Bivariate CGBCR Final

8/13/2019 Bivariate CGBCR Final

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A Bivariate Autoregressive Probit Model:Predicting U.S. Business and Growth Rate Cycle

Recessions

Henri Nyberg

Department of Economics, HECER, and

Department of Mathematics and Statistics,

University of Helsinki.

Preliminary version - work in progress

June 4, 2009

Abstract

We propose new bivariate binary time series models to predict the current state of the

U.S. business and growth rate cycles. The proposed bivariate autoregressive probit model

outperforms the independent univariate models built for both cycle indicators. It is shown

that business cycle recessions, and also growth rate cycle recessions can be predicted with

financial variables, especially with the changes in the Federal funds rate and stock market

returns.

JEL Classification: C32, C35, E32.

Keywords: Bivariate probit model, Autoregressive model, Business cycle, Growth

rate cycle, Recession.

The author would like to thank Markku Lanne, Antti Ripatti and Pentti Saikkonen for cons-

tructive comments. The author is responsible for remaining errors. The financial support from the

Academy of Finland, the Research Foundation of the OP Group and the Finnish Foundation for

Advancement of Securities Markets is gratefully acknowledged.

Homepage: http://blogs.helsinki.fi/hknyberg/. E-mail: [email protected].


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1 Introduction

In the previous literature of time series models for binary dependent variables, the

models have typically been univariate. Given the importance of vector autoregressive

models for continuous dependent variables, it is of interest to study multivariate

binary time series models, where the probabilities of different binary outcomes are

modeled jointly.

In the following sections, we present a bivariate autoregressive probit model which

is an extension to the traditional static bivariate probit model suggested by Ashford

and Sowden (1970). These two models and recently proposed new dynamic univariate

(see, e.g., Chauvet and Potter, 2005) and bivariate (see, e.g., Mosconi and Seri, 2006)

binary time series models have been based on a latent variable formulation, where

the values of binary time series are realizations of corresponding continuous latent

variables. The same approach is used in the qual VAR model of Dueker (2005), where

a vector autoregressive model is assumed for the latent variables behind the binary

and continuous dependent variables.

In this paper the latent variable approach described above will not be used. In-

stead, we extend the idea of the univariate autoregressive probit model structure

proposed by Kauppi and Saikkonen (2008) to the case of a bivariate model. An ad-

vantage of this approach is that parameter estimation can conveniently be carried out

by the method of maximum likelihood and forecasts can be computed using explicit

formulae. Our bivariate model is somewhat similar to the model proposed by Ana-

tolyev (2009). However, we model the dependence between two binary time series in

a different way.

We use different specifications of the proposed bivariate autoregressive probitmodel to assess the probabilities of the current state of the U.S. economy. We measure

the state of the economy by using recession periods occurred in the business cycle

and growth rate cycle. Predicting business cycle recession periods with univariate

probit models have attracted considerable attention in recent years (see, e.g., Estrella

and Mishkin, 1998; Chauvet and Potter, 2005). Given the fact that some economic

slowdown periods do not turn into business cycle recessions, it is also of interest to

consider values of binary growth rate cycle indicator.

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A growth rate cycle consists of period of increasing and decreasing growth rate

in economic activity (see details in Banerji and Hiris, 2001; Osborn, Sensier and van

Dijk, 2004). Classical business cycle recession periods are associated with the level of

economic activity whereas a growth rate recession may occur without a decline in thelevel of economic activity. Therefore, growth rate cycle recessions are more numerous

than classical business cycle recessions, but from the viewpoint of economic policy,

they may be at least equally important and informative. For example, monetary

policy decisions made by central banks are based on the real time assessment of the

current, and also future, economic conditions using the data available at the decision

time. As Osborn et al. (2004) suggest, growth rate cycles are closely related to the

estimated output gap, which is an important variable for monetary policy. However,

it is difficult to assess the output gap in real-time (see, e.g., Orphanides and van

Norden, 2002), and therefore it is of interest to consider probability forecasts for the

growth rate cycle periods.

In recent years various bridge models (see, e.g., Baffigi, Golinelli and Parigi, 2004)

and factor models (see, e.g., Giannone, Reichlin and Small, 2008; Aruoba, Diebold,

and Scotti, 2008), for example, have been used to assess real-time economic activity.

In this study we concentrate on the probabilities of business cycle and growth rate

cycle recession and expansion periods using the predictive power of financial vari-

ables, which are available on continuous basis. A difficulty with using macroeconomic

predictive variables, such as initial estimates of the real GDP, is that they face sub-

stantial revisions during subsequent months and observations of some macroeconomic

variables are not even available on monthly bases.

The results of the paper demonstrate that it is advantageous to model the prob-

abilities of business cycle and growth rate cycle recessions jointly. The proposed bi-

variate autoregressive probit model yields the best in-sample and out-of-sample pre-

dictions. Changes in the Federal funds rate and stock market returns are the best

predictive variables for the current value of the U.S. growth rate cycle indicator. The

U.S. term spread is an important predictive variable for predicting business cycle

recessions, as suggested in many previous studies, but its predictive power for growth

rate cycle periods is limited.

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The remainder of the paper is organized as follows. Univariate and, especially, the

proposed bivariate autoregressive probit model are introduced in Section 2. Issues of

parameter estimation, testing, and forecasting are discussed in Section 3. Section 4

provides the results of the empirical application. Section 5 concludes.

2 Time Series Models for Binary Responses

2.1 Univariate Probit Models

Consider the univariate binary valued time series yt, t = 1, 2,...,T. Conditional on

the information set t1,yt has a Bernoulli distribution

yt|t1 B(Pt). (1)

LetEt1()and Pt1()signify the conditional expectation and the conditional prob-

ability, given the information set t1. In the probit model we then have

Et1(yt) =Pt1(yt= 1) = (t) =Pt, (2)

where () is a standard normal cumulative distribution function and t is a linearfunction of variables included in the information set. In the previous literature, the

static specification

t = + x

tk, (3)

which contains the constant term and the lagged explanatory variables included

in the vector xtk, has been the most commonly used model. The ith component of

the vector xtk, denoted by xi, tk, is assumed to satisfy the condition k h, where

h is a forecast horizon. Using the same lag k in all explanatory variables is only for

notational convenience and can easily be relaxed in practice.

The static model (3) has been used by Estrella and Hardouvelis (1991) and Es-

trella and Mishkin (1998), among others, to predict business cycle recession periods.

