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NCER Working Paper SeriesApplying shape and phase restrictions ingeneralized dynamic categorical models ofthe business cycleDon HardingWorking Paper #58July 2010

Applying shape and phase restrictions ingeneralized dynamic categorical models of thebusiness cycleDon Harding 1;2July 26, 20101 School of Economics and Finance, La Trobe University, Melbourne, Australia.2 Support from a La Trobe University Faculty of Law and Management researchgrant is gratefully acknowledged. Helpful comments were provided by AdrianPagan, John Geweke and participants at the 15 th Australasian macroeconomicsworkshop. Jack Mullumby provided useful research assistance. All errors are myresponsibility.

1 IntroductionThe business cycle is one of many cases in macroeconomics and …nance wheresingle index dynamic discrete choice (DDC) models are invalid and multipleindex DDC models are required. Harding and Pagan (2009) provide a detaileddiscussion of this issue.They introduce a second order generalizeddynamic categorical (GDC) model as a parsimonious framework to approximatethe DGP of NBER business cycle states. The GDC model is set outin section 2 and the form of the transition probabilities derived in section2.1 where it is shown that boundedness and monotonicity, in the forcingvariables, is exhibited by this class of models.The non parametric estimation method developed by Harding and Pagan(2009) imposes phase restrictions necessary to approximate the NBERbusiness cycles, respects boundedness of transition probabilities but does notimpose monotonicity. 1 This paper makes four contributions that develop andextend the framework they develop.The …rst contribution is to develop parametric models that simultaneouslyimpose phase restrictions and shape features.Two particular parametricversions are developed using the Normal and Logistic distributions. Theseare named GDC Probit and GDC Logit respectively. Maximum likelihoodestimation (MLE) of these models is discussed in section 3.The second contribution of this paper, in section 4, is to develop nonparametric methods that simultaneously impose the GDC phase and shaperestrictions. The particular method applied is the constraint weighted boot-1 Henceforth I refer to monotonicity in the forcing variables and boundedness of thetransition probabilities as shape features of GDC models.1

strap of Hall and Huang (2001).The intuition of this method is that it“involves tilting the empirical distribution to the least possible amount, subjectto the constraint being enforced”.The non-parametric method requires choice of the window width in thekernel estimator and a parameter in the distance function. Procedures aredeveloped to determine both of these parameters as functions of the data.The third contribution is made in section 5 where I evaluate the parametricand non parametric procedures in an application that uses the yield spreadto predict the state of the business cycle. Here it is shown that the GDCProbit produces a large improvement in …t over the static Probit developedby Estrella and Mishkin (1998). It is also shown that the non parametricestimator with phase and shape restrictions delivers further improvement in…t over the GDC Probit. The shape restricted non parametric estimator alsoreveals an important feature of United States business cycles that is relevantfor the conduct of monetary policy.The fourth contribution of the paper, made in section 6, is to show thatone needs to be very careful in translating statements about improvementsin …t to statements about policy relevant improvements in prediction aboutpeaks and troughs of the business cycle.Conclusions are in section 7.2

2 A second order Generalized Dynamic ChoicemodelThe variable of interest is S t ; a binary variable constructed using NBERprocedures, it has the properties that, S t = 1 when the economy is in expansion; S t = 0 when the economy is in recession. completed phases have a minimum duration of at least two periods.The GDC model of order 2, introduced in Harding and Pagan (2009), tomodel such a variable is 2P r (S t = 1jS t 1 ; S t 1 ; x t ) = 00 (x t ) S 00t 1 + 01 (x t ) S 01t 1+ 10 (x t ) S 10t 1 + 11 (x t ) S 11t 1 (1)where x t is a vector of conditioning variables. As discussed in Harding andPagan (2009) the NBER method of constructing the S t imposes the restrictionsthat 10 (x t ) = 1 and 01 (x t ) = 0 (2)2 For later use in estimating these models it is useful to de…ne the following variablesand sets where i; j = f0; 1g: S ijt 1 = 1 if S t 1 = i and S t 2 = j: Otherwise S ijt 1 = 0: n ij is number of cases where S t 1 = i and S t 2 = j: I ij is subset of f1; :::; T g where S t 1 = i and S t 2 = j3

The ij (x t ) also have the property of boundedness, ie 0 ij (x t ) 1:For later use it is convenient to write the static Probit model asPr (S t=1 = 1jx t ) =Z 1x t1p2e 1 2 v2 dv 1 ( x t ) (3)where (:) is the standard normal cumulative distribution function.2.1 Obtaining the form of the transition probabilitiesfrom …rst principlesIt is useful to obtain the 2nd order GDC model from …rst principles so as toestablish where the properties of that model come from.The data generating process for S t is assumed to be such that the economystays in an expansion that has lasted at least two quarters provided thelatent variable 11t is positive ( 11t > 0) : The economy shifts from a recessionthat has lasted at least two periods to an expansion if the latent variable 00t is positive ( 00t > 0). Re‡ecting the NBER method of constructing S t ,the economy stays in recession if the recession has lasted only one periodand stays in expansion if the expansion has lasted only one period. Thusthe binary variable S t representing the state of the business cycle evolvesaccording to:S t = S 11t 11 ( 11t > 0) + S 00t 11 ( 00t > 0) + S 10t 1 (4)The two latent variables ( 1t ; 1t ) have the following data generatingprocesses (DGPs)4

