Read an Excerpt
Methods for Applied Macroeconomic Research
By Fabio Canova Princeton University Press
Copyright © 2007 Princeton University Press
All right reserved. ISBN: 978-0-691-11504-7
Chapter One
Preliminaries This chapter is introductory and intended for readers who are unfamiliar with time series concepts, with the properties of stochastic processes, with basic asymptotic theory results, and with the principles of spectral analysis. Those who feel comfortable with these topics can skip directly to chapter 2.
Since the material is vast and complex, an effort is made to present it at the simplest possible level, emphasizing a selected number of topics and only those aspects which are useful for the central topic of this book: comparing the properties of dynamic stochastic general equilibrium (DSGE) models to the data. This means that intuition rather than mathematical rigor is stressed. More specialized books, such as those by Brockwell and Davis (1991), Davidson (1994), Priestley (1981), or White (1984), provide a comprehensive and in-depth treatment of these topics.
When trying to provide background material, there is always the risk of going too far back to the basics, of trying to reinvent the wheel. To avoid this, we assume that the reader is familiar with simple concepts of calculus such as limits, continuity, anduniform continuity of functions of real numbers, and that she is familiar with distributions functions, measures, and probability spaces.
The chapter is divided into six sections. The first defines what a stochastic process is. The second examines the limiting behavior of stochastic processes introducing four concepts of convergence and characterizing their relationships. Section 1.3 deals with time series concepts. Section 1.4 deals with laws of large numbers. These laws are useful to ensure that functions of stochastic processes converge to appropriate limits. We examine three situations: a case where the elements of a stochastic process are dependent and identically distributed; one where they are dependent and heterogeneously distributed; and one where they are martingale differences. Section 1.5 describes three central limit theorems corresponding to the three situations analyzed in section 1.4. Central limit theorems are useful for deriving the limiting distribution of functions of stochastic processes and are the basis for (classical) tests of hypotheses and for some model evaluation criteria.
Section 1.6 presents elements of spectral analysis. Spectral analysis is useful for breaking down economic time series into components (trends, cycles, etc.), for building measures of persistence in response to shocks, for computing the asymptotic covariance matrix of certain estimators, and for defining measures of distance between a model and the data. It may be challenging at first. However, once it is realized that most of the functions typically performed in everyday life employ spectral methods (frequency modulation in a stereo, frequency band reception in a cellular phone, etc.), the reader should feel more comfortable with it. Spectral analysis offers an alternative way to look at time series, translating serially dependent time observations into contemporaneously independent frequency observations. This change of coordinates allows us to analyze the primitive cycles which compose time series and to discuss their length, amplitude, and persistence.
Whenever not explicitly stated, the machinery presented in this chapter applies to both scalar and vector stochastic processes. The objects of interest in this book are defined on a probability space ([??], F, P), where [??] is the space of possible state of nature x, F is the collection of Borel sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [[??].sup.m] psi]] x [[??].sup.m]psi]] x ..., and P[??]s a probability function for x that determines the joint distribution of the vector of stochastic processes of interest. The notation [{[y.sub.t]](x)}.sup.[infinity].sub.t] = -[infinity] indicates the sequence {..., [y.sub.0](x), [y.sub.t] (x), ..., [y.sub.t] (x), ...}, where, for each t, the random variable [y.sub.t] (x)[psi] is a measurable function of the state of nature x, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [??] is the real line. We assume that at each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] belongs to [F.sub.t], so that any function h([y.sub.[tau]]) will be "adapted" to [F.sub.t]. To simplify the notation, at times we write {[y.sub.t] (x)} or [y.sub.t]. A normal random variable with zero mean and variance [[summation].sub.y] psi]] is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a random variable uniformly distributed over the interval [a.sub.1], [a.sub.2]] is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII][??][a.sub.1], [a.sub.1]. Finally, "i.i.d." indicates identically and independently distributed random variables and a white noise is an i.i.d. process with zero mean and constant variance.
1.1 Stochastic Processes
Definition 1.1 (stochastic process). A stochastic process [{[y.sub.t](x)}.sup.[infinity].sub.t=1[psi]] is a probability measure defined on sets of sequences of real vectors (the "paths" of the process).
