Forecasting Quarterly Brazilian GDP: Univariate Models Approach
Kleyton Vieira Sales da Costa, Felipe Leite Coelho da Silva, Josiane da Silva Cordeiro Coelho
FForecasting Quarterly Brazilian GDP: UnivariateModels Approach
Kleyton Vieira Sales da Costa ∗ Felipe Leite Coelho da Silva † Josiane da Silva Cordeiro Coelho ‡ October 27, 2020
Abstract
Gross domestic product (GDP) is an important economic indicator that aggregatesuseful information to assist economic agents and policymakers in their decision-making process. In this context, GDP forecasting becomes a powerful decisionoptimization tool in several areas. In order to contribute in this direction, weinvestigated the efficiency of classical time series models and the class of state-spacemodels, applied to Brazilian gross domestic product. The models used were: aSeasonal Autoregressive Integrated Moving Average (SARIMA) and a Holt-Wintersmethod, which are classical time series models; and the dynamic linear model, a state-space model. Based on statistical metrics of model comparison, the dynamic linearmodel presented the best forecasting model and fit performance for the analyzedperiod, also incorporating the growth rate structure significantly.
Keywords : GDP, Forecasting, SARIMA, Dynamic Linear Model
JEL Codes : C22, C32, C53, E37, E27 ∗ Department of Economics at Federal Rural University of Rio de Janeiro † Department of Mathematics at Federal Rural University of Rio de Janeiro ‡ Department of Mathematics at Federal Rural University of Rio de Janeiro a r X i v : . [ ec on . E M ] O c t Introduction
The economic activity of a country can be influenced by several factors that subjecteconomic agents to change their consumption and investment decisions, in addition toimpacting other results, such as inflation and unemployment. Such factors, or shocks,result from the modification of economic policies, in the level of production technology,through meteorological changes etc. The gross domestic product (GDP) is one of the mainindexes for measuring the level of economic activity, and the forecast of its trajectoryprovides useful information concerning the future economic trend in the short term, actingas an object for the expectation of economic behavior.Significant impacts on economic activity arise through crises. They are a dysfunctioninherent in the free market system. Through the development of information transmissiontechnologies and the global integration of markets, the scope and frequency of thesedysfunctions have been expanded. Beginning in the second quarter of 2014, the Brazilianeconomic crisis is still the subject of many analyzes, with no consensus on the generatingvariables, as well as their consequences. In the second quarter of 2016, the GDP growthrate accumulated in four quarters had reached the lowest level of the last two decades(-4.6 %). The data show that the recovery (after a significant drop) was not complete,followed by a period of stagnation in the country’s growth rate. Paula and Pires (2017)analyzed the ineffectiveness of counter-cyclical policies - between 2011 and 2014 - as aresult of problems in the coordination of macroeconomic policy; and also by the occurrenceof exogenous shocks, such as the deterioration of trade terms and the water crisis thatoccur in period. Filho (2017) argues that the origin of the Brazilian economic crisis wasdue to a series of supply and demand shocks that (mostly) were caused by wrong publicpolicies, contributing to the reduction of growth potential the Brazilian economy and tothe increase in tax cost.According to Feijó and Ramos (2013), the most relevant aggregates that derive from theSystem of National Accounts are the measures of product, income and expenditure. TheMacroeconomic Aggregates are statistical constructions that synthesizes the productiveeffort of a given country or region and its possible consequences on the generation ofincome and expenditure for a specific period of time. By definition, the GDP of a countryor region represents production of all production units of the economy - government,self-employed workers, companies etc. - in a given period, usually quarterly or annually,at market prices .Blanchard and Johnson (2017) presents two ways of interpreting GDP. The so-callednominal GDP is defined as the sum of quantities of final goods multiplied by the current The socially organized economic activity that aims to create goods and services to be traded on themarket and/or they are achieved by means of factors production (land, capital and labor) traded on themarket(IBGE, 2016). Economic transactions with observed or imputed market value. ; Migon et al. (1993) developed a study about the performance of BayesianDynamic Models applied to a set of Brazilian macroeconomics time series (industrialproductivity index, the balance of trade, components of GDP and others) between theperiod 1970 to 1990. The comparison was made between the dynamic models and classicalstructured models and obtained results indicate that the Bayesian approach was similarto the classical approach. Another applied study was