two periods isPr (S t = 1jS t 1 = 0; S t 2 = 0; x t ) = E (S t jS t 1 = 0; S t 2 = 0; x t ) (8)= E f1 ( 00t > 0jx t )g= Pr ( 00t > 0jx t )=Z 1x tg (v) dv 1 G ( x t )where F (:) and G (:) are cumulative distribution functions corresponding tothe densities f (:) and g (:) respectively.The transition probabilities are monotonic in the elements of x 0 t: Thiscan be seen by noting that since 11 (x t ) = 1 F ( x t ) and 00 (x t ) =1 G ( x t ) it follows that@ 11 (x t )@x jt= f ( x t ) j (9)and@ 00 (x t )@x jt= g ( x t ) j (10)Now since f (.) and g (:) are densities they are non negative and thus 11 (x t )and 00 (x t ) are weakly monotonic. Moreover, the direction(s) of the monotonicityare determined by the signs of the coe¢ cients j and j respectively. Theissue of whether the monotonicity is weak or strict depends …rstly, on whetherthe densities f (.) and g (:) are strictly positive on their support and, secondly,on whether j 6= 0 and j 6= 0.6

2.2 GDC Probit and LogitThe double index GDC-Probit model, used in the application, is the specialcase of (7) and (8) where " 11t and " 00t have independent standard normaldistributions so that, f (v) = g (v) = (v) 1 p2e v2 2 and F (z) = G (z) = (z) R (v) dv the cumulative distribution function of the standardz1normal distribution. The GDC Probit model is written asP r (S t = 1jS t 1 ; S t 1 ; x t ) = S 10t 1 + [1 ( x 0 t)] S 00t 1 (11)+ [1 ( x 0 t)] S 11t 1The single index GDC Probit model involves the restriction that = in (11). Importantly, the single index GDC Probit di¤ers from the staticProbit because the former does not include the term St101 which is requiredif the model is to be consistent with the NBER method for constructing theS t . As discussed, in Harding and Pagan (2009) this di¤erence means thatthe static Probit model can never match the features of the NBER businesscycle.Other distribution functions such a the logit might be considered so thatF (z) = G (z) =1 + exp p3z 1(12)yielding a model that might be designated as GDC-Logit.7

3 Maximum likelihood estimationThe conditional likelihood functions areL 11 = Y[1 F ( x t )] S11 t[F ( x t )] 1 S11 t(13)t2I 11L 00 = Y[1 G ( x t )] S00 t[G ( x t )] 1 S00 t(14)t2I 00To implement the maximum likelihood procedures one selects parametricdensities f (:) and g (:) : Estimation proceeds by choosing and to maximizethe log of the likelihoods. In the case of the GDC Probit (11) the MLestimators of and are:b = arg max X S11t ln [1 ( x 0 t)] + 1 St11 ln ( x0t ) (15)t2I 11b = arg max X S00t ln [1 ( x 0 t)] + 1 St00 ln ( x0t ) (16)t2I 004 Non parametric estimationNon parametric methods do not automatically impose the monotonicity thatwas shown above to be central to the GDC class of models.Henderson and Parmenter (2009) survey a range of methods for imposingconstraints on non parametric estimators of conditional means.Some of8

these methods introduce unattractive features such as reduced smoothnessof the estimator. For example, isotonic regression (Friedman and Tibshirani(1984)) is unattractive because they introduce jump discontinuities in theestimator.Hall and Huang’s (2001) method, in contrast, is designed to produce a curve that satis…es the constraints but exhibits the samesmoothness (ie the same number of derivatives exist and are continuous)as for its unconstrained counterpart; be applicable to general kernel methods; mainly modify the unconstrained estimator in the regions where theconstraints are binding; and require little additional computational e¤ort over the unconstrainedestimator.4.1 Hall and Huang’s methodTo implement Hall and Huang’s (2001) method for the problem at handobserve that (1) implies that 3 11 (x t ) = E (S t jS t 1 = 1; S t 2 = 1; x t ) (17)and3 Here x t is assumed to be a scalar. The approach readily generalizes to the case wherethere is a vector of forcing variables.9

00 (x t ) = E (S t jS t 1 = 0; S t 2 = 0; x) (18)For a wide range of non parametric methods, estimators of 11 (x) and 00 (x) can be written asc 00 (x) = 1n 00Xi2I 00A 00i (x) S 00i (19)where the A 11i and A 00ic 11 (x) = 1n 11Xi2I 11A 11i (x) S 11i (20)are weighting functions. For example, if a localconstant kernel method is used then let jji (x) = x xjj ias 4A jjh jand A jji(x) is de…nedi (x) = K jji (x) P Kjji (x) (21)Hall and Huang (2001) suggest a very convenient method for imposinga wide range of constraints on the estimators of 00 (x) and 11 (x) : Forimposing monotonicity and boundedness this involves the following steps.First estimate c 00 (x) ; c 11 (x) saving A 11i (x) and A 00i (x) : Second, checkwhether the boundedness and monotonicity constraints are violated. If theyare not violated then c 00 (x) ; c 11 (x) are the …nal estimators.Otherwise introduce observation speci…c weights p 11i and p 00iwith theproperties that P i2I 0p 00i = P i2I 1p 11i = 1: De…ne new estimators f 00 (x) and4 The window width used to compute E(S t jx t ; S t 1 = j; S t 2 = j) is c j x n 1 5jj wheren jj is the number of cases where (S t 1 = j; S t 2 = j); x is the standard deviation of thex 0 s and c j is a constant:10

