Detecting and predicting recessions

Transcription

1 ISSN EWP 2011/012 Euroindicators working papers Detecting and predicting recessions Don Harding

2 This paper was presented at the 6th Eurostat Colloquium on Modern Tools for Business Cycle Analysis: the lessons from global economic crisis, held in Luxembourg, 26th - 29th September Click here for accessing the collection of the 6th Eurostat Colloquium papers Click here for accessing the full collection of Euroindicators working papers Europe Direct is a service to help you find answers to your questions about the European Union Freephone number (*): (*) Certain mobile telephone operators do not allow access to numbers or these calls may be billed. More information on the European Union is available on the Internet ( Luxembourg: Publications Office of the European Union, 2012 ISSN EWP 2011/012 Doi: / Theme: General and regional statistics European Union, 2012 Reproduction is authorised provided the source is acknowledged. The views expressed in this publication are those of the authors, and do not necessarily reflect the official position of Eurostat.

3 Detecting and Predicting Recessions Don Harding September 26, 2010 Abstract Skill in detecting and predicting recessions is essential to the success of macroeconomic stabilization policy. This paper develops a framework for detecting recessions and shows howit can be applied to evalute the capacity of models to forecast recessions. The framework comprises two indicator functions that de ne the beginning and end of recessions together with a recursive equation that generates the binary business cycles states S t. By choosing the indicator function one speci es how turning points are related to measures of economic activity y t. The user can also specify model(s) M j that specify the DGP of fy t ; x t g in terms of F t i the history of fy t i ; x t i ; S t i g : The DGP of S t is then induced from the DGP of fy t ; x t g : This framewor is adequate for dealing with normal business cycle events, an extension is provided that can deal with great recessions and great depressions and issues related to double dips. The system described above is applied to study the US business cycle. One application looks at the capacity of an AR(1) in growth rates of GDP to correctly predict the state of the business cycle. A second application investigates how capacity to predict S t varies with the information set used in prediction. The third application is to understand the correct way to predict recessions using Markov switching models and to compare the quanitative importance of this correct method with the incorrect method that is currently used when Markov switching methods are used to predict recessions. The fourth application is to evaluate what Markov switching models add to prediction over a simple AR(1) in growth rates of GDP. School of Economics and Finance, La Trobe University, Melbourne Australia. d.harding@latrobe.edu.au. 1

4 Key Words: Recession, Business cycle; forecasting. JEL Code C22, C53, E32 2

5 Contents 1 Introduction 4 2 De ning and detecting recessions The core of dating algorithms Approximating the NBER and BBQ quarterly chronologies Advanced censoring Minimum completed cycle lengths Allowing for double dips in great recessions Approximating the DGP of S t for prediction Nowcasting the NBER business cycle states using GDP Example 1: Gaussian AR(1) Example 2: Markov switching Predicting a recession based typical information Forecasting recessions based on F t Application to the US business cycle Parameters Example 1: AR(1) in growth rate of GDP Case 1: Perfect now casting of y t Case 2: Typical information set F t Case 3: Forecasting recessions based on F t Example 2: Markov switching in growth rate of GDP Case 1 Comparison of P r (S t = 1jF t ) with P r (Z t = 1jy t : : :) Case 2: Comparison of P r (S t = 1jF t ) for the AR(1) and MS models Forecasting in great recessions 20 6 Conclusions 20 3

6 1 Introduction In the wake of the Global Financial Crisis and subsequent North Atlantic great recession, the IMF-FSB, governments, central banks and many academic economists are engaged in research aimed at improved prediction of such events. Behind these activities is the view that recessions and crises can be detected and predicted with a lead that is greater than the policy lag thereby allowing policy makers to design and implement counter cyclical macroeconomic policy. This paper develops a framework for comparing models in terms of their capacity to predict recessions. The framework allows the investigator to choose the particular de nition of recession that they prefer. Models can then be compared in terms of their predictive lead for that de nition of a recession. Issues that arise in de ning and detecting recessions are canvassed in section 2. Particular attention is paid to committees such as the NBER dating committee and algorithms such as that developed by Bry and Boschan (1971). All methods for detecting recessions generate a binary variables S t which take the value zero in recessions and the value one otherwise. It is shown in section 2 that the S t produced by committees and algorithms can be approximated by a set of recursive equations. It is useful in forecasting applications to work with approximations to the DGP for S t that captures important features while reducing the degree of complexity in the DGP of S t. Several useful approximations are discussed in section 3. These approximations are made by taking the conditional expectation of the recursive equations developed in section 2 and have the advantage that they can be applied to S t irrespective of whether that chronology is constructed by committee or by algorithm. The approximations to the DGP of S t obtained in section 3 are applied to the analysis of the United States business cycle in section 4. Issues that arise in forecasting within great recessions are discussed in section 5. Conclusions are presented in section 6. 2 De ning and detecting recessions The causes recessions and how they are propagated remains unresolved. For this reason it is important to de ne the business cycle in a way that does 4

