TIME SERIES MODELS FOR FORECASTING NEW ONE-FAMILY HOUSES SOLD IN THE U.S.
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Date
INTRODUCTION
Since 2005, the general economy has not been consistent. Resultantly, it has adversely affected the housing market. It is important to note that over the last one decade, there has been sharp drop in the housing prices that have resulted in the rise on the loan prices. The year 2007 is a remarkable period that proved to be adversely affected by the drop in the house prices.
In this paper, examination and forecast of one house prices is to be dealt in depth with time series forecast being used as the major analytical tool. The research is employed to predict the future growth and development of the housing market in the state.
THE DATA PATTERN OF NEW ONE-FAMILY HOUSES SOLD IN THE U.S.
The figure below gives a time series data for new one-family houses in the US that were sold since January 1975-January 2010. The time period was selected since it was the most recent data that was available. The data is for non-seasonally adjusted for all the selected months. It is imperative to note that the period was settled because it is adequate enough to give economic fluctuations. It is important to note that the major components of the economic fluctuations are; trend, seasonality, cycles and irregularity. Trend is the long term change in the level of data, whereas seasonality is the regular variation of data. It is imperative to note that with regular data variation, the given data repeats itself around the same period annually. Cycles on the other hand are the wavelength fluctuations of data and it revolves around the trend. Lastly, irregularity is the random and unpredictable data variation that is observed under the area of interest.
FIGURE 1
New One-Family Houses Sold (NHS), in thousands, in the U.S. January 1975 to January 2010
Data source: National Association of Realtors
Data analysis is a complex process and this entails integration of various analytical processes and tools. Autocorrelation is one of the processes and this entails measuring of the correlation between two data sets or observation over time. A k- period of autocorrelation is known as the autocorrelation function, a 12-period plot of autocorrelation function for the NHS data and 24-period plot of ACF for the first differenced NHS data from January 1975 to June 2009, the historical period in our analysis, is shown in two figures. In figure 2, all ACFs are significantly different from zero because they are above the upper limit level and slowly decreasing, this implies there is a trend in the NHS data. Likewise, figure 3, ACFs for lag 12 and 24 are above the upper limit level which means they are significantly different from zero. This affirms seasonality in this monthly data.
FIGURE 2
12-period plot of autocorrelation functions (ACF) for NHS
FIGURE 3
24-period plot of autocorrelation functions (ACF) for first differenced NHS
TIME-SERIES MODELS FOR NEW ONE-FAMILY HOUSES SOLD
The NHS data has all the aspects of cyclonical data and from that; in our research we have employed three of the time series models in our forecasting activity. Additionally, we have employed use of exponential smoothing, the autoregressive integrated moving average and the decomposition. In our use for the data from January 1975 to the year 2009 June, it is employed as a model specification information while as for the data from July 2009 to December 2009 is taken as the holdout time for which the real data is on hand that it can be used for comparison with ex-post forecast to determine the precision of the models.
Additionally, in our research, we have employed the use of excel-based forecast X software that is employed in the estimation and forecasting purposes. It is important when we get estimates, measure the level of accuracy and reliability, in so doing; we use absolute percentage error measurement as well as root-mean-squared error. Both error measurements are shown in the table below. In order to calculate MAPE, we divide the forecast error by the actual value to get the total percentage error and then get the absolute percentage errors. Similarly, the method employed to get RMSE is first dividing the forecast error and then get the squared root of the mean of the squared errors.
TABLE 1
MAPE and RMSE
Models
Historical period
Holdout period
Jan. 1975-June 2009
July 2009-Dec. 2009
MAPE
RMSE
RMSE/Mean*
MAPE
RMSE
RMSE/Mean*
Winter’s exponential smoothing
6.46%
4.68
7.61%
8.08%
2.68
8.61%
Decomposition with exponential smoothing trend
5.27%
3.97
6.46%
6.72%
2.54
8.16%
ARIMA(1,0,0)(2,0,0)
6.94%
5.08
8.27%
8.56%
2.85
9.14%
*Mean of NHS for the historical period is 61.46 and for the holdout period is 31.17
Most importantly, the most reliable means is the one that has the smallest error. Therefore, from our chosen methods, the one that has the smallest error is the decomposition with the exponential smoothing for the recent period while RMSE has the smallest for both the holdout period as well as for the historical period. Resultantly, these methods are chosen for our data analysis since January 1975 to December 2009 in our ex-ante analysis and our interval forecast for the six months from July 2010 to December 2010 as shown in table 2 and figure 4 respectively.
TABLE 2
Point and interval (5% to 95%) forecasts of NHS for the first six months of 2010
Month
Point forecast
5%
95%
January
27.56
21.27
33.84
February
31.21
24.92
37.49
March
36.46
30.17
42.75
April
35.15
28.86
41.44
May
34.92
28.63
41.21
June
33.90
27.61
40.19
FIGURE 4
Actual and forecast values of NHS
IS THE WORST OVER?