f 11 (x) as followsf 00 (x) = X i2I 00p 00i A 00i (x) S 00i (22)f 11 (x) = X i2I 11p 11i A 11i (x) S 11i (23)The idea in the Hall and Huang (2001) method, as applied to the problemhere, is to chose the weights so as to minimize the distances between p 00iand1n 00and p 11i and 1n 11respectively. The distance metric D (p) introduced byCressie and Read (1984) proves to be useful for imposing the monotonicityconstraint 5D p jj ==="1n (1 )nXnXi=1i=1p jjinXi=1ln np jji ; = 0ln np jji ; = 1#np jj i ; jj < 1; 6= 0; 6= 1 (24)If we sought to minimize D (p 00 ) and D (p 11 ) with respect to p 00 andp 11 without imposing any constraints then the solution would be p 00i = 1n 00and p 11i = 1n 11respectively. The relevant constraints when estimating f jj (x t )are: Weights sum to oneXi2I jjp jji = 1 (25)5 Montotonicity can be imposed with weights that are positive thereby making Cressieand Read’s (1984) distance metric appropriate. See Racine, Parmenter and Du (2009) fora discussion of the distance metric that is appropriate with more general shape restrictionsthat may require negative weights.11

Monotonicity 6 @ f jj (x)= X p 00i@x ki2I 00@A jji@x k(x)S jji > 0 8 k (26) Boundedness0 f jj (x) (27)f jj (x) 1 (28)The constrained estimator is obtained by choosing p jj to minimize (24)subject to constraints (25) to (28). This constrained minimization is easilyachieved using a procedure such as fmincon in Matlab.Using the same arguments as in Harding and Pagan (2009) the followingcentral limit holdspnjj h j f jj (x) jj (x) ! N 0;RK ( ) 2 d jj (x) jj (x) 1 jj (x) ! (29)Where jj (x) is the density of x on the relevant sub sample.4.2 Choosing h and There are two data driven techniques available to choose the smoothing parameterh and the parameter in the distance function.The …rst of these is least squares cross validation. It involves choosing h6 Here monotonicity is imposed as a local property via the slope one could also requirethat f jj (x) f jj (x ) for x x .12

and to minimise the objective functionCV (h; ) = 1 nnX(y i bm i (h; )) 2 (30)i=1Where bm i (h; ) is the leave one out non parametric estimate of E (S i jS i 1 ; S i 2 ; sp i 2 ) :Least squares cross validation is expensive in terms of computer time becausethe leave one out estimator must be computed n times. This approach is evenmore expensive here because the optimal observation speci…c weights mustalso be calculated at each iteration.An attractive alternative discussed in Li and Racine (2009, p72) is providedby the improved Aiaike information criterion AIC c criterion developedby Hurvich, Simono¤ and Tsai (1998). Here the objective functions are de-…ned asAIC c (h; ) = ln b 2 +1 + tr (H r ) =n1 ftr (H r ) + 2g =nr = 0; 1 (31)Where H r is an n n weighting matrix and 7b 2 = 1 nnX(Y i bm i (h; )) 2 = Y 0 (I H r ) 0 (I H r ) Y=n (32)i=1Li and Racine (2009) report studies that show that choosing h that minimizesAIC c yields a window width that performs well compared to plug-in methods,and generalized cross validation methods. Li and Racine (2004) show using7 Letting H r i;j denote the (i; j)th element of H r , the weighting matrices H r are relatedto those used in estimation as follows:Hi;j r = p rri A rri (x j ) :13

simulation methods that AIC coutperforms least squares cross validationin small samples and that in large samples the two methods yield similarwindow widths.An alternative version of the AIC criterion that is suitable for workingwith discrete choice models involves replacing ln b 2 in (31) with minus thelog probability score so that the criterion to be minimized isAIC c (h; ) = LP S (h; ) +1 + tr (H r ) =n1 ftr (H r ) + 2g =n(33)WhereLP S (h; ) =nXln fS t bm t (h; ) + (1 S t ) (1 bm t (h; ))gi=1and bm t (h; ) is the non parametric estimator of E (S t jS t 1 ; S t 2 ; sp t 2 ) :5 ApplicationThis application explores the value of the parametric and non parametricmethods discussed above in studying the value of the yield spread (sp t ) inpredicting the state of the business cycle. To achieve comparability with theliterature, these questions are addressed using the same sample of data as inEstrella and Mishkin (1998) and Harding and Pagan (2009). 8To maintaincomparability with these earlier papers I maintain Estrella and Mishkin’s(1998) speci…cation that x t = sp t 2 :8 Estrella and Mishkin (1998) use the spread between the 10 year bond and 3 monthTreasury Bill.14

Table 1: Summary statistics: United States NBER business cycle states,1959.Q3 to 1995.Q1S t S t jS 11t 1 = 1 S tjSt 001= 1 S t jSt 101= 1 S t jSt 011= 1Mean 0.85 0.95 0.40 1 0Standard Deviation 0.36 0.22 0.51 0 0Summary statistics for the NBER business cycle date are presented inTable 1. The unconditional mean of S t in the second column and the twoconditional means in the third and fourth columns will be used later in thepaper as a reference point against which the various models are evaluated.The …fth and sixth columns of Table 1 con…rm that the data does havethe NBER feature identi…ed in Harding and Pagan (2009) that the secondquarter of each phase can be predicted with certainty. They show that failureto allow for this feature in dynamic discrete choice models results in a seriousviolation of the assumptions required for maximum likelihood estimation tobe valid.Inspection of the means in Table 1 makes it apparent that the staticProbit model (3) cannot be a good …t to this data since that model impliesthat all of the means in Table 1 should be equal.5.1 Maximum likelihoodThe static Probit (3) estimated by Estrella and Mishkin (1998) is reportedin column 1 of table 2. It can be compared with the GDC model in column 3where the parameters are constrained so that 0 = 0 and 1 = 1 : This lattermodel produces a substantial improvement over the static Probit in terms of15

