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This 16 page Class Notes was uploaded by Ebba Kessler Jr. on Wednesday September 23, 2015. The Class Notes belongs to FIN642 at Drexel University taught by ThomasChiang in Fall. Since its upload, it has received 38 views. For similar materials see /class/212262/fin642-drexel-university in Finance at Drexel University.
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Date Created: 09/23/15
FIN 6427Winter2004 Lecture 4 Homework 3 Dependent Variable PHS Method Least Squares Date 113003 Time 1538 Sampleadjusted 1990419984 Included observations 33 after ad39usting enonints Variable Coefficient Std Error tStatistic Prob C 4590801 8331449 5510206 00000 USTBR1 1783114 2173983 8202061 00000 DPl1 0041294 0004485 9207624 00000 Q2 9325184 7635678 1221265 00000 Q3 7671742 7424575 1033290 00000 Q4 2265513 7638356 2965970 00062 Rsquared 0930376 Mean dependent var 2708727 Adjusted Rsquared 0917482 SD dependent var 5310294 SE of regression 1525430 Akaike info criterion 8450566 Sum squared resid 6282733 Schwarz criterion 8722659 Log likelihood 1334343 Fstatistic 7215902 DurbinWatson stat 1189063 Prob Fstatistic 0000000 PHS459080 17831 USTBR1 0041 DPI1 93252Q2 76717Q3 22655Q4 b and c PERIOD PHS USTBR DPI 02 Q3 Q4 PHSforecast At Ft AtFt2 31Dec98 3046 431 20194 0 0 1 31 Mar99 2941 4 42 20377 0 0 0 291961 2139 4 574 30Jun99 3771 446 20472 1 0 0 390737 13637 185955 30Sep99 3556 470 20756 0 1 0 377448 21848 477339 31Dec99 3081 506 21124 0 0 1 330810 22710 515734 MSE 295900 RMSE 17202 Double click on any point on above table to activate excel and view calculation formula Business Conditions amp Forecasting 7 Exponential Smoothing Dr Thomas C Chiang LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecture introduces timeseries smoothing forecasting methods Various models are discussed including methods applicable to nonstationary and seasonal timeseries data These models are viewed as classical timeseries model all of them are univariate LEARNING OBJECTIVES Moving averages Forecasting using exponential smoothing Accounting for data trend using Holt s smoothing Accounting for data seasonality using Winter39s smoothing Adaptiveresponserate single exponential smoothing p t Forecasting with Moving Averages The naive method discussed in Lecture 1 uses the most recent observations to forecast future values That is FM Y Since the outcomes of Y are subject to variations using the mean value is considered an alternative method of forecasting In order to keep forecasts updated a simple movingaverage method has been widely used 11 The Model Moving averages are developed based on an average of weighted observations which tends to smooth out shortterm irregularity in the data series They are useful if the data series remains fairly steady over time Notations M t E If Moving average at time t which is the forecast value at time H K Observation at time t e Y If Forecast error A moving average is obtained by calculating the mean for a specified set of values and then using it to forecast the next period That is MKY1Kn1n 111 Mi1K1KzKn 112 Business Conditions amp Forecasting Dr Thomas C Chiang Subtracting Equation 112 from Equation 111 we obtain M2MHYiYn 113 Equation 113 allows us to update the data making the forecasting process much easier This equation states that the moving average can be updated by using a previous moving average plus the average changes in actual value from time tto t n Using either Equation 111 or 113 should yield the same result 12 A Numerical Example To illustrate how a moving average is used consider Table 31 which contains the exchange rate between the Japanese yen and the US dollar from 1983Q1 through 1998Q4 To calculate the threequarter moving average requires first that we sum the first three observations 2393 2398 and 2361 This threequarter total is then divided by 3 to obtained 410 as shown in the third cell of column 4 in Table 1 This smoothed number 23840 becomes the forecast for 1983Q4 displayed in the fourth cell of column 5 of 3Q MAF By the same token we can obtain the forecast for 1984Q1 by moving one quarter ahead and dropping the most distant quarter That is Y Y Y Y3 assume n 3 232 2361 2398 3 23597 The last value of the moving average is 13029 which is the forecast for 1999Q1 11521357213995 3 13029 It is of interest to calculate the squared errors SE and the sum of squared errors SSE The squared errors of using moving average are presented in column 8 labeled by SEMA The resulted meansquared error MSE is 24421 The last row of Table 1 This figure 24421 appears to be larger than the MSE of 21894 obtained by a naive model the randomwalk process Not surprisingly if you are familiar with