Furthermore, Osborn et al. (2004) have considered the static model in the case of

growth rate cycle periods of the European countries. Recently, however, several au-

thors have proposed new univariate extensions of the static model. In this paper, we

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consider the model proposed by Kauppi and Saikkonen (2008) and include a lagged

value oft in the static model (3). Thus, we consider the autoregressive model

t = + 1t1+ x

tk, (4)

where|1|< 1 is assumed. An interesting property of this model is that, by recursive

substitutions, it can be written as

t =i=1

i11 +i=1

i11 x

tki+1. (5)

This representation shows that the autoregressive model (4) is an infinite order

extension of the static model (3) in which the effect of the whole infinite history of

explanatory variables is taken into account in a parsimonious way.

2.2 Bivariate Autoregressive Probit Model

In the following, the aim is to study bivariate models of two binary dependent time

series,y1tand y2t,t = 1, 2,...,T. Conditional on the information sett1, the random

vector (y1t, y2t) follows a bivariate Bernoulli distribution stated formally as (cf. the

univariate case given in (1))

(y1t, y2t)|t1 B2(P11,t, P10,t, P01,t, P00,t). (6)

Here the conditional probabilities of different outcomes ofy1tand y2tare non-negative

and sum up to one

P11,t+ P10,t+ P01,t+ P00,t= 1. (7)

The conditional probability of the outcome (y1t= 1,y2t= 1), for example, is denoted

by P11,t = Pt1(y1t = 1, y2t = 1) and the conditional marginal probabilities of the

separate outcomes y1t= 1 and y2t= 1 are obtained as

P1t = P11,t+ P10,t,

P2t = P11,t+ P01,t. (8)

A bivariate probit model was first proposed by Ashford and Sowden (1970) for

analyzing cross-sectional data. Following their notation, the conditional joint proba-

bility of the outcome (y1t = 1, y2t= 1) is determined as

P11,t = Pt1(y1t = 1, y2t= 1) = 2(1t, 2t, ), (9)

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where 2() is the cumulative distribution function of the bivariate normal distri-

bution with zero means, unit variances and correlation coefficient , || < 1. Fur-

thermore, 1t and 2t are assumed to be linear functions of variables included in the

information sett1. The static bivariate model is obtained by using 1t

2t

=

1

2

+

x

1,tk 0

0 x

2,tk

12

. (10)

Equations (9) and (10) together define the static bivariate probit model.1

The corresponding conditional probabilities for the remaining outcomes of the

vector(y1t, y2t) are obtained from

P10,t = Pt1(y1t = 1, y2t= 0) = 2(1t,2t,),

P01,t = Pt1(y1t = 0, y2t= 1) = 2(1t, 2t,),

P00,t = Pt1(y1t = 0, y2t= 0) = 2(1t,2t, ). (11)

The sign changes in the arguments of the cumulative normal distribution function

are needed to guarantee that condition (7) holds (see, for example, Greene, 2000,

849850). These conditional probabilities are needed, for instance, to construct the

likelihood function of the model described in more detail in Section 3.

Dynamic extensions of the static model described in (10) can be obtained in

various ways. In this paper, we extend the univariate autoregressive probit model (4)

to the bivariate autoregressive probit model

1t

2t

=

1

2

+

11 12

21 22

1,t1

2,t1

+

x

1,tk 0

0 x

2,tk

12

. (12)

In this model, 1t and 2t are specified as linear functions of their lagged values and

lagged values of explanatory variables included in the vectors x1,tkand x2,tk. Model

(12) can also be written as

t = + At1+ x

tk, (13)

1 The corresponding multivariate model is considered, among others, by Ashford and Sowden

(1970), and Chib and Greenberg (1998).

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where t=

1t 2t

, xtk= diag

x

1,tk x

2,tk

, =

1 2

, =

1

2

,

and

A=

11 12

21 22

.

In this bivariate autoregressive model we explicitly allow for the possibility that dif-

ferent explanatory variables affect1tand 2t. On the other hand, since the parameter

matrix A is unrestricted the variable 1t, for example, can depend on the lagged value

of2t.

In the bivariate autoregressive probit model the correlation coefficient is em-

ployed in (9) as in its static counterpart obtained from (13) with the restriction

A= 0. It should be pointed out, however, that even if = 0, the coefficients 12 and21 provide a linkage between the variables 1t and 2t in model (12). In the special

case where = 0and 12 =21 = 0our bivariate autoregressive probit model reduce

to two independent univariate autoregressive probit models (4).

Model (12) is somewhat similar to the multivariate dynamic binary model pro-

posed by Anatolyev (2009). However, because of the characteristics of our empirical

application (see Section 4.2), lagged values of the dependent variables used by Ana-

tolyev (2009) are not included in our model, although that would be a possible ex-

tension of model (12). This extension can be based on the univariate model proposed

by Kauppi and Saikkonen (2008), where the lagged value ofyt,yt1, is also included

in the right hand side of univariate model (4).

Instead of using the lagged values ofy1tand y2tas predictors we use the predictive

power of lagged explanatory variables included in x

tk. As in the case of the univariate

autoregressive model (see (5)), if the roots of the determinant det(I2Az)lie outside

the unit circle, as the usual stationarity condition in VAR models for continuous

dependent variables also require, one can use recursive substitutions and write the

bivariate system (13) as

t =

j=1

Aj1 +

j=1

Aj1x

tkj+1. (14)

This shows that 1t and 2t depend on the infinite history of lagged explanatory

variables included in x

tk. Furthermore, assuming that explanatory variables included

in xtk are stationary then also t is stationary.

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Anatolyev (2009) also suggests using the so called dependence ratios (cf. Ekholm,

Smith, and McDonald, 1995) between dependent variables to construct the condi-

tional joint probabilities for different outcomes of (y1t, y2t). In parameter estimation

the dependence ratios and marginal probabilities for the variablesy1tandy2tare han-dled separately by using a logistic function. In our model, the dependence between

y1t and y2t is instead modeled by using the autoregressive specification (12) and the

bivariate cumulative normal distribution function 2(), where the conditional corre-

lation coefficient is allowed to be nonzero and parameter estimation can be carried

out in the same system without dependence ratios.

3 Parameter Estimation, Testing and Forecasting

3.1 Maximum Likelihood Estimation

As in corresponding univariate models, parameter estimation in the general bivari-

ate autoregressive model described in (9) and (12), as well as its special cases, can

conveniently be carried out by the method of maximum likelihood (ML). Using the

conditional probabilities defined in (9) and (11), one can write the likelihood function

and obtain the maximum likelihood estimate by using numerical methods. In bivari-

ate autoregressive probit model the parameter vector is =

vec(A)

.