the log likelihood and the standard error of the regression. 9Since the twomodels have the same number of parameters we would always prefer the onewith the better …t. In this case the better …t comes solely from recognizingthat, as discussed above and in Harding and Pagan (2009), the method ofconstruction employed by the NBER imposes a particular structure on thetransition probabilities for the NBER states.The results for double index GDC Probit model is shown in column 5of table 2. It produces a large improvement in the log likelihood. Thelikelihood ratio statistic for the null hypothesis that there is a single index is26.12. Since this is distributed 2 (2) the 1% critical value is 9.21 and we canreject the null hypothesis in favour of the alternative that there is a doubleindex.Also shown in columns 2, 4 and 6 of table 2 are the results for the logitmodel which …ts a little worse than the Probit in each case.The GDC single index Probit conditions on the economy being in thesame state in periods t 1 and t 2 (S t 1 = S t 2 ). This is informativebecause the NBER method of constructing S t imposes the restrictions thatE (S t jS t 1 = 1; S t 2 = 0) = 1 and E (S t jS t 1 = 0; S t 2 = 1) = 0: Panel A ofFigure 1 compares the probability being in expansion for the static Probitmodel and the single index GDC model, this Figure clearly demonstratesthe e¤ect of conditioning on (S t 1 = S t 2 ). The dashed lines are con…dencebands with 95% coverage probability. 109 In table 2 the various loglikehods on the subsample and full sample are de…ned asl 00 = log L 00 l 11 = log L 11 and l = l 00 + l 11 :10 The standard errors used in constructing the con…dence intervals are obtained usingthe delta method.16

Table 2: Results from static and GDC-Probit modelsStaticLogitStaticProbitGDC:SingleindexProbitGDC:SingleindexLogitGDC:DoubleindexProbitGDC:DoubleindexLogitCol 1 Col 2 Col 3 Col 4 Col 5 Col 60:624 0:588 0:804 0:760 1:338 1:348 0(0:013) (0:013) (0:015) (0:015) (0:023) (0:026)0:596 0:588 0:534 0:540 0:724 0:763 1(0:011) (0:011) (0:012) (0:013) (0:021) (0:022) 0 na na = 0 na0:699 0:613(0:121) (0:112) 1 na na = 1 na0:558 0:484(0:099) (0:091)l 00 na na na na 8:79 8:84l 11 na na na na 15:71 15:82l 45:89 46:05 37:57 37:70 24:51 24:66SE 00 na na na na 0:455 0:458SE 11 na na na na 0:193 0:208SE 0:313 0:330 0:282 0:296 0:228 0:23917

Probability of expansionProbability of expansionFigure 1: Probability in expansion conditional on yield spead, static Probitand various GDC Probit models1Panel A1Panel B0.90.9GDC Probit→Pr(S =1|S =1,S =1,sp )t t­ 1 t­ 2 t­ k0.8GDC single index Probit→Pr(S t=1|S t­ 1=S t­ 2,sp t­ 2)0.80.70.70.6← Static Probit Pr(S t=1|sp t­ 2)0.6← Static ProbitPr(S t=1|sp t­ k)0.50.50.40.40.30.30.20.2← GD C ProbitPr(S t=1|S t­ 1=0,S t­ 2=0,sp t­ k)0.10.10­4 ­3 ­2 ­1 0 1 2 3 4Yield spread lagged two quarters0­4 ­3 ­2 ­1 0 1 2 3 4Yield spread lagged two quartersPanel B of Figure 1 shows the probability of being in an expansion conditionalon the yield spread lagged two periods. The static Probit conditionsonly on this variable. The GDC Probits also condition on the state of thebusiness cycle at t 1 and t 2. The dashed lines are con…dence bands with95% coverage probability. The clear point made by …gure 1 is that conditioningon whether the economy is in recession or expansion at t1 andt1 matters a great deal for the probability that the economy will be inexpansion at period t.18

5.2 Non parametricThe non parametric procedure described in section 4 above was implementedusing a Nadaraya-Watson kernel estimator with Gaussian kernel. Four procedureswere considered to obtain h and viz ’plug-in’, least squares crossvalidation 11 , AIC c and AIC c :The plug-in window width, involves choosing the constants c 0 = c 1 = 1:When using plug-in values the coe¢ cient in the distance function (24)was set equal to 1 : The results for the non parametric estimator with phase2restrictions, plug-in h but no shape restrictions are reported in table 3.Table 3: Results: plug in h, monotonicity not imposedAIC c AICc l 00 l 11 l-2.07 25.93 -8.14 -14.83 -22.97The AIC c and AIC c criteria proved to be computationally simple to implementand used little computer time. The results for AIC c are reported intable 4 and those for AIC care in table 5. The results in both tables showthat for any practical purpose AIC c is ‡at in something that is reassuringas it would be undesirable if the way in which distance is measured materiallya¤ected the estimates obtained when monotonicity is imposed. For thisreason the discussion below focuses on the choice of c 0 and c 1 which a¤ectthe window width h.The AIC c criterion selects c 0 = 4:57 for recessions which has the e¤ect11 The least squares cross validation method was tried but proved to have two maindisadvantages …rst it consumed a large amount of computer time. Second, it producedestimates of c 0 and c 1 that were implausibly low being one …fth of the plug in values citedabove. Thus, least squares cross validation was not used in any of the empirical resultsreported below.19

