Forecasting US recessions with various risk factors and dynamic probit models

Journal of Macroeconomics 34 (2012) 112–125

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Forecasting US recessions with various risk factors and dynamic probit models Eric C.Y. Ng ⇑ Department of Economics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

a r t i c l e

i n f o

Article history: Received 25 November 2010 Accepted 15 November 2011 Available online 3 December 2011 JEL classification: C22 C25 E32 E37 Keywords: Recession forecasts Recession risks Dynamic probit models

a b s t r a c t This paper extends probit recession forecasting models by incorporating various recession risk factors and using the advanced dynamic probit modeling approaches. The proposed risk factors include financial market expectations of a gloomy economic outlook, credit or liquidity risks in the general economy, the risks of negative wealth effects resulting from the bursting of asset price bubbles, and signs of deteriorating macroeconomic fundamentals. The model specifications include three different dynamic probit models and the standard static model. The out-of-sample analysis suggests that the four probit models with the proposed risk factors can generate more accurate forecasts for the duration of recessions than the conventional static models with only yield spread and equity price index as the predictors. Among the four probit models, the dynamic and dynamic autoregressive probit models outperform the static and autoregressive models in terms of predicting the recession duration. With respect to forecasting the business cycle turning points, the static probit model is as good as the dynamic probit models by being able to flag an early warning signal of a recession. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The recent recession in the US during 2007–2009 reignite considerable discussions in the press and academia about whether the peaks and troughs of business cycles could have been predicted in advance, say a few months or quarters ahead. These are important questions for households, firms and investors, whose current decisions on consumption, production and investment are conditional on what they expect the economic outlook to be. For central bankers and policy makers, an accurate and reliable forecast of the state of the economy facilitates the implementation of appropriate and preemptive polices. An earlier approach for addressing these questions has been to construct an empirical model to forecast economic variables such as GDP, consumption and inflation.1 As noted in Estrella and Mishkin (1998), the models for forecasting economic variables generally suffer from more serious endogeneity issues than the models for predicting probabilities of recessions. As a result, forecasting the binary recession indicator with probit models has attracted much attention in the recent literature (Dueker, 1997; Estrella and Mishkin, 1998; Chauvet and Potter, 2005; Kauppi and Saikkonen, 2008; Nyberg, 2010). Under this approach, the probability of a recession is modeled as a function of lagged values of potential explanatory variables. A general finding from the literature is that the interest rate spread between 10-year Treasury bonds and 3-month Treasury bills, is the most statistically significant predictor of US recessions for 2–6-quarter forecast horizons. ⇑ Tel.: +852 39438007; fax: +852 26035805. E-mail address: [email protected] Papers that examine the predictability of future real activity include Harvey (1988), Chen (1991), Estrella and Hardouvelis (1991), and Estrella and Mishkin (1995). Papers that examine potential factors for predicting future inflation include Mishkin (1990a,b, 1991). 1

0164-0704/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2011.11.001

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Earlier studies of modeling recession probabilities focused on using the static probit model (Estrella and Mishkin, 1998), in which the probability of a recession in the current period is determined by the past values of the explanatory variables. A major drawback of this approach is the lack of a dynamic mechanism to capture how the probabilities of recession may be influenced by the current or past state of business cycle. For example, if the economy is in a recessionary state, the probability of future recession or recovery may be affected. Dueker (1997) strengthens the traditional static probit model by including the lagged recession dummy indicator as an explanatory variable. Kauppi and Saikkonen (2008) use a similar model of Rydberg and Shephard (2003) to develop a more unified modeling framework for dynamic probit models. In their model, the conditional probability of a recession not only depends on the lagged recession dummy indicator, but also on the lagged probability function itself. They also modify the method of using an ‘‘iterated’’ forecasting approach to make multiperiod-ahead forecasts for the dynamic probit models. Their findings suggest that the more advanced dynamic probit models with the iterated forecasting approach generate more accurate forecasts than the static model with the direct forecasting procedure.2 In general, the literature focuses on evaluating the predictive ability of the yield spread using different probit specifications. Though some papers also consider other indicators such as an equity market index and leading indicators, they only evaluate their predictive power separately as a single explanatory variable (e.g., Dueker, 1997; Estrella and Mishkin, 1998). The one-variable approach implies that the models have limitations in providing a complete assessment of upcoming recession risks. A recent paper by Nyberg (2010) extends the dynamic probit models with multiple explanatory variables including domestic and foreign yield spreads, and stock market returns. Although these financial variables are good indicators of the stance of monetary policy and general financial conditions, they cannot capture other potential risk factors for recessions. In this paper, we extend probit recession forecasting models by incorporating a more complete set of recession risk factors and using the advanced dynamic probit modeling approaches. The proposed risk factors include financial market expectations of a gloomy economic outlook, credit or liquidity risks in the general economy, the risks of negative wealth effects resulting from the bursting of asset price bubbles, and signs of deteriorating macroeconomic fundamentals. We use various financial and macroeconomic indicators to measure the extent of the four risk factors. The indicators include the interest rate spread between 10-year US Treasury bonds and 3-month Treasury bills (yield spread), the interest rate differential between 3-month LIBOR and 3-month T-bills (TED spread), the equity price index, the housing price index, and the composite index of eight macroeconomic leading indicators (macro-leading index). In terms of modeling framework, we follow Kauppi and Saikkonen (2008) to consider four different model specifications, including the static, dynamic, autoregressive, and dynamic autoregressive probit models. Our empirical analysis suggests that the proposed four risk factors captured by various lagged values of the 6-month moving average of the yield spread, TED spread, changes in the real equity price index, the real housing price index and the macro-leading index are statistically significant US recession predictors. The out-of-sample analysis further suggests that the four probit models with the proposed risk factors can generate more accurate forecasts for the duration of recessions than the conventional static models with only yield spread and equity price index as the predictors. Among the four probit models, the dynamic and dynamic autoregressive probit models outperform the static and autoregressive models in terms of predicting the recession duration. With respect to forecasting the turning points of business cycles, interestingly, the static probit model is as good as the dynamic probit models by being able to generate an early warning signal of a recession. Last but not least, the stability tests indicate that there is a stable predictive relationship between the recession probability and the risk factors of the probit models. The rest of the paper is organized as follows. Section 2 describes the probit models, the estimation methodology and forecasting procedures. Section 3 describes the recession risk factors and the corresponding variables and presents the in-sample and out-of-sample estimation results, and the results of stability tests. Section 4 concludes the paper. 2. The models: estimation methodology and forecasting procedures 2.1. Model specifications and estimation methodology We follow Kauppi and Saikkonen (2008) to consider four types of probit models: (1) static probit model; (2) dynamic probit model; (3) autoregressive probit model; and (4) dynamic autoregressive probit model. The dependent variable in our models is a binary recession indicator and takes on two possible values depending on the state of the economy:

Yt ¼



1; if the economy is in a recession at time t 0;

otherwise

Conditional on the information set Xt1 at time t  1, Yt has a Bernoulli distribution:

Y t jXt1  Bðpt Þ

2

Their modified version of ‘‘iterated’’ forecasting approach is built from Chauvet and Potter (2005).

