Economic Models - Convex Optimization

184 Fabrizio Iacone and Renzo Orsi
the AD equation, but we re-wrote the PC as
π t = α 0 + ρ 0 π t−1 + ρ 1 π t−1 + ρ 2 π t−2 + ρ 3 π t−3 + ρ y y t−1
+ α q (q t−1 − q t−5 ) + ε π,t PC
(the term q t−1 − q t−5 does not appear in the equation for Germany). This
is a re-parametrization of the original equation, with
ρ 0 = (α π1 + α π2 + α π3 + α π4 − 1)
ρ 1 =−(α π2 + α π3 + α π4 ), ρ 2 =−(α π3 + α π4 ), ρ 3 =−α π4
Being simply a re-parametrization, the residual sum of squares that
has to be minimized is the same, so the OLS estimates are not different
than the estimates that one would obtain in the original model in levels.
The advantage of this specification is that long and short dynamics are
explicitly separated, ∑ 4
j=1 α πj = 1 corresponding to p 0 = 0, and that,
more importantly, the multi-collinearity due to the terms π t−1 to π t−4 is
much less severe.
Correct specification was primarily checked by looking at the tests for
normality (Jarque and Bera), autocorrelation (up to the fourth lag: Breusch
.Godfrey LM) and heteroskedasticity (white without cross terms) in the
residuals: we referred to these tests using N for the normality, AC for the
autocorrelation, and H for the heteroskedasticity. If the model appeared to
be correctly specified, we then analyzed the stability of the parameters over
time. If we did not reject the hypothesis of stability either, we moved to
test specific restrictions on the parameters, and to estimate a more efficient
version of the model. In all the tests, we used the conventional 5% size. In
the tables in the Appendix, we presented both the P-values associated to
the tests and a summary of the equation estimated when the restriction was
imposed.
We considered the normality test first, because we used its result to
choose between the small or large sample version of the other tests. Since
the small and large sample statistics are asymptotically equivalent in large
samples, the asymptotic interpretation of the results is not affected. Admittedly,
due to the asymptotic nature of the Jarque and Bera test, and to the
small sample lower order bias in the estimation of autoregressive coefficients,
the reader may be cautious in the interpretation of the results, especially,
when only portions of the dataset are used to estimate parameters
(like in the Chow stability test or in the estimates on sub-samples): notice
anyway that this is a problem that depends on the dimension of the sample;
so, indeed the same caveat applies to all the empirical analyzes present in