neural networks

USING MACRO-FINANCIAL VARIABLES TO FORECAST RECESSIONS. (AN ANALYSIS OF CANADA, 1957-2002) KIANI, Khurshid M.^∗

KASTENS, Terry L.

Abstract: We employ artificial neural networks using macro-financial variables to predict recessions. We model the relationship between indicator variables and recessions

to 10 periods into the future and employ a procedure that penalizes a misclassified recession more than a misclassified non-recession. Our results reveal that among the 16 models that we constructed from 9 indicator variables and their combinations, the indicator variables Spread, 3-year bond rates, 10-year bond rates, monetary base, industrial production are candidates variables for predicting recessions ranging to 10 periods in the future. However, most indicator variables become candidate for predicting recessions when misclassified recessions are penalized more heavily than misclassified non-recessions.

1

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Keywords: business cycles; neural networks; out-of-sample forecasts; recession; real GDP;

JEL codes: C22, C32, C45 1. Introduction

Prediction of recession is an important task because policymakers would like to anticipate future recessions so that they have enough time for remedial policy actions to combat them. For this reason, careful selection of forecasting models and the underlying estimation algorithms is mandatory for obtaining appropriate forecasts of recessions.

Since empirical evidence from business cycle literature reveals¹ that business cycles fluctuations are asymmetric, linear models cannot be used to forecast recessions when the underlying data generating process is nonlinear. Therefore, it is imperative to employ empirical models that are in conformance with underlying data generating process for predicting recessions.

Estrella and Mishken (1998) employed nonlinear time series models to predict USA business cycles regimes. Similarly, some other nonlinear time series models can be employed to predict future recessions but this study is intended to employ a number of macro financial time series which might have different underlying data gearing processes requiring building a number of nonlinear time series models for forecasting recessions.

However, Qi (2001) successfully used neural networks to predict USA recessions using data from Estrella and Mishken (1998) study and came up with similar results. Therefore,

∗ Khurshid M. Kiani, Department of Economics, The University of the West Indies, Mona, Kingston 7, Jamaica; e-mail address: [email protected] and Terry L. Kastens, Department of Agricultural Economics, Kansas State University, Manhattan KS 66506, USA, e-mail:

[email protected]

1 Neftci (1984), Brunner (1992, 1997), Beaudry and Koop (1993), Potter (1995), and Ramsey and Rothman (1996), Bidarkota (1999, 2000), Kiani and Bidarkota (2004), Anderson and Vahid (1998), and Anderson and Ramsey (2002), including many others concluded that business cycles fluctuations are asymmetric.

we feel that artificial neural networks can be used successfully provided the model selection as well as the selection of the underlying algorithm is adequate.

Neural Networks have been used in a number of disciplines including business, and economics. In this context, Kuan and White (1994), Swanson and White (1995, 1997a, 1997b), Hutchinson, Lo, and Poggio (1994), Garcia and Gencay (2000), Gencay (1998), and Qi and Madala (1999), Gencay (1999) employed neural networks in business and economics. Although neural networks have demonstrated some success in financial and economics applications, only a few studies, for example Vishwakarma (1995), Qi (2001), Kiani (2005), and Kiani, Bidarkota, and Kastens (2005) employed neural networks in business cycles. We believe that the study of business cycles might benefit from additional nonlinear models, especially neural networks. We, therefore, employ artificial neural networks that are considered to be highly flexible functional forms of nonlinear models to find possible predictability of Canadian recessions using a number of macro financial (indicator variables). The remaining paper is organized in the following sections. Section provides a brief description of neural network models employed in this study. Section 3 discusses out-of-sample forecasts and their scoring, section presents empirical results, and section 5 incorporates brief conclusions.

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2. Empirical Model

2.1. Data Sources. We work with quarterly data on nominal and real GDP series;

industrial production, Treasury bill rates, 1−3-year bond rates, 10-year bond rates, and Bank of Canada rates for the Canadian economy that were obtained from the September 2002 version of the International Financial Statistic (IFS)’s CD-ROM. We also obtained data on money supply, Toronto Stock Exchange (TSE) index, real money, and monetary base from the Canadian Socioeconomic, Information, and Management Database (CANSAM). The 0−1 recession indicator dependent variable used in all models was generated from real GDP growth rates in a manner consistent with previous studies (Qi 2001; Estrella and Mishkin 1998), which consider the economy in recession when GDP deviates negatively from its growth trend for two or more consecutive quarters.