the research in international finance this result is consistent with most empirical findings It has been shown that not many models can beat the randomwalk process since the current exchange rate contains all the historical information pertinent to predict exchange rate movements as stated by the efficient market hypothesis Business Conditions amp Forecasting Dr Thomas C Chiang Table 1 The Japanese Yen US Dollar Rate 1983Q1 1998Q4 Period Actual 11Q RW 3Q M A 3 Q MAF XSG0 8 SERW SEMA SEXS Mar83 Missing Missing Missing Missing Jun83 2393 Missin Missing 23930 Sep83 2398 Missin 23970 Dec83 2361 23597 23682 1681 4096 2323 Mar84 232 23095 23597 23296 5256 12581 6747 Jun84 22475 23140 23095 22639 16129 4225 12226 Sep84 23745 23587 23140 23524 6320 19600 10325 Dec84 2454 24481 23587 24337 3819 24691 6744 Mar85 25158 24923 24481 24994 077 3469 058 Jun85 2507 25041 24923 25055 306 008 55 Sep85 24895 23855 25041 24927 108570 118405 110686 Dec85 216 22185 23855 22265 23716 144020 48637 Mar86 2006 19875 22185 20501 43890 178084 64317 Jun86 17965 181 40 19875 18472 24649 121104 43148 Sep86 16395 16574 18140 16810 10650 77117 20951 Dec86 15363 15923 16574 15652 4186 3185 1278 Mar87 1601 15313 15923 15938 20880 18433 18865 Jun87 14565 15083 15313 14840 121 4066 271 Sep87 14675 14625 15083 14708 016 2010 053 Dec87 14635 13837 14625 14650 59292 58806 60005 Mar88 122 13095 13837 12690 625 19228 576 Jun88 1245 12623 13095 12498 5929 156 5213 Sep88 1322 13033 12623 13076 441 6507 1256 Dec88 1343 13080 13033 13359 7056 1965 5915 Mar89 1259 13092 13080 12744 4422 306 2613 Jun89 13255 13413 13092 13153 12996 16987 15431 Sep89 14395 13862 13413 14147 2116 2721 448 Dec89 13935 14223 13862 13977 1640 2288 1315 Mar90 1434 14680 14223 14267 20306 23767 22426 Jun90 15765 15130 14680 15465 2304 3660 326 Sep90 15285 14948 15130 15321 22201 17822 23290 Dec90 13795 14207 14948 14100 650 19834 3138 Mar91 1354 13797 14207 13652 2652 230 1624 Jun91 14055 13803 13797 13974 576 003 254 Sep91 13815 13722 13803 13847 2704 2584 3046 Dec91 13295 13212 13722 13405 5929 14320 7751 Mar92 12525 13042 13212 12701 6084 087 3647 Jun92 13305 12795 13042 13184 5625 2368 3959 Sep92 12555 12595 12795 12681 3969 7569 5713 Dec92 11925 12315 12595 12076 2916 169 1512 Mar93 12465 11975 12315 12387 8649 6084 7263 Jun93 11535 11550 11975 11705 7815 17530 11119 Sep93 10651 10899 11550 10862 199 10823 1238 Dec93 1051 10783 10899 10580 4610 843 3704 Mar94 11189 10660 10783 11067 8263 2533 6198 Jun94 1028 10455 10660 10437 1482 5847 2943 Sep94 9895 10011 10455 10003 013 3548 209 Dec94 9859 9912 10011 9888 154 008 090 Mar95 9983 9560 9912 9964 13110 11542 12678 Jun95 8838 9099 9560 9063 1303 11729 3436 Sep95 8477 9044 9099 8594 17983 165 14976 Dec95 9818 9529 9044 9573 2237 15542 5152 continued Business Conditions amp Forecasting Dr Thomas C Chiang Table 3 1 continued Period Actual 1e va 3Q MA 3 9 MAF XS0iO8 sERvvg sSEMA SEX Mar96 10649 10291 10253 9529 10147 1282 12551 2516 Jun96 10988 10649 10643 10253 10549 1149 5407 1930 Sep96 11145 10988 10927 10643 10900 246 2523 600 Dec96 11598 11145 11244 10927 11096 2052 4498 2520 Mar97 12397 11598 11713 11244 11498 6384 13302 8089 Jun97 1143 12397 11808 11713 12217 9351 803 6196 Sep97 12144 1143 11990 11808 11587 5098 1127 3098 Dec97 12992 12144 12189 11990 12033 7191 10033 9203 Mar98 13339 12992 12825 12189 12800 1204 13233 2904 Jun98 13995 13339 13442 12825 13231 4303 13689 5833 Sep98 13572 13995 13635 13442 13842 1789 169 730 Dec98 1152 13572 13029 13635 13626 42107 44746 44354 Mar99 1152 Missing 13029 11941 MSE 21894 24421 23339 Notes 1Q RW 1quarter random walk process 3Q MA 3quarter moving average 3Q MAF 3quarter moving average forecast XSq08 1quarter exponential smoothing with 0 08 SERW Squared errors by using random walkforecast SEMA Squared errors by 3quarter movingaverage forecast SEXS Squared errors by using exponentialsmoothing forecast MSE Mean squared errors 13 Remarks on Moving Average Method The movingaverage method provides an efficient mechanism for obtaining a value for forecasting stationary time series The technique is simply an arithmetic average as time passes with some laglength determined optimally by an underlying cycle present in the data Thus movingaverages and movingaverage lines are frequently derived by technicians on Wall Street to generate market expectations one of the most important input variables used by fund managers to allocate portfolios The difficulty in using moving averages is their inability to capture the peaks and troughs of the series When the market actual data are moving down persistently the moving average forecast tends to produce overpredicted valued while when the market is moving up continually the movingaverage