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					Progress Energy Florida: Customer Forecast
                 Ryan Speaks

                  ECO 6433

        Business Cycles and Forecasting

           Thursday, June 23rd, 2005
Introduction

         The topic I chose for my paper is the time series of the number of customers my

company has served from 1982 to 2005. This topic is interesting to me for several reasons: 1) it

is similar to population forecasting in that it deals with a time series with an inherent trend 2)

forecasting my company’s customer base is important for several business reasons and 3)

working with actual data present within my company adds a sense of legitimacy to my work and

the project. This data is interesting to study because it is a time series related to the movement

of people, meaning the movement of people in and out of the state and in and out of my

company’s service territory. I find it interesting to examine the ups and downs in the trend and

compare them to Florida’s general population trend. Customer forecasting is important in that it

helps my company to budget appropriately for the coming periods for several types of work

including new customer installation expenses as well as outage restoration and streetlight

construction.

         All of the relevant literature I found on population forecasting talks about the cohort-

method. For example, this exert from the U.S. census bureau: “the projections use the cohort-

component method. The cohort-component method requires separate assumptions for each

component of population change: births, deaths, internal migration, and international migration.”

(1) This is a much more in-dept, lower-level method of forecasting based on demographics.

For the puposes of my paper it is too grandular and complicated. I am more concerned with

learning the process I have learned during the course than delving into demographics.

         Quoting Stanley Smith from the University of Florida, he speaks further about the efforts

of statisticians to better model population trends using advanced cohort-components, ARIMA

models and causal models: “Although no standard definition distinguishing simple from complex

mathematical structure has been developed, there seems to be some consensus in practice; for

example, linear and exponential extrapolations are generally classified as simple, whereas
2 of 2
cohort-component and ARIMA time series models are generally classified as complex. Causal

models are those in which demographic variables are affected by economic and/or other

variables and noncausal models are those in which demographic variables are affected solely

by their own historical values. This two-by-two matrix yields four categories of projections:

simple causal, simple noncausal, complex causal, and complex noncausal. Most of the authors

in the special issue followed this typology.” (2)

         He then addresses the method in which he evaluates different models, “Furthermore, as

pointed out by Rogers (1995), the simple vs. complex classification is really a continuum rather

than a dichotomy. The issues can best be understood in relative rather than absolute terms, or

“simpler vs. more complex” rather than “simple vs. complex”. That is the approach I take in the

present discussion. Following the criteria mentioned above, models are classified as relatively

simple or complex according to their mathematical structures, number of variables, and level of

disaggregation. They are classified as causal or noncausal according to whether they are

affected by other variables or only by their own 7 historical values. Under this approach, a given

model could be classified as relatively complex when compared to one model and as relatively

simple when compared to another.” (2)

         I can conclude by saying that my paper is not even making a minute scratch on the

surface of population forecasting. However, reviewing the literature I found gave me a better

understanding of some of the advanced methods used to accurately perform population

forecasts.

         The six considerations basic to successful forecasting forms a framework for my forecast.

A forecast is made to add value and guide a decision. The forecast in my paper of number of

customers has many business management impacts, budgets could be set to high, to low, or

even not changed when appropriate. Take note of the loss function in Graph XVI based upon

the assumption that over-budgeting is not a bad as under-budgeting because funds can be

3 of 3
redistributed more easily than they can be requisitioned. This would graph as an asymmetric

loss in which the under-budgeting side has a greater loss than the over-budgeting side. The

forecast object in my paper is a time series of changes in customer count over time. It may

have been possible to use additional time series to aid in my forecast such as Florida state

population or individual county population. However, this was not undertaken due to limited

time and to keep my project’s scope and size within the guidelines provided. My forecast

statement is a range meaning a range of numbers into which the future value can be expected

to fall a certain percentage of the time. In this case we use two standard deviations from the

forecast’s mean to establish a range. My chosen forecast horizon (with my actual data ending

in 2005:05) is to the end of 2006 because we budget annually and I used a 1-step-ahead

forecast beginning one observation after my actual data (IE: 2005:06) in that I need to forecast

to the end of 2005 in order to forecast all of 2006. My information set has substantial history,

dating back to 1982:01 but it is somewhat unreliable in that the methods and systems used to

collect the data have change over the years creating a unstable time series. Following the

