Econometrics II

We have defined and been occupied with the definition of the GARCH(1,1) process, the existence and uniqueness of a stationary and ergodic solution to the relevant system of recursion, 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. You can find notes on this issue here. We have also begun our occupation on the limit theory of the OLSE for the autoregressive parameter in the context of an

We were occupied with a linear model with instrumental variables where the orthogonality conditions involved the residuals and a matrix of instrumental variables. Under appropriate assumptions we derived the GMM estimator and its asymptotic properties in two cases. Firstly, in the case where the errors are uncorrelated and secondly in the case where the errors are serially correlated. We observed that the asymptotic variance of the GMME depends on the choice of the weighting matrix, so we addres

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.  

We studied issues concerning the ARMA(1,1) process using Matlab. Semi-parametric estimation of ARMA models when the orders are known was addressed, evaluating the Gaussian QMLE and discussing the numerical nature of its derivation. In the case where the order of the process was unknown we were occupied with the use of the Bayesian Information Criterion (BIC) in order to select between candidate models. You can find notes on the latter here. A Matlab code for all the above can be found here.

We were occupied further with issues concerning statistical inference in ARMA models in the framework of correct statistical specification and of known unit variance for the white noise process. We pointed out that in the context of general AR models the extraction of the asymptotic properties of the OLSE can be similar to the one we have taken in the case of the AR(1) model modulo technical details of essentially multivariate nature that are not present in the latter case. We have studied issue

We examined the MA(1) process built on a stationary ergodic white noise. We were occupied with a GMM estimator using the sample first order autocovariance. In the case that the white noise process is comprised by i.i.d. variables with unit variance and finite fourth moment we examined the asymptotic properties of the GMM estimator. We finally noticed that in the case that we can't prove that the process we are interested in, is a s.m.d. in order to use the appropriate CLT for s.m.d. processes,

We were further occupied with the example of the ARMA(1,1) process by deriving the solution and the relevant properties which were shown to comply with the results of our general theorem.

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 pr

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 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 and short memory, and further conditions f

We occupied ourselves with an example of a linear model with endogeneity problem, in the context of which we examined the inconsistency of the OLS estimator and the consistency and asymptotic distribution of the IV estimator. You can find notes here.

We used 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 has the usual rate and it is asymptotically normal. We verified the conditions and evaluated the asymptotic variance in the context of the aforementioned example, which showed that the presence of conditional heteroskedasticity may affect the asymptotic properties of the estimator, and thereby may also affect properties of inferential procedur