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Define Et1() and Pt1() as the conditional expectation and probability, respectively, given Xt1. The conditional expectation of Yt can then be specified as:

Et1 ðY t Þ ¼ P t1 ðY t ¼ 1Þ ¼ pt

ð1Þ

In probit models, pt refers to the standard normal cumulative distribution function:

Uðpt Þ ¼

Z ðpt Þ 1

1 2 pffiffiffiffiffiffiffi eðz Þ=2 dz 2p

ð2Þ

where pt is a linear function of explanatory variables. The four probit models in this paper vary only in terms of the formulation of pt. The static probit model takes the form:

Pt1 ðY t ¼ 1Þ ¼ Uðpt Þ ¼ Uða þ bX tk Þ

ð3Þ

where X denotes a set of explanatory variables. The probability of recession at time t will be forecasted at time t  1 using the set of information on X at time t  k where k P 1. The parameters are estimated by maximizing the full-sample log-likelihood function:

logðLðY; a; bÞÞ ¼

T X ½Y t logðUða þ bX tk ÞÞ þ ð1  Y t Þ logð1  Uða þ bX tk ÞÞ

ð4Þ

t¼1

As discussed in Dueker (1997), the drawback of static probit models is the lack of a dynamic structure for the dependent variable in applications to time series data. The recession predictor, Xt, is a time series variable with its own autocorrelation structure, but the model uses no information contained in the autocorrelation structure of the dependent variable to form predictions. Therefore, the model may be misspecified without including the lagged dependent variable as an additional explanatory variable.3 The earlier studies such as Dueker (1997) and Moneta (2005) therefore suggest including the lagged value of Yt on the right-hand side to form the dynamic probit model:

pt ¼ a þ bX tk þ dY t1

ð5Þ

Kauppi and Saikkonen (2008) extend the dynamic probit model in two ways. First, they consider the lagged value of the probability function pt on the right-hand side of the model. This allows for richer dynamics in the process pt, i.e., the previous probability is a factor for forecasting future recession probabilities. This type of dynamic probit model is called an ‘‘autoregressive’’ probit model and takes the form:

pt ¼ a þ bX tk þ cpt1

ð6Þ

The other extension is referred to the ‘‘dynamic autoregressive’’ model, and it basically combines both the dynamic and autoregressive probit models. The model is given by:

pt ¼ a þ bX tk þ dY t1 þ cpt1

ð7Þ

Eq. (7) can be viewed as a unified framework for other dynamic probit models. It becomes the simple dynamic model when c = 0 or the autoregressive model when d = 0. In this paper, we consider only the first lag of pt and Yt. Although other lags can be included in the models, Kauppi and Saikkonen (2008) find no evidence that longer lags of pt or Yt can improve the predictive performance of the model. The parameters of the dynamic probit models can be estimated similarly by the maximum likelihood method (ML) described above. To estimate Eqs. (6) and (7), we need to choose the initial value of p0. We set p0 equal to the unconditional mean of pt, i.e., p0 ¼ ða þ bXÞ=ð1  cÞ for the autoregressive model, and p0 ¼ ða þ bX þ dYÞ=ð1  cÞ for the dynamic autoregressive model, where a bar over variables denotes the sample mean of the indicated variables. For the dynamic probit model in Eq. (5), de Jong and Woutersen (forthcoming) show that under appropriate regularity conditions the conventional large-sample theory holds for the ML parameter estimators. For the autoregressive and dynamic autoregressive probit models, we assume the same asymptotic properties apply to the two models provided that the regularity conditions hold. de Jong and Woutersen (forthcoming) also propose a robust standard error of the ML estimator to correct the potential misspecification errors when making multiperiod-ahead forecasts. We use the proposed robust standard error.4 Estrella (1998) proposes a statistic to evaluate the goodness-of-fit for probit models:

Adjusted Pseudo-R2 ¼ 1 

 2 logðLc Þ=T logðLu Þ  k logðLc Þ

ð8Þ

3 Univariate time-series modeling of macroeconomic data has clearly demonstrated the relevance of a variable’s own history in generating forecasts (Dueker 1997). 4 To conserve space, we skip the technical details on the asymptotic properties of ML estimators for the dynamic models and the proposed robust standard error. See Kauppi and Saikkonen (2008) for details.

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where k is the number of explanatory variables, T is the sample size, Lu is the unconstrained maximum value of the likelihood function of the estimated model, and Lc is the maximum value under the constraint that all coefficients except the constant are zero. A higher adjusted pseudo-R2 suggests that the inclusion of explanatory variables can increase the likelihood function of the estimated model. However, this statistic is not suitable for comparing models with different specifications. It fails to indicate the percentage of the variance that the model can explain because the underlying latent variable, the state of being a recession, cannot be observed. For instance, a pseudo-R2 of 0.5 indicates a 50% increase in the log-likelihood function of the estimated model relative to the model with only a constant term – a figure without an obvious meaning that cannot be used for model comparison. Because of the caveat of using the adjusted pseudo-R2, we also consider other criteria for evaluating the predictive power of the probit models. Specifically, we compare the Akaike information criterion, the Bayesian information criterion, the percentage of correct prediction of recession months, and the percentage of overall correct prediction of both recession and nonrecession months. 2.2. Out-of-sample forecasting procedures To make out-of-sample forecasts, we follow Kauppi and Saikkonen (2008) by using the direct forecasting method for the static probit model and the iterated forecasting procedures for the dynamic probit models. In the static probit model, the h-period-ahead forecast for the recession indicator variable (Yt), based on information available at time t  h is the conditional expectation Eth(Yt) = Pth(Yt = 1). By the law of iterated conditional expectations and Eq. (1), the forecast equation can be specified as:

Eth ðY t Þ ¼ Eth ðUða þ bX tk ÞÞ

ð9Þ

For dynamic probit models, we use an ‘‘iterated’’ forecasting method. Under this approach, a one-period-ahead model is always estimated, and forecasts are made iteratively for a given forecast horizon. The forecast equation for the dynamic probit model is represented by:

Eth ðY t Þ ¼ Eth ðUða þ bX tk þ dY t1 ÞÞ

ð10Þ

Given the binary nature of the recession indicator variable Yt, it is possible to compute the forecasts explicitly. For example, when h = 2, Eq. (10) can be specified as:

Et2 ðY t Þ ¼

X

X

Pt2 ðY t1 ÞPt1 ðY t ¼ 1Þ ¼

Y t1 2f0;1g

Pt2 ðY t1 ÞUða þ bX tk þ dY t1 Þ

ð11Þ

Y t1 2f0;1g

where Pt2 ðY t1 Þ ¼ Uða þ bX tk1 þ dY t2 ÞY t1  ½1  Uða þ bX tk1 þ dY t2 Þð1Y t1 Þ In this case, the iterative forecast takes into account two possible outcomes of Yt1 in which there are two different paths that can lead to a recession at time t: recessions at both time t  1 and t, or no recession at time t  1 but a recession at time t.5 For the autoregressive probit model, the h-period-ahead forecast is given by:

