Enter Values
Formula Reference
t = (bj − βj0) / se(bj)
bj = Coefficient estimate |
se = Standard error |
df = n − k − 1
Test Results
Decision
Reject H₀
Statistically Significant
t-statistic
2.500
p-value
0.0142
t critical
1.985
Degrees of Freedom
96
Two-Sided 95% Confidence Interval
0.1030
≤ βj ≤
0.8970
t-Distribution
Formula Breakdown
t = (bj − βj0) / se(bj)
Step-by-step calculation
Model Assumptions
- MLR.1 (Linear in Parameters): y = β₀ + β₁x₁ + … + βkxk + u
- MLR.2 (Random Sampling): Data is a random sample from the population
- MLR.3 (No Perfect Collinearity): No independent variable is a perfect linear function of others
- MLR.4 (Zero Conditional Mean): E(u|x₁,…,xk) = 0
- MLR.5 (Homoskedasticity): Var(u|x₁,…,xk) = σ²
- MLR.6 (Normality): u ~ Normal(0, σ²) — required for exact t and F distributions in finite samples
- For large samples, MLR.6 can be relaxed (Central Limit Theorem applies for approximate inference)
- F-test assumes both restricted and unrestricted models are correctly specified
- This calculator assumes conventional OLS standard errors. Robust/clustered SE inference is out of scope.
For educational purposes. Not financial advice. Statistical assumptions simplified for pedagogical clarity.
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Frequently Asked Questions
A t-test in regression examines whether an individual coefficient is statistically different from a hypothesized value (usually zero). The test computes t = (bj − hypothesized value) / se(bj), which follows a t-distribution with n − k − 1 degrees of freedom under the null hypothesis. If the p-value is below the chosen significance level, you reject the null and conclude the coefficient is statistically significant.
The p-value is the probability of observing a test statistic as extreme as (or more extreme than) the one computed in the direction implied by the alternative hypothesis, assuming the null hypothesis is true. A small p-value (e.g., < 0.05) provides evidence against the null hypothesis. It does not measure the probability that the null is true, nor does it measure the economic importance of the coefficient.
The overall F-test examines whether all slope coefficients in a regression are jointly equal to zero. It uses F = (R²/k) / ((1 − R²)/(n − k − 1)). A significant F-statistic means that at least one independent variable has a non-zero effect on the dependent variable. This test is reported in standard regression output and assumes the model includes an intercept.
Use a one-sided test when economic theory or prior evidence gives you a strong expectation about the direction of the effect (e.g., education positively affects earnings). Use a two-sided test when you have no strong prior about the sign. One-sided tests have more power to detect effects in the predicted direction but cannot detect effects in the opposite direction. The choice should be made before seeing the data.
Statistical significance means the coefficient is unlikely to be zero given the data — it depends on the standard error and sample size. Economic (or practical) significance refers to whether the magnitude of the coefficient is large enough to matter in real-world terms. A large dataset can yield statistically significant but economically trivial estimates. Always examine the coefficient's magnitude alongside its p-value.
A 95% confidence interval is computed as bj ± tcrit × se(bj), where tcrit is the critical value from the t-distribution with n − k − 1 degrees of freedom at the 2.5% level. If the interval does not contain the hypothesized value (e.g., zero), you reject H₀ at the 5% level in a two-sided test. The confidence interval corresponds specifically to the two-sided test and provides a range of plausible values for the true parameter.
Disclaimer
This calculator is for educational purposes only and assumes conventional OLS inference under the Classical Linear Model assumptions (MLR.1–MLR.6). It does not account for heteroskedasticity, serial correlation, or clustered standard errors. For robust inference in applied research, consult appropriate econometric software. This tool should not be used as the sole basis for research conclusions.