Following the notation used in Greene (2000, 849850), the conditional log-likelihood

function can be constructed as follows. Define qjt = 2yjt1and jt =qjtjt,j = 1, 2,

so that

qjt =

1 ifyjt = 1,

1 ifyjt = 0,

and

jt =

jt ifyjt = 1,

jt ifyjt = 0.

Furthermore, set

t =q1tq2t.

With this notation, the conditional probabilities of different outcomes of (y1t,y2t) can

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be expressed as (see (9) and (11))

Pt1(y1t, y2t) = 2(1t, 2t, ).

The log-likelihood function, conditional on initial values, is the sum of the individual

log-likelihood functionslt(),

l() =Tt=1

lt() =Tt=1

log

2(1t, 2t,

t )

=Tt=1

y1ty2tlog(P11,t) + y1t(1y2t)log(P10,t)

+ (1 y1t)y2tlog(P01,t) + (1 y1t)(1 y2t)log(P00,t)

. (15)

It is worth noting that if the correlation coefficient in (9) is zero, the conditionalprobabilities of the four different alternatives are products of the marginal proba-

bilities (8). For instance, in that case the conditional probability of the outcome

(y1t= 1, y2t = 1) is

P11,t = Pt1(y1t = 1, y2t= 1) =Pt1(y1t)Pt1(y2t) = (1t)(2t). (16)

The score vector of the log-likelihood function (15) is

s() =l()

=

T

t=1

st() =

T

t=1

lt()

, (17)

where

lt()

=

1

2(1t, 2t, t )

2(1t, 2t,

t )

.

At this point it is convenient to split the parameter vector into three disjoint com-

ponents, namely = (

1

2 )

, where the parameters in 1 and 2 are related

to the specifications of1t and 2t, respectively. The score vector can be partitioned

accordingly as

st() =

s1t(1)

s2t(2)

s3t()

. (18)

The first component ofst() can be written as

s1t(1) =lt()

1=

1

2(1t, 2t, t )

2(1t, 2t,

t )

1

= 1

2(1t, 2t, t )

2(1t, 2t,

t )

1t

1t1

= 1

2(1t, 2t, t )(

1t)

2t1t

t1 2t

q1t

1t

1,

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Maximization of the log-likelihood function (15) yields the maximum likelihood es-

timate, which solves the first order condition s() = 0. Under appropriate regularity

conditions, including the stationarity of explanatory variables and the correctness of

the probit model specification, the conventional large sample theory of ML estimationgives the usual asymptotic distribution

T1/2( ) LN(0,I()1), (19)

where I() = plim T12l()/

.

A practical difficulty with the bivariate autoregressive probit model (12) is that the

number of parameters can become large if many explanatory variables are included.

ML estimation is considerably simplified if the correlation coefficient is restricted to

zero because then the bivariate probabilities in the log-likelihood function (15) factor

into products of marginal probabilities, as in (16). Thus, it is of interest to test for

the hypothesis= 0. In the next section a LM test is developed for this purpose.

3.2 LM Test for the Correlation Coefficient

The Lagrange multiplier test is attractive because it only requires ML estimation

under the null hypothesis = 0. Kiefer (1982) has proposed a corresponding LM test

for the static bivariate probit model (10). In this section, the test is extended to the

bivariate autoregressive model (12).

Let = (1 2 0) be the restricted ML estimate of obtained by assuming

H0: = 0. (20)

The general form of the LM test statistic (see, for instance, Engle, 1984) is

LM=s() I()1s(), (21)

where I() is a consistent estimate of the information matrix I() and s() is the

score vector (17) evaluated at the restricted ML estimates . Under the null hypoth-

esis (20) the test statistic has an asymptotic 21 distribution.

Due to the complexity of second derivatives of the log-likelihood function (15), an

outer-product of the score is an attractive estimator of the information matrix I().

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The resulting test statistic is

LM =

S()

S()

S()1

S()

, (22)

where is a vector of ones and the matrix S() is given by

S() =

s1() s2() ...sT()

.

As shown in (18), the score vectorst(), evaluated at , consists of three components.

The bivariate densities and probabilities factor into products of marginals and, con-

sequently, the components of the score reduce to

s1t(

1) =

(1t)

(1t)q1t

pi1t

1 ,

s2t(2) = (2t)

(2t)q2t

2t2

,

and

s3t(0) = (1t)(2t)

(1t)(2t)q1tq2t,

where on the right hand side means that the corresponding quantities are eval-

uated at = . The derivatives 1t/1 and 2t/2 depend on the considered

specifications of1t and 2t (see Section 3.1).

3.3 Forecasting

As shown by Kauppi and Saikkonen (2008) explicit formulae can be used to ob-

tain one-period and multiperiod forecasts in the case of the univariate model (4). In

the following we show that the same principles can also be applied in the proposed

bivariate model.In the mean-square sense, the optimal h period forecast based on the given infor-

mation available at forecast time t h,h 1, is the conditional expectation

Eth(y1t, y2t) =Eth

2(1t, 2t,

t )

. (23)

For example, the forecast for outcome (y1t = 1,y2t= 1) is given by

Eth(y1t = 1, y2t= 1) = 2((h)1t ,

(h)2t , ),

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where, as shown in (14), by recursive substitutions the bivariate system (12) can be

written as

(h)t = A

hth+h

j=1

Aj1 + x

tkj+1

, (24)

where k h, the vector th is a function of values of the explanatory variables and

the initial values of1,0 and 2,0, and (h)t =

(h)1t

(h)2t

. The right hand side gives

the h step forecast for the outcome (y1t= 1, y2t = 1)directly using the information

up to the forecast time t h. In practice the unknown parameters and the vector

th have to be replaced by estimates. In this approach, forecasts are horizon specific

so that the employed lags of the predictive variables are always tailored to match the

information available at the time of forecasting.

Both in-sample and out-of-sample predictive performance of the employed models

can be evaluated with goodness-of-fit measures commonly used for binary dependent

variables. The value of the maximized log-likelihood function (15), denoted by logL,

enables to make comparisons between different models. It can also be used to compute

values of model selection criteria, such as the Schwarz information criterion (Schwarz,

1978) defined in this study as

BIC=logL + Klog(T)

2 , (25)

where Kis the number of parameters in and Tis the number of observations. An-

other goodness-of-fit measure is the quadratic probability score, QP S, suggested by

Diebold and Rudebusch (1989). Using the marginal conditional probability forecasts

Pjt obtained from (8), the quadratic probability score for variable yjt is

QPSj = 1T

Tt=1

2

yjt Pjt)2

, (26)

where j = 1, 2. The values of the QPSj are on the interval [0,2] with the value 0

indicating a perfect fit. It can be seen as a counterpart for the mean square error

used in models for continuous variables.