Before making investment decisions, the potential investors current and future take deep analysis to analyze the market trend hence make their investment decisions. Mostly, in order to come up with a stable decision, it is important to analyze all the cyclical events of the data in the respective periods. This is not only limited to the estimated but also the forecasted cyclical factors in reference to the decomposition model as presented in the figure 5. Therefore, it is evident that CF time series does not show any trend or seasonality but only show cycle. Resultantly, the figure shows that the new one family housing market has undergone the trough of the cycle in 2008 and now it is on a recovery stage.
FIGURE 5
Cyclical factors
CONCLUSION
Based on our focus in this research, despite the economic challenges it is evident that the economy and the market for the one family houses is taking the right path and it’s on its way to recovery. However, the pace with which the economy will move to the market peak will be determined by the economic policies that would be put in place to ensure that the micro and macro economic factors for the economy are working right to promote market growth and expansion.
BUSI 405
APPENDIX
ForecastX output for forecastingNHS for the second half of 2010 based on the decomposition model with exponential smoothing trend.
Forecast — Decomposition Selected
Forecast
95% – 5%
95% – 5%
Date
Monthly
Quarterly
Annual
Upper
Lower
Jan-2010
27.56
33.84
21.27
Feb-2010
31.21
37.49
24.92
Mar-2010
36.46
95.22
42.75
30.17
Apr-2010
35.15
41.44
28.86
May-2010
34.92
41.21
28.63
Jun-2010
33.90
103.97
40.19
27.61
Accuracy Measures
Value
Forecast Statistics
Value
Mean Absolute Percentage Error (MAPE)
5.28%
Mean
61.03
R-Square
95.86%
Standard Deviation
19.51
Root Mean Square Error
3.96
Method Statistics
Value
Method Selected
Decomposition
Basic Method
Exponential Smoothing
Alpha
1.00
Decomposition Type
Multiplicative
REFERENCES
Di Martino, D. and Duca, J, (2007, Nov. 11). The Rise and Fall of Subprime Mortgages, Economic
Letter- Insights from the Federal Reserve Bank of Dallas.
Forecast X 6.0. John Galt Solutions, Inc.
Wheelock, D. (2007) Housing Slump Could Lean Heavily on Economy. Federal Reserve Bank
Of St. Louis.
Wilson, H. and Keating, B. (2007) Business Forecasting. New York, NY: McGraw Hill/Irwin.
ACF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 9.5881482673618373E-2 5.0520672976158525E-2 -0.18207190148025995 -0.29537238896011853 3.5075363166233497E-2 -0.21077402820557267 3.6081508549781482E-2 -0.27015679724890246 -0.15635081733155667 8.4457865819060216E-2 0.15484652968002791 0.59155049271953986 0.16079653145689998 8.8988053160963046E-2 -0.18948449871364256 -0.2004180414811442 2.2622095501736116E-2 -0.21224634043501131 1.2804211982514042E-2 -0.29514723894333711 -0.11768493178966823 5.4979715483665446E-2 0.13866482417221571 0.53156226409158447 Upper Limit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 9.6445294935804088E-2 Lower Limit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 -9.6445294935804088E-2 NHS
NHS 27395 27426 27454 27485 27515 27546 27576 27607 27638 27668 27699 27729 27760 27791 27820 27851 27881 27912 27942 27973 28004 28034 28065 28095 28126 28157 28185 28216 28246 28277 28307 28338 28369 28399 28430 28460 28491 28522 28550 28581 28611 28642 28672 28703 28734 28764 28795 28825 28856 28887 28915 28946 28976 29007 29037 29068 29099 29129 29160 29190 29221 29252 29281 29312 29342 29373 29403 29434 29465 29495 29526 29556 29587 29618 29646 29677 29707 29738 29768 29799 29830 29860 29891 29921 29952 29983 30011 30042 30072 30103 30133 30164 30195 30225 30256 30286 30317 30348 30376 30407 30437 30468 30498 30529 30560 30590 30621 30651 30682 30713 30742 30773 30803 30834 30864 30895 30926 30956 30987 31017 31048 31079 31107 31138 31168 31199 31229 31260 31291 31321 31352 31382 31413 31444 31472 31503 31533 31564 31594 31625 31656 31686 31717 31747 31778 31809 31837 31868 31898 31929 31959 31990 32021 32051 32082 32112 32143 32174 32203 32234 32264 32295 32325 32356 32387 32417 32448 32478 32509 32540 32568 32599 32629 32660 32690 32721 32752 32782 32813 32843 32874 32905 