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(Czech Republic, Hungary, Slovakia, and Slovenia) that only experience one or two breaks. The estimated breaks are associated with political or institutional events of relevance in the transition process. Some of these events are common to all the countries, whereas others are idiosyncratic. For example, the degree of openness is larger for the small economies, so they are more exposed to external shocks and therefore exhibit more breaks. In the case of Poland, the higher number of breaks is due to the special transition strategy followed from the very beginning of the 1990s. In any event, structural breaks in the NAIRU of transition countries can be with a high degree of confidence associated with institutional changes coming from the implementation of market-oriented reforms. Further, the combination of techniques introduced in sections 3.4–3.6 has been exploited by Uctum, Thurston, and Uctum (2006) who assess fiscal performances in G7 and selected Latin American and Asian countries. In the paper questions related to fiscal performance and sustainability are considered. The authors find that traditional ADF tests of sustainability overwhelmingly fail to reject the non-stationarity of public debt in the troubled areas of the world. However, these tests suffer a major weakness as they are sensitive to structural breaks, which bias results towards not rejecting unit roots. For this reason the authors control for structural breaks by using Bai and Perron (1998, 2003a) methodology; it is found that the outcomes change drastically for a majority of countries. In sum, they find that: (i) The traditional unit root tests often overlook the corrective actions taken by many governments. Controlling for structural breaks changes the non-stationarity results dramatically among the countries. (ii) The estimation of a reaction function for governments, expanded by incorporating structural breaks, provides further evidence for significant active anti-debt policies among G7 countries, and to a lesser extent in the other regions. Finally, testing for multiple structural changes has recently taken also multivariate approach. Relevant techniques are covered for example by Perron and Qu (2007), and Kejriwal and Perron (2010). A detailed coverage of these technique is beyond the scope of this text, though.
3.7 NON-LINEAR STRUCTURE AND CONDITIONAL HETEROSKEDASTICITY In the preceding sections we presented various univariate time series models and tests. Although often quite diverse in their nature, all the presented models have one common feature. We have always assumed the variance of the error term conditional on the information revealed before the time t realizations (at the time t – 1), denoted as vart −1 [ε t ] = Et −1 [ε t2 ] , to be constant and equal to the Ukázka elektronické knihy, UID: KOS213979
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unconditional variance of εt, denoted as var[ε t ] = E[ε t2 ] . In the following sections devoted to non-linear structure and conditional heteroskedasticity this assumption will be relaxed. Instead of a constant conditional variance we will assume that the conditional variance is governed by an autoregressive, possibly combined with a moving average, process. This means that besides the parameters of the mean equation that govern the behavior of yt, we will have to also estimate the parameters of the AR or ARMA process that will govern the behavior of the conditionally heteroskedastic errors. This richer specification of the model does not occur at no cost. Instead of OLS we will need to apply the maximum likelihood estimation (MLE) procedure to estimate the autoregressive conditional heteroskedastic (ARCH) models. The MLE estimation requires distributional assumptions to be imposed on the error process εt. Such assumptions are obviously much stronger than the orthogonality condition needed for the OLS estimates to be unbiased and consistent. ARCH models are very useful, particularly in the field of finance. Financial time series are prone to exhibit periods of high and low volatility. Exactly such behavior can be modeled using conditional heteroskedastic disturbances. Moreover, it is the variance of financial time series that is of a great importance. If you consider a time series of asset returns, then the returns are represented by the mean, and the risk associated with holding a particular asset is measured by the variance of the series. The optimal portfolio is often chosen within the mean-variance framework, as for example in the Capital Asset Pricing Model (CAPM) of Sharpe (1964), Lintner (1965), and Mossin (1966). Therefore, the capacity of ARCH models to model and predict the changing variance of financial time series is of great importance. This is even more true when this type of model is used by analytical departments of financial firms where vast amounts of money are at stake, unlike in purely academic exercises. As a result, we typically see time series of asset returns, exchange rates, or other high frequency data estimated with ARCH models. Nevertheless, ARCH models began to be used in other areas as well due to their popularity in finance. After all, in the seminal paper by Engle (1982), where ARCH models were introduced, the author used an ARCH model to estimate a time series of U.K. inflation. Below, we will first review the concept of conditional and unconditional expectations and then introduce a simple ARCH model and its generalized extensions (GARCH models). Following this, we will present procedures and tests that enable the detection of conditional heteroskedasticity, propose an algorithm that enables the detection of conditional heteroskedasticity and then identify and estimate the most parsimonious ARCH or GARCH model. Finally, we will briefly list various popular extensions and modifications of ARCH models.
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3.7.1 CONDITIONAL AND UNCONDITIONAL EXPECTATIONS
Understanding the difference between conditional and unconditional expectations is crucial for understanding ARCH models. For simplicity, let us focus on the conditional and unconditional expectations of yt time series governed by a simple AR(1) process. The identical technique can be used to form conditional and unconditional expectations of the variance of an ARCH error term εt. Consider the following AR(1) process:
yt = a0 + a1 yt −1 + ε t ,
Et −1 [ yt=] E[ yt | yt −1 , yt −2 ,...=] E[ yt | yt −1=] E[a0 + a1 yt −1 + ε t=] a0 + a1 yt −1 ,
with 0 < a1 < 1 and εt independent disturbances with mean 0 and variance σ2. Clearly yt depends only on its past values and on the current realization of εt. Thus, the information needed to form conditional expectations is only that of the past values of yt. Moreover, with an AR(1) process the information about all past realizations of yt is fully comprised in the realization of yt–1. The conditional expectation of the mean and variance of yt can therefore be written as vart −1 [ yt =] E[( yt − E( yt ))2 | yt −1=] E[( yt − (a0 + a1 yt −1 ))2 | yt −1=] E[ε t 2=] σ 2 .
With unconditional expectations the situation is quite different. When forming unconditional expectations, we must act as if the only information we have is that about the mean and variance of the error process εt, whose realizations are independent. We do not know anything about the past realizations of yt. Therefore, to form unconditional expectations we must first solve the AR(1) equation in terms of εt. The unconditional expectation of the mean and variance of yt can be written as
∞
E[ yt ] = E[(1 − a1 L)−1 (a0 + ε t )] = E a0 / (1 − a1 ) + ∑ a1i ε t −i = a0 / (1 − a1 ) ,
2 ∞ i var[ yt ] = E[( yt − E( yt )) ] = E ∑ a1ε t −i = σ 2 / (1 − a12 ) . i =0
i =0
2
Note that in the computation of unconditional variance we have used the fact that errors εt are independent and thus uncorrelated across time. In the expressions for the unconditional mean and variance of yt we had to use all the past realizations of εt. That is why the unconditional expectations are sometimes Ukázka elektronické knihy, UID: KOS213979