Econometrics II

You can find here the correction to the erratum appearing in the notes entitled “An Example: GARCH(1,1) Process” concerning part of the ARMA(1,1) representation.

You can find here and here corrections and derivations related to the notes entitled “Notes on the EGARCH(1,1) Model”. Those concern typos, as well as an erratum found in part of the derivation of the auto-covariance of the squares. Specifically, in the case where κ=1, the derivation should take into account that the product is empty

We have been occupied with further issues concerning the GARCH(1,1)processes, as for example that they cannot simultaneously have symmetric marginal distribution and exhibit phenomena of negative dynamic asymmetry (under the relevant moment existence assumptions), their extension to finite and arbitrary orders and the plausibility of the semi-parametric Gaussian QMLE in their context.

We have been occupied with the examination of the limit theory of the OLSE for the autoregressive parameter for

We have been occupied with the definition of the GARCH(1,1) process, the existence and uniqueness of a stationary and ergodic solution to the relevant stochastic recurrence equation, the issue of the weak stationarity of the solution, and the issue of the ARMA(1,1) representation of the squared process under the appropriate restrictions. We thus obtained a non-linear process for which the properties of strict stationarity and ergodicity stem from weaker conditions compared to them that imply wea

We considered the example of a simple indirect inference estimator for an invertible MA(1) model, via the use of a stationary AR(1) model and the corresponding OLSE as auxiliary model and estimator respectively. You can find notes on the above here. You can also find notes for our brief comment on the extension of the ARMA type models here.

We have begun the study of conditional heteroskedasticity by providing a general definition. You can also find notes on the above here.

Exercise

How would th

We have examined issues concerning the semi-parametric estimation of ARMA type models introducing the Gaussian Quasi Maximum Likelihood Estimator (Gaussian QMLE), in cases where the MA component is non trivial (exercise: show that the Gaussian QMLE coincides with the OLSE when the MA component is trivial, the orders of the MA and AR components are known and this information is used). We also abstractly discussed the numerical nature of its derivation, and were briefly and not rigorously occupied

Given a well defined ARMA process (e.g. the solution of the relevant recursion when the UDC holds for the Φ polynomial), we were occupied with the issue of what property is implied when the Θ polynomial satisfies the UDC, thus obtaining the notion of invertibility, which is equivalent to that the white noise process is specified as a linear causal process with respect to the ARMA process with absolutely summable coefficients that are absolutely bounded from above by the coefficients of a geometr

We were occupied with algebraic properties of the ring of formal power series w.r.t. the lag operator as well as some analytic properties that emerge when given relevant properties of the sequences of coefficients. These along with the preparations in the previous lectures on the class of causal linear processes, allowed us to easily describe conditions for the existence and properties of ARMA models as linear processes with absolutely summable coefficients, their weak stationarity, regularity a

Given the use of the afore-examined CLT in order to derive conditions that ensure that the OLSE in the particular context of the stationary and ergodic AR(1) model (i.e. stationarity, ergodicity, smd property, and moment and dependence properties that ensure the convergence of particular series for the white noise process) has the usual rate and it is asymptotically normal, we were occupied with the verification of the conditions and evaluated the asymptotic variance in the context of (a stricte

Even in the context of an AR(1) process with iid white noise, we have seen that the Lindeberg-Levy CLT is not applicable for the derivation of the rate and limiting distribution of the OLSE for the relevant coefficient (why?). Hence in order to tackle such a problem we require the establishment of a CLT applicable in such-like cases. Stationarity and ergodicity are not sufficient for such an establishment without further control of the rate of asymptotic independence of the elements of the under

Given the previous we have (descriptively) been occupied with the notion of ergodicity and the subsequent corollary of Birkhoff's LLN in the framework of stationarity and ergodicity. Given the introduction of the ergodic property in ithe innovations process of our example resulted to the string consistency of the OLSE via the use of the aforementioned LLN and the CLT (remember that unbiasness is not generally the case due to the failure of the strong exogeneity condition).

We have provided the d