(b) Consider the generalized linear regression model. State the model in matrix algebra form. Show that there exists a transformation of the model that satisfies all the Gauss Markov assumptions. Derive the GLS estimator and compute its variance-covariance matrix.
Question
(b) Consider the generalized linear regression model. State the model in matrix algebra form. Show that there exists a transformation of the model that satisfies all the Gauss Markov assumptions. Derive the GLS estimator and compute its variance-covariance matrix.
Solution
(b) The generalized linear regression model can be stated in matrix algebra form as:
Y = Xβ + u
where Y is the dependent variable, X is the matrix of independent variables, β is the vector of parameters to be estimated, and u is the error term.
The Gauss-Markov assumptions for the generalized linear regression model are:
- E(u|X) = 0
- Var(u|X) = σ^2 * I
where I is the identity matrix.
To satisfy these assumptions, we can transform the model by pre-multiplying both sides by Σ^(-1/2), where Σ is the covariance matrix of u. This gives us:
Σ^(-1/2)Y = Σ^(-1/2)Xβ + Σ^(-1/2)u
This transformed model satisfies the Gauss-Markov assumptions because the expected value of the transformed error term is zero and its variance is the identity matrix.
The Generalized Least Squares (GLS) estimator of β in this transformed model is:
β̂GLS = (X'Σ^(-1)X)^(-1)X'Σ^(-1)Y
The variance-covariance matrix of β̂GLS is:
Var(β̂GLS) = (X'Σ^(-1)X)^(-1)
Similar Questions
(a) Consider the classical linear regression model: y = Xβ + u under the Gauss Markov assumptions, such that E uu = σ 2 I . Consider the estimator for σ 2 given by s2∗ = e e/(T − 10) where e is the OLS residual vector: is this an unbiased estimator? If not, how can we correct it to get an unbiased estimator of σ 2 . (b) Consider the generalized linear regression model. State the model in matrix algebra form. Show that there exists a transformation of the model that satisfies all the Gauss Markov assumptions. Derive the GLS estimator and compute its variance-covariance matrix.
Consider the simple linear regression model with no intercept yt = xt β + εt . Assume all the Gauss Markov assumptions hold. Furthermore assume that εt ∼ iidN(0, σε^2 ). Derive the ML estimator of β and discuss its asymptotic properties, constructing the asymptotic covariance matrix.
2. (a) Consider the generalized linear regression model 11 = a + Bat + Et. Assume Eep = 0 and that we have no serial correlation among the disturbances but var(41) =0°1?. where we assume that 2+ ‡ 0 for all t. Derive the feasible GLS estimator of a and B Now consider the following regression, setting &$ = =: 4t Ut It 1 u* = a- =+B+E. It and estimate a and 8 again. How would the estimates differ from GLS. Explain.
(a) Consider the classical linear regression model: y = Xβ + u under the Gauss Markov assumptions, such that E uu = σ 2 I . Consider the estimator for σ 2 given by s2∗ = e e/(T − 10) where e is the OLS residual vector: is this an unbiased estimator? If not, how can we correct it to get an unbiased estimator of σ 2 .
4.1 The Process ModelThe state equations described by (8) can be rewritten by𝑞˙=𝛺[𝜔]𝑞(11)where 𝛺[𝜔]=12[0−𝜔𝑇𝜔[𝜔×]] .(12)ω = [ωx ωy ωz]T denotes the tri-axis gyroscope output; [ω ×] is a 3 × 3 skew symmetric matrix given by [𝜔×]=[0𝜔𝑧−𝜔𝑦−𝜔𝑧0𝜔𝑥𝜔𝑦−𝜔𝑥0] .(13)The next step is to convert continuous-time model represented by (11) into a discrete-time model. Let the system sampling time be Ts, then the discrete-time model is given by
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