Parsimony Principle, I kept my forecast efforts as simple as possible in order to increase the

accuracy of my forecast, more easily identify anomalies, make the forecast more intuitive and

lessen the “data mining” scope in that I want my forecast to accurately forecast future

observations, not just past historical observations. (3)

Plan

         I plan to work through the applied material I have learned in the class step-by-step in

order to see the affect on my time series. Assuming my time series is stationary; I am proposing

to use a deterministic regression process to accurately forecast it. I will approach the project

with the basics moving up to the more advanced analysis techniques.

         Beginning with unit root tests on my data to determine stationarity, I will take the LOG of

my time series to reduce its scale to make my forecasts make more sense. I will then move on

4 of 4
to simpler linear, quadratic and exponential regressions and then examine the results. Next, I

will employ seasonal dummy variables in my regression equations and examine the results.

Next, I will explore ARMA models to see what effect auto-regressive and moving average terms

have on my forecast accuracy. Next, I will perform in-sample and out-of-sample forecasts using

my best model. Finally, I will examine Chow Breakpoint and CUSUM tests to determine if my

time series is stable or not.

Analysis

         First, I ran a 12 lag unit root test including both a trend and intercept and examined the

Augmented Dickey-Fuller statistic in relation to the 5.0% confidence interval. As you can see in

Table VII the unit root test calculated a -1.99 Augmented Dickey-Fuller statistic and a -5.0%

confidence interval value of -3.43 which tells me that my time series is stationary and that I may

proceed to apply deterministic regression models on it. To take it a couple steps further I ran 24

lag and 36 lag unit root tests which had ADFs of -2.47 and -2.43 respectively on a

corresponding -5.0% confidence interval value of -3.43.

         Now that I was confident that my time series was stationary, I ran a simple linear

regression including a time trend variable (called “trend”) and a constant and got the results in

Table I. The Akaike Information Criterion (AIC) was -4.08 and the Schwarz Information Criterion

was -4.05. The graph of the actual data, fitted regression line and regression residuals is

located just below Table I. The graph leads me to believe there is a lot more information hidden

within the regression’s residuals and the plot appears to have some seasonality.

         Next I ran a quadratic regression including my trend variable, the square of my trend

variable and a constant, the results of which are in Table II. The AIC and SIC statistics

decreased to -5.08 and -5.04 respectively. The graph that follows the table did not look much

better than the graph of the linear regression graph.



5 of 5
          Next, I created twelve monthly dummy variables to include in my regressions. Including

the seasonal dummy variables in my linear and quadratic regressions increased my AIC and

SIC statistics to -4.07 and -3.90 respectively for the linear and decreased my AIC to -5.26 and

my SIC stayed flat at -5.08 for the quadratic as detailed in Tables III and IV. This puzzled me

quite a bit because when I looked at the data in a simple X-Y scatter plot it appears that there

are predominant seasonal trends in the data, see Graphs III and IV. However, the seasonal

dummy variables did not have as great of an effect on my quadratic regression model’s AIC and

SIC statistics as I would have hoped. In addition, the dummy variables coefficients are very

similar which supports the notion that there is little seasonality in the time series, at least not

monthly seasonality.

          Finally, I worked through the process of running all combinations of my (best) quadratic

model with seasonal dummies and ARMA(p,q) terms (Table V) from ARMA(O,O) to ARMA (4,4)

and found that the quadratic ARMA(2,4) model (Table VI) decreased my AIC and SIC the

greatest to -5.93 and -5.67 respectively. This seems to be the best model to forecast my time

series.

          The inverted AR roots of this regression indicate that the series in invertible because they

are less than one. Using this “best model” I forecasted both an in-sample forecast (Graph IX

and Graph X) and an out-of-sample (Graph VII and Graph VIII) forecast.

