using ncss in busi460...ncss tip: log files. when you run an ncss procedure, it displays the output...

© 2005. UBC Real Estate Division

SUPPLEMENT: Using NCSS 2004 In BUSI 460 Table of Contents Introduction..................................................................................................................................... 1 LESSON 7 ...................................................................................................................................... 1

Page 7.3: Accessing NCSS Help menus ..................................................................................... 1 Page 7.7: Autocorrelation functions ........................................................................................... 1 Page 7.9: Graphing in NCSS ...................................................................................................... 1 Page 7.10: Generating autocorrelation functions........................................................................ 2 Page 7.11-12 ............................................................................................................................... 4 Page 7.22: Creating a Time Series (Differencing)...................................................................... 9 Page 7.29: Moving average....................................................................................................... 12 Page 7.45-46: Exponential Smoothing ..................................................................................... 14 Page 7.47: Figure A2 ................................................................................................................ 16 Page 7.47: Figure A3 ................................................................................................................ 17 Page 7.49-50: Partial Autocorrelation Function ....................................................................... 18 Page 7.51-52: Creating Seasonal ACF and PACF functions and graphs ................................. 21 Page 7.53-54 Creating an ARIMA Forecast in NCSS.............................................................. 24

LESSON 8 .................................................................................................................................... 26

Page 8.9: Generating Correlation Coefficients in NCSS.......................................................... 26 Page 8.12-13: Generating OLS Regression in NCSS (Model One) ......................................... 26 Page 8.16-17: Multicollinearity Statistics................................................................................. 28 Page 8.17 and 8.18: Generating OLS Regression in NCSS (Model Two) ............................... 29 Page 8.19-20: Generating OLS Regression in NCSS (Model Three)....................................... 30 Page 8.20: Generating OLS Regression Residuals in NCSS (Model Three) ........................... 31 Page 8.21-22: Autocorrelation and Partial Autocorrelation Coefficients for Residuals........... 32 Page 8.41: Generating Stepwise Regression in NCSS.............................................................. 38

© 2005. UBC Real Estate Division 1

Introduction The BUSI 460 course workbook explains and illustrates statistical forecasting using the SPSS software program. This document explains equivalent instructions for students who wish to instead use the NCSS software program. The document follows the BUSI 460 course workbook, explaining the equivalent NCSS commands wherever SPSS commands are given. It also provides and interprets the resulting NCSS output. LESSON 7 Page 7.3: Accessing NCSS Help menus

NCSS Instructions: Help Files

The steps required for the NCSS software will be illustrated in text boxes throughout this NCSS supplement. As a first step, you may wish to review the NCSS help menu. The help menu provides detailed information on all statistics generated by the software. In particular the tutorials may be extremely useful for understanding some of the output generated by the various commands. Choose Help from the top menu and select Help… Select Index and type Tutorial. A list of all tutorials available in NCSS is shown. Select the tutorial you are interested in and all necessary instructions will be explained. Page 7.7: Autocorrelation functions See the Page 7.10 instructions below, “Generating an Autocorrelation Function”. Page 7.9: Graphing in NCSS

NCSS Instructions: Generating a Scatter Graph in NCSS Choose Graphics from the top menu and select Scatter plots From the Scatter Plots window, select the Variable tab: For Horizontal variable(s), choose Year (type in variable name or click the down arrow at the right of the box to select variable from list) For Vertical variable(s), choose or type Pool_Permits From Scatter Plots window, select the Titles tab: For Top Title Line 1, type the appropriate title of your chart: Number of Pool Permits Issued by Year From Scatter Plots window, select the Vertical tab: For Label Text, type the appropriate title of for your vertical axis: # of Permits To run procedure, click on “Play” icon at top left of screen (sideways triangle) or press F9 key.


A separate NCSS Output window will open with the graphic. You may print this, save it, or cut and paste it into another document. You may also wish to save this using a Log file (see NCSS tip below). If you are finished with the Scatter Plots window, close it by clicking the X in the upper right corner. NCSS Tip: Log Files. When you run an NCSS procedure, it displays the output in an output window. Each new run clears this output and replaces it, so if you want to save a report you must do so before running a new procedure. In the “NCSS Output” window, if you select File in the top menu, you will find the “Log” features. You can save the existing data into a log file by selecting File…Save As and then specifying a name (e.g., “L7output.RTF”) and browsing to where you want the file stored. For further output, you can select File…Add Output to Log. Or, you may wish to select File…Toggle Auto-Log which will automatically save all output to the current log file. Page 7.10: Generating autocorrelation functions Once you have examined the graph of the data, the next step in the analysis is to generate the autocorrelation function and the Portmanteau Test statistic (usually referred to as the Box-Ljung statistic), to see if the data is indeed random. This is done using NCSS commands:

NCSS Instructions: Generating an Autocorrelation Function To generate an autocorrelation function and the Portmanteau Test statistic in NCSS: Choose Analysis from the top menu, select Forecasting/Time Series and select ARIMA (Box-Jenkins)* From the ARIMA Fit (Box-Jenkins) window, select the Variable tab: For Time Series Variable, choose Pool_Permit (either click on down arrow at right of line or type in variable name). Set Regular AR to 0, Regular Diff to None, and Regular MA to 0. Set Seasonal AR to 0, Seasonal Diff to None, and Seasonal MA to 0. The remaining boxes can stay with default values (e.g., Remove Mean checked, Seasons = 12, etc.). If you are uncertain you have the default values, you can press “Reset” button at bottom right of window and start again. From the ARIMA Fit (Box-Jenkins) window, select the Reports tab: For Forecast Report, select Omit For Number of Forecasts, select 1 (this is required to generate the autocorrelation function; no forecasts will be produced) Select Autocorrelation Report, Autocorrelation plot, and Portmanteau Test Report. Deselect the other options. Press F9 or click “Play” icon to run the procedure. Relevant excerpts are shown below.


* Note: autocorrelations and partial autocorrelations can also be found by selecting Analyze… Forecasting/Time Series…Autocorrelations. We have used “ARIMA (Box-Jenkins)” here in order to produce the additional significance tests, which will be explained further below. NCSS Tip: Saving Templates. If you are carrying out repeated commands with the same settings, you may wish to save a “Template”. This will allow you to load the template later and easily re-create output or run the same test on different databases in future. Click on Template tab. Under File Name, type in descriptive title you will remember, e.g., “autocorr&port no arima” Click Save Template. To load this template in future, you only need to open the ARIMA window, select Template tab, and click Load Template.

Table 2: Autocorrelations for Pool Permits

Autocorrelations of Residuals of Pool_Permit Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 -0.031901 4 -0.244124 7 -0.058093 10 0.098388 2 -0.213902 5 -0.186702 8 -0.038952 11 -0.043318 3 0.111148 6 -0.064473 9 0.215581 12 -0.040631 Significant if |Correlation|> 0.516398 Autocorrelation Plot Section

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Portmanteau Test Section Pool_Permit Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 0.02 0.891706 Adequate Model 2 2 0.92 0.632541 Adequate Model 3 3 1.18 0.758156 Adequate Model 4 4 2.56 0.633909 Adequate Model 5 5 3.45 0.631123 Adequate Model 6 6 3.57 0.735070 Adequate Model 7 7 3.67 0.816433 Adequate Model 8 8 3.73 0.880660 Adequate Model 9 9 5.70 0.769063 Adequate Model 10 10 6.20 0.798322 Adequate Model 11 11 6.32 0.851320 Adequate Model 12 12 6.46 0.891235 Adequate Model Page 7.11-12 The NCSS ARIMA (Box-Jenkins) command produces the following output: Autocorrelations of Residuals of Pool_Permit

• Lag • Correlation: autocorrelation coefficients

Autocorrelation Plot Section

• The plot of the autocorrelation function Portmanteau Test Section Pool_Permit

• Lag • DF: Degrees of freedom of the Portmanteau Test Value • Portmanteau Test Value (Box-Ljung statistic) • Prob Level: Level of significance value [called "Sig.(b)" in SPSS] of the Box-Ljung statistic • Decision (0.05): Conclusion for model significance at a 95% confidence level.

