some nonlinear threshold autoregressive time series models for ...

SOME NONLINEAR THRESHOLD AUTOREGRESSIVE TIME SERIES MODELS FOR ACTUARIAL USE Wai-Sum Chan,* Albert C. S. Wong,† and Howell Tong‡

ABSTRACT This paper introduces nonlinear threshold time series modeling techniques that actuaries can use in pricing insurance products, analyzing the results of experience studies, and forecasting actuarial assumptions. Basic “self-exciting” threshold autoregressive (SETAR) models, as well as heteroscedastic and multivariate SETAR processes, are discussed. Modeling techniques for each class of models are illustrated through actuarial examples. The methods that are described in this paper have the advantage of being direct and transparent. The sequential and iterative steps of tentative specification, estimation, and diagnostic checking parallel those of the orthodox Box-Jenkins approach for univariate time series analysis.

1. INTRODUCTION As mentioned by Rosenberg and Young (1999), discrete time series models are useful in analyzing actuarial assumptions (such as nonissue rates, lapse rates, investment rates, incidence rates, and severity rates) for pricing and reserving insurance products. Time series modeling is also important to actuaries for generating economic scenarios in a dynamic financial analysis model or in a cash-flow testing model. In addition to analyzing time-dependent variables that are specific to the pricing, reserving, or dynamical analyzing of insurance products, advanced time series models have been used for the estimation of value-at-risk (VaR) and other relevant measures of market risk (see, e.g., Embrechts, Resnick, and Samorodnitsky 1999; Longin 2000; Lucas 2000; Hardy 2001). Panjer (1999) provides an actuary’s perspective on VaR. Stochastic time series modeling has attracted considerable interest from actuaries around the world in recent years. Wilkie (1986, 1995) develops linear time series asset models for U.K. data. Wright (1998) proposes an alternative model based on vector autoregression. Chan and Wang (1998) refine the price inflation component of the Wilkie model by performing a time series outlier analysis. Whitten and Thomas (1999) build a nonlinear time series model for U.K. investment series. In the United States, Foster (1994) uses classical linear autoregressive integrated moving average (ARIMA) processes for modeling U.S. Social Security economic series. Frees et al. (1997) extend Foster’s work to multivariate autoregressive conditionally heteroscedastic (ARCH) models (Engle 1982). Rosenberg and Young (1999) propose using a Bayesian approach to analyze one of the U.S. Social Security data series. Carlin (1992) employs a state space time series process for modeling the number of deaths due to ischemic heart disease. Lai and Frees (1995) examine changes in actuarial reserves using discrete time series interest rate models. Hardy (2001) considers regime-switching lognormal models for monthly total returns of S&P 500 and TSE 300 indices.

* Wai-Sum Chan, FSA, PhD, is an Associate Professor in the Department of Statistics and Actuarial Science at the University of Hong Kong, Pokfulam Road, Hong Kong, China, e-mail: [email protected]. † Albert C. S. Wong, FSA, PhD, is an Associate Professor in the Department of Finance at the Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China, e-mail: [email protected]. ‡ Howell Tong, Hon FIA, PhD, holds a Chair in Statistics in the Department of Statistics, London School of Economics, London WC2A 2AE, United Kingdom, e-mail: [email protected].

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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 8, NUMBER 4

Actuarial time series models have also been developed in other countries. These models include those of Metz and Ort (1993) for Switzerland, Deaves (1993) for Canada, Daykin, Pentika¨inen, and Pesonen (1994) for Finland, Thomson (1996) for South Africa, and Sherris, Tedesco, and Zehnwirth (1999) for Australia. Many actuarial time series models are developed using the Box and Jenkins (1976) linear modeling techniques. In recent years statistical research in nonlinear time series analysis has grown rapidly. Substantial empirical evidence of nonlinearities in economic time series fluctuations is reported in the literature (see, e.g., Hsieh 1991; Potter 1995; Brooks 2001). Nonlinear time series models have the advantage of being able to capture asymmetries, jumps, and time irreversibility, which are “stylized” facts in many observed financial and economic time series (Granger and Tera¨svirta 1993; Franses and van Dijk 2000; Tsay 2002). They provide a much wider range of possible dynamics for the economic and actuarial time series data than do linear models. In this paper we explore the use of nonlinear time series models to analyze economic/actuarial time series data. In particular, we concentrate on the family of threshold autoregressive (TAR) models. Since the introduction of the basic class of “self-exciting” TAR models by Tong (1978) and Tong and Lim (1980), many classes of nonlinear models that stem from the threshold autoregressive framework have been proposed. In this paper we shall discuss several commonly used classes of TAR models and illustrate their corresponding modeling procedures step by step through real economic/actuarial examples.

2. BASIC SETAR MODELS 2.1 The Model The class of self-exciting threshold autoregressive (SETAR) models (Tong 1978, 1983) has been widely employed in the literature to explain various empirical phenomena in an observed time series. See, for example, the work of Tong and Yeung (1991) for beach water pollution, Yadav, Pope, and Paudyal (1994) for futures markets, Watier and Richardson (1995) for epidemiological applications, Lewis and Ray (1997) for sea surface temperatures, Montgomery et al. (1998) for U.S. unemployment, Fuecht et al. (1998) for medical studies, and Clements and Smith (2001) for exchange rate variables. Tong (1990) lists many more examples from diverse fields. The statistical properties and forecasting performance of SETAR models have been extensively examined: see, for example, the work of Tong (1990), Hansen (1996, 1999, 2000), Clements and Smith (1999), Kapetanios (2000), and De Gooijer (2001). The popularity of SETAR models is due to their being relatively simple to specify, estimate, and interpret as compared to many other nonlinear time series models. A k-regime SETAR(d; p1, p2, . . . , pk) model has the form

Yt ⫽

冦

冘␾ p1

␾ 0共1兲 ⫹

共1兲 j

Y t⫺j ⫹ ε t共1兲 , if Yt⫺d ⱕ r1

j⫽1

冘␾ p2

共2兲 0

␾ ⫹

共2兲 t⫺j j

Y

⫹ εt共2兲,

if r1 ⬍ Yt⫺d ⱕ r2 (1)

j⫽1

· · ·

· · ·

冘␾

· · ·

· · ·

pk

共k兲 0

␾ ⫹

共k兲 t⫺j j

Y

⫹ εt共k兲,

if rk⫺1 ⬍ Yt⫺d,

j⫽1

where k is the number of regimes in the model, d is the delay parameter, and pi is the autoregressive order in the ith regime of the model. The threshold parameters satisfy the constraint ⫺⬁ ⫽ r0 ⬍ r1 ⬍ r2 ⬍ . . . ⬍ rk⫺1 ⬍ rk ⫽ ⬁. The innovation within the ith regime εt(i) is a sequence of i.i.d. normal random variables with zero mean and constant variance ␴i2 ⬍ ⬁(i ⫽ 1, 2, . . . , k). If homoscedasticity is assumed

SOME NONLINEAR THRESHOLD AUTOREGRESSIVE TIME SERIES MODELS

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across the regimes (i.e., ␴12 ⫽ ␴22 ⫽ . . . ⫽ ␴k2 ⫽ ␴ε2), the common variance ␴ε2 can be estimated by the sample pooled variance in the data. The superscripts in the model indicate states of the world or regimes. Within each regime, it is assumed that the dynamical behavior of the time series variable follows a linear autoregressive process. The regime that is operative at any time t depends on the observable past history of {Yt} itself—in particular, on the value of Yt⫺d. Tong and Lim (1980, p. 249) therefore call the process in equation (1) a self-exciting threshold autoregressive model.