The binary nature of the dependent variable leads one to ask what is the percent-

age of correct predictions, CR, between the realized values and forecasts. Because

univariate and bivariate probit models produce probability forecasts, it is desirable

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to specify a threshold value that translates the probability forecasts into signal fore-

casts (yjt = 1 or yjt = 0, j = 1, 2). The most commonly used and natural threshold

choice is 0.50, which is also used in this paper.

When the signal forecasts are constructed, a test proposed by Pesaran and Tim-mermann (1992) is available for the evaluation of the directional predictive perfor-

mance of a model. The null hypothesis of the test is that the value of the correct

prediction ratio, CRj , does not differ significantly from the ratio that would be ob-

tained in the case of non-predictability, where the forecasts and realized values ofyt

are independent. Under the null hypothesis of non-predictability, the test statistic

has an asymptotic standard normal distribution.

4 Empirical Application: Predicting the Current State

of the U.S. Economy

We apply different bivariate probit model specifications to predict the state of the

U.S. economy. In this application, we predict, or more specifically nowcast, values

of the dependent business cycle and growth cycle indicators, discussed in more detail

in Section 4.1, using the real-time information available from the financial markets.

The assessment of the current state of the economic activity is important for many

economic agents, for example, in business and finance, and also for policymakers, such

as central banks and government organizations. However, because of informational

lags and revisions of the important macroeconomic variables, such as real GDP, the

current state of the economy is always uncertain to some extent. In our nowcasting

exercise the forecast horizon will therefore be one month, h= 1. Thus, we are inter-ested in predicting the probabilities of business cycle and growth cycle recessions for

monthtusing the information up to the end of the previous month t 1, indicating

that the nowcasts are constructed at the beginning of month t.

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4.1 Binary Indicators for the Business and Growth Rate Cyc-

les

Forecasting the recession periods of the economy with various binary dependent uni-

variate time series models has attracted considerable attention in many previous

studies (see, among others, Estrella and Mishkin, 1998, Chauvet and Potter, 2005,

Kauppi and Saikkonen, 2008, and Nyberg, 2008). These recession periods are related

to business cycle fluctuations in the level of the economic activity. Thus, the binary

recession indicator for the business cycle is

y1t=

1, if the economy is in a recession at time t,

0, if the economy is in an expansion at time t.

(27)

According to the most commonly used definition of the U.S. recession periods, pro-

posed by the National Bureau of Economic Research (NBER), a recession is a sig-

nificant decline in economic activity spread across the economy, lasting more than

a few months, normally visible in real GDP, real income, employment, industrial

production, and wholesale-retail sales. 2 It is important to note that the NBER

uses a broader array of economic indicators than just, for example, the real GDP to

determine the recession periods.

In this study, we are mainly interested in predicting the growth rate cycles of

the U.S. economy, which, to the best of our knowledge, has so far been studied with

binary time series models only by Osbornet al.(2004). In contrast to classical business

cycles characterized by the recession indicator (27), growth rate cycles are related to

the growth rate of the economic activity. We adopt the growth rate cycle periods

defined by the Economic Cycle Research Institute (ECRI). Based on their definition

of periods of cyclical upswings and downswings in growth, we introduce the binary

indicator

y2t=

1, if the growth rate cycle is in a downswing state at time t,

0, if the growth rate cycle is in an upswing state at time t.(28)

In this paper, the downswing state of the growth rate indicator (y2t= 1) is referred

to as a growth rate recession. 3 Note that in a growth rate recession, the economy

2 See details on http://www.nber.org/cycles/recessions.html.

3 Osborn et al. (2004) refer these periods as growth regimes.

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can still be in expansion in terms of the business cycle indicator (27) (i.e. y1t = 0),

and vice versa. It is expected that the growth rate cycle indicatory2twill exhibit more

regime switches than the business cycle indicatory1t. The reason is that a period of a

lower growth rate may be sufficiently long to be classified as a growth rate recession,but not long enough to become a business cycle recession.

ECRI determines the turning points in the growth rate indicator by using an

analog of the NBER approach, where the co-movements and cyclical turns in various

measures of the aggregate macroeconomic activity are taken into account. Banerji and

Hiris (2001) provide a more detailed discussion about the business cycle and growth

rate cycle periods (see also Layton and Moore, 1989).

Figure 1 depicts the ECRI growth rate recession periods with shaded areas along

with the NBER recession periods. Table 1 shows the cross-tabulation of the realized

values of the two dependent variables. As expected, growth rate cycles are more

numerous than classical business cycles. All slowdowns in the growth of the economic

activity do not involve business cycle recessions. As expected, the growth rate cycle

recessions seem to lead the business cycle recessions: the U.S economy has typically

been in growth rate recession for some months before a business cycle recession period

has begun.

4.2 Data Set and Predictive Models

The data set includes the values of the U.S. business cycle indicator y1t, growth rate

cycle indicator y2t, and financial explanatory variables xt, covering the period from

January 1971 to December 2005. This data set is employed in Sections 4.3 and 4.4.

In Section 4.5, the best bivariate probit models, according to model selection criteria

and in-sample predictions, are used to assess real-time conditions of the U.S. economy

from 2006 to 2008. These real-time probabilities are of considerable interest because,

according to the NBER, the peak in the economic activity occurred in December 2007

indicating that the business cycle recession (y1t = 1) began in January 2008.

The employed financial explanatory variables consist of interest rates and stock

market returns. 4 Levels and first differences of various interest rates are considered.

4 The variables are described in more detail in Table 2.

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Assuming that the monetary policy has an impact on the real economic activity

and its growth rate, it is of interest to study which interest rate variable is the most

informative predictor. The Federal funds rate (FFt) is closely related to the monetary

policy in the U.S., so that it is a natural candidate variable. For example, Bernankeand Blinder (1992) have shown that the Federal funds rate is an appropriate short-run

indicator of monetary policy in the U.S.

In the predictive models for business cycle recessions, the term spread (SPt) be-

tween the long-term interest rate and the short-term interest rate has been found to

be the most important predictor (see, e.g., Estrella and Mishkin, 1998). Hence, it is

of interest to examine whether the term spread is also an important predictor for the

growth rate cycle periods. Furthermore, as a forward-looking variable and depending

on the expectations of future dividends and profitability of firms, stock market re-

turns (rt) should also have additional predictive power along with interest rates and

the term spread.