In the Autocorrelation of Residuals of Pool_Permit table, the "Lag" shows the selected or default number of lags for the autocorrelation function. NCSS provided only 12 lags, compared to 13 in SPSS. By default, NCSS produces three less lags than the total number of observations, such that there will always be a minimum of three observations to rely upon to determine the autocorrelation relationship (see next paragraph for an example). The “correlation” shows the values of the autocorrelation coefficients (r1 to r12) that form the autocorrelation function. Recall that autocorrelation at lag 1 (r1) is an average of the relationship between each pair of consecutive annual number of permits data throughout the entire data series. For our example, it is the average of the relationship between 2004 and 2003, 2003 and 2002, 2002 and 2001, 2001 and 2000, …, and 1991 and 1990. Autocorrelation at lag 2 is the average of the relationship between 2004 and 2002, 2003 and 2001, 2002 and 2000,…, and 1992 and 1990. Autocorrelation at lag 12 is the average relationship between 2004 and 1992, 2003 and 1991, and 2002 and 1990.


The pool permits data autocorrelation function show little relationship between any two data points. The autocorrelation at lag 1 shows that at any two successive years, there seems to be, on average, only a very slight drop in pool permits (r1=-0.032). If we looked the function at lag 2, the number of permits seem to be again negatively related (r2=-0.214); however, at lag 3, the number of permits seem to be positively related (r3=0.111). The correlation appears to switch between positive and negative relationships over the years, apparently randomly. As well, note that all of the coefficients are very low in value, indicating a weak relationship between the numbers of pool permits issued in successive years. The table in the Portmanteau Test Section Pool_Permit section provides information to test the randomness of a time series. Portmanteau Test Value (PTV) is the same as Box-Ljung Statistic (BLS) – sometimes also called the Box-Pierce-Ljung statistic. If the Portmanteau Test Value (PTV) is not significant at lags of approximately one-quarter of the sample size, then the series can be viewed as random. In other words, for this example there are 14 lags, so you would examine the PTV at 3 or 4 lags and verify its significance. The generally acceptable cut-off for a significance level is equal to or less than 0.05 (95% confidence level, CL). If the significance level is less than 0.05, the forecaster has less than a 5 percent chance of being wrong in stating that there is autocorrelation between the two time periods. In other words, you can reasonably assume there is a positive trend in the data if the lag autocorrelation is positive and the Prob Level value is less than 0.05. In our example, the PTV shows high numbers (0.6, 0.7, and 0.8) at all lags, including those one-quarter through the sample, indicating there is no significant relationship between permits data at all lags or time intervals (Prob Level greater than 0.05). This means that the correlation coefficients (“Correlations”) are not statistically significant – our conclusion is that the permits data are random over time, confirming our guess in viewing the graph. The significance levels definitively confirm there is no trend or cyclicality in this data. The pattern is stationary and random: white noise. We advise Anna-Marie that past pool sales are a poor predictor of future pool sales, and that she needs to identify more relevant and reliable predictors before proceeding to invest further. Page 7.14: Graph of Housing Under Construction -- see page 7.9 instructions above. Page 7.14: Creating Autocorrelations for July Housing data -- see page 7.10 instructions above.

Table 4: Autocorrelations For July Housing Construction Data Autocorrelations of Residuals of Housing_Under_Construction Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.781202 3 0.241675 5 -0.294550 7 -0.448593 2 0.533126 4 -0.070343 6 -0.433006 8 -0.376870 Significant if |Correlation|> 0.603023



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Portmanteau Test Section Housing_Under_Construction Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 8.73 0.003135 Inadequate Model 2 2 13.24 0.001331 Inadequate Model 3 3 14.29 0.002539 Inadequate Model 4 4 14.39 0.006154 Inadequate Model 5 5 16.46 0.005656 Inadequate Model 6 6 21.82 0.001306 Inadequate Model 7 7 29.01 0.000144 Inadequate Model 8 8 35.78 0.000019 Inadequate Model

.15: The PTV indicates that all lags are significant for the construction data, since the Prob Level is less than 0.05 for all. This confirms that the construction data exhibits a trend. Page 7.17: Creating bar chart in NCSS -- select Graphics…Charts…Bar Charts. Page 7.17: Creating Autocorrelations for Marriages data -- see page 7.10 instructions above.

Table 6: Autocorrelations for Marriages from 1995 to 2004 Autocorrelations of Residuals of Number_of_Marriages Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 -0.061126 11 -0.029145 21 -0.033717 31 0.000948 2 -0.851753 12 0.702855 22 -0.402967 32 0.193152 3 -0.027407 13 -0.042902 23 -0.008053 33 -0.017919 4 0.893673 14 -0.591982 24 0.383815 34 -0.135206 5 -0.057797 15 -0.021705 25 -0.028166 35 0.009990 6 -0.761754 16 0.606701 26 -0.305807 36 0.100998 7 -0.030113 17 -0.039651 27 -0.005479 37 -0.018021 8 0.798099 18 -0.494668 28 0.287490 9 -0.049791 19 -0.014426 29 -0.020517 10 -0.676436 20 0.496647 30 -0.221256


Significant if |Correlation|> 0.316228 Autocorrelation Plot Section

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Portmanteau Test Section Number_of_Marriages Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 0.16 0.688281 Adequate Model 2 2 32.23 0.000000 Inadequate Model 3 3 32.27 0.000000 Inadequate Model 4 4 69.54 0.000000 Inadequate Model 5 5 69.70 0.000000 Inadequate Model 6 6 98.37 0.000000 Inadequate Model 7 7 98.42 0.000000 Inadequate Model 8 8 131.86 0.000000 Inadequate Model 9 9 131.99 0.000000 Inadequate Model 10 10 157.62 0.000000 Inadequate Model 11 11 157.67 0.000000 Inadequate Model 12 12 187.31 0.000000 Inadequate Model 13 13 187.42 0.000000 Inadequate Model 14 14 210.06 0.000000 Inadequate Model 15 15 210.10 0.000000 Inadequate Model 16 16 235.86 0.000000 Inadequate Model 17 17 235.98 0.000000 Inadequate Model 18 18 254.66 0.000000 Inadequate Model 35 35 350.37 0.000000 Inadequate Model 36 36 354.65 0.000000 Inadequate Model 37 37 354.83 0.000000 Inadequate Model

Page 7.18: The PTV shows all autocorrelations as significant except at lag 1, which has a 68.8% [Prob Level = 0.688] chance of being zero rather than negative. Page 7.20: Creating Autocorrelations for Housing Under Construction (Monthly) -- see page 7.10 instructions above.

Table 7: Housing under Construction in Toronto, Monthly


Autocorrelations of Residuals of Housing Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.990410 13 0.800089 25 0.524823 37 0.245736 2 0.979369 14 0.779315 26 0.500523 38 0.225108 3 0.967889 15 0.757337 27 0.476285 39 0.205371 4 0.955233 16 0.735195 28 0.451976 40 0.185982 5 0.941456 17 0.712946 29 0.428489 41 0.166495 6 0.927264 18 0.689917 30 0.403694 42 0.147159 7 0.911197 19 0.666275 31 0.378739 43 0.128665 8 0.895092 20 0.642707 32 0.353447 44 0.110579 9 0.877518 21 0.619190 33 0.329884 45 0.093463 10 0.858560 22 0.595710 34 0.307628 46 0.076060 11 0.839796 23 0.572665 35 0.286784 47 0.059345 12 0.820425 24 0.548929 36 0.266334 48 0.042212 Significant if |Correlation|> 0.099875 Autocorrelation Plot Section

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Portmanteau Test Section Housing Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 396.30 0.000000 Inadequate Model 2 2 784.78 0.000000 Inadequate Model 3 3 1165.16 0.000000 Inadequate Model 4 4 1536.59 0.000000 Inadequate Model 5 5 1898.29 0.000000 Inadequate Model 6 6 2250.06 0.000000 Inadequate Model 7 7 2590.61 0.000000 Inadequate Model 8 8 2920.06 0.000000 Inadequate Model 9 9 3237.51 0.000000 Inadequate Model 10 10 3542.17 0.000000 Inadequate Model 11 11 3834.41 0.000000 Inadequate Model 12 12 4114.03 0.000000 Inadequate Model 13 13 4380.65 0.000000 Inadequate Model 14 14 4634.26 0.000000 Inadequate Model 15 15 4874.39 0.000000 Inadequate Model 16 16 5101.27 0.000000 Inadequate Model 17 17 5315.18 0.000000 Inadequate Model 36 36 7300.91 0.000000 Inadequate Model