          However, after removing the ARMA terms from my model, rerunning it and performing a

CUSUM test on the regression, the resulting graph (Graph XV) indicates that the series is

unstable. Furthermore, after rerunning the regression with the ARMA(2,4) terms included, the

Chow Breakpoint Test on two segments and three segments (broken at 1995:03 and 2002:06)

results in statistics that (if I understand the F-statistic correctly) concur with the CUSUM test by

indicating that the series is indeed unstable. This supports my contention that the time series I

have is not consistent or stable, I address this issue in the following conclusions.

6 of 6
Conclusions

         I learned that an eye ball metric is useless when visually examining graphed data. What

may appear to be a strong seasonal pattern may be very weak or not be present at all. Only a

thorough statistical modeling process will tell you for sure what your data is doing and how to

forecast it. This paper informs the reader about some possible steps a beginning forecaster can

take in order to forecast population/customer data including important auto-regressive and

moving-average components. Readers will take away the message that a step-by-step

structured process for modeling a time series is effective and produces good results...improving

as you go along. They will also learn that a lot of information is hidden with in a time series and

that in running each model some of that information is unlocked and the forecaster can use it to

improve his modeling approach.

         Considering what should be done to better model this time series, when looking at the

simple X-Y scatter plot of the data, it appears that there are three distinct sections, or patterns,

in the data, one from 1982 to approximately the beginning of 1995 (Graph IV and Graph XI), one

from 1995 to 2001 (Graph V and Graph XII) and one from 2002 to 2005 (Graph VI and XIII).

The Chow Breakpoint and CUSUM tests indicate there is instability. Puzzled by this, I contacted

the coworker who supplied me the data and her explanation was that we counted customers in

difference ways (customers billed vs. customers served) over the years using several different

computer systems and methods.

         The disrupted appearance from 1995 to 2002 is caused by a change in meter reading

cycles for large numbers of customers which caused some customer’s electric meters to not be

read one month only to be read twice the next! This caused my company’s customer counts to

jump up and down as the months progressed. Their fix back then was to average year to date

customers. However, this caused a spike every January since the early months of each year

(lower customer counts) kept the average from growing appropriately towards the end of each

7 of 7
year. Then, when the moving average was begun again each January, the number of

customers jumped up to the actual previous year final number. Not the best practice, but it

appears this is what they did. I did not want to alter or massage the data artificially to smooth

out these numbers so I left them as is. In practice a company needs to develop a better way for

recording accurately the number of customers it serves.

         Business implications for forecasting customer numbers reside in the ability to accurately

plan the scope of several different types of work functions as well as material sourcing, fleet

purchases, etc. This data is of utmost importance when it comes around to budgeting season

for finance and accounting departments in that they will be able to better explain their financial

needs and restrictions.




8 of 8
Appendix

I sourced my data from my company through a coworker in our reliability department. It was
collated from several different systems. Here is an Excel file containing the raw data as
provided to me:




Graph I                                                                                                                                 Graph II
                                                             Customers                                                                                                                                            Inc/(Dec)


 1,800,000                                                                                                                               200,000

 1,600,000
                                                                                                                                         150,000
 1,400,000

 1,200,000                                                                                                                               100,000

 1,000,000                                                                                                                                50,000
                                                                                                                            Customers
  800,000                                                                                                                                                                                                                                                                                Inc/(Dec)
                                                                                                                                              -




                                                                                                                                                                                    Nov




                                                                                                                                                                                                                            Nov




                                                                                                                                                                                                                                                                Nov
                                                                                                                                                      Jan




                                                                                                                                                                                          Jan




                                                                                                                                                                                                                                  Jan




                                                                                                                                                                                                                                                                       Jan
                                                                                                                                                                        Jul




                                                                                                                                                                                                              Jul




                                                                                                                                                                                                                                                    Jul
  600,000


                                                                                                                                                            Mar




                                                                                                                                                                                                 Mar




                                                                                                                                                                                                                                        Mar




                                                                                                                                                                                                                                                                             Mar
                                                                                                                                                                              Sep