Osbornet al. (2004) find that the output gap between the potential and realized

level of output is an important predictive variable when predicting growth rate cy-

cle periods in European countries. The long-term and short-term interest rates with

stock market returns have also some predictive power, but their importance is sec-

ondary to the output gap. However, as they note, the final estimate of the output

gap, which was employed as a predictor, is not available on real-time basis (see, e.g.,

Orphanides and van Norden, 2002). This is also the case for some other macroeco-

nomic variables. For example, initial GDP estimates are revised later on. Therefore,

the main interest in this paper is to consider the potential predictive information of

financial variables, which are available with no revisions or information lags at the

monthly data frequency used in the paper

Another issue, which affects the specification of the considered models, is the fact

that the values of the NBER business cycle phases (y1t), and apparently also the

growth cycle periods defined by the ECRI (y2t), become available with very long

delays. We call these delays publication lags. Without explicit assumptions about

the lengths of the publication lags it is difficult to use lagged values of the cycle

indicators in predictive model Overall, it is expected that the publication lags are so

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long that their lagged values are statistically insignificant in estimated models (see

the evidence from univariate models by Nyberg, 2008). 5 Therefore, we only consider

models which do not contain lagged values of the dependent variables.

In the following sections, we consider four different model specifications obtainedfrom the bivariate autoregressive probit model given in (9) and (12). The models are

defined as follows.

Model 1 : 12= 0, 21 = 0, = 0,

model 2 : 12= 0, 21 = 0,

model 3 : = 0,

model 4 : model (12).

Model 1 corresponds the case of two independent autoregressive probit models (4).

Model 2 is obtained from model 1 by removing the restriction that the correlation

coefficient is zero. Note that models 1 and 2 are already extensions of the static

bivariate model because 1t and 2t are modeled by the univariate autoregressive

model (4). Model 4 is the bivariate autoregressive probit model (12) without any

restrictions whereas only the correlation coefficient is restricted to zero in model 3.

4.3 Model Selection and In-Sample Results

Model selection considered in this section is based on models estimated over the entire

sample period from January 1971 to December 2005. The first 12 observations are

used as initial values in estimation. This section is also a starting point for the real-

time assessment of the state of the U.S. economy in 20062008 presented in Section

4.5. Another out-of-sample predictive exercise is considered in Section 4.4.

As mentioned in Section 4.1, unlike for the business cycle recession periods, there

are not much previous findings on the predictability and useful predictive variables

for the growth rate cycle periods. In model selection, therefore, we concentrate on

5 The most recent decisions of business cycle peak and trough months defined by the NBER

(see http://www.nber.org/cycles/cyclesmain.html), which determine values of y1t, reflect that the

publication lag, indicating the time that it has taken before the business cycle turning point was

identified, is varied from five up to twenty months.

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predicting the growth rate cycle recessions with various financial variables. For sim-

plicity, we employ the same explanatory variables as Nyberg (2008) for the predictive

models of the U.S. business cycle recession periods. The explanatory variables used

in that paper were the U.S term spread (SPt), the monthly stock market return (rt),and the foreign German term spread (SPGEt ), which are included in the vector

x1,tk =

SPt6 rt1 SPGEt6

. (29)

Here the employed lags are also the same as in Nyberg (2008) and, because the

forecast horizon is one month (h= 1), the condition k h is satisfied.

The applied model selection procedure concerning the variables included in x2,tk

is as follows. At the first stage, we use model 1, or equivalently, independent univariate

autoregressive probit models (4) for both cycle indicators. We consider models with

one predictor for growth rate cycle periods described in Table 2. When the best

single predictor is found, it will be kept in the model and different models with two

predictors are estimated. We mainly restrict ourselves to models with two predictors,

but make some experiments with models containing the U.S. term spread (SPt) as

a third predictor as well. In the second stage of model selection we consider more

general bivariate models, especially model 4 by using predictors selected at the first

stage. To facilitate comparisons, the same predictive variables are used in different

bivariate probit models (models 14).

Table 3 shows values of the Schwarz information criterion (25) for model 1 with

different explanatory variables, when the employed lag varies from one (k = 1) to

six (k = 6). 6 Especially in more general bivariate probit models, such as model 4,

the number of parameters is quite large, indicating that ML estimation may become

difficult (see Section 3.1). Therefore, we prefer parsimonious models which makes

BICa suitable model selection criterion.

According to Table 3 the best single predictive variable seems to be the first dif-

ference of the Federal funds rate lagged by two or three months (FFt2or FFt3).

The first difference of the short-term interest rate (itk) also performs quite well,

6 It appears that the evidence of predictive power of different explanatory variables is the same

when goodness-of-fit measures other than the BI Care used.

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but it is rather strongly correlated with the differenced Federal funds rate. Further-

more, as seen from Table 4, when models with two predictive variables are considered,

the lagged stock market return rt1 combined withFFt2 provides the lowest value

ofBIC. Thus, the vector x2,tk is selected as

x2,tk =

FFt2 rt1

. (30)

This selection is also meaningful from the viewpoint of economic theory because

variables reflecting the effect of both the monetary policy (FFt2) and the stock

market returns (rt1) are included.

According to Tables 3 and 4, the U.S. term spread has some ability to predict

the growth rate cycle periods. However, it is outperformed by several other variables.

The term spread is also found to be a statistically insignificant predictor when used

in models 14 together with the explanatory variables given in (30).

Table 5 shows the estimation results for models 14 with the explanatory vari-

ables given in (29) and (30). The signs of the estimated coefficients are as expected.

In the case of the growth rate cycle periods, increasing values of the differenced Fed-

eral funds rate and negative stock market returns increase the probability of growth

rate recession. The results indicate that the U.S. monetary policy has a statistically

significant predictive impact for the current state of the growth rate cycle via the

first difference of the Federal funds rate. Stock market returns also have predictive

power for the growth rate cycles as well as for business cycle recession periods for

which the U.S. term spread (SPt6) and the foreign German term spread (SPGEt6 ) are

statistically significant predictors.

In model 2, the estimate of the correlation coefficient is statistically significant.

An application of theLM test based on the restricted model 1 yields the same con-

clusion. A positive estimate of is expected as the correlation between the dependent

variables is positive. Further, in model 3, the estimates of the off-diagonal elements

of the matrix A,12and21, are statistically significant, and according to the values

ofBIC, model 3 outperforms model 2 as an extension of model 1. However, both the

LM based on model 3 and the Wald test based on model 4 point to a nonzero value

of the correlation coefficient . Thus, Model 4 clearly yields the best in-sample fit,

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and this is especially the case when the models for the growth rate cycle periods are

compared with the values ofQPS2 and CR2.

