01:220:322 Chap.3 Simple Linear Regression
Simple Linear Regression
The model and its error term
Simple regression writes the outcome as an intercept plus a slope times one explanatory variable plus an error. The error collects every other influence on the outcome.
Whether the slope has a causal reading depends on the zero conditional mean assumption: the average of the error must not change with the explanatory variable.
Ordinary least squares
OLS picks the intercept and slope that minimise the sum of squared residuals. The slope is the sample covariance of x and y divided by the sample variance of x, and the intercept makes the line pass through the point of means.
The residuals always sum to zero and are uncorrelated with x in the sample, which are facts about the arithmetic, not evidence about the population.
Goodness of fit
The course names the total sum of squares SST, the explained sum of squares SSE and the residual sum of squares SSR, with SST equal to SSE plus SSR. R-squared is the explained share.
A low value is normal with individual data and does not signal bias.
Units and logarithms
Rescaling a variable rescales the coefficients without changing the finding.
Logarithms change the interpretation: a log outcome turns the slope into an approximate percentage change, a log regressor makes the slope divided by one hundred the effect of a one percent change, and logs on both sides give an elasticity.
Statistical properties
Under the SLR assumptions of linearity, random sampling, variation in x and zero conditional mean, OLS is unbiased.
Adding homoskedasticity gives the usual variance formula, which falls as x becomes more spread out and rises with the error variance.
The lecture on simple regression introduces the error-variance estimate as the average squared residual, while software divides by the degrees of freedom.
Rescaling in practice
If a return on equity is recorded as a decimal instead of a percentage, its slope is one hundred times larger, but a one percentage point change still has the same predicted effect.
Always restate the change in the units the question uses before interpreting, and remember that the R-squared never changes when units change.
Stata output to recognise
The regress command prints the sums of squares in the top left, labelled Model, Residual and Total, and the coefficients with standard errors, t statistics, p-values and 95% intervals below.
Exam questions often ask you to read the sample size, the R-squared and the fitted equation straight from that table.
What this chapter covers
- 01
Population regression function and the error term
- 02
OLS slope and intercept
- 03
Residuals and their algebraic properties
- 04
SST, SSE, SSR and R-squared
- 05
Level and log interpretations
- 06
SLR assumptions, unbiasedness and standard errors
Worked example · free
Reading a log-level regression
- 2Slope: one more year of schooling is associated with about 7.2% higher hourly pay on average.
- 2Intercept: the predicted log pay at zero schooling, an extrapolation with little economic meaning.
- 2R-squared: schooling accounts for 14% of the sample variation in log pay; the rest sits in the error.
Key terms
- Error Term
- The part of the outcome explained by factors other than the included regressor.
- Residual
- The actual value minus the fitted value for one observation.
- Explained Sum of Squares
- The variation of fitted values around the mean, named SSE in this course.
- R-Squared
- The share of the sample variation in the outcome that the fitted line accounts for.
- Elasticity
- The percentage change in one variable for a one percent change in another, the slope of a log-log model.
Simple Linear Regression FAQ
Does a low R-squared mean the regression is wrong?
No. It means many factors other than x move the outcome, which is typical for data on individuals. The slope can still be unbiased; bias depends on whether the omitted factors are correlated with x.
How should I interpret a slope when x is in logs and y is not?
Divide the slope by one hundred to get the change in y, in its own units, for a one percent increase in x. Forgetting that division is one of the most frequent interpretation slips on exams.
Why do residuals always sum to zero?
The first-order condition for the intercept forces it. With an intercept in the model, the fitted line balances positive and negative residuals exactly, whatever the true errors look like.
Is the intercept always meaningful?
Only when zero is a realistic value of the regressor and lies near the data. Otherwise the intercept simply positions the line; demeaning the regressor turns the intercept into the prediction at the average instead.
Exam move
Compute one regression entirely by hand from a small table, then practise translating slopes under all four log forms into single sentences with units.
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