ECMT6002 Chap.1 Regression Setup and OLS
Regression Setup and OLS
The opening chapter turns a verbal economic relationship into a population regression, a sample regression and a residual. It separates the unknown disturbance from the observable residual, derives ordinary least squares as a squared error minimisation rule, and interprets slope coefficients in levels and logs.
The central habit is to attach every interpretation to units, ceteris paribus language and the assumptions that justify the claim. Start by distinguishing the population conditional mean from an estimated line. The population disturbance records the gap between an outcome and its conditional mean, whereas a residual is constructed only after coefficients have been estimated on a sample.
OLS chooses the intercept and slope jointly, so its normal equations make residuals orthogonal to the included regressor and make their sample average zero when an intercept is present. Those are algebraic sample properties, not proof that the disturbance has zero conditional mean.
The same distinction prevents a common causal mistake: a fitted line describes conditional association unless the regressor assignment and omitted determinants justify a stronger interpretation. Functional form controls the units of the answer.
A level-level slope is a unit change, a log-level slope gives an approximate percentage response, a level-log slope rescales a percentage change in the regressor, and a log-log slope is an elasticity. Fit measures close the chapter by asking a different question.
R squared describes how much sample variation is captured by the fitted values, but it cannot certify exogeneity, correct specification or useful out-of-sample prediction.
What this chapter covers
- 01
Population regression and disturbance
- 02
Sample regression and residual
- 03
Least squares objective
- 04
Slope and intercept interpretation
- 05
Levels and logarithms
- 06
Goodness of fit versus causal credibility
Worked example · free
Recovering an OLS line from summary statistics
- +1Divide covariance by the variance of x: 6 divided by 3 gives a slope of 2.
- +1Use the fact that the fitted line passes through the sample means.
- +1Compute the intercept as 10 minus 2 times 4, which equals 2.
Key terms
- Population regression
- A model for the conditional mean of an outcome in the population.
- Disturbance
- The unobserved population component collecting influences on the outcome not represented by the systematic part.
- Residual
- The sample difference between an observed outcome and its fitted value.
- Least squares
- The rule that chooses coefficients by minimising the sum of squared residuals.
Regression Setup and OLS FAQ
Why does the fitted line pass through the means?
With an intercept, the first order condition makes the sample residuals sum to zero. Rearranging the average fitted equation therefore gives mean y equal to the intercept plus the slope times mean x.
Is a high R squared enough?
No. It describes sample fit, not whether the regressor is exogenous, the functional form is credible or the coefficient has a causal interpretation. A biased model can fit the sample very closely.
What must a slope interpretation contain?
Name the regressor change, the outcome change, their units and what is held fixed. If the model uses logarithms, translate the coefficient using the correct level log or log log rule.
How can I check an OLS calculation quickly?
Verify that the fitted line passes through the sample means and that residuals sum to zero when an intercept is included. Check the slope units and substitute a simple x value back into the equation. These checks catch arithmetic errors, but they do not establish exogeneity or turn the fitted association into a causal effect.
Exam move
Rebuild the OLS formulas from covariance, variance and the mean point rather than memorising isolated expressions. For each coefficient, write one precise sentence with units and ceteris paribus language. Then make a two column list separating algebraic OLS properties from assumptions needed for unbiasedness, consistency and causal interpretation.
Use a small hand-calculated dataset to verify that residuals sum to zero and that the fitted line passes through the sample means. Next, change one observation and predict which coefficient will move before recalculating. Create interpretation drills covering all four common level and logarithm combinations, and always state whether a percentage reading is exact or approximate.
When reviewing an empirical claim, underline the target outcome, regressor, comparison group and omitted influences. Ask whether the equation describes a conditional mean, a best linear approximation or a causal effect. Practise explaining why a large R squared may coexist with bias and why a low R squared need not make an unbiased slope useless.
On a blank page, reproduce the objective function, first order conditions, slope formula and intercept formula in a connected derivation. Finish each problem with a units audit and an assumption sentence. That routine turns OLS from a collection of formulas into the benchmark against which every later estimator is evaluated.
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