# Predicting U.S. recessions through a combination of probability forecasts

## Authors

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DOI: 10.1007/s00181-012-0671-4

- Cite this article as:
- De Luca, G. & Carfora, A. Empir Econ (2014) 46: 127. doi:10.1007/s00181-012-0671-4

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## Abstract

Recently De Luca and Carfora (Statistica e Applicazioni 8:123–134, 2010) have proposed a novel model for binary time series, the Binomial Heterogenous Autoregressive (BHAR) model, successfully applied for the analysis of the quarterly binary time series of U.S. recessions. In this work we want to measure the efficacy of the out-of-sample forecasts of the BHAR model compared to the probit models by Kauppi and Saikkonen (Rev Econ Stat 90:777–791, 2008). Given the substantial indifference of the predictive accuracy between the BHAR and the probit models, a combination of forecasts using the method proposed by Bates and Granger (Oper Res Q 20:451–468, 1969) for probability forecasts is analyzed. We show how the forecasts obtained by the combination between the BHAR model and each of the probit models are superior compared to the forecasts obtained by each single model.

### Keywords

Binary response modelRecession forecasting Forecasts combinationDiebold–Mariano test### Jel Classifications:

E32E37C53## 1 Introduction

The capacity of forecasting if an economy will be in recession or not, a month, a quarter, or a year ahead, is an important question for researchers and for all the subjects (banks, investors, policy makers) that take important decisions with regard to the business cycle. Actually, many authors have presented empirical researches about this matter. The approaches adopted in literature to predict recessions can be divided into two classes.

A first class indicates that there are important financial and macroeconomic variables, such as interest rate, spread, stock prices, and growth of GDP, able to explain the phenomenon (see e.g., Dueker 1997; Estrella and Mishkin 1998; Kauppi and Saikkonen 2008; Nyberg 2010). These studies are based on models where the probability of a recession is expressed in function of these variables as regressors. In most cases, such probabilities are obtained from the applications of probit models.

A second stream of researches uses purely autoregressive models, in the sense that the probability of a recession is determined by the past behavior of the occurrences of the recessions. A relevant reference is the article of Startz (2008) where the Binomial ARMA (BARMA) model has been introduced as a generalization of the Binomial AR model (Cox 1981; Zeger and Qaqish 1988). The importance of Startz (2008) is also given by the definition of an innovative graphical tool, the autopersistence graph, which is thought as an aid for the researcher in the stage of identification of the appropriate model for the analyzed time series. The approach proposed by Startz (2008) has been extended by De Luca and Carfora (2010). They have proposed a novel model denoted as Binomial Heterogeneous Autoregressive Model (BHAR) for the analysis of binary time series with a long-memory feature. The BHAR model can be interpreted as an autoregressive model including some cumulative lagged components, obtained summing the relative observations. The better in-sample performance of the BHAR model with respect to the BARMA model has been shown using the quarterly U.S. recession time series (De Luca and Carfora 2010).

The aim of this work is twofold. First, we intend to investigate the capacity of forecasting the presence or the absence of recession some time periods ahead in the two approaches. For the first approach, the comparison is realized focusing on the four models studied by Kauppi and Saikkonen (2008). For the second approach we make use of the BHAR model.

The second aim is that of overcoming the duality between the two approaches. A combination of probability forecasts is recommended to obtain new predictions able to combine the strengths of the original models.

In this work, we have considered a largely popular binary time series, that is the quarterly U.S. recessions. The series is constructed using the information of National Bureau of Economic Research (NBER). In particular, each quarter is coded with a 1 for recession, if any month in the quarter is identified by the NBER as being in a recession, with 0 for expansion otherwise. It is worth mentioning that the NBER recession dates are set well after the events of the day, so that forecasting a recession event with lagged data is a formidable task.