7 not prejudge either its nature or it cause(s); Harding and Pagan (2008). Recognition of this was central to the work of Burns and Mitchell (1946 p3) who proceed by associating the business cycle with uctuations in aggregate economic activity which they recognized could be de ned via turning points in an aggregate such as GDP or by aggregating turning points as discussed in Harding and Pagan (2002,2006). Turning these de nitions into operational procedures requires judgement which can be exercised through a committee or it can be embodied in a algorithm that is subsequently applied to date the business cycle. The former approach is taken by the NBER business cycle dating committee. 1 The latter approach is taken in the Bry Boschan (1971) algorithm. The aim of this section is to develop a better understanding of what algorithms and dating committees actually do so that researchers can embed that understanding in applied work using the data constructed by committees and algorithms. An important step in applied work is to develop a system of recursive approximations to the dating procedures used by committees and embedded in algorithms. The purpose of such approximations is not to replace the algorithm or committee rather it is to inform the speci cation of econometric models that are su ciently sophisticated as to yield a reasonable approximation to the DGP for S t. The decisions made by business cycle dating committees and algorithms can be separated into ve component parts 2 : 1. location of candidate dates for the beginning and end of recessions (turning points); 2. censoring to ensure that phases alternate; 3. censoring to ensure that recessions (and expansions) last longer than a minimum period of time; 4. censoring to ensure that completed cycles have a minimum duration; and 5. censoring to ensure that each peak is higher than the previous peak and each trough is higher than the previous trough. The rst three parts form the core of business cycle dating procedures and are discussed in section 2.1 below. The last two parts (points 4 and 5) are part of advanced censoring procedures discussed section The CEPR also has a business cycle dating committee for the Euro area. 2 Some of the decisions related to censoring are not necessarily consiously made by the committees but are an artifact of their procedures. 5

8 2.1 The core of dating algorithms All business cycle dating algorithms (and committees) have at their core procedures that can be written mathematically as the three equation recursion relation ^t = S t 1 1fA t g (1) q 2 _ t = (1 S t 1 ) 1fB t g (2) q Y 1 S t = (1 ^t 1 ) S t j + _ t Y 1 (1 S t j ) j=1 j=1 +f (S t 1 ; : : : ; S t q1 ) (3) where 1fX t g is an indicator function that takes the value one if the logical statement X t is true at date t and the value zero otherwise. The parameters q 1 and q 2 control the minimum duration of expansion phases and recessions respectively. In equation (1) A t is a statement that the expansion is continuing at date t and in equation (2) B t is a statement that the recession ends at date t. The term f (S t 1 ; : : : ; S t q1 ) ensures that expansions have minimum duration q 1 periods. It takes the following form f (S t 1 ; : : : ; S t q1 ) = 0 if q 1 = 1 (4) = q 1 1 Y j=1 S t j (1 S t q1 ) What this core of dating algorithms does is rst select candidate turning points, ensure that expansion and contraction phases of the business cycle alternate and ensure that completed phases have a minimum duration. It is worth noting that while the censoring described above is encoded in the BB and BBQ algorithms it also arises naturally out of the procedures used by dating committees. To understand this latter point let r represent the maximum of the data publication lags for the series used by the committee. Then dating committees must wait k + r periods before they have the minimum information necessary to determine a turning point using (1), (2) and (3). Thus committees will have looked a at monthly data at least k periods into the future and daily nancial data k + r periods into the future. This is because committees are likely to view double dip recessions with recessions shorter than k + r periods to be part of a single recession event. This means that, even if they don t have a speci c censoring rule, committees 6

