FUNCTIONAL COEFFICIENT AUTOREGRESSIVE NONLINEAR TIME-SERIES MODEL FOR DESCRIBING INDIA S LAC EXPORT DATA USING SAS VERSION 9.3

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1 FUNCTIONAL COEFFICIENT AUTOREGRESSIVE NONLINEAR TIME-SERIES MODEL FOR DESCRIBING INDIA S LAC EXPORT DATA USING SAS VERSION 9.3 Ranit Kumar Paul and Himadri Ghosh I.A.S.R.I., Library Avenue, Pusa, New Delhi ranitstat@gmail.com, him_adri@iasri.res.in INTRODUCTION Box Jenkins linear autoregressive integrated moving average (ARIMA) methodology is widely used for analyzing time-series data. Beyond linear domain, there are many nonlinear forms to be explored. In fact, nonlinear time-series analysis has been one of the maor areas of research in Time-series analysis for more than two decades now. These models are generally more appropriate than linear models for accurately describing dynamics of the series and for making multistep-ahead forecasts. Early development of nonlinear time-series analysis focused on various parametric forms. Engle (1982), in a path breaking work, proposed Autoregressive conditional heteroscedastic (ARCH) model for modelling volatility present in a data set. However, the conditional variance of ARCH(q) model, where q indicates the order of lag, has the property that the unconditional autocorrelation function of squared residuals, if it exists, decays very rapidly, unless q is large. To overcome this limitation of ARCH model, Bollerslev (1986) proposed the Generalized ARCH (GARCH) model, in which the unconditional autocorrelation function of squared residuals has slow decay rate. Unlike ARIMA model, these models are able to capture the presence of heteroscedasticity of conditional error variances. Another important family is bilinear timeseries model, proposed by C. W. G. Granger in 1978, which is capable of modelling data sets in which outliers appear in random epochs (Ghosh et al., 2006b). A heartening feature of the third important family, viz. Self exciting threshold autoregressive (SETAR) family, proposed by H. Tong, is that it is able to describe cyclical data quite efficiently (Tong, 1995). Quite often it is not possible to postulate appropriate parametric form for the underlying phenomenon and, in such cases; Nonparametric approach is called for. In the versatile model, viz. Functional-coefficient autoregressive (FCAR) model, introduced by Chen and Tsay (1993), the coefficient function changes gradually rather than abruptly. Fan and Zhang (2000) applied this model for the analysis of longitudinal data, in which different coefficient functions share a single smoothing variable time t, and the two-step estimates of coefficient functions in the model were given. In the present study, an attempt is made to apply FCAR model to country s yearly export data of lac from 1900 to Comparison of the performance of FCAR model with the SETAR and ARIMA models for modelling as well as forecasting is also carried out. DESCRIPTION OF FCAR MODEL Functional-coefficient autoregressive (FCAR) nonparametric nonlinear time-series model, introduced by Chen and Tsay (1993), admits the form X a ( X ) X... a ( X ) X ε (1) t 1 t-d t-1 where { t } is a sequence of independent random variables with zero mean and unit variance, and t is independent of X t-1, X t-2,. The coefficient- functions a 1 (.), a 2 (.),, a p (.) p t-d t-p t

2 are unknown and change gradually rather than abruptly. It is a direct extension of linear AR model, but allows the coefficients to vary according to a threshold variable X t-d. Nonparametric procedures are used to estimate the functions in the model, hence allowing data to speak for themselves regarding the model to be used. FCAR model can be regarded as a Stochastic regression model by introducing dependent variable Y as current observation X t, the i-th independent variable X i as lag i variable X t-i, and U as lag d variable X t-d. With induced variables, FCAR model (Cai et al., 2000) can be written as Y t = a 1 (U) X t-1 + a 2 (U) X t-2 + +a p (U) X t-p + t. (2) It is noted that many of the successful parametric nonlinear models belong to FCAR family. For example, if the functions (X ) a ( X ) I X c, it a i in (1) are step functions 2 x reduces to Threshold autoregressive (TAR) model. When ai ( X ) i i e, the model becomes an Exponential AR (EXPAR) model. SETAR and many other models also belong to this class. Hence, nonparametric determination of functional forms in the model may provide obective guidelines on choosing an appropriate parametric model. It also allows researchers to develop new models that are useful in their applications by specifying a parametric form for the coefficient functions based on nonparametric estimates. Estimation of parameters In order to apply FCAR model to data, the coefficient-functions may be estimated by using a local linear regression technique. These coefficient-functions are expanded by Taylor s series expansion in which unknown coefficients are estimated by the Method of weighted least squares, weights being the kernel density function (Fan and Yao, 2003). For any given u 0 and u in a neighborhood of u 0 and using Taylor s series expansion: a u) a ( u ) a ( u )( u u ) a b ( u ) (3) ( u0 where a and b are local intercept and slope. Using the data with U i around u 0 and local model (3), the following expression is minimized: n p Yi a b U i u0 X i K h U i u0 (4) i1 1 where K h (.) = h -1 K(./h), K(.) is a kernel function and h is the bandwidth. Then, the local linear regression estimator is simply aˆ ( u ) aˆ 0. The local linear regression estimator a and b can be easily obtained. Let e, 2 p be the 2p 1 unit vector with 1 at the th position, X ~ denote an n 2p matrix with X X( u ) as its i th row, and W diag K minimizing h i, i U i 0 T Y ( Y 1,..., Y n ), where superscript T indicates transpose. Set ( U1 u0 ),..., K h ( U n u0 ). Then the local regression problem reduces to ~ T ~ Y Xβ WY Xβ, where β a..., a, b,..., T 1. The local least square estimator is β XW X XW Y 1, 1 p b p 2 i ˆ ~ i i 1 ~ 1 ~ 2

