II. Bivariate vector bilinear model: Xi;t = i;t + 0:6X2;t-2 i;t-i. X2;t = 2;t + 0:4X2;t-i i;t-2. III. Bivariate vector nonlinear moving average model: Xi;t = i;t + 0:6 2;t-i i;t-2.
LAG IDENTIFICATION FOR VECTOR NONLINEAR TIME SERIES Jane L. Harvill and Bonnie K. Ray Mississippi State University Mississippi State, Mississippi 39762 and New Jersey Institute of Technology Newark, New Jersey 07102 Keywords: Mutual information criterion; Kendall's .
ABSTRACT
Exploratory methods for determining appropriate lagged variables in a vector nonlinear time series model are investigated. The rst is a multivariate extension of the R statistic considered by Granger and Lin (1994), which is based on an estimate of the mutual information criterion. The second method uses Kendall's and partial statistics for lag determination. Both methods provide nonlinear analogs of the autocorrelation and partial autocorrelation matrices for a vector time series. Simulation studies indicate the methods reliably identify appropriate lags for many types of vector nonlinear time series. For illustration, the methods are applied to set of annual temperature and tree ring measurements at Campito Mountain in California.
1. INTRODUCTION
Recently interest has developed in techniques for modeling nonlinear dynamics in a vector time series. For example, Tsay (1998) discusses testing and modeling multivariate threshold autoregressive models, with application to modeling returns on the S&P 500 stock index and associated futures contracts. Hipel and McLeod (1994) hypothesize that the relationship between
daily river ows and precipitation may be nonlinear, and so require a vector nonlinear model. A test for nonlinearity in the multivariate framework was developed by Harvill and Ray (1999). Granger and Terasvirta (1993) give examples of vector nonlinear autoregressive processes, vector nonlinear moving average processes, and multiple bilinear time series models. While examination of the sample autocorrelation (ACF) and partial autocorrelation (PACF) matrices is useful for preliminary model speci cation in a vector linear autoregressive moving-average (VARMA) model (see, for example, Reinsel, 1991, Ch. 4), there are cases when it is not sucient for lag selection in the nonlinear framework. Well-known examples include some particular bilinear processes and deterministic chaotic series. Each of these have an ACF that is identical to the ACF of a white noise or linear process (Granger, 1983). In order to model vector nonlinear time series systematically, general techniques must be developed for model speci cation, analogous to those used for linear VARMA modeling. Granger and Lin (1994) use two statistics { the rst based on the mutual information coecient, and the second on rank correlation statistics { to identify signi cant lags in a univariate nonlinear time series model. In this paper, we extend these techniques to the multivariate framework and investigate their power for identifying signi cant lags in a multivariate nonlinear time series model. In Section 2, we de ne the statistics used in our study and discuss computational issues. Section 3 presents results of a simulation study in which the model identi cation tools are applied to various vector nonlinear time series models. In Section 4, we apply the techniques to a set of annual tree ring widths and temperatures. Other methods for lag identi cation and model selection of nonlinear time series in the univariate framework have been suggested in the literature (Auestad and Tjostheim, 1990; Cheng and Tong, 1992; Chen and Tsay, 1993;
Yao and Tong, 1994). All of these methods are based on approximating the nonlinear structure using a nonlinear autoregressive model estimated using a nonparametric regression technique. The number of lags used in the autoregression is chosen by minimizing some function of the mean squared error, such as the Final Prediction Error (FPE) statistic or the cross-validation (CV) statistic. While these methods are useful in the univariate framework, their extension to the multivariate framework, especially at the exploratory model identi cation stage, can be problematic. The explosion in the number of possible lagged predictors in the vector framework results in the \curse of dimensionality" for kernel regression methods used in estimating the nonlinear autoregressions. A method for initial weeding out of nonsigni cant variables is useful before the model tting stage. Additionally, identi cation of nonlinear moving average behavior may simplify the model tting, as moving average terms can be included in the model directly, without resorting to using a large number of autoregressive terms to approximate the moving average behavior.
2. IDENTIFYING LAGS IN VECTOR NONLINEAR MODELS A variety of methods exist for measuring dependence between lagged values of a time series. Two measures that are particularly useful in the linear models framework are the autocorrelation function (ACF) and partial autocorrelation function (PACF). Unfortunately, these well-understood measures are not always sucient for measuring nonlinear dependence. Speci c examples include some particular bilinear processes and deterministic chaotic series which have an ACF that is identical to the ACF of a white noise or linear process (Granger, 1983). In light of this, it is bene cial to have alternative statistics for measuring nonlinear dependence which are analogous to the ACF and PACF.