9 are unlikely to construct chronologies with phases shorter than k + r periods Approximating the NBER and BBQ quarterly chronologies To approximate the NBER (and hence BBQ) quarterly chronology using equation (1) to (3) set k = 2 and q 1 = q 2 = 2 and assume that the chronology is constructed as the reference cycle in GDP. 3 A t corresponds to the event that the expansion continues at date t which is equivalent to not the event that y t 1 > (y t ; ; y t+1 ): It is convenient to break y t 1 > (y t ; ; y t+1 ) up into two parts y t < 0 and y t + y t+1 < 0:Then, we can write the indicator function 1 (A t ) = 1 1 (y t < 0) 1 (y t + y t+1 < 0) (5) In (2), B t is the event that there is a trough at time t function for this event is 1, the indicator 1 (B t ) = 1 (y t > 0) 1 (y t + y t+1 > 0) (6) And S t is generated via equations (1), (2) and (3) with the indicator functions (5) and (6). 2.2 Advanced censoring The BB and BBQ algorithms are not recursive, rather they can look at the whole time series if necessary when making censoring decisions. 4 Obviously, it is not possible to exactly embed these features in a system of recursive equations such as (1) to (3) Minimum completed cycle lengths There is a very simple way to achieve an approximation that imposes minimum cycle lengths which is to choose q 1 and q 2 so that q 1 + q 2 is equal to the imposed minimum completed cycle length. For example, when studying the 3 If one wants to use the clustering of turning points de nition of a business cycle then one can use the reference cycle algorithm developed by Harding and Pagan (2006). In this case A t is the statement that the centre of the cluster of speci c cycle peaks is not at date t. Formally, let d t be the median distance to the nearest speci c cycle peaks then A t is the statement that it is not the case that d t < min(d t1 ; ; d th ) where h is the window width used in the reference cycle. Similarly, let d t be the median distance to the nearest speci c cycle troughs then B t is the statement that d t < min(d t1; ; d th ). 4 This allows the algorithm to choose which pair of ajoining turning points to remove when imposing the minimum cycle length restriction. 7

10 classical cycle one could choose to set q 1 = 6 months (2 quarters) and q 2 = 9 months (3 quarters) Allowing for double dips in great recessions Double dip recessions are those where economic activity follows the pattern shown in Figure which graphs real United States GDP over the great depression and subsequent recovery. There are candidate peaks at p 1 through p 4 and candidate troughs at t 1 through t 4. Without modi cation the recursive equations (1) to (3) will accept all of the candidate turning points in Figure 1. This stands in stark contrast with the decisions of Burns and Mitchel who placed peaks at p 1 and p 4 and troughs at t 2 and t 4. Figure 1: United States real GDP over the course of the great depression p 1 p p 2 p 3 t t t t The reason that Burns and Mitchell made this choice is that they viewed the great depression as a single event rather than a series of recessions and recoveries. The criteria that they imposed to determine what is a single recession event is this: the level of economic activity must rise above the previous peak before a trough is called. The NBER business cycle dating committee refers to this rule on its web page stating that. The NBER does not de ne a special category called a doubledip recession. Two periods of contraction either will be two separate recessions or parts of the same recession. One criterion that the Committee applies is whether economic activity surpassed its previous peak between the two contractions. For example, the Committee s determination that the recession that began in 8

