Abstract
In this chapter we deal with econometric applications of (vector) difference equations with constant coefficients, as well as with aspects of the statistical theory of time series and their application in econometrics.
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- 1.
The superscript H in Eq. (5.3) indicates that it is the the solution to the homogeneous part.
- 2.
In the statistical and, more generally, the mathematical literature this operator is termed the backward operator and is denoted by B. But lag operator is a more convenient term and has a long tradition in the econometrics literature, so we retain it and shall use it exclusively wherever appropriate.
- 3.
The dimension of A ∗ is mr × mr, and that of ζ t is mr × 1.
- 4.
By convention \(\mathrm{Cov}(x_{t+h},x_{t}) = E(x_{t+h} - \mu )(x_{t} - \mu ) = c([t + h] - t) = c(h)\). But since \(E(x_{t+h} - \mu )(x_{t} - \mu ) = E(x_{t} - \mu )(x_{t+h} - \mu ) = c(t - [t + h]) = c(-h)\) the definition is justified.
- 5.
Here R is the real line. Although the coefficients can be allowed to be complex as well, this is very uncommon in economics. The notation ∼ generally means “is equivalent”; in this particular use it is best for the reader to read it as “behaves like” or “has the distribution of”.
- 6.
In the future we shall use exclusively the econometric terminology.
- 7.
The convergence in the right member of the equation below is, depending on the assumptions made, either absolute convergence of the sequence of the coefficients of the errors and finiteness of their first absolute moment, E|ε t |, for all t ∈ T, or is convergence in mean square, or quadratic mean a concept that will be introduced at a later chapter dealing with the underlying probability foundations of econometrics.
- 8.
In the interest of simplicity of presentation we have omitted or ignored the exogenous variable component, containing p t⋅ C 0, and lags thereof; the latter, however, can be easily added at the cost of a bit more complexity in the presentation.
- 9.
The notation \(E_{u}[E[u_{t}\vert u_{t-1}]\) means that, because u t 2 is both a function of \(u_{t-1}^{2}\) and ε t 2, in taking the expectation \(E(u_{t}^{2})\) we first take the expectation conditional on u t−1 and then take the expectation with respect to u t .
- 10.
This feature is responsible for the name of the model ARCH, which stands for autoregressive conditional heteroskedasticity.
- 11.
The property in question is that if ξ n is a sequence of random variables converging in probability to ξ, written \(\mathop{\mathrm{plim}} _{n\rightarrow \infty }\xi _{n} = \xi \), and if g is a continuous (or even a measurable) function then \(\mathop{\mathrm{plim}} _{n\rightarrow \infty }g(\xi _{n}) = g(\xi )\), provided the latter is defined. For a proof of this see Dhrymes (1989), p. 144 ff. These issues will also discussed in a later chapter.
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Dhrymes, P.J. (2013). DE Lag Operators GLSEM and Time Series. In: Mathematics for Econometrics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8145-4_6
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