Lesson M04.L05: Reporting Regression Results
| Field | Detail |
|---|---|
| Module | Module 04 — Statistical Inference in OLS |
| Level | Graduate (ECMT5001, S1 2027) |
| Duration | 35 min |
| Learning Objective | Write up regression results in correct academic style; read and produce standard output tables; avoid common reporting errors; apply asymptotic justifications for dropping MLR.6 in large samples |
| Provenance | Wooldridge, Introductory Econometrics, 7e, Chapters 4–5 (pp. 117–200) |
Explanation
Standard Regression Output Table
Software such as R, Stata, and Python (statsmodels) produces a standard output format. Here is a representative table with annotations:
Dependent variable: log(wage)
Observations (n): 526
─────────────────────────────────────────────────────────────────
Coeff. Std. Err. t-stat p-value
─────────────────────────────────────────────────────────────────
Intercept 0.284 0.104 2.73 0.007 **
educ 0.092 0.007 13.14 <0.001 ***
exper 0.004 0.001 4.00 <0.001 ***
female −0.297 0.036 −8.25 <0.001 ***
─────────────────────────────────────────────────────────────────
R²: 0.316
Adjusted R²: 0.312
F-statistic: 80.7 (df = 3, 522) p-value: <0.001
─────────────────────────────────────────────────────────────────
Significance: * p<0.10 ** p<0.05 *** p<0.01
Column guide: | Column | Meaning | |---|---| | Coeff. | OLS estimate \(\hat{\beta}_j\) | | Std. Err. | \(se(\hat{\beta}_j) = \hat{\sigma}/\sqrt{SST_j(1-R_j^2)}\) | | t-stat | \(\hat{\beta}_j / se(\hat{\beta}_j)\) | | p-value | Two-sided: \(2 \times P(T > |t|)\) where \(T \sim t_{n-k-1}\) | | Stars | Summary indicator of evidence against \(H_0: \beta_j = 0\) |
The Adjusted \(R^2\)
The standard \(R^2 = 1 - SSR/SST\) always increases (weakly) when you add regressors, even irrelevant ones. The adjusted \(R^2\) penalises for additional regressors:
where \(s_y^2 = SST/(n-1)\) is the sample variance of \(y\). \(\bar{R}^2\) can decrease when you add a regressor if its contribution to fit is outweighed by the loss of one degree of freedom. Use \(\bar{R}^2\) (not \(R^2\)) when comparing models with different numbers of regressors.
The Overall F-Statistic
The overall F-statistic tests \(H_0: \beta_1 = \cdots = \beta_k = 0\) (all slope coefficients jointly zero). From Lesson M04.L04:
A large F (with small p-value) tells you the regression explains significantly more variation than a model with only the intercept. This is a minimum bar for any regression model — if overall \(F\) is insignificant, the model as a whole adds nothing.
How to Write Up Results: Academic Style
The standard form for reporting a single coefficient in text is:
"A one-unit increase in \(X_j\) is associated with a \(\hat{\beta}_j\)-unit change in \(Y\) (SE \(= se(\hat{\beta}_j)\), \(t = t_j\), \(p \leq \alpha\)), holding all other variables constant."
For log-transformed dependent variables (where coefficients are approximate percentage changes):
"An additional year of education is associated with an approximately 9.2% increase in hourly wages (SE \(= 0.007\), \(t = 13.14\), \(p < 0.001\)), controlling for experience and gender."
Key elements of a complete write-up: 1. State the dependent variable and its units (or log units). 2. Report \(\hat{\beta}_j\), its SE, t-statistic, and p-value (or CI). 3. Interpret the coefficient in context: direction, magnitude, units. 4. State what is held constant (ceteris paribus). 5. Comment on economic significance — not just statistical significance. 6. Report overall model fit: \(n\), \(R^2\) (or \(\bar{R}^2\)), and overall F-statistic.
Common Reporting Errors to Avoid
| Error | Why it's wrong | Correct practice |
|---|---|---|
| Reporting stars without SEs or t-stats | Stars alone give no sense of precision | Always report \(\hat{\beta}_j\) and \(se(\hat{\beta}_j)\) |
| Omitting sample size \(n\) | Effect sizes and precision are meaningless without \(n\) | Always state \(n\) |
| Confusing log-coefficient interpretation | \(\hat{\beta}_j\) in log-level model is not a "unit change" | State it as "approximately \(100\hat{\beta}_j\%\)" |
| Using \(R^2\) to compare models with different \(k\) | \(R^2\) always rises with more regressors | Use \(\bar{R}^2\) for model comparison |
| Claiming causality from observational OLS | OLS identifies association, not causation, unless exogeneity holds | Use "associated with" or "predicts", not "causes" |
| Over-reporting stars: "more stars = more important" | Stars only measure evidence against \(H_0: \beta_j = 0\), not magnitude | Discuss coefficient size and economic relevance separately |
Asymptotic Justification: Large Samples Without MLR.6
MLR.6 (normality of \(u\)) is required for the exact t and F distributions to hold. In practice, most economic data (wages, prices, returns) do not have normally distributed errors.
