ECMT6002 Chap.2 Multiple Regression and Joint Tests
Multiple Regression and Joint Tests
Multiple regression adds controls so that each slope describes a partial relationship. This chapter uses the partialling out interpretation to explain omitted variable bias, then develops individual and joint hypothesis tests. It also distinguishes R squared from adjusted R squared and statistical significance from economic magnitude.
The aim is to connect every test statistic to a null hypothesis, a reference distribution and a decision stated in context. Partialling out gives the cleanest interpretation: remove the linear contribution of the other regressors from both the target regressor and the outcome, then relate the remaining movements. This makes clear why adding a control can change a coefficient even when sample size stays fixed.
The omitted variable bias sign depends on two relationships: how the omitted factor affects the outcome and how it covaries with the included regressor. A control belongs because of the data-generating argument, not simply because it increases fit. Inference begins by writing the null as parameter restrictions.
A t statistic addresses one linear restriction, while an F statistic measures the fit cost of imposing several restrictions together relative to unrestricted noise. Separate t decisions do not reproduce a joint test because coefficient estimates may be correlated. Economic magnitude remains distinct from statistical evidence: a precisely estimated trivial effect can reject zero, while an important effect can be imprecise.
Adjusted R squared introduces a complexity penalty, yet neither version of R squared resolves endogeneity or establishes that the chosen regressors answer the intended economic question.
What this chapter covers
- 01
Partial effects and controls
- 02
Omitted variable bias
- 03
Individual t tests
- 04
Joint F tests
- 05
R squared and adjusted R squared
- 06
Statistical versus economic significance
Worked example · free
Testing two restrictions jointly
- +1The numerator is the increase in SSR per restriction: (204 minus 180) divided by 2 equals 12.
- +1The unrestricted error variance estimate is 180 divided by 94.
- +1Divide 12 by 180 over 94 to obtain an F statistic of about 6.27.
Key terms
- Partial effect
- The association between one regressor and the conditional mean while included regressors are held fixed.
- Omitted variable bias
- Bias arising when an omitted determinant belongs in the model and is correlated with an included regressor.
- Joint null
- A hypothesis that imposes more than one parameter restriction at the same time.
- Adjusted R squared
- A fit measure that penalises the addition of regressors through degrees of freedom.
Multiple Regression and Joint Tests FAQ
Why can two insignificant t tests produce a significant F test?
The joint test accounts for the covariance between coefficient estimates and asks whether restrictions hold together. Separate t tests answer different marginal questions and do not control the same joint rejection event.
Does adding a regressor always improve the model?
Ordinary R squared cannot fall, but that mechanical fact does not establish better specification. The new variable may be irrelevant, endogenous, measured badly or included in a way that obscures the target interpretation.
How do I diagnose omitted variable bias direction?
Determine how the omitted factor affects the outcome and how it is correlated with the included regressor. The signs of those two relationships determine the usual bias direction under the simple projection logic.
When should restrictions be tested jointly?
Use a joint test when the economic claim imposes several coefficient restrictions at once, such as excluding a category or testing a group of lag terms. Write every restriction and count them before forming the statistic. Separate decisions do not reproduce the same joint error rate or account for covariance among estimates.
Exam move
Practise moving between a verbal null, its coefficient restrictions and the correct test. For omitted variable bias, draw a sign diagram instead of relying on a memorised slogan. For model fit, explain what each statistic can and cannot establish. Always finish a test by returning to the economic wording of the null.
Build a restriction table with one row for the verbal claim, one for the coefficient equation, one for the number of restrictions and one for the reference distribution. Recalculate an F statistic from restricted and unrestricted residual sums of squares until the numerator scaling and unrestricted denominator are automatic. Then compare it with an equivalent single-restriction t test to see when the squared statistics agree.
For partial effects, take a three-regressor equation and explain exactly what is held fixed for each slope. Vary the sign of an omitted factor's outcome effect and regressor covariance to cover all bias directions. When reading output, report the estimate and confidence interval before looking at stars, and compare the range with a substantively meaningful effect size.
Keep a short list of variables that may be bad controls because they follow treatment or open a selection path. End each revision session by choosing one control and defending its inclusion through the economic mechanism rather than through a change in R squared.
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