Common mistake 1 with Multiple regression (OLS)

Chasing a high R². You can always raise it by adding variables; it says nothing about whether your estimate is causal.

Common mistake 2 with Multiple regression (OLS)

Reading "not significant" as "no effect". It means the data cannot rule out zero.

Common mistake 3 with Multiple regression (OLS)

Interpreting a coefficient causally when an omitted variable is obviously lurking. Robust standard errors do not fix bias - nothing about the standard error fixes bias.

Common mistake 4 with Multiple regression (OLS)

Including a variable that is itself an outcome of your treatment. That is a "bad control" and it biases everything.

EconStatsCraft › Multiple regression (OLS)

Multiple regression (OLS)

Ordinary least squares with your choice of standard errors, plus the full diagnostic battery: Breusch–Pagan, White, Breusch–Godfrey, Durbin–Watson, RESET, Jarque–Bera and VIF.

What it needs

Dependent variable - The outcome you are explaining.
Independent variables - Text columns become dummy variables automatically.

The intuition

OLS chooses the coefficients that leave the residuals orthogonal to every regressor - no linear information about y is left unused. The multivariate coefficient has a precise meaning captured by the Frisch–Waugh–Lovell theorem: β̂ⱼ is what you get by stripping the other regressors out of BOTH y and xⱼ, then regressing one residual on the other. "Holding constant" is not a metaphor; it is a residual.

When to use it

The default starting point for a continuous outcome. Everything else in econometrics is a repair for a specific way OLS breaks.

Where the estimator comes from

Minimise S(\beta)=(y-X\beta)'(y-X\beta). The first-order condition -2X'(y-X\hat{\beta})=0 gives the normal equations X'X\hat{\beta}=X'y, hence \hat{\beta}=(X'X)^{-1}X'y whenever X has full column rank. Substituting y=X\beta+u gives \hat{\beta}=\beta+(X'X)^{-1}X'u - so the entire question of bias reduces to whether E[X'u]=0.

The model

y_i=\beta_0+\beta_1x_{1i}+\dots+\beta_kx_{ki}+u_i \qquad \hat{\boldsymbol{\beta}}=(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{y}

What it assumes

Linear in the parameters (you can still use logs and squares of the variables).
The regressors are not perfectly collinear.
E[u | X] = 0 - no omitted variable that is correlated with a regressor. This is the one that matters, and it is untestable.
Homoskedastic errors - needed only for the classical standard errors, and easily fixed with HC3.
No serial correlation - matters for time series; fixed with HAC.
Normal errors - needed only for exact small-sample inference.

Statistical properties

Unbiased: E[β̂ | X] = β requires only E[u | X] = 0. Nothing about variance shape.
Gauss–Markov: adding homoskedasticity and no autocorrelation makes OLS BLUE - best in the class of linear unbiased estimators.
Consistent and asymptotically Normal under far weaker conditions than the finite-sample results need.
Var(β̂) = σ²(X′X)⁻¹ under homoskedasticity; the White/Huber sandwich (X′X)⁻¹(X′ΩX)(X′X)⁻¹ replaces it otherwise.
Heteroskedasticity and autocorrelation break EFFICIENCY and the standard errors, never unbiasedness. Endogeneity breaks unbiasedness, and no standard error can repair it.

How to read the output

Coefficient: the effect of a one-unit change, holding the other regressors constant.
If both y and x are in logs the coefficient is an elasticity: a 1% rise in x gives a β% change in y.
If only y is logged, multiply by 100 to read the coefficient as a percentage change.
p-value < 0.05 means you can reject "this coefficient is zero" at the 5% level.
R² is the share of variance explained. A low R² is not a failure - in cross-section 0.3 is normal.
The F-statistic tests whether all your slopes are zero together.

A worked reading

β̂_educ = 0.0791 (SE 0.0036, HC3). Because the dependent variable is log wage, read it as: one more year of schooling is associated with about 7.9% higher wages, holding experience, sex and union status fixed. "Associated with" is doing real work - it is causal only if nothing omitted moves both schooling and wages, which is exactly why the IV module exists.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Breusch–Pagan or White rejects	Heteroskedastic errors	HC3 robust standard errors - coefficients are already fine
Breusch–Godfrey rejects (time series)	Serially correlated errors	HAC/Newey–West errors, or model the dynamics with lags of y
RESET rejects	Functional form is wrong	Add logs, squares or interactions - a robust SE will not save you
VIF above 10	Near-collinearity	Individual coefficients unreliable; joint tests and predictions still fine
R² very high with trending time series	Likely a spurious regression	Test for unit roots first; difference or use cointegration
Coefficient flips sign when a control is added	Confounding, or a bad control	Ask whether the new variable is a cause or an outcome of your regressor

Common mistakes

Chasing a high R². You can always raise it by adding variables; it says nothing about whether your estimate is causal.
Reading "not significant" as "no effect". It means the data cannot rule out zero.
Interpreting a coefficient causally when an omitted variable is obviously lurking. Robust standard errors do not fix bias - nothing about the standard error fixes bias.
Including a variable that is itself an outcome of your treatment. That is a "bad control" and it biases everything.

Where to read this properly

Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Cengage, chapters 3-8
Angrist and Pischke (2009), Mostly Harmless Econometrics: An Empiricist's Companion, Princeton University Press, chapter 3
Frisch & Waugh (1933) and Lovell (1963) - the partialling-out theorem
White (1980) - the heteroskedasticity-robust covariance matrix

Related Regression

Logit / Probit · Tobit (censored) · Truncated regression · Multinomial logit · Ordered logit / probit · Quantile regression · Poisson / Negative binomial

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Data & variables

Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

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Drag a variable into a slot on the right - or just click it.

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and

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Rules

Base period = 100

Which part

Percent at each tail

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Multiple regression (OLS)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Regression

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