37 37 7327.71 0.000000 Inadequate Model 38 38 7350.27 0.000000 Inadequate Model 39 39 7369.10 0.000000 Inadequate Model 40 40 7384.59 0.000000 Inadequate Model 41 41 7397.03 0.000000 Inadequate Model 42 42 7406.78 0.000000 Inadequate Model 43 43 7414.25 0.000000 Inadequate Model 44 44 7419.79 0.000000 Inadequate Model 45 45 7423.75 0.000000 Inadequate Model 46 46 7426.39 0.000000 Inadequate Model 47 47 7427.99 0.000000 Inadequate Model 48 48 7428.81 0.000000 Inadequate Model

Page 7.22: Creating a Time Series (Differencing)

NCSS Instructions: Creating a Time Series (Differencing) Note: the procedure below has already been run in the Monthly Housing database, and this database can be viewed for illustration. Transformation Method 1: Click on the Variable Info tab at the bottom left of the NCSS window. Under Name, for the next available variable type in name desired, e.g., laghousing. Under Transformation, type the formula desired: e.g., lag(housing) This formula saves all values for the variable lagged one period – in the data view window, all data will be moved down by one row. Now go to next variable available below laghousing, and type in the variable name desired: e.g., HousingDiff Under Transformation, type the formula desired: housing-laghousing or C2-C3 (if these variables were not renamed). This formula subtracts the current housing data from its lagged value one period prior. Choose Data from the top menu and select Recalc All to run the transformations (or click yellow calculator icon). Click on “Sheet1” at the bottom left of the NCSS window, and the lagged values and differenced values will appear in the result columns specified. Transformation Method 2: Select Data…Enter Transform. (Note: you can open the "Enter Transform" window by double clicking on the Transformation cell in the “Variable Info” window). For Result Variable, choose next available variable number ***important, otherwise you may transform over existing variables***. Note that you cannot name the resulting variable in the Transforming Window. Therefore, the laghousing variable will be C3. You can later rename the variable on the Variable Info page. Under Transformation Formula, type lag(housing) OR


Under Available Functions, scroll down to LAG and double click on it: Lagk(X) appears under Transformation Formula. Under Argument Variables, choose housing, and the X will be replaced with housing variable. Under Transformation Formula, change k to 1 (one). ***Ensure you have desired variable selected under Result Variable*** Click OK. Click on Variable Info tab at the bottom left of the NCSS window; you will see the formula beside the result variable selected. (if formula is beside C1 or C2, you forgot to specify Result Variable; delete the formula and start again!) Under Name, for the result variable type in name desired, e.g., laghousing. Return to Data…Enter Transform. For Result Variable, choose next available variable number. Under Transformation Formula, type housing-laghousing OR Under Argument Variables, choose housing. Under Transformation Formula, type – (minus) beside housing. Under Argument Variables, choose laghousing. The formula should now read: housing-laghousing or C2-C3 (if you are using numbers instead of names). ***Again, ensure you have desired variable selected under Result Variable*** Click OK. Click on Variable Info tab at the bottom left of the NCSS window to view the formula beside the result variable selected. Under Name, for the result variable type in name desired, e.g., housingdiff. Choose Data from the top menu and select Recalc All to run the transformations. Click on “Sheet1” at bottom left of NCSS window to view data. The lag values and differenced values will appear in the result columns specified.

Page 7.22: Creating a Line Graph in NCSS – Graphics…Charts… Xbar-R (Variables) Charts. Page 7.23: Creating an Autocorrelation Function for Differenced Data -- see page 7.10 instructions above.


Table 8: Housing under Construction in Toronto, Monthly Difference Autocorrelations of Residuals of HousingDiff Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.258394 13 0.096497 25 0.032072 37 0.064848 2 0.156045 14 0.092272 26 -0.023452 38 -0.048451 3 0.060803 15 -0.014988 27 0.026443 39 -0.060933 4 0.051378 16 -0.005051 28 -0.005997 40 -0.032683 5 0.016846 17 -0.037797 29 0.008349 41 -0.006483 6 0.091097 18 0.031699 30 0.030754 42 -0.044254 7 0.050010 19 0.019892 31 -0.002040 43 -0.037799 8 0.072690 20 -0.001445 32 -0.078256 44 -0.108268 9 0.096287 21 0.078414 33 -0.062545 45 -0.083272 10 0.071667 22 -0.011179 34 -0.049876 46 -0.080469 11 0.126028 23 0.097660 35 0.044282 47 0.010544 12 0.196617 24 0.089671 36 0.088838 48 0.079428 Significant if |Correlation|> 0.099875 Autocorrelation Plot Section

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Portmanteau Test Section HousingDiff Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 26.97 0.000000 Inadequate Model 2 2 36.84 0.000000 Inadequate Model 3 3 38.34 0.000000 Inadequate Model 4 4 39.41 0.000000 Inadequate Model 5 5 39.53 0.000000 Inadequate Model 6 6 42.92 0.000000 Inadequate Model 7 7 43.95 0.000000 Inadequate Model 8 8 46.12 0.000000 Inadequate Model 9 9 49.94 0.000000 Inadequate Model 10 10 52.07 0.000000 Inadequate Model 11 11 58.65 0.000000 Inadequate Model 12 12 74.71 0.000000 Inadequate Model 13 13 78.59 0.000000 Inadequate Model 14 14 82.14 0.000000 Inadequate Model 15 15 82.24 0.000000 Inadequate Model


41 41 110.00 0.000000 Inadequate Model 42 42 110.88 0.000000 Inadequate Model 43 43 111.52 0.000000 Inadequate Model 44 44 116.83 0.000000 Inadequate Model 45 45 119.98 0.000000 Inadequate Model 46 46 122.92 0.000000 Inadequate Model 47 47 122.97 0.000000 Inadequate Model 48 48 125.86 0.000000 Inadequate Model Page 7.24: Assigning Periodicity. This command is not needed in NCSS. Page 7.29: Moving average

NCSS Instructions: Creating a Moving Average Note: the procedure below has already been run in the Monthly Housing database, and this database can be viewed for illustration. Method 1: Click on the Variable Info tab at the bottom left of the NCSS window. In the new variable column, under Name type in the variable name desired, e.g., Moving4. Under Transformation, type the formula desired: e.g., mav4(housing) Choose Data from the top menu and select Recalc Current to run the selected transformation. Click on “Sheet1” at the bottom left of the NCSS window, and the differenced values will appear in the result column specified. Method 2: Choose Data from the top menu and select Enter Transform… In the Transforming window, click on the Transformation Formula box. Type mav4(housing) OR type mav4( and then select the desired variable under “Argument Variables”, and then type ) to close the equation. Make sure in the Result Variable box the next column name is shown, for example C5 Choose Data from the top menu and select Recalc Current to run the transformation. The moving average values will appear in the result column specified. Click on the Variable Info tab at the bottom left of the NCSS window. In the new variable column, under Name type in the variable name desired, e.g., Moving4.

Page 7.38-39: Autocorrelations for Second Naïve and Moving Average -- see page 7.10 instructions above.


Table 12: Autocorrelations Second Naïve Forecast

ARIMA Report Variable NaiveDiff Autocorrelations of Residuals of NaiveDiff Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 -0.524573 2 0.291738 3 -0.134005 4 -0.164900 Significant if |Correlation|> 0.755929 Portmanteau Test Section NaiveDiff Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 2.89 0.089167 Adequate Model 2 2 3.96 0.137948 Adequate Model 3 3 4.24 0.236236 Adequate Model 4 4 4.82 0.306745 Adequate Model

Table 14: Autocorrelations 3 Span Moving Average ARIMA Report

Variable threespan Autocorrelations of Residuals of threespan Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.477469 2 -0.072047 3 -0.396760 Significant if |Correlation|> 0.816497 Portmanteau Test Section threespan Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 2.19 0.139038 Adequate Model 2 2 2.25 0.324512 Adequate Model 3 3 4.77 0.189471 Adequate Model

Page 7.43: Detecting Heteroskedacity. Heteroskedacity can easily be detected by examining the line chart of the residuals from a regression model. First we need to develop a forecast model and generate the corresponding residuals. Next, we graph the residuals to see if the residuals have a pattern. The two graphs in Figure A1 were provided for illustration only – the SPSS steps were not shown, and the NCSS steps will not be shown here either. Page 7.44: Correcting Heteroskedacity.


