Why Regressors Can Be Jointly Significant but Individually Insignificant

Summary

The document presents a regression question about two predictors whose individual coefficient tests produce relatively large p-values, while a joint test of both coefficients being zero produces a smaller p-value. It asks why this can happen and seeks references, but provides no answer or supporting analysis.

The example distinguishes testing each coefficient separately from testing a combined restriction on multiple coefficients. These tests address different hypotheses, so their outcomes need not match. The document supplies the specific regression setup and reported p-values as an illustration, but it does not explain possible mechanisms such as correlation between predictors, give the joint test procedure, or discuss assumptions and inference choices. It is therefore a prompt for understanding joint hypothesis testing rather than a complete treatment or evidence that either predictor has a useful trading relationship.

Key ideas

• An individual coefficient test and a joint test assess different null hypotheses.
• Two coefficients can fail separate significance tests while their joint restriction is rejected.
• The example reports separate p-values of 0.56 and 0.27 and a joint p-value of 0.03.
• The document poses the statistical question but does not provide an explanation or method.

Tags

# Joint hypothesis tests


# Joint hypothesis tests












When running regressions,

$$Y_t=\alpha+\beta_{9}x_{9,t-1}+\beta_2x_{2,t-1}+\beta_3x_{3,t-1}+\beta_4x_{4,t-1}+\varepsilon_t (1)$$

$$Y_t=\alpha+\beta_1x_{1,t-1}+\beta_2x_{2,t-1}+\beta_3x_{3,t-1}+\beta_4x_{4,t-1}+\varepsilon_t (2)$$

In equation (1), I test $$\beta_{9}=0$$, and find a p-value of 0.56

In equation (2), I test $$\beta_{1}=0$$, and find a p-value of 0.27

I then run a third regression

$$Y_t=\alpha+\beta_9x_{9,t-1}+\beta_1x_{1,t-1}+\beta_2x_{2,t-1}+\beta_3x_{3,t-1}+\beta_4x_{4,t-1}+\varepsilon_t (3)$$

In equation (3), I test, $$\beta_{9}=\beta_{1}=0$$, and get a p-value of 0.03.

Is there a reason why two variables may be insignificant individually but significant together. Does anyone have any useful books, references etc.

Thanks.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.