Matrix Algebra for OLS Estimator - Miami University

Matrix Algebra for OLS Estimator

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Big Picture

? Matrix algebra can produce compact notation. ? Some packages such as Matlab are matrix-oriented. ? Excel spreadsheet is just a matrix.

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Dependent Variable

? Suppose the sample consists of n observations.

? The dependent variable is denoted as an n ? 1 (column) vector

Y =

y1

y2 ...

yn

? The subscript indexes the observation. ? We use boldface for vector and matrix.

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Independent Variables

? Suppose there are k independent variables and a constant term. In the spreadsheet there are k + 1 columns and n rows.

? Mathematically that spreadsheet corresponds to an n ? (k + 1)

matrix, denoted by X :

X

=

1

1 ...

x11

x21 ...

... ... ...

x1k

x2k ...

1 xn1 . . . xnk

where xij is the i-th observation of the j-th independent variable.

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Linear Regression Model

? Define as a (k + 1) ? 1 vector of coefficients

=

0 ...

k

and U as an n ? 1 vector of error terms. The linear multiple regression model in matrix form is

Y = X + U

? Read Appendix D of the textbook. ? The key to work with matrix is keeping track of the dimension.

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