In Empirical Analysis, Can You Directly Compare the Magnitudes of Coefficients Across Different Variables? When Can You? How Should You Compare Correctly?

1. Opening: Two Coefficients Side by Side—0.08 and 0.03, Which Is "Larger"?

You run a wage equation:

ln_wage = 0.080 × education + 0.030 × experience + 0.120 × tenure + controls

Looking at these three coefficients, a perfectly natural question pops into your head: "Among education, work experience, and job tenure, which has the largest effect on wages?"

Then you scan the coefficients—0.120 > 0.080 > 0.030. You are about to write in your paper: "Job tenure has the largest effect on wages, followed by education, with work experience having the smallest effect."

But just before you put pen to paper, a voice chimes in: the units of measurement for these three variables are completely different.

• The unit of education is years (typically ranging from 0–22)
• The unit of experience is also years (typically ranging from 0–45)
• The unit of tenure is also years (typically ranging from 0–40)

Wait—all three have the same unit, "years." So can we compare them directly?

Same units ≠ directly comparable. Even if all three variables are measured in "years," their degree of variation in your sample is completely different. The sample standard deviation of education could be 4 years, that of work experience could be 12 years, and that of job tenure could be 8 years. The coefficient 0.120 is the effect of "one additional year of job tenure," but "one additional year of job tenure" is a common change in the sample, whereas "one additional year of education" may be a larger change than you think (education is a discrete variable concentrated at a few nodes—12 years, 16 years, 19 years).

Core message: Directly comparing the magnitudes of two raw coefficients is meaningless whenever the units of measurement differ—it is like comparing "1 kilometer" and "1 hour" to see which is longer. But even when the units are the same, direct comparison is still incomplete—because a coefficient reflects "the effect of a one-unit change," and the typical meaning of "one unit" (i.e., the degree of dispersion) can differ completely across variables in the sample. To compare "which variable has a larger effect on Y," you need to first define what "larger effect" means, then choose the corresponding method—standardized coefficients, Shapley decomposition, and dominance analysis each provide their own answer, and none is a universally "best answer."

2. When Can You Compare Directly?—Three Conditions, None Dispensable

2.1 Condition 1: Identical Units of Measurement

Identical units are the minimum threshold for two coefficients to be comparable.

• "Years of education (years)" and "years of work experience (years)"—same units ✅
• "Annual household income (yuan)" and "years of education (years)"—different units ❌ (cannot compare directly)
• "GDP (100 million yuan)" and "population (10,000 persons)"—different units ❌ (cannot compare directly)
• "Firm R&D expenditure (thousand yuan)" and "firm advertising expenditure (thousand yuan)"—same units ✅

Identical units are the eligibility criterion—without it, you cannot enter the competition. But it is not the medal—with it, your comparison is not necessarily fair either.

2.2 Condition 2: Comparable Degrees of Sample Dispersion

Even with identical units, the "degree of variation" of different variables in the sample can be vastly different. A coefficient represents "the change in Y for a one-unit change in the variable," but for different variables, one unit can mean very different levels of "typicality."

Example: Suppose you study "factors affecting student performance":

• study_hours (hours of study per week): coefficient = 2.0, sample standard deviation = 12
• class_attendance (attendance rate, 0–100%): coefficient = 1.5, sample standard deviation = 8

The units differ (hours vs. percentage), so you cannot compare directly. OK, you knew that.

Let us change the example: both variables are measured in "hours"—

• study_hours (hours of study per week): coefficient = 2.0, sample standard deviation = 12
• sleep_hours (hours of sleep per day): coefficient = 3.0, sample standard deviation = 1.2

Both coefficients are on the "hour" scale, with identical units ✅. But can you say "sleep has a larger effect on performance than study"?

No, you cannot. Because weekly study hours vary greatly in the sample (some study 5 hours, others 40 hours), while sleep duration is almost universally between 6.5 and 8.5 hours. "One additional hour of study" is a common, small change in the sample; "one additional hour of sleep" may be an extremely large, atypical change. The coefficients do not reflect this difference.