32933 32964 32994 33025 33055 33086 33117 33147 33178 33208 33239 33270 33298 33329 33359 33390 33420 33451 33482 33512 33543 33573 33604 33635 33664 33695 33725 33756 33786 33817 33848 33878 33909 33939 33970 34001 34029 34060 34090 34121 34151 34182 34213 34243 34274 34304 34335 34366 34394 34425 34455 34486 34516 34547 34578 34608 34639 34669 34700 34731 34759 34790 34820 34851 34881 34912 34943 34973 35004 35034 35065 35096 35125 35156 35186 35217 35247 35278 35309 35339 35370 35400 35431 35462 35490 35521 35551 35582 35612 35643 35674 35704 35735 35765 35796 35827 35855 35886 35916 35947 35977 36008 36039 36069 36100 36130 36161 36192 36220 36251 36281 36312 36342 36373 36404 36434 36465 36495 36526 36557 36586 36617 36647 36678 36708 36739 36770 36800 36831 36861 36892 36923 36951 36982 37012 37043 37073 37104 37135 37165 37196 37226 37257 37288 37316 37347 37377 37408 37438 37469 37500 37530 37561 37591 37622 37653 37681 37712 37742 37773 37803 37834 37865 37895 37926 37956 37987 38018 38047 38078 38108 38139 38169 38200 38231 38261 38292 38322 38353 38384 38412 38443 38473 38504 38534 38565 38596 38626 38657 38687 38718 38749 38777 38808 38838 38869 38899 38930 38961 38991 39022 39052 39083 39114 39142 39173 39203 39234 39264 39295 39326 39356 39387 39417 39448 39479 39508 39539 39569 39600 39630 39661 39692 39722 39753 39783 39814 39845 39873 39904 39934 39965 39995 40026 40057 40087 40118 40148 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 29 34 44 54 57 51 51 53 46 46 46 39 41 53 55 62 55 56 57 59 58 55 49 47 57 68 84 81 78 74 64 74 71 63 55 51 57 63 75 85 80 77 68 72 68 70 53 50 53 58 73 72 68 63 64 68 60 54 41 35 43 44 44 36 44 50 55 61 50 46 39 33 37 40 49 44 45 38 36 34 28 29 27 29 28 29 36 32 36 34 31 36 39 40 39 33 44 46 57 59 64 59 51 50 48 51 45 48 52 58 63 61 59 58 52 48 53 55 42 38 48 55 67 60 65 65 63 61 54 52 51 47 55 59 89 84 75 66 57 52 60 54 48 49 53 59 73 72 62 58 55 56 52 52 43 37 43 55 68 68 64 65 57 59 54 57 43 42 52 51 58 60 61 58 62 61 49 51 47 40 45 50 58 52 50 50 46 46 38 37 34 29 30 40 51 50 47 47 43 46 37 41 39 36 48 55 56 53 52 53 52 56 51 48 42 42 44 50 60 66 58 59 55 57 57 56 53 51 46 58 74 65 65 55 52 59 54 57 45 40 47 47 60 58 63 64 64 63 54 54 46 45 54 68 70 70 69 65 66 73 62 56 54 51 61 69 81 70 71 71 69 72 67 62 61 51 64 75 81 82 82 83 75 75 68 69 70 61 67 76 84 86 80 82 78 78 65 67 61 57 67 78 88 78 77 71 76 73 70 71 63 65 72 85 94 84 80 79 76 74 66 66 67 66 66 84 90 86 88 84 82 90 82 77 73 70 76 82 98 91 101 107 99 105 90 88 76 75 89 102 123 109 115 105 96 102 94 101 84 83 92 109 127 116 120 115 117 110 99 105 86 87 89 88 108 100 102 98 83 88 80 74 71 71 66 68 80 83 79 73 68 60 53 57 45 44 44 48 49 49 49 45 43 38 35 32 27 26 24 29 31 32 34 37 38 36 30 33 26 24 Forecast of NHS 27395 27426 27454 27485 27515 27546 27576 27607 27638 27668 27699 27729 27760 27791 27820 27851 27881 27912 27942 27973 28004 28034 28065 28095 28126 28157 28185 28216 28246 28277 28307 28338 28369 28399 28430 28460 28491 28522 28550 28581 28611 28642 28672 28703 28734 28764 28795 28825 28856 28887 28915 28946 28976 29007 29037 29068 29099 29129 29160 29190 29221 29252 29281 29312 29342 29373 29403 29434 29465 29495 29526 29556 29587 29618 29646 29677 29707 29738 29768 29799 29830 29860 29891 29921 29952 29983 30011 30042 30072 30103 30133 30164 30195 30225 30256 30286 30317 30348 30376 30407 30437 30468 30498 30529 30560 30590 30621 30651 30682 30713 30742 30773 30803 30834 30864 30895 30926 30956 30987 31017 31048 31079 31107 31138 31168 31199 31229 31260 31291 31321 31352 31382 31413 31444 31472 31503 31533 31564 31594 31625 31656 31686 31717 31747 31778 31809 31837 31868 31898 31929 31959 31990 32021 32051 32082 32112 32143 32174 32203 32234 32264 32295 32325 32356 32387 32417 32448 32478 32509 32540 32568 32599 32629 32660 32690 32721 32752 32782 32813 32843 32874 32905 32933 32964 32994 33025 33055 33086 33117 33147 33178 33208 33239 33270 33298 33329 33359 33390 33420 33451 33482 33512 33543 33573 33604 33635 33664 33695 33725 33756 33786 33817 33848 33878 33909 33939 33970 34001 34029 34060 34090 34121 34151 34182 34213 34243 34274 34304 34335 34366 34394 34425 34455 34486 