NCSS Instructions: Correcting Heteroskedacity Choose Analysis from the top menu, select Forecasting/Time Series and select ARIMA Fit (Box-Jenkins). From the ARIMA Fit (Box-Jenkins) window, select the Variables tab: Under Time Series Variable, choose Marriages Under Regular AR, choose 1 Under Seasonal AR, choose 1 Make sure the other buttons are not activated under Variables. For example, Regular Diff and Seasonal Diff should be None; Regular MA and Seasonal MA should be 0 (zero). Ignore the Seasons value at 12 and the other values. Select the Reports tab: Under Forecast Report, select Omit Under Number of Forecasts, select 1 Make sure the other buttons are not activated under Report. Select the Storage tab: Under Forecast, select C3 Press F9 or play icon to run procedure The forecast will be generated in Column C3 (you can rename the variable if desired)

Page 7.45-46: Exponential Smoothing

NCSS Instructions: Exponential Smoothing (Holt’s Method) To generate an exponential smoothing forecast using NCSS: Choose Analysis from the top menu, select Forecasting/Time Series and select Exponential Smoothing - Trend Click on the Variable tab: For Time Series Variable(s), click or type Housing Under Forecast Method, select Holt’s Linear Trend. Under Search Method, select Search on MAPE. This will select the best alpha and beta using the minimum sum of squared error criterion. Under Number of Forecasts, type 19, since 19 months will take the series to December 2006. Click on the Reports tab: For Forecast Report, select Forecasts Only Click on Summary Report, Data Plot, and Residual Plot. If you want to store the forecast: Click on the Storage tab:


In the Forecasts box, select the next available column to store your forecast, e.g., C7. Either type in the variable or use the down arrow icon at the right of the box. Press F9 or “Play” icon to run the procedure.

Table A1: Exponential Smoothing (Holt’s Method) Using Lowest Mean Squared Error

Trend Report Forecast Summary Section Variable housing Number of Rows 401 Mean 25794.34 Pseudo R-Squared 0.991212 Mean Square Error 992023.2 Mean |Error| 760.5784 Mean |Percent Error| 3.296461 Forecast Method Holt's Linear Trend Search Iterations 93 Search Criterion Mean |Percent Error| Alpha 0.9999972 Beta 0.1826293 Intercept (A) 44761.74 Slope (B) -11.47068 Forecast and Residuals Plots

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411 40047.29 412 40035.82 413 40024.35 414 40012.88 415 40001.41 416 39989.94 417 39978.47 418 39967 419 39955.53 420 39944.05 Page 7.47: Figure A2

NCSS Instructions: Exponential Smoothing (Holt’s Method) with predetermined Alpha (1.0) and Beta Values (0.2)

Instructions continued from page 7.45. Return to Exponential Smoothing – Trend window. Entries are the same as previous, except for: On Variable tab: Under Search Method, select Specified Value. Under Alpha, type 1.0. Under Beta, type 0.2 If you want to store the forecast, then click on the Storage tab: In the Forecasts box, select the next available column to store your forecast, e.g., C8. Press F9 or “Play” icon to run the procedure.

Figure A2: Exponential Smoothing (Holt’s Method) Using Alpha = 1.0 and Beta =0.2 Forecast Summary Section Variable housing Number of Rows 401 Mean 25794.34 Pseudo R-Squared 0.991172 Mean Square Error 996571.4 Mean |Error| 761.1559 Mean |Percent Error| 3.297229 Forecast Method Holt's Linear Trend Search Iterations 0 Search Criterion None Alpha 1 Beta 0.2 Intercept (A) 45344.95 Slope (B) -12.92507 Forecast and Residuals Plots


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Forecasts Section Row Forecast No. housing 402 40149.07 403 40136.15 404 40123.23 405 40110.3 406 40097.38 407 40084.45 408 40071.52 409 40058.6 410 40045.68 411 40032.75 412 40019.82 413 40006.9 414 39993.97 415 39981.05 416 39968.13 417 39955.2 418 39942.27 419 39929.35 420 39916.42

Page 7.47: Figure A3

NCSS Instructions: Exponential Smoothing (Holt’s Method) with predetermined Alpha (0.3) and

Beta Values (0.3) Repeat instructions above, with exception of: Under Search Method, select Specified Value. Under Alpha, type 0.3. Under Beta, type 0.3. If you want to store the forecast, then click on the Storage tab: In the Forecasts box, select the next available column to store your forecast, e.g., C9.

Figure A3: Exponential Smoothing (Holt’s Method) Using Alpha = 0.3 and Beta =0.3


Forecast Summary Section Variable housing Number of Rows 401 Mean 25794.34 Pseudo R-Squared 0.979023 Mean Square Error 2367984 Mean |Error| 1199.885 Mean |Percent Error| 5.314497 Forecast Method Holt's Linear Trend Search Iterations 0 Search Criterion None Alpha 0.3 Beta 0.3 Intercept (A) 213561 Slope (B) -433.442 Forecast and Residuals Plots

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Forecasts Section Row Forecast No. housing 402 39317.31 403 38883.87 404 38450.42 405 38016.98 406 37583.54 407 37150.1 408 36716.66 409 36283.21 410 35849.77 411 35416.33 412 34982.89 413 34549.45 414 34116 415 33682.56 416 33249.12 417 32815.68 418 32382.23 419 31948.79 420 31515.35 Page 7.49-50: Partial Autocorrelation Function


NCSS Instructions: Generating a Autocorrelation Function (ACF) and a Partial Autocorrelation Function (PACF)

To generate an ACF and a PACF using NCSS: Under the Analysis menu, select Forecasting/Time Series and Autocorrelations. Under the Variable tab: For Time Series Variable(s), click or type Housing. For Regular Differencing, select None. For Seasonal Differencing, select None. For Seasons, leave at default setting 12. Ensure none of the three checkboxes on the right are selected. Under the Reports tab: For Autocorrelations, select 50. For Number of PACs, select 50. Click on Autocorrelation Report, Autocorrelation Plot, Partial Autocorrelation Report, and Partial Autocorrelation Plot. Press F9 or “Play” icon to run the procedure.

Figure A4 and A5: Autocorrelation Function and Partial Autocorrelation Function Data and Plot

for Monthly Construction in Toronto Autocorrelation Plot Section

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Autocorrelations of housing (0,0,12,0,0) Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.993971 14 0.903231 27 0.798879 40 0.704480 2 0.987938 15 0.895550 28 0.790946 41 0.698148 3 0.982088 16 0.887781 29 0.783326 42 0.691865 4 0.975900 17 0.879957 30 0.775218 43 0.685820 5 0.969389 18 0.871739 31 0.767095 44 0.679853 6 0.962744 19 0.863353 32 0.758839 45 0.674024 7 0.955576 20 0.855040 33 0.751076 46 0.667904 8 0.948496 21 0.846757 34 0.743672 47 0.661873 9 0.941013 22 0.838660 35 0.736699 48 0.655703 10 0.933207 23 0.830712 36 0.729973 49 0.649878 11 0.925648 24 0.822719 37 0.723382 50 0.644366 12 0.918127 25 0.814783 38 0.716919 13 0.910657 26 0.806846 39 0.710700 Significant if |Correlation|> 0.099875 Partial Autocorrelations of housing (0,0,12,0,0) Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.993971 14 0.000256 27 -0.006290 40 -0.010127 2 -0.003343 15 -0.022877 28 -0.004871 41 -0.019558 3 0.012167 16 -0.010592 29 0.019301 42 -0.008155 4 -0.031127 17 -0.011306 30 -0.046204 43 0.010868 5 -0.029711 18 -0.035058 31 -0.007201 44 0.003097 6 -0.015337 19 -0.018803 32 -0.016983 45 0.009907 7 -0.047028 20 -0.000119 33 0.039286 46 -0.026931 8 0.004734 21 0.001133 34 0.027854 47 0.004686 9 -0.037291 22 0.013725 35 0.034984 48 -0.017421 10 -0.027750 23 0.011360 36 0.020915 49 0.025111 11 0.017424 24 -0.003380 37 0.002342 50 0.019884 12 0.000674 25 0.000470 38 0.005017 13 0.005956 26 -0.004818 39 0.010953 Significant if |Correlation|> 0.099875 Data Plot Section

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Page 7.51-52: Creating Seasonal ACF and PACF functions and graphs

NCSS Instructions: Evaluating Seasonal ACF and PACF Functions and Graphs Generating ACF and PACF functions and the Portmanteau Test Statistics requires two separate steps in NCSS: Step 1: Generate ACF and a PACF functions: Under the Analysis menu, select Forecasting/Time Series and Autocorrelations Under the Variable tab: For Time Series Variable(s), click or type Housing. For Regular AR, select 0; for Regular Diff, select First; for Regular MA, select 0. For Seasonal AR, select 0; for Seasonal Diff, select none; for Seasonal MA, select 0. For Seasons, leave at default setting 12. Under the Reports tab: For Autocorrelations, select 72. For Number of PACs, select 72. Click on Autocorrelation Report, Autocorrelation Plot, Partial Autocorrelation Report, and Partial Autocorrelation Plot Press F9 or “Play” icon to run the procedure. Step 2: Generate Portmanteau Test Statistic: Under the Analysis menu, select Forecasting/Time Series and ARIMA (Box-Jenkins). Under the Variable tab: Click Reset in bottom right corner, to start with default settings. For Time Series Variable(s), click or type Housing. For Regular AR, select 0; for Regular Diff, select First; for Regular MA, select 0. For all other boxes, leave the default settings. Under the Reports tab: For Forecast Report, select Omit For Number of Forecasts, set to 1 (this is required to generate the autocorrelation function; no forecasts will be produced). For checkboxes, check only Portmanteau Test Report By default NCSS only provides 48 observations in the Portmanteau Test Report. As a result, the output from SPSS for the 60th and the 72nd month cannot be replicated. Press F9 or “Play” icon to run the procedure.