The definition implies, among other things, that the set [??] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for arbitrary [??] [member of] [??] and t] psi] fixed, has well-defined probabilities. In other words, choosing different [??] [member of] [??] for a given t, and performing countable unions, finite intersections, and complementing the above set of paths, we generate a set of events with proper probabilities. Note that [y.sub.t] [psi]] is unrestricted for all [tau][psi][less than or equal to] t: the realization need not exceed [??] only at t. Observable time series are realizations of a stochastic process {[y.sub.t](x)}, given [x.sup.2]. Two simple stochastic processes are the following.
Example 1.1. (i) {[y.sub.t](x)} = [e.sub.1] cos (t x[e.sub.2]), where [e.sub.1[psi]] and [e.sub.2[psi]] are random variables, [e.sub.1[psi]] > 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], t > 0. Here [y.sub.t]] psi]] is periodic: [e.sub.1[psi]] controls the amplitude and [e.sub.2[psi]] the periodicity of [y.sub.t].
(ii) {[y.sub.t](x)}is such that P][y.sub.t]]psi]] = [+ or -] 1][psi]=0.5 [psi]for all t. Such a process has no memory and flips between -1 and 1 as t] psi] changes.
Example 1.2. It is easy to generate complex stochastic processes from primitive ones. For example, if, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [??](0, 1), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [??](0, 1), and [e.sub.1t]psi]] and [e.sub.2t]psi]] are independent of each other, [y.sub.t] [psi]] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a stochastic process. Similarly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i.i.d. (0, 1) is a stochastic process.
1.2 Convergence Concepts
In a classical framework the properties of estimators are obtained by using sequences of estimators indexed by the sample size, and by showing that these sequences approach the true (unknown) parameter value as the sample size grows to infinity. Since estimators are continuous functions of the data, we need to ensure that the data possess a proper limit and that continuous functions of the data inherit these properties. To show that the former is the case, one can rely on a variety of convergence concepts. The first two deal with convergence of the sequence, the next with its moments, and the last with its distribution.
1.2.1 Almost Sure Convergence
The concept of almost sure (a.s.) convergence extends the idea of convergence to a limit employed in the case of a sequence of real numbers.
As we have seen, the elements of the sequence [y.sub.t](x)[psi] are functions of the state of nature. However, once x = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] drawn, {[y.sub.1]([bar.x]), ..., [y.sub.t]([bar.x]), ...} looks like a sequence of real numbers. Hence, given x=[bar.x], convergence can be similarly defined.
Definition 1.2 (a.s. convergence). [y.sub.t](x)[psi] converges almost surely to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for x[psi][member of] [[??].sub.1][[subset].bar] [??], and every [epsilon] > 0. [psi]
According to definition 1.2 {[y.sub.t](x)} converges a.s. if the probability of obtaining a path for [y.sub.t]] psi] which converges to y(x) after some T] ??] 1. The probability is taken over x. The definition implies that failure to converge is possible, but it will almost never happen. When [??] is infinite dimensional, a.s. convergence is called convergence almost everywhere; sometimes a.s. convergence is termed convergence with probability 1 or strong consistency criteria.
Next, we describe the limiting behavior of functions of a.s. convergent sequences.
Result 1.1. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let h be an n x 1 vector of functions, continuous at y(x). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [psi]
Result 1.1 is a simple extension of the standard fact that continuous functions of convergent sequences are convergent.
Example 1.3. Given x, let {[y.sub.t](x)} = 1 - 1/t] psi] and h([y.sub.t](x))[psi]=[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then h([y.sub.t](x)) is continuous at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Exercise 1.1. Suppose {[y.sub.t](x)} = 1/t]psi] with probability 1-1/t]psi] and {[y.sub.t](x)}=t] psi] with probability 1/t. Does {[y.sub.t](x)} converge a.s. to 1? Suppose h([y.sub.t])[psi]= (1/T)x [left arrow] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. What is its a.s. limit?
