2014120213이선우

1. Generalized Linear Squares (GLS)

2. Feasible Generalized Linear Squares (FGLS)

3. Example

The case of a known covariance matrix of the error terms

Use generalized least squares (GLS) estimator.

�̂�𝛽𝑔𝑔𝑔𝑔𝑔𝑔 = 𝑋𝑋′𝑊𝑊𝑋𝑋 −1𝑋𝑋′𝑊𝑊𝑊𝑊

,𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑊𝑊 = 𝐶𝐶𝐶𝐶𝐶𝐶(𝜀𝜀)−1

Generalized Least Squares (GLS)

The case of an unknown covariance matrix of the error terms

Estimate covariance matrix of the error terms.

Use feasible generalized least squares (FGLS) estimator.

�̂�𝛽𝑓𝑓𝑔𝑔𝑔𝑔𝑔𝑔 = 𝑋𝑋′ �𝑊𝑊𝑋𝑋 −1𝑋𝑋′ �𝑊𝑊𝑊𝑊

Feasible Generalized Least Squares (FGLS)

Estimation of covariance matrix

1. Serial Correlation (Autocorrelation)

2. Seemingly unrelated regression models

3. Heteroscedasticity (Weighted Least Squares)

4. Error components models

5. Random coefficients models

Feasible Generalized Least Squares (FGLS)

Serial Correlation (Autocorrelation)

The error terms follows AR(1).

Feasible Generalized Least Squares (FGLS)

𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀 = 𝜎𝜎2

1−𝜌𝜌2

1𝜌𝜌

𝜌𝜌1

⋮𝜌𝜌𝑛𝑛−1

⋮𝜌𝜌𝑛𝑛−2

⋯⋯

𝜌𝜌𝑛𝑛−1

𝜌𝜌𝑛𝑛−2⋱⋯

⋮1

Serial Correlation (Autocorrelation)

The error terms follows AR(1).

Feasible Generalized Least Squares (FGLS)

�𝜌𝜌 =∑𝑖𝑖=2𝑛𝑛 𝑤𝑤𝑖𝑖−1𝑤𝑤𝑖𝑖∑𝑖𝑖=2𝑛𝑛 𝑤𝑤𝑖𝑖−12

Heteroscedasticity (Weighted Least Squares)

Covariance Matrix of the error terms is a diagonal matrix.

Assume that the diagonal elements is at least two different values.

Feasible Generalized Least Squares (FGLS)

𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀 =𝜎𝜎120

0𝜎𝜎22

⋮0

⋮0

⋯⋯

00

⋱⋯

⋮𝜎𝜎𝑛𝑛2

Example

Example - OLS

�̂�𝛽𝑜𝑜𝑔𝑔𝑔𝑔 = 𝑋𝑋′𝑋𝑋 −1𝑋𝑋′𝑊𝑊

�̂�𝛽𝑜𝑜𝑔𝑔𝑔𝑔 =�̂�𝛽0�̂�𝛽1

=156.35−1.19

Example - FGLS (Heteroscedasticity)

�̂�𝛽𝑓𝑓𝑔𝑔𝑔𝑔𝑔𝑔 = 𝑋𝑋′ �𝑊𝑊𝑋𝑋 −1𝑋𝑋′ �𝑊𝑊𝑊𝑊

�̂�𝛽𝑓𝑓𝑔𝑔𝑔𝑔𝑔𝑔 =�̂�𝛽0�̂�𝛽1

=158.39−1.22

Example - Residual Plots