We investigate sample periods synonymous to Qi (2001) to validate our model via out-of- sample testing for the two Canadian recessions in the 1980s and 1990s. However, our forecasts of future recessions are based on accuracy measure of forecasts from our models from the sample period that spans from 1957Q2 to 2002Q1 for predicting future recessions. Table1 shows the structure of all the economic and financial variables (indicators) employed in this study.

2.2. Neural Network Models

An artificial neural network (ANN) consists of a number of interconnected elements called neurons or nodes. ANN are powerful computational devices because they can process information based on learnings from examples and generalize them to solve problems never seen before (Reilly and Cooper 1990). ANN are treated as nonlinear, nonparametric statistical methods due to which these are independent of distributions of the underlying data generating processes (White 1989). Although researchers developed numerous dissimilar ANN models since 1980s, one of the most influential neural

networks models is multilayer perceptron (MLP). This type of neural network is used for variety of problems especially forecasting because of its intrinsic capability to use arbitrary input output mapping. There are three types of nodes in this model. The input node stores the data, output node are where solutions to the problems are realized, and the hidden nodes separate input and output nodes (Zhang, Patuwo, and Hu (1998). Equation 1 presents the general form of artificial neural networks models employed in this study:

0 0

1 1

( ) , (1)

n k

j ij i j

j i

f x sig

α α

sig

β

x

β ε

= =

⎡ ⎛ ⎞⎤

= ⎢ + ⎜ + ⎟⎥+

⎝ ⎠

⎣

∑ ∑

⎦

Where n is the number of hidden nodes in the network, k is the number of explanatory variables in the network, sig (x) = 1/(1+e^-x),

α

_j represents a vector of parameters or weights that link the hidden node to the output layers’ units,

β

_ij (i = 1 , … , k; j = 1 , … , n) denotes a matrix of parameters linking the input to the hidden layers’ units, andε is the error term.

Due to relatively large number of parameters and nonlinearity inherent in the ANN specification, the objective function is unlikely to be globally convex and thus can have many local minima. Thus, for sum of squared errors (SSE) minimization we employed genetic algorithm (GA) ² which is considered to be among the most reliable algorithms to estimate a nonlinear functional form, however, it is slower than other algorithms that could be used to approximate neural networks. We considered four independent runs of genetic algorithm each with a fixed number of iterations. Out of these runs, the parameter vector that had the smallest sum of squared errors (SSE) was next used as the starting conditions for a fixed-iterations back-propagation (BP) algorithm, which is a reliable algorithm, but again slow. Finally, the BP output was used as starting conditions for Matlab’s fminsearch algorithm that worked well for closing on the final optima.

3. Out-of-Sample Forecasts

Using indicator variables shown in Table1, we approximate neural network models using Equation1. We first estimate a model from the start of a sample to a particular period usingx-data lagged three periods from the dependent variable. We next use our model along different lags to predict the th, th, and th observation, however, we update our forecasts every three periods to save computer time. For a data set with

4 5 6

Tobservations, the out-of-sample forecast for a given horizon h for the parameters (αˆ ,_t^h βˆ_t^h

) from vantage point of t is computed from Equation 2:

∑ ∑

= =

+ = + ² +

1

, , 1

,

0 ˆ ( ˆ * ˆ )]

[ˆ ˆ

j

h t ij t i k

j h

t ij jt

t k

t sig sig x

y α α β β (2)

2Initially, De Jong (1975) applied genetic algorithms, to mathematical optimization. Later, this algorithm was used in biology, engineering and operation research (Goldberg, 1989). The first economic application of genetic algorithms was implemented by Axelord (1987) and later by Marimon, McGratten and Sergeant (1990).

