Estimating Brazilian Monthly GDP: a State-Space Approach

Issler, João Victor; Notini, Hilton Hostalacio

doi:10.5935/0034-7140.20160003

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Articles • Rev. Bras. Econ. 70 (1) • Jan-Mar 2016 • https://doi.org/10.5935/0034-7140.20160003 copy

Estimating Brazilian Monthly GDP: a State-Space Approach¹ 1 We gratefully acknowledge the comments of participants at CIRET 2008 in Santiago, Chile. We also thank Regis Bonelli and Silvia Matos for allowing us to reproduce the results of their nowcast exercise. All errors are ours. We thank Marcia Waleria Machado, Marcia Marcos, and Rafael Burjack for excellent research assistance and thank CNPq-Brazil, FAPERJ and INCT for financial support.

Authorship SCIMAGO INSTITUTIONS RANKINGS

This paper has several contributions. The first is to employ a superior interpo lation method that enables to estimate, nowcastandforecast monthly Brazilian GDP for 1980-2012 in an integrated way-see Bernanke, Gertler, & Watson (1997)Bernanke, B. S., Gertler, M., & Watson, M. (1997).Systematic monetary policy and the effects of oil price shocks (Brookings Papers in Economic Activity No. 1). Washington, DC: Brookings Institution. Retrieved from http://www.brookings.edu/about/projects/bpea/papers/1997/effects-of-oil-price-shocks-bernanke
http://www.brookings.edu/about/projects/... [Systematic monetary policy and the effects of oil price shocks(Brookings Papers in Economic Activity No.1)]. Second, along the spirit of Mariano & Murasawa (2003)Mariano, R. S., & Murasawa, Y. (2003). A new coincident index of business cycles based on monthly and quarterly series. Journal of Applied Econometrics, 18(4), 427-443. doi: 10.1002/jae.695
https://doi.org/10.1002/jae.695... [A new coincident index of business cycles based on monthly and quarterly series.Journal of Applied Econometrics, 18(4), 427-443], we propose and test a myriad of interpolation models and interpolation auxiliary series-all coincident with GDP from a business-cycle dating point of view. Based on these results, we finally choose the most appropriate monthly indicator for Brazilian GDP. Third, this monthly GDP estimate is compared to an economic ac tivity indicator widely used by practitioners in Brazil-the Brazilian Economic Activity Index (IBC-Br). We found that our monthly GDP tracks economic ac tivity better than IBC-Br. This happens by construction, since our state-space approach imposes the restriction (discipline) that our monthly estimate must add up to the quarterly observed series in any given quarter, which may not hold regarding IBC-Br. Moreover, our method has the advantage to be easily im plemented: it only requires conditioning on two observed series for estimation, while estimating IBC-Br requires the availability of hundreds of monthly series. Third, in a nowcasting and forecasting exercise, we illustrate the advantages of our integrated approach. Finally, we compare the chronology of recessions of our monthly estimate with those done elsewhere.

Keywords:
GDP Interpolation; State-Space Representation; Kalman Filter; Composite and Leading Indicators; Nowcasting; Forecasting

Model	ϕ	ρ
Mi Static model in levels with IID residuals	0	0
M₂: Static model in levels with AR(1) residuals (Chow & Lin, 1971)	0	free
M₃: Static model in 1st differences with IID residuals (Fernández, 1981)	0	1
M₄: Dynamic model in levels with IID residuals (Mitchell & Jones, 2005)	free	0
M₅: Dynamic model in 1st differences with IID residuals	free	1
M₆: Dynamic model in levels with AR(1) residuals	free	free

Variables	H₀:z is I(1)	H₀:z is I(2)
GDP^* *()** indicates that a linear trend was included in the test equation; (iii) figures in brackets are p-values.	-0.898	-4.098
	[0.95]	[0.01]
Steel	-0.792	-5.514
	[0.82]	[0.00]
Vehicles	0.522	-6.208
	[0.99]	[0.00]
Cement	-0.220	-3.392
	[0.93]	[0.01]
Energy	-0.208	-26.672
	[0.93]	[0.00]
IP	0.690	-6.270
	[0.99]	[0.00]
Sales	-0.118	-6.156
	[0.95]	[0.00]

