Dating US business cycles with macro factors

Sebastian Fossati

doi:10.1515/snde-2015-0037

Article

Dating US business cycles with macro factors

Sebastian Fossati

Published/Copyright: February 6, 2016

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 20 Issue 5

Abstract

Latent factors estimated from panels of macroeconomic indicators are used to generate recession probabilities for the US economy. The focus is on current (rather than future) business conditions. Two macro factors are considered: (1) a dynamic factor estimated by maximum likelihood from a set of 4 monthly series; (2) the first of eight static factors estimated by principal components using a panel of 102 monthly series. Recession probabilities generated using standard probit, autoregressive probit, and Markov-switching models exhibit very different properties. Overall, a simple Markov-switching model based on the big data macro factor generates the sequence of out-of-sample class predictions that better approximates NBER recession months. Nevertheless, it is shown that the selection of the best performing model depends on the forecaster’s relative tolerance for false positives and false negatives.

Keywords: business cycle; factors; forecasting; Markov-switching model; probit model

JEL Classification: E32; E37; C01; C22

Corresponding author: Sebastian Fossati, Department of Economics, University of Alberta, Edmonton, AB T6G 2H4, Canada. Web: http://www.ualberta.ca/~sfossati/

Acknowledgments

I thank two anonymous referees, the editor Bruce Mizrach, Jeremy Piger, Eric Zivot, Drew Creal, Dante Amengual, German Cubas, Ana Galvão, and Byron Tsang for helpful comments.

Appendix A. Autoregressive probit model estimation

The regression equation for the factor-augmented autoregressive probit model is

(A.1)yt∗=γ′zt+θyt−1∗+ϵt,

where γ=(α, δ)′ and zt=(1, h^t)′, and the likelihood function for the model is

(A.2)L(y|z, γ, θ, y0)=∏t=1T[Φ(γ′zt+θyt−1∗)]yt[1−Φ(γ′zt+θyt−1∗)]1−yt.

The implementation of the Gibbs sampler is similar to that of Dueker (1999) and Chauvet and Potter (2005, 2010). After generating initial values of the latent variable yt∗, the sampler proceeds as follows: (i) generate draws of the latent variable yt∗ conditional on (γ′, θ) and the observed data; (ii) generate draws of γ′ conditional on (yt∗, θ) and the observed data; (iii) generate draws of θ conditional on (yt∗, γ′) and the observed data. Prior and posterior distributions are discussed next.

A.1 Generating draws of the latent variable

Initial values of the latent variable, yt∗(0) for t=1, …, T, are drawn from f(yt∗(0)|yt−1∗(0), yt) with y0∗(0)=0. Conditional on yt−1∗ and y_t, yt∗ has a truncated normal distribution where yt∗≥0 if y_t=1 and yt∗<0 if y_t=0. The truncation imposes a sign condition on yt∗ based on the observed value y_t. Then, potential values of yt∗(0) are drawn from yt∗(0)∼N(γ′zt+θyt−1∗(0), 1). Draws are discarded if the sign condition is not satisfied.

Obtaining subsequent draws of the latent variable yt∗ conditional on the parameters and the observed data requires the derivation of the the conditional distribution yt∗|yt−1∗, yt+1∗. As the vector (yt+1∗, yt∗, yt−1∗) has a joint normal distribution, the conditional distribution yt∗|yt−1∗, yt+1∗ is also normal. Starting with (A.1) and substituting backwards for lagged y^*’s on the right side, the following results can be derived:

yt∗=∑s=0t−1θsγ′zt−s+∑s=0t−1θsϵt−s,E(yt∗)=At=∑s=0t−1θsγ′zt−s=γ′zt+θAt−1,Var(yt∗)=Bt=∑s=0t−1θ2s=1+θ2Bt−1,Cov(yt∗, yt−1∗)=θBt−1.

The joint distribution of the vector (yt+1∗, yt∗, yt−1∗) is then

[yt+1∗yt∗yt−1∗]∼N([At+1AtAt−1], [Bt+1θBtθ2Bt−1BtθBt−1Bt−1]).

Using standard results for the multivariate normal distribution, yt∗|yt+1∗, yt−1∗∼N(μ˜t, Σ˜t) for t=2, …, T−1, with truncation such that yt∗≥0 if y_t=1 and yt∗<0 if y_t=0 and

μ˜t=At+θ(BtBt−1)′(Bt+1θ2Bt−1Bt−1)−1(yt+1∗−At+1yt−1∗−At−1),Σ˜t=Bt−θ2(BtBt−1)′(Bt+1θ2Bt−1Bt−1)−1(BtBt−1).

