Empirical Economics

, Volume 46, Issue 1, pp 127–144

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

Authors

    • Department of Statistics and Mathematics for Economic ResearchUniversity of Naples Parthenope
  • Alfonso Carfora
    • Department of Statistics and Mathematics for Economic ResearchUniversity of Naples Parthenope
Article

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

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 \(t\).

The probit models by Kauppi and Saikkonen (2008) assume that the expected value at the time t conditionally on the information at the time \( t-1\) is given by
$$\begin{aligned} E_{t-1}(Y_{t})= \phi (\pi _{t}) \end{aligned}$$
where \(\phi (\cdot )\) is the cdf of a standard Normal distribution.
Four different equations for \(\pi _t\) give rise to four different models.
  1. 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,
    $$\begin{aligned} \pi _t = \omega + \beta x_{t-k} \end{aligned}$$
    (1)
    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,
    $$\begin{aligned} E_{T}(Y_{T+h}) = \phi (\omega + \beta x_{T+h-k}). \end{aligned}$$
    In particular when \(h\le k\), the explicative variable is known.
     
  2. 2.
    The dynamic model includes the one-lagged recession indicator,
    $$\begin{aligned} \pi _t = \omega + \beta x_{t-k} + \delta y_{t-1}. \end{aligned}$$
    (2)
    The one-step-ahead prediction is simply given by
    $$\begin{aligned} E_{T}(Y_{T+1}) = \phi (\omega + \beta x_{T+1-k} + \delta y_{T}) \end{aligned}$$
    while for \(h>1\) we need \(E_T(Y_{T+h-1})\). For \(h=2\),
    $$\begin{aligned} E_T(Y_{T+2})&= E_T(\phi (\omega + \beta x_{T+2-k} + \delta Y_{T+1})) \\&= \phi (\omega + \beta x_{T+2-k} + \delta )P_T(Y_{T+1}=1)\\&\quad +\phi (\omega + \beta x_{T+2-k})(1-P_T(Y_{T+1}=1)) \\&= \phi (\omega + \beta x_{T+2-k} + \delta )\phi (\omega + \beta x_{T+1-k} + \delta y_{T}) \\&\quad + \phi (\omega + \beta x_{T+2-k})(1-\phi (\omega + \beta x_{T+1-k} + \delta y_{T})). \end{aligned}$$
    For a generic \(h\),
    $$\begin{aligned} E_{T}(Y_{T+h})&= E_T(\phi (\omega + \beta x_{T+h-k} + \delta Y_{T+h-1})) \\&= \phi (\omega + \beta x_{T+h-k} + \delta ) P_T(Y_{T+h-1}) \\&\quad + \phi (\omega + \beta x_{T+h-k})(1-P_T(Y_{T+h-1})) . \end{aligned}$$
     
  3. 3.
    In the autoregressive probit model the one-lagged value of \(\pi _t\) is encountered,
    $$\begin{aligned} \pi _t = \omega + \beta x_{t-k} + \alpha \pi _{t-1}. \end{aligned}$$
    (3)
    The one-step-ahead prediction is given by
    $$\begin{aligned} E_{T}(Y_{T+1}) = \phi (\omega + \beta x_{T+1-k} + \alpha \pi _{T}). \end{aligned}$$
    For \(h=2\),
    $$\begin{aligned} E_T(Y_{T+2})&= E_T(\phi (\omega + \beta x_{T+2-k} + \alpha \pi _{T+1})) \\&= \phi (\omega + \beta x_{T+2-k} + \alpha (\omega + \beta x_{T+1-k} + \alpha \pi _{T})) . \end{aligned}$$
    For a generic \(h\),
    $$\begin{aligned} E_T(Y_{T+h})&= E_T(\phi (\omega + \beta x_{T+h-k} + \alpha \pi _{T+h-1})) \\&= \phi \left(\omega + \beta x_{T+h-k} + \sum _{j=0}^{h-1} \alpha ^j(\omega + \beta x_{T-j}) + \alpha ^h \pi _T \right). \end{aligned}$$
     
  4. 4.
    Finally, in the dynamic autoregressive probit model,
    $$\begin{aligned} \pi _t = \omega + \beta x_{t-k} + \delta y_{t-1} + \alpha \pi _{t-1}. \end{aligned}$$
    (4)
    It is the more general model, including all the explicative variables. Its predictions can be easily obtained from the previous models.
     
