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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: 16 of 16

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