Table 4: AIC c resultsMethod Value AIC c l 00 l 11 lh c 0 c 1 0 1Plug-in Plug-in 1 1 0.5 0.5 -2.02 -8.13 -15.45 -23.58opt Plug-in 4.57 1.00 0.5 0.5 -2.32 -9.85 -15.26 -27.58Plug-in Opt 1 1 0.34 0.56 -2.02 -8.13 -15.39 -23.51Opt Opt 4.57 1.11 0.28 0.47 -2.32 -9.85 -15.59 -25.44of smoothing the yield spread out of the probability of expansion. However,the log probability score is -27.58 which is substantially worse than the -23.58achieved when plug-in values of the smoothing parameters are used. 12TheAIC c criterion also selects c 1 = 1 (it also selects c 1 =1.11) which gives theyield spread a role in predicting the onset of recessions.Table 5: AICc results 0 1 0 1MethodhValuecAICc c l 00 l 11 lPlug-in Plug-in 1 1 0.5 0.5 26.54 -8.13 -15.45 -23.58opt Plug-in 0.88 0.50 0.5 0.5 25.74 -8.13 -14.67 -22.60Plug-in Opt 1 1 0.45 0.57 26.40 -8.13 -15.31 -23.43Opt Opt 0.84 0.77 0.24 0.19 25.93 -7.94 -14.88 -22.82The AIC ccriterion selects c 0 = 0:88 and c 1 = 0:5 which gives the yieldspread a role in determining the probability of being in expansion. Overallthe the log probability score is -22.60 which is slightly better than the modelwithout monotonicity imposed but with plug-in values of the smoothing parameters.This result suggests that the additional smoothness achieved viaimposition of monotonicity allows an improved …t to be achieved through asmaller window width.12 This seemingly strange result arises because the AICc is not directly related to thelog probability score.20

Probability of expansionProbability of expansion5.2.1 Probability exit a recession that has lasted at least two quartersBoth panels of Figure 2 plot, against the yield spread lagged two quarters,non parametric estimates of the probability of exiting a recession that haslasted at least two quarters. The heavy line in both panels are estimatedwith phase and shape restrictions imposed. The dashed line in panel A is theprobability obtained using the method in Harding and Pagan (2009) whichdoes not impose monotonicity. Clearly, imposing monotonicity matters forthe estimate of the probability of exiting a recession.Figure 2: Probability exit a recession that has lasted for at least two quarters1Pa ne l A1Panel BMonotonicity imposedMonotonicity imposedunconstrainedGDC probit0 .90 .90 .80 .80 .70 .70 .60 .60 .50 .50 .40 .40 .30 .30 .20 .20 .10 .10­2 ­1 0 1 2 3 4Yield spread lagged two quarters0­2 ­1 0 1 2 3 4Yield spread lagged two quartersPanel B of Figure 2 compares the shape constrained non parametric estimatorwith that from the GDC dual index Probit model. The importantdi¤erences here are that the GDC dual index Probit model misses the ‘‡atspot’that occurs when the slope of the yield curve is between 0 and 2.5 per21

cent. A consequence of this is that the GDC dual index Probit is a poorapproximation to the non parametric estimate of the probability of exitinga recession. The “‡at spot” cited above is of considerable policy relevancebecause it says that unless they can get the slope of the yield curve above2.5 per cent, policy makers in the United States have little chance of raisingthe probability that the economy exits a recession above 50 per cent.5.2.2 Probability of continuing in an expansion that has lasted atleast two quartersFigure 3 plots the probability of continuing in an expansion that has lastedat least two quarters conditional on the yield spread lagged two quarters.Panel A compares the estimated constrained to be monotonic with the unconstrainedestimate. There are relatively minor di¤erences between thesetwo methods. The most notable di¤erence is that the unconstrained methodoverestimates the probability of remaining in an expansion when the slopeof the yield curve is below1 per cent.Panel B of Figure 3 compares the constrained non parametric estimatewith that from the GDC dual index Probit model it is clear that the lattermodel is too restrictive. Over the range of yield spreads experienced in theUnited States the GDC Probit Over predicts the probability of remaining in an expansion for yieldspreads of less than -1 per cent; Under predicts the probability of remaining in an expansion for yieldspreads of between -1 and 0.5 per cent; .22

Probability of expansionProbability of expansionFigure 3: Probability continue in an expansion that has lasted at least twoquarters1Pa ne l A1Panel B0 .90 .90 .80 .80 .70 .70 .60 .60 .50 .50 .40 .40 .30 .30 .20 .20 .1Monotonicity imposedunconstrained0­2 ­1 0 1 2 3 4Yield spread lagged two quarters0 .1Monotonicity imposedGDC probit0­2 ­1 0 1 2 3 4Yield spread lagged two quarters6 Predicted probability of recessionAnalysts use dynamic categorical models to predict the probability that theeconomy will go into recession. Figure 4 is typical of the graphical devicesused in the literature to compare the predictions of several competing models.Panel A shows that the static Probit model misses the 1960, 1970 and 1990recessions, provided a weak signal of the mid 1970s recession and provided asomewhat stronger signal for the two recessions of the early 1980s.The single index GDC Probit shown in panel B …ts the data a little betterbut tends to provide a de…nitive recession signal after the recession has ended.The double index GDC Probit shown in panel C does better than the twoother Probits at …tting the business cycle states but this mainly comes aboutthrough better …tting the business cycle state away from turning points. The23