The asymptotic justification (MLR.1–MLR.5 only):
Under MLR.1–MLR.5 (no normality required), as \(n \to \infty\):
by the CLT (since \(\hat{\beta}_j\) is a sample average of terms involving \(u_i\), which have finite variance by MLR.5). Therefore, for large \(n\):
- t-statistics are asymptotically standard normal → t-critical values are approximately valid.
- F-statistics divided by \(q\) are asymptotically chi-squared → F-critical values are approximately valid.
- The normal approximation improves as \(n\) grows.
Practical implication: The \(t\) and \(F\) inference tools you learned are valid for large samples even without the normality assumption. For small samples (say \(n < 50\)), the validity of inference depends more heavily on whether errors are approximately normal — check residual plots.
Worked Example
Setup. A researcher estimates a wage regression on \(n = 526\) CPS workers. Write up the results in academic report style.
The model: $\(\log(\text{wage}_i) = \beta_0 + \beta_1\,\text{educ}_i + \beta_2\,\text{exper}_i + \beta_3\,\text{female}_i + u_i\)$
OLS output:
| Variable | \(\hat{\beta}_j\) | \(se\) | \(t\) | \(p\) |
|---|---|---|---|---|
| Intercept | 0.284 | 0.104 | 2.73 | 0.007 |
| educ | 0.092 | 0.007 | 13.14 | <0.001 |
| exper | 0.004 | 0.001 | 4.00 | <0.001 |
| female | −0.297 | 0.036 | −8.25 | <0.001 |
\(n = 526\), \(R^2 = 0.316\), \(\bar{R}^2 = 0.312\), \(F_{3,\,522} = 80.7\) (\(p < 0.001\)).
Academic write-up:
Table 1 presents OLS estimates of the determinants of log hourly wages for \(n = 526\) workers from the 2019 Current Population Survey. The overall model is statistically significant (\(F_{3,\,522} = 80.7\), \(p < 0.001\)) and explains approximately 31.6% of the variation in log wages (\(R^2 = 0.316\); adjusted \(R^2 = 0.312\)).
Education has a precisely estimated positive return: each additional year of schooling is associated with approximately a 9.2% increase in hourly wages (SE \(= 0.007\), \(t = 13.14\), \(p < 0.001\)), holding experience and gender constant. This estimate is both statistically significant at the 1% level and economically meaningful — a four-year university degree is associated with roughly a 37% wage premium over a high school diploma (\(\approx 4 \times 9.2\%\)).
Experience also has a significant positive effect: each additional year of experience is associated with approximately a 0.4% increase in wages (SE \(= 0.001\), \(t = 4.00\), \(p < 0.001\)). While statistically significant, the magnitude is small relative to education, suggesting diminishing economic importance of raw experience after controlling for schooling.
Gender has a large and statistically significant coefficient: female workers earn approximately 29.7% lower log wages than male workers with equivalent education and experience (SE \(= 0.036\), \(t = -8.25\), \(p < 0.001\)). This is both statistically significant at the 1% level and economically substantial, representing a sizeable unexplained wage gap after controlling for observable human capital characteristics.
What this write-up does well: - Reports \(\hat{\beta}_j\), SE, \(t\), \(p\) for each coefficient. - Interprets log coefficients as approximate percentage changes. - States the ceteris paribus condition explicitly. - Comments on economic significance (magnitude, not just stars). - Reports \(n\), \(R^2\), \(\bar{R}^2\), and overall F. - Avoids causal language (uses "associated with", not "causes").
Common Misconception
"Three asterisks () means the variable is more important than one asterisk ()."**
Stars indicate statistical significance — the strength of evidence against \(H_0: \beta_j = 0\). They are primarily driven by sample size and precision, not by the economic importance or magnitude of the effect.
Example. Suppose a study with \(n = 100{,}000\) observations finds that a firm's energy efficiency rating is associated with 0.1% higher stock returns (\(t = 6.2\), \(p < 0.001\), ***). Meanwhile, a study with \(n = 80\) finds that a CEO's previous industry experience is associated with 15% higher returns (\(t = 1.9\), \(p = 0.06\), not significant at 5%).