Finally, assuming y0∗=0, y1∗|y2∗∼N(μ˜1, Σ˜1), with truncation such that y1∗≥0 if y₁=1 and y1∗<0 if y₁=0 and

μ˜1=A1+θB1B2−1(y2∗−A2)=A1+θ1+θ2(y2∗−A2),Σ˜1=B1−θ2B1B2−1B1=1−θ21+θ2.

Based on these results, subsequent draws of the latent variable, yt∗(i) for t=1, …, T, are taken from f(yt∗(i)|yt−1∗(i−1), yt+1∗(i), yt) for t=1, …, T−1 and f(yt∗(i)|yt−1∗(i−1), yt) for t=T where i denotes the ith cycle of the Gibbs sampler. As in Chauvet and Potter (2005, 2010), I start drawing a value of yT∗ conditional on a value of yT−1∗ and y_T from yT∗(i)∼N(γ′zT+θyT−1∗(i−1), 1), with truncation such that yT∗(i)≥0 if y_T=1 and yT∗(i)<0 if y_T=0. With this value of yT∗, I generate draws of yt∗ for t=1, …, T−1 backwards using the results described above. Potential draws of yt∗ are discarded if the sign condition is not satisfied.

A.2 Prior and posterior for γ

Following Albert and Chib (1993) and Dueker (1999), I use a flat non-informative prior for γ. Initial values for γ in the first cycle of the Gibbs sampler are the least squares estimates from a regression on the observed variable y_t without autoregressive terms. Let Wtγ=yt∗−θyt−1∗, then draws of γ are generated from the multivariate normal distribution γ|y∗, θ, y∼N(γ^, (z′z)−1) where γ^=(z′z)−1z′Wγ.

A.3 Prior and posterior for θ

Similarly, I use a flat non-informative prior for the autoregressive parameter θ. The initial value of θ to start the Gibbs sampler is set at 0.5. Let Wtθ=yt∗−γ′zt and Wty=yt−1∗, with W1y=0. Then, potential draws of θ are generated from θ|y∗, γ, y∼N(θ^, (Wy′Wy)−1) where θ^=(Wy′Wy)−1Wy′Wθ. Draws are discarded if the stationarity condition |θ|<1 is not satisfied.

A.4 Recession probabilities

Conditional recession probabilities are generated at each draw of the Gibbs sampler such that

(A.3)pt(i)=Φ(γ(i)′zt+θ(i)yt−1∗(i)),

where i denotes the ith cycle of the Gibbs sampler. The posterior mean probability of recession is given by

(A.4)p^t=1I∑i=1Ipt(i),

where I denotes the total number of draws.

Appendix B. Data appendix

A data set of the four indicators used to estimate the dynamic factor (industrial production, real manufacturing sales, real personal income less transfer payments, and employment) corresponding to the February 2011 vintage was provided by Jeremy Piger. Real-time vintage data for the dynamic factor is from Camacho, Perez-Quiros, and Poncela (2013).

The table below lists the 102 time series included in the balanced panel. The table lists the short name of each series, the transformation applied (number of months to be lagged in parentheses), and a brief data description. All series are from FRED – St. Louis Fed –, unless the source is listed as ECON (Economagic), GFD (Global Financial Data), or AC (author’s calculation) and correspond to the February 2011 vintage. The transformation codes are: 1=no transformation; 2=first difference; 3=second difference; 4=logarithm; 5=first difference of logarithms; 6=second difference of logarithms.

