Homework answers / question archive / Consider a generalised linear model y = X β2 the best predictor of yθ y^θ'θβ^gls ' -1^ + ε    with E(ε) = 0 and V(ε) = σV, for an observation θ outside the sample is given as follows : = x+vVεgls                                                    (2) where xθ is the explanatory variable for the observation θ, E(εθε') = σ2 v'  and β^gls = (X'V-1X)-1X'V-1y = y - X β^gls   Let us apply this result to the AR 1 (Autocorrelation Regression) model: if yt = x'tβ + εt   t=1,

Consider a generalised linear model y = X β2 the best predictor of yθ y^θ'θβ^gls ' -1^ + ε    with E(ε) = 0 and V(ε) = σV, for an observation θ outside the sample is given as follows : = x+vVεgls                                                    (2) where xθ is the explanatory variable for the observation θ, E(εθε') = σ2 v'  and β^gls = (X'V-1X)-1X'V-1y = y - X β^gls   Let us apply this result to the AR 1 (Autocorrelation Regression) model: if yt = x'tβ + εt   t=1,

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Consider a generalised linear model y = X β2 the best predictor of yθ y^θ'θβ^gls ' -1^ + ε    with E(ε) = 0 and V(ε) = σV,

for an observation θ outside the sample is given as follows :

= x+vVε_{gls                                                    (2)}

where x_θ is the explanatory variable for the observation θ,

E(ε_θε') = σ² v^'  and

β^{^}_gls = (X^'V^-1X)^-1X^'V^-1y

₌ y - X β^{^}_gls

 

Let us apply this result to the AR 1 (Autocorrelation Regression) model: if y_t = x^'_tβ + ε_t   t=1,....T

ε_{t =} ρ ε_t-1 +u_t

 

Suppose that we want to predict y_T+1

Note: Do the following calculations assuming that ρ is known, and then replace ρ by ρ^ in the end.

 

1. Give the expression of V(ε_t).

2. Give σ² u^'  = E(ε _T+1 ε^') using the general formula of E(ε_tε_s), t ≠ s for the AR(1) process.

3. Derive u^' V^-1 (Hint : express u^' in terms of V and then use VV^-1 = I' to get the result).

4. Finally obtain u^' V^-1ε^{^} _gls  and substitute in (1) to get y^ _T+1