                                                                                                                                                                                                                     Sep




                                                                                                                                                                                                                                                          Sep
                                                                                                                                                                  May




                                                                                                                                                                                                       May




                                                                                                                                                                                                                                              May




                                                                                                                                                                                                                                                                                   May
  400,000                                                                                                                                 (50,000)

  200,000
                                                                                                                                         (100,000)
       -
             Jan




                                    Jan




                                                           Jan




                                                                                  Jan




                                                                                                          Jan
                         Jul




                                                Jul




                                                                       Jul




                                                                                              Jul




                                                                                                                      Jul
                   Apr




                                          Apr




                                                                 Apr




                                                                                        Apr




                                                                                                                Apr
                               Oc




                                                      Oc




                                                                             Oc




                                                                                                    Oc




                                                                                                                                         (150,000)



Graph III                                                                                                                               Graph IV
                                                       Observations 1-281                                                                                                                               Observations 1-159


                                                                                                                                         1,600,000
 1,600,000
                                                                                                                                         1,500,000
 1,500,000
                                                                                                                                         1,400,000
 1,400,000
                                                                                                                                         1,300,000
 1,300,000
                                                                                                                                         1,200,000                                                                                                                                       Customers

 1,200,000                                                                                                                  Customers    1,100,000

 1,100,000                                                                                                                               1,000,000

                                                                                                                                          900,000
 1,000,000
                                                                                                                                          800,000
  900,000                                                                                                                                            0            20           40           60               80            100          120         140          160           180

  800,000
             0                 50               100              150              200               250               300



Graph V                                                                                                                                 Graph VI
                                                      Observations 160-242                                                                                                                             Observations 242-281


                                                                                                                                         1,600,000
 1,600,000
                                                                                                                                         1,500,000
 1,500,000
                                                                                                                                         1,400,000

 1,400,000                                                                                                                               1,300,000

                                                                                                                                         1,200,000                                                                                                                                       Customers
 1,300,000
                                                                                                                                         1,100,000
 1,200,000                                                                                                                  Customers
                                                                                                                                         1,000,000

 1,100,000                                                                                                                                900,000

                                                                                                                                          800,000
 1,000,000
                                                                                                                                                     240          245         250          255           260               265          270         275          280           285

  900,000

  800,000
             0           10         20          30         40          50         60          70          80          90




9 of 9
Table I

Linear Regression

  Dependent Variable: LDATA
  Method: Least Squares
  Date: 06/21/05 Time: 21:19
  Sample: 1982:01 2005:05
  Included observations: 281
                            Variable                                        Coefficient   Std. Error    t-Statistic       Prob.
                              C                                              13.67847     0.003739      3658.601        0.0000
                            TREND                                            0.002193     2.31E-05      94.91680        0.0000
  R-squared                                                                  0.969962     Mean dependent var           13.98552
  Adjusted R-squared                                                         0.969854     S.D. dependent var           0.180963
  S.E. of regression                                                         0.031420     Akaike info criterion       -4.075663
  Sum squared resid                                                          0.275430     Schwarz criterion           -4.049767
  Log likelihood                                                             574.6306     F-statistic                  9009.198
  Durbin-Watson stat                                                         0.343288     Prob(F-statistic)            0.000000

                                                                     14.4

                                                                     14.2

                                                                     14.0

0.10
                                                                     13.8

0.05
                                                                     13.6

0.00

-0.05

-0.10
        82   84   86   88   90    92   94   96   98   00   02   04

                       Residual         Actual         Fitted




Table II

Quadratic Regression

  Dependent Variable: LDATA
  Method: Least Squares
  Date: 06/21/05 Time: 21:20
  Sample: 1982:01 2005:05
  Included observations: 281
                            Variable                                        Coefficient   Std. Error    t-Statistic       Prob.
                            C                                                13.62325     0.003376      4035.314        0.0000
                          TREND                                              0.003381     5.57E-05      60.69449        0.0000
                         TREND^2                                            -4.24E-06     1.93E-07     -22.02468        0.0000
  R-squared                                                                  0.989057     Mean dependent var           13.98552
  Adjusted R-squared                                                         0.988978     S.D. dependent var           0.180963
  S.E. of regression                                                         0.018998     Akaike info criterion       -5.078296
  Sum squared resid                                                          0.100342     Schwarz criterion           -5.039452
  Log likelihood                                                             716.5005     F-statistic                  12562.97
  Durbin-Watson stat                                                         0.941688     Prob(F-statistic)            0.000000




