Gets modelling is not restricted to time-series data, and the purpose of this chapter is to illustrate the use of PcGets for model selection from cross-section data.
Start GiveWin, and load the XsVars.in7 and XsVars.bn7 files in GiveWin, following the procedure explained earlier. The data set comprises 92 observations on 27 random variables, Y, Xa,..., Xz. The first has been generated by a linear function of the first ten Xs, namely:
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where ei~IN[ 0,se2] with se2=1. However, there are 16 additional, but irrelevant, variables in the GUM, as well as an unnecessary constant. All the Xs,i were randomly generated, independently of ei, so there is a valid conditional model. The objective is to select the DGP in (eq:6.1), commencing from the GUM:
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Select the Formulate command on the Model menu of PcGets. Select every variable, and add them all to the model, as shown in the capture below.
Once the model formulation is complete, click on OK or press Enter to bring up the Model Settings dialog.
Mark Outlier correction and unmark the F presearch testing (lag order selection) as that is irrelevant here, select Expert user's strategy, and Report only the finally selected model, then accept (you can also use the Liberal default, but some of the diagnostic tests will be irrelevant).
Select Testimation on the Estimate Model dialog (also referred to as GETS):
Press the Options button to choose a set of mis-specification tests relevant to cross-section data. Thus, check the number of residual standard deviations for outliers, unmark any autocorrelation or ARCH tests, and mark normality, constancy, and heteroscedasticity as shown:
Press the OK button to accept, and return to the Estimate Model dialog and press OK (or Enter) to estimate the GUM, conduct automatic mis-specification tests, and select the model -- in an instant.
The regression results are written to the Results window. As you know from Chapter 2, this window does not reside in PcGets, but in GiveWin. After estimation, focus switches automatically to the Results window in GiveWin.
Even with the `condensed' selection, the output is huge, so we focus on the salient aspects.
GUM( 1) Modelling Y by GETS (using XsVars.in7), 1 - 92
Coeff StdError t-value t-prob
Constant 0.17078 0.31893 0.535 0.5941
Xa 0.33996 0.14013 2.426 0.0180
Xb 0.30224 0.13038 2.318 0.0236
Xc 0.56260 0.14950 3.763 0.0004
Xd 0.31337 0.14256 2.198 0.0315
Xe 0.26910 0.12955 2.077 0.0417
Xf 0.52000 0.13006 3.998 0.0002
Xg 0.36789 0.13694 2.686 0.0092
Xh 0.40299 0.13020 3.095 0.0029
Xi 0.44040 0.11823 3.725 0.0004
Xj 0.60144 0.15515 3.877 0.0002
Xk 0.09302 0.13702 0.679 0.4996
Xl -0.05680 0.13088 -0.434 0.6658
Xm -0.14386 0.15228 -0.945 0.3483
Xn -0.05930 0.11800 -0.503 0.6170
Xo 0.09159 0.13129 0.698 0.4879
Xp 0.08099 0.14566 0.556 0.5801
Xq 0.01730 0.13879 0.125 0.9012
Xr -0.03138 0.12337 -0.254 0.8000
Xs 0.17276 0.11562 1.494 0.1399
Xt -0.10239 0.12738 -0.804 0.4244
Xu 0.01181 0.14404 0.082 0.9349
Xv 0.02257 0.11568 0.195 0.8459
Xw -0.31468 0.51655 -0.609 0.5445
Xx -0.01746 0.03032 -0.576 0.5667
Xy -0.02813 0.13810 -0.204 0.8392
Xz 0.06407 0.07973 0.804 0.4245
RSS 77.72806 sigma 1.09353 R^2 0.70128 Radj^2 0.58179
LogLik 7.75432 AIC 0.41838 HQ 0.71709 SC 1.15847
T 92 p 27 FpNull 0.00000 FpConst 0.00000
Significance levels set: 0 Test(s) to be excluded.
value prob alpha
Chow(47:1) 0.6274 0.9028 0.0100
Chow(83:1) 0.5069 0.8634 0.0100
normality test 0.3064 0.8579 0.0100
hetero test 0.4991 0.9573 0.0100
Stage-0 (Step 0): Outlier correction
No outliers found:
Largest residual = 1.96 sigma;
Critical value = 3.00 sigma.
