Econometrics Mini-Course: Foundations
OLS intuition, variable handling, endogeneity, discrete & truncated models
All Models, but What's the Difference Between Econometric Models, Mathematical Models, and Economic Models?
You've probably heard phrases like: \"Your model isn't accurate,\" \"This model's assumptions are too idealized,\" or \"Let's run the model on the data.\"
What Is One Actually Doing When Conducting Empirical Research?
\"Is this conclusion empirically supported?\" \"We need to conduct empirical research.\" \"This paper is purely theoretical and lacks empirical testing.\"
With So Many Data Classifications, Do We Need to Learn Them All? What Are They For?
For beginners in econometrics, the first wave of shock often comes from the way data are classified.
When You Receive a Dataset, How Should You Scrutinize Its Representativeness and Validity from a Sampling Perspective?
Anyone who has taken an econometrics methods course has likely studied sampling—simple random sampling, stratified sampling, cluster sampling, systematic sampling, and PPS sampling, memorizing the formulas and characteristics of each. For exams, you know stratified sampling reduces variance and how to calculate the design effect for cluster sampling.
Everyone Cares About Causality, So What Is the Point of "Correlational" Data Analysis?
If you follow methodological debates in the social sciences in recent years, you will surely feel a powerful trend—causal inference has come to dominate the discourse almost entirely.
How Should We Understand \"Ceteris Paribus\" in Multiple Regression?
Open any empirical paper that uses multiple regression, and you will most likely see this sentence in the section interpreting the coefficients:
Potential Outcomes Framework vs. Multiple Regression: How Do the Two "Causal" Languages Differ? Which Is More Credible?
If you read both types of papers, you will find that the term 'causal relationship' is used in two seemingly completely different contexts—
Why Does OLS Produce Just One Number, Yet We Call It \"Unbiased\" and \"Efficient\"?
Anyone who has studied introductory econometrics has experienced this cognitive moment—
What Is the Relationship Between Regression Coefficients, Partial Regression Coefficients, and Correlation Coefficients?
Students of econometrics typically encounter these three concepts at three different points in time and from three different directions:
What Is the Relationship Between \"Residuals\" and \"Errors\" in a Linear Regression Model?
If you flip through the chapters on linear regression in any econometrics textbook, you will repeatedly encounter two terms that look almost identical but have completely different meanings—
Variance Decomposition in Regression Models: What Lies Behind R² and the F-test?
Variance Decomposition in Regression Models: What Lies Behind R² and the F-test?
Why Are Most Regression Coefficients in Empirical Research Between -1 and 1?
Open the empirical section of any economics journal, scan the regression tables, and you will observe a pattern—
What Is \"Economic Significance\"? How Does It Differ from Statistical Significance?
If you have ever sat in on an economics seminar, you have surely seen this scene—
In Panel Data Regression, What Do Within R², Between R², and Overall R² Mean?
You ran a panel data fixed effects model in Stata:
How to Understand Linear Regression? Are \"Linear in X\" and \"Linear in Parameters\" the Same Thing?
\"Linear regression\" — at first glance, the name seems self-evident — isn't it just drawing a straight line?
The Magic of Degrees of Freedom: Why Are OLS Residuals n-2? Where Do the Degrees of Freedom for t-tests and F-tests Come From?
If you open any econometrics textbook, degrees of freedom (df) appear as a 'ghost parameter' in almost every chapter—
When We Say \"OLS Estimators Are Unbiased,\" What Do We Actually Mean?
Beginners in econometrics often feel a strange sense of reassurance after learning about the unbiasedness of OLS.
How to Choose Control Variables in Empirical Research? — Not Everything That Affects Y Belongs in the Model
In methodological discussions of empirical research, one question is asked with remarkable frequency yet answered systematically with remarkable rarity—
What Is the Relationship Between R² and Adjusted R²? Why Can Adjusted R² Sometimes Be Less Than 0?
Everyone who has run a regression has seen these two numbers side by side in Stata output:
What Is Multicollinearity? How to Detect, Test, and Address It?
Suppose you run a wage regression with years of education as the key variable. All coefficients look normal. Then a reviewer suggests you 'add another control variable'—you do so, adding 'mother's years of education.'
