Time-Frequency Functional Models: An Approach for Identifying and Predicting Economic Recessions in Real-Time David S. Matteson Department of Statistical Science Cornell University [email protected]www.stat.cornell.edu/ ~ matteson Joint work with: Scott H. Holan & Christopher K. Wikle (Missouri Statistics) and Wen-Hsi Yang (CSIRO) Sponsorship: National Science Foundation–Census Research Network 2014 May 17 David S. Matteson ([email protected]) Time-Frequency Functional Models 2014 May 17 1 / 37
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Time-Frequency Functional Models:An Approach for Identifying and Predicting
Economic Recessions in Real-Time
David S. MattesonDepartment of Statistical Science
I (CPI) percentage change in the U.S. consumer price index
I (UNEMP) seasonally adjusted U.S. national unemployment ratepercentage
I (GDP) percentage change in U.S. real gross domestic product
I (GDI) percentage change in U.S. real gross domestic income
I (IP) percentage change in the U.S. industrial production index
I (SPREAD) difference between yields on 10-year Treasury bonds and3-month Treasury bills
David S. Matteson ([email protected]) Time-Frequency Functional Models 2014 May 17 16 / 37
Recessionary Periods ShadedFFR
year
1975 1980 1985 1990 1995 2000 2005
0
5
10
15
20
GDP
year
1975 1980 1985 1990 1995 2000 2005
−10
0
10
20
CPI
year
1975 1980 1985 1990 1995 2000 2005
−4
−2
0
2
4
UNEMP
year
1975 1980 1985 1990 1995 2000 2005
3
6
9
12
IP
year
1975 1980 1985 1990 1995 2000 2005
−6
−3
0
3
SPREAD
year
1975 1980 1985 1990 1995 2000 2005
−2
0
2
4
GDI
year
1975 1980 1985 1990 1995 2000 2005
−3
0
3
6
David S. Matteson ([email protected]) Time-Frequency Functional Models 2014 May 17 17 / 37
Preliminary Analysis
I Recessionary periods are associated with peaks in the FFR series.
I Troughs are evident in the GDP, GDI, IP and SPREAD series duringperiods of recession.
I Peaks in the UNEMP series are evident immediately following aneconomic recession.
I The relationship between the CPI series and recessions appearsinconclusive.
David S. Matteson ([email protected]) Time-Frequency Functional Models 2014 May 17 18 / 37
Preliminary Analysis
We also note:
1. When recessions occur, they last multiple quarters. Hence, thelikelihood of a recession in a given period may be a useful predictor ofthe likelihood in future periods.
2. The number of quarters since the previous period of recession mayhave predictive power for the current level of economic activity.
We consider including both of these features in our model implementation.
David S. Matteson ([email protected]) Time-Frequency Functional Models 2014 May 17 19 / 37
Model Implementation
I Daily log NASDAQ index return signals from two quarters(current and previous quarter)
I Begin Q4 of 1971. End Q2 of 2010.
I T = 155 quarters; 29 were recessions
I In R, STFT functions available in e1071, RSEIS,...
I Moving Hamming window, length 16, with overlap of 14 (insensitive?)
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Model Implementation
I Spectrogram and EOFs are calculated
I First 40 standardized EOFs considered (95% variation)
I We evaluated the model performance using various hyperparameters
I Fixed π = 0.5, various combinations of (τ, c) considered
I For both identification & prediction, τ = 0.01 & c = 10 selected
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Model Implementation
I SSVS sampler: 40,000 iterations with 5,000 burn-in
I Classification was set to 1 (recession) if posterior probability ≥ 0.5
I This classification represents “model averaging” over covariatecombinations
I Accounts for their relative importance via the stochastic search
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Model Implementation
In order to define the specific models for Zt , consider the following twogeneral models:
Zt+h = δVt + β′xt + εt , (w/count) (4)
andZt+h = θZt + β′xt + εt (w/lag) (5)
I h = 0, 1, 2, 3 denotes a nowcast, 1-, 2-, and 3-step ahead forecasts
I εt ∼ N(0, 1)
I Vt equals the number of quarters since the previous recession
I β′xt includes both macroeconomic and EOF covariates(time-frequency daily NASDAQ log return predictor)
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Model Implementation
Seven models total:
I M1 macroeconomic covariates ONLY
I M2 version of Equation (4) with no T-F NASDAQ covariate
I M3 version of Equation (5) with no T-F NASDAQ covariate
I M4 (EOF) T-F daily NASDAQ log return covariate ONLY
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Identifying and Predicting Recessions – Results
I Reconstruct the posterior mean and standard deviation for the differencedspectrogram (e.g., the mean spectrogram for the recessions minus the expansions)one-step ahead.
I We see that the volatility in the daily NASDAQ log returns roughly lagged onequarter is a key predictor of recession.
I More recent recessionary periods, timing is approximately the same, but thehigh-frequency behavior becomes more pronounced(i.e., exhibits stronger energy in the spectrogram).
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Identifying and Predicting Recessions – Results
Fre
quency
Period 1: mean difference
20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
Day
Fre
quency
Period 1: standard deviation
20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
Period 2: mean difference
20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
Day
Period 2: standard deviation
20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
Period 3: mean difference
20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
−5
0
5
10x 10
−4
Day
Period 3: standard deviation
20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
2
4
6
8
10x 10
−5
Posterior mean and standard deviation of the reconstructed differenced-spectrograms that
results from back projecting each quarter in the training and forecast period on the EOFs.
David S. Matteson ([email protected]) Time-Frequency Functional Models 2014 May 17 34 / 37
Summary
I Nowcasting/predicting recessions extensively researched & still challenging
I We propose an approach based on T-F functional models of daily NASDAQindex log returns.
I Model M7 (AR model w/ macroeconomic and T-F predictors) 85% and 80%out-of-sample forecasting accuracy for recessions and expansionsrespectively, even 3 quarters ahead.
I The proposed approach may be viewed as a Bayesian mixed data frequencymodel (MIDAS)
I Many possible extensions!
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Selected References:
Holan, S.H., Yang, W.H., Matteson, D.S., and Wikle, C.K. (2012) An Approach forIdentifying and Predicting Economic Recessions in Real-Time Using Time-FrequencyFunctional Models. (with discussion) Applied Stochastic Models in Business andIndustry. 28: 485–499.
Yang, W.-H., Wikle, C.K., Holan, S.H., and Wildhaber, M.L. (2013) NonlinearMultivariate Time-Frequency Functional Data Models for Ecological Prediction. Journalof Agricultural, Biological, and Environmental Statistics. 18: 450–474.
Wikle, C.K. and Holan, S.H. (2011) Polynomial Nonlinear Spatio-TemporalIntegro-Difference Equation Models. Journal of Time Series Analysis. 32: 339–350.
Holan, S.H., Wikle, C., Sullivan-Beckers, L. and Cocroft, R. (2010) Modeling ComplexPhenotypes: Generalized Linear Models Using Spectrogram Predictors of AnimalCommunication Signals. Biometrics. 66: 914–924.
Wikle, C.K., (2010) Low Rank Representations for Spatial Processes. In: Handbook ofSpatial Statistics. A.Gelfand, P. Diggle, M. Fuentes, P. Guttorp (eds.), Chapman &Hall. 107–118.
David S. Matteson ([email protected]) Time-Frequency Functional Models 2014 May 17 37 / 37