10 of 10
                                                                     14.4

                                                                     14.2

                                                                     14.0
0.10
                                                                     13.8
0.05
                                                                     13.6
0.00

-0.05

-0.10
        82   84   86   88   90    92   94   96   98   00   02   04

                       Residual         Actual         Fitted




Table III

Linear Regression w/ dummies

 Dependent Variable: LDATA
 Method: Least Squares
 Date: 06/21/05 Time: 21:21
 Sample: 1982:01 2005:05
 Included observations: 281
                   Variable                                 Coefficient     Std. Error    t-Statistic       Prob.
                       TREND                                    0.002193    2.27E-05      96.46268        0.0000
                         M1                                     13.68616    0.007044      1942.838        0.0000
                         M2                                     13.69165    0.007055      1940.821        0.0000
                         M3                                     13.68635    0.007065      1937.266        0.0000
                         M4                                     13.68489    0.007075      1934.248        0.0000
                         M5                                     13.67246    0.007085      1929.675        0.0000
                         M6                                     13.66974    0.007156      1910.191        0.0000
                         M7                                     13.66467    0.007166      1906.839        0.0000
                         M8                                     13.67117    0.007176      1905.092        0.0000
                         M9                                     13.67320    0.007186      1902.715        0.0000
                        M10                                     13.67213    0.007196      1899.897        0.0000
                        M11                                     13.68485    0.007206      1898.986        0.0000
                        M12                                     13.68418    0.007217      1896.206        0.0000
 R-squared                                                      0.972092    Mean dependent var           13.98552
 Adjusted R-squared                                             0.970843    S.D. dependent var           0.180963
 S.E. of regression                                             0.030900    Akaike info criterion       -4.070941
 Sum squared resid                                              0.255894    Schwarz criterion           -3.902619
 Log likelihood                                                 584.9672    Durbin-Watson stat           0.325744

                                                                     14.4

                                                                     14.2

                                                                     14.0
0.15

0.10                                                                 13.8

0.05
                                                                     13.6
0.00

-0.05

-0.10
        82   84   86   88   90    92   94   96   98   00   02   04

                       Residual         Actual         Fitted




11 of 11
Table IV

 Dependent Variable: LDATA
 Method: Least Squares
 Date: 06/21/05 Time: 21:22
 Sample: 1982:01 2005:05
 Included observations: 281
                   Variable                                 Coefficient     Std. Error    t-Statistic       Prob.
                   TREND                                         0.003394   4.99E-05      68.04866        0.0000
                  TREND^2                                       -4.29E-06   1.72E-07     -24.88223        0.0000
                     M1                                          13.63168   0.004450      3063.372        0.0000
                     M2                                          13.63716   0.004455      3061.084        0.0000
                     M3                                          13.63186   0.004460      3056.480        0.0000
                     M4                                          13.63040   0.004465      3052.844        0.0000
                     M5                                          13.61798   0.004470      3046.858        0.0000
                     M6                                          13.61287   0.004551      2991.071        0.0000
                     M7                                          13.60778   0.004556      2986.572        0.0000
                     M8                                          13.61426   0.004561      2984.715        0.0000
                     M9                                          13.61629   0.004566      2981.981        0.0000
                    M10                                          13.61522   0.004571      2978.669        0.0000
                    M11                                          13.62796   0.004575      2978.473        0.0000
                    M12                                          13.62730   0.004580      2975.447        0.0000
 R-squared                                                      0.991591    Mean dependent var           13.98552
 Adjusted R-squared                                             0.991182    S.D. dependent var           0.180963
 S.E. of regression                                             0.016993    Akaike info criterion       -5.263434
 Sum squared resid                                              0.077104    Schwarz criterion           -5.082163
 Log likelihood                                                 753.5124    Durbin-Watson stat           1.078483