Eth ðY t Þ ¼ Eth ðUða þ bX tk þ cpt1 ÞÞ

ð12Þ

By repetitive substitution in Eq. (6), the forecast equation is modified as follows:

Eth ðY t Þ ¼ U ch pth þ

h X

!

cj1 ða þ bX tkþ1j Þ

ð13Þ

j¼1

where pth is basically a function of past values of Xt and the initial value p0. In the dynamic autoregressive probit model, the forecast equation is given by:

Eth ðY t Þ ¼ Eth ðUða þ bX tk þ dY t1 þ cpt1 ÞÞ

ð14Þ

For illustration, consider h = 2. Using Eq. (7) and by repetitive substitution, we can obtain the forecast equation as follows:

Et2 ðY t Þ ¼ Et2 Uða þ bX t2 þ dY t1 þ c2 pt2 Þ þ cða þ bX t3 þ dY t2 Þ

ð15Þ

Given that Yt1 only takes two possible values, Eq. (15) can be solved explicitly. When the forecast horizon increases, the number of possible paths leading to a recession at time t also increases. The formula for iterated forecasting will be more complicated.6 5 6

See Kauppi and Saikkonen (2008) for details about computing iterated multiperiod forecasts for h > 2. See Kauppi and Saikkonen (2008) for details on the formula for dynamic autoregressive model.

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125 Table 1 Monthly recession reference dates for the US. Source: National Bureau of Economic Research. From peak

To trough

December 1969 November 1973 January 1980 July 1981 July 1990 March 2001 December 2007

November 1970 March 1975 July 1980 November 1982 March 1991 November 2001 June 2009

3. Empirical analysis 3.1. Definition of recession dummy indicator The dependent variable refers to the recession indicator variable with a value of one standing for a recession month, and zero for a non-recession month. The Business Cycle Dating Committee of the National Bureau of Economic Research (NBER) officially dates the beginning and end of US recessions. The committee defines a recession as a significant decline in economic activity across the economy, being visible in a set of indicators and lasting more than a few months. The indicators include real income and GDP, employment, industrial production, manufacturing and trade sales. We use the NBER business cycle reference dates to construct the recession indicator data. Table 1 lists the monthly recession reference dates during the last 40 years. There are seven recession episodes over 1969–2009. The duration of each recession period varies substantially, from a short duration of 6–8 months in the 1980, 1990–1991 and 2001 recessions to a long duration of 16–18 months in the 1973–1975, 1981–1982 and 2007–2009 recessions. 3.2. Recession risk factors and corresponding indicators We consider four risk factors as warning signals for recessions in the near future: (1) financial market expectations of a gloomy economic outlook, (2) credit or liquidity risks in the general economy, (3) risks of negative wealth effects resulting from the bursting of asset price bubbles, and (4) signs of deteriorating macroeconomic fundamentals. The first risk factor is commonly captured by the interest rate spread between 10-year US Treasury bonds and 3-month Treasury bills (yield spread). The high predictive ability of the yield spread is because the yield curve contains useful information about financial market expectations of future economic activity.7 In general, the yield curve is upward-sloping with the long-term rates being higher than the short term ones. Investors holding long-term securities bear the risk that future interest rates will be higher than expected, so they demand a positive risk premium in terms of higher yields. The yield curve therefore displays a positive slope during the early to mid economic expansion period. When the economy is expected to slowdown or enter a recession in the near future, the short-term interest rates are anticipated to decline gradually during the recession period until economic conditions improve. Investors then desire to purchase financial assets with a longer maturity, such as long-term bonds, to hedge against the recession and receive higher yields until the economy rebounds. The rising demand for long-term bonds causes their prices to increase and their yields to fall. To purchase long-term bonds, investors may sell off short-term assets such as short-term bonds. The prices of the short-term bonds will then fall and their yields will rise. Consequently, prior to a recession, the slope of the yield curve will become flat or inverted. Chart 1 indicates that the yield spread turned negative prior to almost all past recessions. The second risk factor is reflected by the interest rate differential between 3-month LIBOR and 3-month Treasury bills (TED spread). The TED spread is commonly regarded as an indicator of perceived credit risk in the general economy. The LIBOR reflects the credit risk of lending to commercial banks in the inter-bank market while T-bills are considered as risk-free assets. When the TED spread increases, this is a sign that lenders perceive the risk of default on inter-bank loans (or the counterparty risk) is increasing. To compensate for the increased risk, lenders demand higher interest rates in the inter-bank market and accept lower returns on risk-free assets such as T-bills. Chart 2 illustrates that in most of the past recessions (except the 1990–1991 and 2001 recessions), the TED spread increased significantly several months before the onset of each recession. This may reflect that most industries tend to have lower profits during recessions so that the perceived credit risk in the general economy is likely to increase. The high TED spread also reflects the tight liquidity condition in the inter-bank markets, which eventually spills over to other money markets. Lending and borrowing activities will therefore decline. The subsequent credit crunch results in declines in economic activity and increases the probability of a recession. Hence, the TED spread is expected to be positively related to the recession probability. The risks of negative wealth effects are captured by changes in the equity price index (e.g., S&P 500 equity index) and the housing price index (e.g., S&P/Case-Shiller Home Price Index, Composite-10). Suppose that asset bubbles have been formed 7

See Estrella (2005) for formal discussion on the predictive power of yield curve for output and inflation.

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125 Recession Period

Yield_Spread

4.5 3.5 2.5 1.5 0.5 -0.5

Jun-09

Jun-07

Jun-05

Jun-03

Jun-01

Jun-97

Jun-99

Jun-95

Jun-93

Jun-91

Jun-89

Jun-85

Jun-87

Jun-83

Jun-79

Jun-81

Jun-77

Jun-75

Jun-73

Jun-71

Jun-67

Jun-69

Jun-65

-2.5

Jun-63

-1.5

Chart 1. Monthly performance of yield spread (6-month moving average).

Recession Period

TED_Spread

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Jun-09

Jun-07

Jun-05

Jun-03

Jun-01

Jun-99

Jun-97

Jun-95

Jun-93

Jun-89

Jun-91

Jun-87

Jun-85

Jun-83

Jun-81

Jun-79

Jun-77

Jun-75

Jun-73

Jun-71

Jun-69

Jun-67

Jun-65

Jun-63

-0.5

Chart 2. Monthly performance of TED spread (6-month moving average).

with the equity and housing prices rising persistently at levels above their intrinsic values determined by fundamental factors. A subsequently large fall in equity and housing prices (in real terms) may reflect that the asset bubbles have burst. Households, firms and investors who held these over-valued assets experience substantial reductions in net worth. The negative wealth effect may lead to lower demand and therefore increases the risk of a recession. Charts 3 and 4 illustrate that there were often substantial and persistent downward corrections in equity and housing prices before the start of most recessions in the past. Finally, the signs of deteriorating economic fundamentals are reflected by changes in the composite index of eight macroeconomic leading indicators (macro-leading index). The eight leading indicators include average weekly hours worked in manufacturing industries, average weekly initial jobless claims, manufacturers’ new orders of consumer goods and materials, index of supplier deliveries (vendor performance), manufacturers’ new orders of non-defence capital goods, building permits for new private housing units, real money supply, index of consumer expectations (University of Michigan consumer sentiment index).8 The composite index of the eight leading indicators is constructed by using the same methodology as the Conference Board to calculate the leading economic index. It is obtained by summing the eight leading indicators’ normalized month-to-month changes. The normalization is done using standard deviations of the leading indicators’ history.9 In general, these indicators tend to lead business cycles. Hence, declines in these indicators show signs of weak economic fundamentals in the near future. These variables are therefore expected to be negatively related to the recession probability.