2.2 Modeling Procedures A number of strategies have been proposed for SETAR modeling in the literature (see, e.g., Tong 1983; Tsay 1989). The Tsay (1989) procedures are relatively simple and easy to understand. Actuaries and actuarial students who are not expert in this area should still be able to implement the procedures. Therefore, this paper employs Tsay’s strategy, which follows the systematic process for constructing actuarial and other statistical models (Box and Jenkins 1976; Hickman 1997). It involves iterative stages of model specification, estimation, and diagnostic checking. Computer codes (Tong 1983, pp. 288–302) and software packages (Tong 1990; Whitten and Thomas 1999) for modeling and simulating SETAR processes are readily available. 2.2.1 Testing for Nonlinearity and Model Specification

Linear time series models (e.g., the Wilkie model in the United Kingdom) have been reasonably successful as practical tools for actuarial modeling. The computation time required for obtaining a parsimonious linear ARMA model for most economic/actuarial time series data sets is well within the reach of actuaries. Ready-made computer packages are widely available. There are certainly tradeoffs between linear ARMA and nonlinear SETAR models in analyzing and forecasting time series observations. Therefore, it is important to have a test capable of telling a practitioner whether linear ARMA models or nonlinear SETAR models are better in describing the dynamics of the series under study. Consider the general SETAR model in equation (1). When p1 ⫽ p2 ⫽ . . . ⫽ pk ⫽ p and ␾i(1) ⫽ (2) ␾i ⫽ . . . ⫽ ␾i(k), for i ⫽ 0, 1, . . . , p, the SETAR model collapses into a linear autoregressive process of order p. Tsay (1989) proposes a novel test for threshold nonlinearity, which is based on the concept of ordered autoregression (Petruccelli and Davies 1986). Given p ⫽ max{ p1, . . . , pk} and d ⱕ p, we observe the time series {Y1, . . . , Yn}. Let (i) be the time index of the ith smallest observation of {Yp⫹1⫺d, . . . , Yn⫺d}. Rolling ordered autoregressions of the form

冢冣冢

Y 共1兲⫹d 1 Y 共1兲⫹d⫺1 · · · Y 共1兲 · · · Y 共1兲⫹d⫺p Y 共2兲⫹d 1 Y 共2兲⫹d⫺1 · · · Y 共2兲 · · · Y 共2兲⫹d⫺p ⫽ · · · · · ·· ·· · · · · · · · · · · · · Y 共 j兲⫹d 1 Y 共 j兲⫹d⫺1 · · · Y 共 j兲 · · · Y 共 j兲⫹d⫺p

冣冢冣冢冣 ␾0 a 共1兲⫹d ␾1 a 共2兲⫹d · ⫹ · · · · · ␾p a 共 j兲⫹d

(2)

are fitted successively, where j ⫽ m, m ⫹ 1, . . . , n ⫺ p, and m is the number of start-up observations in the ordered autoregression. Tsay (1989) suggests using m ⬇ (n/10) ⫹ p. A simple example is employed to illustrate the key steps of arranging the ordered autoregression. Table 1 gives a hypothetical time series with n ⫽ 24 observations. For illustrative purposes, we assume p ⫽ 3 and d ⫽ 1. The symbol (i) represents the time index of the ith smallest observation of {Y3, . . . , Y23}. The values of (i) are obtained for this example; see Table 1. The full-length (i.e., j ⫽ n ⫺ p ⫽ 21) ordered autoregression data matrices can be easily arranged using computer spreadsheet packages. The results are given in Table 2. The dependent variable Yt in Table 2 is arranged according to t ⫽ (i) ⫹ d, for d ⫽ 1 and i ⫽ 1, . . . , 21. The regressors {1, Yt⫺1, Yt⫺2, Yt⫺3} are arranged, correspondingly, by {1, Y(i)⫹d⫺1, Y(i)⫹d⫺2, Y(i)⫹d⫺3} ⫽ {1, Y(i), Y(i)⫺1, Y(i)⫺2} for d ⫽ 1 and i ⫽ 1, . . . , 21. After the arrangement, the data matrices are ordered by the regressor Yt⫺1 (see the sixth column in Table 2).

40


Table 1 Hypothetical Time Series Data t

Yt

1 2 3 4 5 6 7 8

101 82 66 35 31 7 20 92

(i )

t

Yt

(i )

t

Yt

(i )

(10) ⫽ 3 (7) ⫽ 4 (6) ⫽ 5 (1) ⫽ 6 (3) ⫽ 7 (16) ⫽ 8

9 10 11 12 13 14 15 16

154 125 85 68 38 23 10 24

(21) ⫽ 9 (18) ⫽ 10 (14) ⫽ 11 (12) ⫽ 12 (8) ⫽ 13 (4) ⫽ 14 (2) ⫽ 15 (5) ⫽ 16

17 18 19 20 21 22 23 24

83 132 131 118 90 67 60 47

(13) ⫽ 17 (20) ⫽ 18 (19) ⫽ 19 (17) ⫽ 20 (15) ⫽ 21 (11) ⫽ 22 (9) ⫽ 23

In general, the ordered autoregressions in equation (2) are sorted by the variable Yt⫺d, which is the regime indicator in the SETAR model. For each j in equation (2), we can compute the one-step-ahead standardized forecast error, eˆ( j⫹1)⫹d. If the underlying model is a linear AR( p) process, the standardized forecast errors are not only independently and identically distributed, but then also orthogonal to the regressors {Y( j⫹1)⫹d⫺1, . . . , Y( j⫹1)⫹d⫺p}. If the true model is a nonlinear SETAR process, then the orthogonality will be destroyed. Tsay (1989) utilizes this property and considers the regression e ⫽ Y␤ ⫹ ␩,

(3)

where e ⫽ (eˆ(m⫹1)⫹d, . . . , eˆ(n⫺p)⫹d)⬘, Y is the data matrix for the regressor {Y( j⫹1)⫹d⫺1, . . . , Y( j⫹1)⫹d⫺p} for j ⫽ m, . . . , n ⫺ p ⫺ 1, ␤ is the p-dimension parameter vector, and ␩ is the error vector. The usual F-statistic for H0 : ␤ ⫽ 0 in the above regression can be used to test the orthogonality, and thus SETAR-type nonlinearity. To perform the F-test in equation (3), the values of both p and d must be given. However, in practice, p and d are seldom known. As a quick method, Tsay (1989) selects p by the sample partial autocorrelation function (PACF) of Yt. Once p is selected, d is chosen so that it gives the most significant F-statistic. For the number of regimes k and the threshold parameters (i.e., the r values in model 1), he

Table 2 Ordered Autoregression Data Matrices (p ⴝ 3, d ⴝ 1)

i

(i )

(i ) ⴙ d

Independent Variable Y(i )ⴙd

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

6 15 7 14 16 5 4 13 23 3 22 12 17 11 21 8 20 10 19 18 9

7 16 8 15 17 6 5 14 24 4 23 13 18 12 22 9 21 11 20 19 10

20 24 92 10 83 7 31 23 47 35 60 38 132 68 67 154 90 85 118 131 125

Time Indices

Regressors 1

Y(i )ⴙdⴚ1

Y(i )ⴙdⴚ2

Y(i )ⴙdⴚ3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

7 10 20 23 24 31 35 38 60 66 67 68 83 85 90 92 118 125 131 132 154

31 23 7 38 10 35 66 68 67 82 90 85 24 125 118 20 131 154 132 83 92

35 38 31 68 23 66 82 85 90 101 118 125 10 154 131 7 132 92 83 24 20


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proposes using various scatter plots to locate them. The t ratios for the lth (l ⱕ p) AR coefficient in the rolling ordered autoregression (2) can be computed for j ⫽ m, . . . , n ⫺ p. When the coefficient is significant, the t ratios gradually and smoothly converge to a fixed value as the recursion moves. However, once a threshold is reached, the estimated AR coefficient starts to change (toward the new coefficient in the next regime), and the t ratio begins to turn and, sometimes, changes direction. Therefore, scatter plots of these t ratios versus the regime indicator variable Yt⫺d often provide useful hints about locations of the threshold values. Finally, the Akaike Information Criterion (AIC; Akaike 1974) is used to refine the AR order ( pk ⱕ p) in each regime. 2.2.2 Model Estimation and Diagnostic Checking