E(S t|sp t­ 2)E(S t|sp t­ 2)E(S t|sp t­ 2)E(S t|sp t­ 2)Figure 4: Predicted probability of recession various models, 1959.3 to 1995.11Panel A: Static Probit1Panel B: GDC single index Probit0.80.80.60.60.40.40.20.201960 1970 1980 1990 2000Year01960 1970 1980 1990 2000Year1Panel C: GDC double index Probit1Panel D: GDC Non parametric monotonic0.80.80.60.60.40.40.20.201960 1970 1980 1990 2000Year01960 1970 1980 1990 2000Year24

double index GDC model estimated non parametrically with shape restrictionsis shown in panel D. It …ts the business cycle states, away from turningpoints, a little better than the double index GDC Probit.The reason for the quali…cation “away from turning points” above isshown in table 6 which shows Pr(S t = 0jsp t2 ) for the static Probit modeland Pr(S t = 0jS t 1 = 1; S t 2 = 1; sp t 2 ) for various GDC models in the …rstquarter of each recession .Except for the two recessions of the 1980s allmodels fail to achieve a probability of recession greater than one-half in the…rst quarter of the recession.Table 6: P r(S t = 0jS t 1 = 1; S t 2 = 1; sp t 2 ): First quarter of recessionDate Probits Non parametricStaticGDCSingle index Double index1960.Q3 0.18 0.14 0.04 0.051970.Q1 0.35 0.28 0.13 0.091974.Q1 0.59 0.48 0.36 0.51980.Q2 0.70 0.59 0.50 0.601981.Q4 0.71 0.60 0.51 0.601990.Q4 0.15 0.12 0.03 0.04Mean 0.45 0.37 0.26 0.31Std dev 0.25 0.22 0.22 0.28The average predicted probability of recession varies from 0:22 for theGDC Probits to 0:28 for the GDC estimated non parametrically with phaseand shape restrictions. Although these probabilities are low they are substantiallyhigher than Pr(S t = 0jS t 1 = 1; S t 2 = 1) of 0:05 in Table 1demonstrating that the yield spread plays a useful role in predicting recessions.25

The predicted probability of S t = 0 in the second quarter of recessionis shown in Table 7. On average, the static Probit yields a probability ofrecession of 0.37 per cent even though the economy is known to be in thesecond quarter of a recession.All of the GDC models yield a predictedprobability of being in recession of 1 — this is because they are designed tohave this feature.Table 7: P r(S t = 0jS t 1 = 1; S t 2 = 0; sp t 2 )Date Probits Non parametricStaticGDCSingle index Double index1960.Q4 0.09 1 1 11970.Q2 0.33 1 1 11974.Q2 0.49 1 1 11980.Q3 0.72 1 1 11982.Q1 0.50 1 1 11991.Q1 0.11 1 1 1Mean 0.37 1 1 1Std dev 0.25 0 0 0The four recessions with durations of three quarters or more provide usefulinformation on the various models studied here. As is shown in Table 8the static Probit yields an average predicted probability of recession of 0.21for these cases which is a little better than the single index GDC Probit.The double index GDC Probit and non parametric estimate yield predictedprobabilities of recession of 0.62 and 0.66 respectively.These should becompared with the P r (S t = 0jS t 1 = 0; S t 2 = 0) = 0:6 in Table 1. Thisneeds to be interpreted with some caution and sophistication as it does notmean that the GDC model is useless at predicting or understanding how theyield spread in‡uences the probability of exit from recession. Inspection of26

Figure 2 shows that there is a “‡at spot”in the exit probability curve whichcoincides with the range of yield spreads observed during recessions. It isonly when policy makers push the yield spread is above this range that theeconomy has a high probability of exiting from recession.Table 8: P r(S t = 0jS t 1 = 0; S t 2 = 0; sp t 2 ): Third quarter of recessionDate Probits Non parametricStaticGDCSingle index Double index1961.Q1 0.07 0.06 0.46 0.551970.Q3 0.28 0.22 0.77 0.651974.Q3 0.45 0.36 0.88 0.911982.Q2 0.04 0.04 0.38 0.53Mean 0.21 0.17 0.62 0.66Std dev 0.19 0.15 0.24 0.17In summary, once a recession is underway the GDC models do very wellat getting the second quarter of the recession but this is because the modelshave been constructed to capture that feature of the data. The double indexGDC models do well in the third and subsequent quarters of the recessionand this is where much of the improvement in …t comes from. The lessonthat I take away from this is an old one that improved …t of a model doesnot necessarily produce improvement in economically relevant or valuableforecasts.The economically relevant events are the turning points rather than thestates S t and thus a lesson from the analysis above is that to achieve improvedforecasts of these events it may be worth focusing attention on predictingquantities such as S t 1 (1 S t ) and (1 S t 1 ) S t which take the values 1 ifthere is a turning point at t and zero otherwise.27

It is also instructive to compare the predictions of the various modelsin the …rst quarter of an expansion as is done in Table 9. The static probitmodel and the single index GDC probit yield predictions of the probability ofexpansion that average 0.88 and 0.90 respectively. This does not mean thatsuch models are useful as they also yielded these predictions in the earlierstages of the recession — indeed these predictions are not far away fromthe unconditional probability of expansion of 0.85 shown in Table 1. Theparametric and non parametric GDC models yield average probabilities thatthe economy is in expansion of 0.49 and 0.53 which should be compared withthe 0.40 in Table 1. These results suggest that the yield spread makes onlya modest contribution to the predicting the on set of an expansion.Table 9: P r(S t = 1jS t 1 = 0; S t 2 = 0; sp t 2 ):First quarter of expansionDate Probits Non parametricStaticGDCSingle index Double index1961.Q2 0.94 0.95 0.56 0.461971.Q1 0.88 0.90 0.43 0.431975.Q2 0.75 0.80 0.25 0.391980.Q4 0.82 0.85 0.33 0.431983.Q1 1.00 1.00 0.89 1.001991.Q4 0.91 0.92 0.48 0.44Mean 0.88 0.90 0.49 0.53Std dev 0.09 0.07 0.22 0.237 ConclusionAllowing for the phase restrictions imposed by the method in which the datais constructed can improve the …t of DDC models of the business cycle. Tak-28