Assuming 10 periods for model initialization, the first h-specific vantage point is 10+h and the first period forecasted is10+2h. For example, forh=3, the first model estimation involves x-data from periods 1 to10 and y-data from periods to13; whereupon, period 16 is forecasted from a vantage point of period13. The next model would use x-data from periods 1 to 11 and y-data from periods (predicting period17), and so on, expanding data with additional recursive estimation.

4

=3 h 14 4−

In model comparisons it is useful to consider suitable benchmark. For example, Qi (2001) used a benchmark for prediction equal to the historical frequency of recession. Similarly, we consider such a static benchmark forecast, which is in our sample period (1957Q2-2002Q1). However, we also considered a dynamic benchmark forecast for our sample period, which is based on the historical frequency of recessions only up to the time a forecast is made.

1389 . 0

3.1. Mapping Thresholds for Out-of-Sample Forecasts

As specified in Equation 1 and2, the ANN output computed from Equations 1 and above is a continuous variable on the [ ] interval. Such a continuous variable needs to be mapped according to some mapping threshold rule (

2 1

, 0

MTR) “unless probability type forecasts are sufficient” either to1 when recession is predicted or otherwise. Generally, the

0

MTR is assumed to be , which is reasonable if the ANN continuous output is considered because the SSE minimizing routine used in model estimation, otherwise some other type of estimation procedure would have been appropriate.

5 . 0

Due to data infrequency, the event of interest (i.e. recessions) can be missed in prediction when using discreet dependent variables. That is, a forecast of all 0’s might be deemed an accurate forecast by most moment-based statistics (e.g., SSE, or MSPE (mean squared prediction error)) when only a few1’s actually occur in the data. In such situations, decision makers would often wish to penalize a Type I error of missing a recession more heavily than a Type II error (predicting a recession when it does not exist). Medical and financial experts compute receiver-operator curves (ROC), which make trade-off between Type I and II errors associated with each possible MTR (Reiser and Faraggi, 1997).

However, one can also consider the usual MTR = rule in generating predictions which is consistent with SSE minimization, but follow up with consideration of different Type I and II error costs. We use this approach since it is more intuitive than the ROC method.

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3.2. SCORING Out-of-Sample Forecasts

We compute SCOREs for each model according to Equation to evaluate ANN and benchmark predictions using weighting of Type I and II errors:

3

( ) ( ) ( ) ( )

( )

0 .5 1 1 0 .5

1 , (

1

t t t t t t

t t t t

t t

t t

C G G I Y G G I Y

S C O R E

C G G

⎡ − ≥ ⎤ ⎡+ − − − < ⎤

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

= − + −

∑ ∑ ∑ ∑

∑ ∑

³⁾

Where Gt is1 when t shows a recession, else0, and Yt is the continuous prediction of the model, I(.) is an indicator which is1 if its argument is true otherwise 0, and C is a

constant that weights Type I errors for missed recessions. The left- and right-bracketed term in the numerator show the number of incorrect recession and non-recession predictions, respectively. The denominator, however, shows maximum error cost assuming that every prediction were wrong. The fraction is subtracted from one so that higher SCORE show a better model synonymous to R-square.

4. Empirical Results

4.1.Out-of-Sample Forecasts. We validate our models using three sample periods as in earlier studies (Qi 2001; Estrella and Mishkin 1998) and our results based on mean square predition errors calculated from these samples (not shown for brevity) show that most indicator variables employed in this study are candidate variables for predicting Canada recessions. This necessitated measuring accuracy of forecasts from our models based on some measure of accuracy. Therefore, we introduced a measure of prediction accuracy for forecasts from our neural network models for predicting future recessions. Our study results provide evidence that we predicted recessions up to 10-periods in future introducing a measure of prediction accuracy, which is an improvement over previous studies (Qi 2001; Estrella and Mishkin 1998). This is because we carefully selected our neural network models and underlying estimation algorithms. For example our neural network models that we employed in this study incorporate two hidden nodes compared to Qi (2001) who used three hidden nodes as well as our estimation process is a combination of genetic algorithm, back propagation, and Matlab’s fminsearch algorithm, as against Qi (2001), who employed the Levenberg-Marquardt algorithm.