8 The composite index of eight leading indicators is similar to the Conference Board Leading Economic Index except that we remove the interest rate spread and stock price index. These two variables are removed because they are proxies for the first and third risk factors, respectively. 9 The detailed methodologies and procedures can be found at the Conference Board’s website.

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125 Recession Period

Equity_Price_Index

8.0 6.0 4.0 2.0 0.0 -2.0 -4.0 -6.0

Jun-09

Jun-07

Jun-03

Jun-05

Jun-01

Jun-97

Jun-99

Jun-95

Jun-91

Jun-93

Jun-89

Jun-85

Jun-87

Jun-81

Jun-83

Jun-79

Jun-77

Jun-75

Jun-73

Jun-69

Jun-71

Jun-67

Jun-63

Jun-65

-8.0 -10.0

Chart 3. Monthly percentage changes in equity price index (6-month moving average).

Recession Period

Housing_Price_Index

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 Jun-09

Jun-07

Jun-05

Jun-03

Jun-01

Jun-99

Jun-97

Jun-95

Jun-93

Jun-91

Jun-89

Jun-87

Jun-85

Jun-83

Jun-81

Jun-79

Jun-77

Jun-75

Jun-73

Jun-71

Jun-69

Jun-67

Jun-65

Jun-63

-3.0

Chart 4. Monthly percentage changes in housing price index (6-month moving average).

Recession Period

Macro_Leading_Index

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5

Jun-09

Jun-07

Jun-05

Jun-03

Jun-01

Jun-99

Jun-95

Jun-97

Jun-93

Jun-91

Jun-89

Jun-87

Jun-85

Jun-83

Jun-81

Jun-79

Jun-77

Jun-73

Jun-75

Jun-71

Jun-69

Jun-67

Jun-65

Jun-63

-3.0

Chart 5. Monthly percentage changes in macro-leading index (6-month moving average).

Chart 5 suggests that the past recessions were often preceded by several months with persistently negative growth in the macro-leading index.

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125 Table 2 Data definitions, sources and sample periods. Variables

Descriptions

Periods

Data sources

Y

Recession indicator variable: 1 for a recession month or 0 for a non-recession month

National Bureau of Economic Research

Yield_Spread

Yield spread between 10-year US Treasury bonds and 3-month Treasury bills

TED_Spread

Interest rate spread between 3-month LIBOR (London Inter-bank Offered Rate) and 3-month Treasury bills

Monthly, 1963:M1– 2010:M6 Monthly, 1963:M1– 2010:M6 Monthly, 1963:M1– 2010:M6

Equity_Price_Index

Real equity price index. The nominal S&P 500 equity index is deflated by the consumer price index

Housing_Price_Lndex

Real property price index. The nominal S&P/CaseShiller Home Price Index (Composite-10) is deflated by the consumer price index

Macro_Leading_Index

A composite index of eight macroeconomic leading indicators, including average weekly hours worked in manufacturing industries, average weekly initial jobless claims, manufacturers’ new orders of consumer goods and materials, index of supplier deliveries, manufacturers’ new orders of non-defence capital goods, building permits for new private housing units, real money supply, index of consumer expectations

Monthly, 1963:M1– 2010:M6 Monthly, 1963:M1– 2010:M6 Monthly, 1963:M1– 2010:M6

Federal Reserve Bank of St. Louis Economic Data

The LIBOR data during 1963–1984 are obtained from International Financial Statistics, and the data during 1985–2010 are taken from Bloomberg Database. The Treasury bill data are obtained from Federal Reserve Bank of St. Louis Economic Data The equity index data are obtained from Bloomberg Database. The consumer price index data are taken from Federal Reserve Bank of St. Louis Economic Data The nominal housing price index data are obtained from Standard & Poor’s. The consumer price index data are taken from Federal Reserve Bank of St. Louis Economic Data The data for the eight leading indicators are taken from Bloomberg Database. The composite index is constructed by the author. See Section 3.2 for details

Table 3 In-sample estimation of static probit model with one factor: pseudo-R2 measures. Explanatory Variables

Model: Pt1(Yt = 1) = a + bXtk k: months lag

Xtk

1

2

3

4

5

6

7

8

9

Yield_Spread TED_Spread Equity_Price_Index Housing_Price_Index Macro_Leading_Index

0.085 0.187 0.298 0.139 0.293

0.114 0.185 0.281 0.134 0.255

0.143 0.177 0.223 0.130 0.218

0.169 0.167 0.175 0.124 0.193

0.194 0.154 0.141 0.119 0.172

0.214 0.139 0.104 0.114 0.138

0.230 0.123 0.070 0.108 0.101

0.239 0.108 0.045 0.101 0.064

0.240 0.096 0.029 0.088 0.040

10

101

12

13

14

15

16

17

18

0.237 0.086 0.016 0.076 0.020

0.231 0.077 0.007 0.063 0.010

0.219 0.069 0.001 0.053 0.005

0.202 0.061 0.000 0.046 0.002

0.182 0.052 0.001 0.039 0.001

0.162 0.044 0.003 0.034 0.001

0.141 0.036 0.006 0.029 0.000

0.120 0.028 0.011 0.025 0.000

0.101 0.022 0.015 0.021 0.000

Yield Spread TED Spread Equity Price Index Housing_Price_Index Macro_Leading_Index

Note: The models are estimated using monthly data of recession indicators and each of the explanatory variables from 1963 M1 to 2010 M6.

3.3. In-sample estimation results In-sample estimation results are obtained by estimating each of the four probit specifications Eqs. (3), (5), (6), and (7) using monthly data from January 1963 to June 2010. The five explanatory variables described above are used as proxies for the four recession risk factors. The levels of the yield spread and TED spread refer to their 6-month moving average values. The changes in the equity price index, the housing index and the macro-leading index refer to their month-to-month growth rates of their corresponding 6-month moving average values.10 Table 2 summarizes the definitions, sample periods and data sources for the dependent and the five explanatory variables. Before evaluating the performance of the probit models with five explanatory variables, we first estimate the static probit model (Eq. (3)) with only a single explanatory variable at a time for each of the five explanatory variables. We consider dif-

10 We also consider using other moving average values of the explanatory variables (e.g., 3-month, 9-month or 12-month). The estimation results are available upon request. Overall, the 6-month moving average values of most explanatory variables have the highest predictive power for different forecast horizons.

