Your Textbook Used a Distributed Lag Model with Only Current$\hat { \phi }$

Question

Examlex · Accepted Answer

The Answer is (a)The quasi-difference model is derived by multiplying the equation by φ₁ and lagging it, then subtracting the resulting equation from the first equation, and using the AR(1)equation of the error term for simplification of the resulting specification.
Y_t = β₀ + β₀X_t + u_t -[φ₁Y_t_-₁ = φ₁β₀ + φ₁β₁X_t_-₁ + φ₁u_t_-₁]
which results in Y_t - φ₁Y_t_-₁ = β₀(1 - φ₁)+ β₁X_t_-₁ - φ₁β₁X_t_-₁ + (u_t - φu_t_-₁).
Using the quasi-difference notation then yields

ilde { Y } _ { t } = \alpha _ { 0 } + \beta _ { 1 } \widetilde { X } _ { t } + ilde { u } _ { t }

If φ₁ was known, then it would be possible to generate the quasi-difference variables in a statistical package and then estimate the coefficients using the transformed variables using OLS.
(b)In this case, nonlinear least squares has to be used to estimate the three parameters. One possible feasible GLS estimator in this case is the Cochrane-Orcutt estimator. In the first step, φ₁ is set to zero, in which case β₀ and β₁ can be estimated by OLS. The resulting residuals are then used to calculate the OLS estimator for φ₁. This, in return, can then generate the quasi-differenced variables and OLS is then employed to get the estimate of β₀ and β₁.
(c)The iterated Cochrane-Orcutt estimator continues the process described in (a). For example, in the next step, a new set of residuals is used to update the previous estimate of φ₁, which will generate a new set of quasi-differenced variables and new estimates of β₀ and β₁. The iterations stop when the differences in the estimates from one round to the next differ by less than a very small number, which can be chosen by the econometrician. This is then called convergence.
(d)Under the Hildreth-Lu method, the sum of squared residuals is computed for various values of φ₁, using quasi-differenced variables. For example, initially a coarse grid is chosen of -0.9, -0.8, -0.7, …, 0.7, 0.8, 0.9. For the value of φ₁ which yields the smallest SSR, say 0.7, a new finer grid is chosen, such as 0.65, 0.66, 0.67, …, 0.73, 0.74, 0.75, and again the SSR is calculated for each of these values. The value of φ₁ which has the smallest SSR is retained and yet a finer grid around it is chosen, etc.

Your Textbook Used a Distributed Lag Model with Only Current $\hat { \phi }$

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Your Textbook Used a Distributed Lag Model with Only Current ϕ^\hat { \phi }ϕ^​

Definitions:

Your Textbook Used a Distributed Lag Model with Only Current $\hat { \phi }$