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Forecasting Monthly Prices and Quantities: A Study of Apparel Cottons Export Lau Chi-Keung, Marco* To Kin-Man, Chester Zhang Zhi Ming The Hong Kong Polytechnic University Keywords: Apparel Cottons; bilateral trade; Seasonal ARIMA model; Forcasting. JEL codes: F14, C15, Last version: March-2007 Abstract The aim of this paper is to construct a seasonal ARIMA forecasting model for the prices and quantities of MFA apparel cottons imported from HK and China to the U.S. during 1989-2005 by following Box-Jenkins technique, and the predictive power was satisfactory. We believed that the empirical findings are useful to international textile and clothing buyers and sellers as well as the trade policy makers. I. Introduction The aim of this paper is to build up a forecasting model for MFA apparel cottons imported from HK and China to the U.S. Monthly data from Jan-1989 to June-2005 was collected from the Office of Textiles and Apparel of the U.S. Department of Commerce (OTEXA, Jan, 2006)1. The forecasting model was estimated though Box- Jenkins (1976) seasonal Autoregressive Integrated Moving Average (ARIMA) modeling technique, which involves four steps: identification, estimation, diagnostic checking and finally forecasting. Detailed forecasting technique can also found in Bowerman and O’Connell(1993). Original data on prices and quantities are plotted in figure 2, from the line graph; two preliminary observations are found. First, the unit prices in Hong Kong are generally higher and less volatile than that in Mainland China. Second, before Year 2000 the import quantity was roughly the same for Hong Kong and Mainland China, however, Mainland China increased its exports tremendously after year 2000 and the gap is widening afterwards because of the removal of MFA quota. Recently, Lau, To & Zhang (2007) found evidence of structural break in the year 2000 for time series * Corresponding author: Lau Chi Keung, Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hung Hum, Hong Kong. Email: 06901279r@polyu.edu.hk 1 Original Data are available at http://otexa.ita.doc.gov/msrpoint.htm 1 examined in this study. Therefore, we believed that a seasonal forecasting model based on the entire sample period may prevail useful information to international textile and clothing buyers and sellers as well as the trade policy makers. Figure 1- Apparel Cottons (Jan/1989 - Sept/2005) Figure 1.a-Apparel cottons quantity Figure 1.b-Apparel cottons price 400,000,000 8 350,000,000 7 300,000,000 6 250,000,000 5 200,000,000 4 150,000,000 3 100,000,000 50,000,000 2 0 1 90 92 94 96 98 00 02 04 90 92 94 96 98 00 02 04 APPAREL COTTONS QUANTITY-China APPAREL COTTONS PRICE-China APPAREL COTTONS QUANTITY-HK APPAREL COTTONS PRICE-HK The remainder of the paper is organized as follows: Section 2 provides methodology and empirical findings for building up the model. Section 3 concludes. 2. Seasonal ARIMA Modeling 2.1 Methodology In general, forecasting future price and trade volume is the estimation of the expected value of a dependent variable for observations that are not part of the same data set. This goal can be archived by running linear regression equations to forecast the dependent variable by plugging likely value of the independent variables into the estimated equations and calculating its predicted value. However, the ARIMA approach is appropriate when little or nothing is known about the dependent variable being forecasted, or lacking of theoretical underpinnings of particular regression equations. Among others, Leuthod et al. (1970) considered the forecasting of daily hog prices and quantities using the same technique we adopted here. In order to identify an appropriate model the first step is to determine the tentative form of the ARIMA model. The order of the ARIMA is to look at the correlation properties of the series that is under investigation. We observed from the original data that the time series is likely to have variance and level non-stationarty. Hence, we generate a logarithm transformation of the original series to make it variance stationary. Stationarity of a time series can be tested statistically by Augmented Fuller (ADF) unit root test pioneered by Dickey and Fuller (1979), the logarithm transformation of the original series