When the values of (d, k, p1, . . . , pk, r1, . . . , rk⫺1) are specified, the full-length ordered autoregression matrices in equation (2) can be divided into k regimes. We use the data matrices in Table 2 as an illustration. We assume d ⫽ 1, k ⫽ 3, ( p1, p2, p3) ⫽ (3, 1, 2), r1 ⫽ 50 and r2 ⫽ 100. The full-length ordered autoregression matrices can be divided into three regimes:

冢冣冢冣冢冣冢冣冢冣 20 1 7 31 35 24 1 10 23 38 92 1 20 7 31 10 1 23 38 68 ⫽ 83 1 24 10 23 7 1 31 35 66 31 1 35 66 82 23 1 38 68 85 47 1 60 35 1 66 60 1 67 38 1 68 ⫽ 132 1 83 68 1 85 67 1 90 154 1 92

冢冣冢冣共1兲 a 共1兲⫹1

␾ 0共1兲

␾ 1共1兲

␾ 2共1兲

␾ 3共1兲

冉冊 ␾ 0共2兲

␾ 1共2兲

共1兲 a 共2兲⫹1 ⫹ , · · · 共1兲 a 共8兲⫹1

共2兲 a 共9兲⫹1

⫹

共2兲 a 共10兲⫹1 , · · · 共2兲 a 共16兲⫹1

冢冣冢冣冢冣 90 1 118 131 85 1 125 154 118 ⫽ 1 131 132 131 1 132 83 125 1 154 92

(4)

冢冣 ␾

共3兲 0

␾ 1共3兲 ⫹ ␾ 2共3兲

(5)

共3兲 a 共17兲⫹1

共3兲 a 共18兲⫹1 . · · · 共3兲 a 共23兲⫹1

(6)

In general, let ␲l be the largest value of l such that {rl⫺1 ⬍ Y(␲l) ⱕ rl} for l ⫽ 1, . . . , k ⫺ 1. We also define ␲0 ⫽ 0 and ␲k ⫽ n ⫺ p. For the jth regime of data, we have a linear model of the form Y j ⫽ Aj⌽共j兲 ⫹ aj,

(7)

where Yj and Aj are the vector of observations and the data matrix of the jth regime ordered autoregression, respectively, ⌽ 共j兲 ⫽ 共␾0共 j兲, ␾1共 j兲, . . . , ␾p共 j兲j 兲⬘,

(8)

42


and 共 j兲共 j兲共 j兲 , a共␲ , . . . , a共␲ 兲⬘. a j ⫽ 共a共␲ j⫺1⫹1兲⫹d j⫺1⫹2兲⫹d j兲⫹d

(9)

The least squares estimate of ⌽(j), for j ⫽ 1, 2, . . . , k, can then be obtained by the ordinary least squares method: ˆ 共j兲 ⫽ 共AⴕA 兲⫺1共Aⴕ Y 兲. ⌽ j j j j

(10)

After the model estimation, usual time series diagnostic checking techniques (such as the examination of histograms and autocorrelations of the residuals) can be used to check the model adequacy.

2.3 An Application The long-term interest rate, an important variable for many actuarial applications, is one of the four key variables in the original Wilkie (1986) model. In the Request for Proposals for the Modeling of Economic Series Coordinated with Interest Rate Scenarios project jointly sponsored by the Society of Actuaries and the Casualty Actuarial Society (www.casact.org/research/ctordfac.htm), the long-term interest rate is one of the major variables. In this section we illustrate SETAR modeling techniques using U.S. government long-term interest rate series. Quarterly government 10-year bond yield (rt) data were obtained from the International Monetary Fund (2001) under the series code 11161ZF. The time frame covers from the first quarter of 1957 to the fourth quarter of 2000, with a total of 176 observations. The effective quarterly rate for period t is defined as it ⫽

冋冉

1⫹

rt 100

冊

1/4

册

⫺ 1 ⫻ 100.

(11)

Following Frees et al. (1997, p. 71), the analysis will be based on the difference of logarithmic interest rates defined as

冋冉

Y t ⫽ ln 1 ⫹

冊冉

it it⫺1 ⫺ ln 1 ⫹ 100 100

冊册

⫻ 100.

(12)

Figure 1 gives a time series plot of Yt. The sample PACF for the Yt series is first computed in Table 3. It suggests that p ⫽ 5 for the F-tests of nonlinearity. We perform the tests with p ⫽ 5 and d ⫽ 1, . . . , 5; the results are given in Table 4. The combination of ( p, d) ⫽ (5, 1) gives the most significant F-statistic. The next step is to specify the number of regimes k and the threshold value(s). A scatter plot of t ratios of the significant AR coefficient (lag 5) in the rolling ordered autoregression versus the regime indicator variable Yt⫺1 is given in Figure 2. The other AR coefficients are not statistically significant. Their corresponding scatter plots of t ratios do not contain useful information about the thresholds, and hence they are not shown here. At the beginning of Figure 2, the t ratio moves gradually and smoothly. However, when Yt⫺1 approaches the value 0.0, we observe an abrupt drop in the t ratios. Therefore, we tentatively specify k ⫽ 2 and r ⫽ 0.0 for the data. Next, we proceed to the estimation stage. The data set is divided into regimes according to equations (7)–(9) as discussed in Section 2.2.2. The least squares estimate of ⌽(j) can be computed. The AR order within each regime is obtained by minimizing the AIC (see Akaike 1974) for the regime linear model in equation (7) over the set {0 ⱕ pj ⱕ p}. The resulting fitted SETAR model for the changes in the logarithmic U.S. long-term interest rate series is

Yt ⫽

冦

0.2685 Y t⫺1 ⫺ 0.3294 Y t⫺5 ⫹ ε t共1兲 , 共0.1096兲共0.1234兲

if Yt⫺1 ⱕ 0.0

0.3180 Yt⫺1 ⫺ 0.2088 Yt⫺2 ⫹ εt共2兲, 共0.1031兲共0.1016兲

if Yt⫺1 ⬎ 0.0.

(13)


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Figure 1 Time Plot of Changes in Logarithmic U.S. Long-Term Interest Rates

The AR orders are 5 and 2, the number of effective observations are 73 and 97, and the residual variances are 0.0141 and 0.0102 for the two regimes of the fitted model. The value of ␴ε2 can be estimated by pooling the residual variances, and we have ␴ˆ ε2 ⫽ 0.0119. The histogram and sample autocorrelation plot of the standardized residuals are given in Figure 3. Diagnostic checking of the fitted model does not show any model inadequacy. The above model has quite a simple economic implication. The data are divided into two regimes at the threshold r ⫽ 0.0. Regime 1 corresponds to a decreasing long-term interest rate environment. Regime 2 corresponds to a state of increasing interest rate. The long-term interest rate dynamic has a longer memory (up to lag 5) in the downward environment (Regime 1). On the other hand, during an upward tendency (Regime 2), the dynamic carries shorter lag dependence (up to lag 2). The dynamics of the estimated model (13) can be deduced. The most prominent pair of complex roots of the autoregressive process in Regime 1 is (0.7077 ⫾ 0.4655i), which has a modulus of 0.8471 and a period of 10.8 quarters (Box and Jenkins 1976, p. 55). On the other hand, the autoregressive model in Regime 2 is completely characterized by a complex pair of roots (0.1590 ⫾ 0.4284i), which has a modulus of 0.4570 and a period of 5.2 quarters. This asymmetry is the most striking feature of the model. The claim that major economic time series are asymmetric over different phases of the cycle arises in almost all major works on classical business cycle analysis (see, e.g., Keynes 1936; Neftci 1984; Franses and Paap 1998). The fitted SETAR model (13) is able to provide a possible mechanism to describe the asymmetric behavior of the long-term interest rate over various phases of the business cycle.