The essence of Condition 2: even with identical units, coefficients do not account for the "actual range of variation" of variables in the sample. Directly comparing coefficients is equivalent to assuming that "one unit represents an equivalent change for all variables"—but this default assumption is fragile in most data.

2.3 Condition 3: No Severe Collinearity Among Variables

If two variables are highly correlated (e.g., "annual household income" and "annual household consumption"), their coefficient estimates are unstably "splitting" the shared explanatory power. In this case, comparing the magnitudes of their coefficients is essentially comparing an unstable "division of spoils"—change the sample, and the two coefficients may swap positions.

In models with severe collinearity, any comparison of "whose coefficient is larger" should be viewed with great suspicion—because the two coefficients themselves are unstable.

2.4 A Rare Real-World Scenario Where All Three Conditions Hold

If you have a strictly designed experiment in which:

• All treatment variables use the same units (e.g., all administered at a uniform standard dose)
• The range of variation for each treatment in the sample is designed to be consistent
• There is no collinearity among variables (orthogonal design)

Then you can directly compare regression coefficients. But this holds almost exclusively in experimental data and a few carefully designed simulation datasets. In empirical analysis with observational data, these three conditions are rarely satisfied simultaneously.

3. Comparing "Which Variable Has a Larger Effect on Y"—Four Methods, Four Answers

Since directly comparing coefficients is unreliable, how should you compare? This question has no single correct answer—different methods define "larger effect" differently and yield different rankings. You need to choose the method that matches your research question.

3.1 Method 1: Standardized Coefficients (Beta Coefficients)—"For a One-Standard-Deviation Change in X, How Many Standard Deviations Does Y Change?"

Procedure: Standardize all variables (Y and all X) via Z-scores, then run the regression. The resulting coefficients are called standardized coefficients or Beta coefficients.

$\beta_j^* = \hat{\beta}_j \cdot \frac{s_{X_j}}{s_Y}$

Question answered: "When Xⱼ undergoes a 'typical change' (one standard deviation), by how many standard deviations is Y expected to change?"

Advantages: Simple to implement—in Stata, a single command reg y x1 x2, beta suffices. All variables are placed on the "standard deviation" scale, partially resolving the issue of differing units.

Disadvantages:

• The interpretation is unintuitive ("How many years is one standard deviation of education?"—readers must translate).
• Not comparable across samples (different samples have different standard deviations → Beta coefficients cannot be compared across samples).
• Beta coefficients depend on the dispersion of variables in this particular sample—mixing "effect size" with "sample characteristics."

When to use: You want to roughly assess the relative importance of different variables within the same sample and the same model, and your readers do not mind seeing expressions like "standard deviation." Typically reported in the appendix or footnotes as supplementary information.

3.2 Method 2: Variance Decomposition / Shapley Value Decomposition—"How Much of the R² Does This Variable Contribute?"

Procedure: Decompose the model's R² (or goodness of fit) across each independent variable to see what percentage of R² each variable "contributes."

Shapley value decomposition is the gold standard among such methods. Its core idea comes from cooperative game theory: treat R² as the "outcome achieved by all variables cooperating to explain Y," then allocate credit to each variable according to its marginal contribution across all possible "orders of entry."

Intuition: If education is first added to the model → R² goes from 0 to 0.20. If experience and tenure are already in the model, adding education → R² goes from 0.15 to 0.22 (marginal contribution = 0.07). The Shapley value averages over all possible entry orders—each variable's final allocation equals its average marginal contribution across all orders.

In Stata:

reg y x1 x2 x3
rego y x1 x2 x3   // Install: ssc install rego

Question answered: "Of the total variation in Y, what proportion can be attributed to the contribution of Xⱼ?"

Advantages:

• The decomposition results sum exactly to R² (a complete variance decomposition).
• Based on the intuition of "marginal contribution," with a solid axiomatic foundation in game theory.
• Not affected by units of measurement—because it decomposes R², not coefficients.