was proved to be level non-stationary. Results are not shown here to save space, but available upon request. The autocorrelations is the correlation coefficient of the current value of the series with the series lagged a certain number of periods. The partial autocorrelations measure the correlation of the current and lagged series after taking account of the predictive power of all the values of the series with smaller lags. Our decision rule was as follows: 2 1) If the autocorrelation function dies out smoothly at a geometric rate, and the partial autocorrelations were zero after one lag, then a first-order autoregressive model would be suggested. 2) Alternatively, if the autocorrelations were zero after one lag and the partial autocorrelations declined geometrically, a first-order moving average process would be suggested. In short, the criterions to judge for the best model are as follows: Relatively small of BIC /AIC Relatively small of SEE Relatively high adjusted R2 Q- statistics and correlogram show that there is no significant pattern left in the ACFs and PACFs of the residuals, it means the residuals of the selected model are white noise. To protect against disastrous forecasting errors, the least we can do is to check that the fitted model is a satisfactory one. If we had a large amount of data, it would be feasible to break the data into two parts, identify and estimate the model on the first part and check the quality of the forecasts on the second part. This method, known as cross validation, gives one of the few ways of obtaining an honest estimate of forecasting error. Unfortunately, there is typically not enough data for cross- validation to be used, so that models must be identified, estimated, and diagnostically checked just as what we proposed in this paper. For diagnostic checking the most commonly used method is to examine the correlogram of the residuals from the fitted model to see if the residuals are white noise. 2.2 Empirical Findings Figure. 3 presents the estimated correlogram for imported prices and quantities difference in logarithm for China and HK. Since there is seasonal effect at lags 12 in the correlogram for all case, we therefore use seasonal autoregressive (SAR) and seasonal moving average (SMA) terms as Box and Jenkins (1976) recommend for monthly data with systematic seasonal movements to capture the holiday and seasonal effect. Figure 2- Correlograms (ACF & PACF) Figure 2.a-Apparel cottons quantity (China) DLNAC DLNAC 1.0 1.0 .5 .5 0.0 0.0 Partial ACF -.5 Confidence Limits -.5 Confidence Limits ACF -1.0 Coefficient -1.0 Coefficient 1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Lag Number Lag Number 3 Figure 2.b-Apparel cottons quantity (HK) DLNHAC DLNHAC 1.0 1.0 .5 .5 0.0 0.0 Partial ACF -.5 Confidence Limits -.5 Confidence Limits ACF -1.0 Coefficient -1.0 Coefficient 1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Lag Number Lag Number Figure 2.c-Apparel cottons prices (China) DLNACP DLNACP 1.0 1.0 .5 .5 0.0 0.0 Partial ACF -.5 Confidence Limits -.5 Confidence Limits -1.0 Coefficient ACF -1.0 Coefficient 1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Lag Number Lag Number Figure 2.d-Apparel cottons prices (HK) DLNHACP DLNHACP 1.0 1.0 .5 .5 0.0 0.0 Partial ACF Partial ACF -.5 Confidence Limits -.5 Confidence Limits -1.0 Coefficient -1.0 Coefficient 1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Lag Number Lag Number In our several trial and error procedures, the best model selected which best fitted with the selecting criteria are the following specifications. Apparel Cotton quantities from China: DLNAC = 0.04 +μt [0.0181] ** 2 (1+0.18L+0.05L -0.75L3+0.19L4)(1-0.93L12)μt=(1-0.11L2-0.86L3+0.22L4)(1-0.89L12)εt (1) [ 0.0743]*** [ 0.0683] [0.0747]*** [0.0910]** [ 0.0128]*** [ 0.0451]*** [0.0487]*** [ 0.0420] *** [ 0.0214] *** Adjusted R2= 0.6879 where DLNAC is the difference of the logarithm of the apparel cotton quantities imported from China ; L is the lag operator; R2 is adjust R square; μt is the white noise error term while εt is the moving average term. Standard errors are in parentheses. *,**, and *** denotes significant at the 10,5 and 1 per cent level, respectively. 