MFIN6201 Chap.4 Simple Regression and OLS
Simple Regression and OLS
This chapter explains how ordinary least squares estimates a one-regressor linear relationship. It distinguishes population coefficients, fitted values and residuals, then interprets the slope with units. R-squared and the standard error of the regression describe fit but do not establish an unbiased or causal coefficient. Every answer separates sample association from causal interpretation.
The population equation places all other outcome determinants inside an error term. OLS chooses the intercept and slope that minimise squared vertical residuals in the observed sample. A fitted value sits on the estimated line, while a residual measures actual minus fitted at the same regressor value. The slope is a change in predicted Y per unit of X and must be translated with both measurement scales.
The intercept predicts Y at zero, which may be algebraically necessary but economically unhelpful outside the observed support. R-squared reports the fraction of sample outcome variation explained by the line; the regression standard error reports typical residual spread in outcome units. Neither checks whether X is related to omitted determinants.
The chapter therefore ends every numerical calculation with a data-generating question about the contents of the error.
What this chapter covers
- 01
Population regression model
- 02
OLS objective
- 03
Fitted values and residuals
- 04
Slope and intercept interpretation
- 05
R-squared
- 06
Standard error of regression
Worked example · free
Prediction and residual from a fitted line
- +1Compute the fitted value 4.2-0.6(3)=2.4.
- +1Compute residual as actual minus fitted: 2.9-2.4=0.5.
- +1State that the observation lies 0.5 Y-units above the fitted line.
Key terms
- Fitted value
- The outcome predicted by the estimated regression at an observed regressor value.
- OLS residual
- The observed outcome minus its fitted value.
- R-squared
- The fraction of sample outcome variation explained by the fitted regression.
Simple Regression and OLS FAQ
What does an OLS slope mean?
It is the predicted change in the outcome associated with a one-unit increase in the regressor. A complete interpretation gives both variable units and avoids causal wording unless the identifying assumptions or research design justify it.
Is a high R-squared evidence of causality?
No. R-squared measures in-sample explained variation. A model can fit extremely well while its slope captures omitted influences, reverse causality or selection. Causal credibility comes from the source of regressor variation and the error assumption.
When is the intercept meaningful?
The intercept predicts the outcome at a regressor value of zero. It has a useful economic meaning only when zero is substantively sensible and supported by the data. Otherwise it remains necessary for fitting the line but should not be overinterpreted.
What does a residual tell me?
A residual is the observed outcome minus its fitted value. Its sign locates the point above or below the line and its magnitude measures that vertical miss. It is not automatically a data error or omitted variable.
Why is extrapolation risky?
The fitted line summarises relationships over observed regressor support. Far beyond that range, the mechanism may become nonlinear or comparable observations may disappear. A numerical prediction can be calculated while its empirical support remains weak.
Why does OLS square residuals?
Squaring prevents positive and negative misses from cancelling and penalises large vertical errors more strongly. It also produces a tractable optimisation problem, but that sensitivity means extreme observations and high-leverage regressor values deserve careful diagnosis.
How are fitted values and population means related?
A fitted value is the sample model's predicted outcome at a given X. Interpreting it as the population conditional mean requires the linear specification and error assumptions to be credible; the algebraic prediction alone does not establish that interpretation. State those assumptions before giving the fitted line a population meaning.
Exam move
Practise every fitted-line question in the same order: substitute for the fitted value, calculate actual minus fitted, interpret the slope with units, and state the association boundary. Compare R-squared with residual spread without turning either into an identification test. Check whether zero lies inside the observed regressor range before discussing the intercept.
Draw a small scatterplot and mark the vertical residual for at least one observation. Verify that fitted plus residual returns the actual outcome. Change the regressor scale from dollars to thousands of dollars and rewrite the slope interpretation rather than merely changing the number. Compare two nearby X values to recover the same slope difference.
For fit questions, give one sentence on R-squared and a separate sentence on residual spread. Then name an omitted factor that could move with X, making clear why a tidy fitted line is not a causal design. Work predictions inside and outside the observed range to identify extrapolation. Finish revision by explaining the population coefficient, its sample estimate and a fitted value without using those terms interchangeably.
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