11 1980 was separate from the one that began in 1981 was based in part on the fact that major indicators bounced back in late 1980 and early 1981 su ciently far that economic activity had surpassed its 1980 peak before the recession of 1981 began. The Committee has not encountered any other episode since its inception in 1978 that involved two consecutive contractions. The Committee does not apply xed formulas in this and other determinations, but rather forms judgments based on the underlying concepts of recessions and expansions and the goal of preserving historical continuity in the NBER business cycle chronology. Source accessed 13 September To incorporate this de nition in an approximation to NBER procedures we need to extend the three equation recursion in the following way. First let lp t represent the the distance, in time, from t to the last peak. Also let np t represent the the distance, in time, from t to the next peak. De ne lp t and np t as follows lp t = lp t (1 ^t) (7) Then modify equation (2) to np t = min fj^t+ = 1g (8) _ t = (1 S t 1 ) 1fB t g 1 n o y t+ np > y t t lp t These modi cations introduce forward and backward looking components into the recursive equations (7), (8), (1), (9) and (3). One important insight from this is that if one is undertaking simulation based forecasts working with algorithms that have this advanced censoring feature then it is important to ensure that the forecast horizon is su ciently large as to ensure that the censoring procedures are not in uencing the next peak. A useful way to do this is to calculate T which is de ned as n o T = min jy T + > y T (10) The distribution of T can then guide the choice of the forecast horizon for the particular data generating process. An important point to make is that the recursive equations (7), (8), (1), (9) and (3) do not represent substitutes for dating algorithms such as BB and BBQ or for committees such as NBER BCDC rather they are approximations 9 lp t (9)

12 that may be useful in forecasting exercises and in making the operations of such committees more transparent. 3 Approximating the DGP of S t for prediction An advantage of the approach to approximating the rules for detecting business cycles embodied in equations (1) to (3) and equations (7), (8), and (9) is that predictions about the probability S t+i = 1 conditional on information set F t j = fs t j 1 ; S t j 2 ; x t j g can be made by taking the expectation of S t+i conditional on F t j : This set up provides a common framework for discussing nowcasting (i = j = 0), forecasting (i > 0; j > 0) and backcasting (i < 0) applications. In most cases E(S t+i jf t j ) will need to be calculated via numerical methods. However, in certain cases it is possible to obtain analytic results these include a Gaussian AR(1) in y t and a simple Markov switching model. These analytic examples are set out in some detail below to serve as illustrations of how the method works. They also provide the information necessary to discuss the statement it is only within a regime-switching framework that the concept of a turning point has intrinsic meaning. In linear frameworks, by way of contrast, there are no turning points (Diebold and Rudebusch 1999, p15). The analytic examples provided below allow for a precise evaluation of the merits of Diebold and Rudebusch s stance. 3.1 Nowcasting the NBER business cycle states using GDP Here attention is focused on Pr (S t = 1jF t ) where the maximum feasible information is available at time t about the histories of fs t 1 ; y t ; x t g where x t is a vector of relevant variables: That is F t = fs t 1 ; S t 2 ; : : : ; y t ; y t 1 ; : : : ; x t ; : : :g. One might think of F t as the information set that would apply if one had perfect nowcasting of y t so this it represents an idealized situation. In this case the optimal forecast of S t is E (S t jf t ) which is E [1 (A t ) jf t ] = 1 1 (y t < 0) E [1 (y t + y t+1 < 0) jf t ] (11) 10

13 E [1 (B t ) jf t ] = 1 (y t > 0) E [1 (y t + y t+1 > 0) jf t ] (12) E (S t jf t ) = S t 1 S t 2 E [1 (A t ) jf t ] + S t 1 (1 S t 2 ) + (1 S t 1 ) (1 S t 2 ) E [1 (B t ) jf t ] (13) Equations (11) to (13) can be used with any model of y t it could be univariate or multivariate. The examples below relate to the cases where y t follows an AR(1) and a Markov switching model Example 1: Gaussian AR(1) Here y t is a Gaussian AR process parameterized as follows y t = + y t 1 + " t " t ~iid N (0; 1) (14) Then E [1 (y t + y t+1 0) jf t ] = Pr (y t+1 y t ) (15) (1 + ) yt = and E [1 (y t + y t+1 > 0) jf t ] = Pr (y t+1 > y t ) (16) (1 + ) yt = 1 Combing these yields the equation for E (S t jf t ) when y t Gaussian AR(1). contains a (1 + ) yt E (S t jf t ) = S t 1 S t (y t < 0) (17) +S t 1 (1 S t 2 ) (1 + ) yt + (1 S t 1 ) (1 S t 2 ) 1 (y t > 0) 1 Notice that equations (15) and (16) when combined with (11) to (13) generate a set of turning points de ned for a gaussian AR(1) process. Thereby 11

