Chapter 1
Introduction

1.1 Motivation

In the economy, the relations between economic agents are subject to change. As the knowledge of production techniques improves, as the means of transportation allow for long-distance trade, as the society changes its preferences for certain goods or services, the structure of the economy varies accordingly. One of the goals of econometrics is to describe this structure of the economy, and to track possible changes in the relationships between the economic variables. If the basic structure of (a part of) the economy is understood, this structure can be used to construct a decision-theoretic framework for policy analysis.

Ever since econometrics started as a separate discipline in the 1930s, it has tried to describe the relations between economic agents. Originally, only small data sets with linear relations could be handled, but quickly knowledge and technology improved, allowing for the analysis of larger regression models, or of time series using e.g. stochastic difference equations and distributed lag models.

Gradually models became more involved, more flexible, and, as a result, they were better able to describe more intricate relations between economic variables. A simpler model might e.g. assume a constant exchange rate between the currencies of two countries, and break down if after a number of months one currency would depreciate strongly with respect to the other currency. A more elaborate model could be able to faithfully describe a larger part of the economic structure, and be robust against changes in the value of the currency of a country.

One can distinguish between two types of changes in a model of the economy: First, there can be a change within the structure of the model, necessitating the use of a more elaborate model, to describe correctly this more general structure. Second, there can be a change in the data space, meaning that a parameter in the model changes in value, due to a change in a variable. In this thesis, the focus is on changes in time series models, of the structural type in the first chapters, and in the data space in chapter 5.

The next sections provide a short introduction to the research questions of the thesis. They also outline why more standard models and methods of analysing may not be sufficient for the question at hand, and why the choices proposed in this thesis promise to provide better, more detailed, results.

1.2 Inflation: Long memory, changing means and changing variances

In recent decades a vast literature has been published on the subject of deciding whether a macroeconomic time series is stationary (also called `integrated of order 0, I(0)') or non-stationary (often meaning `integrated of the first order, I(1)'). The discussion started by the article by Nelson and Plosser [1982], and continues in e.g. Perron [1989] and Perron and Vogelsang [1992]. This discussion is closely connected to the Box-Jenkins [Box and Jenkins, 1970,Box et al., 1994] methodology of modelling time series. In this methodology, the time series y_t is modelled as a linear function of past observations y_t-1, .., y_t-p and present and past disturbances e_t, .., e_t-q (p, q ³ 0). If the time series y_t is non-stationary, first differences are taken as often as necessary to get to a stationary series.

Part of the success of the ARIMA framework can be attributed to the fact that it provides a common framework with clear notation which can be used for many kinds of time series. In the continuation of this introduction, and in the first chapters of the thesis, the notation is used extensively. Therefore we introduce it here briefly. For a more detailed explanation, see e.g. Franses [1998].

The lag operator L is central in the notation. It serves to lag an observation y_t one time period, i.e. L y_t = y_t-1. With the lag operator, taking first differences of the data is written as y_t - y_t-1 = y_t - L y_t = (1-L) y_t. The first difference operation can be applied d times to get dth order differences: (1-L)^d y_t. The autoregressive (AR) part of the model filters the observations in a (linear) manner such that a function of only disturbances results, e.g. f(e_t, .., e_t-q) = y_t - f₁ y_t-1 - .. - f_py_t-p = (1-f₁ L - .. - f_p L^p)y_t. Past and present disturbances influence the observation through the so-called moving average (MA) part of the model, with f(e_t, ..,e_t-q) = e_t + q₁ e_t-1+ .. + q_qe_t-q = (1 + q₁ L + .. + q_q L^q)e_t. Lastly, the basic assumption is that the disturbances e_t are independently and identically distributed with variance s_e². With all of the elements taken together, the notation of an ARIMA(p,d,q) model is

F(L) (1-L)^d y_t

= Q(L) e_t,
F(L)

= 1-f₁ L - ¼- f_p L^p,
Q(L)

= 1 + q₁ L + ¼+ q_q L^q,
(1.1)
e_t

~ i.i.d. (0, s_e²).