34516 34547 34578 34608 34639 34669 34700 34731 34759 34790 34820 34851 34881 34912 34943 34973 35004 35034 35065 35096 35125 35156 35186 35217 35247 35278 35309 35339 35370 35400 35431 35462 35490 35521 35551 35582 35612 35643 35674 35704 35735 35765 35796 35827 35855 35886 35916 35947 35977 36008 36039 36069 36100 36130 36161 36192 36220 36251 36281 36312 36342 36373 36404 36434 36465 36495 36526 36557 36586 36617 36647 36678 36708 36739 36770 36800 36831 36861 36892 36923 36951 36982 37012 37043 37073 37104 37135 37165 37196 37226 37257 37288 37316 37347 37377 37408 37438 37469 37500 37530 37561 37591 37622 37653 37681 37712 37742 37773 37803 37834 37865 37895 37926 37956 37987 38018 38047 38078 38108 38139 38169 38200 38231 38261 38292 38322 38353 38384 38412 38443 38473 38504 38534 38565 38596 38626 38657 38687 38718 38749 38777 38808 38838 38869 38899 38930 38961 38991 39022 39052 39083 39114 39142 39173 39203 39234 39264 39295 39326 39356 39387 39417 39448 39479 39508 39539 39569 39600 39630 39661 39692 39722 39753 39783 39814 39845 39873 39904 39934 39965 39995 40026 40057 40087 40118 40148 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 0 24.750637325971969 27.555751941429602 31.205302855762465 36.462447122760956 35.147996281808837 34.919132644911038 33.898784855257148 32.190917608038312 32.872967277115393 29.941183115606734 29.651285569905507 26.281947584673549 24.743966450611495 Upper 27395 27426 27454 27485 27515 27546 27576 27607 27638 27668 27699 27729 27760 27791 27820 27851 27881 27912 27942 27973 28004 28034 28065 28095 28126 28157 28185 28216 28246 28277 28307 28338 28369 28399 28430 28460 28491 28522 28550 28581 28611 28642 28672 28703 28734 28764 28795 28825 28856 28887 28915 28946 28976 29007 29037 29068 29099 29129 29160 29190 29221 29252 29281 29312 29342 29373 29403 29434 29465 294 95 29526 29556 29587 29618 29646 29677 29707 29738 29768 29799 29830 29860 29891 29921 29952 29983 30011 30042 30072 30103 30133 30164 30195 30225 30256 30286 30317 30348 30376 30407 30437 30468 30498 30529 30560 30590 30621 30651 30682 30713 30742 30773 30803 30834 30864 30895 30926 30956 30987 31017 31048 31079 31107 31138 31168 31199 31229 31260 31291 31321 31352 31382 31413 31444 31472 31503 31533 31564 31594 31625 31656 31686 31717 31747 31778 31809 31837 31868 31898 31929 31959 31990 32021 32051 32082 32112 32143 32174 32203 32234 32264 32295 32325 32356 32387 32417 32448 32478 32509 32540 32568 32599 32629 32660 32690 32721 32752 32782 32813 32843 32874 32905 32933 32964 32994 33025 33055 33086 33117 33147 33178 33208 33239 33270 33298 33329 33359 33390 33420 33451 33482 33512 33543 33573 33604 33635 33664 33695 33725 33756 33786 33817 33848 33878 33909 33939 33970 34001 34029 34060 34090 34121 34151 34182 34213 34243 34274 34304 34335 34366 34394 34425 34455 34486 34516 34547 34578 34608 34639 34669 34700 34731 34759 34790 34820 34851 34881 34912 34943 34973 35004 35034 35065 35096 35125 35156 35186 35217 35247 35278 35309 35339 35370 35400 35431 35462 35490 35521 35551 35582 35612 35643 35674 35704 35735 35765 35796 35827 35855 35886 35916 35947 35977 36008 36039 36069 36100 36130 36161 36192 36220 36251 36281 36312 36342 36373 36404 36434 36465 36495 36526 36557 36586 36617 36647 36678 36708 36739 36770 36800 36831 36861 36892 36923 36951 36982 37012 37043 37073 37104 37135 37165 37196 37226 37257 37288 37316 37347 37377 37408 37438 37469 37500 37530 37561 37591 37622 37653 37681 37712 37742 37773 37803 37834 37865 37895 37926 37956 37987 38018 38047 38078 38108 38139 38169 38200 38231 38261 38292 38322 38353 38384 38412 38443 38473 38504 38534 38565 38596 38626 38657 38687 38718 38749 38777 38808 38838 38869 38899 38930 38961 38991 39022 39052 39083 39114 39142 39173 39203 39234 39264 39295 39326 39356 39387 39417 39448 39479 39508 39539 39569 39600 39630 39661 39692 39722 39753 39783 39814 39845 39873 39904 39934 39965 39995 40026 40057 40087 40118 40148 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 0 33.84428430305 0017 37.493835217382973 42.750979484381425 41.436528643429362 41.207665006531457 40.187317216877737 38.479449969658774 39.161499638735918 36.229715477227217 35.939817931526008 