Disadvantages:

• Computationally expensive (with k variables, 2ᵏ submodels are needed. But for k ≤ 15, modern computers handle this easily).
• Decomposition may be unstable among highly collinear variables—facing similar issues as directly comparing coefficients.
• Shapley values measure "contribution to explanatory power," not "magnitude of causal effect"—a variable may receive a high Shapley value because of strong correlation (rather than causation) with Y.

When to use: You want to decompose "among the variables' combined ability to explain Y, who contributes the most." Especially suitable for exploratory analysis where there is no severe collinearity among variables.

3.3 Method 3: Dominance Analysis—"Who 'Dominates' Whom Across All Submodels?"

Dominance analysis, proposed by Budescu (1993), is an "enhanced version" of Shapley decomposition. It not only provides a decomposition value for each variable but also a partial order (dominance ranking).

Core idea: For any two variables Xⱼ and Xₖ, compare their incremental R² across all submodels of the same size:

• In all submodels containing 1 other variable, is the incremental R² from adding Xⱼ always greater than that from adding Xₖ?
• In all submodels containing 2 other variables?
• ...

If Xⱼ consistently provides a larger incremental R² than Xₖ at all submodel sizes, then Xⱼ is defined as having complete dominance over Xₖ. If it dominates only at most sizes, it is conditional dominance. If it dominates only in the average sense, it is general dominance (in which case it is equivalent to the Shapley value ranking).

Question answered: "Across all possible model specifications, is Xⱼ consistently superior to Xₖ?"

Advantages: More robust than a standalone Shapley value—it is not just a single number but a comparison across submodel sizes. If Xⱼ consistently dominates Xₖ, this conclusion is robust to the specific choice of model specification.

Disadvantages: Higher computational cost; with a large number of variables (k > 10), the output and interpretation become complex.

When to use: You want not only to know "who is most important" but also to ensure that "the most important conclusion does not depend on a particular model choice." When there is some collinearity among variables, the ranking from dominance analysis is more reliable than directly comparing coefficients.

3.4 Method 4: Partial and Semi-Partial Correlation Coefficients—"After Removing All Other Variables, How Much Association Remains Between Xⱼ and Y?"

• Partial correlation coefficient: The correlation between the residuals of Xⱼ and Y after simultaneously removing the linear effects of all other X variables from both. It measures the strength of the "net association" between Xⱼ and Y after all other variables have been "controlled for."

• Semi-partial (or part) correlation coefficient: The linear effects of other X variables are removed only from Xⱼ (Y remains as is), and then the correlation between the "purified Xⱼ" and the original Y is computed. The square of the semi-partial correlation equals the incremental R² of that variable—that is, the increase in R² when Xⱼ is added last to the model.

The relationship between the two: semi-partial correlation ≤ partial correlation. One purifies only X; the other purifies both X and Y.

Questions answered:

• Partial correlation: "After all other variables are linearly removed, how 'correlated' are Xⱼ and Y?"
• Semi-partial correlation: "When Xⱼ is added last to the model, how much of the variation in Y can it explain on its own?"

Advantages: Intuitive (they are essentially "purified correlation coefficients"), and can be directly output in Stata (the pcorr command).

Disadvantages: They provide only the magnitude of "association," not the "share" in a variance decomposition. Partial correlation coefficients are not robust when variables are highly correlated.

4. A Framework for Choosing Among the Four Methods—Which Question Are You Actually Trying to Answer?

An important warning: All of the above methods measure the "strength of statistical association," not the "magnitude of causal effects." A variable may receive a high Shapley value or Beta coefficient because it is highly correlated with other important X variables—but this is not the causal conclusion you want. If your research goal is causal inference, the "relative importance" of variables is a concept that requires more careful definition within a causal framework—typically drawing on causal mediation analysis or causal attribution methods (such as variable importance measures in causal forests), rather than relying solely on methods based on R² decomposition in regression.

5. Same Units = Directly Comparable?—An Eye-Opening Example

Returning to the opening example—three variables all measured in "years":

ln_wage = 0.080 × education + 0.030 × experience + 0.120 × tenure + controls

Can you say tenure is the most important?