For ease of notation, the autoregressive and moving average polynomials are denoted by F(L) (of degree p) and Q(L) (of degree q) respectively. These polynomials have their roots outside the unit circle, see e.g. Box et al. [1994] for details.

The parameter d governs the stationarity of the series. In the original framework of Box and Jenkins [1970], d was restricted to be integer, with d=0 for stationary series or d=1 for non-stationary, I(1) series. In a simple ARIMA(0,d=0,0) model, y_t = e_t, and clearly the model is stationary. When d=1, the model becomes (1-L)y_t = e_t, or y_t = y_t-1 + e_t = å_i=0^te_i. Such a model is clearly nonstationary, as the effect of a shock never dies out. In Granger and Joyeux [1980] the generalization to non-integer values of d was introduced. It can be derived that, for both integer¹ and non-integer d, the dth order difference operator can be written as

(1-L)^d = ¥
å
k=0
æ
ç
è

d

k
ö
÷
ø (-1)^k L^k = ¥
å
k=0
G(d+1)
G(k+1) G(d-k+1)
(-1)^k L^k.
(1.2)
In Beran [1994] the effect of fractional integration on the long term autocorrelation is given: If 0 < d < [ 1/2], the autocorrelation of an ARFIMA(0,d,0) model at long lags k can be approximated as

r(k) ~ G(1-d)
G(d)
|k|^2d-1, (k ® ¥).
(1.3)
The autocorrelations at long term lags go down to zero at a (slow) hyperbolic rate. This contrasts with the original case of the stationary ARIMA(p,0,q) model, where the autocorrelations for large k decrease at a (quicker) exponential rate,

r(k) ~ c g^k (k ® ¥),
(1.4)
see e.g. Greene [1990].

Figure 1.1: U.S. monthly inflation, all items, 1959-1999, with autocorrelation

Figure 1.2: Theoretical and simulated autocorrelation function of I(1), ARMA(1,1) and I(d) series

For inflation series such as the U.S. inflation² in the top panel of figure 1.1, it is difficult to decide if the series is stationary (in the sense that d=0 in (1.1)) or not. If it were stationary, then observing a prolonged period with higher inflation as in the seventies is not likely. On the other hand, if inflation were I(1), then the implication is that (the logarithm of) prices are integrated of the second order. Except for periods of hyperinflation, such behaviour is usually not observed for price indices (see Hassler and Wolters [1995]).

The autocorrelation function (ACF) of the series (see the bottom panel of figure 1.1) can often help in getting an impression whether the series is stationary. Clearly the underlying inflation series is not integrated of the first order, as in that case the autocorrelation function stays at one perpetually, at least in theory. This theoretical ACF is drawn in the top panel of figure 1.2, together with the empirical sample autocorrelation function of a sample of length T=2000 of an I(1) model.³ Short memory ARMA models (that is, models which are integrated of order d=0) display autocorrelation functions which go down to zero at an exponential rate, see equation (1.4). By picking special values for f and q the autocorrelation function may diminish rather slowly. The second panel of figure 1.2 displays both the theoretical and a simulated sample ACF for an ARMA model with parameters [^(f)] = 0.98 and [^(q)] = -0.78 as estimated from the inflation data. Comparing this second panel of figure 1.2 to the sample autocorrelation function in figure 1.1 we see that the shape of the autocorrelation function is not yet quite right: At short horizons, both figures seem to display reasonably similar correlation, but at larger horizons, the correlation according to the ARMA model decreases too fast.