ing into account the fact that double index models are required to matchNBER data also improves …t. Shape restrictions implied by theory are automaticallyimposed in parametric models because they are inherited fromthe properties of probability distributions. Non parametric methods do notautomatically impose these shape restrictions. Procedures were developed toimpose these shape restrictions on non parametric estimators. In the applicationit was shown that imposing these shape restrictions did not materiallyworsen the …t of the model. The double index non parametric GDC modelwas shown to have the best …t to the business cycle states and was superiorto the various Probit models. However, it was shown that this improved …twas achieved, primarily, through better …t away from turning points and didnot bring as large an improvement in the prediction of turning points.ReferencesCressie, N.A.C., T.R.C. Read. (1984). Multinomial goodness-of-…t tests.Journal of the Royal Statistical Society, Series B 46, 440-464.Estrella, A. and F.S. Mishkin. (1998). “Predicting US Recessions: FinancialVariables as Leading Indicators”, Review of Economics and Statistics,LXXX, 28-61.Friedman, J. H. and R. J. Tibshirani. (1984). The monotone smoothingof scatterplots. Technometrics 26 243–250.Hall, P., and L.,S. Huang. (2001) "Nonparametric Kernel RegressionSubject to Monotonicity Constraints", The Annals of Statistics, Vol 29 No.3 pp 624-647.29

Harding, D., and A.,R. Pagan (2009). "An Econometric Analysis of SomeModels for Constructed Binary Time Series", NCER working paper No 39,July.Henderson, D.J., and C.,F., Parmenter (2009). “Imposing Economic Constraintsin Nonparametric Regression: Survey, Implementation and Extension,IZA Discussion Paper No 4103, March.Hurvich, C.M., J.S. Simono¤ and C.L. Tsai. (1998). Smoothing parameterselection in non parametric regression using an improved Akaikeinformation criterion. Journal of the Royal Statistical Society Series B 60,271-293.Li, Q. and J.S. Racine. (2004). “Cross-validated local linear non parametricregression”, Statistica Sinica, 14(2), 458-512.Li, Q. and J.S. Racine. (2009). Nonparametric Econometrics, PrincetonUniversity Press, Princeton.Racine, J.S., Parmeter, C.F., and P. Du. (2009). “Constrained NonparametricKernel Regression: Estimation and Inference, mimeo, Department ofEconomics, University of Victoria, Canada.30

List of NCER Working PapersNo. 57 (Download full text)Renee Fry and Adrian PaganSign Restrictions in Structural Vector Autoregressions: A Critical ReviewNo. 56 (Download full text)Mardi Dungey and Lyudmyla HvozdykCojumping: Evidence from the US Treasury Bond and Futures MarketsNo. 55 (Download full text)Martin G. Kocher, Marc V. Lenz and Matthias SutterPsychological pressure in competitive environments: Evidence from a randomized naturalexperiment: CommentNo. 54 (Download full text)Adam Clements and Annastiina SilvennoinenEstimation of a volatility model and portfolio allocationNo. 53 (Download full text)Luis Catão and Adrian PaganThe Credit Channel and Monetary Transmission in Brazil and Chile: A Structured VARApproachNo. 52 (Download full text)Vlad Pavlov and Stan HurnTesting the Profitability of Technical Analysis as a Portfolio Selection StrategyNo. 51 (Download full text)Sue Bridgewater, Lawrence M. Kahn and Amanda H. GoodallSubstitution Between Managers and Subordinates: Evidence from British FootballNo. 50 (Download full text)Martin Fukac and Adrian PaganStructural Macro‐Econometric Modelling in a Policy EnvironmentNo. 49 (Download full text)Tim M Christensen, Stan Hurn and Adrian PaganDetecting Common Dynamics in Transitory ComponentsNo. 48 (Download full text)Egon Franck, Erwin Verbeek and Stephan NüeschInter‐market Arbitrage in Sports BettingNo. 47 (Download full text)Raul CarusoRelational Good at Work! Crime and Sport Participation in Italy. Evidence from Panel DataRegional Analysis over the Period 1997‐2003.

No. 46 (Download full text) (Accepted)Peter Dawson and Stephen DobsonThe Influence of Social Pressure and Nationality on Individual Decisions: Evidence from theBehaviour of RefereesNo. 45 (Download full text)Ralf Becker, Adam Clements and Christopher Coleman‐FennForecast performance of implied volatility and the impact of the volatility risk premiumNo. 44 (Download full text)Adam Clements and Annastiina SilvennoinenOn the economic benefit of utility based estimation of a volatility modelNo. 43 (Download full text)Adam Clements and Ralf BeckerA nonparametric approach to forecasting realized volatilityNo. 42 (Download full text)Uwe Dulleck, Rudolf Kerschbamer and Matthias SutterThe Economics of Credence Goods: On the Role of Liability, Verifiability, Reputation andCompetitionNo. 41 (Download full text)Adam Clements, Mark Doolan, Stan Hurn and Ralf BeckerOn the efficacy of techniques for evaluating multivariate volatility forecastsNo. 40 (Download full text)Lawrence M. KahnThe Economics of Discrimination: Evidence from BasketballNo. 39 (Download full text)Don Harding and Adrian PaganAn Econometric Analysis of Some Models for Constructed Binary Time SeriesNo. 38 (Download full text)Richard DennisTimeless Perspective Policymaking: When is Discretion Superior?No. 37 (Download full text)Paul Frijters, Amy Y.C. Liu and Xin MengAre optimistic expectations keeping the Chinese happy?No. 36 (Download full text)Benno Torgler, Markus Schaffner, Bruno S. Frey, Sascha L. Schmidt and Uwe DulleckInequality Aversion and Performance in and on the FieldNo. 35 (Download full text)T M Christensen, A. S. Hurn and K A LindsayDiscrete time‐series models when counts are unobservable

