Regression Inference & Diagnostics Calculator
Extend simple OLS with an intercept using coefficient standard errors, t/F inference, confidence intervals and residual plots.
Intercept OLS · conventional standard errors · no robust or HAC estimates
Results
Review or adjust your inputs, then calculate to see your results and interpretation.
Understand this calculation
The fitted line describes the sample's linear pattern. Conventional standard errors quantify coefficient uncertainty under the model assumptions. Residuals are observed minus fitted values. The slope t usually tests β₁=0; for this one-predictor model with an intercept, the overall F agrees with the two-sided slope test.
See calculation method and assumptions
Formula
Regression Inference & Diagnostics
Model and residual standard error
Slope test and coefficient interval
Fit and overall test
Variables, rates and periods
- β₀, β₁, b₀, b₁
- Population coefficients and intercept OLS estimates.
- ŷᵢ, eᵢ
- Fitted value and residual yᵢ−ŷᵢ.
- SSE, SSR, SST
- Residual, model and total sums of squares.
- RSE, SE(bⱼ)
- Residual standard error and conventional OLS coefficient SE.
- ν, t, F
- ν=n−2; F=t² for the same zero-slope test.
- ε
- Error; conventional inference requires independence, constant variance and appropriate normality.
Read the residuals
- Segments connect each observation to its fitted value at the same X.
- The second plot places fitted values on the horizontal axis and residuals on the vertical axis.
- Curvature may suggest a wrong functional form;
- a funnel may suggest changing variance.
- The display itself performs no formal diagnostic test.
Assumptions of conventional inference
- The conditional mean should be linear, errors should have zero conditional mean given X, and observations should be independent with equal error variance.
- Normal errors justify exact small-sample t/F inference.
- Financial series often have heavy tails, dependence and clustered volatility;
- a significant slope alone does not resolve these issues.
When F equals t²
- With the same data, one slope, an intercept, conventional OLS assumptions and H₀:β₁=0, the overall F equals the squared slope t, and right-tail F agrees with two-sided t.
- This does not extend to every regression, directional test or standard-error method.
Beyond fitting
- R² is the share of sample variation explained by the fit, not out-of-sample forecast accuracy.
- Prediction needs new-data validation and care with extrapolation;
- causation needs a study design and treatment of confounding.
- This tool adds no multiple regression, causal identification or formal time-series diagnostics.
Extend simple OLS with an intercept using coefficient standard errors, t/F inference, confidence intervals and residual plots.
Worked example
- Enter X/Y in their original pair order; use at least three pairs with variation in X.
- Set α and fit OLS with an intercept; read coefficients, intervals, R² and t/F.
- Inspect fitted and residual plots. Check constant variance, independence and model form; significance does not establish causation.
Frequently asked questions
Are robust or Newey–West standard errors included?
No. This version uses conventional constant-variance OLS standard errors, without heteroskedasticity-robust, HAC or Newey–West estimates. Heteroskedasticity, serial dependence or volatility clustering can invalidate nominal p-values and intervals.
Does a perfect straight line give p=0?
With no residual variation the SE is zero, so a finite t ratio is undefined. The tool preserves fit information and marks undefined inference N/A instead of inventing Infinity or an exact p=0.
Is an outlier always influential?
A large residual means Y is far from the fitted line; leverage concerns an X position far from other X values. Influence also depends on how much removing that point changes the fit. They are not synonyms.
Related calculators
Simple Linear Regression Calculator
Fit the relationship between paired returns.
Pearson Correlation Calculator
Inspect linear co-movement in paired returns.
Student’s t-test Calculator
Compare mean differences, uncertainty and test direction.
Statistical Distributions & Critical Values Calculator
Explore normal, t, χ² and F tails and critical values.