We have used maximum likelihood to estimate the parameters of all the models with \(k=4\), that is with the interest rate spread at time \(t-4\), for quarterly U.S. recession data from 1961 (II quarter) to 2009 (II quarter). The time interval is restricted due to the availability of data of the interest spread. Table 1 contains the estimates and the standard errors of the four models. All the parameters are significant, except the coefficient \(\alpha \) of the dynamic autoregressive model.
Table 1

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

In the analysis of continuous time series, Box–Jenkins procedure suggests to use the global and partial correlograms to identify the model to estimate. For a binary time series, Startz (2008) defines the autopersistence functions
$$\begin{aligned} {\text{ APF}}^0(l) = P(Y_{t+l} =1| Y_t = 0) \end{aligned}$$
and
$$\begin{aligned} {\text{ APF}}^1(l) = P(Y_{t+l}= 1 | Y_t = 1) \end{aligned}$$
simply obtained as estimated conditional probabilities from the time series. The graphical representation of these functions is an important graphical tool suggesting the type of model that is needed to describe a specific binary time series. In particular, the speed of convergence of the conditional probabilities APF\(^0(l)\) and APF\(^1(l)\) to the marginal probability \(P(Y_t = 1)\) is an indication for the model selection. The empirical autopersistence functions of quarterly U.S. recession data from 1855 (I quarter) to 2009 (II quarter) are represented in Fig. 1 with a number of lags equal to 20.
https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig1_HTML.gif
Fig. 1

Estimated APF\(^0(l)\) (solid line) and APF\(^1(l)\) (dotted line) for quarterly U.S. recession data from 1855 (I quarter) to 2009 (II quarter)

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.

In the Binomial Heterogeneous Autoregressive (BHAR) model the conditional probability is defined as
$$\begin{aligned} P(Y_t|Y_{t-1},\ldots ,Y_{t-4\cdot 2^{k}}) = \mu _t \end{aligned}$$
where
$$\begin{aligned} \mu _t&= \frac{\exp (\eta _t)}{1+\exp (\eta _t)}, \end{aligned}$$
(5)
$$\begin{aligned} \eta _t&= \phi _0 + \phi _1 Y_{t-1} + \sum _{j=0}^k \gamma _{j} Y^{(2^{j})}_{t-1} \end{aligned}$$
(6)
with \(k\in \{0,1,2,\ldots \}\) and
$$\begin{aligned} Y^{(i)}_t = \sum _{j=0}^{4i-1} Y_{t-j} \end{aligned}$$
is a cumulative sum.
For quarterly U.S. recession binary time series, we set \(k=2\) obtaining
$$\begin{aligned}&P(Y_t|Y_{t-1},\ldots ,Y_{t-16}) = \mu _t\\&\mu _t = \frac{\exp (\eta _t)}{1+\exp (\eta _t)} \end{aligned}$$
with expression (6) becoming
$$\begin{aligned} \eta _t = \phi _0 + \phi _1 Y_{t-1} + \gamma _{0} Y^{(1)}_{t-1} + \gamma _{1} Y^{(2)}_{t-1} + \gamma _{2} Y^{(4)}_{t-1}. \end{aligned}$$
(7)
In the logit link function, \(\mu _t\) depends on \(\eta _t\) which, in turn, depends on \(Y_{t-1}\), then on the cumulative 1-, 2-, and 4-years lagged components. Finally, defined \(\theta \) the parameter vector, the conditional likelihood function of the BHAR model is
$$\begin{aligned} \ell (\theta ) = \sum _{t=17}^T\left(y_t\log \mu _t + (1-y_t)\log (1-\mu _t)\right) \end{aligned}$$
(8)
where \(T\) is the length of the time series. The estimates and standard errors are reported in Table 2. The only \(p\) value above the traditional significance levels concerns \(\gamma _{1}\) (0.1724).
Table 2

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)

The autopersistence functions of the estimated BHAR model produced by simulation (Fig. 2) is very similar to the empirical counterpart.
https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig2_HTML.gif
Fig. 2

Estimated APF\(^0(l)\) (solid line) and APF\(^1(l)\) (dotted line) of the BHAR model with \(k=2\) for quarterly U.S. recession data

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.