2.1. MEASURING NONLINEAR DEPENDENCE
One well-known measure of association that can identify certain nonlinear relationships is Kendall's (1938) statistic, based on ranks. It is de ned as follows. For pairs of observations (x ; y ); i = 1; 2; : : : ; n, de ne d(i; j ) = sign(x , x )(y , y ). Let N be the number of pairs (i; j ) that are nonzero, C be the number of pairs for which d(i; j ) is positive, and D be the number of pairs for which d(i; j ) is negative. Then i
i
j
i
i
j
^ =
C ,D N
(1:1)
is a measure of the dependence between x and y. Note that, although is able to measure some types nonlinear dependence between x and y, it cannot capture non-monotonic nonlinear relationships. A variation of Kendall's , commonly referred to as Kendall's partial (Quade, 1967), is a measure of the association between two variables x and y while controlling for a third variable, z . Kendall's partial is de ned by ^ = p
C
p
,D ; N p
p
where N denotes the number of pairs (x ; y ; z ) such that jz , z j T for T a prede ned tolerance; that is, N is the number of pairs (x ; y ) such that z is relatively constant. C and D are de ned similarly. Both and have asymptotically normal limiting distributions. Granger and Lin (1994) develop a statistic for measuring nonlinear dependence based on a the mutual information coecient, also referred to as relative entropy (Joe, 1989a). Let x and y be two random variables with joint probability density function f (x; y) and marginal density functions f (x) and f (y ), respectively. Then the relative entropy is de ned by p
i
z
i
i
p
i
p
i
j
i
z
i
p
p
x
y
(x; y ) =
Z Z y
x
f (x; y ) log
f (x; y ) dx dy; f (x)f (y ) x
y
(1:2)
The function measures the distance between a joint distribution and the distribution when x and y are independent. Shannon entropy is often interpreted as the amount of information present in a system. The relative entropy, , is related to the Shannon entropy, H , by
R
(x; y ) = H (x) + H (y ) , H (x; y );
(1:3)
where H (x) = , x logff (x)gf (x) dx for any p-dimensional random variable x. The function is also easily de ned for discrete x and y with the integrals in (1.2) replaced by sums. The statistic R is then de ned by x
x
R(x; y ) = [1 , expf,2 (x; y )g]1 2: =
(1:4)
Granger and Lin (1994) show that R has a number of attractive properties, among them 1. R is well-de ned for both continuous and discrete variables, 2. R takes values in the range [0; 1], with values increasing with ; R takes value zero if x and y are independent; it is unity if there is an exact nonlinear relationship between x and y, e.g., x = h(y), 3. if x and y have a bivariate normal distribution with correlation , then R = jj. Similarly, a partial mutual information coecient, and hence a partial R statistic, R , for variables x and y may be de ned as Z Z f (x; y jz ) dx dy; (x; y ) = f (x; y jz ) log f (xjz )f (y jz ) where f (jz) denotes the conditional distribution given variable z. In the time series framework, R (and ) can be used to measure the strength of association between lagged values of a series. For a k-variate time series X , the analogue to the (i; j ) element of the ACF matrix at lag l based on R is de ned by p
z
x
x
y
y
th
t
R(X ; X , ) = R(X ; X , ): t
t
l i;j
i;t
j;t
l
If, for example, X1 = :622 ,1 + 1 , where f g denotes a white noise process, the value of R(X ; X , )1 2 is nonzero at lag l = 1 and zero thereafter. Note that the value of in this case is zero, since the relationship between X1 and X2 ,1 is non-monotonic. The analogue to the PACF matrix at lag l for a k-variate time series based on R ( ) is ;t
t
;t
;t
;t
t
l
t
;
;t
p
p
R (X ; X , ) = R (X ;X , jX1 ,1 ; : : : ; X ,1 ; : : : ; p
t
t
l i;j
p
i;t
j;t
l
;t
k;t
X1 , +1 ; : : : ; X , + ); ;t
l
k;t
l
l
that is, z is the vector (X1 ,1; : : : ; X ,1; : : : ; X1 , +1; : : : ; X , +1). The following subsection discusses how to estimate the \shadow correlation coecient" R for a vector time series. ;t
k;t
;t
l
k;t
l
2.2 ESTIMATION AND COMPUTATION