The long term correlation can be adjusted by allowing for fractional integration (see Granger and Joyeux [1980]), with d taking non-integer values. For values of d < [ 1/2] the resulting ARFIMA(p, d, q) model is still stationary (meaning that the series has a finite mean and variance), but with longer lasting correlations than in the short memory ARIMA(p, 0, q) model. When d ³ [ 1/2] the model is non-stationary. The third panel in figure 1.2 shows the autocorrelation a sample of an I(d) model, with d estimated at 0.36. Especially on longer horizons this model fits the sample ACF of inflation well; for the short-term correlation, the fit can be improved using AR and MA parameters.

In chapter 2 the ARFIMA framework is used for analysing inflation the G7 countries. Indeed the parameter d is found to be significantly different from zero. However, in these series changes in the mean seem to have occurred. The contribution of the chapter is the investigation into the effect of these possible changes in the mean, due to the oil crises, on the amount of integration present in the data.

The chapter starts with a simulation exercise, using a simple AR(1) and ARFIMA(1,d,0) model as data generating process, to investigate what could be the effect of a neglected break in the generated series on the estimate for the parameter d. It is found that neglecting a break leads to a positive bias in the estimate [^d]. As a consequence, when we take the changes in the mean of inflation during the 1970s into account for the G7 countries, we find less evidence for fractional integration.

The importance of estimating the degree of integration correctly lies in the consequence for predictions. Policy makers are in general interested in getting an impression of future inflation, both at short (e.g. 1 month) and longer (say two-year) horizons. With a stationary short memory series one can be quite sure that even at longer horizons the series will stay close to the overall mean: Deviations are temporary as the effect of shocks dies out quickly. For non-stationary series, the uncertainty grows with the horizon: The series may well wander off indefinitely, moving further away from the present value for longer horizons. Long memory time series, with 0 < d < 1, take up intermediate positions between I(0) and I(1) models, with the variance increasing unboundedly with the horizon if d > [ 1/2].

This effect of the degree of integration on predictions of inflation is the topic of chapter 3. U.S. core inflation is predicted up to 24 months ahead, using a range of ARFIMA models, with and without explanatory variables serving as leading indicators. The predictions are compared both on the basis of the precision of the point forecasts, as on the widths of the forecast intervals. We find that the width of forecast intervals derived from ARFIMA models appears to be correct, though we have to allow for temporary shifts in the mean inflation in the seventies and for lower variance of inflation in the Volcker-Greenspan period, after 1983. Non-stationary I(1) models result in forecast intervals which are too wide.

1.3 Exchange rate: Varying trend and variance

From inflation and the internal devaluation of money it is a small step towards exchange rates, governing the external value of money [Bos, 1969]. The exchange rates of the German DMark and Japanese Yen vis-a-vis the U.S. Dollar (1982 º 1) are displayed in figure 1.3, together with the cumulative interest rate differentials. Where there is a great controversy about the degree of integration of inflation rates, the opposite is true for exchange rates: Economists agree that (the logarithms of) exchange rates are integrated of the first order. Also, as with many financial time series, it is accepted that exchange rate returns are heteroskedastic, with periods of high volatility followed by more tranquil periods. There is less agreement on the correlation structure of the returns. Theoretically, the expected return should equal the interest rate differential between the home and foreign countries, with no further predictability. However, this uncovered interest rate parity is not found to hold convincingly on daily return data; also in figure 1.3 no clear link between the evolution of the exchange rates and the interest rate differential is found. Does this mean that there is no correlation in the exchange rate returns at all?

Figure 1.3: Scaled DM/USD and Yen/USD exchange rates, together with the cumulative return of the interest rate differential

Table 1.1: Estimation results for daily DM/USD exchange rate return data


Parameter	AR		FI		ARMA		ARFIMA
f	0.016	(0.015)			0.857	(0.200)	0.993	(0.802)
d			0.021	(0.012)			0.016	(0.037)
q					-0.844	(0.209)	-0.991	(0.832)
s_e²	0.460		0.460		0.460		0.460
r(f, q)					-0.9992		-0.9840
Note: Maximum likelihood estimation results on demeaned daily DM/USD exchange rate returns 1/1/1982-31/12/1997, with standard errors, calculated using the ARFIMA package [Doornik and Ooms, 1999]. Last row reports the correlation between estimates of f and q.