The expected value for a generic quarter \(T+h\) conditionally on the information at time \(T\) is associated to the conditional probability that is
$$\begin{aligned} E_{T}(Y_{T+h})= P_{T}(Y_{T+h}=1) = P_T(Y_{T+h}). \end{aligned}$$
In the probit models by Kauppi and Saikkonen (2008), with \(h=1\), we obtain
$$\begin{aligned} P_{T}(Y_{T+1}) = \phi (\pi _{T+1}). \end{aligned}$$
When \(h>1\) the lags included between \(T+1\) and \(T+h-1\) are to be estimated by one-step-ahead iterative procedure. So, defined the vector notation
$$\begin{aligned} Y^{T+h}_{T+h-k} = (Y_{T+h-k},Y_{T+h-k+1},\dots , Y_{T+h}) \end{aligned}$$
for \(k=0,1,2,\ldots \) and the Cartesian product
$$\begin{aligned} B_k = \{0,1\} ^k \end{aligned}$$
for \(k =1,2,\ldots , h-1\),
$$\begin{aligned} P_{T}(Y_{T+h}) = \sum _{Y^{T+h-1}_{T+1}\in B_{h-1}} P_{T}\left(Y^{T+h-1}_{T+1}\right) \phi (\pi _{T+h}) \end{aligned}$$
with
$$\begin{aligned} P_{T}\left(Y^{T+h-1}_{T+1}\right) = \prod _{j=1}^{h-1} (P_{T}(Y_{T+j}))^{Y_{T+j}} (1-(P_{T}(Y_{T+j})))^{1-Y_{T+j}}. \end{aligned}$$
The same procedure applies for the predictions of the BHAR model, considering
$$\begin{aligned} P_{T}(Y_{T+1}) = F(\eta _{T+1}) \end{aligned}$$
where \(F\) is the logistic \(cdf\).

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).

Figures 3 and 4 illustrate the estimated probabilities of recession one quarter ahead of the probit models and the BHAR model, respectively. Figures 5 and 6 report the estimated probabilities of recession for the forecast horizon of four quarters.
https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig3_HTML.gif
Fig. 3

Actual recessions (bars) and estimated probabilities of recession (solid lines) from the probit models (from top-left to bottom-right Static, Dyn., AR, Dyn.AR), \(h=1\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig4_HTML.gif
Fig. 4

Actual recessions (bars) and estimated probabilities of recession (solid line) from the BHAR model, \(h=1\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig5_HTML.gif
Fig. 5

Actual recessions (bars) and estimated probabilities of recession (solid lines) from the probit models (from top-left to bottom-right Static, Dyn., AR, Dyn.AR), \(h=4\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig6_HTML.gif
Fig. 6

Actual recessions (bars) and estimated probabilities of recession (solid line) from the BHAR model, \(h=4\)

For each model and for each \(h\)-ahead forecast, the predictions are compared with the actual recessions. Table 3 reports the root mean square error,
$$\begin{aligned} \text{ RMSE} = \frac{\sum _{j=T}^{T+157-h}(y_{j+h}-P_j(Y_{j+h}))^2}{157-h+1}, \end{aligned}$$
the mean absolute error,
$$\begin{aligned} \text{ MAE} = \frac{\sum _{j=T}^{T+157-h}|y_{j+h}-P_j(Y_{j+h})|}{157-h+1}, \end{aligned}$$
and the proportion of correct forecast (PCF) computed as the proportion of times a forecasted recession (the probability forecast is greater than 0.50) is associated to an actual recession and viceversa. The results are not unambiguous. The MAE indicates different probit models according to the forecast horizon, while the RMSE favors the BHAR model for \(h=1\), the dynamic probit model for \(h=2\) and the static probit model for \(h=3,4\). Also, the PCF shows a tendency to favor the BHAR model for \(h=1,2\) and the static probit for \(h>2\).
Table 3

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

Moreover, the predictions of the BHAR model and of each probit model are compared through the Diebold–Mariano test of predictive accuracy (Diebold and Mariano 1995; Harvey et al. 1997). Let us define \(g(e_{Bt})\) and \(g(e_{it})\) the loss functions for the BHAR model and for one of the probit models \((i=1,2,3,4)\), respectively, and
$$\begin{aligned} d_t=g(e_{Bt})-g(e_{it}) \end{aligned}$$
the loss differential. We have considered the null hypothesis of the Diebold–Mariano test that the models have the same forecast accuracy
$$\begin{aligned} H_0:E(d_t) = 0 \end{aligned}$$
opposed to the two-sided alternative hypothesis that the forecasts obtained by one of the two models are more accurate
$$\begin{aligned} H_1:E(d_t) \ne 0 \end{aligned}$$
using the quadratic function \(g(a)=a^2\).
Table 4 reports the test statistics and the \(p\) values of the test. The null hypothesis of equal predictive accuracy of the BHAR model and the probit models is generally accepted. The superiority of long-memory model is found only in two cases for \(h=1\), while for \(h=4\) both the static and the dynamic probit models seem to be better.
Table 4

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.