The estimation of R requires two steps: rst, the density functions f (x; y ), f (x), and f (x) must be estimated; evaluation of the integral in (1.2) is then required to obtain ^. To estimate the univariate densities, we use a kernel-based method applied to the standardized data with kernel a Student's t distribution with four degrees of freedom, bandwidth h = 0:85n,1 5, and no boundary correction. The bivariate density is estimated using a product kernel of Student's t distributions with bandwidth h = 0:85(1:0 , 2 )5 12 (1 + 2=2),1 6 n,1 6 (see, e.g., Scott, 1992). We use the t-distribution kernel based on results of Hall and Morton (1993) concerning estimation of entropy. Fifty point Gaussian quadrature was used to evaluate the integral in (1.2) and obtain ^(x; y). Joe (1989b) and Hall and Morton (1993) show that a summation-based estimator of H (x), and thus of and R, is root-n consistent up to the three dimensional case under certain regularity conditions on the underlying distribution, particularly conditions concerning the tails of the x
y
=
=
=
=
underlying distribution. However, Granger and Lin (1994) nd that the numerical integration-based estimator of R performs better in nite samples. No distributional theory is currently available for R^ . In the vector time series case, the estimation of R is problematic due to the \curse of dimensionality" when trying to estimate the conditional densities. Other density estimation methods, such as projection pursuit, could applied as an attempt to mitigate this problem. However, we leave this for future research and consider only R in this paper. Kendall's partial is computed for the rank transformed data with tolerance T = 0:2n to examine partial nonlinear dependence relationships. p
z
In the next section, we investigate the ecacy of these measures for lag selection in various vector nonlinear time series models.
3. SIMULATION RESULTS A simulation study was conducted to investigate the size and power of R^ , and to compare its performance to that of Kendall's and Kendall's partial in selecting important lags. Data from a variety of bivariate time series models, each of length n = 300, was generated. The models considered included a (linear) vector moving average model (Model I), a vector bilinear model (Model II), two vector nonlinear moving average models (Models III and IV), a vector threshold autoregressive model with delay d = 2 (Model V), and a vector exponential autoregressive model (Model VI). For each model, the white noise disturbances were generated from a bivariate Gaussian distribution with standard marginals and cross-correlation = 0:0 and 0.5. The empirical mean and standard deviation of R^ , ^, and ^ were found for each of lags one through ve based on 200 replications. Speci cally the models were p
I. Bivariate vector moving average model of lag two:
, 0:61 ,1 , 0:22 ,1 , 0:41 ,2 X2 = 2 + 0:41 ,1 , 0:31 ,2 + 0:22 ,2 X1 = 1 ;t
;t
;t
;t
;t
;t
;t
;t
;t
;t
II. Bivariate vector bilinear model: X1 = 1 + 0:6X2 ,2 1 ,1 ;t
;t
;t
;t
X2 = 2 + 0:4X2 ,1 1 ,2 ;t
;t
;t
;t
III. Bivariate vector nonlinear moving average model: X1 = 1 + 0:62 ,1 1 ,2 ;t
;t
;t
;t
X2 = 2 + 0:61 ,1 2 ,2 ;t
;t
;t
;t
IV. Bivariate vector nonlinear moving average model:
, 0:41 ,1 + 0:31 ,2 + 0:521 ,1 X2 = 2 , 0:42 ,1 + 0:32 ,2 + 0:521 ,1 X1 = 1 ;t
;t
;t
;t
;t
;t
;t
;t
;t
;t
V. Bivariate vector threshold autoregressive model:
1 + 0:6X2 ,1 1 2 + 0:6X1 ,1 X2 = 2 X1 = ;t
;t
;t
;t
;t
;t
;t
;t
if X1 if X1 if X2 if X2
,2 > 0 ,2 0 ,2 > 0 ,2 0
;t ;t ;t ;t
VI. Bivariate vector exponential autoregressive model: X1 = 0:4X1 ,1 + expf,0:5(0:3X12 ,2 + 0:7X22 ,2 )g0:6X2 ,1 + 1 X1 = 0:6X2 ,1 + expf,0:5(0:8X12 ,2 + 0:2X22 ,2 )g0:4X1 ,1 + 1 ;t
;t
;t
;t
;t
;t
;t
;t
;t
;t
;t
;t
Distributional properties of R^ under the null hypothesis of no association (i.e., X is a multivariate white noise series) are not known in general, thus Monte Carlo simulation was used to obtain the null distribution. Onethousand Gaussian white noise series of lengths n = 66 and 300 were generated and R^ (l); l = 1; : : : ; 5 was computed for each series. Table I shows the t
empirical mean, standard deviation, and 90%, 95%, and 99% critical values for this case.