No. 34 (Download full text)Adam Clements, A S Hurn and K A LindsayDeveloping analytical distributions for temperature indices for the purposes of pricingtemperature‐based weather derivativesNo. 33 (Download full text)Adam Clements, A S Hurn and K A LindsayEstimating the Payoffs of Temperature‐based Weather DerivativesNo. 32 (Download full text)T M Christensen, A S Hurn and K A LindsayThe Devil is in the Detail: Hints for Practical OptimisationNo. 31 (Download full text)Uwe Dulleck, Franz Hackl, Bernhard Weiss and Rudolf Winter‐EbmerBuying Online: Sequential Decision Making by Shopbot VisitorsNo. 30 (Download full text)Richard DennisModel Uncertainty and Monetary PolicyNo. 29 (Download full text)Richard DennisThe Frequency of Price Adjustment and New Keynesian Business Cycle DynamicsNo. 28 (Download full text)Paul Frijters and Aydogan UlkerRobustness in Health Research: Do differences in health measures, techniques, and timeframe matter?No. 27 (Download full text)Paul Frijters, David W. Johnston, Manisha Shah and Michael A. ShieldsEarly Child Development and Maternal Labor Force Participation: Using Handedness as anInstrumentNo. 26 (Download full text)Paul Frijters and Tony BeattonThe mystery of the U‐shaped relationship between happiness and age.No. 25 (Download full text)T M Christensen, A S Hurn and K A LindsayIt never rains but it pours: Modelling the persistence of spikes in electricity pricesNo. 24 (Download full text)Ralf Becker, Adam Clements and Andrew McClellandThe Jump component of S&P 500 volatility and the VIX indexNo. 23 (Download full text)A. S. Hurn and V.PavlovMomentum in Australian Stock Returns: An Update

No. 22 (Download full text)Mardi Dungey, George Milunovich and Susan ThorpUnobservable Shocks as Carriers of Contagion: A Dynamic Analysis Using IdentifiedStructural GARCHNo. 21 (Download full text) (forthcoming)Mardi Dungey and Adrian PaganExtending an SVAR Model of the Australian EconomyNo. 20 (Download full text)Benno Torgler, Nemanja Antic and Uwe DulleckMirror, Mirror on the Wall, who is the Happiest of Them All?No. 19 (Download full text)Justina AV Fischer and Benno TorglerSocial Capital And Relative Income Concerns: Evidence From 26 CountriesNo. 18 (Download full text)Ralf Becker and Adam ClementsForecasting stock market volatility conditional on macroeconomic conditions.No. 17 (Download full text)Ralf Becker and Adam ClementsAre combination forecasts of S&P 500 volatility statistically superior?No. 16 (Download full text)Uwe Dulleck and Neil FosterImported Equipment, Human Capital and Economic Growth in Developing CountriesNo. 15 (Download full text)Ralf Becker, Adam Clements and James CurchinDoes implied volatility reflect a wider information set than econometric forecasts?No. 14 (Download full text)Renee Fry and Adrian PaganSome Issues in Using Sign Restrictions for Identifying Structural VARsNo. 13 (Download full text)Adrian PaganWeak Instruments: A Guide to the LiteratureNo. 12 (Download full text)Ronald G. Cummings, Jorge Martinez‐Vazquez, Michael McKee and Benno TorglerEffects of Tax Morale on Tax Compliance: Experimental and Survey EvidenceNo. 11 (Download full text)Benno Torgler, Sascha L. Schmidt and Bruno S. FreyThe Power of Positional Concerns: A Panel Analysis

No. 10 (Download full text)Ralf Becker, Stan Hurn and Vlad PavlovModelling Spikes in Electricity PricesNo. 9 (Download full text)A. Hurn, J. Jeisman and K. LindsayTeaching an Old Dog New Tricks: Improved Estimation of the Parameters of StochasticDifferential Equations by Numerical Solution of the Fokker‐Planck EquationNo. 8 (Download full text)Stan Hurn and Ralf BeckerTesting for nonlinearity in mean in the presence of heteroskedasticity.No. 7 (Download full text) (published)Adrian Pagan and Hashem PesaranOn Econometric Analysis of Structural Systems with Permanent and Transitory Shocks andExogenous Variables.No. 6 (Download full text) (published)Martin Fukac and Adrian PaganLimited Information Estimation and Evaluation of DSGE Models.No. 5 (Download full text)Andrew E. Clark, Paul Frijters and Michael A. ShieldsIncome and Happiness: Evidence, Explanations and Economic Implications.No. 4 (Download full text)Louis J. Maccini and Adrian PaganInventories, Fluctuations and Business Cycles.No. 3 (Download full text)Adam Clements, Stan Hurn and Scott WhiteEstimating Stochastic Volatility Models Using a Discrete Non‐linear Filter.No. 2 (Download full text)Stan Hurn, J.Jeisman and K.A. LindsaySeeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate theParameters of Stochastic Differential Equations.No. 1 (Download full text)Adrian Pagan and Don HardingThe Econometric Analysis of Constructed Binary Time Series.