Table A2 and A3: Autocorrelation, Partial Autocorrelation and Function and Portmanteau Test Statistics


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Autocorrelations of housing (1,0,12,0,0) Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.258408 19 0.019947 37 0.064843 55 -0.042302 2 0.156061 20 -0.001377 38 -0.048458 56 -0.046371 3 0.060791 21 0.078501 39 -0.060948 57 -0.004884 4 0.051356 22 -0.011095 40 -0.032706 58 0.014414 5 0.016823 23 0.097754 41 -0.006494 59 0.048455 6 0.091062 24 0.089780 42 -0.044247 60 0.128568 7 0.049998 25 0.032184 43 -0.037790 61 0.008443 8 0.072666 26 -0.023340 44 -0.108257 62 -0.047699 9 0.096285 27 0.026558 45 -0.083271 63 -0.018348 10 0.071704 28 -0.005865 46 -0.080448 64 0.002338 11 0.126067 29 0.008463 47 0.010558 65 -0.038252 12 0.196645 30 0.030869 48 0.079459 66 0.018711 13 0.096516 31 -0.001950 49 0.035147 67 -0.017377 14 0.092294 32 -0.078153 50 -0.001242 68 -0.073482 15 -0.014939 33 -0.062477 51 -0.000974 69 -0.073903 16 -0.004992 34 -0.049834 52 -0.032337 70 -0.058435 17 -0.037760 35 0.044303 53 -0.031612 71 0.016761 18 0.031744 36 0.088844 54 0.010231 72 0.100398 Significant if |Correlation|> 0.100000 Partial Autocorrelations of housing (1,0,12,0,0) Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.258408 19 0.001011 37 0.007145 55 -0.030090 2 0.095675 20 -0.052444 38 -0.064832 56 0.002265 3 -0.000434 21 0.074453 39 -0.057058 57 0.030860 4 0.024130 22 -0.069580 40 0.029899 58 0.035209 5 -0.006655 23 0.088147 41 0.020219 59 0.011566 6 0.086973 24 0.036274 42 -0.064972 60 0.089483 7 0.008766 25 -0.036256 43 -0.002006 61 -0.075841 8 0.041448 26 -0.036359 44 -0.085627 62 -0.061818 9 0.066555 27 0.039924 45 -0.019294 63 0.031363 10 0.019031 28 0.006894 46 -0.020762 64 0.005955 11 0.094900 29 0.013151 47 0.038685 65 -0.027248 12 0.141303 30 0.007246 48 0.065051 66 0.005783 13 -0.006073 31 -0.015521 49 -0.017851 67 -0.000351 14 0.031882 32 -0.111420 50 0.007325 68 -0.071593


15 -0.076932 33 -0.052087 51 0.027014 69 -0.036331 16 -0.012211 34 -0.007477 52 -0.002127 70 -0.027183 17 -0.046227 35 0.038854 53 0.018093 71 0.009280 18 0.020135 36 0.080831 54 0.024274 72 0.058814 Significant if |Correlation|> 0.100000 Data Plot Section

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Portmanteau Test Section housing Portmanteau Prob Lag DF Test Value Level Decision (0.05) 1 1 26.97 0.000000 Inadequate Model 2 2 36.84 0.000000 Inadequate Model 3 3 38.34 0.000000 Inadequate Model 4 4 39.41 0.000000 Inadequate Model 5 5 39.53 0.000000 Inadequate Model 6 6 42.92 0.000000 Inadequate Model 7 7 43.95 0.000000 Inadequate Model 8 8 46.12 0.000000 Inadequate Model 9 9 49.94 0.000000 Inadequate Model 10 10 52.07 0.000000 Inadequate Model 11 11 58.65 0.000000 Inadequate Model 12 12 74.71 0.000000 Inadequate Model 13 13 78.59 0.000000 Inadequate Model 14 14 82.14 0.000000 Inadequate Model 15 15 82.24 0.000000 Inadequate Model 16 16 82.25 0.000000 Inadequate Model 17 17 82.85 0.000000 Inadequate Model 18 18 83.27 0.000000 Inadequate Model 19 19 83.44 0.000000 Inadequate Model 20 20 83.44 0.000000 Inadequate Model 21 21 86.05 0.000000 Inadequate Model 22 22 86.11 0.000000 Inadequate Model 23 23 90.19 0.000000 Inadequate Model 24 24 93.63 0.000000 Inadequate Model 25 25 94.07 0.000000 Inadequate Model 26 26 94.31 0.000000 Inadequate Model 27 27 94.61 0.000000 Inadequate Model 28 28 94.63 0.000000 Inadequate Model 29 29 94.66 0.000000 Inadequate Model 30 30 95.07 0.000000 Inadequate Model 31 31 95.07 0.000000 Inadequate Model


32 32 97.76 0.000000 Inadequate Model 33 33 99.47 0.000000 Inadequate Model 34 34 100.57 0.000000 Inadequate Model 35 35 101.43 0.000000 Inadequate Model 36 36 104.93 0.000000 Inadequate Model 37 37 106.80 0.000000 Inadequate Model 38 38 107.84 0.000000 Inadequate Model 39 39 109.50 0.000000 Inadequate Model 40 40 109.98 0.000000 Inadequate Model 41 41 110.00 0.000000 Inadequate Model 42 42 110.88 0.000000 Inadequate Model 43 43 111.52 0.000000 Inadequate Model 44 44 116.83 0.000000 Inadequate Model 45 45 119.98 0.000000 Inadequate Model 46 46 122.92 0.000000 Inadequate Model 47 47 122.97 0.000000 Inadequate Model 48 48 125.86 0.000000 Inadequate Model Page 7.53-54 Creating an ARIMA Forecast in NCSS

NCSS Instructions: Creating an ARIMA Forecast To generate a forecast for a single time series in NCSS: Under the Analysis menu, select Forecasting/Time Series and ARIMA (Box-Jenkins) Under the Variable tab: For Time Series Variable(s), click or type Housing For Regular AR, select 1; for Regular Differencing, select First. For Regular MA, set to 0; for Seasonal AR, select 1. For Seasonal Differencing, select None. For all other boxes, leave the default settings. Under the Reports tab: For Forecast Report, select Forecasts Only. For Number of Forecasts, set to 19 (this is required to generate the forecast to December 2006). Press F9 or “Play” icon to run the procedure.