3 Selection of bandwidth and model dependent variable As discussed in Cai et al. (2000), optimal bandwidth h in the local linear regression methodology is selected by Modified multifold cross-validation criterion. Taking m and Q as two positive integers such that n > mq, the parameters are estimated using various bandwidth values h and the Q-fitted models are used for carrying out one-step forecasting error of the next section of the time-series of length m based on the estimated models. Let a ˆ,q. be the estimated coefficients using q th, q = 1, 2,, Q subseries {(U i, X i, Y i ), 1 i n-qm}. The average prediction error using q th subseries is given by 1 APE q m The overall average prediction error is given by APE 2 nqmm p i, q i X i, 1 1 i n qm h Y aˆ U h Q 1 Q q1 APE q h. (5) The proposed data driven bandwidth is the one that minimizes APE(h). In practice, generally m = [0.1n] and Q = 4 are considered. The selected bandwidth does not depend critically on the choice of m and Q so long as mq is reasonably large, thereby ensuring that evaluation of prediction errors is stable. Choosing an appropriate model dependent variable U is also very important. Knowledge of physical background of data may be very helpful. Without any prior information, it is pertinent to choose U in terms of some data driven methods, such as Akaike information criterion (AIC), cross - validation and other criteria. Let APE(h,d) be the average prediction error defined by (2.5) using lagged variable U = X t-d. A simple and practical approach is to minimize APE(h,d) simultaneously for h in a certain range and d over the set {1, 2,, p}. The order p can also be chosen to minimize the APE. An Illustration (Ghosh et al, 2010) Lac is a resinous protective secretion of the tiny lac insect. Conventionally, it is used for ewellery, scaling wax, wood polish, mirror coating, leather finishing, etc. It is also used for coating of citrus fruits, chocolate chewing gum, printing ink, nail polish, solar cell, etc. Climatic factor have a profound influence on the development, life-cycle, larval emergence, etc. of this insect. Naturally, therefore, crop production is determined to a large extent by temperature, humidity and rainfall. These factors along with presence of interactions among three-species system involving host trees biomass and populations of lac insect and its predators are mainly responsible for cyclical variations in production as well as in export quantity of lac. The graph of country s lac export data during the period 1900 to 2000 (Fig. 1), obtained from Shellac and Forest Products Export Promotion Council, Kolkata, India, depicts this type of pattern. In general setup of time series observations the above data can be taken to be observed against chronological evolution of time like 1,2,3, Further, the ascent periods tend to exceed descent periods by approximately 50%, thereby indicating asymmetricity of the data set. 3

4 Fitting of ARIMA model by SAS Partial sas output Fig. 1. Indian Lac export data data lac; input y; /* y denotes the lac export*/ cards; ; proc arima data= lac; identify var=y(2) ; /* differencing the series twice*/ estimate p=2 q=4; forecast lead=4 out=flac ; run; proc print data=flac; run; Name of Variable = export Period(s) of Differencing 2 Mean of Working Series Standard Deviation Number of Observations 99 Observation(s) eliminated by differencing 2 Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq Autocorrelations