                                                                     14.4

                                                                     14.2

                                                                     14.0
0.10
                                                                     13.8
0.05
                                                                     13.6
0.00

-0.05

-0.10
        82   84   86   88   90    92   94   96   98   00   02   04

                       Residual         Actual         Fitted




12 of 12
Table V

ARMA(p,q) models Akaike and Schwarz Information Criterions.

           Akaike:                     Schwarz:

   R1        -5.263434           R1        -5.082163
   R2        -5.487894           R2        -5.293173
   R3        -5.734990           R3        -5.526749
   R4        -5.800736           R4        -5.578903
   R5        -5.796435           R5        -5.560940
   R6        -5.377179           R6        -5.182961
   R7        -5.539962           R7        -5.332796
   R8        -5.630006           R8        -5.409892
   R9        -5.705839           R9        -5.472777
   R10       -5.843579           R10       -5.635877
   R11       -5.844743           R11       -5.624060
   R12       -5.874755           R12       -5.641090
   R13       -5.882439           R13       -5.635793
   R14       -5.846978           R14       -5.625721
   R15       -5.845989           R15       -5.611718
   R16       -5.861966           R16       -5.614680
   R17       -5.933141           R17       -5.672839
   R18       -5.873389           R18       -5.638507
   R19       -5.892826           R19       -5.644895
   R20       -5.883167           R20       -5.622187
   R21       -5.885547           R21       -5.611518
   R22       -5.889352           R22       -5.640773
   R23       -5.881058           R23       -5.619397
   R24       -5.876801           R24       -5.602056
   R25       -5.918988           R25       -5.631160

Table VI

Dependent Variable: LDATA
Method: Least Squares
Date: 06/21/05 Time: 21:26
Sample(adjusted): 1982:03 2005:05
Included observations: 279 after adjusting endpoints
Convergence achieved after 45 iterations
Backcast: OFF (Roots of MA process too large for backcast)
         Variable        Coefficient    Std. Error     t-Statistic    Prob.
       TREND              0.002468      0.001465      1.684809       0.0932
      TREND^2            -1.93E-06      2.94E-06     -0.655277       0.5129
         M1               13.72500      0.199445      68.81608       0.0000
         M2               13.73077      0.199418      68.85418       0.0000
         M3               13.72470      0.199397      68.83094       0.0000
         M4               13.72351      0.199366      68.83572       0.0000
         M5               13.71061      0.199361      68.77289       0.0000
         M6               13.70657      0.199412      68.73486       0.0000
         M7               13.70107      0.199416      68.70606       0.0000
         M8               13.70779      0.199409      68.74217       0.0000
         M9               13.70927      0.199425      68.74395       0.0000
         M10              13.70849      0.199402      68.74808       0.0000
         M11              13.72083      0.199431      68.79976       0.0000
         M12              13.72031      0.199396      68.80934       0.0000
        AR(1)             0.076934      0.035240      2.183116       0.0299

13 of 13
                        AR(2)                                           0.896315                           0.033244                          26.96183                0.0000
                        MA(1)                                          -0.004728                           0.070114                         -0.067429                0.9463
                        MA(2)                                          -0.650476                           0.066144                         -9.834243                0.0000
                        MA(3)                                           0.200680                           0.065796                          3.050009                0.0025
                        MA(4)                                          -0.238648                           0.068600                         -3.478861                0.0006
  R-squared                                                            0.995794                        Mean dependent var                                     13.98806
  Adjusted R-squared                                                   0.995485                        S.D. dependent var                                     0.179097
  S.E. of regression                                                   0.012034                        Akaike info criterion                                 -5.933141
  Sum squared resid                                                    0.037509                        Schwarz criterion                                     -5.672839
  Log likelihood                                                       847.6732                        Durbin-Watson stat                                     1.913160
  Inverted AR Roots                                                   .99       -.91
  Inverted MA Roots                                                   .86     .09 -.51i     .09+.51i                                                                 -1.03
                                                                  Estimated MA process is noninvertible

