# ECOM30002/90002 Econometrics 2 – University of Australia

**QUESTION 1**

(a) Plot c t and y t over time on the same graph and describe both variables’ main feature(s).

(b) Based on a maximum lag of eight, use VARselect (as in tutorial 10) to choose an appropriate lag length for a VAR model for c t and y t . Report the chosen lag length.

(c) (*) Write out in equation form your chosen estimated equation for c t from (b) above.

(d) Does the residual correlogram (autocorrelation function) suggest that your chosen equation for c t constitutes a valid forecasting model, and why (or why not)?

(e) Use your estimated equation for c t to forecast log consumption two periods ahead. Report these forecasts.

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**QUESTION 2**

(a) Generate the new variable, r t = c t − y t, which is the log of the consumption to income ratio. Plot r t over time and describe its main feature(s).

(b) Using a maximum lag of eight, test for a unit root in each of c t, y t, rt , ∆c t, ∆y t, and ∆r t. For each variable, report the p-value for the unit root test and draw an appropriate conclusion.

(c) Based on the results obtained in (b) above, which variables are stationary, and which are nonstationary, and why?

(d) (*) Based on the CADFtest input and ADF test output for r t, estimate the regression used to test r t for a unit root directly (i.e., using dynamic), and write out your results in equation form.

(e) Based on all the results from this question, explain the difficulty of specifying an appropriate bivariate time series model for c t and y t .

**QUESTION 3**

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 characterized 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 characterization 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)

Notes.

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 reparameterization 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.

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**APPENDIX QUESTION**

(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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