Table 3 Sample PACF for Yt Lag

Sample PACF S.E.

1

2

3

4

5

6

7

8

0.27 0.08

⫺0.09 0.08

0.11 0.08

⫺0.06 0.08

⫺0.19 0.08

⫺0.00 0.08

⫺0.13 0.08

0.07 0.08

44


Table 4 Results of F-tests for Nonlinearity for Yt p

d

F-statistics

p-value

5 5 5 5 5

1 2 3 4 5

3.93 1.54 2.16 2.05 0.36

0.0011 0.1682 0.0503 0.0633 0.9029

One of the main objectives of actuarial modeling is to provide realistic simulation of the variables. In Figure 4 we show a set of 30 simulated paths of the quarterly U.S. long-term interest rate variable (rt) from 2001 to 2020, along with the record since 1957 using the fitted SETAR model. Empirical forecast medians, 80% forecast limits, as well as 95% forecast bounds based on 1,000 simulations are also displayed in Figure 4.

3. HETEROSCEDASTIC DTARCH MODELS 3.1 The Model Much recent evidence shows that the volatility (␴t) of a financial or economic time series is often not constant. Since the introduction by Engle (1982), ARCH models have firmly established themselves among the foremost techniques for modeling volatility in economic activities and financial markets. The class of ARCH models provides a simple and appealing scheme in which the conditional variance of a stationary time series follows a restricted linear autoregressive process. For example, an AR(1) ⫺ ARCH(1) model is defined as Y t ⫽ ␾ 0 ⫹ ␾ 1 Y t⫺1 ⫹ ε t

Figure 2 Scatter Plot of t Ratios versus Ytⴚ1

(14)


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Figure 3 Sample ACF Plot (a) and Histogram of Standardized Residuals (b) from Model (13)

for 兩␾1兩 ⬍ 1, and assume that conditional on the information available at time (t ⫺ 1), Ᏺt⫺1, the disturbance term in equation (14) is distributed as 兵ε t 兩Ᏺ t⫺1 其 ⬃ N共0, ␴ t2 兲,

Figure 4 U.S. Long-Term Interest Rates, 1957–2000, and Forecast Medians, 80% Forecast Limits, 95% Forecast Bounds, and 30 Simulations Using Fitted SETAR Model, 2001–20

Note: 80% forecast limits ⫽ dashed lines; 95% forecast limits ⫽ bold solid lines.

(15)

46


where 2 ␴ t2 ⫽ ␤ 0 ⫹ ␤ 1 ε t⫺1

(16)

for ␤0 ⬎ 0 and ␤1 ⱖ 0. There are numerous applications of ARCH-type models in economic and financial time series analysis (see, e.g., Granger and Tera¨svirta 1993; McKenzie 1997; Franses and van Dijk 2000). Wilkie (1995) and Frees et al. (1997) use ARCH processes for actuarial modeling. A major drawback of the standard ARCH-type models is that the estimated model coefficients are assumed to be fixed throughout the observed period, and they fail to take into account the possibility of asymmetrical regime switching. As we have seen, such a possibility is built into the “threshold” framework in the last section. Moreover, in the basic SETAR model the conditional variances for the different regimes need not be the same. Li and Li (1996) combine the features of ARCH and SETAR models and propose the class of double-threshold autoregressive conditional heteroscedastic (DTARCH) models. A general k-regime DTARCH (d; p1, p2, . . . , pk; m1, m2, . . . , mk) model for the time series {Yt} is defined as

Yt ⫽

冦

冘␾ p1

␾ 0共1兲 ⫹

共1兲 j

Y t⫺j ⫹ ε t , if Yt⫺d ⱕ r1

j⫽1

冘␾ p2

␾0共2兲 ⫹

共2兲 t⫺j j

Y

⫹ εt,

if r1 ⬍ Yt⫺d ⱕ r2

j⫽1

· · ·

· · ·

冘␾

· · ·

(17)

· · ·

pk

共k兲 0

␾ ⫹

共k兲 t⫺j j

Y

⫹ εt,

if rk⫺1 ⬍ Yt⫺d,

j⫽1

where the conditional mean of {Yt} follows a basic SETAR process as defined in equation (1). The innovation εt is assumed to have zero mean and conditional variance ht ⫽ E[εt2兩Ᏺt⫺1]. In the ith regime, the process ht follows an ARCH(mi) model for i ⫽ 1, 2, . . . , k, that is,

ht ⫽

冦

冘␤

m1

␤

共1兲 0

⫹

共1兲 2 j t⫺j

ε

, if Yt⫺d ⱕ r1

j⫽1

冘␤

m2 共2兲 0

␤ ⫹

if r1 ⬍ Yt⫺d ⱕ r2

共2兲 2 j t⫺j

ε ,

j⫽1

· · ·

· · ·

冘␤

· · ·

(18)

· · ·

mk

␤0共k兲 ⫹

共k兲 2 j t⫺j

ε ,

if rk⫺1 ⬍ Yt⫺d .

j⫽1

Furthermore, we assume that all the parameters in the conditional variance equation (18) are positive or non-negative, that is, ␤0(i) ⬎ 0 and ␤j(i) ⱖ 0 for i ⫽ 1, . . . , k and j ⫽ 1, . . . , mi.

3.2 Modeling Procedures Li and Li (1996) suggest a procedure for modeling DTARCH processes. We summarize their proposed steps as follows:


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47

a. Employ the modeling procedures that are outlined in Section 2.2 to build a basic SETAR model for the conditional mean equation (17). b. Calculate the squared residuals, εˆ t2, from the tentative model that is obtained in step (a). c. Identify the ARCH order in each regime using the squared residuals that are computed in step (b). d. Once the structures of the conditional mean in equation (17) and the conditional variance in equation (18) are specified, fit the entire DTARCH model by the maximum likelihood estimation method via the iterative weighted least squares (IWLS) algorithm. Details of the IWLS algorithm for DTARCH models are given by Mak, Wong, and Li (1997). e. Use AIC (Akaike 1974) to refine the AR order (pi) and the ARCH order (mi) in each regime. f. Perform diagnostic checking for the fitted model in step (e), refine the fitted model by repeating steps (a)–(e), if necessary. To check the adequacy of the fitted conditional mean model (17), examine the residual autocorrelations. The lag l residual autocorrelation is defined as

冘 n

␳ˆ l ⫽

冑

冒冘 n

冑

共εˆ t/ hˆt ⫺ ␥兲共εˆ t⫺l/ hˆt⫺1 ⫺ ␥兲

t⫽l⫹1

冑

共εˆ t / hˆt ⫺ ␥兲2,

(19)

t⫽1

where hˆt is the fitted conditional variance from the DTARCH model, and ␥⫽

1 n

冘 ε /冑hˆ . n

t

t

t⫽1

To check the adequacy of the fitted conditional variance model (18), compute the autocorrelations of the squared residuals. The lag l squared residual autocorrelation is defined as

冘 n

ˆ ⫽ ᏼ l

冒冘共εˆ /hˆ ⫺ ␦兲 , n

2 共εˆ t2 /hˆ t ⫺ ␦兲共εˆ t⫺l /hˆ t⫺l ⫺ ␦兲

t⫽l⫹1

2 t

t

2

(20)

t⫽1

where ␦⫽

1 n

冘 ε /hˆ . n

2 t

t

t⫽1

Analogous to the portmanteau test that is proposed by Box and Pierce (1970) for checking the overall goodness of fit in linear time series modeling, Li and Li (1996) derive portmanteau statistics based on the residual autocorrelation function and squared residual autocorrelation of the DTARCH process. These statistics are useful in the diagnostic checking of fitted DTARCH models.