Table I
Critical Values of R for n = 300. Lag 1 2 3 4 5
90% 0.21800 0.21771 0.21832 0.21870 0.21510
95% 0.22712 0.22520 0.22780 0.22780 0.22170
99% 0.24341 0.23920 0.24490 0.24261 0.23820
Mean 0.18980 0.18852 0.18980 0.19032 0.18976
Standard Deviation 0.022037 0.021897 0.022000 0.021811 0.020385
The statistic has positive bias, due to the relation between R and . A very small bias in ^, e.g. 0.02, results in an R^ value of 0.19802. A density estimate of the distribution of R^ (1) is shown in Figure 1. The distribution is right-skewed. FIGURE 1 ABOUT HERE. For each lag, the rst two rows of Tables II through VII give the empirical averages and standard deviations (in parentheses) for the R statistic, Kendall's , and Kendall's partial , for Models I through VI, respectively. TABLES II THROUGH VII ABOUT HERE Mean values of the computed statistic signi cantly greater than that for a white noise process at level = 0:10 are denoted by a single asterick (*), by two astericks at level = 0:05, and by a dagger (y) at level = 0:01. To determine whether R^ , ^, and ^ were signi cantly better than their linear counterparts at detecting possibly nonlinear dependence relationships, the sample ACF and PACF we also computed for each model. Those results are not included here for reasons of space, but are referred to in the following discussion. p
For the linear VMA(2) model, Model I, R^ and ^ perform comparatively to their linear counterparts, taking values signi cantly di erent from zero at the correct lags. The cross-correlation between the white noise disturbances has little e ect on the results. For a VMA(2) model, we expect to see partial correlation values slowly decrease as l increases. The ^ statistic goes to zero faster than the PACF as l increases. Results for Model II, a bivariate bilinear model, indicate that R^ successfully identi es appropriate lags, whereas ^ and ^ do not. This is a case in which the nonlinear dependence is not monotonic. Model III is a bivariate nonlinear MA model. This is another case in which R^ is signi cantly di erent from zero at the appropriate lags, while ^ and ^ do not detect the (nonmonotonic) association between X and , ; i 6= j; l = 1; 2. The sample ACF and PACF for both Model II and Model III mimic that of a white noise process. p
p
p
i;t
j;t
l
Model IV is another MA model, containing both linear and nonlinear terms. The R^ statistic is the only one to detect the nonlinear dependence between X1 and 1 ,1 . Correlation between the disturbances causes values of R^ for X1 and 2 ,1 and X1 and 2 ,2 to take signi cant values as well. ;t
;t
;t
;t
;t
;t
Model V represents a bivariate TAR model with delay d = 2 and threshold variables X ,2 ; i = 1; 2. In this case, all three statistics detect the linear AR component X ,1 for the rst regime of each X ; i 6= j; i; j = 1; 2. Furthermore, R^ and ^ pick up the delay lag d = 2 in the threshold variable X ,2 . Analogous to the ACF, the statistics R^ and ^ slowly decay past lag one, with R^ appearing to decay a bit faster than ^. The values of ^ (an analog of the PACF) cut o after lag 1. i;t
j;t
i;t
i;t
p
The last model we consider, Model VI, is a vector exponential AR model (Granger and Terasvirta, 1993, p.38). Interestingly, values of R^ for X1 with X2 , and X2 with X1 , exhibit a slow decay as l increases, indicative ;t
;t
l
;t
;t
l
of the nonlinear relationship. The ^ statistic is not as powerful in detecting this relationship. The ACF detects only the linear relationship between X1 and X1 , and X2 and X2 , . Both the PACF and ^ cut o after lag one, failing to detect the nonlinear relationship at lag two. ;t
;t
l
;t
;t
l
p
For each model, we also recorded the empirical rejection rates of the di erent statistics for (X ; X , ); i; j = 1; 2; l = 1; : : : ; 5 for = 0:05. The pattern of the empirical rejection rates, matches those of results summarized in the preceeding paragraphs. In general, the empirical power of R^ was higher in detecting important lags. A complete listing of these empirical rejection rates, along with result of of similar simulations on other models not presented in this paper may be downloaded in postscript les called lagselsim.ps format from http://www2.msstate.edu/harvill/ftp/. t;i
