How to read regression results: coefficients, uncertainty and fit

To read regression results, begin with the outcome, sample and units. Then interpret the coefficients alongside their uncertainty, examine model fit and ask whether the research design supports the conclusion. Stars beside an estimate answer only a small part of that exercise.

This article focuses on interpreting a table you already have. For the model and a runnable fitting example, start with what is regression analysis?.

Read the table note first

Before interpreting a number, establish:

• What is the dependent variable, and is it measured in levels, logs or percentages?
• Which observations entered the model, and which were excluded?
• What controls, fixed effects or weights were used?
• How were standard errors calculated?
• Which confidence level and reference distribution were used?

A table without these details leaves key parts of the result undefined. A change in the sample can matter as much as a change in the estimator.

A hypothetical regression table

The following numbers are invented for interpretation practice. They are not findings about education or a fitted OpenEconometrics result. Assume a wage regression on 1,000 hypothetical workers, with hourly wage in dollars, education and experience in years, and large-sample normal intervals.

For workers with the same experience in this model, an additional year of education is associated with $1.50 higher hourly wage on average. Four additional years correspond to a fitted difference of $6.00. That comparison holds experience fixed; it does not describe two otherwise identical real people.

The experience estimate is $0.10 per hour per additional year. Its interval includes zero and also includes positive associations that might matter. “No statistically significant association at 5%” is more precise than “experience has no relationship with wages.”

The intercept predicts wage at zero education and zero experience. If that combination is outside the observed data, its substantive meaning is limited.

Read precision alongside magnitude

A standard error describes uncertainty in an estimate under the model's sampling and covariance assumptions. It has the same units as the coefficient. In the hypothetical education row, 1.50 divided by 0.25 gives a test statistic of 6 for a zero-coefficient null.

The displayed interval uses 1.50 ± 1.96 × 0.25, giving 1.01 to 1.99. A small-sample t interval or a different covariance estimator can change the endpoints. Penn State's regression-inference lesson explains the estimate-plus-or-minus-multiplier construction.

A 95% confidence procedure covers the fixed population coefficient in 95% of repeated samples when its assumptions hold. It is not a statement that 95% of individual workers have an effect inside this interval. An interval for a conditional mean also differs from a prediction interval for a new individual observation.

Keep p-values in their place

A p-value assesses how unusual the test statistic would be under a specified null model and its assumptions. It is not the probability that the null hypothesis is true, or the chance that the finding happened “by accident”.

The American Statistical Association's statement emphasises that statistical significance does not measure effect size or practical importance. Report the estimate and interval, and explain their units. For a fuller discussion, see p-values in economics.

Compare models on a common basis

For ordinary least squares with an intercept, R-squared reports the share of in-sample outcome variation accounted for by the fitted model. It does not measure causal credibility or future forecast accuracy. Penn State's introductory regression lesson explains these cautions.

Suppose one hypothetical table uses 1,000 workers and another uses 650 after adding a variable with missing observations. A coefficient change could come from the added control, the changed sample or both. Refit the simpler specification on the same 650 observations to separate those explanations.

Also keep the outcome transformation consistent. An R-squared for wages and one for log wages do not describe variation in the same quantity. For prediction, compare errors on held-out observations with the same target and evaluation period.

Finish with a defensible sentence

A suitable report of the hypothetical education row is: “In this sample and specification, one additional year of education is associated with $1.50 higher hourly wage, holding experience constant; the approximate 95% interval is $1.01–$1.99.”

A claim that schooling caused that increase needs an identification argument addressing confounding and selection. A precise coefficient cannot supply that argument by itself. Link the table back to the question, the causal assumptions and the standard-error choice.

References

• Penn State Department of Statistics. STAT 501, Lesson 2: SLR model evaluation.
• Penn State Department of Statistics. STAT 501, Lesson 1: Simple linear regression.
• American Statistical Association (2016). Statement on statistical significance and p-values: release and principles.