Michio Hatanaka

Several efficient two-step estimators for the dynamic simultaneous equations model with autoregressive disturbances

Abstract Several asymptotically efficient methods are suggested on both the full and the limited information approach to estimate the simultaneous equations model in which the lagged endogenous variables and the autoregressive disturbances coexist. They are two-step procedures and do not involve iterations. A method is suggested also for the case where any portion of the autoregressive parameter matrix is specified to be zero. Since the consistency and efficiency depend upon the asymptotic, local identifiability, the necessary and sufficient condition is derived for it. It does not depend on the exclusion of the lagged endogenous variables.

On the efficient estimation methods for the macro-economic models nonlinear in variables

Abstract All the macro-economic models have the nonlinearity in variables within their simultaneous equations systems. I propose a full information estimation method for such models. The method is (i) asymptotically efficient, (ii) feasible in the contemporary computer technology as it consists of calculations very much like the nonlinear multipliers, and (iii) hopefully applicable to the undersized sample case which prevails in the macro-economic model building. Though two other methods are also investigated, one is found to be asymptotically inefficient, and another turns out to be inapplicable to the undersized sample case.

Multicollinearity and the Estimation of Low-Order Moments in Stable Lag Distributions

Publisher Summary This chapter presents arguments that for typical economic time series data, low-order moments of lag distributions in many cases can be estimated more precisely than short-run effects. The transformation from short-run effects to moments, again for typical time series data in economics, orders the precision inherent in the design matrix so that one can expect to estimate the long-run effect (sum of ordinates in the lag distribution) with greatest precision, the first-order moment (mean lag) with next greatest precision, and so on. Thus, it is argued that much can be learned with available data by redirecting attention away from short-run effects to a different parameter space. There are three reasons for this redirection of interest. First, low-order moments are of considerable policy interest. Often the long-run effect and mean lag are sufficient summary statistics for the investigators purpose. Second, with the focus on estimation of short-run effects, priors of one kind or another on the shape of the lag distribution typically have been imposed on data, usually in ad hoc ways. Many quite flexible probability density functions require knowledge of only a few low-order moments for complete specification. Third, in cases where smoothness priors have been imposed a priori, least squares estimation of low-order moments of lag distributions serves as an additional check on specification.

Statistics from the Data Covariance Matrix

We shall utilise the information contained in the data covariance matrix to investigate the co-trending relations. Given an eigenvalue of the matrix, (i) the eigenvalue itself, (ii) the eigenvector associated with it, and (iii) the principal component associated with it form one set of statistics. Altogether n sets of statistics are available, and they are ordered in descending order in terms of the eigenvalues. In Section 3.1 below, the relations among the deterministic trends are classified into those that hold among the stochastic trends as well and those that do not hold among the stochastic trends. Let r1 and r2, respectively, represent the numbers of the former and the latter. By definition we have that r = r1 + r2. Section 3.2 will show that the first n - r sets of statistics provide the information on the common deterministic trends. It will be shown in Section 3.3 that the next r2 sets of statistics represent the relations that hold among the deterministic trends but do not hold among the stochastic trends. Section 3.4 will show that the last r1 sets of statistics are related to the relations that hold among both the deterministic trends and the stochastic trends. Sections 3.5 and 3.6 will indicate how these results provide the bases for the testing procedures which will be given after Chapter 4. Section 3.6 contains an overview of the rest of this book.

Unit Root Tests

We shall prove that the minor, nonstandard terms in principal components do not affect the applicability of the unit root tests. Both the univariate and the multivariate unit root tests are considered. The univariate unit root test, which is the well known method in Perron (1989), is expected to be applied to each individual principal component in Group ⊥, one principal component in Group 2 when r2 = 1, and each individual component in Group 1. The test will be described in Section 5.1. It will be followed in Sections 5.2 and 5.3 by our proofs of its applicability to the principal components in Group ⊥, and in Section 5.4 by its applicability to Group 2 and Group 1. The multivariate unit root test is a test for zero cointegration rank in Johansen et al. (2000). It is expected to be applied to the entire set of principal components in Group 2 when r2 ≥ 2. The method will be described in Section 5.5, and our proof of its applicability to Group 2 will be given in Section 5.6. So far we have assumed that the correct division among Groups ⊥, 2, and 1 is known, and that the division is used to assign the correct unit root tests to the principal components. In Section 5.7, we shall analyse the effects that incorrect divisions among the three Groups would bring about.

Sequential Decision Rule

We are now ready to consider how to determine n - r, r2, and r1. In Chapter 3 i = 1, ..., n is the running index to denote the descending order of eigenvalues of the data covariance matrix. i is also associated with the principal component that corresponds to the i-th eigenvalue. Given n - r, r2, and r1, the principal components with i = 1, ..., n - r, with i = n - r + 1, ..., n - r + r2 ≡ n - r1, and with i = n - r1 + 1, ..., n each reveal asymptotically distinctive features. Thus i = 1, ..., n has been divided in three groups, Group ⊥ that consists of i = 1,..., n - r, Group 2 that consists of i = n - r + 1, ..., n - r + r2 ≡ n - r1, and Group 1 that consists of i = n - r1+1,...,n.

Confidence judgment of the extrapolation from a dynamic money demand function

Abstract In order to check diagnostically the specification of a dynamic equation with autoregressive disturbances, methods are shown to construct a confidence interval on the extrapolated value at a time distant from the sample period, and also to construct a confidence belt on the extrapolated path. The methods are applied to investigate the stability of a demand function for money, which has been a controversial issue in the monetary economics.