14 showing that it is indeed possible to de ne turning points within a linear model Example 2: Markov switching The simplest form of the Markov switching process is y t = Z t + " t " t ~iid N (0; 1) Z t is binary Markov process of order one. Pr (Z t = 1jZ t 1 = 1) = p 11 and Pr (Z t = 0jZ t 1 = 0) = p 00 Can be written as Z t = Z t 1 + t 0 = 1 p 00 ; 1 = p 11 + p 00 1 So to construct E (S t jf t ) via () () using the MS model de ne E [1 (A t ) jf t ] = 1 1 (y t < 0) P t and E [1 (B t ) jf t ] = 1 (y t > 0) (1 P t ) where P t = E [1 (y t + y t+1 0) Now using the features of the MS model yt P t = p 11 Pr (Z t = 1jF t ) yt (1 p 00 ) Pr (Z t = 0jF t ) yt + 0 p 00 Pr (Z t = 0jF t ) yt + 0 (1 p 11 ) Pr (Z t = 1jF t ) (18) Then E (S t jf t ) where y t is generated by a Markov switching model is given in equations (19) and (18) E (S t jf t ) = S t 1 S t 2 (1 1 (y t < 0) P t ) + S t 1 (1 S t 2 ) + (1 S t 1 ) (1 S t 2 ) 1 (y t > 0) (1 P t ) (19) Note that (19) is evidently di erent from the rule that S t = 1 Pr (Z t = 1jF t ) > 0:5): The results above complement those in Harding and Pagan (2004) who used approximations to show the dating rule implied by a particular markov switching model. 12

15 3.2 Predicting a recession based typical information Typically, information on fy t ; S t 1 ; x t g is not available when making predictions about S t. A more typical information set available at time t is F t 1 = fy t 1 ; : : : ; S t 2 ; S t 3 ; : : : ; x t g It is useful to proceed by lagging S t to obtain the equation for S t 1 in (20) and substituting into the equation for S t S t 1 = S t 2 S t 3 1 (A t 1 ) + S t 2 (1 S t 3 ) (20) After some rearranging we obtain + (1 S t 2 ) (1 S t 3 ) 1 (B t 1 ) S t = S t 2 S t 3 (21) S t 2 S t 3 [1 (A t ) + 1 (A t 1 )] +S t 2 S t 3 1 (A t ) 1 (A t 1 ) +S t 2 (1 S t 3 ) 1 (A t ) + (1 S t 2 ) (1 S t 3 ) 1 (B t 1 ) + (1 S t 2 ) 1 (B t ) (1 S t 2 ) (1 S t 3 ) 1 (B t 1 ) 1 (B t ) Now taking expectations conditional on F t 1 we obtain E (S t jf t 1 ) = S t 2 S t 3 S t 2 S t 3 [E (1 (A t ) jf t 1 ) + E (1 (A t 1 ) jf t 1 )] (22) +S t 2 S t 3 E [1 (A t ) 1 (A t 1 ) jf t 1 ] +S t 2 (1 S t 3 ) E [1 (A t ) jf t 1 ] + (1 S t 2 ) (1 S t 3 ) E [1 (B t 1 ) jf t 1 ] + (1 S t 2 ) E [1 (B t ) jf t 1 ] (1 S t 2 ) (1 S t 3 ) E [1 (B t 1 ) 1 (B t ) jf t 1 ] (23) Again consider the example where y t follows the AR(1) in (14). Then in (22) E (1 (A t ) jf t 1 ) = Z 0 1 (1 + ) yt 1 yt y t 1 (1 + ) yt 1 E (1 (A t 1 ) jf t 1 ) = 1 (y t 1 0) E [1 (A t ) 1 (A t 1 ) jf t 1 ] = 1 (y t 1 0) E (1 (A t ) jf t 1 ) 13 dy t

