01:220:322 Chap.4 Multiple Regression Estimation and Omitted Variables
Multiple Regression Estimation and Omitted Variables
Why add regressors
Multiple regression holds observed factors fixed explicitly. Each slope becomes a partial effect: the change in the predicted outcome when that regressor changes and the others stay put.
Interpretation sentences should name the variables held fixed.
The MLR assumptions
Unbiasedness needs linearity in parameters, random sampling, no perfect collinearity and a zero conditional mean of the error given all regressors. Perfect collinearity means an exact linear relation among regressors, such as two spending categories that add up to a total also included; software then drops a variable.
High but imperfect correlation is allowed and only costs precision.
Omitted variable bias
If a relevant variable is left out and it is correlated with an included regressor, the short regression slope is biased.
The bias equals the omitted variable's coefficient times the slope from regressing the omitted variable on the included one, so its sign follows from two signs: the effect of the omitted variable and its correlation with the included one.
Exams repeatedly ask for this sign and for a comparison of short and long regression estimates.
Precision
The variance of a slope rises with the error variance, falls with the total variation in its regressor, and rises with the R-squared from regressing that regressor on the others.
That last term is multicollinearity; it has no universal cutoff and is not a violation of any assumption.
Gauss-Markov
With homoskedasticity added, OLS is the best linear unbiased estimator.
Adding an irrelevant variable never biases the other slopes but can inflate their variances.
Comparing simple and multiple estimates
When a regressor is added, the slope on the original variable changes by exactly the new coefficient times the slope of the new variable on the original one. If the two regressors are uncorrelated in the sample, the simple and multiple slopes coincide.
Exam items give both regressions and the correlation between the regressors and ask you to explain the change, so practise stating the direction before computing anything.
One bad regressor spreads its bias
If a single regressor is correlated with the error, the other slopes are generally biased too, because OLS partials each variable out of the others.
Endogeneity is therefore a property of the whole model, not of one coefficient.
What this chapter covers
- 01
Partial effects and ceteris paribus interpretation
- 02
MLR.1 to MLR.4 and unbiasedness
- 03
Perfect collinearity versus multicollinearity
- 04
Omitted variable bias and its sign
- 05
Variance of a slope and its three ingredients
- 06
Gauss-Markov and BLUE
Worked example · free
Short versus long regression
- 2Write the identity: short slope = long slope + size coefficient times the slope of size on advertising.
- 2The size coefficient is positive and the short slope exceeds the long one, so size and advertising are positively correlated.
- 2Conclude that the short regression credited advertising with part of the effect of store size: upward bias.
Key terms
- Partial Effect
- The change in the expected outcome from changing one regressor while the others are held fixed.
- Perfect Collinearity
- An exact linear relationship among regressors, which makes the OLS estimates undefined.
- Omitted Variable Bias
- The bias in a slope caused by leaving out a relevant variable that is correlated with the included regressor.
- Multicollinearity
- High but imperfect correlation among regressors, which enlarges standard errors without causing bias.
- Endogenous Regressor
- An explanatory variable correlated with the error term.
- BLUE
- Best linear unbiased estimator, the property of OLS under the Gauss-Markov assumptions.
Multiple Regression Estimation and Omitted Variables FAQ
Is a correlation of 0.9 between two regressors a violation?
No. The no perfect collinearity assumption rules out only exact linear relations. A correlation of 0.9 leaves the estimates unbiased but makes them less precise, which shows up as larger standard errors.
How do I sign omitted variable bias quickly?
Multiply two signs: the effect of the omitted variable on the outcome and its correlation with the included regressor. Positive times positive or negative times negative gives upward bias; mixed signs give downward bias.
Does adding an irrelevant variable bias the other coefficients?
No. Its true coefficient is zero, so including it leaves the other estimates centred on their true values. The cost is a possible increase in their variances when it is correlated with them.
Why does R-squared always rise when I add a variable?
Adding a regressor can never increase the sum of squared residuals, because OLS could always set its coefficient to zero. R-squared therefore creeps up even for useless variables and cannot tell you whether a variable belongs.
Exam move
Keep a two-by-two sign table on your note sheet and practise one short-versus-long comparison from every multiple-regression example you meet.
Working through Multiple Regression Estimation and Omitted Variables in 01:220:322? Sia is AskSia’s AI Economics tutor — ask any 01:220:322 Multiple Regression Estimation and Omitted Variables question and get a clear, step-by-step explanation grounded in how 01:220:322 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
