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Weekly UpdatesAbstract This paper presents a methodology for estimating the Brazilian GDP quarterly series in the period between 1960-1996. Firstly, an Engle-Granger's static equation is estimated using GDP yearly data and GDP-related variables. The estimated coefficients from this regression are then used to obtain a first estimation of the quarterly GDP, with unavoidable measurement errors. The subsequent step is entirely based on benchmarking models estimated within a state space framework and consists in improving the preliminary GDP estimation in order to both eliminate as much as possible the measurement error and that the sum of the quarterly values matches the annual GDP.
Keywords Benchmarking * Engle-Granger's equation * Kalman's filter * State space models * GDP
JEL C32 * C51 * C52 * E01
Classification Numbers C10
Introduction
This paper describes and applies a methodology for estimating the quarterly Brazilian GDP from 1960 to 1996. The methodology is based on the initial estimation of the quarterly GDP, obtained through an Engle and Granger (1991) static equation using GDP annual data and GDP-related variables, namely: automobile and cement production, industrial consumption of electricity in the Rio de Janeiro-Sao Paulo axis, and the national treasury tax revenue. The estimated coefficients of this regression are then used to build a quarterly equation between GDP and those variables.
This equation produces a first quarterly GDP estimation, henceforth, referred to as dirty GDP--due to unavoidable measurement errors. The next step consists of improving the estimation, ridding it as much as possible from measurement errors and making it consistent with the annual GDP calculated by the Brazilian Institute of Geography and Statistics (IBGE), an official agency; to clarify, the sum of the quarterly estimations must be equal to the year total.
The process of harmonizing quarterly and annual estimations is called benchmarking, constantly used by official data agencies. In this paper, the benchmarking procedure adopted is anchored on the state space modeling, in which a time series is broken into unobserved and directly interpretable components (cf. Harvey 1990; Durbin and Koopman 2004).
The paper is organized as follows: the next section briefly discusses the benchmarking process and reviews the literature on the subject. In the following, we describe the quarterly GDP estimation methodology. Then we introduce and discuss the estimated models' results, and finally we present our conclusions. The appendix describes the construction of the series used and shows the results of the unit root tests employed.
Benchmarking
A common problem with official statistics is the adjustment of monthly and quarterly observations obtained through surveys or sampling and therefore subject to errors. The adjustment is made with annual data from censuses or more detailed surveys, hypothetically presumed to be free of sampling errors. The annual total is called the benchmark; the process of harmonizing estimations with the year total is called benchmarking.
More specifically, benchmarking--or harmonization--consists of conveniently matching two measurement sources of the same time series, usually obtained from distinct frequencies. The lowest frequency series--the benchmark series--is assumed as having a more reliable registry. Benchmarking is the process of trying to adjust the highest frequency series to the benchmark one. This is done by breaking down the series into its structural elements--trend, seasonality, cycle, irregularity and measurement error--and from the sum of those elements excluding the error.
There are two major methodologies to apply benchmarking to a time series: a purely numeric approach and a statistic modeling one. The numerical approach differs from the statistic modeling by not specifying a statistical model to the studied series. The numerical approach encompasses the family of methods based on the minimization of a squared sum proposed by Denton (1971)--following the principle of movement preservation--Bassie (1958), and Ginsburgh (1973). An application of such procedure can be found in Di Fonzo and Marini (2003). The statistic modeling method, in turn, involves models based on ARIMA processes proposed by Hillmer and Trabelsi (1987), state space models proposed by Durbin and Queenneville (1997), and models that use a group of regressions such as Cholette and Dagum (1994), Mian and Laniel (1993) as well as the references cited by these authors.
Paper Methodology
First Step: Obtaining Dirty GDP
In the lack of official statistics collected for computing GDP, the assembly of quarterly estimates for the actual accrued product can adopt three criteria: (1) annual data interpolation; (2) own survey from samples of goods and services; and (3) a combination of the first two criteria; cf. Contador and Santos Filho (1987). In Brazil, researchers preferably adopt interpolation with GDP-related series. In this paper, we shall follow such procedure by using the series used by Cardoso (1981), namely: automobile and cement production, industrial consumption of electricity in the Rio de Janeiro-Sao Paulo axis, and the national treasury tax revenue.
Computation of the dirty quarterly GDP is based on the estimation of a GDP regression against the mentioned series, with annual-frequency series being expressed in 1980-based indexes with the purpose of obtaining long-term relationship coefficients between the variables. The estimation method used was Engle-Granger's two-stage cointegration procedure. Having estimated a cointegrating vector by OLS, we form a linear combination by using the previously listed quarterly frequency series in order to obtain the interpolated series of the quarterly GDP. The estimated quarterly series is chained with IBGE's series in 1980, producing the dirty GDP series, which shall be perfected in the second half of the procedure.