4.2.Out-of-Sample Forecast Evaluation. The neural network and benchmark forecasts developed in this study were scored using the Equation 3 procedure. Table reports the SCOREs for the case where Type I and Type II errors are assumed to have the same cost ( in Equation ). With this selection, Table ranks models identical to using MSPE calculated using model predictions that have been mapped to 0 or1. Given our choice of mapping threshold rule (MTR

2

=1

C 3 2

5 .

=0 ), the static and dynamic benchmarks discussed earlier have the same accuracy. Unlike Qi (2001), however, combining macro financial (indicator) variables did not seem to improve our accuracy over single indicators. The results for our out-of-sample forecasts based on SCORE measure of forecast for each indicator variable are shown in Table . In this Table, column 1 shows the names of the indicator variables and columns

2 11

2− show out-of-sample forecasts based on SCORE accuracy measure of prediction for each single indicator variable as well as combined indicator variables for 1−10 horizons in future. The SCOREs presented in this Table for all variables are calculated assuming that the Type I error has the same cost as Type II (C =1). This shows that while predicting recessions a policymaker can set desired penalty (error costing) on candidate variables to predict recessions. In this Table, for example, column row 1 shows SCORE for indicator variable in out-of-sample forecast horizon1. Rows

2

Spread 2−11 for this variable show accuracy measure SCORE for forecast horizons2−10. Out-of-sample forecasts for the remaining variables are presented in a similar manner. We use dynamic benchmarks of predictions that are shown in the last row of the Table. These benchmark predictions are different from predictions by the indicator variables. For example, for Bond1t3y, SCORE

(accuracy measure of prediction) for horizon 2 is , which is more than the benchmark prediction at horizon ( ) showing that the indicator variable Bond1t3y can be the useful predictor of recessions. The numbers greater than dynamic benchmark beat the benchmark and hence, the underlying variable is considered suitable for predicting Canada recessions.

8683 . 0 2 0.8623

Table shows out-of-sample forecasts based on SCORE accuracy measure of prediction for each indicator variable for forecast horizons

3

10

1− . In this Table, for example, column row 1 shows SCORE for indicator variable in out-of-sample forecast horizon1. Rows

2 Spread

11

2− for this variable show accuracy measure SCORE for forecast horizons . Out-of-sample forecasts for the remaining variables in this Table are presented in a similar manner. We use dynamic benchmark for predictions that are shown in the last row of the Table. These benchmark predictions are different at different forecast horizons. For example, for Bond1t3y, SCORE (accuracy measure of prediction) for horizon 2 is , which is more than the benchmark perdition at horizon ( ) showing that the indicator variable Bond1t3y can be a useful predictor of recessions. The numbers greater than the dynamic benchmark beats the benchmark hence the underlying variable is considered suitable for the prediction of Canada recessions.

10 2−

4118 .

0 2

3850 . 0

To provide some indication of how our models might perform in a situation where Type I and II errors are assigned different costs, Table 3 reports the SCORE’s using an arbitrary selection of C = 10 in Equation . Now, many models beat the benchmark. For example, the single indicators Spread, 3-year bond rates ( ), 10-year bond (Bond10y) and monetary base (MB), along with the combined stock market and industrial production indicators ( ) each beat the benchmark in 6 of 10 horizons and thus are likely candidates to help predict recessions. This is encouraging because it seems highly likely that policy makers would want to penalize a misclassified recession more than a misclassified non-recession. Moreover, such error costing may not necessarily have to be incorporated directly into the model estimation procedure since it is an ex post exercise.