Table 1.1 reports the maximum likelihood results of estimating basic ARFIMA models on the exchange rate returns. From the estimate of f for the AR model it is seen that there is very little autocorrelation in the returns. Estimating the parameter d of an ARFIMA(0,d,0) model shows that indeed the order of integration of the returns is approximately 0, no clear sign of long memory is found in the data. The results of the ARMA model deserve some extra attention. At first sight, results are significant with large values for the AR and MA parameters. At second sight, it may strike the researcher that this is a clear case of (near) root cancellation, with the AR polynomial (1-fL) (nearly) cancelling against the MA polynomial (1+qL).⁴ Allowing for fractional integration as in chapters 2- does not alter this result.

In chapter 5 (chapter 4 is referred to below), exchange rates and the interest rates of the home and foreign countries are taken as input for deciding if a company is expected to be better off hedging its currency exposure or not. Hedging is advisable for companies with large exposures in a foreign currency when the expected return of the currency is low (taking the uncertainty of the return into account); when a positive return is expected, it is often better not to hedge. For building such a decision framework, the AR(F)IMA models do not serve well, as we saw from the imprecise parameter estimates in table 1.1. Therefore, two alterations in the modelling strategy are made:

We use a structural model to model the varying local mean of the exchange rate returns (or the local trend of the logarithm of the exchange rate);

We use a Bayesian approach to incorporate the large parameter uncertainty concerning the parameters modelling the local mean/trend.

The first change is meant to provide a better description of the important characteristics of the data: For a hedging decision, we want to separate underlying trending⁵ behaviour from the irregular component of the disturbances. Also, in the structural model it is possible to have only the variance of the observation equation varying over time, leaving the transition equation (1.6) untouched. The exchange rate returns are modelled as

s_t

= m_t + e_t, Return = Expected return + Disturbance,

(1.5)

m_t

= rm_t-1 + h_t, Expected return = r× Previous expectation + Disturbance.

(1.6)

With a so-called structural model [Harvey, 1989], the time varying mean return contains information on the trending of the exchange rate levels. When the disturbances are Gaussian with constant variance, the structural model is an alternative representation for an ARIMA model.⁶

In chapter 5 we allow for time variation of the variance of the disturbance term e_t; details are provided in the chapter. With such varying variance, or with a heavy-tailed disturbance distribution, the direct link between the state space model and the ARIMA model no longer holds.

Figure 1.4: Uncertainty for the parameter r governing autocorrelation in equation (1.6). The figure displays the likelihood of the parameter r marginal of parameters s_e and s_h, and the marginal posterior density of r.

The second alteration in comparison to chapters 2-3 is the type of statistical analysis that is used. Where in the earlier chapters the data contained enough information to find estimates for the parameters, in chapter 5 the exchange rate data is little informative about the parameters in equation (1.6) describing the local trend. In section 5.6.1 a signal-to-noise ratio is reported, which describes the strength of the signal m_t in comparison to the noise e_t. This ratio is computed as around 2%, indicating the scarcity of information on m_t in the data.

In a classical statistical analysis, inference based on a vector of parameter estimates (possibly adjusted for results of a sensitivity analysis). The parameters are often estimated through the method of maximum likelihood. Figure 1.4 displays the likelihood⁷ of the parameter r governing the autocorrelation in (1.6). The classical analysis would use the top of the likelihood function, disregarding the large spread of the likelihood over values of r.

With a Bayesian analysis, a posterior density of the parameters is constructed (see the histogram in figure 1.4). The uncertainty about the value of r is taken up in the procedure for deciding whether we want to hedge or not, as the decision is made over all possible values of the parameters, integrating them out with respect to their posterior density.⁸

In chapter 5 extensive use is made of a range of Bayesian simulation techniques. These are explained in detail in chapter 4. Apart from the simulation techniques, the chapter provides information on methods useful for comparing models in a Bayesian fashion, for assessing convergence, and especially on implementing the algorithms. The theory in this chapter is put to action in an elaborate example.

