Equations · Probability and statistics
Ordinary least squares estimator
The coefficients of the linear regression line that minimise the sum of squared errors.
β̂ = (XᵀX)⁻¹Xᵀy
Aethel says
beta hat equals the quantity X transpose X, inverse, times X transpose y
What it says
X holds the data, one row per observation and one column per predictor, and y holds the outcomes. The formula gives the coefficients that make the predictions X times beta as close as possible to y, in the sense of the smallest sum of squared differences.
It requires XᵀX to be invertible, which fails when one predictor is an exact combination of others.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| β̂ | beta hat | the estimated coefficients; a hat marks an estimate |
| Xᵀ | X transpose | the matrix X with rows and columns swapped |
| ⁻¹ | inverse | the inverse of the matrix |
| y | y | the vector of outcomes |
More onx̂HatᵀTransposeβBeta
Where you meet it
Econometrics, statistics and machine learning courses.
Questions
How do you say Ordinary least squares estimator out loud?
β̂ = (XᵀX)⁻¹Xᵀy is read: “beta hat equals the quantity X transpose X, inverse, times X transpose y.” That is exactly what Aethel says when it meets the formula in a book.
What does β̂ mean in Ordinary least squares estimator?
β̂ (said “beta hat”) is the estimated coefficients; a hat marks an estimate.
What does Xᵀ mean in Ordinary least squares estimator?
Xᵀ (said “X transpose”) is the matrix X with rows and columns swapped.
Can I listen to the equations in my own textbook like this?
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