No, you cannot. The reason is not the units—they are indeed all in years. The reason lies in the "range of variation":

• education: In the sample, years of education are concentrated between 9 and 19, with a standard deviation of about 3.5 years. The coefficient 0.08 means "one more year of schooling raises wages by 8%." But in this sample, for each "natural" one-standard-deviation change in education (3.5 years), wages change by approximately 0.08 × 3.5 = 0.28 (i.e., 28%).
• tenure: In the sample, job tenure is concentrated between 0 and 15 years, with a standard deviation of about 7 years. The coefficient 0.12 means "one more year of tenure raises wages by 12%." But for each natural one-standard-deviation change (7 years), wages change by approximately 0.12 × 7 = 0.84 (i.e., 84%).

The coefficient magnitudes you see (0.12 > 0.08) mask the differences in the variables' ranges of variation. If you are comparing "the effect of a typical change in the variable," you need standardized coefficients—not raw coefficients.

In Stata:

reg ln_wage education experience tenure, beta

The output will additionally provide Beta coefficients. If Beta(tenure) > Beta(education) → in this sample, under a "typical change," tenure's effect is indeed larger. But if Beta(education) > Beta(tenure) → your intuition from looking at the raw coefficients 0.08 < 0.12 is wrong—once the range of variation is taken into account, education's effect is actually larger.

This is why concluding based solely on "the coefficient looks larger" is dangerous—you are comparing two numbers on different scales without calibrating for the "range of variation."

6. Common Pitfalls

6.1 Pitfall 1: Using the Magnitude of t-Statistics to Judge "Who Is More Important"

❌ "education's t-statistic = 8, experience's t-statistic = 3.75 → education is more important."

The t-statistic = coefficient / standard error. It measures "how precisely the coefficient is estimated"—not "how large the coefficient is." An extremely precisely estimated 0.0001 (t = 10) is not "larger" than a less precisely estimated 10.0 (t = 2).

The t-statistic answers "is this effect precisely different from zero," not "how large is this effect." If you want to compare importance, use standardized coefficients or Shapley decomposition, not t-statistics.

6.2 Pitfall 2: Using the Magnitude of p-Values to Judge "Who Is More Important"

❌ "education's p < 0.001, experience's p = 0.03 → education is more important."

The p-value is a monotonic function of the t-statistic—like the t-statistic, it answers "significance" rather than "magnitude." A very small effect can have an extremely small p-value in a very large sample (p < 0.001), but this does not mean it is "more important" than a medium-sized effect (p = 0.03, in a small sample).

6.3 Pitfall 3: Standardized Coefficients Solve Everything

Standardized coefficients eliminate differences in units but introduce new problems:

• Standardized coefficients from different samples are not comparable. Because standardization depends on the sample mean and standard deviation.
• The meaning of dummy variables is lost after standardization.
• They still measure only the strength of linear association, not causal importance.

Standardized coefficients are "the simplest method for comparing importance," but also "the most easily misused method." If you report them in a paper, place them in a footnote or appendix as robustness information—do not let them replace the discussion of raw coefficients.

6.4 Pitfall 4: Equating "Explanatory Power" with "Causal Importance"

Shapley value decomposition tells you how much of R² each variable contributes. But R² analyzes correlation and prediction—a variable may receive a high Shapley value because of its strong correlation with Y, but this does not mean it has a large causal effect on Y.

Classic example: "Mother's years of education" may receive a non-negligible Shapley value in a wage equation—because it is highly correlated with one's own education, which in turn affects wages. But the causal effect of "mother's years of education" on wages may be very weak—its high Shapley value arises because it is indirectly associated with wages through "one's own education," and in the variance decomposition this indirect association is counted as the mother's "contribution."

7. Summary

Five core takeaways for comparing variable importance:

1. Directly comparing raw coefficients is nonsense when units differ, and incomplete even when units are the same. Identical units are only the minimum threshold—if the sample ranges of variation differ across variables, the coefficient magnitudes are not comparable.

2. "Importance" is a concept that needs definition—different methods define it differently and yield different rankings. Standardized coefficients define "standardized effect under a typical change," Shapley values define "average marginal contribution to R²," and dominance analysis defines "consistent advantage across submodels." Choosing a method = choosing a definition of "what importance means."