3.3 An Application Modeling exchange rate variables is useful in pricing and reserving mixed-currency insurance products (see, e.g., Law 1995 and Mange 2000). The Society of Actuaries provides a Professional Actuarial Specialty Guide (PASG No. V-1-97) on the actuarial aspects of currency risk. To link local stochastic actuarial models for a number of countries, it is also necessary to model exchange rates (Wilkie 1992). In this section we illustrate DTARCH modeling procedures using the U.S. dollar versus Australia dollar exchange rate time series. We use monthly observations (Et) from January 1971, when currencies started floating freely after the breakdown of postwar Bretton Woods exchange rate system, to March 2002. The data are available from FRED威, which is an economic time series database maintained by the Federal Reserve Bank of St. Louis (www.stls.frb.org/fred/data/exchange/exusal). Raw exchange rate data are likely to be nonstationary (Bleaney and Mizen 1996). The results from the standard unit root tests (Dickey and Fuller 1979) confirm that our raw Et series is nonstationary.

48


Figure 5 U.S. Dollar versus Australia Dollar Exchange Rate Returns, January 1971–March 2002

Therefore, we transform the raw series into log-returns R t ⫽ 关ln共Et兲 ⫺ ln共Et⫺1兲兴 ⫻ 100

(21)

for model building. This transformation has become standard in the finance and economic literature (see, e.g., Brooks 1996; Andersen et al. 2000). Figure 5 gives a time series plot of Rt. The sample PACF for the Rt series is given in Table 5. It suggests that p ⫽ 4 for the F-tests for nonlinearity. We perform the tests with p ⫽ 4 and d ⫽ 1, . . . , 4; the results are given in Table 6. The combination of ( p, d) ⫽ (4, 1) gives the most significant F-statistic. Scatter plots of recursive t ratios of the two significant AR coefficients (at lag 0 and lag 3) in the ordered autoregression versus the regime indicator variable Rt⫺1 are given in Figure 6. In Figure 6a, the t ratio moves gradually and smoothly at the beginning toward the value ⫺2 after a short warm-up period. However, when Rt⫺1 is approaching the value 0.0, we observe an abrupt jump in the t ratios. Figure 6b shows the recursive t ratio plot for the lag-3 AR coefficient. It clearly shows a “U-turn” at Rt⫺1 ⫽ 0.0. Both figures indicate k ⫽ 2 and r ⫽ 0.0 for the data. A basic two-regime SETAR model with p ⫽ 4, d ⫽ 1, and r ⫽ 0.0 is first fitted. Table 7 (upper panel) presents the sample autocorrelation function (ACF) of the residuals. It does not show any model inadequacy. However, when we further examine the sample ACF of the squared residuals (Table 7, lower panel), heteroscedasticity is found at the first lag. Therefore, a two-regime DTARCH (1; 4, 4; 1,

Table 5 Sample PACF for Rt Lag

Sample PACF S.E.

1

2

3

4

5

6

7

8

0.275 0.052

⫺0.081 0.052

⫺0.033 0.052

⫺0.088 0.052

0.056 0.052

0.039 0.052

0.038 0.052

0.078 0.052


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Table 6 Results of F-tests for Nonlinearity for Rt p

d

F-statistics

p-value

4 4 4 4

1 2 3 4

4.151 0.988 3.370 1.650

0.0011 0.4248 0.0055 0.1464

1) model is tentatively suggested for the series. Maximum likelihood estimates of the specified DTARCH model are computed using the IWLS algorithm (Mak et al. 1997). The AR order ( pi) and the ARCH order (mi) within each regime are further fine-tuned using the AIC. The resulting final fitted DTARCH model for the exchange rate return series is

Rt ⫽

冦

0.2154 Rt⫺1 ⫺ 0.2024 Rt⫺2 ⫹ 0.0823 Rt⫺3 ⫺ 0.0842 Rt⫺4 ⫹ εt, if Rt⫺1 ⱕ 0.0 共0.0753兲共0.0547兲共0.0477兲共0.0461兲 0.5119 Rt⫺1 ⫹ 0.0961 Rt⫺2 ⫺ 0.0812 Rt⫺3 ⫹ εt, 共0.0621兲共0.0621兲共0.0633兲

if Rt⫺1 ⬎ 0.0

(22)

and

ht ⫽

冦

2 1.8695 ⫹ 0.3359 ε t⫺1 , 共0.2925兲共0.1042兲

if Rt⫺1 ⱕ 0.0

2 1.6830 ⫹ 1.9102 εt⫺1 , 共0.3126兲共0.3566兲

if Rt⫺1 ⬎ 0.0.

(23)

The portmanteau statistics (see, e.g., Li 2004, p. 10) for the residuals and squared residuals from the 2 variate (e.g., fitted model are Q24(εˆ t) ⫽ 25.12 and Q24(εˆ t2) ⫽ 19.30. They should be compared with a ␹24

Figure 6 Scatter Plots of t Ratios versus Rtⴚ1

50


Table 7 Sample ACF for Residuals and Squared Residuals Lag Sample ACF of ␧ˆ t S.E. Sample ACF of ␧ˆ t2 S.E.

1

2

3

4

5

6

7

8

⫺0.01 0.05 0.18 0.05

⫺0.00 0.05 0.01 0.05

0.03 0.05 0.01 0.05

⫺0.07 0.05 ⫺0.03 0.05

0.02 0.05 0.00 0.05

0.06 0.05 ⫺0.01 0.05

0.03 0.05 0.03 0.05

0.09 0.05 0.03 0.05

2 ␹24,␣ ⫽ 31.41 at ␣ ⫽ 5%). Based on the portmanteau tests, the fitted DTARCH model is adequate for the exchange rate return series. Based on the fitted DTARCH model, we simulate the U.S. dollar versus Australia dollar exchange rates for the four months after the end of the sampling period (i.e., En(1), . . . , En(4)). Figure 7 gives histograms of them using 10,000 realizations. The empirical distributions of En(3) and En(4) are highly skewed, while the histograms of En(1) and En(2) are fairly normal. It should be noted that the skewness of En(4) in Figure 7, in particular, is much larger than one would expect from a logarithmic random walk model. Tong and Moeanaddin (1988) suggest that although the predictive distribution of the one-stepahead SETAR forecast is normal, the predictive distributions of L-step-ahead (L ⱖ 2) forecast can be skewed and nonnormal, especially when L increases. On the other hand, in simulating ARCH models, Wilkie (1995, p. 903) reports that excessive large values might occasionally be generated. Figure 7 shows that simulated values from DTARCH processes have combined SETAR and ARCH properties. Asymmetries and nonlinearities are commonly found in economic variables (Fornari and Mele 1997). A DTARCH framework is employed to model the conditional mean and the conditional variance of the U.S. dollar versus Australia dollar exchange rate return variable. The fitted DTARCH process captures most of the asymmetric nonlinearity and heteroscedasticity in the data. The threshold is found at r ⫽ 0.0, which implies that the exchange rate dynamic could behave differently depending on whether we are in a bull market or a bear market.

Figure 7 Simulated Values from Fitted DTARCH Model, En(1)–(4)


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4. MULTIVARIATE SETAR MODELS 4.1 The Model Tsay (1998) generalizes the univariate threshold principle to a multivariate framework. Consider an s-dimensional time series Yt ⫽ ( y1t, y2t, . . . , yst)⬘. An s-dimensional k-regime multivariate SETAR (d; p1, . . . , pk) model is defined as

Yt ⫽

冦

冘⌽ p1

C0共1兲 ⫹

共1兲 j

Yt⫺j ⫹ ␧t共1兲, if zt⫺d ⱕ r1

j⫽1

冘⌽ p2

共2兲 0

C ⫹

共2兲 j

Yt⫺j ⫹ ␧t共2兲, if r1 ⬍ zt⫺d ⱕ r2

j⫽1

· · ·

· · ·

· · ·

冘⌽

(24)

· · ·

pk

C0共k兲 ⫹

共k兲 j

Yt⫺j ⫹ ␧t共k兲, if rk⫺1 ⬍ zt⫺d ,

j⫽1

where C0(i) are (s ⫻ 1)– dimensional constant vectors and ⌽j(i) are (s ⫻ s)– dimensional matrix parameters for i ⫽ 1, . . . , k. The innovational vectors in the ith regime satisfy ␧t(i) ⫽ ⌺i1/ 2at, where ⌺i1/ 2 are symmetric positive definite matrices and {at} is a sequence of serially uncorrelated normal random vectors with mean 0 and covariance matrix I, the (s ⫻ s)– dimensional identity matrix. The threshold variable zt⫺d is assumed to be stationary; it depends on the observable past history of Yt⫺d. For example, we can set z t⫺d ⫽ ␻⬘Y t⫺d, where ␻ is a prespecified (s ⫻ 1)– dimensional vector. When ␻ ⫽ (1, 0, . . . , 0)⬘, the threshold variable 1 1 1 is simply zt⫺d ⫽ y1,t⫺d. When ␻ ⫽ (s , s , . . . , s )⬘, the threshold variable is the average of all of the elements in Yt⫺d.