The paper is organized as follows. Section 2 contains a description of some classic probit binary response models and an innovative model (BHAR) to forecast recession. The forecasting procedures and the related comparison of predictive accuracy of the models are described in Sect. 3. In Sect. 4 the authors present a method for combining probability forecasts with the aim of improving the predictive accuracy of the single models. Finally, Sect. 5 concludes.

## 2 The models

### 2.1 The probit models by Kauppi and Saikkonen (2008)

Let us define \(Y_t\in \{0,1\}\) the binary time series at time \(t\).

*t*conditionally on the information at the time \( t-1\) is given by

- 1.The static model considers the lagged interest rate spread between the 10-year Treasury bond rate and the 3-month Treasury bill rate as explicative variable,where \(x_{t-k}\) is the interest spread at lag \(k\). Given the observed time series \(y_t\) for \(t=1,\ldots , T\), the generic \(h\)-step-ahead prediction is easy to compute,$$\begin{aligned} \pi _t = \omega + \beta x_{t-k} \end{aligned}$$(1)In particular when \(h\le k\), the explicative variable is known.$$\begin{aligned} E_{T}(Y_{T+h}) = \phi (\omega + \beta x_{T+h-k}). \end{aligned}$$
- 2.The dynamic model includes the one-lagged recession indicator,The one-step-ahead prediction is simply given by$$\begin{aligned} \pi _t = \omega + \beta x_{t-k} + \delta y_{t-1}. \end{aligned}$$(2)while for \(h>1\) we need \(E_T(Y_{T+h-1})\). For \(h=2\),$$\begin{aligned} E_{T}(Y_{T+1}) = \phi (\omega + \beta x_{T+1-k} + \delta y_{T}) \end{aligned}$$For a generic \(h\),$$\begin{aligned} E_T(Y_{T+2})&= E_T(\phi (\omega + \beta x_{T+2-k} + \delta Y_{T+1})) \\&= \phi (\omega + \beta x_{T+2-k} + \delta )P_T(Y_{T+1}=1)\\&\quad +\phi (\omega + \beta x_{T+2-k})(1-P_T(Y_{T+1}=1)) \\&= \phi (\omega + \beta x_{T+2-k} + \delta )\phi (\omega + \beta x_{T+1-k} + \delta y_{T}) \\&\quad + \phi (\omega + \beta x_{T+2-k})(1-\phi (\omega + \beta x_{T+1-k} + \delta y_{T})). \end{aligned}$$$$\begin{aligned} E_{T}(Y_{T+h})&= E_T(\phi (\omega + \beta x_{T+h-k} + \delta Y_{T+h-1})) \\&= \phi (\omega + \beta x_{T+h-k} + \delta ) P_T(Y_{T+h-1}) \\&\quad + \phi (\omega + \beta x_{T+h-k})(1-P_T(Y_{T+h-1})) . \end{aligned}$$
- 3.In the autoregressive probit model the one-lagged value of \(\pi _t\) is encountered,The one-step-ahead prediction is given by$$\begin{aligned} \pi _t = \omega + \beta x_{t-k} + \alpha \pi _{t-1}. \end{aligned}$$(3)For \(h=2\),$$\begin{aligned} E_{T}(Y_{T+1}) = \phi (\omega + \beta x_{T+1-k} + \alpha \pi _{T}). \end{aligned}$$For a generic \(h\),$$\begin{aligned} E_T(Y_{T+2})&= E_T(\phi (\omega + \beta x_{T+2-k} + \alpha \pi _{T+1})) \\&= \phi (\omega + \beta x_{T+2-k} + \alpha (\omega + \beta x_{T+1-k} + \alpha \pi _{T})) . \end{aligned}$$$$\begin{aligned} E_T(Y_{T+h})&= E_T(\phi (\omega + \beta x_{T+h-k} + \alpha \pi _{T+h-1})) \\&= \phi \left(\omega + \beta x_{T+h-k} + \sum _{j=0}^{h-1} \alpha ^j(\omega + \beta x_{T-j}) + \alpha ^h \pi _T \right). \end{aligned}$$
- 4.Finally, in the dynamic autoregressive probit model,It is the more general model, including all the explicative variables. Its predictions can be easily obtained from the previous models.$$\begin{aligned} \pi _t = \omega + \beta x_{t-k} + \delta y_{t-1} + \alpha \pi _{t-1}. \end{aligned}$$(4)