Variables	Cement	Vehicles	Steel	Energy	IP	Sales
GDP	0.49	0.33	0.29	0.21	0.66	0.57

Parameter	M ₁	M ₂	M ₃	M ₄	M ₅	M ₆
ϕ				0.94	-0.67	-0.60
Intercept	11.03	33.51	51.60	-0.029	62.01	45.77
I. Production	0.693	0.198	0.212	0.0480	0.349	0.36
Sales	0.0002	0.0001	0.0001	0.0000	0.0001	0.0001
ρ		1.0041			1.0045
σ	6.5881	0.8955	0.9439	1.1463	1.2570	1.3259
Log likelihood	205.6269	88.3283	93.7001	109.0752	84.9365	92.7201

Notes: (1) The models are described in the text and in the ^Appendix APPENDIX. DETAILED INTERPOLATION MODELS The literature appoint as the main problem of the modelsM1 and M2 is the fact that they extract signals only from the presumed stochastic process of the original series in a way that no new information is added. We believe that we would be better off enriching the model with additional information contained in the related series. Model M1 This is the simpler model which incorporates information from related series. So it can be used as a benchmark when we estimate the more sophisticated models. It can be stated as: where: x′t is a vector of related series and ϵt a iid error term with distribution N (0;σ2). Model M2 This is the Chow & Lin (1971)model. This model is very used in the literature because they are the first to show how to include related series in the interpolation procedure. They suggest a regression model without lagged dependent variables, but autoregressive errors, so we can obtain the Chow & Lin model by fixing ϕ = 0 and by letting ρ to be estimated freely: It is important to note that in their seminal paper, Chow & Lin do not use the Kalman filter but it can be shown that the Kalman filter above and the Chow & Lin regression yield the same estimates by maximum likelihood (Cuche & Hess, 2000). Model M3 This is a variation of the Chow & Lin model, suggested by Fernández (1981), where instead of the regression of yt in levels, it uses the first differences of yt in order to account for non-stationarity. This model is obtained by letting regression residuals to be a random walk, i.e. ϕ = 0 andρ = 1. Model M4 This model was suggested by Mitchell & Jones (2005) and it is a dynamic version of model M1. Model M5 A modified version of model M4 where we impose that ρ = 1. Model M6 It is the most general version of the Mitchell & Jones (2005) model, where we let ϕ andρ to be estimated freely. In our interpolation exercise, we estimate the six models summarized above via Maximum Likelihood. We then perform a set of bilateral likelihood ratio tests to discover whether the imposed restrictions are borne by the data or not and select the most appropriate model accordingly. .
(2) Industrial Production and Sales are used as auxiliary series_t.

	R ² _diffs
Model	GDP_filtered	GDP_smoothed
M ₁	0.0273	0.0346
M ₂	0.4652	0.8097
M ₃	0.4382	0.7923
M ₄	0.3609	0.7098
M ₅	0.4011	0.5806
M ₆	0.3766	0.5736

Dated Peaks			Dated Throughs
Brazilian Monthly GDP	Duarte et al.	CODACE	Brazilian Monthly GDP	Duarte et al.	CODACE
1980:11	NA	1980:10	1981:10	NA
1982:07			1983:02		1983:02
-	1987:04	1987:02	-	1989:02	1988:10
1989:06	1989:08	1989:06	1991:12	1991:12	1991:12
1994:12	1994:12	1994:12	1995:08	1995:09	1995:09
1997:10	1997:10	1997:10	1999:02	1999:02	1999:02
2000:12	2000:12	2000:12	2001:09	2001:09	2001:09
-	NA	2002:10	-	NA	2003:06
2008:07	NA	2008:07	2008:12	NA	2009:01

Nowcast Period	IBC-Br	Monthly GDP Model
2007.4-2011.4	0.46%	0.40%
2010.4-2011.4	0.35%	0.16%

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E-mail: rbe@fgv.br

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[TFN1] Notes: (i) All series are in logs. The specification of the test equation was chosen on the basis of the Schwarz Information Criterion; (ii) the asterisk

[TFN2] ^(*)
indicates that a linear trend was included in the test equation; (iii) figures in brackets are p-values.