504 H. White Table 8 Artificial data: Modified nonlinear least squares – Logistic Summary goodness of fit Hidden units Estimation MSE CV MSE Hold-out MSE Estimation R-squared CV R-squared Hold-out R-squared 01.30098 1.58077 0.99298 0.23196 0.06679 0.06664 11.30013 1.49201 0.99851 0.23247 0.11919 0.06144 21.30000 1.50625 1.00046 0.23255 0.11079 0.05961 30.91397 1.10375 0.84768 0.46044 0.34840 0.20321 40.86988 1.05591 0.80838 0.48647 0.37665 0.24016 50.85581 1.03175 0.80328 0.49478 0.39091 0.24495 60.85010 1.01461 0.80021 0.49815 0.40102 0.24783 70.84517 1.00845 0.79558 0.50105 0. 40466 0.25219 80.83541 1.00419 ∗ 0.75910 0.50681 0.40718 0.28648 ∗ 90.80738 1.07768 0.75882 0.52336 0.36379 0.28674 10 0.79669 1.03882 0.73159 0.52967 0.38673 0.31233 11 0.79664 1.04495 0.73181 0.52971 0.38312 0.31213 12 0.79629 1.05454 0.72912 0.52991 0.37745 0.31466 13 0.79465 1.06053 0.72675 0.53088 0.37392 0.31688 14 0.78551 1.04599 0.71959 0.53628 0.38250 0.32361 15 0.78360 1.07676 0.72182 0.53740 0.36433 0.32152 16 0.76828 1.09929 0.70041 0.54645 0. 35103 0.34165 17 0.76311 1.08872 0.70466 0.54950 0.35727 0.33765 18 0.76169 1.11237 0.70764 0.55034 0.34332 0.33484 19 0.76160 1.13083 0.70768 0.55039 0.33242 0.33481 20 0.76135 1.13034 0.70736 0.55054 0.33271 0.33511 . . . . . . . . . . . . . . . . . . . . . 41 0.68366 1.14326 0.65124 0.59640 0.32508 0.38786 ∧ results as good as those seen in Table 9. Nevertheless, we observe quite good perfor- mance. The best CV MSE performance occurs with 50 hidden units, corresponding to a respectable hold-out R 2 of 0.471. Moreover, CV MSE appears to be trending downward, suggesting that additional terms could further improve performance. Table 11 shows analogous results for the polynomial version of QuickNet. Again we see that additional polynomial terms do not improve in-sample fit as rapidly as do the ANN terms. We also again see the extremely erratic behavior of CV MSE, arising from precisely the same source as before, rendering CV MSE useless for polynomial model selection purposes. Interestingly, however, the hold-out R 2 of the better-performing models isn’t bad, with a maximum value of 0.390. The challenge is that this model could never be identified using CV MSE. We summarize these experiments with the following remarks. Compared to the fa- miliar benchmark of algebraic polynomials, the use of ANNs appears to offer the ability to more quickly capture nonlinearities; and the alarmingly erratic behavior of Ch. 9: Approximate Nonlinear Forecasting Methods 505 Table 9 Artificial data: QuickNet – Logistic Summary goodness of fit Hidden units Estimation MSE CV MSE Hold-out MSE Estimation R-squared CV R-squared Hold-out R-squared 01.30098 1.58077 0.99298 0.23196 0.06679 0.06664 11.21467 1.44012 0.93839 0.28292 0.14983 0.11795 21.00622 1.16190 0.86194 0.40598 0.31407 0.18982 30.87534 1.02132 0.81237 0.48324 0.39706 0.23641 40.82996 0.94456 0.71615 0.51004 0.44238 0.32685 50.79297 0.91595 0.67986 0.53187 0.45927 0.36096 60.76903 0.89458 0.67679 0.54600 0.47188 0.36384 70.72552 0.84374 0.62678 0.57169 0. 50190 0.41085 80.68977 0.81835 0.58523 0.59280 0.51689 0.44991 90.66635 0.80670 0.55821 0.60662 0.52376 0.47530 10 0.63501 0.79596 0.55889 0.62512 0.53010 0.47466 . . . . . . . . . . . . . . . . . . . . . 29 0.49063 0.62450 0.49194 0.71036 0.63133 0.53759 30 0.47994 0.61135 0.49207 0.71667 0.63909 0.53747 31 0.47663 0.61293 0.48731 0.71862 0.63816 0. 54195 32 0.47217 0.60931 0.48532 0.72125 0.64029 0.54382 33 0.46507 0.59559 ∗ 0.48624 0.72545 0.64840 0.54295 ∗ 34 0.46105 0.59797 0.48943 0.72782 0.64699 0.53995 35 0.45784 0.60633 0.48603 0.72971 0.64206 0.54315 36 0.45480 0.60412 0.48765 0.73151 0.64336 0.54163 37 0.45401 0.60424 0.48977 0.73198 0.64329 0.53964 . . . . . . . . . . . . . . . . . . . . . 