Estimates (standar errors) of the probit models by Kauppi and Saikkonen (2008) for the interval 1961.2–2009.2

Parameters | Static | Dyn. | AR | Dyn.AR |
---|---|---|---|---|

\(\omega \) | -0.4379 | -1.3103 | -0.0633 | -1.5956 |

(0.1438) | (0.2227) | (0.0792) | (0.3310) | |

\(\beta \) | -0.6698 | -0.3962 | -0.3654 | -0.4745 |

(0.1269) | (0.1577) | ( 0.1021) | (0.1989) | |

\(\delta \) | - | 2.2292 | - | 2.6715 |

(0.3329) | (0.4987) | |||

\(\alpha \) | - | - | 0.6402 | -0.2129 |

(0.1063) | (0.1599) |

### 2.2 The BHAR model

The graphic suggests that quarterly U.S. recessions time series is a typical time series with long-memory effects in the sense that the conditional probabilities do not rapidly converge to the marginal probabilities, but keeps memory of the past. Approximately at lags 5, 10, and 20 the autopersistence functions cross the horizontal line of the marginal probability, creating a wave motion in the graph. We believe that the message of the graph should be acknowledged and an appropriate model including a relevant part of the past information should be proposed. In other words, we would like to use the long memory effects of the series to produce recession forecasts.

Estimates (standard errors) of the BHAR model for the interval 1855.1–2009.2

Parameters | Estimate |
---|---|

\(\phi _0\) | -2.4559 |

(0.2643) | |

\(\phi _1\) | 7.8342 |

(1.1680) | |

\(\gamma _{0}\) | -1.0771 |

(0.3571) | |

\(\gamma _{1}\) | -0.1617 |

(0.1185) | |

\(\gamma _{2}\) | 0.1103 |

(0.0554) |

## 3 Comparison of the predictive accuracy

The first aim of this work is to compare the predictive accuracy for the U.S. recessions time series between the probit models proposed by Kauppi and Saikkonen (2008) and the BHAR model. We have computed the predictions \(P_T(Y_{T+h})\), for \(T\) starting in 1970.2 for \(h=1\) (for a total of 157 one-step-ahead forecasts), in 1970.3 for \(h=2\) (156 two-step-ahead forecasts), in 1970.4 for \(h=3\) (155 three-step-ahead forecasts), in 1971.1 for \(h=4\) (154 four-step-ahead forecasts), and ending in 2009.2.

A sequential estimation procedure has been adopted for the out-of-sample forecasts. Let us focus on the BHAR model. It is estimated using data from the beginning of the sample up to a particular quarter. The estimated coefficients are used to form predictions from one quarter ahead to four quarters ahead. Then, one observation is added to the sample and the model is estimated to form new predictions. In this way the procedure ensures that the forecasts are obtained using only the information available in the past, that is data available after the forecast are not used to make it. For example, to make a forecast four quarters ahead in the first quarter of 2005, we have estimated the model using the observations from the beginning until the first quarter of 2004, then we have used the estimated coefficients to make the prediction four quarters ahead. In order to make a prediction for the second quarter of 2005 we estimate the model using the data until the second quarter of 2004 and so on. This type of iterative procedure leads to a fairer and more realistic test of the predictive abilities of the different models than the in-sample results (Kauppi and Saikkonen 2008).

