimate, SE, t-stat, and CI for a linear combination
lincom size - 50                         // same test via linear combination

* =============================================================================
* STEP 5: Joint F-test — are groups of coefficients jointly significant?
* =============================================================================
* Overall F-test is shown automatically in regress output (top-right corner)
* It tests H0: all slope coefficients = 0

// Subset F-test: are the 5 extra variables jointly significant?
// H0: beta_bedrooms = beta_bathrooms = beta_lotsize = beta_age = beta_monthsold = 0
test bedrooms bathrooms lotsize age monthsold

// Stata reports F(5, 22) and p-value
// If p > 0.05 -> fail to reject -> the extras are not jointly significant

* =============================================================================
* STEP 6: Model comparison — restricted vs unrestricted
* =============================================================================
* Estimate the restricted model: only size as predictor
regress price size
estimates store model_restricted         // save results under a name

* Estimate the full (unrestricted) model
regress price size bedrooms bathrooms lotsize age monthsold
estimates store model_full               // save results under a name

// Compare models side by side with esttab (requires estout package)
// ssc install estout                    // run once to install
esttab model_restricted model_full,                                  ///
    se r2 ar2 stats(N r2 r2_a F, labels("N" "R²" "Adj. R²" "F"))   ///
    title("Model Comparison: Restricted vs Full")

// Formal F-test: does the full model fit significantly better?
// lrtest compares two nested models via likelihood ratio
lrtest model_restricted model_full

* =============================================================================
* STEP 7: Robust standard errors — valid under heteroskedasticity
* =============================================================================
* Add ", robust" (or ", vce(robust)") to get HC1 robust standard errors
* Coefficients stay the same; only SEs, t-stats, and p-values change
regress price size bedrooms bathrooms lotsize age monthsold
estimates store model_standard

regress price size bedrooms bathrooms lotsize age monthsold, vce(robust)
estimates store model_robust

// Compare standard vs robust SEs side by side
esttab model_standard model_robust,                                  ///
    se title("Standard vs Robust (HC1) Standard Errors")             ///
    mtitles("Standard" "Robust")

* =============================================================================
* STEP 8: Coefficient plot — visual summary of significance
* =============================================================================
* coefplot draws coefficient estimates with confidence intervals
* Error bars crossing zero = not significant at 5%
regress price size bedrooms bathrooms lotsize age monthsold
coefplot, drop(_cons) xline(0, lcolor(red) lpattern(dash))          ///
    title("Coefficient Estimates with 95% CIs")                      ///
    xtitle("Coefficient Value")                                      ///
    levels(95)

Paste into your Stata do-file editor

# =============================================================================
# CHAPTER 11 CHEAT SHEET: Statistical Inference for Multiple Regression
# =============================================================================

# --- Libraries ---
library(haven)           # read Stata .dta files
library(fixest)          # fast OLS estimation with feols()
library(dplyr)           # data manipulation
library(ggplot2)         # grammar of graphics

# =============================================================================
# STEP 1: Load data directly from a URL
# =============================================================================
url <- "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_HOUSE.DTA"
data_house <- read_dta(url)

cat("Dataset:", nrow(data_house), "observations,", ncol(data_house), "variables\n")

# =============================================================================
# STEP 2: Estimate the full multiple regression model
# =============================================================================
model_full <- feols(price ~ size + bedrooms + bathrooms + lotsize + age + monthsold,
                    data = data_house)
summary(model_full)

cat("Size effect: $", round(coef(model_full)["size"], 2), "per sq ft\n")
cat("R-squared:", round(r2(model_full), 4), "\n")

# =============================================================================
# STEP 3: Confidence intervals — does the CI exclude zero?
# =============================================================================
# confint() computes 95% CIs; if the interval excludes zero, significant at 5%
confint(model_full, level = 0.95)

# =============================================================================
# STEP 4: Hypothesis test on a single coefficient
# =============================================================================
# Test H0: beta_size = 50 vs H1: beta_size != 50
# wald() from fixest tests linear hypotheses
wald(model_full, "size = 50")

# =============================================================================
# STEP 5: Joint F-test — are groups of coefficients jointly significant?
# =============================================================================
# Overall F-test is shown in summary output
# Subset F-test: are the 5 extra variables jointly significant?
wald(model_full, c("bedrooms", "bathrooms", "lotsize", "age", "monthsold"))

# =============================================================================
# STEP 6: Model comparison — restricted vs unrestricted
# =============================================================================
model_restricted <- feols(price ~ size, data = data_house)

etable(model_restricted, model_full,
       headers = c("Size only", "Full model"),
       fitstat = c("r2", "ar2", "n"))

# =============================================================================
# STEP 7: Robust standard errors — valid under heteroskedasticity
# =============================================================================
# vcov = "HC1" provides HC1 (White's correction) robust SEs
model_robust <- feols(price ~ size + bedrooms + bathrooms + lotsize + age + monthsold,
                      data = data_house, vcov = "HC1")

etable(model_full, model_robust,
       headers = c("Standard", "Robust (HC1)"))

# =============================================================================
# STEP 8: Coefficient plot — visual summary of significance
# =============================================================================
# coefplot() from fixest draws coefficient estimates with CIs
# Error bars crossing zero = not significant at 5%
coefplot(model_full, drop = "(Intercept)")

Paste into your R console or RStudio

Statistical Inference for Multiple Regression

Confidence Interval Explorer

Hypothesis Test for a Single Coefficient

Multicollinearity & SE Inflation

Joint F-Test

Model Comparison: R² vs Parsimony

Robust vs Standard Errors

Code Summary