                                                                              14.4

                                                                              14.2

                                                                              14.0
  0.10
                                                                              13.8
  0.05
                                                                              13.6
  0.00

 -0.05

 -0.10
              84   86   88    90    92    94    96    98    00    02    04

                        Residual            Actual           Fitted




 Graph VII                                                                                                                                                Graph VIII
 Out of sample level forecast 2005:06 to 2006:12                                                                                                          Out of sample LOG forecast 2005:06 to 2006:12
1800000                                                                                                                                                    14.4

                                                                                                                                                                                                                     14.4
1600000
                                                                                                                                                           14.2

1400000                                                                                                                                                                                                              14.2


                                                                                                                                                           14.0
1200000                                                                                                                                                                                                              14.0



1000000                                                                                                                                                    13.8
                                                                                                                                                                                                                     13.8



 800000                                                                                                                                                                                                              13.6
                                                                                                                                                           13.6
          82 84 86 88 90 92 94 96 98 00 02 04 06                                                                                                                                                                            82   84   86   88   90    92   94   96   98   00   02   04   06
                                                                                                                                                                  82 84 86 88 90 92 94 96 98 00 02 04 06
                                     DATA             FCST                                                                                                                                                                                           LDATA           UPPER
                                                                                                                                                                                   YHAT         LDATA                                                FCST            LOWER




 Graph IX                                                                                                                                                 Graph X
 In sample LOG forecast 2004:01 2005:05                                                                                                                   In sample level forecast 2004:01 2005:05
  14.4                                                                                                                                                   1600000


                                                                                     14.4

  14.2                                                                                                                                                   1400000

                                                                                     14.2


  14.0                                                                                                                                                   1200000
                                                                                     14.0



  13.8                                                                                                                                                   1000000
                                                                                     13.8



                                                                                     13.6                                                                 800000
  13.6
                                                                                            82   84   86   88   90   92   94   96    98   00   02   04             82 84   86 88   90 92   94   96   98 00   02 04
         82   84   86    88    90    92    94    96    98    00    02    04
                                                                                                                 LDATA              UPPER                                           DATA        FCST2
                                    LDATA            FCST                                                        FCST               LOWER




 14 of 14
 Graph XI                                                                                             Graph XI
 Line graph of level data                                                                             Line graph of LOG data
1600000                                                                                               14.4



1400000                                                                                               14.2



1200000                                                                                               14.0



1000000                                                                                               13.8


 800000
                                                                                                      13.6
          82 84   86 88   90 92   94   96   98 00   02 04
                                                                                                             82    84   86    88   90   92   94   96   98   00   02   04
                              CUSTOMER
                                                                                                                                             LDATA




 Graph XI                                                            Graph XII                                                                         Graph XIII
 Line graph of level data 1982:01                                    Line graph of level data                                                          Line graph of level data
 1995:03                                                             1995:04 2002:06                                                                   2002:07 2005:05
1300000                                                             1500000                                                                          1580000

                                                                    1450000                                                                          1560000
1200000

                                                                    1400000                                                                          1540000
1100000
                                                                    1350000                                                                          1520000
1000000
                                                                    1300000                                                                          1500000

 900000
                                                                    1250000                                                                          1480000

 800000                                                             1200000                                                                          1460000
          82 83 84 85 86 87 88 89 90 91 92 93 94 95                           96   97     98     99   00          01     02                                 02:07     03:01   03:07    04:01   04:07   05:01

                              CUSTOMER                                                         CUSTOMER                                                                               CUSTOMER




 Table VII

  ADF Test Statistic                            -1.999590         1% Critical Value*                              -3.9956
                                                                  5% Critical Value                               -3.4279
                                                                  10% Critical Value                              -3.1370
  *MacKinnon critical values for rejection of hypothesis of a unit root.