5 Conditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > t Lag MU MA1, < MA1, MA1, MA1, AR1, AR1, < A perusal of Fig. 1 indicates that the country s lac export data is not stationary. Therefore, the original series is differenced repeatedly unless it becomes stationary. It is found that this is achieved after second differencing. On investigating the autocorrelation functions (acf) and partial autocorrelation functions (pacf) of the resultant series, as well as minimum AIC and BIC values computed respectively by (6) and (7), ARIMA(2,2,4) model is selected. AIC = - 2 log(l) + 2p (6) BIC = - 2 log(l) + p log(n) (7) where L is the likelihood function, p denotes the number of parameters, and N is the sample size. The parameters of this model, which is a particular case of Cyclical trend model given in Harvey (1996), are estimated by using SPSS, Ver software package and the same are reported along with other related statistics in Table 1. For selected model, the AIC and BIC values are respectively computed as and Table 1: Parameter estimates along with other related statistics Parameters Estimate Standard t-statistic Probability Error Constant AR <.0001 AR MA MA MA MA <.0001 Fitting of FCAR model Optimum FCAR model is selected by the methodology discussed in Section 2.2. Estimates of parameters of this model are obtained by using computer program written in SAS- IML and appended as an Annexure. To ease the calculation, logarithm of the data is considered for analysis. The optimum values for p and d are found as p = 4 and d = 3. The estimated coefficient functions are plotted in Fig. 2, which clearly shows that the coefficient functions a i (u), i = 1, 2, 3, 4 are no longer constants but are changing as u varies. The function APE against the bandwidth over a grid of point h = 0.1 ( = 1, 2,, 10) is computed. The selected bandwidth along with autoregressive order p and delay parameter d which minimizes the APE defined in (5) is found as h =

6 Finding optimum values of (p, d, h) on the basis of APE criterion in SAS IML data lac; input export; cards; ; Proc iml; use lac; read all into Y;*/specify the time-series vector; pdh=(1,4,999); */Initializing the vector; APEh=(4,1,0); u0=sum(y)/nrow(y); do p=1 to 8; do d= 1 to p; x0=(nrow(y),p,0); do i = p+1 to nrow(y); x0[i,]=y[i-1:i-p]`; do h=0.1 to 1 by 0.1; do =1 to 4; t=nrow(y)-7*; n=t-p; x=(t,p,0); /*forming the matrix of the lagged values*/ u=(t,1,0); do i = p+1 to t; x[i,]=y[i-1:i-p]`; u[i]=y[i-d]; x1=x[p+1:t,]; U1=u[p+1:t,]; w=(n,n,0); w1=(n,1,0); /* kernel function is k(x)=1/h*k(x/h)*/ /*kx=(1/(h*sqrt(2*3.14))*exp(-x*x/(2*h*h)));*/ /*calculating the weight matrix by using kernel function*/ x2=(n,p,1); do i = 1 to n; ind=(u1[i]-u0)/h; /* ind is the indicator variable*/ if ind<=1 then in=1;else in=0; w[i,i]=(0.75/h)*(1-((u1[i]-u0)*(u1[i]-u0))/(h*h))*in; w1[i,]=w[i,i]; x2[i,]=x1[i,]*(u1[i]-u0); Xcurl=(n,2*p,0); xcurl=x1 x2; y1=y[p+1:t,]; beta=inv(xcurl`*w*xcurl)*(xcurl`*w*y1); beta1=beta[1:p,]; 6

7 yhat=xcurl*beta; error=y1-yhat; apeerror=(7,1,0); apeerror=error[n-7*+1:n-7*+7,]; APEh[]=(1/7)*apeerror`*apeerror; APE=sum(APEh)/4; pdh=pdh//(p d h APE); pdh=pdh[2:nrow(pdh),]; minape=min(pdh[,4]); do i=1 to nrow(pdh); if (pdh[i,4]=minape) then rw=i; result=pdh[rw,]; print beta1; run; Partial SAS Output p d h APE

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9 FCAR nonlinear time-series model for describing Indian lac export data minape p d h Beta p and d are the order of FCAR model where p denotes AR lag and d is the delay parameter. h is the optimal bandwidth. minape denotes the minimum Average Prediction Error a 1 (u) a 2 (u) a 3 (u) a 4 (u) Fig. 2. The functional forms of a 1 (u), a 2 (u), a 3 (u) and a 4 (u) 9