49 0.43136 0.64107 0.47770 0.74535 0.62154 0.55098 ∧ CV MSE for polynomials definitely serves as a cautionary note. In our controlled en- vironment, QuickNet, either with logistic cdf or ridgelet activation function, performs well in rapidly extracting a reliable nonlinear predictive relationship. Naïve NLS is bet- ter than a simple linear forecast, as is modified NLS. The lackluster performance of the latter method does little to recommend it, however. Nor do the computational com- plexity, modest performance, and somewhat erratic behavior of naïve NLS support its routine use. The relatively good performance of QuickNet seen here suggests it is well worth application, further study, and refinement. 7.2. Explaining forecast outcomes In this section we illustrate application of the explanatory taxonomy provided in Sec- tion 6.2. For conciseness, we restrict attention to examining the out-of-sample pre- dictions made with the CV MSE-best nonlinear forecasting model corresponding to 506 H. White Table 10 Artificial data: QuickNet – Ridgelet Summary goodness of fit Hidden units Estimation MSE CV MSE Hold-out MSE Estimation R-squared CV R-squared Hold-out R-squared 01.30098 1.58077 0.99298 0.23196 0.06679 0.06664 11.22724 1.43273 0.87504 0.27550 0.15419 0.17750 21.17665 1.39998 0.83579 0.30537 0.17352 0.21439 31.09149 1.30517 0.75993 0.35564 0.22949 0.28570 40.98380 1.22154 0.75393 0.41922 0.27887 0.29134 50.88845 1.13625 0.73192 0.47550 0.32922 0.31203 60.85571 1.03044 0.71145 0.49483 0.39168 0.33126 70.83444 1.02006 0.69144 0.50739 0. 39781 0.35008 80.81150 0.98440 0.64753 0.52093 0.41886 0.39135 90.78824 0.99417 0.67279 0.53467 0.41309 0.36761 10 0.77323 0.96053 0.70196 0.54352 0.43295 0.34018 . . . . . . . . . . . . . . . . . . . . . 27 0.56099 0.82982 0.55838 0.66882 0.51012 0.47515 28 0.55073 0.80588 0.53706 0.67488 0.52425 0.49518 29 0.54414 0.82178 0.51536 0.67877 0.51487 0. 51559 ∧ 30 0.54103 0.81704 0.53229 0.68060 0.51766 0.49967 31 0.53545 0.80240 0.53970 0.68390 0.52630 0.49271 32 0.53222 0.80171 0.55080 0.68581 0.52671 0.48227 . . . . . . . . . . . . . . . . . . . . . 47 0.47173 0.75552 0.56503 0.72152 0.55398 0.46890 48 0.46773 0.74575 0.55972 0.72388 0.55975 0.47389 49 0.46531 0.73767 0.55892 0.72530 0.56452 0.47464 50 0. 46239 0.73640 ∗ 0.56272 0.72703 0.56527 0.47107 ∗ Table 9. This is an ANN with logistic cdf activation and 33 hidden units, achieving a hold-out R2 of 0.5493. The first step in applying the taxonomy is to check whether the forecast function ˆ f is monotone or not. A simple way to check this is to examine the first partial derivatives of ˆ f with respect to the predictors, x, which we write D ˆ f = (D 1 ˆ f, ,D 9 ˆ f),D j ˆ f ≡ ∂ ˆ f/∂x j . If any of these derivatives change sign over the estimation or hold-out sam- ples, then ˆ f is not monotone. Note that this is a necessary and not sufficient condition for monotonicity. In particular, if ˆ f is nonmonotone over regions not covered by the data, then this simple check will not signal nonmonotonicity. In such cases, further ex- ploration of the forecast function may be required. In Table 12 we display summary statistics including the minimum and maximum values of the elements of D ˆ f over the hold-out sample. The nonmonotonicity is obvious from the differing signs of the max- ima and minima. We are thus in Case II of the taxonomy. Ch. 9: Approximate Nonlinear Forecasting Methods 507 Table 11 Artificial data: QuickNet – Polynomial Summary goodness of fit Hidden units Estimation MSE CV MSE Hold-out MSE Estimation R-squared CV R-squared Hold-out R-squared 01.30098 1.58077 0.99298 0.23196 0.06679 0.06664 11.20939 1.42354 ∗ 0.96230 0.28604 0.15962 0.09547 ∗ 21.13967 1.54695 0.93570 0.32720 0.08676 0.12048 31.09208 2.26962 0.93592 0.35529 −0.33987 0.12027 41.03733 2.14800 0.89861 0.38761 −0.26807 0.15534 51.00583 4.26301 0.87986 0.40621 −1.51666 0.17297 60.98113 4.01405 0.86677 0.42079 −1.36969 0.18527 70.95294 3.34959 0.85683 0.43743 −0.97743 0.19461 80.93024 3.88817 0.86203 0.45083 −1.29538 0 .18972 90.90701 4.35370 0.84558 0.46455 −1.57020 0.20519 10 0.89332 3.45478 0.84267 0.47263 −1.03953 0.20792 . . . . . . . . . . . . . . . . . . . . . 