developed by Baurle et al. (2020),with the aim of forecasting GDP in the euro area and Switzerland with a Bayesian vectorautoregressive structure (BVAR) and a factor model structure. He found evidence thatthe factor model structure performs satisfactorily.Another accepted approach to GDP forecasting is macroeconomic projections based onleading indicators. Garnitz et al. (2019) applied this strategy to forecast GDP growth inforty-four countries, including Brazil. One of the results found indicates that the forecastscan be improved by adding World Economic Survey (WES) indicators of the three maintrading partners by country.The aim of this work is to investigate a suitable time series model to describe andforecast Brazilian GDP, also investigating the fit of these models to dynamics betweenperiods of economic growth and recession. For this purpose, it is compared different classesof time series models. Thus, the chosen models were the Holt-Winters method, SARIMAand dynamic linear model. In the literature, there are some applications regarding thesemodels but no comparative studies were found using the models adopted in this work.This work is organized as follows: Section 2 describes the methodology. Section3 presents the results and discussion, and, finally, the last section provides the mainconclusions and some possibilities for future research. To follow we outline the data and the empirical approach used to fitted and forecast thetime series of Brazilian gross domestic product between the years 1996 and 2019, at 1995prices. This section also defines the models that were investigated.Care was also taken that the references used in the definition of models and metricsalso correspond to studies and authors with wide use and quality proven by the academiccommunity. Monthly indicator of national economic activity published by Central Bank of Brazil. .1 Data The quality of the data used in empirical analysis is a fundamental element for the qualityof the results. A factor that contributes to the empirical analysis of GDP is the vastdocumentation made available by government agencies. For that, we obtained the timeseries in the IBGE Automatic Recovery System (IBGE, 2020).Data used for the analysis are quarterly and comprise the first quarter of 1996 untilthe fourth quarter of 2019. Data values used in this article are in Brazilian Real (BRL).Statistical analyzes, as well as graphic representations, were built using open-sourcesoftware R Core Team (2020).
The United Nations (2010) says that GDP derives from the concept of value added.Therefore, GDP is the sum of gross value added of all resident producer units plus thatpart of taxes on products, fewer subsidies on products. GDP is also equal to the sum offinalizes of goods and services measured at purchasers’ prices, less the value of importsgoods and services. And GDP is too equal the sum of primary incomes distributed byresident producer units.According to Feijó and Ramos (2013) GDP can be calculated in three different ways,but are part of the Accounting Identity (
Production = Income = Expenditure ), guidingNational Accounts. The perspective of production is calculated by sum the added valuesof economic activities plus taxes, net of subsidies, on products. That is, Y production = GV A − IC + ( T − Sub ) , (1)where GVA it is gross value added, IC is the intermediate consumption, T are taxes onproducts and Sub are subsidies on products.The income perspective is obtained by adding the remunerations of factors of production.Labor is remunerated by wages, loan capital is remunerated by interest, venture capital isremunerated by profit, and ownership of production goods ("land") is remunerated by rent.That is, Y income = W + GOS + ( T − Sub ) , (2)where W are wages, GOS are gross operating surplus (sum of interest, profit e rent), T are taxes on products and Sub are subsidies on products.The time series constructed in this work was built from the perspective of expenditure.It is calculated by the sum of household consumption, investment, government spendingand net exports. That is, 5 expenditure = C + G + I + ( N E ) , (3)where C it is household consumption, I it is investment (gross fixed capital formation lessstock variation), G it is government consumption, and NE it is net exports (Exports lessImports). As described in Cowpertwait and Metcalfe (2009), the Holt-Winters method was proposedby Holt (1957) and Winters (1960), using exponentially weighted moving averages toupdate those needed for seasonal adjustment of the mean (trend) and seasonality.The method has two variations with four equations: one forecast equation and threesmoothing equations. Hyndman and Athanasopoulos (2018) describes that in the additivemethod, the seasonal component is defined in absolute terms on the scale of the observedseries. In the level equation, the series is seasonally adjusted by subtracting the seasonalcomponent. Within each year, the seasonal component sum up to approximately zero.With the multiplicative method, the seasonal component is defined in percentage termsand the series is