4 Apparel Cotton quantities from HK: DLNHAC = -0.02 +μt [0.0144] ** 3 (1-0.18L-0.23L +0.26L4)(1-0.97L12)μt=(1-0.75L-0.17L3) (1-0.83L12)εt (2) [ 0.144]** [ 0.0952]** [0.0891]*** [0.0136]*** [ 0.1196]*** [ 0.1178] [0.0439]*** Adjusted R2= 0.7633 Where DLNHAC is the difference of the logarithm of the apparel cotton quantities imported from HK. Apparel Cotton Prices from China: DLNHACP = -0.001 +μt [0.0038] (1-0.57L2-0.49L3+0.57L4)(1-0.46L12)μt=(1-0.84L2-0.59L3+0.59L4)εt (3) [ 0.0494]*** [ 0.0604]*** [0.0542]*** [0.0703]*** [ 0.0328]*** 0.0348]*** [0.0263]*** Adjusted R2= 0.351 DLNACP is the difference of the logarithm of the apparel cotton prices from China Apparel Cotton Prices from HK: DLNACP = -0.006 +μt [0.0038] (1+0.85L+0.51L2-0.24L3)(1-0.92L12)μt=(1+0.47L-0.58L3)εt (1-0.80L12) (4) [ 0.0494]** [ 0.0604]*** [0.0542]*** [0.0703]*** [ 0.0328]*** [ 0.0348]*** [0.0263]*** Adjusted R2= 0.362 where DLNHACP is the difference of the logarithm of the apparel cotton prices from HK. The estimated ARMA structure was accessed using three diagnostic views namely roots, correlogram, and impulse response. These views are reported in Figure 3. The root view displays the inverse roots of the AR and MA characteristic polynomial which indicated that the estimated ARMA process is (covariance) stationary and invertible since all roots lie inside the unit circle. The correlogram view compares the autocorrelation pattern of the structural residuals and that of the estimated model for a specified number of periods. The view indicated that our model is a properly specified model since the residual and theoretical autocorrelations and partial autocorrelation is very close. Finally, the ARMA impulse response view traces the response of the ARMA part of the estimated equation to shocks in the innovation. An impulse response function traces the response to a one-time shock in the innovation. The accumulated response is the accumulated sum of the impulse responses. It can be interpreted as the response to step impulse where the same shock occurs in every period from the first. Our estimated ARMA model is stationary because the impulse responses asymptote to zero, while the accumulated responses asymptote to its long-run value, where these asymptotic values was shown as dotted horizontal lines in the graph view. 5 Figure 3- ARMA Structure Diagnostic View Figure 3.a-Apparel cottons quantity (China) Inverse Roots of AR/MA Polynomial(s) Impulse Response ± 2 S.E. .8 1.5 .2 Autocorrelation .4 .1 1.0 .0 .0 -.4 0.5 -.1 2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 16 18 20 22 24 Actual Theoretical AR roots 0.0 MA roots Partial autocorrelation .50 Accumulated Response ± 2 S.E. -0.5 .20 .25 .15 .00 -1.0 .10 -.25 .05 -.50 2 4 6 8 10 12 14 16 18 20 22 24 -1.5 .00 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 12 14 16 18 20 22 24 Actual Theoretical Figure 3.b-Apparel cottons quantity (HK) Inverse Roots of AR/MA Polynomial(s) Impulse Response ± 2 S.E. 1.5 .2 .8 Autocorrelation .1 .4 1.0 .0 .0 0.5 -.1 -.4 2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 16 18 20 22 24 Actual Theoretical AR roots 0.0 MA roots Accumulated Response ± 2 S.E. Partial autocorrelation -0.5 .50 .15 .25 .10 .00 -1.0 .05 -.25 .00 -.50 -1.5 2 4 6 8 10 12 14 16 18 20 22 24 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -.05 2 4 6 8 10 12 14 16 18 20 22 24 Actual Theoretical Figure 3.c-Apparel cottons prices (China) Inverse Roots of AR/MA Polynomial(s) .6 1.5 Autocorrelation Impulse Response ± 2 S.E. .4 .10 .2 1.0 .05 .0 -.2 .00 0.5 2 4 6 8 10 12 14 16 18 20 22 24 -.05 Actual Theoretical 2 4 6 8 10 12 14 16 18 20 22 24 AR roots 0.0 MA roots Partial autocorrelation .4 -0.5 Accumulated Response ± 2 S.E. .2 .12 .0 .08 -1.0 -.2 .04 -.4 .00 -1.5 2 4 6 8 10 12 14 16 18 20 22 24 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Actual Theoretical -.04 2 4 6 8 10 12 14 16 18 20 22 24 Figure 3.d-Apparel cottons prices (HK) Inverse Roots of AR/MA Polynomial(s) .4 1.5 Autocorrelation .2 Impulse Response ± 2 S.E. .08 .0 1.0 .04 -.2 -.4 .00 0.5 2 4 6 8 10 12 14 16 18 20 22 24 Actual Theoretical -.04 2 4 6 8 10 12 14 16 18 20 22 24 AR roots 0.0 MA roots Partial autocorrelation .2 -0.5 Accumulated Response ± 2 S.E. .0 .06 -1.0 -.2 .04 .02 -.4 -1.5 2 4 6 8 10 12 14 16 18 20 22 24 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Actual Theoretical .00 2 4 6 8 10 12 14 16 18 20 22 24 6 Finally, 1-step ahead static forecasting was