4.2 Modeling Procedures Analogous to the Tsay (1989) procedures for univariate SETAR modeling, Tsay (1998) extends the method to multivariate situation. The method has been applied successfully to many data sets, ranging from U.S. interest rates to Icelandic river flow series. This article uses the Tsay (1998) strategy for multivariate SETAR modeling. 4.2.1 Testing for Nonlinearity

Given p ⫽ max{ p1, . . . , pk} and d ⱕ p, we observe the vector time series {Y1, . . . , Yn}. Tsay (1998) considers the multivariate generalization of the ordered regression arrangement in equation (2). It should be noted that the threshold variable zt⫺d in equation (24) can assume values only in ᐆ ⫽ { zp⫹1⫺d, . . . , zn⫺d}. Let (i) be the time index of the ith smallest observation in ᐆ. Rolling ordered multivariate autoregressions of the form

冢冣冢

Y⬘共1兲⫹d 1 Y⬘共1兲⫹d⫺1 Y⬘共2兲⫹d 1 Y⬘共2兲⫹d⫺1 ⫽ · · · · · · · · · Y⬘共 j兲⫹d 1 Y⬘共 j兲⫹d⫺1

· · · Y⬘共1兲⫹d⫺p · · · Y⬘共2兲⫹d⫺p · ·· · · · · · · Y⬘共 j兲⫹d⫺p

冣冢冣冢冣 c⬘0 ␧⬘共1兲⫹d ⌽⬘1 ␧⬘共2兲⫹d · ⫹ · · · · · ⌽⬘p ␧⬘共 j兲⫹d

(25)

can be arranged successively, where j ⫽ m, m ⫹ 1, . . . , n ⫺ p, and m is the number of start-up observations in the ordered autoregression. Tsay (1998) suggests a range of m (between 3公n and 5公n).

52


Different values of m can be used to investigate the sensitivity of the modeling results with respect to the choice. It should be noted that the ordered autoregressions in equation (25) are sorted by the variable zt⫺d, which is the regime indicator in the multivariate SETAR model. Let eˆ(m⫹1)⫹d denote the one-step-ahead standardized predictive residual from the least squares fitted multivariate regression (25) for j ⫽ m. Tsay (1998, p. 1190) provides the direct computational formula for eˆ(m⫹1)⫹d. Alternatively, they can be easily obtained from many commonly used statistical software packages (e.g., Timm and Mieczkowski 1997). Analogous to the univariate case, if the underlying model is a linear vector autoregressive process, then the predictive residuals are white noise, and they are uncorrelated to the regressor X⬘t ⫽ {1, Y⬘t⫺1, Y⬘t⫺2, . . . , Y⬘t⫺p}. However, if Yt follows a threshold process, then the predictive residuals are correlated with the regressor. Tsay (1998) utilizes this property again and considers the multivariate regression eˆ 共⬘l 兲⫹d ⫽ X⬘共l 兲⫹d ␤ ⫹ w⬘共l 兲⫹d

(26)

for l ⫽ m ⫹ 1, . . . , n ⫺ p. The problem of testing nonlinearity is then transformed to test the hypothesis H0 : ␤ ⫽ 0 in the above regression. Tsay (1998) employs the test statistic C共d兲 ⫽ 共n ⫺ p ⫺ m ⫺ kp ⫺ 1兲兵ln兩S0兩 ⫺ ln兩S1兩其,

(27)

where 兩A兩 denotes the determinant of the matrix A, and

冘

n⫺p 1 S0 ⫽ eˆ eˆ⬘ , n ⫺ p ⫺ m l⫽m⫹1 共l 兲⫹d 共l 兲⫹d

S1 ⫽

冘

n⫺p 1 w ˆ w ˆ⬘ , n ⫺ p ⫺ m l⫽m⫹1 共l 兲⫹d 共l 兲⫹d

where w ˆ t is the least squares residual of regression (26). Under the null hypothesis that Yt is linear, Tsay (1998) shows that C(d) is asymptotically a ␹2 random variable with ( pk2 ⫹ k) degrees of freedom. 4.2.2 Model Specification, Estimation, and Diagnostic Checking

To perform the C(d) test for nonlinearity in equation (27), both values of p and d must be given. In practice, we can select p by the partial autoregression matrix (PAM) of Yt. Tiao and Box (1981) define the PAM at lag l, which is denoted by ⌸(l ), to be the last matrix coefficient when the data are fitted to a vector autoregressive process of order l. This is a direct extension of the Box and Jenkins (1976, p. 64) definition of the partial autocorrelation function for univariate time series. The partial autoregression matrices ⌸(l ) of a linear vector AR( p) process are zero for l ⬎ p. This “cut-off” property provides very useful information for identifying the order p. Once p is selected, d is chosen so that it gives the most significant C(d) statistic. In univariate SETAR modeling, we use various scatter plots for specifying the number of regimes k and the threshold parameters (i.e., the r values). Unfortunately, these plots are not applicable to high-dimensional multivariate SETAR analysis. Following Tong (1983, p. 186), we use AIC to search for these parameters. Given p, d, k, and ᏾k ⫽ {r1, . . . , rk⫺1}, the full-length ordered multivariate autoregression in equation (25) can be divided into regimes. For the jth regime of data, we have a general linear model of the form Y j ⫽ Aj⌽共j兲 ⫹ ␧j, (28) where (29) Y j ⫽ 共Y⬘共␲j⫺1⫹1兲⫹d, Y⬘共␲j⫺1⫹2兲⫹d, . . . , Y⬘共␲j兲⫹d兲⬘, ⌽ 共j兲 ⫽ 共c⬘0, ⌽⬘1共 j兲, . . . , ⌽⬘p共 j兲兲⬘,

(30)


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53

␧ j ⫽ 共␧⬘共␲j⫺1⫹1兲⫹d, ␧⬘共␲j⫺1⫹2兲⫹d, . . . , ␧⬘共␲j兲⫹d兲⬘,

冢

1 Y⬘共␲j⫺1⫹1兲⫹d⫺1 1 Y⬘共␲j⫺1⫹2兲⫹d⫺1 Aj ⫽ · · · · · · 1 Y⬘共␲j兲⫹d⫺1

(31)

冣

· · · Y⬘共␲j⫺1⫹1兲 · · · Y⬘共␲j⫺1⫹2兲 · ·· · · · · · · Y⬘共␲j兲

· · · Y⬘共␲j⫺1⫹1兲⫹d⫺p · · · Y⬘共␲j⫺1⫹2兲⫹d⫺p , · ·· · · · · · · Y⬘共␲j兲⫹d⫺p

(32)

where ␲j is the largest value of j such that {rj⫺1 ⬍ z( j) ⱕ rj} for j ⫽ 1, . . . , k ⫺ 1. We define ␲0 ⫽ 0 and ␲k ⫽ n ⫺ p. The number of observation in the jth regime is nj ⫽ ␲j ⫺ ␲j⫺1. The least squares estimate of ⌽(j) can be obtained by the ordinary multivariate least squares method: ˆ 共j兲 ⫽ 共AⴕjAj兲⫺1共Aⴕj Yj兲, ⌽