Mean absolute error (MAE), root mean square error (RMSE), and proportion of correct forecasts (PCF) for \(h\)=1,2,3,4

BHAR | Static | Dyn. | AR | Dyn.AR | |
---|---|---|---|---|---|

\(h=1\) | |||||

MAE | 0.151 | 0.182 | 0.120 | 0.172 | 0.113 |

RMSE | 0.262 | 0.342 | 0.268 | 0.337 | 0.266 |

PCF | 0.924 | 0.815 | 0.898 | 0.828 | 0.904 |

\(h=2\) | |||||

MAE | 0.266 | 0.181 | 0.172 | 0.177 | 0.180 |

RMSE | 0.351 | 0.343 | 0.333 | 0.346 | 0.343 |

PCF | 0.846 | 0.821 | 0.840 | 0.821 | 0.833 |

\(h=3\) | |||||

MAE | 0.353 | 0.178 | 0.192 | 0.187 | 0.205 |

RMSE | 0.408 | 0.338 | 0.355 | 0.361 | 0.374 |

PCF | 0.819 | 0.826 | 0.806 | 0.806 | 0.800 |

\(h=4\) | |||||

MAE | 0.388 | 0.172 | 0.181 | 0.201 | 0.210 |

RMSE | 0.420 | 0.331 | 0.341 | 0.380 | 0.378 |

PCF | 0.812 | 0.831 | 0.818 | 0.786 | 0.792 |

Diebold–Mariano test-statistics and \(p\) values (in parenthesis) for the predictive accuracy of the BHAR model against the four probit models

BHAR vs. Static | BHAR vs. Dyn. | BHAR vs. AR | BHAR vs. Dyn.AR | |
---|---|---|---|---|

\(h=1\) | -2.8499 | -1.2358 | -2.8097 | -1.1809 |

(0.0044) | (0.2165) | (0.0049) | (0.2376) | |

\(h=2\) | -0.271 | 0.0442 | -0.4053 | -0.3866 |

(0.7864) | (0.9647) | (0.6853) | (0.699) | |

\(h=3\) | 1.6879 | 1.3314 | 0.7766 | 0.4735 |

(0.09143) | (0.1831) | (0.4374) | (0.6359) | |

\(h=4\) | 2.5538 | 2.1304 | 0.4877 | 0.6081 |

(0.0106) | (0.0331) | (0.6257) | (0.5431) |

On the whole, the comparison between the models proves the substantial indifference in the predictive accuracy.

## 4 Combination of forecasts

The equal predictive accuracy of the BHAR model against the probit models has suggested to extend the analysis considering a combination of forecasts. Our aim is to test if a combination of probability forecast improves upon the individual forecasts.

In economic literature, an extensive number of studies has verified the usefulness of combining forecasts.

The pioneering article by Bates and Granger (1969) applied linear combinations of forecasts to the monthly time series of airline passengers. Granger and Ramanathan (1984) approached the issue of determining the weights of the combination after running a regression where the actual value of the time series is the dependent variable and the individual forecasts are the regressors. This procedure is also known as regression-based method. In more detail it is a constrained regression, because the sum of the weights has to be constrained to be one. Diebold (1998) implemented the regression-based method taking into account the possible serial correlation in the combined prediction errors. A comprehensive review is Clemen (1989). Finally, Clements and Harvey (2011) considered the combination of probability forecasts. They present different combination methods, the linear opinion pool, the logarithmic opinion pull, and the Kamstra and Kamstra (1998) procedure advocating log odds ratios to be combined through logit regressions.

In our study, we have considered the simplest method of combining forecasts, that is the linear combination *à la* Bates and Granger.