Specific model of Y, 1 - 92
Coeff StdError t-value t-prob Split1 Split2 reliable
Xa 0.33348 0.11705 2.849 0.0055 0.0217 0.0004 1.0000
Xb 0.30901 0.11124 2.778 0.0068 0.0134 0.0006 1.0000
Xc 0.55896 0.12318 4.538 0.0000 0.0001 0.0000 1.0000
Xd 0.33407 0.11775 2.837 0.0057 0.0455 0.0001 1.0000
Xe 0.27778 0.10959 2.535 0.0132 0.1074 0.0000 0.6000
Xf 0.50679 0.11708 4.329 0.0000 0.0516 0.0000 0.6000
Xg 0.36973 0.11757 3.145 0.0023 0.0027 0.0046 1.0000
Xh 0.45753 0.11145 4.105 0.0001 0.0072 0.0001 1.0000
Xi 0.44382 0.10388 4.272 0.0001 0.0000 0.0009 1.0000
Xj 0.58389 0.12501 4.671 0.0000 0.0002 0.0000 1.0000
RSS 85.62644 sigma 1.02187 R^2 0.67092 Radj^2 0.63480
LogLik 3.30255 AIC 0.14560 HQ 0.25623 SC 0.41970
T 92 p 10 FpNull 0.00000 FpGUM 0.98362
value prob
Chow(47:1) 0.5158 0.9826
Chow(83:1) 0.6001 0.7929
normality test 0.3852 0.8248
hetero test 0.8920 0.5974
These results are far better than one might ordinarily hope for -- a perfect selection, in fact; but serve to illustrate PcGets' performance.
Next, graph the fitted values, namely y^i, and the outcomes, a cross-plot of the same two variables, and the scaled residuals:
| ûi= |
| . |
Select Graphic analysis using the fourth button on the toolbar). Both Actual and fitted values, and Residual analysis should be marked: accepting produces Figures Figure:6.1 and Figure:6.2 in GiveWin (the second graph has been edited to delete the correlogram and spectrum). As before, any graphs can be saved, edited or printed.
The first plot shows the `track' of the outcome by the fitted model. The overall tracking is fair. This is perhaps easier to see from the cross-plot, where there is considerable scatter on either side of the regression line. Finally, the scaled residuals and squared residuals seem random and homoscedastic. Figure Figure:6.2 confirms that they are approximately normal.
This single sample of artificial data can also be used to illustrate some differences between Liberal, Conservative and Expert selection strategies. Of course, the last two cannot do any better -- we have already found the correct answer as the Liberal strategy already selected all ten Xa--Xj, with no irrelevant variables!
The Expert strategy with default settings also correctly selects all ten Xa--Xj, with no irrelevant variables: again, we stress that this is exceptional, and represents an upper bound rarely achieved. In one sense, therefore, the Liberal strategy performed in the direction in which it should, relative to the `expert', by retaining at least as many variables. The Conservative strategy under-selects by omitting Xd, Xe and Xg, but again keeps no irrelevant variables: while this is an appropriate operating characteristic, it well illustrates the dangers of overly penalizing selection of relevant variables, even when there are many potentially-irrelevant regressors.
In many analyses, some variables are deemed to be so central that an investigator wishes to ensure they are retained by any model selection. As discussed in Chapter 5, this is easily achieved by marking them as Fixed during model formulation. To illustrate, we Fix Xs, which delivers the selected equation:
Specific model of Y, 1 - 92
Coeff StdError t-value t-prob Split1 Split2 reliable
Xa 0.34732 0.11718 2.964 0.0040 0.0067 0.0004 1.0000
Xb 0.30023 0.11109 2.703 0.0084 0.0024 0.0007 1.0000
Xc 0.54664 0.12316 4.438 0.0000 0.0000 0.0000 1.0000
Xd 0.30441 0.11974 2.542 0.0129 0.0438 0.0001 1.0000
Xe 0.26855 0.10947 2.453 0.0163 0.1009 0.0001 0.6000
Xf 0.50161 0.11676 4.296 0.0000 0.0550 0.0000 0.6000
Xg 0.38321 0.11767 3.257 0.0016 0.0011 0.0054 1.0000
Xh 0.42873 0.11345 3.779 0.0003 0.0307 0.0001 1.0000
Xi 0.45000 0.10365 4.342 0.0000 0.0000 0.0011 1.0000
Xj 0.57409 0.12483 4.599 0.0000 0.0003 0.0000 1.0000
Xs 0.12322 0.09882 1.247 0.2160 0.0034 0.9417 0.0000
RSS 84.01362 sigma 1.01843 R^2 0.67712 Radj^2 0.63726
LogLik 4.17724 AIC 0.14832 HQ 0.27002 SC 0.44984
T 92 p 11 FpNull 0.00000 FpGUM 0.99200
value prob
Chow(47:1) 0.6211 0.9351
Chow(83:1) 0.5108 0.8621
normality test 0.0193 0.9904
hetero test 1.0905 0.3829
In this instance, no harm -- nor good -- is done; the forced variable is insignificant, and shown to be completely unreliable. Chapter 5 discusses the consequences of fixing variables more generally.
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