Prior to 1980, GDP data were available only in annual frequency, which explains why the estimation was performed in this frequency. At first, the estimation aims to recover GDP data for the 1964-1979 period. However, given the few observations, the period was increased to 1960-1996. These dates are associated to the series availability and to the change in IBGE's quarterly GDP investigation methodology.
Although at first sight the variables selected for initial estimation might seem questionable, further analysis shows that they represent the chief economically productive sectors in an adequate way. Since 1960, the automobile industry has been Brazil's most important economic segment. Cement consumption, in turn, is a suitable representation of the rank of civil construction activity, an area that generates a large portion of employment in the economy. Likewise, industrial electricity consumption in the country's major economic region (the Rio de Janeiro--Sao Paulo axis) is an added relevant index of economic activity. Lastly, it seems obvious that the federal tax revenue should suitably represent governmental activities, especially when it is considered that a part of it is passed on to states and cities, being entirely spent in a wide portion of the studied period.
Thus, dirty GDP is the first approximation of the quarterly GDP that recovers missing data prior to 1980 and that has thenceforth a seasonal pattern identical to that of the IBGE's series.
Second Step: State Space Models for Benchmarking and Obtaining the Clean GDP
The general linear Gaussian state space model (SS) is defined by the following equations:
[y.sub.t] = [Z.sub.t][[alpha].sub.t] + [[epsilon].sub.t], [[epsilon].sub.t] ~ N(0,[H.sub.t]), [[alpha].sub.t + 1] = [T.sub.t][[alpha].sub.t] + [R.sub.t][[eta].sub.t], [[eta].sub.t] ~ N(0,Q.sub.t), [[alpha].sub.1], [P.sub.1]). (1)
[y.sub.t] is a of p x 1 vector of observations; [[alpha].sub.t] is called the state vector, it is unobservable and its dimensions are m x 1; [[epsilon].sub.t] and [[eta].sub.t] are independent error terms; matrices of system [Z.sub.t], [T.sub.t], [R.sub.t], [H.sub.t] and [Q.sub.t], general, contain unknown parameters, which are assembled in a [psi] parameters vector. The model estimation in Eq. 1 is done with Kalman filter (KF) (state vector) combined with maximum likelihood (parameters vector). KF is formed by a set of equations that estimate, recursively in time, the average and conditional variances of the state vector. For further details about those equations, their deductions and their combination with maximum likelihood estimation, see Harvey (1990, chapter 3) and Durbin and Koopman (2004), chapters 4 and 7.
State space models for benchmarking problems have been studied quite broadly by Durbin and Queenneville (1997). One of the corresponding forms in SS proposed by Durbin & Queenneville was later reviewed by Durbin and Koopman (2004), chapter 3. In this paper, a probabilistically equivalent version of Durbin and Queenneville's model is presented. Taking [y.sub.t] as the dirty quarterly GDP (theoretically with measurement errors) and [x.sub.t] as IBGE's annual GDP (theoretically free of measurement errors), observations must be [y.sub.t] if time t is not a multiple of four, or ([y.sub.t], [x.sub.t]) in case time t is a multiple of four (due to the quarterly frequency, this indicates the turning of a new year). The state vector, entirely inspired in the structural model approach for time series (cf. Harvey, 1990) is given by the expression [[alpha].sub.t] [equivalent to] [[[mu].sub.t] [[mu].sub.t-1] [[mu].sub.t-2] [[mu].sub.t-3] [[gamma].sub.t] [[gamma].sub.t-1] [[gamma].sub.t-2] [[gamma].sub.t-3] [[epsilon].sub.t] [[epsilon].sub.t-1] [[epsilon].sub.t-2] [[epsilon].sub.t-3] [[xi].sub.t]]', where [[mu].sub.t] is a local level, [[gamma].sub.t] stands for stochastic seasonality, [[epsilon].sub.t] is an irregular component and [[xi].sub.t] is a measurement error associated to the dirty GDP, which admittedly follows an AR(1) stationary process. Matrices [Z.sub.t] must be [Z.sub.t] = [1 0 0 0 1 0 0 0 1 0 0 0 1] if t is not a multiple of four, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if t is a multiple of four. Matrices [H.sub.t] are null for all moments in time. The other matrices associated to the state vector equation and time-invariant for this model must follow the postulated evolutions for the state vector components; their full expressions are presented in Durbin and Queenneville (1997). Notice that, given the SS formulation, the consistency between clean GDP (defined by [y*.sub.t] = [y.sub.t] - [[xi].sub.t]) and IBGE's annual GDP always occurs in multiples of four periods.
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