Omitted Variable Bias: If We Know a Variable Is Omitted, Why Not Include It? How Can We Analyze Its Impact?
In the previous article, we distinguished between multicollinearity and omitted variable bias—the former makes estimates \"blurry,\" while the latter makes estimates \"wrong.\" Between the two, omitted variable bias (OVB) is the most fundamental and intractable problem in econometrics.
What Is the Variance Inflation Factor (VIF)? How to Use It to Diagnose Multicollinearity?
Suppose you use three independent variables—years of education, work experience, and age—to explain wages. Intuitively, these three variables must be related: older people typically have more work experience. But specifically, what impact does this \"correlation among independent variables\" have on your regression estimates?
What is the relationship between endogenous and exogenous explanatory variables? When should you worry about endogeneity?
$$ \\text{Wage} = \\beta0 + \\beta1 \\cdot \\text{Education} + \\beta2 \\cdot \\text{Experience} + \\varepsilon $$
What is the FWL Theorem? How to Understand It Intuitively? What Important Methods Can It Help You Understand?
Everyone who has studied econometrics will say one thing: \"In multiple regression, the coefficient is the partial effect of X on Y, holding other variables constant.\"
How to Interpret Coefficient Confidence Intervals in Stata Regression Output? What Is Their Relationship with p-values?
wage | Coefficient Std. err. t P|t| [95% conf. interval] + education | .0800000 .0100000 8.00 0.000 .060399 .0996001 ...
In Empirical Analysis, Can Data Be Standardized? How Do You Interpret Coefficients After Standardization? Why Do Researchers Prefer Taking ln?
After running your regression, your Stata output shows two coefficients:
When Should You Add a Quadratic Term in Empirical Research? How to Interpret It? How to Justify Its Inclusion?
You study the effect of income on happiness. Your intuition is: when poor, an extra dollar greatly boosts happiness; but beyond a certain level of wealth, additional money has little effect on happiness—perhaps even slightly reducing it due to comparison and stress. This implies that the relationship between income and happiness is not a straight line—but a curve that first rises then falls (or first rises then flattens)...
When Should Moderation Effects Be Considered in Empirical Analysis? Why Can Interaction Terms Examine Moderation? How Should They Be Interpreted?
You ran a standard Mincer wage equation:
What Are Residuals Used for in Empirical Research? — From Model Validation to Special Problem Analysis
You run your regression, glance at the R², coefficients, p-values, and significance stars—then copy the results table into your paper and close Stata.
Standard Deviation, Standard Error, Standardization—All So 'Standard,' What Exactly Are They Talking About?
If you learned statistics and econometrics in a Chinese context, you must have experienced this subtle confusion—
In Empirical Analysis, Can You Directly Compare the Magnitudes of Coefficients Across Different Variables? When Can You? How Should You Compare Correctly?
lnwage = 0.080 × education + 0.030 × experience + 0.120 × tenure + controls
Must Dummy Variables Be One Fewer Than the Number of Categories? When Are Exceptions Allowed? Is This the Same as One-Hot Encoding in Machine Learning?
You run a regression in Stata that includes \"region\" (East/Central/West, three categories):
How to Understand Interactions of Dummy Variables? How Does This Understanding Map to Difference-in-Differences (DID)?
The general meaning of an interaction term is that the effect of X on Y varies with the value of a third variable M. When M is continuous (e.g., income), the interaction term tells you \"how the marginal effect of X changes at different levels of M.\" But when M is a dummy variable (e.g., gender), the interpretation of the interaction term has a more intuitive and concrete geometric understanding...
Can You Run OLS Directly with a Dummy Variable as the Dependent Variable? — The 'Three Sins' of the Linear Probability Model and Why People Still Use It
You are studying 'what kind of students choose to take the postgraduate entrance exam.' Your dependent variable is kaoyan—taking the exam = 1, not taking it = 0. You have a set of explanatory variables: GPA, school tier, family income, gender……
Heteroskedasticity: Whose Variance Differs? How Does It Manifest Differently in Cross-Sectional vs. Panel Data? How to Test and Address It?
You study \"the effect of education on personal income.\" You run an OLS regression:
Using Lagged Variables as Instrumental Variables—When Is It Justified, and When Is It Self-Deception?