(33)

and the residual variance-covariance matrix for the jth regime can be obtained by

冘

nj ˆ ⫽1 兵␧ˆ ␧ˆ ⬘ 其. ⌺ j nj t⫽1 共␲j⫺1⫹t兲⫹d 共␲j⫺1⫹t兲⫹d

(34)

The AIC of a multivariate fitted SETAR model in equation (24) is defined as

冘兵n ln兩⌺ˆ 兩 ⫹ 2k共kp ⫹ 1兲其. k

AIC共 p, d, k, ᏾k兲 ⫽

j

j

(35)

j⫽1

Given p and d, we can search the parameters k and ᏾k by minimizing the AIC. Due to the computational complexity and possible interpretations of the final model, we usually restrict k to a small number, such as 2 or 3. For the threshold parameters ᏾k, we divide the data into subgroups according to the empirical percentiles of zt⫺d, and use the AIC to select the r values. Finally, the AIC is used to refine the AR order ( pk ⱕ p) in each regime. To guard against incorrectly specifying the model, a detailed diagnostic analysis of the residuals is required. This includes an examination of the plots of standardized residuals and the sample crosscorrelation matrices of the residuals (Tiao and Box 1981).

4.3 An Application In this section we build a monthly multivariate SETAR model for the U.S. investment series. This stochastic investment model consists of three key variables: the inflation rate, the long-term interest rate, and the S&P 500 Total Return Index. Consumer Price Index (CPI) data for All Urban Consumers without seasonal adjustment, which are obtained from the FRED威 database, are employed to compute the U.S. inflation rate. The inflation rate for period t is defined as y 1t ⫽ 100 ⫻

冋

册

CPI t ⫺1 . CPI t⫺1

(36)

For the long-term interest rate, annualized 10-Year Treasury Constant Maturity rates (rt) from FRED威 are used. The effective monthly rate for period t is defined as i t ⫽ 100 ⫻

冋冉

1⫹

rt 100

冊

1/12

册

⫺1 .

(37)

The analysis will be based on the difference of logarithmic transformation of it, that is,

冋冉

y 2t ⫽ 100 ⫻ ln 1 ⫹

冊冉

冊册

it it⫺1 ⫺ ln 1 ⫹ . 100 100

(38)

54


Monthly log returns on the S&P 500 index, with dividends reinvested, are computed from the Total Return Index: y 3t ⫽ 100 ⫻ 关ln共S&P 500 TR兲t ⫺ ln共S&P 500 TR兲t⫺1兴.

(39)

The time frame of the study is January 1956 to December 2000, with 539 monthly observations of Yt ⫽ ( y1t, y2t, y3t)⬘. Figure 8 shows time series plots of these variables. Retail price inflation is the most important driving force of the Wilkie (1995) investment model. Directly or indirectly, it provides inputs to other components of the model. In this section we also use the inflation rate as the threshold indicator variable, which is the driving force of the multivariate SETAR model. Therefore, we set zt ⫽ y1t. We first examine the partial autoregression matrices (PAMs) of the observed vector time series. Tiao and Box (1981) suggest summarizing the PAM using indicator symbols ⫹, ⫺, and 䡠, where ⫹ denotes a value that is greater than twice the estimated standard error, ⫺ denotes a value that is less than twice the estimated standard error, and 䡠 denotes an insignificant value based on the above criteria. The resulting indicator matrices for the PAM are given in Table 8. The likelihood ratio statistic, M(l ), can be used to test the null hypothesis that a PAM is a zero matrix (i.e., H0 : ⌸(l ) ⫽ 0). Bartlett (1938) shows that the M(l ) statistic is asymptotically ␹2 distributed with s2 degrees of freedom if the null hypothesis is true, where s is the dimension of the PAM. In Table 8 we observe that the M(l ) statistics drop significantly after l ⫽ 2. This suggests that p ⫽ 2 for the C(d) test for nonlinearity. We perform the C(d) test with p ⫽ 1, 2, d ⱕ p and various values of m (the number of start-up observations in the ordered autoregression); the results are given in Table 9. The combination ( p, d) ⫽ (2, 2) consistently gives the most significant C(d) statistic under different values of m. Therefore, we tentatively specify ( p, d) ⫽ (2, 2) for the data. With 539 observations, we entertain only the possibilities of multivariate SETAR models with two or three regimes, that is, k ⫽ 2 or 3. Given p, d, and k, we use a grid search method and select the thresholds by minimizing the AIC values that are defined in equation (35). Let ᏼ␣(zt⫺d) be the empirical ␣th percentile of zt⫺d. For two-regime models, we assume that r 僆 [ᏼ10( zt⫺d), ᏼ90( zt⫺d)]. For three-regime models, we assume that r1 僆 [ᏼ10(zt⫺d), ᏼ45( zt⫺d)], and r2 僆 [ᏼ55(zt⫺d), ᏼ90( zt⫺d)]. Table 10 shows the selected threshold values under different combinations of ( p, d, k). It indicates that the overall minimum AIC is ⫺4217.59 when k ⫽ 2, p ⫽ 2, d ⫽ 2, and r ⫽ 0.5208. We further refine the model by allowing different AR orders for different regimes. The AIC selects ( p1, p2) ⫽ (2, 2). Therefore, our final specified model is a two-regime multivariate SETAR model with the following form:

Yt ⫽

再

C0共1兲 ⫹ ⌽1共1兲Yt⫺1 ⫹ ⌽2共1兲Yt⫺2 ⫹ ␧t共1兲, if zt⫺2 ⱕ 0.5208 C0共2兲 ⫹ ⌽1共2兲Yt⫺1 ⫹ ⌽2共2兲Yt⫺2 ⫹ ␧t共2兲, if zt⫺2 ⬎ 0.5208.

(40)

Least squares estimation results of the above specified model are given in Table 11. The indicator matrices for the residual sample cross-correlations and the residual PAM are given in Table 12 and do not show any model inadequacy. The stochastic modeling of real (i.e., inflation-adjusted) investment returns is useful for pricing and designing long-term indexing insurance contracts (Wilkie 1981). Our fitted multivariate SETAR model in Table 11 can be used for generating inflation-adjusted returns. As an illustration, we generate 10 years of S&P 500 total log returns ( y3t) and inflation rates ( y1t) from the fitted model. To start the simulation, we need to specify a set of initial conditions (i.e., Y⫺1 and Y0). In our simulation, starting values are set at their sample means (Y⫺1 ⫽ Y0 ⫽ Y៮ ). These are called neutral initial conditions by Wilkie (1995, p. 902). A compound annual real rate of return is computed over the simulated 10-year period. The experiment is repeated 5,000 times. A histogram of the simulated returns is plotted in Figure 9. The average inflation adjusted total return of the S&P 500 from the simulation is around 7.32% per annum.


