Abstract
In this chapter, we take up two important applications, involving simultaneous equation, AR, and ARMA models, which are very important in several disciplines, such as economics, other social sciences, engineering and statistics. We partially discussed such topics in Chap. 4, in the context of difference equations, since simultaneous equation and AR models involve the use of difference equations and it is important to establish the nature and the properties of their solutions.
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Notes
- 1.
In econometrics, the least squares procedure is termed “Ordinary Least Squares” (OLS), for reasons that will become evident presently.
- 2.
In the special case where D is upper triangular and Σ is diagonal, it may be shown that L 1i ′ Dσ⋅i = 0 and, in this case only, the OLS estimator of the structural parameters is indeed consistent. Such systems are termed simply recursive.
- 3.
In order to avoid excessive notational clutter we shall use, in the following discussion, the notation Π i to indicate not C i D as defined in Eq. ( 12.3 ), but to mean ΠL 1i , as well as in all discussions pertaining to the estimation, limiting distributions and tests of hypotheses. In this usage Π i refers to the coefficients of X in the reduced form representation of Y i , i.e. the endogenous variables that appear as explanatory variables in the ith structural equation. We shall use the definition C i D , of Eq. ( 12.3 ), only in connection with issues of forecasting from the GLSEM.
- 4.
We shall discuss identification issues at some length at a later section.
- 5.
Notice that, in this alternative representation, consistency of the 2SLS and 3SLS estimators is obvious.
- 6.
Some readers may be confused by the terminology. To clarify, note that if q > 0, this means that we have specified that more parameters in B and C are zero than is allowable consistent with identification, i.e. relative to the number of restrictions that render the equation just identified. To test for their validity we need to allow for fewer restrictions, i.e. to increase m i and/or G i ; thus we would then be estimating a model with more parameters than before!
- 7.
As we shall see later this is not possible.
- 8.
This means that L 1i ∗ is what remains after we eliminate form I m the columns of L 1i and from I G the columns of L 2i .
- 9.
Note, incidentally, that S i ∗′ S i ∗ is the probability limit of \(L_{i}^{{\prime}}{Q}^{{\ast}{\prime}}{Q}^{{\ast}}L_{i}/T\).
- 10.
The symbol L i ∗ in this discussion corresponds to the symbol L i ∗∘ in Proposition 6.
- 11.
Noting that \(V _{11}^{-1} = \left (\mathop{\mathrm{plim}} _{T\rightarrow \infty }\frac{{Q}^{{\prime}}Q} {T} \right ) + L_{i}^{{\ast}}L_{i}^{{\ast}{\prime}}\), and that \(M_{xx} =\mathop{ \mathrm{plim}} _{T\rightarrow \infty }\frac{{X}^{{\prime}}X} {T}\), and their frequent use in the immediately ensuing discussion, we shall not, in our usage, distinguish between these probability limits and their definition for finite samples, letting the context provide the appropriate meaning. We do so only in order to avoid notational cluttering.
- 12.
Note that Θ i is an \(m + G \times m + G\) matrix of rank G.
- 13.
This section may be omitted, if desired, without any loss of ability to deal with the remainder of this volume.
- 14.
The expression in the left member of Eq. (12.49) is also referred to in the literature as the (asymptotic) Kullback information of θ0 on θ.
- 15.
We also include below the result for the unrestricted reduced form for ease of reference.
- 16.
In the forecasting literature, at least the one found in public discussions, the specification of the exogenous variables is referred to as “scenarios”.
- 17.
In the following discussion Π1 and Π2 refer to these entities as defined in Eq. (12.3).
- 18.
We use the symbol \(\mathcal{T}\) to denote the linear index set to avoid confusion with T, which was used repeatedly in this chapter to denote the length of the sample.
- 19.
If the sequence does not have zero mean one uses, instead of X t , X t − μ, where μ is the constant mean of the sequence, or \(x_{t} -\bar{ x}\) if we are using a realization to estimate autocovariances.
- 20.
There is a convention in notation the reader may not be aware of, viz. the lag k is, by this convention, the difference between the index of the first factor, X t+k and the second factor X t . Thus the autocovariance at lag − k is EX t X t+k or, of course, EX t−k X t . Since the two terms under expectation commute it is evident that \(c(k) = c(-k)\). This is of course obvious also by the very definition of stationary sequences. To avoid such minor confusions one generally defines \(c(k) = EX_{t+\vert k\vert }X_{t}\) whether k is positive or negative.
- 21.
Even though dividing by N, instead of N − k, renders this estimator biased, it is the preferred practice in this literature because it preserves the positive semi-definiteness of the autocovariance matrix or function.
- 22.
Note that when the u sequence is also N(0,σ2), Eu t 4 = 3σ4!
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Dhrymes, P.J. (2013). GLSEM and TS Models. In: Mathematics for Econometrics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8145-4_12
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