Table A4: ARIMA Model Parameters Model Description Section Series housing Model Regular(1,1,0) Seasonal(1,0,0) Seasons = 12 Observations 401 Iterations 3 Pseudo R-Squared 99.188503 Residual Sum of Squares 3.673375E+08 Mean Square Error 922958.6 Root Mean Square 960.7073 Model Estimation Section Parameter Parameter Standard Prob Name Estimate Error T-Value Level AR(1) 0.2406292 4.849292E-02 4.9622 0.000001 SAR(1) 0.197394 4.863813E-02 4.0584 0.000049

Table A5: Housing Under Construction: ARIMA Forecast Data

Forecast Section of housing Row Date Forecast Lower 95% Limit Upper 95% Limit 402 34 6 40243.5 37243.1 43243.9 403 34 7 40284.8 36414.3 44155.4 404 34 8 40814.5 36222.3 45406.7 405 34 9 40785.6 35567.7 46003.6 406 34 10 40820.6 35043.7 46597.5 407 34 11 40757.4 34470.9 47044.0 408 34 12 40673.2 33915.2 47431.1 409 35 1 40371.9 33173.5 47570.4 410 35 2 40252.7 32639.1 47866.3 411 35 3 39760.6 31753.4 47767.8 412 35 4 40239.3 31857.0 48621.6 413 35 5 40281.1 31427.1 49135.1 414 35 6 40297.2 30967.6 49626.8 415 35 7 40305.4 30516.8 50094.0 416 35 8 40409.9 30181.4 50638.4 417 35 9 40404.2 29753.6 51054.8 418 35 10 40411.1 29354.5 51467.8 419 35 11 40398.7 28950.3 51847.0 420 35 12 40382.0 28554.9 52209.2


LESSON 8 Page 8.9: Generating Correlation Coefficients in NCSS

NCSS Instructions: Generating Correlation Coefficients To generate a correlation coefficient table in NCSS: Choose Analysis from the top menu, select Multivariate Analysis and select Correlation Matrix. Under the Variable tab: For Correlation Variables, choose Housing, Mort5yr. DispInc, tbill, and unemp_rt For Correlation Type, choose Pearson Product Moment For Missing Value Removal, choose Row Wise Under Reports tab: For Report Format, choose Short For Variable Names, choose Labels Press F9 or “Play” icon to run the procedure. Table 1: Correlation Coefficients Pearson Correlations Section (Row-Wise Deletion) Housing Starts 5 Yr Mortgage Disp Income T-Bill Rate UnempRate Housing Starts 1.000000 -0.873582 0.918211 -0.820555 -0.051211 5 Yr Mortgage -0.873582 1.000000 -0.693752 0.957384 0.207593 Disp Income 0.918211 -0.693752 1.000000 -0.647923 -0.117374 T-Bill Rate -0.820555 0.957384 -0.647923 1.000000 0.072144 Unemploy Rate -0.051211 0.207593 -0.117374 0.072144 1.000000 Cronbachs Alpha = 0.018920 Standardized Cronbachs Alpha =- 0.903782 Page 8.12-13: Generating OLS Regression in NCSS (Model One)

NCSS Instructions: Generating OLS Regression To generate an OLS regression in NCSS: Choose Analysis from the top menu, select Regression/Correlation and select Multiple Regression Under the Variable tab: For Y: Dependent Variables(s), choose Housing For X’s: Numeric Independent Variable(s), choose Mort5yr, DispInc, tbill, and unemp_rt Make sure the other buttons are not activated.


Under the Reports tab: Under Select a Group of Reports and Plots, choose Display only those items that are checked below. Select Run Summary, Durbin-Watson, ANOVA Summary, and Coefficient boxes. You may deselect Show All Rows. Under the Storage tab: Make sure none of the buttons are activated Press F9 or “Play” icon to run the procedure. Note: To fully appreciate all of the information that can be obtained from a multiple regression procedure and the output that is generated, we recommend that you use the "Help" command in NCSS. Choose Help...Help…Index…Tutorial - Multiple Regression.

Table 2: Model One Summary Multiple Regression Report

Run Summary Section Parameter Value Parameter Value Dependent Variable Housing Rows Processed 30 Number Ind. Variables 4 Rows Filtered Out 0 Weight Variable None Rows with X's Missing 0 R2 0.9715 Rows with Weight Missing 0 Adj R2 0.9669 Rows with Y Missing 0 Coefficient of Variation 0.0177 Rows Used in Estimation 30 Mean Square Error 75.28132 Sum of Weights 30.000 Square Root of MSE 8.676481 Completion Status Normal Completion Ave Abs Pct Error 1.320 Serial Correlation of Residuals Section Serial Serial Serial Lag Correlation Lag Correlation Lag Correlation 1 0.1840 9 -0.2782 17 0.2154 2 -0.0786 10 -0.3148 18 0.0126 3 0.1216 11 0.0432 19 -0.0114 4 -0.0818 12 0.0840 20 0.0012 5 0.0968 13 -0.0033 21 -0.0012 6 0.0363 14 0.0083 22 0.0045 7 -0.3964 15 -0.0927 23 -0.0290 8 -0.1708 16 0.1705 24 0.0095 Above serial correlations significant if their absolute values are greater than 0.365148 Durbin-Watson Test For Serial Correlation Did the Test Reject Parameter Value H0: Rho(1) = 0? Durbin-Watson Value 1.6032 Prob. Level: Positive Serial Correlation 0.1350 Yes Prob. Level: Negative Serial Correlation 0.6117 No


Table 3: Model One ANOVA (b) Multiple Regression Report

Analysis of Variance Section Sum of Mean Prob Power Source DF R2 Squares Square F-Ratio Level (5%) Intercept 1 7233060 7233060 Model 4 0.9715 64082.09 16020.52 212.809 0.0000 1.0000 Error 25 0.0285 1882.033 75.28132 Total (Adjusted) 29 1.0000 65964.13 2274.625

Table 4: Model One Coefficients

Multiple Regression Report Regression Coefficient Section Independent Regression Standard Lower Upper Standardized Variable Coefficient Error 95% C.L. 95% C.L. Coefficient Intercept 375.8749 21.6571 331.2713 420.4785 0.0000 DispInc 0.0055 0.0004 0.0046 0.0064 0.5834 mort5yr -12.3640 2.1556 -16.8036 -7.9244 -0.8109 tbill 4.0818 1.6676 0.6473 7.5163 0.3221 unemp_rt 4.9154 1.1734 2.4988 7.3321 0.1624 Note: The T-Value used to calculate these confidence limits was 2.060. Page 8.16-17: Multicollinearity Statistics

NCSS Instructions: Generating OLS Regression Multicollinearity Statistics

Instructions continued from page 8.12 above: Under the Reports tab: Select Multicollinearity box. Press F9 or “Play” icon to run the procedure.

Table 5: Model One Multicollinearity Statistics Multicollinearity Section Variance R2 Diagonal Independent Inflation Versus of X'X Variable Factor Other I.V.'s Tolerance Inverse DispInc 1.9528 0.4879 0.5121 2.621961E-09 mort5yr 17.5155 0.9429 0.0571 6.172447E-02 tbill 15.1724 0.9341 0.0659 3.694038E-02 unemp_rt 1.3164 0.2404 0.7596 1.828916E-02


Page 8.17 and 8.18: Generating OLS Regression in NCSS (Model Two)

NCSS Instructions: Generating OLS Regression: Model Two

Instructions continued from page 8.12 above: Choose Analysis from the top menu, select Regression/Correlation and select Multiple Regression Under Variable tab: For Y: Dependent Variables(s), choose Housing For X’s: Numeric Independent Variable(s), choose Mort5yr. DispInc, and unemp_rt (remove tbill). Under Reports tab: Select Run Summary, Coefficient, ANOVA Summary, Durbin-Watson, and Multicollinearity boxes. Press F9 or “Play” icon to run the procedure.

Table 6, 7, and 8: Model Two Statistics

Multiple Regression Report Run Summary Section Parameter Value Parameter Value Dependent Variable Housing Rows Processed 30 Number Ind. Variables 3 Rows Filtered Out 0 Weight Variable None Rows with X's Missing 0 R2 0.9646 Rows with Weight Missing 0 Adj R2 0.9606 Rows with Y Missing 0 Coefficient of Variation 0.0193 Rows Used in Estimation 30 Mean Square Error 89.73295 Sum of Weights 30.000 Square Root of MSE 9.472748 Completion Status Normal Completion Ave Abs Pct Error 1.525 Regression Coefficient Section Independent Regression Standard Lower Upper Standardized Variable Coefficient Error 95% C.L. 95% C.L. Coefficient Intercept 363.4462 22.9855 316.1987 410.6936 0.0000 DispInc 0.0056 0.0005 0.0046 0.0066 0.5957 mort5yr -7.3964 0.7932 -9.0269 -5.7659 -0.4851 unemp_rt 3.6150 1.1422 1.2671 5.9629 0.1194 Note: The T-Value used to calculate these confidence limits was 2.056. Analysis of Variance Section Sum of Mean Prob Power Source DF R2 Squares Square F-Ratio Level (5%) Intercept 1 7233060 7233060 Model 3 0.9646 63631.07 21210.36 236.372 0.0000 1.0000 Error 26 0.0354 2333.057 89.73295 Total (Adjusted) 29 1.0000 65964.13 2274.625


