Question about `cate_intercept` interpretation when `include_bias=True` and `fit_cate_intercept=False` 🆚 `include_bias=False` and `fit_cate_intercept=True`

[xjtusjtu]

Background

I'm working with a binary treatment variable T and a single outcome, and I'm confused about how to interpret the cate_intercept in the summary results when include_bias is set to True vs False.

Issue Description

When I run the same analysis with different include_bias settings, the summary results show different coefficients, and I'm unclear about the relationship between cate_intercept and the coefficient results.

Case 1: include_bias=True

When include_bias=True, I observe that:

• The cate_intercept point estimate, confidence interval, and other statistics are identical to the coefficient for "1" in the Coefficient Results section
• Both show the same values (e.g., point estimate, CI, etc.)

Questions:

1. What is the meaning of cate_intercept in this case?
2. What does the "1" coefficient in the Coefficient Results represent?
3. For treatment effect calculation, is it: 1.092 - 2.579 + ... + 1.092?
4. Does this mean the 1.092 value gets added twice in the treatment effect calculation?

Case 2: include_bias=False

When include_bias=False, I observe that:

• There is no "1" coefficient in the Coefficient Results section
• The cate_intercept is still reported in the summary

Questions:

1. What is the meaning of cate_intercept when include_bias=False?
2. Is the treatment effect calculated as: sum of all point estimates in Coefficient Results + cate_intercept point estimate?

Any help would be greatly appreciated!

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The following is my code

# Treatment effect function
def exp_te(x):
    return np.exp(2 * x[0])# DGP constants

np.random.seed(123)
n = 1000
n_w = 30
support_size = 5
n_x = 4
# Outcome support
support_Y = np.random.choice(range(n_w), size=support_size, replace=False)
coefs_Y = np.random.uniform(0, 1, size=support_size)
def epsilon_sample(n):
    return np.random.uniform(-1, 1, size=n)
# Treatment support
support_T = support_Y
coefs_T = np.random.uniform(0, 1, size=support_size)
def eta_sample(n):
    return np.random.uniform(-1, 1, size=n)

# Generate controls, covariates, treatments and outcomes
W = np.random.normal(0, 1, size=(n, n_w))
X = np.random.uniform(0, 1, size=(n, n_x))
# Heterogeneous treatment effects
TE = np.array([exp_te(x_i) for x_i in X])
# Define treatment
log_odds = np.dot(W[:, support_T], coefs_T) + eta_sample(n)
T_sigmoid = 1/(1 + np.exp(-log_odds))
T = np.array([np.random.binomial(1, p) for p in T_sigmoid])
# Define the outcome
Y = TE * T + np.dot(W[:, support_Y], coefs_Y) + epsilon_sample(n)

# get testing data
X_test = np.random.uniform(0, 1, size=(n, n_x))
X_test[:, 0] = np.linspace(0, 1, n)

include_bias=False

est2 = LinearDML(model_y=RandomForestRegressor(),
                       model_t=RandomForestClassifier(min_samples_leaf=10),
                       discrete_treatment=True,
                       featurizer=PolynomialFeatures(degree=2, include_bias=False),
                       cv=6)
est2.fit(Y, T, X=X, W=W)
te_pred2 = est2.effect(X_test)
lb2, ub2 = est2.effect_interval(X_test, alpha=0.01)
est2.summary()

include_bias=True

est2 = LinearDML(model_y=RandomForestRegressor(),
                       model_t=RandomForestClassifier(min_samples_leaf=10),
                       discrete_treatment=True,
                       featurizer=PolynomialFeatures(degree=2),
                       cv=6)
est2.fit(Y, T, X=X, W=W)
te_pred2 = est2.effect(X_test)
lb2, ub2 = est2.effect_interval(X_test, alpha=0.01)
est2.summary()

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@2025-07-09 Updated: Now I find that the results of include_bias=True and fit_cate_intercept=False are very similar to those of include_bias=True and fit_cate_intercept=True

The difference between the results of "include_bias=True and fit_cate_intercept=False" 🆚 "include_bias=False and fit_cate_intercept=True" doesn't seem to be significant.

Now, an important question is, which setting is more recommendable and worthy of trying?

[xjtusjtu]

According to the discussion on "Polynomial Features for Heterogeneity" at https://econml.azurewebsites.net/spec/estimation/dml.html#usage-examples, when include_bias = True, does "1" and cate_intercept correspond to $\alpha_0$ below? So can they only be calculated once?

$$ Y=\left(\alpha_0+\alpha_1 X+\alpha_2 X^2+\ldots\right) \cdot T+g(X, W, \epsilon) $$

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@2025-07-09 Updated: Now I think both of the "1" term and the cate_intercept term should be included in the final CATE, which means the identical value should be added twice.