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125

Table 4 In-sample estimation results for various probit models. Factors

Static

Dynamic

Yield_Spread (t6)

0.474** (0.184) 0.290* (0.172)

TED_Spread (t1)

Housing_Price_Index (t1)

0.861*** (0.151) 1.346*** (0.280)

Macro_Leading_Index (t5) Macro_Leading_Index (t6)

0.537*** (0.110) 0.819*** (0.304) 0.287 (0.374)

0.680*** (0.141) 1.106** (0.434) 0.665 (0.504)

0.262* (0.157) 1.387*** (0.456) 57.4 0.67 64.39 79.52 82% 95% 93% 94%

3.088*** (0.383) 0.308*** (0.107) 2.810*** (0.477) 37.1 0.76 45.11 62.40 92% 98% 93% 98%

2.323*** (0.305)

pt1

log-L Adj. ps_R2 AIC BIC Correct_recession (50%) Correct_overall (50%) Correct_recession (25%) Correct_overall (25%)

0.504*** (0.193) 0.703*** (0.167) 0.972*** (0.352) 0.773** (0.385)

1.188*** (0.377)

Yt1

a

0.546** (0.228) 0.391* (0.215)

0.466* (0.240)

TED_Spread (t9) Equity_Price_Index (t1)

Dynamic autoregressive

0.671*** (0.255)

Yield_Spread (t8)

TED_Spread (t6)

Autoregressive 0.551*** (0.177)

Yield_Spread (t5)

1.579*** (0.479) 59.4 0.66 65.44 78.40 80% 96% 92% 94%

2.099*** (0.407) 38.7 0.75 45.69 60.82 90% 97% 94% 97%

Notes: The models are estimated using monthly data of the recession indicator and explanatory variables from 1963 M1 to 2010 M6. Standard errors are in parentheses. Yt1 is the lagged recession indicator; pt1 is the probability function of explanatory variables; a is the constant term. See Table 2 for definitions of other explanatory variables. log-L is the log-likelihood function. Adj. ps_R2 is the adjusted pseudo-R2. AIC and BIC are the Akaike information criterion and Bayesian information criterion, respectively. The last four rows indicate the percentage of correct prediction of recession months and overall correct prediction of both recession and non-recession months using the thresholds of 50% and 25%, respectively, to identify a recession month. * Statistically significant at 10%. ** Statistically significant at 5%. *** Statistically significant at 1%.

ferent lag orders of each explanatory variable, from 1 up to 18. This exercise can tell us how well each of these variables can predict recessions. Table 3 summarizes the in-sample estimation results for the one-variable static probit model. The yield spread tends to have high predictive ability for medium- to long-term forecast horizons (5–13 months) with the pseudo-R2 close to or above 0.2. The TED spread, however, tends to have high predictive power at short- to medium-term horizons (1– 9 months) with the pseudo-R2 close to or above 0.1. The real equity or housing price index tends to exhibit high explanatory power in the short-term forecast horizons (1–4 months). The macro-leading index also indicates satisfactory forecasting performance in the short- and medium-term horizons (1–7 months) but virtually has no predictive power for long-term horizons. We then estimate the four probit models with all the five explanatory variables. Though each of the four probit models contain the same set of explanatory variables, the lag orders of the five explanatory variables are allowed to vary, from 1 up to 18 months. This allows us to search for the optimal lag of each explanatory variable in each model that yields the highest adjusted pseudo-R2. Table 4 summarizes the in-sample estimation results for the four probit models. All the estimated coefficients of the five explanatory variables are statistically significant in the four probit models (except the coefficient of the macro-leading index in the dynamic and dynamic autoregressive models) and have the expected signs.11 Moreover, the coefficient of the lagged recession indicator (Yt1) is significantly positive in the two dynamic models. Because of the potential autocorrelation structure of the dependent variable, the lagged recession indicator is crucial for predicting future recessions. Similarly, the coefficient of 11 The insignificance of the macro-leading index’s coefficient in the two dynamic probit models may be related to the inclusion of the lagged recession indicator. As mentioned before, the macro-leading index contains information on recent economic trends and the short-term outlook. Since the one-month lagged recession indicator also contains information on short-term economic conditions, the inclusion of both variables may make one of these variables become insignificant.

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125 Table 5 In-sample predictions of business cycle turning points from various probit models. Actual

Turning points of business cycles

Leads/lags with respect to actual turning points

Static

Dynamic

Autoregressive

Dynamic autoregressive

Static

Dynamic

Autoregressive

Peaks 1969/12 1973/11 1980/01 1981/07 1990/07 2001/03 2007/12

Dynamic autoregressive

1969/09 1973/09 1979/11 1981/07 1990/03 2001/03 2007/09

1969/12 1973/12 1980/02 1981/08 1990/08 2001/04 2007/12

1969/09 1973/08 1979/11 1981/06 1990/03 2001/03 2007/09

1969/12 1973/12 1980/02 1981/08 1990/08 2001/04 2007/12

3 2 2 0 4 0 3

0 1 1 1 1 1 0

3 3 2 1 4 0 3

0 1 1 1 1 1 0

Troughs 1970/11 1975/03 1980/07 1982/11 1991/03 2001/11 2009/06

1970/12 1975/04 1980/11 1982/10 1991/03 2001/11 2009/07

1970/12 1975/04 1980/08 1982/11 1991/04 2001/12 2009/07

1970/12 1975/04 1980/11 1982/10 1991/03 2001/11 2009/07

1970/12 1975/04 1980/08 1982/10 1991/04 2001/12 2009/07

1 1 4 1 0 0 1

1 1 1 0 1 1 1

1 1 4 1 0 0 1

1 1 1 1 1 1 1

Notes: ‘‘x’’ (‘‘+x’’) means that the predicted turning point is x months before (after) the actual turning point. The specific models and their parameter estimates are based on the in-sample estimation results in Table 4. A threshold of 25% is used to classify a recession month. A peak of a business cycle is identified when the predicted recession probability changes from less than 0.25 in the last month to 0.25 or above in the current month. Once the peak has been identified, a trough is flagged when the predicted probability changes from 0.25 or above in the last month to less than 0.25 in the current month and the predicted probability cannot go back to 0.25 or above at least in the next 3 months.