Serial Correlation of Residuals Section Serial Serial Serial Lag Correlation Lag Correlation Lag Correlation 1 0.4156 9 -0.3738 17 0.0855 2 0.1233 10 -0.4036 18 0.0715 3 0.0269 11 -0.0694 19 0.1336 4 -0.0741 12 0.0570 20 0.0758 5 -0.0069 13 0.0259 21 -0.0293 6 -0.0040 14 -0.0301 22 -0.0537 7 -0.2143 15 -0.2138 23 -0.0217 8 -0.1318 16 0.0210 24 0.0380 Above serial correlations significant if their absolute values are greater than 0.365148 Durbin-Watson Test For Serial Correlation Did the Test Reject Parameter Value H0: Rho(1) = 0? Durbin-Watson Value 1.1296 Prob. Level: Positive Serial Correlation 0.0045 Yes Prob. Level: Negative Serial Correlation 0.9543 No Multicollinearity Section Variance R2 Diagonal Independent Inflation Versus of X'X Variable Factor Other I.V.'s Tolerance Inverse DispInc 1.9306 0.4820 0.5180 2.592224E-09 mort5yr 1.9898 0.4974 0.5026 7.011951E-03 unemp_rt 1.0465 0.0445 0.9555 1.453978E-02

Page 8.19-20: Generating OLS Regression in NCSS (Model Three). Instructions continued from page 8.12 above: Remove unemployment rate from “X’s: Numeric Independent Variable(s)”.

Table 9, 10, and 11: Model Three Statistics Multiple Regression Report

Run Summary Section Parameter Value Parameter Value Dependent Variable Housing Rows Processed 30 Number Ind. Variables 2 Rows Filtered Out 0 Weight Variable None Rows with X's Missing 0 R2 0.9510 Rows with Weight Missing 0 Adj R2 0.9474 Rows with Y Missing 0 Coefficient of Variation 0.0223 Rows Used in Estimation 30 Mean Square Error 119.6985 Sum of Weights 30.000 Square Root of MSE 10.94068 Completion Status Normal Completion Ave Abs Pct Error 1.766


Regression Coefficient Section Independent Regression Standard Lower Upper Standardized Variable Coefficient Error 95% C.L. 95% C.L. Coefficient Intercept 388.7279 24.8927 337.6522 439.8035 0.0000 DispInc 0.0057 0.0006 0.0045 0.0068 0.6018 mort5yr -6.9536 0.9018 -8.8039 -5.1033 -0.4561 Note: The T-Value used to calculate these confidence limits was 2.052. Analysis of Variance Section Sum of Mean Prob Power Source DF R2 Squares Square F-Ratio Level (5%) Intercept 1 7233060 7233060 Model 2 0.9510 62732.27 31366.13 262.043 0.0000 1.0000 Error 27 0.0490 3231.861 119.6985 Total (Adjusted) 29 1.0000 65964.13 2274.625 Serial Correlation of Residuals Section Serial Serial Serial Lag Correlation Lag Correlation Lag Correlation 1 0.5090 9 -0.4075 17 -0.1140 2 0.1980 10 -0.5085 18 -0.0147 3 0.0802 11 -0.3018 19 0.1512 4 0.0505 12 -0.1962 20 0.1290 5 0.2072 13 -0.1042 21 0.0519 6 0.2165 14 -0.1067 22 0.0275 7 -0.0649 15 -0.3178 23 0.0457 8 -0.0979 16 -0.2177 24 0.0875 Above serial correlations significant if their absolute values are greater than 0.365148 Durbin-Watson Test For Serial Correlation Did the Test Reject Parameter Value H0: Rho(1) = 0? Durbin-Watson Value 0.9234 Prob. Level: Positive Serial Correlation 0.0002 Yes Prob. Level: Negative Serial Correlation 0.9974 No Multicollinearity Section Variance R2 Diagonal Independent Inflation Versus of X'X Variable Factor Other I.V.'s Tolerance Inverse DispInc 1.9279 0.4813 0.5187 2.588517E-09 mort5yr 1.9279 0.4813 0.5187 6.793771E-03

Page 8.20: Generating OLS Regression Residuals in NCSS (Model Three) Footnote 9 explained why storing the residuals from a model is necessary (residuals are the “error” from subtracting actual observation from the model prediction).


NCSS Instructions: Generating OLS Regression: Residuals

Instructions continued from page 8.19 above: Open Multiple Regression module. Under the Reports tab: Make sure none of the buttons are activated Under the Storage tab: For Storage Option, select Store in Empty Columns Only In the list of check boxes, select Residuals (for interest, you may wish to select Predicted Y as well, to see how the model predicts values). The Residuals will be stored in the next available Column (C7). If you forget to unclick Residuals in any subsequent runs, NCSS will generate more residual variables in the subsequent columns (C8, C9, etc.). If this occurs, you can simply delete the unnecessary columns in the “Sheet1” window. Press F9 or “Play” icon to run the procedure. Return to “Sheet1” to confirm the variable(s) were added. You may wish to name them, to avoid confusion later, e.g., “Res” and “Pred”.

Model Three Residuals Generated

Multiple Regression Report Storage Variables Section Stored First Last Item Variable Variable Residuals C9 This report lists the variables on the database into which various statistics were stored.

Page 8.21-22: Autocorrelation and Partial Autocorrelation Coefficients for Residuals

NCSS Instructions: Generating Autocorrelation Coefficients and Partial Autocorrelation Coefficients For Residuals (Model Three)

Choose Analysis from the top menu, select Forecasting/Time Series and Autocorrelations. Under the Variable tab: For Time Series Variable, choose C7 (or whichever variable you used to store residuals). Set Regular Differencing and Seasonal Differencing to None. Under the Reports tab: For Autocorrelations, select 16 For Number of PACs, select 16 For the check boxes below, select Autocorrelation Report and Partial Autocorrelation Report and uncheck the rest. Press F9 or “Play” icon to run the procedure.


Tables 12 and 13: Autocorrelation and Partial Autocorrelation Report Autocorrelation Report

Variable C7 (0,0,12,0,0) Autocorrelations of C7 (0,0,12,0,0) Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.509021 5 0.207177 9 -0.407462 13 -0.104245 2 0.197972 6 0.216528 10 -0.508486 14 -0.106656 3 0.080221 7 -0.064912 11 -0.301808 15 -0.317752 4 0.050514 8 -0.097896 12 -0.196237 16 -0.217696 Significant if |Correlation|> 0.365148 Partial Autocorrelations of C7 (0,0,12,0,0) Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.509021 5 0.237518 9 -0.520568 13 0.104904 2 -0.082509 6 0.009375 10 -0.217849 14 0.026126 3 0.017847 7 -0.299056 11 0.008200 15 0.011339 4 0.020920 8 0.079337 12 -0.017137 16 -0.029269 Significant if |Correlation|> 0.365148 Page 8.20-26: Since we have autocorrelation in the residuals, a correction procedure is necessary if the model is to be used for forecasting or for other research. There are several techniques available that will correct for autocorrelation in a regression. Unfortunately, NCSS does not have any AR(1) corrective procedure written into the software. The following instructions to correct for autocorrelation are based on the Cochrane-Orcutt approach. Because this procedure is not available in NCSS, the instructions show a manual method using linear multiple regression technique. The manual method for Cochrane-Orcutt is explained in Lesson 8’s Appendix 2 on page 8.39. Below we will illustrate how to carry out this process in NCSS. Note: on page 8.39 of the course workbook, Equation 1 should start with Y, not X:

ttt XY εβα ++= Step 1. Generate the residuals from the housing starts model that has autocorrelation. This is the term tε in the following formula: ttt XY εβα ++= The instructions earlier in this supplement for Page 8.20 told students to save the model residual values into a variable and name this “Res”. Step 2. Generate a new variable with lagged values of the residual. Click on the “Variable Info” window and go to the next available row (likely C8, below Res). Under “Name”, type in ResLag. In the Transformation column for ResLag, type lag(Res) and run the transformation (either Run Select, Run All, or yellow calculator icon). Step 3. Generate Rho ρ from the regression. Mathematically, Rho ρ is found from this equation:

1t t tε ρε υ−= + In NCSS, this can be calculated by running a regression of Res (ε ) as the Y variable and Reslag ( 1−tε ) as the X variable. Choose Analysis from the top menu and select Regression/Correlation and Linear Regression


From the Linear Regression window, select the Variables tab: Under Y: Dependent Variables(s), choose Res Under X: Dependent Variables(s), choose ResLag Click on Remove Intercept Click on the Reports tab and check only Run Summary. Press F9 or click Play arrow to run the report.