	Short name	Trans.	Description
1	PI	5 (1)	Personal Income (Bil. Chain 2005 $)
2	PILT	5 (1)	Personal Income Less Transfer Payments (AC)
3	CONS	5 (1)	Real Consumption (Bil. Chain 2005 $)
4	IP	5 (1)	Industrial Production Index – Total Index
5	IPP	5 (1)	Industrial Production Index – Products, Total (ECON)
6	IPF	5 (1)	Industrial Production Index – Final Products
7	IPCG	5 (1)	Industrial Production Index – Consumer Goods
8	IPDCG	5 (1)	Industrial Production Index – Durable Consumer Goods
9	IPNDCG	5 (1)	Industrial Production Index – Nondurable Consumer Goods
10	IPBE	5 (1)	Industrial Production Index – Business Equipment
11	IPM	5 (1)	Industrial Production Index – Materials
12	IPDM	5 (1)	Industrial Production Index – Durable Goods Materials
13	IPNDM	5 (1)	Industrial Production Index – Nondurable Goods Materials
14	IPMAN	5 (1)	Industrial Production Index – Manufacturing
15	NAPMPI	1 (0)	Napm Production Index (%)
16	MCUMFN	2 (1)	Capacity Utilization
17	CLFT	5 (1)	Civilian Labor Force: Employed, Total (Thous.,sa)
18	CLFNAI	5 (1)	Civilian Labor Force: Employed, Nonagric. Industries (Thous.,sa) (ECON)
19	U: all	2 (1)	Unemployment Rate: All Workers, 16 Years & Over (%,sa)
20	U: duration	2 (1)	Unempl. By Duration: Average Duration In Weeks (sa)
21	U<5 weeks	5 (1)	Unempl. By Duration: Persons Unempl. less than 5 weeks (Thous.,sa)
22	U 5–14 weeks	5 (1)	Unempl. By Duration: Persons Unempl. 5–14 weeks (Thous.,sa)
23	U 15+ weeks	5 (1)	Unempl. By Duration: Persons Unempl. 15 weeks+(Thous.,sa)
24	U 15–26 weeks	5 (1)	Unempl. By Duration: Persons Unempl. 15–26 weeks (Thous.,sa)
25	U 27+ weeks	5 (1)	Unempl. By Duration: Persons Unempl. 27 weeks+(Thous,sa)
26	UI claims	5 (0)	Average Weekly Initial Claims, Unempl. Insurance
27	Emp: total	5 (1)	Employees On Nonfarm Payrolls: Total Private
28	Emp: gds prod	5 (1)	Employees On Nonfarm Payrolls – Goods-Producing
29	Emp: mining	5 (1)	Employees On Nonfarm Payrolls – Mining
30	Emp: const	5 (1)	Employees On Nonfarm Payrolls – Construction
31	Emp: mfg	5 (1)	Employees On Nonfarm Payrolls – Manufacturing
32	Emp: dble gds	5 (1)	Employees On Nonfarm Payrolls – Durable Goods
33	Emp: nondbles	5 (1)	Employees On Nonfarm Payrolls – Nondurable Goods
34	Emp: serv	5 (1)	Employees On Nonfarm Payrolls – Service-Providing
35	Emp: TTU	5 (1)	Employees On Nonfarm Payrolls – Trade, Transportation, And Utilities
36	Emp: wholesale	5 (1)	Employees On Nonfarm Payrolls – Wholesale Trade
37	Emp: retail	5 (1)	Employees On Nonfarm Payrolls – Retail Trade
38	Emp: fin	5 (1)	Employees On Nonfarm Payrolls – Financial Activities
39	Emp: govt	5 (1)	Employees On Nonfarm Payrolls – Government
40	Avg hrs	2 (1)	Avg Weekly Hrs, Private Nonfarm Payrolls – Goods-Producing
41	Overtime	1 (1)	Avg Weekly Hrs, Private Nonfarm Payrolls – Mfg Overtime Hours
42	Avg hrs mfg	1 (1)	Average Weekly Hours, Mfg. (Hours)
43	NAPM emp	1 (0)	NAPM Employment Index (%)
44	Starts: nonfarm	4 (1)	Housing Starts: Total (Thous.,saar)
45	Starts: NE	4 (1)	Housing Starts: Northeast (Thous.U.,sa)
46	Starts: MW	4 (1)	Housing Starts: Midwest(Thous.U.,sa)
47	Starts: S	4 (1)	Housing Starts: South (Thous.U.,sa)
48	Starts: W	4 (1)	Housing Starts: West (Thous.U.,sa)
49	BP: total	4 (1)	Housing Authorized: Total New Priv Housing Units (Thous.,saar)
50	NAPM new ords	1 (0)	NAPM New Orders Index (%)
51	NAPM vend del	1 (0)	NAPM Vendor Deliveries Index (%)
52	NAPM invent	1 (0)	NAPM Inventories Index (%)
53	M1	6 (1)	Money Stock: M1 (Bil $,sa)
54	M2	6 (1)	Money Stock: M2 (Bil $,sa)
55	MB	6 (1)	Monetary Base, Adj For Reserve Requirement Changes (Mil $,sa)
56	Rsrv tot	3 (1)	Depository Inst Reserves: Total, Adj For Reserve Req Chgs (Mil $,sa)
57	Rsrv nonbor	3 (1)	Depository Inst Reserves: Nonborrowed, Adj Res Req Chgs (Mil $,sa)