3. Standardized coefficients are the simplest and the most easily misused method. They bring different units onto the same "standard deviation" track, but at the cost of making the coefficient interpretation unintuitive and incomparable across samples.

4. Shapley decomposition and dominance analysis provide a more complete "picture of importance"—but they still measure statistical association, not causal effects. If your research goal is causal inference, discussions of "importance" require greater caution.

5. t-statistics and p-values measure "significance" and "precision"—not "importance" and "magnitude." Using t-statistics to argue who is more important is like answering a question that has nothing to do with the t-statistic.

A closing thought:

"Directly comparing the magnitudes of two coefficients is like looking at two numbers and saying '3 is smaller than 50'—but what if 3 is in units of '100 million yuan' and 50 is in units of 'yuan'? Standardized coefficients help you unify the units, but they merely swap one ruler for another—they do not answer whether 'this ruler is reasonable.' Shapley decomposition asks a different question: 'When you all jointly explain this outcome, how should the credit be divided?'—it is fairer, but it still does not measure causation. Before comparing variable importance, ask yourself: which layer of 'importance' am I actually trying to answer?"

8. Presentation Suggestions for Bilibili/WeChat Official Account

• Bilibili video: Consider using an "awards ceremony" scenario as the narrative framework. Opening: three variables (education, experience, tenure) stand on the podium, and the host (you) announces the "Most Important Variable of the Year" award. A member of the audience (a reviewer) raises a hand in objection: "Wait—they are measured with different rulers—you are comparing apples to oranges!" The scene then cuts backstage, where the three variables measure themselves with three different rulers—education's ruler is marked in "years (0–22)," experience's ruler in "years (0–45)," and tenure's ruler in "years (0–40)." Although the scale names are the same, each ruler's "typical notch" differs (shown with bar charts of standard deviations). The host switches to a unified ruler (standardization) and re-announces the results—but the audience says again: "This ruler has different markings in different samples!" Finally, the Shapley value approach is used—each variable "takes turns entering" all submodels, and the average incremental R² from each entry is computed—this is the truly fair basis for the award. Present the core message with three "award titles"—the "Typical Change Award" for standardization, the "Contribution Share Award" for Shapley, and the "Consistent Advantage Award" for dominance analysis—with narration: "Three different awards, three different criteria—none is the uniquely correct definition of 'importance.'"
• WeChat official account: The comparison table of the four methods (what question they answer, how to compute, advantages, disadvantages, Stata commands) is recommended as the central infographic. The "can we compare?" flowchart for Conditions 1/2/3 should be made into a decision-tree diagram. The numerical example for "same units ≠ directly comparable" (raw coefficients vs. standardized coefficients for education vs. tenure) should include a detailed calculation expansion. Pitfall 4 should be made into a "common error card." For the Shapley value decomposition intuition of "averaging over all permutations," consider a simplified three-variable permutation tree diagram.
• Recommended titles:
  • Main title: "Can You Directly Compare Coefficient Magnitudes Across Different Variables in Empirical Analysis?—Three Methods, Three Answers"
  • Alternative title: "0.08 and 0.03—Can These Two Coefficients Be Directly Compared in Magnitude?"
  • New-media title: "Ranking the 'Importance' of Regression Coefficients: Why You Cannot Conclude Just by Looking at Coefficient Magnitudes"
• Key quotes:

  "Directly comparing the magnitudes of two coefficients is equivalent to assuming that '1 unit' represents an equally important change for all variables—and this assumption almost never holds in real data."

  "Standardized coefficients help you convert apples and oranges into 'numbers of standard deviations'—but different orchards have different standard deviations, so your conversion ratio applies only to the current batch."

  "The Shapley value asks: across all possible orders of variable entry, how much additional variation in Y can you explain on average? This is not about 'who is more causal,' but 'who contributes more explanatory power statistically.'"

  "t-statistics and p-values measure 'whether you caught a fish'—the magnitude of the coefficient measures 'how big the fish is.' You cannot claim the fish in the net is bigger just because this cast was more precise (a larger t-statistic)."