constructed for the above model. Figure 4 reports the forecast graph and table of statistical results evaluating the forecast performance. The relative low “variance proportion” indicates the forecasts are tracking the variation in the actual series quite well. Figure 4- Forecast Graph and Table Figure 4.a-Apparel cottons quantity (China) 20 19 21 Forecast: LNACF Actual: LNAC 20 Forecast sample: 1989M01 2005M09 18 Adjusted sample: 1990M06 2005M09 19 Included observations: 184 Root Mean Squared Error 0.163872 17 18 Mean Absolute Error 0.124747 Mean Abs. Percent Error 0.721466 17 Theil Inequality Coefficient 0.004732 16 Bias Proportion 0.001474 Variance Proportion 0.018556 16 Covariance Proportion 0.979970 15 15 90 92 94 96 98 00 02 04 1990 1992 1994 1996 1998 2000 2002 2004 LNACF ± 2 S.E. LNAC LNACF Figure 4.b-Apparel cottons quantity (HK) 18.0 17.8 18.2 17.6 Forecast: LNHACF 18.0 Actual: LNHAC 17.4 Forecast sample: 1989M01 2005M09 17.8 Adjusted sample: 1990M06 2005M09 Included observations: 184 17.2 17.6 Root Mean Squared Error 0.111446 17.0 17.4 Mean Absolute Error 0.083556 Mean Abs. Percent Error 0.481162 17.2 Theil Inequality Coefficient 0.003202 16.8 Bias Proportion 0.010430 17.0 Variance Proportion 0.077364 16.6 Covariance Proportion 0.912207 16.8 16.4 16.6 90 92 94 96 98 00 02 04 1990 1992 1994 1996 1998 2000 2002 2004 LNHACF ± 2 S.E. LNHAC LNHACF Figure 4.c-Apparel cottons prices (China) 2.0 Forecast: LNACPF 1.8 1.8 Actual: LNACP Forecast sample: 1989M01 2005M09 1.6 1.6 Adjusted sample: 1990M06 2005M09 Included observations: 184 1.4 1.4 Root Mean Squared Error 0.085069 1.2 Mean Absolute Error 0.066491 Mean Abs. Percent Error 5.667052 1.2 1.0 Theil Inequality Coefficient 0.033554 Bias Proportion 0.000060 1.0 0.8 Variance Proportion 0.014766 Covariance Proportion 0.985174 0.6 0.8 0.4 0.6 1990 1992 1994 1996 1998 2000 2002 2004 90 92 94 96 98 00 02 04 LNACPF ± 2 S.E. LNACP LNACPF 7 Figure 4.d-Apparel cottons prices (HK) 2.0 2.0 1.9 Forecast: LNHACPF 1.9 Actual: LNHACP Forecast sample: 1989M01 2005M09 1.8 Adjusted sample: 1990M05 2005M09 1.8 Included observations: 185 1.7 1.7 Root Mean Squared Error 0.044055 Mean Absolute Error 0.034433 1.6 Mean Abs. Percent Error 2.145840 1.6 Theil Inequality Coefficient 0.013711 1.5 Bias Proportion 0.001675 Variance Proportion 0.068590 1.5 Covariance Proportion 0.929734 1.4 1.4 1.3 1990 1992 1994 1996 1998 2000 2002 2004 90 92 94 96 98 00 02 04 LNHACPF ± 2 S.E. LNHACP LNHACPF 6. Conclusion In this paper, we estimate a seasonal ARMA model and construct a forecasting model based on the former for MFA apparel cottons prices and quantities imported from HK & China to the U.S. during 1989 and 2005. Using the modeling strategy advocated by Box and Jenkins (1976) we identify and estimate the seasonal ARIMA model using OLS technique. Diagnostic checking which shows that the estimated models are satisfactory assessed while the forecasting model indicates relatively “variance proportion”. In addition, the forecasts are able to track the variation in the actual price series. Further research may undergo by using state space technique instead of OLS so that unobserved components can be recovered within a dynamic setup. References Box, G. E. P., and Jenkins, G. (1976), Time Series Analysis: Forecasting and Control, Holden-Day. Bowerman and O’Connell. (1993): Forecasting and Time Series: An Applied Approach, Third Edition, The Duxbury Advanced Series in Statistics and Decision Sciences, Duxbury Press, Belmont, CA. Lau, C.K., To, K.M., Zhang, C.M. (forthcoming) ‘MFA Fibers & Cotton Imported to the U.S. from China and Hong Kong - A Structural Change Analysis’, Journal of the Textiles Institute Leuthod, R.M., MacCormick, M.A., Schmitz, A., Watts, D.G. (1970) ‘Forcasting Daily Hog Prices and Quantities: A Study of Alternative Forecasting Techniques’, Journal of the American Statistical Association, March: 90-107 OTEXA (Jan, 2006), Preliminary Data and Official Data, U.S. General Imports, Office of Textiles and Apparel, U.S. Department of Commerce, International Trade Administration. Available : http://otexa.itsa.doc.gov. Date : 05 Jan 2006. Said, S. E., and Dickey, D. A. (1984) ‘Testing for unit roots in autoregressive-moving average models of unknown order’, Biometrika, 71, 599-607. 8

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