In some applications we will be interested in examining situations where a.s. convergence does not hold. This can be the case when the observations have a probability density function that changes over time or when matrices appearing in the formula for estimators do not converge to fixed limits. However, even though h([y.sub.1t](x)) does not converge to h(y(x)), it may be the case that the distance between h([y.sub.1t](x)) and h([y.sub.2t](x)) becomes arbitrarily small as t] psi][right arrow][infinity], where {[y.sub.2t](x)} is another sequence of random variables. To obtain convergence in this situation we need to strengthen the conditions by requiring uniform continuity of h (for example, assuming continuity on a compact set).
Result 1.2. Let h be continuous on a compact set [[??].sub.2[psi]][member of][[??].sup.m]. Suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 0 and there exists an [member of] > 0 such that, for all t > T, [y.sub.2t] psi]] is in the interior of [[??].sub.2], uniformly in t. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [psi]
One application of result 1.2 is the following: suppose {[y.sub.1t](x)}is some actual time series and {[y.sub.2t](x)} is its counterpart simulated from a model where the parameters of the model and x are given, and let h be some continuous statistics, e.g., the mean or the variance. Then, result 1.2 tells us that if simulated and actual paths are close enough as t] psi][right arrow] [infinity] statistics generated from these paths will also be close.
1.2.2 Convergence in Probability
Convergence in probability is a weaker concept than a.s. convergence.
Definition 1.3 (convergence in probability). If there exists a y(x) < [infinity] such that, for every [member of] > 0, P]x[psi][parallel][y.sub.t](x)-y(x)[parallel] < [member of] [right arrow] 1 for t] psi][infinity], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [psi]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is weaker than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because in the former we only need the joint distribution of ([y.sub.t](x), y(x)) not the joint distribution of ([y.sub.t](x), [y.sub.[tau]](x), y(x)), [for all][tau] > T. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies that it is less likely that one element of the {[y.sub.t](x)} sequence is more than an [member of] away from y(x)as t] psi][right arrow][infinity]. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies that the T] psi] the path of {[y.sub.t](x)} is not far from y(x) as T] psi][right arrow][infinity]. Hence, it is easy to build examples where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] does not imply [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Examples 1.4. Let [y.sub.t] psi]] and [y.sub.[tau][psi]] be independent [for all]t, [tau], let [y.sub.t]] psi]] be either 0 or 1 and let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] j > 0, so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is because the probability that [y.sub.t] psi]] is in one of these classes is 1/j]psi] and, as t]psi][right arrow][infinity], the number of classes goes to infinity. However, [y.sub.t] psi]] does not converge a.s. to 0 since the probability that a convergent path is drawn is 0; i.e. the probability of getting a 1 for any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], j > 1, is small but, since the streak [2.sup.j-1[psi]]+ 1, ... , [2.sup.j] psi]] is large, the probability of getting a 1 is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which converges to 1 as j] psi] goes to infinity. In general, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 0 is too slow to ensure that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [psi].
Although convergence in probability does not imply a.s. convergence, the following result shows how the latter can be obtained from the former.
Result 1.3. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there exists a subsequence [y.sub.tj](x) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see, for example, Lukacs 1975, p.48).
Intuitively, since convergence in probability allows a more erratic behavior in the converging sequence than a.s. convergence, one can obtain the latter by disregarding the erratic elements. The concept of convergence in probability is useful to show "weak" consistency of certain estimators.
Example 1.5. (i) Let [y.sub.t] [psi]] be a sequence of i.i.d. random variables with E([y.sub.t]]) < [infinity]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Kolmogorov strong law of large numbers).
(ii) Let [y.sub.t] [psi]] be a sequence of uncorrelated random variables, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Chebyshev weak law of large numbers).
In example 1.5 strong consistency requires i.i.d. random variables, while for weak consistency we just need a set of uncorrelated random variables with identical means and variances. Note also that weak consistency requires restrictions on the second moments of the sequence which are not needed in the former case.
The analog of results 1.1 and 1.2 for convergence in probability can be easily obtained.
(Continues...)
Excerpted from Methods for Applied Macroeconomic Research by Fabio Canova Copyright © 2007 by Princeton University Press. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