10 The fitted FCAR (4,3) model for log data is obtained by using equations 2 and 3, and using the parameter estimates Beta1 (SAS output) as follows: log X t = { (log X t )} log X t-1 + { (log X t )} log X t-2 +{ (log X t )} log X t-3 + { (log X t )} log X t-4 Here, X t-3 is the threshold value and is the mean of the data By taking antilog of fitted values by above model, performance of the fitted model is evaluated by computing the AIC and BIC values: AIC = N log(rss/n) + 2p (8) BIC = N log(rss/n) + p log(n) (9) where RSS is the residual sum of squares by minimizing (2.4), p is the total number of parameters to be estimated and N is the sample size. For the present example, the AIC and BIC values computed form (8) and (9) are and respectively. Ghosh et al. (2006a) had earlier fitted the following SETAR (2;1,2) model to the same data: X t X t-1 = X t-1, if X t-1 < = X t X t-2, if X t-1 > with AIC value computed as , which is slightly more than the one for FCAR model. Thus, for describing the data set under consideration, FCAR model is slightly better than SETAR model and both of these are superior to ARIMA model. To get a visual idea, graph of fitted FCAR model along with data points is exhibited in Fig. 3. Fig. 3. The fitted FCAR(4, 3) model along with the data points Forecasting One-step ahead forecasts and iterative two-step ahead forecasts of lac export during the years 2001 to 2004 for ARIMA (2,2,4), SETAR(2;1,2) and FCAR(4,3) models are carried out and reported in Table 2 and Table 3 respectively. For one-step ahead forecast, let T=2000 be the origin of forecast. Let ŷt i1 T i be the one step ahead forecast of yt i1 given information upto time T+i, i=0,1,2,3. Similarly, for two-step ahead forecast, let T=1999 be the origin of forecast. Let ŷt i2 Ti be the one step ahead forecast of yt i2 given information upto time T+i, i=0,1,2,3. The models are compared on the basis of Mean absolute prediction error (MAPE), Mean square prediction error (MSPE) and Relative mean absolute prediction error (RMAPE) given by 10

11 4 MAPE = 1 / 4y t ŷ i1 4 i ti MSPE = 1/ 4 y yˆ i1 4 ti 2 ti RMAPE = 1/ 4y ˆ ti yti / yti100 i1 Computed values of MAPE, MSPE and RMAPE for the above mentioned three models are reported in Table 4. lead time of forecast Table 2. One-step ahead Forecast of lac export Actual Forecast by ARIMA model SETAR model FCAR model lead time of forecast Table 3. Two-step ahead Forecast of lac export Actual Forecast by ARIMA model SETAR model FCAR model Table 4. Comparison of forecast performance One-step ahead Two-step ahead Models MAPE MSPE RMAPE MAPE MSPE RMAPE ARIMA(2,2,4) SETAR(2,1,2) FCAR(4,3) A perusal of Table 4 indicates that FCAR model has performed better than both the SETAR as well as ARIMA models for forecasting. To sum up, for the data under consideration, FCAR model is found to be the best for modelling as well as forecasting. Concluding Remarks FCAR nonparametric nonlinear time-series model is applied for modelling and forecasting of Indian lac export data by using SAS. Superiority of FCAR model over ARIMA and SETAR models for the data under consideration is clearly demonstrated. One-step ahead forecasts and iterative two-step ahead forecasts of lac export during the years 2001 to 2004 for ARIMA (2,2,4), SETAR(2;1,2) and FCAR(4,3) models are carried out. On the basis of three statistics 11

12 namely MSPE, MAPE and RMAPE, it is concluded that FCAR model performs better than the SETAR model and ARIMA model for the data under consideration. References Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. J. Econ. 31, Cai, Z., Fan, J. and Yao, Q. (2000). Functional-coefficient regression models for nonlinear time-series. J. Amer. Statist. Assoc. 95, Chen, R. and Tsay, R. S. (1993). Functional coefficient autoregressive models. J. Amer. Statist. Assoc. 88, Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50, Fan, J., Zhang, J. (2000). Two-step estimation of functional linear models with applications to longitudinal data. J. Roy. Statist. Soc. Ser. B 62, Fan, J. and Yao, Q. (2003). Nonlinear Time-Series: Nonparametric and Parametric Methods. Springer, U.S.A. Ghosh, H., Sunilkumar, G. and Praneshu. (2006a). Self exciting threshold autoregressive models for describing cyclical data. Cal. Statist. Assoc. Bull. 58, Ghosh, H., Sunilkumar, G. and Praneshu. (2006b). Modelling and forecasting of bilinear time-series using frequency domain approach. J. Combi. Info. Sys. Sci. 31, Ghosh, H., Paul, R. K. and Praneshu. (2010). Functional coefficient autoregressive model for forecasting Indian lac export data. Mod. Assist. Statist. Appl. 5, Granger, C. and Terasvirta, T. (1993) Modelling Nonlinear Economic Relationships. Oxford University Press, Oxford Harvey, A. C. (1996). Forecasting Structural Time-Series Models and the Kalman Filter Cambridge University Press, U. K. Tong, H. (1995). Nonlinear Time-series Analysis: A Dynamic Approach. 2 nd Ed. Oxford University Press, Oxford. 12