Consider the ARDL(2,2) model,
c t = β 0 + β 1 c t−1 + β 2 c t−2 + δ 1 y t−1 + δ 2 y t−2 + u t . (3.1)
Let equilibrium be reached in the long run, and let equilibrium be characterised by no change; i.e., c t = c t−1 = c t−2 = c, and y t−1 = y t−2 = y.
(a) Use this characterisation of equilibrium to derive the long-run relationship between c and y implied by (3.1); i.e., derive an equation of the form c = a + by. (3.1a)
Note 1. Set u to its expected value of zero.
Note 2. Your answer needs to explicitly express a and b in terms of the βs and δs.
(b) Use your parameter estimates from (3.1) to compute an estimate of b, and comment on its plausibility based on economic theory and/or common sense.
(c) How does your estimate of b from (b) above help to explain the time series plots in question 1, part (a) and question 2, part (a)?
(d) Derive a restricted version of (3.1) that imposes a long-run elasticity of consumption with respect to income of one (i.e., b = 1) of the form
∆c t = γ 0 + γ 1 (c t−1 − y t−1 ) + γ 2 ∆c t−1 + γ 3 ∆y t−1 + u t . (3.2)
1. Show sufficient derivation steps and show explicitly how the γs relate to the βs and δs.
2. Equation (3.2) is an example of the so-called (single-equation) ‘error-correction’ model (ECM). The ECM in vector form (the VECM) is explored in lecture 20.
(e) (*) Estimate a suitable reparameterisation of (3.1) to test the restriction imposed by the error-correction model (3.2) by using the p-value for the statistical significance of a single estimated coefficient. Report the p-value for the test and draw the appropriate conclusion.
Hint. You should find creating the additional variable z t = r t + ∆y t useful.
(f) Does (3.2) constitute a ‘balanced’ regression (in the terminology of tutorial 11), and why (or why not)?
(g) Create a new variable, e t = c t − b̂y t , where b̂ is your estimate computed in (b) above, and determine the order of integration of this variable through unit root testing. Based on your test result, write out an unrestricted version of (3.2). Does this regression constitute a ‘balanced’ regression, and why (or why not)?
(h) Suggest an additional variable to include in (3.1a) that might address the problem indicated in (b) above, and explain your choice.
(a) Present neatly tabulated regression results for all parts above marked (*) in Appendix A.
(b) Present your R-code in Appendix B.
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