forecast will underpredict the market Obviously this method fails to deal with nonstationary data Moreover since all the data points in the movingaverage process are given equal weight this approach fails to re ect the importance of time ordering with respect to observations For this reason a weighted movingaverage method has been suggested It merely imposes different weights on the observations being used for forecasting The double movingaverage method taking the form of moving average on the first movingaverages gives more weight on the middle point Exponential smoothing methods are the techniques that place more weights on the recent observations Business Conditions amp Forecasting Dr Thomas C Chiang 2 Forecasting with Exponential Smoothing 21 The Model Simple exponential smoothing takes the form of F 1 aY 1 aF 211 aY lt1 agt Notations Ft1 forecast value for period t1 made at time t which can be de ned as 1 1 K actual value in period t Wilson and Keating use X t others use Z t we use Y to maintain notation consistency E forecast value for period i made by t J a smoothing constant 0lt a lt1 By continuing to substitute previous forecasting values back to the starting point of the data 111 aY an 001 a1 00212 a1 ooHr1 1 aim0 212 Writing this equation in compact form a ia WY lt1 am 213 1170 It is clear that the weights aa1 a 01 002 on K YH YE2 as implied in Equation 213 are exponentially declining Two points deserve our attention before we proceed to make our forecasts First we need to decide the initial value Y0 A convenient way to accomplish this is to utilize the value of the initial data point or the average value of the first few observations of the data series Second we must determine the value of 0 Usually this selection can be achieved by minimizing the MSE 0r RMSE based on insample experiments Business Conditions amp Forecasting Dr Thomas C Chiang 22 Numerical Example 39 I d 39 39II Simple r can be J by using quarterly data of the yendollar exchange rate in Table 31 Assuming that 1 08 calculations of the exponential smoothing of the exchange rate are as follows FM 0th 1 aE 082398 108 23930 23970 Forecast for 1983Q3 08236l 108 23970 23682 Forecast for 1983Q4 By calculating the smoothing values in the same manner we obtain the gures presented in column 6 under XS0L 08 Again the squared errors are shown in column 9 the resulting MSE is still higher than the randomwalk process although it is slightly better than the movingaverage smoothing calculation From our exercise the naive model in the form of a random walk is not so naive it is quite appropriate to describe an assetprice behavior As a guide for selecting the smoothing constant it is suggested that 0L values close to 0 are selected if the series has small variations and values close to l are selected if the forecast values appear to depend on recent changes in actual values Usually the MSE or RMSR can be used as the criterion for selecting an appropriate smoothing constant For instance by assigning 0L values from 01 to 099 we select the value that produces the smallest MSE 23 Remarks on Simple Exponential Smoothing The model 1 aY 1 001 can be rewritten as If 1 lf otY l change in forecasting value is proportionate to the forecast error That is YH1 lf ot8 Exponential smoothing provides an effective mechanism for forecasting especially when we have only a few observations in hand for conducting the forecast process This method is appropriate for series that move randomly above and below a constant mean However if the series presents a trend or seasonal pattern some modification is required 3 Exponential Smoothing with Trend Holt s Model Holt s m J r quot 39 quot 39 model extends simple exponential smoothing to include a lineartrend component Accordingly Holt s model is appropriate for non stationary data We shall brie y present the model below Business Conditions amp Forecasting Dr Thomas C Chiang 31 The Holt s Model aY 1aF 11 311 312 Hm FM WM 313 Notations F 1 forecast value for period t1 made at time t which can be denoted as 1 1 K actual value in period t E forecast value for period t made by t J Tl trend a smoothing constant for the data 0lt a lt1 smoothing constant for the trend estimate 0lt lt1 m number of periods ahead to be forecast H Holt s forecast value ofperiod Hm Hm The model proposed by Holt contains two smoothing constants one for the level of the series and one for the trend In equation 311 the smoothing value F 1 is predicted based on the current observation and the previous smoothed value However the latter is adjusted by adding a trend factor The trend in equation 312 evolves by weighting the average of the recent change of the smoothed value and the previous trend