32.570479946294057 31.032498812231943 Lower 27395 27426 27454 27485 27515 27546 27576 27607 27638 27668 27699 27729 27760 27791 27820 27851 27881 27912 27942 27973 28004 28034 28065 28095 28126 28157 28185 28216 28246 28277 28307 28338 28369 28399 28430 28460 28491 28522 28550 28581 28611 28642 28672 28703 28734 28764 28795 28825 28856 28887 28915 28946 28976 29007 29037 29068 29099 29129 29160 29190 29221 29252 29281 29312 29342 29373 29403 29434 29465 29495 29526 29556 29587 29618 29646 29677 29707 29738 29768 29799 29830 29860 29891 29921 29952 29983 30011 30042 30072 30103 30133 30164 30195 30225 30256 30286 30317 30348 30376 30407 30437 30468 30498 30529 30560 30590 30621 30651 30682 30713 30742 30773 30803 30834 30864 30895 30926 30956 30987 31017 31048 31079 31107 31138 31168 31199 31229 31260 31291 31321 31352 31382 31413 31444 31472 31503 31533 31564 31594 31625 31656 31686 31717 31747 31778 31809 31837 31868 31898 31929 31959 31990 32021 32051 32082 32112 32143 32174 32203 32234 32264 32295 32325 32356 32387 32417 32448 32478 32509 32540 32568 32599 32629 32660 32690 32721 32752 32782 32813 32843 32874 32905 32933 32964 32994 33025 33055 33086 33117 33147 33178 33208 33239 33270 33298 33329 33359 33390 33420 33451 33482 33512 33543 33573 33604 33635 33664 33695 33725 33756 33786 33817 33848 33878 33909 33939 33970 34001 34029 34060 34090 34121 34151 34182 34213 34243 34274 34304 34335 34366 34394 34425 34455 34486 34516 34547 34578 34608 34639 34669 34700 34731 34759 34790 34820 34851 34881 34912 34943 34973 35004 35034 35065 35096 35125 35156 35186 35217 35247 35278 35309 35339 35370 35400 35431 35462 35490 35521 35551 35582 35612 35643 35674 35704 35735 35765 35796 35827 35855 35886 35916 35947 35977 36008 36039 36069 36100 36130 36161 36192 36220 36251 36281 36312 36342 36373 36404 36434 36465 36495 36526 36557 36586 36617 36647 36678 36708 36739 36770 36800 36831 36861 36892 36923 36951 36982 37012 37043 37073 37104 37135 37165 37196 37226 37257 37288 37316 37347 37377 37408 37438 37469 37500 37530 37561 37591 37622 37653 37681 37712 37742 37773 37803 37834 37865 37895 37926 37956 37987 38018 38047 38078 38108 38139 38169 38200 38231 38261 38292 38322 38353 38384 38412 38443 38473 38504 38534 38565 38596 38626 38657 38687 38718 38749 38777 38808 38838 38869 38899 38930 38961 38991 39022 39052 39083 39114 39142 39173 39203 39234 39264 39295 39326 39356 39387 39417 39448 39479 39508 39539 39569 39600 39630 39661 39692 39722 39753 39783 39814 39845 39873 39904 39934 39965 39995 40026 40057 40087 40118 40148 40179 40210 40238 40269 40299 40330 40360 40391 40422 40452 40483 40513 0 21.267219579809087 24.916770494142007 30.173914761140509 28.859463920188354 28.630600283290491 27.610252493636754 25.902385246417786 26.584434915494953 23.652650753986251 23.362753208284989 19.993415223053088 18.45543408899097 CF 27576 27607 27638 27668 27699 27729 27760 27791 27820 27851 27881 27912 27942 27973 28004 28034 28065 28095 28126 28157 28185 28216 28246 28277 28307 28338 28369 28399 28430 28460 28491 28522 28550 28581 28611 28642 28672 28703 28734 28764 28795 28825 28856 28887 28915 28946 28976 29007 29037 29068 29099 29129 29160 29190 29221 29252 29281 29312 29342 29373 29403 29434 29465 29495 29526 29556 29587 29618 29646 29677 29707 29738 29768 29799 29830 29860 29891 29921 29952 29983 30011 30042 30072 30103 30133 30164 30195 30225 30256 30286 30317 30348 30376 30407 30437 30468 30498 30529 30560 30590 30621 30651 30682 30713 30742 30773 30803 30834 30864 30895 30926 30956 30987 31017 31048 31079 31107 31138 31168 31199 31229 31260 31291 31321 31352 31382 31413 31444 31472 31503 31533 31564 31594 31625 31656 31686 31717 31747 31778 31809 31837 31868 31898 31929 31959 31990 32021 32051 32082 32112 32143 32174 32203 32234 32264 32295 32325 32356 32387 32417 32448 32478 32509 32540 32568 32599 32629 32660 32690 32721 32752 32782 32813 32843 32874 32905 32933 32964 32994 33025 33055 33086 33117 33147 33178 33208 33239 33270 33298 33329 33359 33390 33420 33451 33482 33512 33543 33573 33604 33635 33664 33695 33725 33756 33786 33817 33848 33878 33909 33939 33970 34001 34029 34060 34090 34121 34151 34182 34213 