You are studying the effect of FDI on economic growth. You know FDI is endogenous—countries with faster growth also attract more foreign capital, so Cov(FDI, ε) ≠ 0. You need an instrumental variable.
Unbalanced Panels and Missing Values—When to Drop? When to Impute? How to Impute Properly?
You are using five years of CGSS tracking survey data. The original sample has 8,000 people. You run a panel fixed effects model, and Stata tells you \"Number of obs = 24,000\"—but you clearly have 8,000 × 5 = 40,000 possible observations. The remaining 1...
Outliers in Panel Data—How to Define Them? What Is the Relationship Between Winsorizing and Trimming? How to Choose? How to Check for Adverse Effects?
You study the effect of education on individual annual income. Your sample has 10,000 people—9,999 of whom earn between 0 and 1 million per year. But one person earns 50 million.
What Is a Non-Random Sample? What Is the Difference Between Endogenous and Exogenous Selection? What Should You Do with a Non-Random Sample?
You use CGSS (Chinese General Social Survey) to study the effect of education on personal income. CGSS is a nationwide sample survey—theoretically, it is a random sample of the population. But think more carefully:
Dependent Variable with Many Zeros—Can You Just Run OLS? What Happens If You Drop the Zeros? What Is the Established Practice in Empirical Work?
You study \"the number of patents held by firms.\" In your sample of 10,000 firms, 7,000 have zero patents in every year—the dependent variable patent = 0. The remaining 3,000 have patent counts ranging from 1 to 5,000.
Adding Lagged Terms of Independent Variables in Panel Data—When Should You? When Shouldn't You? And What Can You Uncover Once You Do?
You study 'the effect of firm R&D investment on patent output.' You run a panel fixed effects model:
Controlling for Time Trends—Add a Single t or a String of Year Dummies? How Should t Be Coded?
You are studying the effect of urbanization rate on household consumption. You use provincial panel data from 2000 to 2020. You run a regression:
What Is Spurious Regression? When Does It Occur? What Are the Signs? How Can It Be Avoided?
Granger and Newbold conducted a famous simulation experiment in 1974. They generated two completely independent random walk processes:
When Is Seasonal Adjustment Needed? Does Panel Data Require It? How to Determine and Implement It?
You are using quarterly GDP data—Q1, Q2, Q3, Q4. You plot a time-series chart and discover that GDP exhibits a regular \"sawtooth\" pattern every year: Q1 is always the annual trough (Spring Festival shutdown), and Q4 is always the annual peak (year-end push + holiday consumption).
Panel Data Contains Time-Series Information—Why Not Routinely Detrend and Deseasonalize?
You have finished learning time-series econometrics—unit root tests, cointegration, detrending, seasonal adjustment. Armed with this toolkit, you open a provincial panel dataset—30 provinces, 1998–2020, one observation per year.
In Empirical Research, How Do You Identify and Diagnose Autocorrelation? What Impact Does It Have on OLS? How Should You Adjust?
You ran a time series regression—using annual macro data from 1980–2020 to study \"the effect of money supply on inflation.\" Your model is:
Panel Data Also Contains Time-Series Information—Why Do Empirical Studies Often Not Need to Consider Whether the Process Is Stationary?
You have just finished time-series econometrics. You learned the ADF test, the KPSS test, unit roots, orders of integration, difference stationarity vs. trend stationarity, cointegration, and error correction models. You remember the classic warning of Granger and Newbold (1974): when two independent random walks are regressed in levels, the spurious significance rate can exceed 70%...
Textbooks Cover Both FE and RE, and the Hausman Test Exists—So Why Do Most Empirical Studies Just Use Fixed Effects Models Directly?
Open the panel data chapter of any intermediate econometrics textbook, and you are sure to encounter this standard narrative arc:
What Categories Do Robust Standard Errors Commonly Refer to in Empirical Work? When Is Each Applicable? How Should You Choose and Justify Your Choice?
You run a simple regression—using firm panel data to study \"the impact of R&D investment on firm productivity.\" You write three lines of code in Stata and obtain three very different sets of t-values:
The Two Core Requirements of Instrumental Variables: How to Argue Relevance and Exogeneity? What Tests Are Needed? From What Angles Should the Argument Be Constructed?