Let us define \(f_T^{B}(1)\) the probability forecast at time \(T+1\), given the information at time \(T\), for the BHAR model, and \(f_T^{(i)}(1)\) the corresponding probability forecast of the probit model \(i\) (\(i=1,2,3,4\)) by Kauppi and Saikkonen (2008). Then, the combination of forecasts at time \(T+1\) is given by
$$\begin{aligned} f_T(1) = \phi _Tf_T^{B}(1) + (1-\phi _T)f_T^{(i)}(1). \end{aligned}$$
The weight \(\phi _T\) is obtained as
$$\begin{aligned} \phi _T = \frac{E_{(i)}}{E_B+E_{(i)}}. \end{aligned}$$
In this formula, \(E_B\) is the sum of the last \(v\) squared forecast error of the BHAR model,
$$\begin{aligned} E_B = \sum _{t=T-v+1}^T (e_{tB})^2 \end{aligned}$$
and \(E_{(i)}\) is the corresponding sum for each of the four probit models.
According to this definition
$$\begin{aligned} 0 \le \phi _T\le 1 \end{aligned}$$
and this ensures that the combined forecast of probability is neither negative nor superior to 1, that is
$$\begin{aligned} 0 \le f_{T}(1) \le 1. \end{aligned}$$
The forecasts obtained by the combination with \(v=1\) are superior compared to the simple models. In fact, the performance of the forecasts combination in terms of measures of accuracy (MAE and RMSE) and PCF shows a clear improvement (see Table 5 compared to Table 3).
Table 5

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

Moreover, the predictive performance has been compared through the Diebold–Mariano test. Defined \(e_{Ct}\) the forecast error of the combination of forecasts, the alternative hypothesis
$$\begin{aligned} H_1: E(d_t) < 0 \end{aligned}$$
means that the combined forecasts are more accurate with respect to the BHAR model,
$$\begin{aligned} d_t = g(e_{Ct})-g(e_{Bt}) \end{aligned}$$
and to each probit model
$$\begin{aligned} d_t = g(e_{Ct})-g(e_{(i)t}), \end{aligned}$$
with \(g(a)=a^2\).

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

Figure 7 illustrates the out-of-sample predictive performances of the combination between the the BHAR model and the probit models when the forecast horizon is one quarter ahead and the parameter \(v=1\). Figure 8 shows the same contents for \(h=4\).
Table 6

\(P\) 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

Table 7

\(P\) 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

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig7_HTML.gif
Fig. 7

Actual recessions (bars) and estimated probabilities of recession (solid lines), \(h=1\). Out of sample predictions of the combination between BHAR model and the probit models (from left to right the combinations BHAR-Static, BHAR-Dyn., BHAR-AR, BHAR-Dyn.AR), \(v=1\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig8_HTML.gif
Fig. 8

Actual recessions (bars) and estimated probabilities of recession (solid lines) \(h=4\). Out of sample predictions of the combination between BHAR model and the probit models (from left to right the combinations BHAR-Static, BHAR-Dyn., BHAR-AR, BHAR-Dyn.AR), \(v=1\)

Finally, Figs. 9, 10, 11 and 12 depict the weights of the BHAR model for the four forecast horizons. The dynamics of the weights does not favor a model in particular. However, it is interesting to note that in correspondence of actual recessions the weights tend to increase, that is the prediction provided by the BHAR model is more relevant. This feature is especially evident for \(h=1\) and tends to smooth when the forecast horizon increases.
https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig9_HTML.gif
Fig. 9

Estimated weights for the combining models (from left to right the combinations BHAR-Static, BHAR-Dyn., BHAR-AR, BHAR-Dyn.AR), \(h=1\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig10_HTML.gif
Fig. 10

Estimated weights for the combing models (from left to right the combinations BHAR-Static, BHAR-Dyn., BHAR-AR, BHAR-Dyn.AR), \(h=2\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig11_HTML.gif
Fig. 11

Estimated weights for the combing models (from left to right the combinations BHAR-Static, BHAR-Dyn., BHAR-AR, BHAR-Dyn.AR), \(h=3\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00181-012-0671-4/MediaObjects/181_2012_671_Fig12_HTML.gif
Fig. 12

Estimated weights for the combing models (from left to right the combinations BHAR-Static, BHAR-Dyn., BHAR-AR, BHAR-Dyn.AR), \(h=4\)

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.

Footnotes
1

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.

 

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© Springer-Verlag Berlin Heidelberg 2012