Equation 313 is a forecast equation which is used to forecast in periods into the future by using both updated smoothing valueFM and trend estimate Tl derived from Equations 311 and 312 To conduct forecasts in this model in periods ahead we can follow the same procedure as that of simple exponentialsmoothing model What we have to add here is a trend variable The initial trend value is usually set at 0 the increment is advanced by 1 over time Here we need to search for two smoothing constants a and by using MSE or RMSE criterion 4 Winters Exponential Smoothing Due to the fact that previous models ignored the seasonal component Winters three parameter exponentialsmoothing model extends Holt s model by adding a seasonality factor which is itself smoothed Accordingly we have three smoothing parameters one for Business Conditions amp Forecasting Dr Thomas C Chiang actual data one for trend and one for seasonal factors Since a new variable is added to the system there are four equations in Winters model 41 The Winters Model E aKSp1aE1T1 411 412 T1 Ft E11 T1 413 Wm E mT1SW 414 Notations E smoothed value of the level of series for period t 11 H smoothed value for period t J K actual value in period t T trend estimate St seasonality estimate a smoothing constant for the data 0lt 0 lt1 smoothing constant for the trend estimate 0lt lt1 y smoothing constant for seasonality estimate 0ltlt1 p number of periods in seasonal cycle in number of periods ahead to be forecast W Hm Winters forecast for in periods into the future A special feature of this model is that the element of a seasonal factor is added to the model Equation 411 is similar to Holt s equation for smoothing the trend A minor difference is that seasonal uctuations in Y have been removed As can be seen in the first term Y is divided by Slip to adjust for seasonality The seasonality estimate and trend are updated in a fashion similar to the simple exponential process as described Equations 412 and 413 Finally Equation 414 is employed to compute the forecast for in periods into the future Business Conditions amp Forecasting Dr Thomas C Chiang S Adaptive Response Approach An alternative to simple exponential smoothing for stationary and nonseasonal time series is the adaptiveresponse approach to single exponential smoothing model in which an adaptive algorithm is used to determine a timevarying smoothing parameter Accordingly adaptive smoothing has the ability to adapt to a changing mean of an otherwise stationary and nonseasonal time series 51 The Adaptive Response model E1 aiY 1aF 511 3 a 512 W S e 1 SH 513 A lel1 z41 514 eYt Ft 515 The basic equation 511 for forecasting is generally the same as the simple exponential smoothing model represented by equation 211 The only difference is that the smoothed term a in Equation 211 is replaced by on and the later is adaptive over time governed by the value of the smoothed error divided by the absolute smoothed error as expressed by Equation 512 Like most exponential smoothing models both S not a notation for seasonality and A are smoothed out by using Equations 513 and 514 The forecasting procedure for this model can be proceeded recursively Given the values of Y andFt we can estimate 6 from 515 The 6 then is plugged into 513 and 514 Given the estimated value of we can obtain a by using 512 Finally we use at Y and E to predict FM by employing equation 511 52 Remarks The advantage of this model is that it allows the smoothing value to change over time However the underlying rationale for time varying is less clear If we want to have a time varying coefficient model to reflect the changing pattern the behavior of at can be specified as a random coefficient or autoregressive form depending on the nature of the process Another drawback for this model is that there is no explicit way to handle seasonality Thus in facing seasonal data the data must first be deseasonalized and then reseasonalized to Business Conditions amp Forecasting Dr Thomas C Chiang generate forecasts Adaptive smoothing is an alternative to Winters39 smoothing when handling seasonal time series A major aw with smoothing models is their inability to predict cyclical reversals in the data since forecasts depend solely on the past Perhaps even more pernicious is the possibility of spurious cycles since all smoothing models produce serially correlated forecasts Threaded question Assume you were to use a values of 01 05 and 095 in a simple exponential smoothing model How would these different a values weight past observations of the variable to be forecasted How would you know which of these a values provides the best forecasting model If the a 099 value provides the best forecast for your data