3

3 1t Bond IPG

TSE300&

5. Conclusions

In this research, we used artificial neural networks to investigate the predictability of Canadian recessions 1 to 10 quarters in the future using a number of economic and financial (indicator) variables and their combinations. We model relationship between the indicator variables and recessions periods in future via out-of-sample testing. Our out-of-sample forecast results with measure of prediction accuracy indicate that among the 16 models constructed from 9 indicator variables and their combinations that we considered, the indicator variables Spread, 3-year bond rates, 10-year bond rates, monetary base, industrial production growth, and the combined indicator variable are the candidate variables for possible predictions of recessions 2 to quarters in the future. However, most indicator variables and in their combination become candidate variables for predicting recessions

10 1−

IPG TSE300&

10

8

6− out of 10 horizons in future when we penalized misclassified recessions more heavily than misclassified non- recessions. We carefully selected estimating algorithm and minimized the usage of hidden nodes (two) in our neural network models to avoid over fitting problem that

helped us predict Canada recessions periods in future as against Qi (2001) who predicted US recessions

10 1−

1−4 periods in future using more hidden nodes (three) and a different estimating algorithm. However, future work might need to include additional models and cross-country data in such studies that might be able to predict recessions even beyond 10-periods (quarters) in the future.

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Annex

Table 1. Structure of Indicator Variables Single

Indicator

Description Combined Indicator Description Interest Rates and Spread T-bill&Bond1t3y T-bill&Bond1t3y Spread 10-year Treasury bond rate less 3-

month Treasury bill rate

Bond10y&Bond1t3y Bond10y&Bond1t3y CPTB 6-month commercial paper rate less

3-month Treasury bill rate

CPTB&Spread CPTB & Spread T-bill 3-month Treasury bill market

equivalent bond rate

T-bill&Spread T-bill & Spread Bond10y 10-year Treasury bond rate Bond1t3y&Spread Bond1t3y & Spread

Bond1t3y 1-3 year bond rate TSE300&MB TSE300 & MB

Monetary Aggregates TSE300&IPG TSE300 & IPG

M1 M1 Money Supply, seasonally adjusted

MB Monetary Base, seasonally adjusted Stock Prices

TSE300 Canadian Stock Market Index Individual Macroeconomics Variable IPG Industrial Production Growth

Table 2: Out-of-Sample SCORES (C=1) For Full-Sample (1957Q2-2002Q1)

Forecast Horizon in Number of Quarters Indicator

Variable

1 2 3 4 5 6 7 8 9 10

Spread 0.8402 0.8204 0.8182 0.8528 0.8571 0.8491 0.8535 0.8452 0.8301 0.8543 CPTB 0.8284 0.8262 0.8061 0.8405 0.8571 0.8491 0.8535 0.8516 0.8431 0.8411 T-bill 0.8343 0.8503 0.8242 0.8466 0.8199 0.8553 0.8280 0.8452 0.8301 0.8146 Bond10y 0.8521 0.8323 0.8364 0.8650 0.8571 0.8428 0.8344 0.8323 0.8431 0.8146 Bond1t3y 0.8402 0.8683 0.8667 0.8405 0.8323 0.7862 0.8280 0.8387 0.8170 0.8344 M1 0.8462 0.8563 0.8364 0.8405 0.8261 0.8239 0.8471 0.8387 0.8235 0.8411 MB 0.8580 0.8443 0.8606 0.8589 0.8634 0.8428 0.8408 0.8452 0.8366 0.8477 TSE300 0.8284 0.8443 0.8485 0.8344 0.8571 0.8491 0.8535 0.8452 0.8497 0.8411 IPG 0.8402 0.8443 0.8364 0.8650 0.8571 0.8553 0.8535 0.8516 0.8562 0.8411 T-bill&