the lagged recession probability (pt1) is positively significant in the autoregressive probit model. This also confirms the importance of incorporating the autocorrelation structure of the recession probability in the model. The lower panel of Table 4 presents several statistics for assessing the predictive power of each model. The different statistics consistently suggest that both the dynamic and dynamic autoregressive probit models outperform the static or autoregressive probit models in predicting the duration of recessions.12 For example, using the threshold of 50% to classify a recession month, the two dynamic models generate 90–92% correct prediction of recession months while the static and autoregressive models only generate 80–82% correct prediction. It should be noted that when using the lower threshold of 25% to identify recessions, the percentage of correct prediction of recession months from both the static and autoregressive probit models improve substantially. However, the percentage of overall correct prediction of both recession and non-recession months generated by the two models decreases. This suggests that while the use of a lower threshold to identify recessions increases their ability to predict recessions, it also increases the false alarm signals. On the other hand, the use of a lower threshold further improves the predictive accuracy of both the dynamic and dynamic autoregressive models without increasing the false alarm signals. We now turn to the performance of predicting the turning points of business cycles. To do so, we compare the predicted dates of peaks and troughs with the actual beginning and ending months of recession. We use the threshold of 25% to classify a recession month.13 A peak of a business cycle is identified when the predicted recession probability changes from less than 0.25 in the last month to 0.25 or above in the current month. Once the peak has been identified, a trough is flagged when the predicted probability changes from 0.25 or above in the last month to less than 0.25 in the current month and the predicted probability cannot go back to 0.25 or above at least in the next 3 months. Table 5 illustrates the ability of various probit models to predict individual business cycles in the past by comparing the predicted peaks and toughs with the actual turning points. The static and autoregressive probit models outperform the other two dynamic probit models in terms of generating an early warning signal for a recession for most of the past business cycles. The former two models often predicted the peak 1–4 months ahead of the actual beginning month of the recession. From the perspective of economic agents and policy makers, an early recession signal is preferable to a late signal. By contrast, the dynamic and dynamic autoregressive models tend to exhibit a 1-month lag in predicting both the peaks and toughs of the past business cycles. This may reflect their ‘‘dynamic’’ property in terms of using the one-period lagged recession dummy indicator as one of the explanatory variables. 3.4. Out-of-sample estimation results It is well known that the in-sample estimation results are not necessarily coincident with the out-of-sample results. To further evaluate the predictive accuracy of the four different probit models, we follow the out-of-sample analysis as in Estrella and Mishkin (1998). First, we consider a series of out-of-sample forecasts obtained from various estimation samples 12 We also use other performance measures such as the root-mean-square error and the quadratic probability score applied by Diebold and Rudebusch (1989), and come to the same conclusion. 13 The use of 25% threshold to identify a recession is motivated by the subsequent out-of-sample estimation results. We find that regardless of the model types and forecast horizons, the use of 25% threshold consistently exhibits better out-of-sample performance than the use of 50% threshold.

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Table 6 Out-of-sample estimation results for various probit models: percentage of correct prediction of recession months (%). Forecast horizon (Months)

1

5

6

7

8

Static (with yield spread only) Static (with yield spread and equity price index) Static (with proposed four risk factors) Dynamic (with proposed four risk factors) Autoregressive (with proposed four risk factors) Dynamic Autoregressive (with proposed four risk factors)

50% Threshold of classifying recessions 0.0 0.0 0.0 0.0 29.4 29.4 29.4 29.4 67.6 67.6 67.6 67.6 91.2 85.3 73.5 67.6 67.6 67.6 67.6 64.7 88.2 79.4 73.5 67.6

2

3

4

0.0 29.4 61.8 58.8 58.8 52.9

0.0 29.4 50.0 41.2 44.1 41.2

0.0 29.4 35.3 35.3 32.4 35.3

0.0 29.4 20.6 29.4 11.8 29.4

Static (yield spread only) Static (yield spread + equity price index) Static (with proposed four risk factors) Dynamic (with proposed four risk factors) Autoregressive (with proposed four risk factors) Dynamic Autoregressive (with proposed four risk factors)

25% Threshold of classifying recessions 32.4 32.4 32.4 32.4 52.9 47.1 47.1 47.1 79.4 76.5 70.6 70.6 94.1 88.2 82.4 76.5 79.4 76.5 70.6 67.6 94.1 88.2 76.5 70.6

32.4 47.1 64.7 67.6 61.8 67.6

32.4 47.1 61.8 61.8 52.9 64.7

32.4 44.1 52.9 41.2 47.1 44.1

32.4 44.1 44.1 41.2 38.2 44.1

Notes: The out-of-sample forecasts are obtained from various estimation samples running from 1963 M1–1986 M12 up to 1963 M1–2010 M5. The explanatory variables for each model are the same as those in Table 4.

running from 1963:M1–1986:M12 up to 1963:M1–2010:M5. Using each sample at a time, we estimate the four probit models, and make forecasts for a given forecast horizon. We then add one more month to the estimation period, and redo the estimation and forecasting procedures. For instance, if the first estimation period is 1963:M1-1986:M12 and the forecast horizon is a period of 4 months ahead, the forecast refers to the predicted recession probability in April 1987. The second estimation period is then 1963:M1-1987:M1 and the forecast will be made for May 1987. These procedures are repeated until the last forecast for June 2010 is obtained. Recall that the financial variables are real-time data and the eight macroeconomic leading indicators used to construct the macro-leading index are available with 1- or 2-month delay before the figures are officially published. By contrast, the dependent variable (recession dummy indicator) is constructed using the NBER business cycle reference dates, which are announced with substantial time delay. The average announcement delay (or publication lag) during 1980–2008 is about 9 months. Thus, it is difficult to identify whether the economy is in a recession in real time. In out-of-sample analysis, we need to consider more practical forecasting situations and make reasonable assumptions about what is known at the time of making forecasts. Recent papers like Kauppi and Saikkonen (2008) and Nyberg (2010) assume that the recession reference dates are announced with a delay of 9–12 months. However, the assumption of a long publication lag implies practical limitation on using current information of explanatory variables. For instance, a publication lag of 9 months means that the explanatory variables are known 9 months ahead of the recession dummy variable. The most recent data of the explanatory variables (in the past 9 months) thus cannot be used for parameter estimation. Recent research such as Chauvet and Piger (2008) suggests alternative procedures for dating business cycle turning points that work well in real time even if they cannot forecast future turning points. Since most of the macroeconomic indicators that the NBER monitors are available with a short announcement delay of one or 2 months, it is plausible to make reasonable assumptions on whether a specific month in the recent past is in a recessionary state even if the official announcement by the NBER is till pending. To balance between recognizing the existence of publication lags of recession reference dates and moving closer to realtime recession forecasting, we assume a publication lag of 3 months.14 Hence, all explanatory variables are known 3 months ahead of the recession dummy variable. With the assumed publication lag, the forecast horizon then consists of two periods. A forecast horizon of less than or equal to 3 months is related to the prediction of recent values of the recession indicator in the period when other explanatory variables are already known. On the other hand, a forecast horizon of more than 3 months is related to the prediction of future values of the recession indicator for which the values of the explanatory variables are unknown. The same probit models in the in-sample analysis are used for the out-of-sample analysis.15 As in the in-sample analysis, we consider the out-of-sample forecasting performance in two aspects. First, we assess the ability of forecasting the duration of recessions by the percentage of correct prediction of recession months. Second, we evaluate the ability of predicting the turning points of business cycles. Table 6 presents the percentage of correct prediction of recession months for the four probit models using the 50% and 25% thresholds to identify recessions. The corresponding results from the conventional static models with the yield spread as the single explanatory variable (Estrella and Hardouvelis, 1991; Boulier and Stekler, 2000) or with the yield spread and equity price index as the predictors (Estrella and Mishkin, 1998) are also reported for comparison. Regardless of the model types

14

Kauppi (2008) also assumed a publication lag of 3 months to conduct the out-of-sample analysis. It should be noted that the lags of some of the explanatory variables may need to be adjusted accordingly when the forecast horizon increases. For instance, the static model contains the equity price index with a lag of 1. Recall that the explanatory variables are assumed to be known three months ahead of the recession data. Hence, for any forecast horizon greater than 4 months, the lag of the equity price index needs to be adjusted to the difference between the forecast horizon and the assumed delay of 3 months. 15