at the current calculation would this imply that you should make forecasts in the future based on availability of new data Does exponential smoothing place more or less weight on the most recent data when compared with the movingaverage method What weight is applied to each observation in a movingaverage model Why is smoothing simple Holt s and Winters also termed exponential smoothing Assignment Consider the monthly stock index data for the US USsp market from l99501200006 available to the class Use Excel to calculate both the 12month movingaverage and simple exponential smoothing a 06 for these data and compare the forecasts of these two methods by calculating the rootmeansquared errors Make brief comments on your findings Note the data file is MonthlyiUSjPiMacroData and variable name is USsp E View Program Description and Forecasting by using Regression Models Thomas C Chiang EView Program Description for Forecasting 1 Data and Basic EView Procedure A Prepare the data in Excel worksheet Eviews can only read Excel worksheet not workbook The rst thing to do is to prepare data in Excel worksheet format 0 Create a new Excel workbook Assuming that the data have been copied from my Web page and stored in the Excel le called MonthlyiUSJPMacrodata in Drive C of your computer Don t forget to copy the date into the rst column next thing is to copy all data including 14 variables such as Trend treasury bill rate 10year government bond yield stock price seasonallyadjusted industrial production seasonallyadjusted M1 and consumer price index for both US and Japan from your le into a new workbook for EView operation B Retrieve data from DiscHard drive CCD 0 Double clicks on Eview to activate the EView program 0 Create a new work le by selecting FileNewVVorkfile 0 Select work le frequency as Monthly start date as 19801 and end date as 20007 this is for the monthly data then click OK 0 In the new work le window it automatically creates two series named c and resid 0 Import data from Excel by selecting ProcsImportRead Text Lotus Excel 0 Choose the le from your hard drive C MonthlyUSJPMacrodataxls 0 Select Order of data as By Observation The upperleft data cell is B2 starting cell of the data in Excel sheet enter the name for series or number of series as 14 since we already have 14 variables stored in Excel le 0 Select Save and save the work le as USJPwfl This is the data le ready for EView format C View the data 0 Select all the series by ViewSelect All except C RESIDshow 0 Or ViewSelect All except C RESID and Right click your mouse and select Openas Group 0 View the data by selecting individual series by using ViewSelect by lter and type series in Object Filter e g ussp uspi ustbr OKShowOK 0 To see the each series in different graphs ViewMultiple GraphsLine 0 To see all the series in one graph ViewGraphLine 0 To see the mean Standard deviation Skewness Kurtosis and Normality test by using JarqueBera ViewSelect filterussp uspi ustbr OKShowOK Viewdescriptive StatsCommon Sample D Transform the data take series ussp as example 0 The graph shows that series ussp displays an upward trend so we need to transform the data One common method is to take natural log or logdifference Double clicks on series ussp and in the window of series ussp select button Genr or go to ProcGenerate by Equation Enter equation lussplogussp and there is the new series lussp F i i I gt View the series and now it only has additive trend You can also generate the variable in the upper blank area You just type Genr lussplogussp You can also take a difference on the logussp to remove the trend or to generate a return on US stock index To do so you just do ProcGenerate by Equation and Enter equation dlussplussp lussp 1 Or type Genr dlussplussp lussp 1 in the upper blank area You should save the le after the new variables have been generated the generated variables will be included in the data base You can also copy the result to the word file Regression Methods Simple Regression Model Lecture 3 Table 35 It is generally believed that production will increase if the short term interest rate declines Since production takes time the production will have a lagged response to the shortterm interest rate changes Following this idea we regress USPI on oneperiod lagged USTBR see Lecture 3 equation 524 1 Model Representation USPI b0 b1USTBR1 s EView Procedures 2 Go to main menu QuickEstimate Equation and type the following variables Enter uspi c ustbr 1 in the Equation Speci cation window The first variable is the dependent variable the c refers to a constant term and the third one is the explanatory variable Note that 1 means oneperiod lag The Method as LS and Sample period is the 198601 199907 OK