34243 34274 34304 34335 34366 34394 34425 34455 34486 34516 34547 34578 34608 34639 34669 34700 34731 34759 34790 34820 34851 34881 34912 34943 34973 35004 35034 35065 35096 35125 35156 35186 35217 35247 35278 35309 35339 35370 35400 35431 35462 35490 35521 35551 35582 35612 35643 35674 35704 35735 35765 35796 35827 35855 35886 35916 35947 35977 36008 36039 36069 36100 36130 36161 36192 36220 36251 36281 36312 36342 36373 36404 36434 36465 36495 36526 36557 36586 36617 36647 36678 36708 36739 36770 36800 36831 36861 36892 36923 36951 36982 37012 37043 37073 37104 37135 37165 37196 37226 37257 37288 37316 37347 37377 37408 37438 37469 37500 37530 37561 37591 37622 37653 37681 37712 37742 37773 37803 37834 37865 37895 37926 37956 37987 38018 38047 38078 38108 38139 38169 38200 38231 38261 38292 38322 38353 38384 38412 38443 38473 38504 38534 38565 38596 38626 38657 38687 38718 38749 38777 38808 38838 38869 38899 38930 38961 38991 39022 39052 39083 39114 39142 39173 39203 39234 39264 39295 39326 39356 39387 39417 39448 39479 39508 39539 39569 39600 39630 39661 39692 39722 39753 39783 39814 39845 39873 39904 39934 39965 39995 40026 40057 40087 40118 40148 40179 40210 40238 40269 40299 40330 1 1.0278776978417299 1.0262467191601099 1.0161977834612101 1.0050335570469768 1.0025041736227001 1.0091590341382231 1.0099009900990066 1.0147058823529398 1.0169082125603861 1.0095011876484561 1.0086274509803899 1.01866251944012 1.0236641221373961 1.0328113348247601 1.034657039711194 1.02930914166085 1.0277966101694807 1.0164907651715001 1.0142764438676199 1.0179142674344137 1.0131992457573837 1.0086848635235701 1.0061500615006138 1.0024449877750599 0.99695121951219678 0.99143730886850157 0.99691548426897003 1.0037128712871299 1.0030826140567231 1.0043023970497778 1.0012239902080768 0.99694376528117401 1.0024524831391799 1.0030581039755431 0.99817073170731496 0.99694563225412647 0.99448529411764519 0.99568699938385696 0.99071782178217749 0.98438475952529658 0.98350253807106358 0.98838709677419301 0.99477806788511802 0.99212598425196696 0.98412698412698196 0.98118279569892319 0.98150684931506549 0.98255408234473096 0.98295454545454497 0.96893063583815064 0.95152870991797156 0.95297805642633404 0.96957236842105132 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1.0007692307692269 0.99000768639508163 0.9891304347826072 0.99450549450549564 1.0007892659826398 1.00236593059937 1.0039339103068399 1.0101880877742899 1.0139643134212566 1.01836266258607 1.0105184072126199 0.9985130111524142 1.0044676098287431 1.0133432171979142 1.0117044623262599 1.0079537237888601 1.01865136298422 1.0323943661971768 1.0231923601637101 1.0073333333333299 0.99669093315685064 0.99003984063744999 0.99798792756539201 1.0053763440860199 0.9993315508021372 0.99933110367892997 1 0.9986613119143235 0.98927613941018799 0.98102981029810576 0.982734806 62983603 0.98524244553759699 0.99286733238231051 1.0014367816091956 0.99713055954089003 0.99280575539568305 0.99492753623188579 0.98761835396941 0.9837758112094408 0.98950524737631196 0.99318181818181805 0.99313501144164751 0.99846390168970756 1.0069230769230799 1.0068754774637101 1.0037936267071266 1.0037792894935798 1.0052710843373498 1.0037453183520568 1.0037313432835768 1.0104089219330945 1.0036791758646098 0.98973607038123157 0.98666666666666658 0.99174174174174157 0.99242997728993199 0.99847444698703258 1.0053475935828899 0.99772036474163861 0.99162223914699199 0.99846390168970756 1.0015384615384599 0.99308755760368705 0.99381283836040202 0.99922178988326549 0.99376947040498564 0.98510971786833801 0.98488464598249759 0.98061389337641403 0.974464 57990115121 0.97802197802197954 0.97839239412273049 0.97614840989399365 0.97828054298642497 0.97594819611471129 0.97630331753554678 0.98349514563106644 0.99111549851924996 0.99501992031872499 0.99399399399399402 0.9939577039274925 0.99696048632218803 0.99898373983739608 1.0030518819939001 1.0091277890466499 1.01206030150754 1.0248262164846054 1.0319767441860499 1.0187793427229965 1.0073732718893968 1.00731930466606 1.0099909173478654 1.0134892086330898 1.01685891748004 1.02094240837696 1.0179487179487199 1.0083963056255199 1.0074937552039942 1.0016528925619799 