In models 3 and 4, statistically significant estimates of12 and21 reflect the fact

that recession probabilities of the two cycle indicators are dependent on both 1,t1

and2,t1. For instance, the positive estimates of12 indicate that the probability of

a business cycle recession is high when the lagged probability of growth rate recession

is high (high value of2t). This is in line with the fact that the growth rate recession

appears to precede occurred business cycle recession periods. Note that in models 3

and 4 the U.S. term spread, which is a statistically significant predictor for business

cycle recession periods, has an effect on the growth rate cycle recession probability

via the coefficient 12.

As an extension of the estimation results presented in Table 5, we consider the

possibility that there has been a structural break in the data generating process. In

the previous literature, it has been suggested that there might have been a structural

change in the U.S. economic activity in the mid-1980s. For example, McConnell and

Perez-Quiros (2000) and Blanchard and Simon (2001) have documented that the

variability of output, and also the variability of inflation, has declined after the mid-

1980s. 7 Sensier and van Dijk (2004) also provide evidence that there have been

structural breaks in the unconditional volatility of many U.S. macroeconomic time

series around the years 1984 to 1986.

In our application, a parsimonious way to allow for the potential effect of a struc-

tural break is to include an additional dummy variable in the models specified for

1t and 2t. This dummy variable, denoted by 17184, takes the value one before the

beginning of the year 1985. It turns out to be a statistically significant predictor in

the models of Table 6 which are also superior to their counterparts in Table 5 accord-

ing to BICvalues. Model 4 with the dummy variable 17184 yields clearly the best

in-sample predictions, especially for growth rate cycle recessions.

Although model 4 seems to outperform its special cases, the estimated coeffi-

cients of the explanatory variables are quite the same in all models. These findings

strengthen the results from our model selection. The changes in the Federal funds

7 This time period after the mid-1980s is often referred to the Great Moderation period.

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rate and stock market returns appears to be the main predictive variables for growth

rate cycles also in more general models.

Figures 2 and 3 depict the in-sample fitted values from models 1 and 4 shown

in Table 6. As the estimation results in Tables 5 and 6 suggest the business cyclerecession periods predicted by these two models are almost identical. For the growth

rate recession periods, model 4 gives somewhat more precise signals around the years

from 1984 to 1987 and before the year 2000.

In conclusion, especially the general bivariate autoregressive model (model 4),

but also its special cases (models 2 and 3), are superior to the independent univariate

autoregressive models for both cycle indicators (model 1). These findings indicate

that additional predictive power can be found by using bivariate models instead of

independent univariate autoregressive probit models.

4.4 Out-of-Sample Performance

Before considering the out-of-sample predictions of the real-time recession probabil-

ities after December 2005, it is of interest to examine the out-of-sample predictive

performance of different models also in another out-of-sample period. The first out-

of-sample forecasts are made for January 2000 and the last ones for December 2005.

Notice that the out-of-sample period contains three growth rate cycle recession peri-

ods, but only one business cycle recession in 2001 (see also Figure 1).

Different predictive models are estimated using the data up to December 1999

assuming that the state of the economy is known at that time. To emulate real-

time forecasting, we should take into account the fact that the latest values of the

dependent variables are not available at the time the forecast is made (see Section

4.2). Thus, the out-of-sample exercise is carried out without updating the parameter

estimates and the values of the estimated parameters are based on the sample period

up to December 1999. The same framework is applied in Section 4.5. It turns out

that the model selection procedure employed in Section 4.3, using the data set up to

December 1999, yields the same conclusions as obtained when using the whole sample

in estimation. Thus, the differenced Federal funds rate and the stock market returns

have predictive power for the growth rate cycles also in this estimation sample.

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Table 7 shows the out-of-sample performance of the employed models. Out-of-

sample forecast accuracy is evaluated with QPSand the percentage of correct fore-

cast CR. In the first four models, the dummy variable 17184 is also included in the

model. As in Section 4.3, the general model 4 yields the best out-of-sample predictionsfor the growth rate cycles. Based on the predictability test of Pesaran and Timmer-

mann (1992) the reported percentages of correct forecasts in Table 7 are statistically

significant at the 1 % level in models with the 17184 variable. Without the additional

dummy variable the percentages of correct forecasts and, consequently, the p-values

are higher and statistically insignificant at 5 % level.

Although the dummy variable 17184 is useful in predicting the growth rate cycle

recessions, this is not the case for nowcasts obtained for the business cycle recession

periods. However, the more general models 3 and 4 outperform models 1 and 2 even

in this case. Note that the percentages of correct forecasts for business cycle recession

and expansion periods are statistically significant in all models presented in Table 7.

Figure 4 illustrates the out-of-sample performance of the general model 4. It ap-

pears that recession probabilities match well with the realized values of both cycle

indicators. As depicted in the left panel, model 4 predicts the business cycle recession

in 2001 really well, but the increase in the recession probability at 2002 weakens then

performance of the model in terms of the values of statistical goodness-of-fit measures

reported. In the right panel, out-of-sample nowcasts match the growth rate cycle pe-

riods well. For instance, using the 50 % threshold value for probabilitity forecasts to

construct signal forecasts for growth rate recessions and expansions, model 4 gives

the correct signal forecast with 85 % accuracy, as reported in Table 7.

In summary, the results confirm that the proposed general bivariate autoregressive

model (12) outperforms its restricted versions and the values of both cycle indicators

appear to be predictable also out-of-sample.

4.5 Predictions for 20062008

There has been great uncertainty about the state of the economy in the United

States during the last couple of years. Therefore, it is of great interest to consider

the probabilities of the business and growth rate cycle periods during 20062008.

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Business cycle recession probabilities are of special interest due to the fact that the

NBER Business Cycle Dating Committee declared that a peak in the U.S. economic

activity occurred in December 2007 indicating that the value of the recession indicator

(27) has been one, at least some months from January 2008.In this section, we examine the real-time probabilities for the state of the economy

using the best models found in Sections 4.3 and 4.4. The first real-time predictions

are made for January 2006 and the last ones for November 2008. Because of the

recent decision of the business cycle recession made by NBER, it is clear that the

U.S. economy has been in business cycle recession in the first part of the year 2008.

In the case of the growth rate cycle periods, it is, however, not evident that there

has been a constant downswing state (i.e. a growth recession, y2t = 1) after January

2006. Thus, at the time this paper is written the realized values ofy2t are uncertain

after January 2006.

Figures 5 and 6 depict the predictions from models 1 and 4. The probabilities in

Figure 6 are based on the models including the additional dummy variable 17184. It

seems that the business cycle recession probabilities in model 1 are higher than in

model 4 at the beginning of the recession in January 2008 (see Figure 5). However,

in both cases, model 1 gives higher false recession risks than model 4 for some

months before the recessions started. Overall, the employed models seem to nowcast

the beginning of the recession period at 2008 reasonably well.