The result is as follows Dependent Variable USPI Method Least Squares Date 012004 Time 2344 Sample 198601 199907 Included observations 163 Variable Coefficient Std Error tStatistic Prob C 1087614 3518161 3091428 00000 USTBR1 2745703 0628115 4371337 00000 Rsquared 0106095 Mean dependent var 9390615 Adjusted Rsquared 0100543 SD dependent var 1225531 SE of regression 1162290 Akaike info criterion 7756026 Sum squared resid 2174978 Schwarz criterion 7793986 Log likelihood 6301161 Fstatistic 1910859 DurbinWatson stat 0005199 Prob Fstatistic 0000022 The above result can also be obtained if we go to the main menu in the top blank area and type smpl 198601 199907 IEnter This line specifies the sample period LS uspi c ustbr1 IEnter This line says that using LS method to regress uspi on c and lagged ustbr 95 0 Another method is that you do QuickEstimate Equation and type the following equation uspic1c2ustbr 1 in the Equation Speci cation window The method as LS and sample period is the 198601 199907 OK The result is as follows Dependent Variable USPI Method Least Squares Date 012004 Time 2330 Sample 198601 199907 Included observations 163 USPIC1C2USTBR 1 Coefficient Std Error tStatistic Prob C1 1087614 3518161 3091428 00000 02 2745703 0628115 4371337 00000 Rsquared 0106095 Mean dependent var 9390615 Adjusted Rsquared 0100543 S D dependent var 1225531 SE of regression 1162290 Akaike info criterion 7756026 Sum squared resid 2174978 Schwarz criterion 7793986 Log likelihood 6301161 DurbinWatson stat 0005199 0 You can see coefficients standard errors tstatistics and pvalues Rsquares DurbinWatson statistics for each independent variable among others 0 By substituting the estimated coefficient into the representation yields USPI 108761 2746USTBR1 s o The Rsquare is 0106 tratio for lagged USTBR is 437 which is rejected at the 1 level The coefficient is statistically significant However the DW is 0005 the absence of serial correlation is rejected Multiple Regression Model Lecture 4 Table 41 The theory predicts that production is not only in uenced by the lagged short term interest rate but also follows a stochastic trend We set up a multiple regression model as follows see equation 113 in Lecture 4 1 The Model Representaiton USPI o A USTBRH p USPIH 5 113 2 EView Procedures 0 Go to main menu QuickEstimate Equation and type uspi c ustbr 1 uspi 1 in the Equation Speci cation window The first variable is the dependent variable the c refers to a constant term and the third and fourth variables are the lagged explanatory variables 0 The Method as LS and Sample period is the 198601 199907 OK o The result is as follows Dependent Variable USPI Method Least Squares Date 012104 Time 1003 Sample 198601 199907 Included observations 163 Variable Coefficient Std Error tStatistic Prob C 0136881 0358230 0382103 07029 USTBR1 0073709 0025626 2876284 00046 USPI 1 1005564 0003068 3277973 00000 Rsquared 0998671 Mean dependent var 9390615 Adjusted Rsquared 0998654 SD dependent var 1225531 SE of regression 0449572 Akaike info criterion 1257190 Sum squared resid 3233835 Schwarz criterion 1314131 Log likelihood 9946103 Fstatistic 6011155 DurbinWatson stat 2009700 Prob Fstatistic 0000000 USPI 0137 00737USTBR110056USPI1 o By substituting the estimated coefficient into the representation yields USPI 0137 00737USTBR71 10056USP1 s o The Rsquare is 099 tratio for lagged USTBR is 288 which is rejected at the 1 level The tratio for the lagged USPI is 32780 which is highly signi cant 3 Procedure for Forecasting based on the estimated model given above 0 After estimating the LS equation you can click on quotforecastquot button In the quotforecastquot window the series name of the forecast quotuspifquot is shown Check the following in appropriate boxes Insert actuals for outofsample Dynamic Do Grap Forecast Evaluation Change Forecast Sample to 199908 200007 Click OK and there are a new forecast chart and a summary of forecasting statistics seeLet747uspiforecastitables41742 and forecast statistics given below A new series is generated in the main window named quotuspifquot You can verify the results by printing out the series of uspi and uspif Forecast Statistics for USPI Forecast USPIF Actual USPI Forecast sample 199908 200007 Included observations 12 Root Mean Squared Error 1118750 Mean Absolute Error 0840086 Mean Abs Percent Error 0669375 Theil Inequality Coefficient 0004536 Bias Proportion 0440628 Variance Proportion 0515408 Covariance Proportion 0043965 Note The Root Mean Squared Error differs slightly from the one in Table 42 due to the revisions of the last three observations by IMF and different decimal points being used in calculating RMSE
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