  Augmented Dickey-Fuller Test Equation
  Dependent Variable: D(DATA)
  Method: Least Squares
  Date: 06/22/05 Time: 21:27
  Sample(adjusted): 1983:02 2005:05
  Included observations: 268 after adjusting endpoints
                  Variable                          Coefficient   Std. Error            t-Statistic                     Prob.
              DATA(-1)                              -0.120505     0.060265         -1.999590                       0.0466
            D(DATA(-1))                             -0.875652     0.081445         -10.75152                       0.0000
            D(DATA(-2))                             -0.686735     0.097155         -7.068423                       0.0000
            D(DATA(-3))                             -0.504503     0.102081         -4.942203                       0.0000
            D(DATA(-4))                             -0.516190     0.100854         -5.118196                       0.0000
            D(DATA(-5))                             -0.562015     0.091494         -6.142662                       0.0000
            D(DATA(-6))                             -0.660683     0.087772         -7.527231                       0.0000
            D(DATA(-7))                             -0.651258     0.086161         -7.558576                       0.0000
            D(DATA(-8))                             -0.816810     0.087103         -9.377537                       0.0000
            D(DATA(-9))                             -0.544031     0.094327         -5.767507                       0.0000
            D(DATA(-10))                            -0.456722     0.093714         -4.873596                       0.0000
            D(DATA(-11))                            -0.217926     0.085617         -2.545364                       0.0115
            D(DATA(-12))                            -0.062004     0.062029         -0.999604                       0.3185
                 C                                   125586.5     50340.48          2.494742                       0.0132

 15 of 15
      @TREND(1982:01)                                           283.1860               154.2208                  1.836238             0.0675
R-squared                                                        0.616708              Mean dependent var                         2615.313
Adjusted R-squared                                               0.595498              S.D. dependent var                         24786.38
S.E. of regression                                               15764.26              Akaike info criterion                      22.22322
Sum squared resid                                                6.29E+10              Schwarz criterion                          22.42421
Log likelihood                                                  -2962.912              F-statistic                                29.07648
Durbin-Watson stat                                               2.012239              Prob(F-statistic)                          0.000000

***Unit Root test of 24 lags ADF was -2.47 and 35 lags was -2.43.

Graph XV                                                             Graph XVI
100
                                                                                                      Loss Function
 80                                                                      10
 60                                                                       9
 40
                                                                             8
                                                                             7
 20
                                                                             6
                                                                      Loss




  0                                                                          5
-20                                                                          4
                                                                             3
-40
                                                                             2
-60
      84   86   88   90   92   94    96   98   00     02   04
                                                                             1
                                                                             0
                                                                                       -8


                                                                                            -6


                                                                                                 -4


                                                                                                         -2


                                                                                                                0


                                                                                                                      2


                                                                                                                          4


                                                                                                                              6


                                                                                                                                  8


                                                                                                                                       10
                                                                                 -10




                     CUSUM          5% Significance
                                                                                                              Error




***ls ldata trend trend^2 m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 (no AR or MA terms)

Sited Sources:

(1) http://www.census.gov/population/www/projections/ppl47.html

(2) http://www.bebr.ufl.edu/Articles/IJF_1997.pdf

(3) Francis X. Diebold (2004) Elements of Forecasting, Third Edition, Thomson South-Western.

Additional Sources:

http://www.geosp.uq.edu.au/qcpr/Homepage/discussion_papers/2004-
04_ProbabilisticRegionPopulationForecast.pdf

http://www.jws.com/pdfs/timberlandreport/v4n3.pdf

http://iussp2005.princeton.edu/download.aspx?submissionId=50327

http://econ.la.psu.edu/~hbierens/POPFORC.PDF

http://www.census.gov/population/www/documentation/twps0057/twps0057.html

http://www.labor.state.ak.us/trends/sep98.pdf

Eviews Workfile and Program:




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