16 E [1 (B t 1 ) jf t 1 ] = 1 E (1 (A t 1 ) jf t 1 ) E [1 (B t ) jf t 1 ] = H (y t 1 ; ; ; ) Where E [1 (B t 1 ) 1 (B t ) jf t 1 ] = 1 (y t 1 > 0) H (y t 1 ; ; ; ) H (y t 1 ; ; ; ) = Z (1 + ) yt 1 yt y t 1 dy t 3.3 Forecasting recessions based on F t 1 Here we are interested in obtaining forecasts of S t+1 ; S t+2 and S t+3 conditional on F t 1 We proceed as before by substituting in for S t+1, S t and S t 1 until we have an equation expressed in terms of fs t 2 ; S t 3 ; :::g : The resulting equations are rather long and is omitted from the text but are used in the applications that follow. 4 Application to the US business cycle Here I apply the framework above to the US business cycle from to and the logarithm of real GDP is used for y t : The parameterization of the AR(1) and Markov switching models are described in section Parameters The estimated parameters of the AR(1) in y t are shown in table 1. Table 1: Estimated coe cients AR(1) in US GDP growth rates Estimate Se t 0:488 0:075 6:5 0:357 0:062 5:7 0:770 The Markov switching parameters are taken from Hamilton (1989). 14

17 4.2 Example 1: AR(1) in growth rate of GDP In this example the assumed DGP is an AR(1) in y t with the estimated coe cients in table 1. The assumed information set is varied in each case below Case 1: Perfect now casting of y t Here conditioning on F t yields the estimated probability of being in expansion shown in Figure 2 Figure 2: Probability of being in expansion: United States to conditional on F t, Model is AR(1) in growth rates Pr(S t =1 F 0 t ) Clearly, even using a model as simple as an AR(1) in growth rates yields estimated probabilities that are very clearly de ned and that match the NBER dates closely provided the information set is F t. There is only one signi cant false signal and that is in the late 1950s Case 2: Typical information set F t 1 Again using the coe cients of the AR(1) model in table 1 I calculate E (S t jf t 1 ) ; using numerical integration, the results are shown in Figure where the reces- 15

18 sion bars are shown in grey. Figure 3: Probability in NBER expansion conditional on F t to Based on AR(1) in growth rates Pr(S t =1 F 1 t ) Evidently, it is far more di cult to predict whether there economy is in a recession at date t when one allows for the information that is typically available at date t Case 3: Forecasting recessions based on F t 1 Here we are interested in obtaining forecasts of S t+1 ; S t+2 and S t+3 conditional on the information that is typically available at date t viz, F t 1 We proceed as before by substituting in for S t+1, S t and S t 1 until we have an equation expressed in terms of fs t 2 ; S t 3 ; :::g : The resulting forecasts of S t+1 ; S t+2 and S t+3 conditional on F t 1 are shown in gures 4, 5 and 6 respectively. The main point to emerge from this discussion is that our capacity to predict recessions one quarters ahead has diminished markedly. Moreover, there is virtually no capacity to predict recessions two or three quarters ahead. 16

19 1 Figure 4: E(S t+1 jf t 1 ) Pr(S t =1 F 2 t ) Monte Carlo integration Figure 5: E(S t+2 jf t 1 ) Pr(S t =1 F 3 t ) Monte Carlo integration

20 1 Figure 6: E(S t+3 jf t 1 ) Pr(S t =1 F 4 t ) Monte Carlo integration Example 2: Markov switching in growth rate of GDP Here there are several interesting comparisons to make. The Case 1 Comparison of P r (S t = 1jF t ) with P r (Z t = 1jy t : : :) This involves comparison of Pr (S t = 1jF t ) obtained using the approximation to BBQ algorithm to date the cycle and y t forecast from the MS model with the Pr (Z t = 1jy t : : :) obtained directly from the markov switching model. The comparison is shown in Figure 7. It is evident from the gure that the Pr (S t = 1jF t ) provides a more accurate signal of the business cycle than does Pr (Z t = 1jy t : : :) : To con rm this impression calculate the log probability scores (LPS) as follows LP S S (F t ) = TX ln Pr (S t = 1jF t ) S t + ln Pr (S t = 0jF t ) (1 S t ) (24) t=1 LP S Z ( t ) = TX ln Pr (Z t = 1j t ) S t + ln Pr (Z t = 0j t ) (1 S t ) (25) t=1 18