The first finding gets three stars; the second gets none. But the second effect (if real) is 150 times larger. Stars measure evidence, not magnitude.
Correct practice: Interpret coefficients by their magnitude and economic meaning, not their star count. A coefficient can be precisely estimated as trivially small (many stars, no economic importance) or imprecisely estimated as large (few stars, potentially large economic importance).
Practice Prompts
Q1. A regression of house prices (\(000s) on square footage, number of bedrooms, distance to CBD, and a dummy for pool (\)k = 4$, \(n = 148\), \(df = 143\)) gives:
| Variable | \(\hat{\beta}\) | SE |
|---|---|---|
| sqft | 0.110 | 0.014 |
| bedrooms | 8.200 | 4.100 |
| dist_CBD | −3.500 | 0.850 |
| pool | 5.100 | 8.200 |
| Intercept | 32.5 | 18.3 |
\(R^2 = 0.682\), \(\bar{R}^2 = 0.673\), \(F_{4,143} = 76.8\) (\(p < 0.001\)).
Write a brief academic-style paragraph reporting these results, including comments on statistical and economic significance.
Answer
Table 1 presents OLS estimates of house prices for $n = 148$ observations ($df = 143$). The model explains 68.2% of price variation ($R^2 = 0.682$; $\bar{R}^2 = 0.673$) and is highly significant overall ($F_{4,143} = 76.8$, $p < 0.001$). Square footage has a positive and statistically significant effect: each additional square foot is associated with \$110 higher price (SE $= 14$, $t = 7.86$, $p < 0.001$), a large and economically meaningful effect. Distance to CBD has a significant negative effect: each additional kilometre from the city reduces price by approximately \$3,500 (SE $= 850$, $t = -4.12$, $p < 0.001$). Number of bedrooms shows a positive coefficient of \$8,200 (SE $= 4,100$, $t = 2.00$, $p \approx 0.047$), marginally significant at 5%. The pool premium is positive (\$5,100) but not statistically significant at conventional levels (SE $= 8,200$, $t = 0.62$, $p \approx 0.53$); the sample is likely underpowered to detect this effect.Q2. A researcher reports: "Education is highly significant (**) while experience is only marginally significant (). Therefore education matters more for wages than experience." Identify the flaw in this reasoning and explain what additional information would be needed.
Answer
The flaw is conflating statistical significance (which measures evidence against $H_0: \beta = 0$) with economic magnitude. The stars reflect the precision of the estimate relative to zero — not the size of the effect. To assess which variable matters more economically, the researcher should: 1. Compare the **magnitude** of the coefficients, standardised to a common scale (e.g., a one standard-deviation change in each variable: $\hat{\beta}_j \times sd(x_j)$). 2. Consider the realistic range of variation in each variable. If most workers have similar experience levels but widely varying education, education's total contribution to wage variation would be larger regardless of which coefficient is more "significant." Stars alone cannot determine economic importance.Q3. A student estimates a model with \(n = 800\) workers and reports that the model has \(R^2 = 0.44\) and \(\bar{R}^2 = 0.43\). They add three more regressors and find \(R^2 = 0.45\) but \(\bar{R}^2 = 0.42\). Should they prefer the extended model? What does this tell you about the three new variables?
Answer
No, they should prefer the original model. The rise in $R^2$ from 0.44 to 0.45 is mechanical — $R^2$ always weakly increases with additional regressors. The fall in $\bar{R}^2$ from 0.43 to 0.42 indicates the three new regressors do not contribute enough explanatory power to justify their cost in degrees of freedom. In other words, the improvement in fit is smaller than what one would expect by chance from adding three arbitrary variables. This also suggests that the three new regressors are likely jointly insignificant — a formal F-test for their joint significance would help confirm this. If $F < F_{3,\,794,\, 0.05} \approx 2.61$, we cannot reject that their true coefficients are all zero.Further Resources
- Wooldridge (2019), §4.2–4.5, §5.1–5.3: Inference and asymptotic theory
- Wooldridge (2019), §4.6: "Reporting Regression Results" (pp. 168–173)
- Angrist & Pischke (2009), Mostly Harmless Econometrics, §3.1.3: Regression anatomy and reporting
- Gelman & Hill (2007), Data Analysis Using Regression, Ch. 3–4: Practical guidance on writing up regression results
- AEA (2020), American Economic Review style guide: standard for coefficient tables in top journals