0.99257425742574301 0.99916874480465223 1.0141430948419301 1.0155865463494698 1.0096930533117898 1.0071999999999965 1.003177124702143 1.0055423594616 1.0110236220472368 1.0147975077881599 1.0153491941673098 1.0083144368858701 1.0074962518740544 1.0163690476190466 1.0095168374817001 1.0043509789702738 1.002166064981953 0.99495677233429403 0.99927588703837955 0.99927536231884095 0.99854967367657976 0.9949164851125607 0.98613138686131196 0.99259807549962997 0.99254287844891897 0.98121712997745691 0.983920367534456 0.99299610894941559 1.00548589341693 1.01636788776306 1.01226993865031 1.0030303030302998 0.99773413897280949 0.99848599545798444 1.00454890068234 1.00905660377358 1.02094240837696 1.02271062271062 1.0157593123209161 1.0126939351198898 1.0048746518105798 1.0020790020789998 1.00829875518672 1.01234567901235 1.0067750677506799 1.0067294751009366 1.0093582887700498 1.0086092715231798 1.0052527905449768 1. 0078380143696866 1.00712896953986 1.0012870012870001 1.0051413881748066 1.0057544757033199 1.0012714558169098 1.0025396825396737 1.0069664344521898 1.00817610062893 1.0043668122270666 1.0018633540372666 1.0055796652200866 1.0036991368680599 1.0073710073710098 1.0140243902438966 1.0138304269392699 1.01067615658363 1.0052816901408466 1.0023350846468233 1.0046592894583599 1.00927536231884 1.01091326823665 1.00738636363636 1.0022560631697701 1.0022509848058545 1.0039303761931468 1.00111856823266 0.99832402234636897 1.0011191941801898 1.0033538289547201 1 0.99721448467966556 0.99385474860335199 0.99269252388982598 0.99773499433748603 1.0011350737797999 1.0034013605442198 0.99774011299434995 0.99377123442808946 0.99202279202279198 0.99253302699597856 0.9959490740740764 1 1.0052295177222466 1.0034682080924837 1.0057603686635899 1.00744558991982 1.0068220579874854 1.0073404856013599 1.0067264573990937 1.0050111358574598 1.0060941828254766 1.0044052863436099 1.0005482456140298 0.99835616438356156 0.99506037321624397 0.999448428019857 1.0027593818984499 0.99724821133736896 0.99613686534216195 0.99722991689750695 0.99888888888888905 1.0055617352613966 1.00719026548673 1.0060406370126298 1.0120087336244499 1.01725997842503 1.0143160127253399 1.0088865656037633 1.0051813471502598 1.0072164948453599 1.0040941658137201 1.0030581039755431 1.0066056910569099 1.0090863200403799 1.0180090045022501 1.01965601965602 1.0154216867469859 1.0109159943046999 1.0089201877934268 1.0065146579804554 1.00369856680536 1.0082911100875198 1.0150753768844198 1.0202520252025233 1.0189677988531098 1.01385281385281 1.0051238257899198 0.99787595581988164 0.99744572158365297 1.0004268032437 1.0072525597269621 1.0088945362134698 1.0067170445004201 1.0045871559633031 1.0041511000415133 1.0045473336089301 1.0045267489711898 1.0049160180253998 1.0061149612719138 1.0125607779578598 1.0116046418567399 1.00514240506329 1.0035419126328198 1.0023529411764733 1.00234741784038 1.0003903200624498 0.99063597346859433 0.98424576604962599 0.98599439775910402 0.98620129870129858 0.98559670781892761 0.97870563674321776 0.97610921501706505 0.98208041958041903 0.97774810858923178 0.97906235776058304 0.98558809855881002 0.98160377358490603 0.97933685728015463 0.97644749754661464 0.97738693467336701 0.97943444730077278 0.97480314960629899 0.97845988152934804 0.97633461750137829 0.96899661781285262 0.97440372309482204 0.97432835820895503 0.96752450980392157 0.96896770107662944 0.97254901960784479 0.96572580645161599 0.95476687543493399 0.95335276967929949 0.9556574923547424 0.95760000000000178 0.96073517126148855 0.96521739130434758 0.96126126126125944 0.95970009372071263 0.96484375000000167 0.96153846153846101 0.95894736842105299 0.95938529088913305 0.95995423340961328 0.96185935637664055 0.97149938042131401 0.98341836734693588 0.99092088197146444 0.99083769633507979 0.99471598414794971 1 0.99601593625498064 0.995703833785062 0.99675030500161244 0.99814415045072202 0.99951216292293921 0.99934622409416696 0.99977275679806399 1.00043100690855 0.99953681923058801 0.99939414797002557 0.99945052546832058 0.99955977842622457 0.99979014353663798