21 Figure 7: Comparison of Pr(S t = 1jF t ) with Pr(Z t = 1jy t ; :::) from Markov switching model Where t = fy t ; y t 1 ; : : : :g On the data used by James Hamilton the relevant values for the Markov switching model are LP S S (F t ) = 15:7 and LP S Z (y t : : :) = 23:7. Thus the data strongly supports the approach developed in this paper over just relying on Pr (Z t = 1jy t : : :) from the MS model Case 2: Comparison of P r (S t = 1jF t ) for the AR(1) and MS models The literature abounds with statements such as that by Diebold and Rudebusch cited earlier claiming that turning points have no intrinsic meaning in linear models but do have such a meaning in regime switching models. Earlier I showed that the this statement is factually wrong. This left open the question of whether regime switching models are superior empirically. This is the question addressed in this section Figure 8 which compares the probability of being in an expansion for both models using the evaluation framework developed earlier. It is evident from the gure that there is little di erence between the t of the two models. This similarity in t is con rmed by the log probability score for the AR(1) model of 17:6 for the same data as used by James Hamilton (1989). This can be compared with the LPS of 15:7 reported above for the Markov switching model. Given that the AR(1) model has 3 parameters and the MS model has 5 parameters some allowance needs to be made for the additional parameters. Using the AIC criteria the relevant values are 41.2 for the AR(1) 19

22 Figure 8: Comparison of P r (S t = 1jF t ) for the AR(1) and MS models model and 41.5 for the MS model indicating that on this criteria parsimony favors the AR(1) model. 5 Forecasting in great recessions As discussed earlier, forecasting the end of great recessions, is particularly di cult because the NBER does not call a trough until economic activity has exceeded its previous known peak. These rules adopted by the NBER and incorporated in the Bry Boschan algorithms help to explain why on 12 April 2010 the NBER business cycle dating committee reported that they had met but were not yet ready to determine the location of trough. As is shown in Figure 9 US real GDP in June quarter 2010 of $13191billion is still below the value at the last peak of $13363billion in December quarter Using the framework developed above it is possible to generate a probability distribution over the event that there is a double dip recession and conditional on there not being a double dip recession a probability distribution over date at which the NBER will nally say that the recession ended in June Quarter Conclusions I have written down an approximation to the procedures for determining turning points and have shown how when combined with a model for y t this leads to an approximation to the DGP for the binary states. I have shown 20

23 Figure 9: US real GDP to that this approximation can be used to evaluate models of the business cycle. In the application the model was used to evaluate the Markov switching model developed by James Hamilton. The later model was found to be inferior to an AR(1) in growth rates for predicting turning points. In a second application it was shown that these models are useful for now casting the state of the business cycle but have little predictive power more than one quarter into the future. Importantly this framework can be used to evaluate models that claim to have predictive power at longer horizons. References Bry, G., Boschan, C., (1971), Cyclical Analysis of Time Series: Selected Procedures and Computer Programs, New York, NBER. Burns, A.F., Mitchell, W.C., (1946), Measuring Business Cycles, New York, NBER. Diebold, F.X. and G. D. Rudebusch (2001), Five Questions About Business Cycles, FRBSF Economic Review Hamilton, J.D., (1989), A New Approach to the Economic Analysis of Non-Stationary Times Series and the Business Cycle, Econometrica, 21

24 57, pp Harding, D., (2008), The equivalence of several methods for extracting permanent and transitory components Harding, D, Pagan, A.R., (2000a), Knowing the Cycle, In: Backhouse, R., Salanti, A., (Eds.) Macroeconomics in the Real World (Oxford University Press) Harding D., and A.R. Pagan, (2002), Dissecting the Cycle: A methodological Investigation, Journal of Monetary Economics. 49 pages Harding D., and A.R. Pagan, (2002), A Comparison of Two Business Cycle Dating Methods, Journal of Economic Dynamics and Control, 27 pages Harding D., and A.R. Pagan, (2006), Synchronisation of Cycles, Journal of Econometrics. Harding D., and A.R. Pagan, (2008), Measurement of Business Cycles, New Palgrave Harding D., and A.R. Pagan, (2007), The Econometric Aanalysis of Some Constructed Binary Time series, Mimeo, University of Melbourne IMF (2002) Global Financial Stability Report A Quarterly Report on Market Developments and Issues March 2002 Thorpe, W. (1926), Business Annals, Monograph Number 8, NBER. 22