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125 Table 7 Out-of-sample predictions of business cycle turning points from various probit models: 4-month forecast horizon. Actual

Turning points of business cycles

Leads/lags with respect to actual turning points

Static

Dynamic

Autoregressive

Dynamic autoregressive

Static

Dynamic

Autoregressive

Dynamic autoregressive

Peaks 1990/07 2001/03 2007/12

1990/10 2001/10 2007/09

1990/11 2001/07 2007/11

1990/10 None 2007/12

1990/11 2001/09 2007/10

3 7 3

4 4 1

3 N/A 0

4 6 2

Troughs 1991/03 2001/11 2009/06

1991/04 2001/11 2009/07

1991/07 2002/03 2009/08

1991/04 None 2009/07

1991/07 2002/03 2009/09

1 0 1

4 4 2

1 N/A 1

4 4 3

Notes: ‘‘x’’ (‘‘+x’’) means that the predicted turing point is x months before (after) the actual turning point. The out-of-sample forecasts are estimated from various samples running from 1963:M1–1986:M12 up to 1963:M1–2010:M5. The explanatory variables for each model are the same as those in Table 4. A threshold of 25% is used to identify a recession month. A peak of a business cycle is identified when the predicted recession probability changes from less than 0.25 in the last month to 0.25 or above in the current month. Once the peak has been identified, a trough is flagged when the predicted probability changes from 0.25 or above in the last month to less than 0.25 in the current month and the predicted probability cannot go back to 0.25 or above at least in the next 3 months.

and forecast horizons, the use of 50% threshold constantly exhibits poorer out-of-sample performance than the use of 25% threshold. These findings suggest that the use of 25% threshold can produce more accurate out-of-sample forecasts for the duration of recessions. The out-of-sample results also suggest that the four models with the four risk factors proposed in this paper outperform the conventional static models (with either the yield spread or the yield spread and the equity price index as the predictors) by generating higher percentage of correct prediction of recession months. Among the proposed four models, at the 25% threshold, the dynamic and dynamic autoregressive models can predict more accurately the duration of recessions than the static or autoregressive models for various forecast horizons up to 8 months. We next evaluate the performance of predicting the turning points of business cycles. Table 7 compares the out-of-sample results from various probit models in forecasting the peaks and troughs of the last three recession periods. As mentioned before, a forecast horizon of more than 3 months is related to the prediction of future values of the recession indicator. As we are interested in assessing the ability of predicting the future recession indicator, we report the results for the 4-month forecast horizon.16 All four probit models failed to flag an early warning signal for the 1990–1991 recession. As for the 2001 recession, the autoregressive probit model underperformed the static and the other two dynamic models as it failed to generate recession signals throughout the entire recession period. For the recent 2007–2009 recession, the dynamic and dynamic autoregressive models as well as the static model outperformed the autoregressive model by generating an early warning recession signal 1–3 months ahead of the actual peak. On balance, the static, the dynamic and dynamic autoregressive probit models outperform the autoregressive model in terms of predicting the peak of business cycles. 3.5. Tests for parameter stability The sample period used in this paper started from 1963:M1 to 2010:M6. In fact, there were major changes in monetary policy regimes during the sample period. Monetary policy during the 1970s focused primarily on targeting short-term growth rates for monetary aggregates. From 1979 to 1982, the Federal Reserve changed its operating techniques by targeting non-borrowed reserves. In the absence of a stable relationship between money and economic activity, the FOMC then placed less weight on monetary aggregates and focused on measures of inflation and economic activity. The operating target was also changed from non-borrowed reserves to borrowed reserves in 1983. Since the late 1980s, the FOMC has focused on targeting the federal fund rate. The changes in monetary policy regimes during the sample period imply that there may be structural changes, which may affect the stability of the predictive relationship between the recession probability and the explanatory variables. We use two standard tests of structural breaks to examine the stability of parameters. The first test assumes that the date of structural change (or breakpoint date) is known a priori. The null hypothesis assumes that there is no structural change over the full sample period while the alternative hypothesis suggests that a structural change after the known breakpoint date. Based on Andrews and Fair (1988), the applied test statistic for a static probit model is the Lagrange multiplier statistic defined by:

LM ¼

1 ^ 0^Jða ^ 1 S1 ða ^ ^ ; bÞ ^ ; bÞ ^ ; bÞ S1 ð a w1 w2

ð16Þ

^ ^ ; bÞ where wi indicates the proportion of the data before (i = 1) or after (i = 2) the breakpoint with w1 + w2 = 1. The vector S1 ða is obtained from the first derivative of the log-likelihood function given in Eq. (4) where the sum is taken over the first por^ The ^ ; b. tion (w1) of the full sample and the parameters a, b are replaced by their full sample maximum likelihood estimates a 16 The out-of-sample results for the 5-month and 6-month forecast horizons are similar to those for the 4-month forecast horizon. Results are available upon request.

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E.C.Y. Ng / Journal of Macroeconomics 34 (2012) 112–125

Table 8 Results of stability tests for various probit models. Model

Known breakpoints LM statistic

Static Dynamic Autoregressive Dynamic autoregressive Model

Critical value at 5%

October/1979

October/1982

5.98 6.33 6.79 6.46

4.66 9.36 5.32 9.52

H0: no structural change

12.59 14.07 14.07 15.51

October/1979

October/1982

Not Not Not Not

Not Not Not Not

rejected rejected rejected rejected

rejected rejected rejected rejected

Unknown breakpoints

Static Dynamic Autoregressive Dynamic autoregressive

Sup LM statistic

Critical value at 5%

H0: no structural change

8.96 9.36 9.50 9.52

19.30 21.09 21.09 22.83

Not Not Not Not

(December 1998) (October 1982) (June 1981) (October 1982)

rejected rejected rejected rejected

Notes: All the four model specifications refer to those in Table 4 with parameters estimated over the sample period from 1963 M1 to 2010 M6. The LM statistic is the Lagrange multiplier test statistic for one breakpoint with a known date. For the test of one breakpoint with an unknown date, the sup LM statistic is taken over an interior portion of the full sample that excludes 25% of the sample at each end. The date beside the sup LM statistic is the implied breakpoint date for the corresponding probit model.

Table 9 Out-of-sample predictions of business cycle turning points from various probit models using great moderation period data: 4-month forecast horizon. Turning points of business cycles

Leads/lags with respect to actual turning points

Actual

Static

Dynamic

Autoregressive

Dynamic autoregressive

Static

Dynamic

Autoregressive

Dynamic autoregressive

Peaks 2007/12

2007/04

2007/08

2008/04

2007/09

8

4

4

3

Troughs 2009/06

2009/06

2009/06

2009/07

2009/06

0

0

1

0

Notes: ‘‘x’’ (‘‘+x’’) means that the predicted turning point is x months before (after) the actual turning point. The out-of-sample forecasts are estimated from various samples running from 1985:M1–2006:M12 up to 1985:M1–2010:M2. The explanatory variables for each model are the same as those in Table 4. A threshold of 25% is used to identify a recession month. A peak of a business cycle is identified when the predicted recession probability changes from less than 0.25 in the last month to 0.25 or above in the current month. Once the peak has been identified, a trough is flagged when the predicted probability changes from 0.25 or above in the last month to less than 0.25 in the current month and the predicted probability cannot go back to 0.25 or above at least in the next 3 months.