NHS 27395 27426 27454 27485 27515 27546 27576 27607 27638 27668 27699 27729 27760 27791 27820 27851 27881 27912 27942 27973 28004 28034 28065 28095 28126 28157 28185 28216 28246 28277 28307 28338 28369 28399 28430 28460 28491 28522 28550 28581 28611 28642 28672 28703 28734 28764 28795 28825 28856 28887 28915 28946 28976 29007 29037 29068 29099 29129 29160 29190 29221 29252 29281 29312 29342 29373 29403 29434 29465 29495 29526 29556 29587 29618 29646 29677 29707 29738 29768 29799 29830 29860 29891 29921 29952 29983 30011 30042 30072 30103 30133 30164 30195 30225 30256 30286 30317 30348 30376 30407 30437 30468 30498 30529 30560 30590 30621 30651 30682 30713 30742 30773 30803 30834 30864 30895 30926 30956 30987 31017 31048 31079 31107 31138 31168 31199 31229 31260 31291 31321 31352 31382 31413 31444 31472 31503 31533 31564 31594 31625 31656 31686 31717 31747 31778 31809 31837 31868 31898 31929 31959 31990 32021 32051 32082 32112 32143 32174 32203 32234 32264 32295 32325 32356 32387 32417 32448 32478 32509 32540 32568 32599 32629 32660 32690 32721 32752 32782 32813 32843 32874 32905 32933 32964 32994 33025 33055 33086 33117 33147 33178 33208 33239 33270 33298 33329 33359 33390 33420 33451 33482 33512 33543 33573 33604 33635 33664 33695 33725 33756 33786 33817 33848 33878 33909 33939 33970 34001 34029 34060 34090 34121 34151 34182 34213 34243 34274 34304 34335 34366 34394 34425 34455 34486 34516 34547 34578 34608 34639 34669 34700 34731 34759 34790 34820 34851 34881 34912 34943 34973 35004 35034 35065 35096 35125 35156 35186 35217 35247 35278 35309 35339 35370 35400 35431 35462 35490 35521 35551 35582 35612 35643 35674 35704 35735 35765 35796 35827 35855 35886 35916 35947 35977 36008 36039 36069 36100 36130 36161 36192 36220 36251 36281 36312 36342 36373 36404 36434 36465 36495 36526 36557 36586 36617 36647 36678 36708 36739 36770 36800 36831 36861 36892 36923 36951 36982 37012 37043 37073 37104 37135 37165 37196 37226 37257 37288 37316 37347 37377 37408 37438 37469 37500 37530 37561 37591 37622 37653 37681 37712 37742 37773 37803 37834 37865 37895 37926 37956 37987 38018 38047 38078 38108 38139 38169 38200 38231 38261 38292 38322 38353 38384 38412 38443 38473 38504 38534 38565 38596 38626 38657 38687 38718 38749 38777 38808 38838 38869 38899 38930 38961 38991 39022 39052 39083 39114 39142 39173 39203 39234 39264 39295 39326 39356 39387 39417 39448 39479 39508 39539 39569 39600 39630 39661 39692 39722 39753 39783 39814 39845 39873 39904 39934 39965 39995 40026 40057 40087 40118 40148 40179 29 34 44 54 57 51 51 53 46 46 46 39 41 53 55 62 55 56 57 59 58 55 49 47 57 68 84 81 78 74 64 74 71 63 55 51 57 63 75 85 80 77 68 72 68 70 53 50 53 58 73 72 68 63 64 68 60 54 41 35 43 44 44 36 44 50 55 61 50 46 39 33 37 40 49 44 45 38 36 34 28 29 27 29 28 29 36 32 36 34 31 36 39 40 39 33 44 46 57 59 64 59 51 50 48 51 45 48 52 58 63 61 59 58 52 48 53 55 42 38 48 55 67 60 65 65 63 61 54 52 51 47 55 59 89 84 75 66 57 52 60 54 48 49 53 59 73 72 62 58 55 56 52 52 43 37 43 55 68 68 64 65 57 59 54 57 43 42 52 51 58 60 61 58 62 61 49 51 47 40 45 50 58 52 50 50 46 46 38 37 34 29 30 40 51 50 47 47 43 46 37 41 39 36 48 55 56 53 52 53 52 56 51 48 42 42 44 50 60 66 58 59 55 57 57 56 53 51 46 58 74 65 65 55 52 59 54 57 45 40 47 47 60 58 63 64 64 63 54 54 46 45 54 68 70 70 69 65 66 73 62 56 54 51 61 69 81 70 71 71 69 72 67 62 61 51 64 75 81 82 82 83 75 75 68 69 70 61 67 76 84 86 80 82 78 78 65 67 61 57 67 78 88 78 77 71 76 73 70 71 63 65 72 85 94 84 80 79 76 74 66 66 67 66 66 84 90 86 88 84 82 90 82 77 73 70 76 82 98 91 101 107 99 105 90 88 76 75 89 102 123 109 115 105 96 102 94 101 84 83 92 109 127 116 120 115 117 110 99 105 86 87 89 88 108 100 102 98 83 88 80 74 71 71 66 68 80 83 79 73 68 60 53 57 45 44 44 48 49 49 49 45 43 38 35 32 27 26 24 29 31 32 34 37 38 36 30 33 26 24 21
ACF 1 2 3 4 5 6 7 8 9 10 11 12 0.93010245893606758 0.84840846728754693 0.76198747479621154 0.70101791422911963 0.67848717778081091 0.64937251096899262 0.64778111637191649 0.64211721763140939 0.67063926452403566 0.71981798830652566 0.75857456014807945 0.77666443953576592 Upper Limit 1 2 3 4 5 6 7 8 9 10 11 12 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 9.6328744687866147E-2 Lower Limit 1 2 3 4 5 6 7 8 9 10 11 12 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2 -9.6328744687866147E-2