1.4 Overview

The previous sections of the introduction provided the gist of what is to come in following chapters. In short, this thesis deals with changing parameter models for economic time series of inflation and exchange rates. In chapter 2 (based on Bos et al. [1999], published in Empirical Economics) the relation between a changing mean and the degree of integration in inflation series is investigated. Building on these results, chapter 3 [Bos et al., 2001,published in the International Journal of Forecasting] introduces explanatory variables in the ARFIMA model, to estimate the uncertainty of predicting inflation along the lines prescribed by the Phillips curve. Chapter 4 prepares the Bayesian methodology to be used in chapter , with an example concerning a state space model with an unobserved changing mean component. Part of the chapter, concerning the Adaptive Polar Sampler (section 4.3.4) is based on Bauwens et al. [2000]. In chapter 5 (part of the material in this chapter is published in the articles of Bos et al. [2000a],Bos et al. [2000b], in the Journal of Applied Econometrics and the proceedings of ISBA 2000), the example is elaborated into a full-fledged range of models fitting time varying trend and volatility behaviour of exchange rates. These models are used for deciding whether it is worthwhile to hedge daily currency risk or not, for DM/USD, Yen/USD, USD/DM and USD/Yen currency exposures. For this purpose, extensive use is made of the Bayesian toolkit of chapter 4.

In the thesis, choices have been made regarding the topics of research. Necessarily many other interesting topics have been left aside. Chapter 6 briefly summarizes what has been done, and provides a list of topics for possible future research.

Footnotes:

¹For integer d, it is not trivial to see that equation (1.2) holds. Courant, [1954,p. 335] elaborates the necessary theorems to show that indeed the equation is correct for ordinary first or second order differences as well.

²Source: Bureau of Labor Statistics, series CUUR0000SA0, consumer price index U.S. city average. The series was transformed to monthly inflation figures adjusted for seasonality, see chapter 2.

³For highly correlated time series, the sample autocorrelation functions tends to underestimate the autocorrelation function of the data generating process, as is seen from the panels in figure 1.2.

⁴Not only the parameter f is estimated close to -q, but also the correlation between the two parameters is extreme, see the bottom row of the table. In such a case, often the estimation procedure is not robust. Using a different optimization method may well lead to different outcomes, e.g. EViews 3.1 reports estimates of f = 0.927 (0.064) and q = -0.917 (0.068) for the same ARMA(1,1) model.

⁵Notice that the (local) trending of the exchange rates S_t in figure 1.3 corresponds to a (temporarily) non-zero mean of the exchange rate returns s_t.

⁶The structural model can be rewritten as s_t = [(h_t)/(1-rL)] + e_t Û (1-rL) s_t = h_t+ (1-rL) e_t. This model displays the same correlation structure as the ARMA(1,1) model (1-fL)s_t = (1+qL) v_t. The ARMA model in table 1.1 corresponds to a GLL model with parameters r = f = 0.857, s_e = 0.673, s_h = 0.044.

⁷Actually, the likelihood displayed is the likelihood marginal of the parameters s_e and s_h in the Generalized Local Level model (1.5)-(1.6), with normally distributed disturbances e_t and h_t. The top of the likelihood function lies at [^(r)] = 0.864,[^(s)] _e = 0.673, [^(s)] _h = 0.044, even though the top of the likelihood function after marginalising over s_e and s_h lies at r = 0.72.

⁸Using bootstrap methods and an elaborate sensitivity analysis, part of the uncertainty concerning the parameters can be taken into account in a classical analysis. However, this tends to be harder than using a Bayesian approach.

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Chapter 1 Introduction