As discussed above, the latest values of the growth rate cycle indicator are un-

certain. So that it is difficult to make comparisons between different models with the

currently available information. It seems that the growth rate recession probability

decreases after the mid of 2006. As the business cycle recession started at the be-

ginning of the year 2008, it is likely that the U.S. economy has been in a growth

recession some months before. At those months, the predicted probabilities in Figure

5 are quite high and exceed the 50% threshold value, indicating a growth rate cycle

recession at that time. Note also that a decreasing probability of growth rate recession

affects the business cycle recession probability in model 4. This is a potential reason

why the business cycle recession probabilities in the left panels of Figures 5 and 6 are

lower than in the independent univariate autoregressive model (model 1).

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5 Conclusions

In this paper we introduce new bivariate time series models for binary dependent

variables. These models are applied to predict the current state of the U.S. economy

using binary indicator variables for the level and the growth rate of the U.S. economic

activity. The proposed bivariate model framework, where predictions for the growth

rate cycle phases are also obtained extends the traditional univariate analysis of

business cycle recession periods examined in the previous literature.

We find strong in-sample and out-of-sample evidence in favour of the bivariate

autoregressive probit model proposed in the paper. The results suggest that it is pos-

sible to gain additional predictive power by modeling the recession probabilities of

the two cycle indicators jointly instead of two independent univariate autoregressive

probit models. Changes in the Federal funds rate are useful predictors for the state of

the growth rate cycle. Also monthly stock market returns are statistically significant

predictors for both cycle indicators. As suggested in previous studies, a term spread

between the long-term and short-term interest rate is an important explanatory vari-

able for business cycle recession periods, but it is not the best predictive variable for

the growth rate cycle periods. We also find evidence that the Great Moderation

time period after the mid-1980s has a statistically significant effect on the levels of

recession probabilities in both cycle indicators.

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Tables and Figures

Table 1: Dependent variables and the cross-tabulation of realized values.

y2t

0 1

y1t 0 162 205

1 5 48

Notes: U.S. business cycle periods y1t (recession/expansion) and growth rate cycle periods y2t

(growth recession/ growth rate expansion) are obtained from

http://www.nber.org/cycles/cyclesmain and http://www.businesscycle.com. The sample is 1972

M12005 M12.

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Table 2: Explanatory variables.

rt Stock market return, log-difference of the S&P500 index

F Ft Federal funds rate

it The 3-month Treasury bill rate, secondary market

Rt The 10-year Treasury bond yield rate, constant maturity

SPt The term spread,Rtit

F Ft The first difference in the Federal funds rate

it The first difference in the 3-month Treasury bill rate

Rt The first difference in the 10-year Treasury bond yield

SPGEt The German term spread between the long-term and short-term interest rate.

Notes: Interest rates are from Federal Reserve Statistical Release Historical Data

(http://www.federalreserve.gov/releases/h15/data.htm). S&P500 stock market index is taken from

Yahoo Finance (http://finance.yahoo.com) and from http://www.econstats.com. German term

spread is constructed as the difference between 10-year Federal security (series WZ9826, the

missing values between 1971 M1-1972 M9 are replaced by the OECD 10-year interest rate) and the

three-month money market rate (series su0107, see http://www.bundesbank.de/statistik/statistik).

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Table 3:BICvalues for model 1 with different explanatory variables for the growth

rate cycle indicator.

k rtk F Ftk itk Rtk SPtk F Ftk itk Rtk

1 308.77 312.76 315.26 331.25 300.52 335.83 335.83 335.412 314.75 312.20 314.66 330.95 300.51 280.41 286.39 335.48

3 319.02 314.11 314.92 330.27 304.30 280.34 282.19 336.05

4 321.64 317.55 318.36 330.21 313.58 289.94 286.34 335.90

5 327.01 321.22 320.82 330.55 318.26 298.25 286.87 319.66

6 331.01 323.71 323.53 330.80 323.52 304.70 308.30 325.52

Notes: Explanatory variable included in x2,tk and its lag k are mentioned in the first row and the

first column of the table. Schwarz information criterion,B IC, is defined in (25).

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Table 4:BICvalues for model 1 with the changes in the Federal funds rate and other

explanatory variables for the growth rate cycle indicator.

k rt F Ft it Rt SPt it Rt

F Ft2 1 269.75 277.56 277.49 276.91 283.39 281.06 278.262 273.18 277.49 277.31 276.54 303.51 283.08 280.49

3 278.27 277.49 277.25 276.26 283.28 282.00 282.94

4 277.05 277.64 277.35 276.11 283.20 280.28 283.30

5 278.38 277.88 277.57 276.12 283.11 276.31 282.87

6 282.23 278.19 277.97 276.23 282.94 281.82 282.66

F Ft3 1 270.27 278.18 277.90 277.49 283.32 282.57 280.85

2 274.09 278.43 278.08 277.22 303.44 281.84 281.95

3 278.64 278.62 278.73 276.84 307.16 282.85 282.64

4 278.61 278.62 278.30 276.83 282.92 283.23 283.24

5 279.22 278.63 278.30 276.55 282.87 280.26 283.34

6 282.07 278.78 278.51 276.53 282.72 282.28 282.93

Notes: The employed predictive variables in x2,tk are the second, or the third, lag of the first

difference of the Federal funds rate and mentioned variable in the first row of the table. See also

Notes to Table 3.

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Table 5: Estimation results of different bivariate models.model variable Model 1 Mo del 2 Mo del 3 Model 4

1t constant1 0.06 0.06 -0.07 -0.16

(0.03) (0.03) (0.05) (0.05)

1,t1 0.85 0.86 0.88 0.86

(0.02) (0.02) (0.01) (0.01)

2,t1 0.17 0.18

(0.05) (0.05)

SPt6 -0.19 -0.16 -0.08 -0.06

(0.03) (0.04) (0.03) (0.02)

rt1 -0.11 -0.10 -0.10 -0.10

(0.02) (0.02) (0.02) (0.02)

SPGEt6 -0.08 -0.07 -0.13 -0.11

(0.03) (0.02) (0.03) (0.03)

2t constant2 0.07 0.07 -0.02 -0.05

(0.01) (0.01) (0.01) (0.01)

2,t1 0.91 0.91 0.92 0.96

(0.02) (0.01) (0.01) (0.01)

1,t1 -0.03 -0.04

(0.01) (0.01)

FFt2 0.51 0.51 0.34 0.31

(0.18) (0.06) (0.06) (0.05)

rt1 -0.03 -0.03 -0.06 -0.06

(0.01) (0.01) (0.01) (0.01)