Combination BHAR-probit: mean absolute error (MAE), root mean square error (RMSE), and proportion of correct forecasts (PCF) for \(h\)=1,2,3,4

BHAR-Static | BHAR-Dyn. | BHAR-AR | BHAR-Dyn.AR | |
---|---|---|---|---|

\(h=1\) | ||||

MAE | 0.149 | 0.119 | 0.142 | 0.115 |

RMSE | 0.272 | 0.261 | 0.273 | 0.261 |

PCF | 0.930 | 0.917 | 0.917 | 0.924 |

\(h=2\) | ||||

MAE | 0.179 | 0.176 | 0.178 | 0.184 |

RMSE | 0.318 | 0.325 | 0.328 | 0.334 |

PCF | 0.859 | 0.859 | 0.853 | 0.840 |

\(h=3\) | ||||

MAE | 0.195 | 0.205 | 0.196 | 0.215 |

RMSE | 0.329 | 0.347 | 0.338 | 0.356 |

PCF | 0.826 | 0.826 | 0.826 | 0.819 |

\(h=4\) | ||||

MAE | 0.194 | 0.198 | 0.203 | 0.217 |

RMSE | 0.326 | 0.331 | 0.340 | 0.346 |

PCF | 0.831 | 0.831 | 0.799 | 0.818 |

The \(p\) values of the tests for the different values of \(h\) are displayed in Tables 6 and 7. We can observe that for \(h=1\) the strategy of combining the forecasts does not provide an evidence of superiority compared to the BHAR model, but is surely winning compared to the probit models, particularly against the static and AR models not including a dynamic component. For \(h>1\) the \(p\) values confirm the superiority of the combined forecasts compared to all the models: there is no \(p\) value greater than 0.10.^{1}

\(P\) values of the Diebold–Mariano test for the predictive accuracy of the combined forecasts models against the BHAR model

BHAR-Static | BHAR-Dyn. | BHAR-AR | BHAR-Dyn.AR | |
---|---|---|---|---|

vs BHAR | vs BHAR | vs BHAR | vs BHAR | |

\(h=1\) | 0.9127 | 0.4216 | 0.9202 | 0.6989 |

\(h=2\) | 0.0028 | 0.0074 | 0.0501 | 0.0879 |

\(h=3\) | 0.0000 | 0.0000 | 0.0001 | 0.0014 |

\(h=4\) | 0.0006 | 0.0004 | 0.0001 | 0.0003 |

\(P\) values of the Diebold–Mariano test for the predictive accuracy of the combined forecats models against the four probit models

BHAR-Static vs. Static | BHAR-Dyn. vs. Dyn. | BHAR-AR vs. AR | BHAR-Dyn.AR vs. Dyn.AR | |
---|---|---|---|---|

\(h=1\) | 0.0021 | 0.3264 | 0.0031 | 0.0794 |

\(h=2\) | 0.0637 | 0.0563 | 0.0624 | 0.0523 |

\(h=3\) | 0.1153 | 0.0420 | 0.0442 | 0.0211 |

\(h=4\) | 0.0594 | 0.0159 | 0.0205 | 0.0165 |

## 5 Conclusions

In this article, we have analyzed the predictions of the binary time series of the quarterly U.S. recessions. In particular, we have measured the accuracy of the out-of-sample forecasts of a novel model, the BHAR model, to investigate if it is possible to find an alternative to the dominant approaches in literature where financial and macroeconomic variables are used as predictors. The forecasts of the BHAR model have proved to be very positive but not superior to the probit models by Kauppi and Saikkonen (2008).

In reply to the substantial indifference of the predictive accuracy of the forecasts of the traditional and innovative approaches, the use of a probability forecast combination has provided statistically significant results both in terms of measures of predictive accuracy (MAE and RMSE) and in terms of percentage of correct forecasts.

Moreover, according to the Diebold–Mariano test, the combined forecasts, enriched by the distinctive elements of the two types of models, that is the capacity of capturing the long memory effects of the BHAR model and the indications provided by some financial indicator of the probit model, ensures a lower expected loss function differential.

We have also computed the same test for different values of \(v\). The general results do not change. However, for \(h=4\) the combination of forecasts seems to suffer from the comparison with the static and the dynamic model when the value of \(v\) increases. This fact can be explained by the presence in the probit models of the variable \(x_{t-4}\) which gives a relevant contribute to the forecasts four period ahead.