t
l;j
4. AN EXAMPLE As an example, the lag identi cation techniques of Section 2 are applied to a series of annual temperatures ( F ) and tree ring widths (0.01 mm) measured at Campito Mountain in California for the years 1907-1972 (n = 66). This data has been used by Fritts et al. (1971) in studying multivariate techniques for specifying tree growth and climatic relationships. The data was produced and assembled at the Tree Ring Laboratory at the University of Arizona, Tucson and is available from o
http://www.stat.cmu.edu/datasets/hipel-mcleod
under the directory lamarche. Figure 2 shows time series plots of the data. FIGURE 2 ABOUT HERE. Recent work, e.g. Arbarbanel and Lall (1996), Tong, Thanoon, and Gudmundsson (1985), indicate that many climatic relationships involve nonlinear dynamics. The multivariate nonlinearity test of Harvill and Ray (1999) applied to the temperature data (X1) and tree ring data (X2) using lagged
values of the series up to lag four indicates nonlinearity at the 3.0% signi cance level. A semi-multivariate nonlinearity test applied to each component separately indicates stronger evidence of nonlinearity in the temperature series than in the tree ring series. Both tree ring growth and temperature have been shown to be highly in uenced by the El Nino/Southern Oscillation (ENSO) (Cole and Cook, 1998). ENSO is a quasiperiodic variation in climate which arises from a complex interaction between the tropical Paci c Ocean and the atmosphere. Additionally, tree ring widths re ect changes in other climatic variables, such as precipitation. The manner in which extratropical circulation, hence regional climate variables, are a ected can be highly nonlinear. Table VIII contains the values of the ACF, PACF, R, , and for this data up to lag four, with values signi cantly di erent from zero marked by asterisks and daggers as in Tables II through VII. p
TABLE VIII ABOUT HERE. Empirical signi cance levels of R^ were found by Monte Carlo simulation for a white noise series of length n = 66 using 1000 replications. These values are given in Table IX.
Table IX
Critical Values of R for n = 66. Lag 1 2 3 4 5
90% 0.29950 0.29805 0.30380 0.29785 0.30515
95% 0.31660 0.31500 0.32480 0.31475 0.32585
99% 0.34835 0.34890 0.37100 0.34500 0.36690
Mean 0.241829 0.240738 0.243390 0.241537 0.243028
Standard Deviation 0.042384 0.044710 0.046060 0.043042 0.048110
The ACF identi es an association between tree ring widths in year t
and tree ring widths up to four years back, while changes in temperature are associated with the previous year's tree growth. The results for ^ match those of the ACF. The PACF takes nonsigni cant values after lag one, suggesting an AR(1) for linearly modeling the data. The R^ statistic, however, indicates a relation between temperature in year t and tree ring width two to three years back that is not identi ed by the ACF, suggesting a nonlinear relationship between temperatures and lagged tree ring width. Additionally, values of ^ are signi cantly di erent from zero for temperatures in relation to temperature and tree ring width at lag four. It is well-known that frequency of ENSO periods in the last century has been between two and ve years. p
5. CONCLUSIONS Initial determination of appropriate lags for inclusion in a vector nonlinear time series model can be made using the R statistic of Granger and Lin (1994), and using Kendall's and partial statistics in certain instances. Much work remains to be done on actual model estimation techniques in the vector nonlinear time series case. Hardle, Tsybakov, and Yang (1998), Rao and Wong (1998), and Tsay (1998) have begun work in this direction. In future research, we plan to investigate nonparametric methods that avoid the curse-of-dimensionality, such as projection pursuit regression (Friedman and Stuetzle, 1981) or a multivariate version of MARS (Friedman, 1991), for modeling vector nonlinear time series.
ACKNOWLEDGEMENTS The research of Bonnie Ray is supported, in part, by a grant from the National Science Foundation.