0.58 0.53

(0.21) (0.21)logL -242.70 -240.31 -223.38 -217.94

BIC 269.75 270.36 256.44 254.01

QPS1 0.061 0.063 0.061 0.062

QPS2 0.340 0.330 0.302 0.284

CR50%1

0.963 0.961 0.961 0.963

CR50%2 0.713 0.723 0.770 0.789

LM 8.58 27.21

p-value 0.000 0.000

Notes: The models are estimated using the data from 1971 M1 to 2005 M12. The first 12 observations are used as

initial values. Standard errors of the estimated coefficients are given in parentheses. Estimated values of the

log-likelihood function (15), logL, and Schwarz (1978) information criteria, BIC, are reported, as well as the values

of quadratic probability scores, QP Sj , where j = 1, 2. Further, CR50%j indicate the ratio of correct prediction with

using the 50% threshold value in the classification of probability forecasts. Lagrange Multiplier test statistics LM

(see (22) for the null hypothesis (20) and the corresponding p-values are also reported.

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Table 6: Estimation results of different models with the additional dummy variable

17184.

model variable Model 1 Mo del 2 Mo del 3 Model 4

1t constant1 0.09 0.07 0.08 0.05

(0.05) (0.05) (0.04) (0.03)

17184 -0.04 -0.00 -0.25 -0.20

(0.06) (0.05) (0.09) (0.08)

1,t1 0.85 0.86 0.89 0.89

(0.02) (0.02) (0.01) (0.01)

2,t1 0.11 0.10

(0.04) (0.03)

SPt6 -0.20 -0.16 -0.11 -0.10

(0.04) (0.04) (0.03) (0.02)

rt1 -0.11 -0.09 -0.09 -0.08

(0.04) (0.02) (0.02) (0.02)SPGEt6 -0.08 -0.08 -0.12 -0.11

(0.03) (0.03) (0.03) (0.02)

2t constant2 0.06 0.06 -0.07 -0.08

(0.01) (0.01) (0.02) (0.02)

17184 0.06 0.06 0.09 0.10

(0.02) (0.02) (0.03) (0.03)

2,t1 0.91 0.91 0.93 0.93

(0.01) (0.01) (0.01) (0.01)

1,t1 -0.04 -0.05

(0.01) (0.01)

FFt2 0.55 0.55 0.40 0.38

(0.07) (0.06) (0.08) (0.07)

rt1 -0.03 -0.03 -0.08 -0.08

(0.01) (0.01) (0.01) (0.01)

0.59 0.71

(0.22) (0.16)

logL -235.15 -232.76 -198.63 -193.38

BIC 268.21 268.83 237.71 235.46

QPS1 0.061 0.063 0.058 0.058

QPS2 0.327 0.310 0.256 0.254

CR50%1 0.963 0.961 0.958 0.961

CR50%2 0.725 0.740 0.816 0.834

LM 8.14 26.21

p-value 0.000 0.000

Notes: See notes to Table 5. Variable 17184 indicates a variable which takes value one at period from 1971 M1 to

1984 M12, and zero otherwise.

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Table 7: Out-of-sample performance of different models.

QP S1 CR1 QP S2 CR2

Model 1 with 17184 0 .395 0.722 0.395 0.708

Model 2 with 17184 0 .435 0.722 0.395 0.708

Model 3 with 17184 0 .167 0.889 0.303 0.833

Model 4 with 17184 0 .110 0.903 0. 307 0. 847

Model 1 0.179 0.833 0.446 0.625

Model 2 0.228 0.819 0.446 0.625

Model 3 0.078 0.944 0.460 0.625

Model 4 0.108 0.944 0.608 0.611

Notes: The out-of-sample values ofQP Sj and CRj , j = 1, 2, based on models on the left.

Explanatory variables included in the model are the same as in the estimation results presented in

Tables 5 and 6. Nowcasts from model 4 with the dummy variable 17184 (the fourth model) are

depicted in Figure 4.

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70 72 75 77 80 82 85 87 90 92 95 97 00 02 05 07

Figure 1: U.S. recession periods (line) and growth rate cycles since January 1972 until

December 2005. Shaded areas indicates growth rate recessions (downswing periods in

the growth rate cycle, i.e. y2t= 1).

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70 72 75 77 80 82 85 87 90 92 95 97 00 02 05 070

0.1

0.2

0.3

0.4

0.5

0.6

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0.8

0.9

1

70 72 75 77 80 82 85 87 90 92 95 97 00 02 05 070

0.1

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

1

Figure 2: In-sample fitted values from the model 1 presented in Table 6. Recession

periods are depicted with shaded areas. Business cycle recession probabilities are

presented in the left panel, probabilities for growth rate cycle periods in the right

panel.

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70 72 75 77 80 82 85 87 90 92 95 97 00 02 05 070

0.1

0.2

0.3

0.4

0.5

0.6

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0.9

1

70 72 75 77 80 82 85 87 90 92 95 97 00 02 05 070

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0.2

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0.5

0.6

0.7

0.8

0.9

1

Figure 3: In-sample fitted values from the model 4 presented in Table 6. Recession

periods are depicted with shaded areas. Business cycle recession probabilities are

presented in the left panel, probabilities for growth rate cycle periods in the right

panel.

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99 00 01 02 03 04 05 060

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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1

99 00 01 02 03 04 05 060

0.1

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0.7

0.8

0.9

1

Figure 4: Out-of-sample nowcasts for business cycle recession (left panel) and growth

rate cycle (right panel) recession periods (shaded areas) using the bivariate autore-

gressive probit model (model 4) with explanatory variables and dummy variable

17184 given in Section 4.3.

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05 06 07 08 090

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1

05 06 07 08 090

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0.9

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Figure 5: Real-time predictive probabilities from model 1 (dashed line) and from

model 4 for the business cycle (left panel) and the growth rate cycle periods (right

panel) using the models described in Table 5. In the left panel, the business cycle

recession that began at 2008 is depicted with shaded area. In the right panel, the

shaded area corresponds to the time period of uncertain values of the growth rate

cycle indicatory2t.

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05 06 07 08 0905 06 07 08 090

0.1

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0.5

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05 06 07 08 090

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0.7

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0.9

1

Figure 6: Real-time recession probabilities from model 1 (dashed line) and from model

4 for the business cycle (left panel) and the growth rate cycle periods (right panel)

using the models described in Table 6. In the left panel the business cycle recession

period that began at 2008 is depicted with shaded area. In the right panel, the

shaded area corresponds to the time period of uncertain values of the growth rate

cycle indicatory2t.

Bivariate CGBCR Final

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