41 0.61881 15.22200 0.67752 0.63468 −7.98627 0.36316 42 0.61305 14.85660 0.67194 0.63809 −7.77057 0.36841 43 0.60894 15.82990 0.67470 0.64051 −8.34518 0.36581 44 0.60399 15 .23310 0.67954 0.64344 −7.99283 0.36126 45 0.60117 13.93220 0.67664 0.64510 −7.22489 0.36399 46 0.59572 15.58510 0.66968 0.64832 −8.20064 0.37053 47 0.59303 15.63730 0.66592 0.64990 −8.23149 0.37407 48 0.58907 16.39490 0.65814 0.65224 −8.67874 0.38137 49 0.58607 15.33290 0.65483 0.65402 −8.05178 0.38448 50 0.58171 16.08150 0.64922 0.65659 −8.49372 0. 38976 ∧ Table 12 Hold-out sample: Summary statistics Summary statistics for derivatives of prediction function x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 mean −8.484 7.638 3.411 −7.371 −9.980 −8.375 0.538 −5.512 −12.267 sd 17.353 19.064 6.313 13.248 18.843 10.144 8.918 7.941 17.853 min −155.752 −5.672 −6.355 −115.062 −168.269 −93.124 −9.563 −68.698 −156.821 max 3.785 166.042 51.985 2.084 4.331 0.219 70. 775 3.177 2.722 Summary statistics for predictions and predictors Prediction x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 mean −0.111 0.046 0.048 0.043 0.580 0.582 0.586 1.009 1.010 1.013 sd 0.775 0.736 0.738 0.743 0.455 0.456 0.457 0.406 0.406 0.408 min −2.658 −1.910 −1.910 −1.910 0.000 0.000 0.000 0.000 0.000 0.000 max 3.087 2.234 2.234 2.234 2.234 2.234 2.234 2.182 2.182 2.182 508 H. White Table 13 Hold-out sample: Actual and standardized values of predictors Order stat. Prediction x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 253 3.087 −1.463 −0.577 −0.835 1.463 0.577 0.835 1.686 0.896 1.132 −2.051 −0.847 −1.183 1.944 −0.010 0.545 1.668 −0.281 0.290 252 1.862 0.014 1.240 0.169 0.014 1.240 0.169 0.303 1.089 0.339 −0.043 1.615 0.169 −1.243 1.444 −0.913 −1.738 0.193 − 1.654 251 1.750 −0.815 −0.315 −1.043 0.815 0.315 1.043 1.093 0.523 1.583 −1.170 −0.492 −1.463 0.517 −0.584 1.001 0.208 −1.198 1.397 2 −2.429 −0.077 0.167 0.766 0.077 0.167 0.766 1.008 1.965 0.786 −0.167 0.161 0.973 −1.107 −0.909 0.394 −0.003 2.349 −0.559 1 −2.658 −0.762 −0.014 1.146 0.762 0.014 1.146 1.483 0.634 1.194 −1.097 −0.084 1.484 0.400 −1.244 1.225 1.167 −0.925 0.444 Note: Actual values in first row, standardized values in second row. Thenextstepistoexamine ˆ δ = ˆ f −Y for remarkable values, that is, values that are either unusual or extreme. When one is considering a single out-of-sample prediction, the comparison must be done relative to the estimation data set. Here, however, we have a hold-out sample containing a relatively large number of observations, so we can con- duct our examination relative to the hold-out data. For this, it is convenient to sort the hold-out observations in order of ˆ δ (equivalently ˆ f ) and examine the distances between the order statistics. Large values for these distances identify potentially remarkable val- ues. In this case we have that the largest values between order statistics occur only in the tail, so the only remarkable values are the extreme values. We are thus dealing with cases II.C.2, II.D.3, or II.D.4. The taxonomy resolves the explanation once we determine whether the predictors are remarkable or not, and if remarkable in what way (unusual or extreme). The comparison data must be the estimation sample if there are only a few predictions, but given the rel- atively large hold-out sample here, we can assess the behavior of the predictors relative to the hold-out data. As mentioned in Section 6.2, a quick and dirty way to check for remarkable values is to consider each predictor separately. A check of the order statis- tic spacings for the individual predictors does not reveal unusual values in the hold-out data, so in Table 13 we present information bearing on whether or not the values of the predictors associated with the five most extreme ˆ f ’s are