Bond1t3y

0.8047 0.8263 0.8121 0.8344 0.8447 0.8491 0.8535 0.8258 0.8301 0.8411 Bond10y&

Bond1t3y

0.8225 0.8323 0.8061 0.8344 0.8447 0.8553 0.8471 0.8452 0.8497 0.8411 CPTB&

Spread

0.8343 0.8443 0.8424 0.8528 0.8447 0.8239 0.8344 0.8452 0.8497 0.8344 T-bill&

Spread

0.8284 0.8383 0.8121 0.8405 0.8323 0.8553 0.8344 0.8323 0.8497 0.8212 Bond1t3y

&Spread

0.8225 0.8323 0.8000 0.8344 0.8571 0.8553 0.8471 0.8452 0.8235 0.8344 TSE300

&MB

0.8047 0.7784 0.8303 0.8405 0.8261 0.8176 0.8408 0.8258 0.8235 0.8477 TSE300

&IPG

0.8402 0.8383 0.8424 0.8282 0.8696 0.8491 0.8344 0.8323 0.8434 0.8477

Benchmark 0.8639 0.8623 0.8606 0.8589 0.8571 0.8553 0.8535 0.8516 0.8497 0.8477 Notes. We use dynamic benchmark for predictions that are shown in the last row of the Table which are different for different forecast horizons. For example, for Bond1t3y, SCORE (accuracy measure of prediction) for horizon 2 is 0.8683, which is more than the benchmark perdition at horizon 2 (0.8623) showing that the indicator variable Bond1t3y can be the useful predictor of recessions. Therefore, the numbers greater than the dynamic benchmark beats the benchmark and hence the underlying variable is considered suitable for the prediction of Canada recessions. Moreover, the numbers (SCORE) presented in this Table for all variables are calculated assuming that the Type I error has the same cost as Type II (C=1).

Table 3: Out-of-Sample SCORES (C=10) For Full-Sample (1957Q2-2002Q1) Forecast Horizon in Number of Quarters

Indicator Variable

1 2 3 4 5 6 7 8 9 10

Spread 0.3777 0.3663 0.3629 0.3757 0.3750 0.3689 0.3681 0.3619 0.3528 0.3855 CPTB 0.3723 0.3690 0.3575 0.4189 0.4239 0.3689 0.3681 0.3646 0.3583 0.3547 T-bill 0.3989 0.4037 0.3656 0.3730 0.3587 0.3962 0.3571 0.3619 0.4028 0.3436 Bond10y 0.4069 0.3957 0.3710 0.4054 0.3995 0.3661 0.3599 0.4061 0.3833 0.3436 Bond1t3y 0.4016 0.4118 0.4086 0.3703 0.3886 0.3415 0.3571 0.3591 0.3472 0.3520 M1 0.3803 0.3824 0.3710 0.3703 0.3614 0.3579 0.3654 0.3591 0.3500 0.3547 MB 0.4814 0.4251 0.4059 0.3784 0.4511 0.3907 0.3626 0.3619 0.3806 0.3575 TSE300 0.3723 0.4251 0.3763 0.3676 0.3750 0.3689 0.3681 0.3619 0.3611 0.3547 IPG 0.3777 0.4011 0.3710 0.4054 0.3995 0.3716 0.3681 0.3646 0.3889 0.3547 -bill&Bond

1t3y

0.3856 0.3690 0.3602 0.3676 0.3696 0.3689 0.3681 0.3536 0.3778 0.3547 Bond10y&

Bond1t3y

0.3936 0.3717 0.3575 0.3919 0.3940 0.3716 0.3654 0.3619 0.3611 0.3547 CPTB&

Spread

0.3750 0.4011 0.3978 0.3757 0.4185 0.3579 0.3599 0.3619 0.3611 0.3520 T-bill&

Spread

0.3963 0.4225 0.3602 0.3703 0.3641 0.3962 0.3599 0.3564 0.3861 0.3464 Bond1t3y&

Spread

0.3697 0.4198 0.3548 0.3919 0.3995 0.3962 0.3654 0.3619 0.3500 0.3520 TSE300

&MB

0.3617 0.4198 0.3683 0.3946 0.3614 0.3552 0.3626 0.3536 0.3500 0.3575 TSE300

&IPG

0.4016 0.3984 0.3978 0.3649 0.4538 0.3934 0.3599 0.3564 0.3583 0.3827

Benchmark 0.3883 0.3850 0.3817 0.3784 0.3750 0.3716 0.3681 0.3646 0.3611 0.3575 Notes. See notes on Table 2. The benchmark predictions for Bond1t3y, SCORE (accuracy measure of prediction) for horizon 2 is 0.4118, which is more than the benchmark perdition at horizon 2 (0.3850) showing that the indicator variable Bond1t3y can be the useful predictor of recessions. The numbers (SCORE) presented in this Table for all variables are calculated assuming that the Type I error has the same cost as Type II (C=10). This shows that while predicting recessions a policymaker can set desired penalty (error costing) on candidate variables to predict recessions.

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