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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
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