FOR

ACTUARIAL USE

Figure 8 Time Series Plots of Inflation Rates, Growth of Long-Term Interest Rates, and Log Returns on S&P 500 Total Return Index

55

56


Table 8 Indicator Matrices for PAM (U.S. Investment Series) Lag (l ) 1

冢

M(l )

2

⫹

⫹

䡠

䡠

⫹

⫹

䡠

⫺ 䡠 252.23

冣

冢

5. OTHER EXTENDED CLASSES

OF

3

⫹

䡠

䡠

䡠

⫺

⫹

䡠

䡠䡠 105.24

冣

冢

4

⫹

䡠

䡠

䡠

⫹

䡠

䡠

䡠 21.94

䡠

冣

冢

⫹

䡠

䡠

䡠

䡠

䡠

䡠

䡠䡠 22.26

冣

MODELS

Many families of nonlinear time series models that stem from the threshold principle have been developed in the literature (see Tong 1990; Tsay 2002 and the references therein). For example, the basic SETAR model that is discussed in equation (1) can be generalized to a more flexible version:

冘␾ pi

Y t ⫽ C 0共i兲 ⫹

j⫽1

冘␪ qi

共i兲 j

Y t⫺j ⫹

共i兲 j

X t⫺j ⫹ ε t ,

if ri⫺1 ⬍ Zt⫺d ⱕ ri

(41)

j⫽1

for i ⫽ 1, . . . , k and ⫺⬁ ⫽ r0 ⬍ r1 ⬍ . . . ⬍ rk ⫽ ⬁. The time series Yt not only is dependent on its own past history, but also depends on some exogenous observable input variables. The threshold indicating variable, Zt, could be a function of {Yt, Yt⫺1, . . . } and {Xt, Xt⫺1, . . . }, or any combinations of other outside exogenous variables, some of which may even be hidden. For this flexible class of threshold models, the most difficult task can be choosing the “appropriate” exogenous and threshold variables for a given time series Yt under study. In actuarial applications, experience and professional judgment are crucial in making such choices. The SETAR model assumes that the movement of Yt among the regimes is controlled by a “switch,” the threshold-indicating variable. A more gradual transition among the different regimes can be obtained by replacing the indicator function inherent in the above description with a continuous function. In the literature it is called the class of smooth threshold autoregressive (STAR) models (Chan and Tong 1986; Tera¨svirta 1994). On the other hand, the transition from one regime to another can be probabilistic. Tong and Lim (1980, p. 285) and Tong (1983, p. 63) discuss the idea of probabilistic switching, in which the switching can be effected by either a hidden mechanism or an observable covariate. Using a similar concept, but emphasizing aperiodic transition among the various states (regimes) of an economy, Hamilton (1989) considers the class of Markov switching autoregressive (MSA) models. This class of models employs a hidden Markov chain to govern the regime transition. Actuarial applications of MSA models can be found in the work of Harris (1997) and Hardy (2001).

Table 9 Results of the C(d) Test for Nonlinearity (U.S. Investment Series) m

p

d

C(d )

p-value

75

1 2 2

1 1 2

25.85 28.77 52.14

0.0113 0.1196 0.0002

100

1 2 2

1 1 2

23.17 28.02 50.15

0.0263 0.1395 0.0003

125

1 2 2

1 1 2

21.86 29.56 48.47

0.0392 0.1012 0.0006


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ACTUARIAL USE

57

Table 10 Selection of k, p, d, and Threshold Values for U.S. Investment Series k

p

d

r1

r2

2 2 2

1 2 2

1 1 2

0.7444 0.7444 0.5208

3 3 3

1 2 2

1 1 2

0.1971 0.1224 0.0669

AIC ⫺4,146.59 ⫺4,193.23 ⫺4,217.58 ⫺4,119.88 ⫺4,163.51 ⫺4,169.75

0.7444 0.7444 0.5208

Recent developments of threshold time series analysis include Bayesian estimation (Chen 1998), threshold cointegration (Balke and Fomby 1997), unit root testing (Tong 1995; Caner and Hansen 2001), and many others.

6. CONCLUSION

AND

FURTHER RESEARCH

We have introduced nonlinear threshold time series modeling techniques that actuaries can use in pricing insurance products, analyzing the results of experience studies, and forecasting actuarial assumptions. Basic SETAR models as well as heteroscedastic and multivariate SETAR processes are discussed. Modeling techniques for each class of models are illustrated through actuarial examples. The methods that are described in this paper have the advantage of being direct and transparent. The sequential and iterative steps of tentative specification, estimation, and diagnostic checking parallel those of the orthodox Box-Jenkins approach for univariate time series analysis. Kemp (2000) discusses the applicability of the Wilkie model (a mean-reverting process) to derivatives. The obvious effect of mean reversion on option values is through its impact on volatility. Threshold models are piecewise linear processes, and Kemp’s findings are also applicable to SETAR models. One relatively easy way to obtain market consistent and arbitrage-free valuations of options and guarantees would be to make the models risk neutral. This is known as the principle of risk-neutral

Table 11 Estimation Results for Model (40) ˆ1 ⌽

ˆ0 C

ˆ2 ⌽

ˆ ⌺

First Regime

冢冣冢

⫺0.0063

0.1576

0.2559

0.9360

(0.0211)

(0.0482)

(0.6298)

(0.0032)

0.0054

0.3468

0.0003

(0.0036)

(0.0468)

(0.0002)

⫺0.0026

(0.0016) 1.6460

(0.3316)

⫺1.5493 (0.7568)

⫺14.6783 (9.8921)

⫺0.0656 (0.0501)

冣冢

0.2386

0.4766

0.0041

(0.0639)

(0.6302)

(0.0031)

0.0006 (0.0047) ⫺0.4081 (1.0038)

⫺0.0943 (0.0468) ⫺3.9314 (9.8988)

0.0003 (0.0002) ⫺0.0406 (0.0483)

冣

冢

冣

冢

0.0544 0.0004

0.0003

⫺0.0560

⫺0.0165

13.4237

冣

Second Regime

冢冣冢 ⫺0.0012

0.4327

1.0841

0.0070

(0.0927)

(0.0800)

(1.0677)

(0.0055)

0.0045

0.0121

0.4448

0.0005

(0.0079)

(0.0068)

(0.0905)

(0.0005)

⫺0.2054

(1.5224)

⫺0.6305 (1.3138)

⫺61.4800 (17.5348)

0.0214 (0.0905)

冣冢

0.3973

1.8818

0.0031

(0.1267)

(1.1295)

(0.0059)

⫺0.0113

⫺0.3955

(0.0107)

(0.0957)

1.6989

14.8558

(2.0813)

(18.5501)

Note: Standard errors of the estimates are given in parentheses.

0.0009 (0.0005) ⫺0.0427 (0.0963)

0.0944 0.0015 ⫺0.2107

0.0007 ⫺0.01530

26.4655

冣

58


Table 12 Indicator Matrices for Residual Sample Cross-correlations and Residual PAM Lag (l ) 1

2

3

4

(a) Cross-correlations

冢

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

冣

冢

⫺

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

冣

冢

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

冣

冢

䡠

䡠

䡠

⫺

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

冣

(b) PAM

冢 M(l )

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

1.98

冣

冢

⫺

䡠

䡠

䡠

䡠

䡠

䡠

䡠

䡠

10.55

冣

冢

䡠

䡠

䡠

䡠

⫹

䡠

䡠

䡠

䡠

冣

16.79

冢

冣

15.00

valuation, which states that it is valid to assume that the world is risk neutral when pricing options and guarantees. The resulting prices are correct not only in a risk-neutral world, but also in the real world (Hull 2000, p. 205). In a world where investors are risk neutral, the expected return on all securities is the risk-free rate. To conduct a risk-neutral valuation, the models in this paper would need to be reparameterized so that all of the assets have the same expected returns as the risk-free rate. If the expected return on the asset classes is the risk-free rate, then the risk-free rate can be used to discount the option and guarantee payoffs (cash flows). Under special types of MSA models, Bollen (1998) and Hardy (2001) discuss methods of valuing options. For general threshold models, this is a nontrivial task and should be an area for further research. The interpolation of the models to continuous time situations is another possible topic for further research. Interested readers can refer to Spahr and Schwebach (1998) and Kemp (1997, 2000).

Figure 9 Simulated Inflation-Adjusted Total Return of S&P 500 from Fitted Multivariate SETAR Model


FOR

ACTUARIAL USE

59

ACKNOWLEDGMENTS The first two authors wish to acknowledge the financial support provided by a grant from the AERF Committee of The Actuarial Foundation. The authors also thank three anonymous referees, an associate editor, and the editor for their helpful comments that improved the presentation. The computer program for DTARCH modeling used in Section 3 was generously provided by Professor W. K. Li of the University of Hong Kong.

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