extreme. We provide both actual values and standardized values, in terms of (hold-out) standard deviations from the (hold-out) mean. The largest and most extreme prediction ( ˆ f = 3.0871) has associated predictor val- ues that are plausibly extreme: x 1 and x 4 are approximately two standard deviations from their hold-out sample means, and x 7 is at 1.67 standard deviations. This first example therefore is plausibly case II.D.4: an extreme forecast explained by extreme predictors. This classification is also plausible for examples 2 and 4, as predictors x 2 , Ch. 9: Approximate Nonlinear Forecasting Methods 509 x 7 , and x 9 are moderately extreme for example 2 and predictor x 8 is extreme for ex- ample 4. On the other hand, the predictors for examples 3 and 5 do not appear to be particularly extreme. As we earlier found no evidence of unusual nonextreme predic- tors, these examples are plausibly classified as case II.C.2: extreme forecasts explained by nonmonotonicities. It is worth emphasizing that the discussion of this section is not definitive, as we have illustrated our explanatory taxonomy using only the most easily applied tools. This is certainly relevant, as these tools are those most accessible to practitioners, and they afford a simple first cut at understanding particular outcomes. They are also help- ful in identifying cases for which further analysis, and in particular application of more sophisticated tools, such as those involving multivariate density estimation, may be war- ranted. 8. Summary and concluding remarks In this chapter, we have reviewed key aspects of forecasting using nonlinear models. In economics, any model, whether linear or nonlinear, is typically misspecified. Con- sequently, the resulting forecasts provide only an approximation to the best possible forecast. As we have seen, it is possible, at least in principle, to obtain superior approx- imations to the optimal forecast using a nonlinear approach. Against this possibility lie some potentially serious practical challenges. Primary among these are computational difficulties, the dangers of overfit, and potential difficulties of interpretation. As we have seen, by focusing on models linear in the parameters and nonlinear in the predictors, it is possible to avoid the main computational difficulties and retain the benefits of the additional flexibility afforded by predictor nonlinearity. Further, use of nonlinear approximation, that is, using only the more important terms of a nonlinear se- ries, can afford further advantages. There is a vast range of possible methods of this sort. Choice among these methods can be guided to only a modest degree by a priori knowl- edge. The remaining guidance must come from the data. Specifically, careful applica- tion of methods for controlling model complexity, such as Geisser’s (1975) delete-d cross-validation for cross-section data or Racine’s (2000) hv-block cross-validation for time-series data, is required in order to properly address the danger of overfit. A care- ful consideration of the interpretational issues shows that the difficulties there lie not so much with nonlinear models as with their relative unfamiliarity; as we have seen, the interpretational issues are either identical or highly parallel for linear and nonlinear approaches. In our discussion here, we have paid particular attention to nonlinear models con- structed using artificial neural networks (ANNs), using these to illustrate both the challenges to the use of nonlinear methods and effective solutions to these challenges. In particular, we propose QuickNet, an appealing family of algorithms for constructing nonlinear forecasts that retains the benefits of using a model nonlinear in the predictors while avoiding or mitigating the other challenges to the use of nonlinear forecasting 510 H. White models. 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