58	Cons credit	6 (2)	Consumer Credit Outstanding – Nonrevolving
59	S&P 500	5 (0)	S&P’s Common Stock Price Index: Composite (1941-43=10) (GFD)
60	S&P indst	5 (0)	S&P’s Common Stock Price Index: Industrials (1941-43=10) (GFD)
61	S&P div yield	5 (0)	S&P’s Composite Common Stock: Dividend Yield (% per annum) (GFD)
62	S&P PE ratio	5 (2)	S&P’s Composite Common Stock: Price-Earnings Ratio (%) (GFD)
63	Fed Funds	2 (0)	Interest Rate: Federal Funds (Effective) (% per annum)
64	Comm paper	2 (0)	Commercial Paper Rate
65	3-month T-bill	2 (0)	Interest Rate: U.S. Treasury Bills, Sec Mkt, 3-Mo. (% per annum)
66	6-month T-bill	2 (0)	Interest Rate: U.S. Treasury Bills, Sec Mkt, 6-Mo. (% per annum)
67	1-year T-bond	2 (0)	Interest Rate: U.S. Treasury Const Maturities, 1-Yr. (% per annum)
68	5-year T-bond	2 (0)	Interest Rate: U.S. Treasury Const Maturities, 5-Yr. (% per annum)
69	10-year T-bond	2 (0)	Interest Rate: U.S. Treasury Const Maturities, 10-Yr. (% per annum)
70	AAA bond	2 (0)	Bond Yield: Moody’s AAA Corporate (% per annum) (GFD)
71	BAA bond	2 (0)	Bond Yield: Moody’s BAA Corporate (% per annum) (GFD)
72	CP spread	1 (0)	Comm paper – Fed Funds (AC)
73	3-month spread	1 (0)	3-month T-bill – Fed Funds (AC)
74	6-month spread	1 (0)	6-month T-bill – Fed Funds (AC)
75	1-year spread	1 (0)	1-year T-bond – Fed Funds (AC)
76	5-year spread	1 (0)	5-year T-bond – Fed Funds (AC)
77	10-year spread	1 (0)	10-year T-bond – Fed Funds (AC)
78	AAA spread	1 (0)	AAA bond – Fed Funds (AC)
79	BAA spread	1 (0)	BAA bond – Fed Funds (AC)
80	Ex rate: index	5 (0)	Exchange Rate Index (Index No.) (GFD)
81	Ex rate: Swit	5 (0)	Foreign Exchange Rate: Switzerland (Swiss Franc per U.S.$)
82	Ex rate: Jap	5 (0)	Foreign Exchange Rate: Japan (Yen per U.S.$)
83	Ex rate: UK	5 (0)	Foreign Exchange Rate: United Kingdom (Cents per Pound)
84	Ex rate: Can	5 (0)	Foreign Exchange Rate: Canada (Canadian$ per US$)
85	PPI: fin gds	6 (1)	Producer Price Index: Finished Goods (82=100,sa)
86	PPI: cons gds	6 (1)	Producer Price Index: Finished Consumer Goods (82=100,sa)
87	PPI: int mat	6 (1)	Producer Price Index: Intermed. Mat. Supplies & Components (82=100,sa)
88	PPI: crude mat	6 (1)	Producer Price Index: Crude Materials (82=100,sa)
89	Spot Mrk Price	6 (2)	Spot market price index: all commodities (GFD)
90	CPI-U: all	6 (1)	Cpi-U: All Items (82-84=100,sa)
91	CPI-U: app	6 (1)	Cpi-U: Apparel & Upkeep (82-84=100,sa)
92	CPI-U: transp	6 (1)	Cpi-U: Transportation (82-84=100,sa)
93	CPI-U: med	6 (1)	Cpi-U: Medical Care (82-84=100,sa)
94	CPI-U: comm	6 (1)	Cpi-U: Commodities (82-84=100,sa) (ECON)
95	CPI-U: dbles	6 (1)	Cpi-U: Durables (82-84=100,sa) (ECON)
96	CPI-U: serv	6 (1)	Cpi-U: Services (82-84=100,sa) (ECON)
97	CPI-U: ex food	6 (1)	Cpi-U: All Items Less Food (82-84=100,sa)
98	CPI-U: ex shelter	6 (1)	Cpi-U: All Items Less Shelter (82-84=100,sa) (ECON)
99	CPI-U: ex med	6 (1)	Cpi-U: All Items Less Medical Care (82-84=100,sa) (ECON)
100	PCE defl	6 (1)	PCE, Implicit Price Deflator: PCE (1987=100)
101	AHE: const	6 (1)	Avg Hourly Earnings – Construction
102	AHE: mfg	6 (1)	Avg Hourly Earnings – Manufacturing

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Keywords for this article

business cycle; factors; forecasting; Markov-switching model; probit model

Dating US business cycles with macro factors

Article

Abstract

Acknowledgments

Appendix A. Autoregressive probit model estimation

A.1 Generating draws of the latent variable

A.2 Prior and posterior for γ

A.3 Prior and posterior for θ

A.4 Recession probabilities

Appendix B. Data appendix

References

Supplemental Material

Supplementary Material

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