BIBLIOGRAPHY Abarbanel, H. and Lall, U. (1996). \Nonlinear Dynamics of the Great Salt Lake: System Identi cation and Prediction," Climate Dynamics, 12, 287-297. Auestad, B. and Tjostheim, D. (1994). \Nonparametric Identi cation of Non-linear Time Series: Selecting Signi cant Lags," J. Am. Statist. Assoc., 89, 1410-1419. Chan, K.S. and Tong, H. (1994). \A Note on Noisy Chaos," J. R. Statist. Soc., B, 56, 301-311. Chen R. and Tsay, R.S. (1993). \Nonlinear Additive ARX Models," J. Am. Statist. Assoc., 88, 955-967. Cheng, B. and Tong, H. (1992). \On Consistent Nonparametric Order Determination and Chaos," J. R. Statist. Soc., B, 54, 427-449. Cole, J. and Cook, E. (1998). \The Changing Relationship Between ENSO Variability and Moisture Balance in the Continental United States," Geophysical Research Letters, 25, 4529-4532. Friedman, J. (1991). \Multivariate Adaptive Regression Splines (with Discussion)," Ann. of Statist., 19, 1-142. Friedman, J. and Stuetzle, W. (1981). \Projection Pursuit Regression," J. Amer. Statist. Assoc., 76, 817-823. Fritts, H., Blasing, T., Hayden, B. and Kutzbach, J. (1971). \Multivariate Techniques for Specifying Tree Growth and Climatic Relationships and for Reconstructing Anomalies in Paleoclimate," J. of Appl. Meteorology, 10, 845-864. Granger, C.W.J. (1983). \Forecasting White Noise," Appl. Time Ser. Anal.
of Economic Data, (Zellner, A., Ed.). U.S. Dept. of Commerce, Washington, D.C.
Granger, C.W.J. and Lin, J. (1994). \Using the Mutual Information Coef cient to Identify Lags in Nonlinear Models," J. Time Ser. Anal., 15, 371-384. Granger, C.W.J. and Terasvirta,T. (1993). Modeling Nonlinear Economic Relationships, New York: Oxford University Press. Hall, P. and Morton, S.C. (1993). \On the Estimation of Entropy," Ann. Inst. Statist. Math., 45, 69-88. Hardle, W., Tsybakov, A., and Yang, L. (1998). \Nonparametric Vector Autoregression," J. Statist. Plann. Inference, 68, 221-245. Harvill, J.H. and Ray, B.K. (1999). \A Note on Tests for Nonlinearity in a Vector Time Series," Biometrika, (to appear). Hipel, K.W. and McLeod, A.I. (1994). Time Series Modeling of Water Resources and Environmental Systems, Amsterdam: Elsevier. Joe, H. (1989a). \Relative Entropy Measures of Multivariate Dependence," J. Am. Statist. Assoc., 84, 157-164. Joe. H. (1989b). \Estimation of Entropy and Other Functionals of a Multivariate Density," Ann. Inst. Statist. Math., 41, 683-697. Kendall, M. (1938). \A New Measure of Rank Correlation," Biometrika, 30, 81-89. Quade, D. (1967). \Rank Analysis of Covariance," J. Am. Statist. Assoc., 62, 1187-1200. Rao, S. and Wong, W. (1998). \Some Contributions to the Multivariate Nonlinear Time Series and to the Bilinear Models," Dept. of Statistics,
University of Manchester Institute of Science and Technology (preprint). Reinsel, G.C. (1993). Elements of Multivariate Time Series Analysis, New York: Springer Verlag. Scott, D. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization, New York: Wiley. Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach, Oxford: Clarendon Press. Tong, H., Thanoon, B., and Gudmundsson, G. (1985). \Threshold Time Series Modeling of Two Icelandic River ow Systems," Water Resources Bulletin, 21, 651-660. Tsay, R.S. (1998). \Testing and Modeling Multivariate Threshold Models," J. Am. Statist. Assoc., 93, 1188-1202. Yao, Q. and Tong, H. (1994). \On Subset Selection in Non-parametric Stochastic Regression," Statist. Sinica, 4, 51-70.
0
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FIG. 1. Empirical Distribution of R(1) for a White Noise Process of Length n = 300 Estimated Using Bandwidth h = 0:02 and Triangular Kernel.
0.15
0.20
0.25
FIG. 2. Annual Temperatures and Tree Ring Widths for the Years 1907-1970 at Campito Mountain, California.
51 52 53 54 55 56
Temperature ( degrees Fahrenheit)
1910
1920
1930
1940
1950
1960
1970
1950
1960
1970
Year
30
40
50
60
70
Tree ring width (0.01 mm)
1910
1920
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1940 Year