^ is a misspecification robust estimator of the covariance matrix of the score function (Kauppi and Saikkonen, ^ ; bÞ matrix ^Jða 2008). The LM statistics for the dynamic probit models can be derived straightforwardly by inserting their corresponding probability function pt and their parameter estimates in Eq. (4). As Estrella et al. (2003) find that October 1979 and October 1982 are two significant breakpoint dates in a yield-curve based forecasting model for US inflation, we test whether the two dates are also significant breakpoints in our models. The second test assumes that the date of structural change is unknown a priori. To test for a structural break with an unknown breakpoint, the sup of LM can be applied where the sup is taken over an interior portion of the full sample that excludes some observations at each end.17 Andrews (1993) derives that the sup LM statistic converges in distribution to the square of a standardized tied-down Bessel process under general conditions. Estrella (2003) tabulates the corresponding critical values of the distribution of this process. Table 8 presents the results of stability tests for the four probit models. There is no evidence that either October 1979 or October 1982 is a significant breakpoint date. Also, the stability tests with unknown breakpoints for the four probit models support the null hypothesis that there is no structural change in the whole sample period. In sum, the two tests of structural changes both suggest that the parameters of the various probit models are stable over the full sample period. Since the mid-1980s, the US economy has entered into a great moderation period in which the volatility of business cycle fluctuations has reduced substantially (except for the recent recession during 2007–2009), and it is also characterized by low inflation and predictable policy. As an additional robustness check to the out-of-sample results, we modify the above out-ofsample analysis by considering a series of out-of-sample forecasts obtained from various estimation samples running from 1985:M1–2006:M12 up to 1985:M1–2010:M2. Using each sample at a time, we estimate the four probit models and make 4month-ahead forecasts.18 We then add one more month to the estimation period, and redo the estimation and forecasting procedures. Table 9 compares the out-of-sample results from various probit models in forecasting the peaks and troughs of the 17 The LM test statistics in Eq. (16) is unbounded in the limit if the potential breakpoints include the endpoints of the sample. There is no consensus in the literature on how many observations should be excluded. We follow Estrella et al. (2003) to exclude 25% of the sample at each end. 18 We obtain similar results for using 5- or 6-month forecast horizons. Results are available upon request.

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recent business cycle. In terms of predicting the peak, the static, the dynamic and dynamic autoregressive probit models outperform the autoregressive model as they could flag an early warning signal for a recession. The results are consistent with the above out-of-sample analysis. 4. Conclusion This paper proposes to use a more complete set of recession risk factors and the advanced dynamic probit models to forecast US recessions. The risk factors include financial market expectations of a gloomy economic outlook, liquidity risk in the general economy, the risk of a negative wealth effect resulting from the bursting of asset price bubbles, and signs of weak macroeconomic fundamentals. There are two main findings from the empirical analysis. First, the proposed risk factors combined with the various probit model specifications can forecast the duration of recessions more accurately than the conventional static models with only yield spread and stock market return as the predictors. Second, while the advanced dynamic probit models outperform the static model in terms of predicting the duration of recessions, the static probit model with the proposed risk factors is as good as the dynamic probit models with respect to forecasting the peaks of business cycles. These findings suggest that the static and dynamic probit models with the proposed risk factors can complement one another to provide a better assessment on the duration and timing of business cycles in the near future. References Andrews, D., 1993. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856. Andrews, D., Fair, R., 1988. Inference in non-linear econometric models with structural change. Review of Economic Studies 55, 615–640. Boulier, B., Stekler, H.O., 2000. The term spread as a monthly cyclical indicator: an evaluation. Economics Letters 66, 79–83. Chauvet, M., Piger, J., 2008. A comparison of the real-time performance of business cycle dating methods. Journal of Business and Economic Statistics 26, 42– 49. Chauvet, M., Potter, S., 2005. Forecasting recessions using the yield curve. Journal of Forecasting 24, 77–103. Chen, N.-F., 1991. Financial investment opportunities and the macro-economy. Journal of Finance 46 (2), 529–554. de Jong, R.M., Woutersen, T.M., forthcoming. Dynamic time series binary choice. Econometric Theory. Diebold, F.X., Rudebusch, G.D., 1989. Scoring the leading indicators. Journal of Business 62, 369–391. Dueker, M., 1997. Strengthening the case for the yield curve as a predictor of US recessions. Federal Reserve Bank of St. Louis Economic Review 79, 41–51. Estrella, A., 1998. A new measure of fit for equations with dichotomous dependent variables. Journal of Business and Economic Statistics 16, 198–205. Estrella, A., 2003. Critical values and p values of bessell process distributions: computation and application to structural break tests. Econometric Theory 19, 1128–1143. Estrella, A., 2005. Why does the yield curve predict output and inflation? The Economic Journal 115, 722–744. Estrella, A., Hardouvelis, G., 1991. The term structure as a predicator of real economic activity. Journal of Finance 46 (2), 555–576. Estrella, A., Mishkin, F.S., 1995. The Term Structure of Interest Rates and its Role in Monetary Policy for the European Central Bank. NBER Working Paper 5279. Estrella, A., Mishkin, F.S., 1998. Predicting US recessions: financial variables as leading indicators. Review of Economics and Statistics 80, 45–61. Estrella, A., Rodrigues, A., Schich, S., 2003. How stable is the predictive power of the yield curve? Evidence from Germany and the United States. Review of Economics and Statistics 85 (3), 629–644. Harvey, C., 1988. The real term structure and consumption growth. Journal of Financial Economics 22, 305–333. Kauppi, H., 2008. Yield-Curve Based Probit Models for Forecasting US Recessions: Stability and Dynamics. Helsinki Center of Economic Research Discussion Papers 221. Kauppi, H., Saikkonen, P., 2008. Predicting US recessions with dynamic binary response models. Review of Economics and Statistics 90 (4), 777–791. Mishkin, F.S., 1990a. What does the term structure tell us about future inflation? Journal of Monetary Economics 25, 77–95. Mishkin, F.S., 1990b. The information in the longer-maturity term structure about future inflation. Quarterly Journal of Economics 55, 815–828. Mishkin, F.S., 1991. A multi-country study of the information in the term structure about future inflation. Journal of International Money and Finance 19, 2– 22. Moneta, F., 2005. Does the yield spread predict recessions in the euro area? International Finance 8 (2), 263–301. Nyberg, H., 2010. Dynamic probit models and financial variables in recession forecasting. Journal of Forecasting 29 (1–2), 215–230. Rydberg, T.H., Shephard, N., 2003. Dynamics of trade-by-trade price movements: decomposition and models. Journal of Financial Econometrics 1 (spring), 2–25.