You are studying the effect of education on income. You know that OLS has a classic endogeneity problem—ability ($u_i$) affects both educational choices and income levels, and omitting ability biases the coefficient on years of education. You need an instrumental variable.
Using \"Higher-Level\" Variables as Instruments — The Exogeneity Dilemma of Industry-Mean IVs and Strategies for Justification
You are studying the effect of corporate digital transformation on productivity. Digitalization is endogenous—high-productivity firms may be more inclined to invest in digitalization, and you cannot tell whether digitalization drives productivity or productivity drives digitalization.
Do Empirical Studies Require Rigorous Statistical Tests to Examine Variable Endogeneity?
You have finished the endogeneity chapter in econometrics. You learned about the Durbin-Wu-Hausman (DWH) test—first run an instrumental variables regression, then compare the difference between the IV and OLS estimates. If the difference is significant → OLS has endogeneity bias → you should use IV. If the difference is not significant → you cannot...
From Linear Regression to Logistic Models—Core Principles, Transitional Reasoning, and Economically Exploitable Implications
You study 'whether an individual starts a business'—Y = 1 indicates entrepreneurship, Y = 0 indicates not. You want to predict an individual's probability of entrepreneurship using age, education, gender, and family wealth.
Binary Dependent Variables—Logit or Probit? What's Special in Panel Data?
You ran a binary choice model—studying \"whether firms export.\" You first used Logit:
Dependent Variable Is Positive—Panel OLS or Tobit? What's the Difference? When Should You Switch?
You are studying the determinants of firm R&D investment. Your dependent variable is firm R&D expenditure—many firms have zero R&D expenditure in a given year (Y = 0), while positive R&D expenditure values span an enormous range (from a few thousand yuan to several billion yuan).
Truncation and Truncated Regression—What Problems Do They Address? Consequences of Misusing OLS? Minimal Necessary Knowledge and a Decision Framework
You study 'the determinants of wages.' You use a micro-survey dataset—but it only contains people who are currently employed. You have no wages for the unemployed (they have no wages), nor information on those who have exited the labor force.
Descriptive Statistics Tables—Which Variables to Include? What Summary Statistics to Report? Raw or Transformed Variables?
In your empirical paper, the first table is usually the \"Descriptive Statistics\" (Summary Statistics) table. You include the dependent variable, the key independent variable, a few control variables, run a summarize command, and paste the output into a table. You may think...
Why Sample Sizes Often Vary Across Regression Models in Empirical Analysis—Causes, Red Flags, and Reviewer Concerns
Your paper has four regression tables. Each table has a different sample size:
Popular articles (by visits)
- Variance Decomposition in Regression Models: What Lies Behind R² and the F-test?
Variance Decomposition in Regression Models: What Lies Behind R² and the F-test?
27 visits - In Empirical Analysis, Can Data Be Standardized? How Do You Interpret Coefficients After Standardization? Why Do Researchers Prefer Taking ln?
After running your regression, your Stata output shows two coefficients:
27 visits - What Is the Variance Inflation Factor (VIF)? How to Use It to Diagnose Multicollinearity?
Suppose you use three independent variables—years of education, work experience, and age—to explain wages. Intuitively, these three variables must be related: older people typically have more work experience. But specifically, what impact does this \"correlation among independent variables\" have on your regression estimates?
25 visits - Textbooks Cover Both FE and RE, and the Hausman Test Exists—So Why Do Most Empirical Studies Just Use Fixed Effects Models Directly?
Open the panel data chapter of any intermediate econometrics textbook, and you are sure to encounter this standard narrative arc:
23 visits - What Is the Relationship Between Regression Coefficients, Partial Regression Coefficients, and Correlation Coefficients?
Students of econometrics typically encounter these three concepts at three different points in time and from three different directions:
20 visits - Adding Lagged Terms of Independent Variables in Panel Data—When Should You? When Shouldn't You? And What Can You Uncover Once You Do?
You study 'the effect of firm R&D investment on patent output.' You run a panel fixed effects model:
20 visits - How to Choose Control Variables in Empirical Research? — Not Everything That Affects Y Belongs in the Model
In methodological discussions of empirical research, one question is asked with remarkable frequency yet answered systematically with remarkable rarity—
19 visits