Linear Regression Report Y = Res X = Reslag Run Summary Section Parameter Value Parameter Value Dependent Variable Res Rows Processed 30 Independent Variable Reslag Rows Used in Estimation 29 Frequency Variable None Rows with X Missing 0 Weight Variable None Rows with Freq Missing 0 Intercept 0.0000 Rows Prediction Only 0 Slope 0.5121 Sum of Frequencies 29 R-Squared 0.2751 Sum of Weights 29.0000 Correlation 0.5245 Coefficient of Variation 19.8259 Mean Square Error 79.2778 Square Root of MSE 8.903809

The slope is the estimated Rho = 0.5121. Note this is very close to the Rho(0)=0.518 from Table 14 on page 8.23 of the workbook.

Step 4. Transform the original forecasting equation using the calculated Rho value (∧ρ ).

Mathematically, this looks as follows: *1 0 1 1[ ] (1 ) [ ]t t t t tY Y b b X Xρ ρ ρ ν

∧ ∧ ∧

− −− = − + − + Since the Model Three forecast has three variables, Housing, Mort5yr, and DispInc, we need to create three new variables:

NewHouse = 1sinsin −

∧− tt gHougHou ρ

NewMort = 155 −

∧− tt yrMortyrMort ρ

NewDInc = 1−

∧− tt DispincDispInc ρ

In NCSS these are generated as follows. First you create the lag of the variable, then the transformed variable using the Rho correction. (Note: if you transform these in “Enter Transform”, you will want to run each separately, because you cannot run a Lag transformation together with others. Alternatively, if you transform them in “Variable Info” you can run them all at once).

HouseLag lag(Housing) NewHouse Housing-(HouseLag*0.5121)

MortLag lag(mort5yr) NewMort Mort5yr-(MortLag*0.5121)

DIncLag lag(dispInc) NewDInc DispInc-(DIncLag*0.5121)


Step 5. Run the housing starts multiple regression using the new variables. Choose Analysis from the top menu, select Regression/Correlation and Multiple Regression From the Multiple Regression window, select the Variables tab: Under Y: Dependent Variables(s), choose NewHouse Under X’s: Numeric Independent Variable(s), choose NewMort and NewDInc Make sure the other buttons are not activated. Select the Reports tab: Check Run Summary, Coefficient, ANOVA Summary, Durbin-Watson and Multicollinearity Select the Storage tab: Make sure none of the buttons are activated Run the regression by pressing F9 or clicking the Play arrow.

Multiple Regression Report Run Summary Section Parameter Value Parameter Value Dependent Variable NewHouse Rows Processed 32 Number Ind. Variables 2 Rows Filtered Out 3 Weight Variable None Rows with X's Missing 0 R2 0.8941 Rows with Weight Missing 0 Adj R2 0.8860 Rows with Y Missing 0 Coefficient of Variation 0.0361 Rows Used in Estimation 29 Mean Square Error 77.15613 Sum of Weights 29.000 Square Root of MSE 8.783855 Completion Status Normal Completion Ave Abs Pct Error 2.886 Regression Equation Section Regression Standard T-Value Reject Power Independent Coefficient Error to test Prob H0 at of Test Variable b(i) Sb(i) H0:B(i)=0 Level 5%? at 5% Intercept 217.1817 17.3052 12.550 0.0000 Yes 1.0000 NewDinc 0.0044 0.0009 5.118 0.0000 Yes 0.9985 NewMort -8.5371 1.0522 -8.113 0.0000 Yes 1.0000 Analysis of Variance Section Sum of Mean Prob Power Source DF R2 Squares Square F-Ratio Level (5%) Intercept 1 1720773 1720773 Model 2 0.8941 16943.98 8471.989 109.803 0.0000 1.0000 Error 26 0.1059 2006.059 77.15613 Total (Adjusted) 28 1.0000 18950.04 676.787


Multiple Regression Report Dependent NewHouse Serial Correlation of Residuals Section Serial Serial Serial Lag Correlation Lag Correlation Lag Correlation 1 0.0261 9 -0.2371 17 0.0817 2 -0.0348 10 -0.3626 18 0.0420 3 0.0972 11 -0.0584 19 0.0314 4 -0.1182 12 0.0042 20 0.0473 5 0.1002 13 -0.0671 21 0.0248 6 0.2071 14 -0.0042 22 0.0761 7 -0.3095 15 -0.2443 23 0.0038 8 0.1301 16 0.0617 24 0.0388 Above serial correlations significant if their absolute values are greater than 0.371391 Durbin-Watson Test For Serial Correlation Did the Test Reject Parameter Value H0: Rho(1) = 0? Durbin-Watson Value 1.9221 Prob. Level: Positive Serial Correlation 0.3273 No Prob. Level: Negative Serial Correlation 0.4923 No

Similar to the discussion on page 8.24, these results show a much improved Durbin Watson statistic at 1.922, compared to the original value of 0.923 for the uncorrected model. This indicates that the first order transformation has reduced the autocorrelation to an extent that the model is good for forecasting. Step 6. Generate a forecast. The final step is to generate the housing forecast – after all of our technical procedures, we must keep in mind this is our ultimate objective! The coefficients were found from estimating the NewHouse variable, not Housing. Since we are

forecasting Housing and not [ 1sinsin −

∧− tt gHougHou ρ ] we have to rearrange the equation so that

Housing is on the left hand side of the equation. This is shown as follows:

][][sinsin 2101 NewDIncNewMortgHougHou tt βββρ +++= −

∧

To create forecasted values in NCSS, we will run a transformation for each term in this formula, based on the regression coefficients found above. We will name these terms X, Y, and Z. Then we will create a new variable that adds these terms to find the predicted housing value: ModHousing = X + intercept + Y + Z In NCSS, these variable can be created as follows:

X = HouseLag*0.5121 Formula notation: 1sin −

∧

tgHouρ intercept = 217.1817 Formula notation: 0β Y = -8.5371*NewMort Formula notation: ][1 NewMortβ Z = 0.0044*NewDInc Formula notation: ][2 NewDincβ ModHousing = X+217.1817+Y+Z


The ModHousing variable shows the model’s predicted values: predictions of what housing would be based on the disposable income and five year mortgage rate variables, and with the autocorrelation in these variables corrected. These predicted values are provided below, compared to the actual observed values:

ModHousing Prediction

Actual Values

1976 434.2 431.5 1977 454.2 448.1 1978 456.6 447.9 1979 445.2 451.4 1980 436.2 432.5 1981 405.8 403.3 1982 407.9 407.9 1983 448.4 446.9 1984 449.8 457.2 1985 471.3 485.3 1986 486.3 475.9 1987 479.6 491.3 1988 488.3 493.2 1989 490.6 487.1 1990 476.3 491 1991 499.3 512.4 1992 516.6 523.1 1993 521.8 533.2 1994 518.4 497.7 1995 508.1 502.6 1996 518.4 522.7 1997 533.2 538.7 1998 542.2 533.6 1999 535.9 531.9 2000 535.7 528.1 2001 544.6 544.9 2002 553.4 547.7 2003 557.6 561.2 2004 574.2 581.5

Page 8.26: Table 19: Forecasted Values We can now apply the ModHousing formula to predict future housing values for 2005 and 2006. We will apply the Table 18 predictions for the 5 Year Mortgage Rate and Disposable Income values to find forecasted values of Housing. These forecasted values are:

2005 577,850 2006 578,390

These values can be found by manually calculating results in the ModHousing formula, or, alternatively, in NCSS as follows:


• Enter the 2005 income and mortgage values into the 2005 row and enter the 2004 Housing value 581.539806 into HouseLag.

• Run the X, Y, Z, and ModHousing transformations again. • Enter the 2006 income and mortgage values into the 2006 row and enter the 2005 predicted value,

577.8500701, into HouseLag. • Run the transformations again. • ModHousing will show the predicted values for 2005 and 2006.

Page 8.41: Generating Stepwise Regression in NCSS

NCSS Instructions: Generating a Stepwise Regression

Choose Analysis from the top menu, select Regression/Correlation, Variable Selection Routines, and Stepwise Regression In the Stepwise Regression window, select the Variable tab: For Y: Dependent Variables, choose desired dependent variable. For X’s: Independent Variables, choose desired independent variable(s). For Selection Method, select Stepwise.

using ncss in busi460...ncss tip: log files. when you run an ncss procedure, it displays the output...

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