seasonally adjusted by dividing through by the seasonal component.Within each year the seasonal component will sum up to approximately m .The additive method equations is describe as following,ˆ y t + h | t = ‘ t + hb t + s t + h − m ( k +1) ,‘ t = α ( y t − s t − m ) + (1 − α )( ‘ t − + b t − ) ,b t = β ( ‘ t − ‘ t − ) + (1 − β ) b t − ,s t = γ ( y t − ‘ t − − b t − ) + (1 − γ ) s t − m , (4)where ˆ y t + h | t is the forecast equation. The ‘ t , b t and s t are respectively level, trendand seasonality equations, with corresponding smoothing parameters α , β and γ . Theparameter m denotes the frequency of seasonality, and for quarterly data m = 4. Finally, k is the integer part of ( h − m ) which ensures that the estimates of the seasonal indices usedfor forecasting come from the final year of the sample.For the multiplicative method the same equations ‘ t , b t and s t are defined. But thechange in structure occurs because instead of sum the equations in ˆ y t + h | t an operation isperformed to multiply the sum of the level and trend equations by the seasonality equation. Box & Jenkins models determine the proper stochastic process to represent a given timeseries by passing white noise through a linear filter (Morettin and Toloi, 2018). The modelused was SARIMA, seeking to incorporate the seasonality component that is present in6he data under analysis.The SARIMA of order ( p, q, d ) × ( P, Q, D ) s is defined by, φ ( B )Φ( B s ) ∇ d ∇ Ds Y t = θ ( B )Θ( B s ) α t , (5)where θ ( B ) is the moving average operator of q order, φ ( B ) is the autoregressive operatorof p order, Φ( B s ) is the seasonal autoregressive operator of P order, Θ( B s ) is the seasonalmoving average operator of Q order, ∇ d is the simple difference operator, ∇ Ds is theseasonal difference operator and α t is the noise. Dynamic linear models are an important class of state-space models. Broadly used in thelast decades, they have a high degree of efficiency for the analysis and forecast of timeseries, providing flexibility and applicability through an elegant and robust probabilisticapparatus.The estimation and inference challenges are solved by recursive algorithms, whichfollow the Bayesian approach, calculating conditional distributions of quantities of interestgiven the observed information. Considering a series affected by time, through dynamicand random deformations, they associate seasonal or regressive components.In this work were used contributions from West and Harrison (1997), Laine (2019),Petris et al. (2009) and Petris (2010). For each time t , the general univariate DLM isdefined by a observational equation, Y t = F t θ t + v t , v t ∼ N m (0 , V t ) , (6)a system equation θ t = G t θ t − + w t , w t ∼ N p (0 , W t ) (7)and initial information given by ( θ | D ) ∼ N ( m , C ) , (8)where F t e G t are known matrices; v t and w t are two sequences of independent noises,with average zero and known covariance matrices V t and W t respectively. D t is the currentinformation set; m and C contains relevant information about the future, accordingusual statistical sense, given D t , ( m t , C t ) is sufficient for ( Y t +1 , θ t +1 , . . . , Y t + k , θ t + k ).To take into account growth and seasonality, it is defined θ t = ( µ t , β t , γ t , γ t − , γ t − ),where µ t is the current level, β t is the slope of the trend, γ t , γ t − and γ t − are the seasonalcomponents. 7 .6 Metrics The selection of most suitable forecasting model was made through the contributions ofHyndman and Koehler (2006), Armstrong (2001), Morettin and Toloi (2018) and Ahlburg(1984) using the following metrics: i. square root of the mean squared error (RMSE); ii.mean absolute error (MAE); iii. mean absolute percentage error (MAPE); and iv. Theil’sinequality coefficient (U-Theil).The first two metrics are widely used for measures whose scale depends on the scaleof the data. The third metric has the advantage of being scale-independent, and so arefrequently used to compare forecast performance across different data sets. And the lastmetric can improve the accuracy of a forecast through Theil’s decomposition of forecasterror into bias, regression and disturbance proportions and his associated linear correctionprocedure.
This section presents the results obtained using the Holt-Winters additive method, SARIMAand dynamic linear models to fit the data of interest. For each model, it was plottedthe observed and predicted values, and also the 95% confidence interval for the predictedvalues. Graphics are effective tools to understand the behavior of the series and whetherthe models generate reasonable fit and predictions in relation to the observed data.
Compared to the multiplicative Holt-Winters method, the additive formulation wasconsidered the most appropriate, taking into account the sum of squared errors. Figure 1shows the adjustment of the additive method for the Brazilian quarterly GDP data in theperiod 1996 to 2016 and the forecast between the years 2017 and 2019.8 D P l l l l l l l l l l l l Figure 1: Holt-Winters additive method fitted (solid line) to the observed Brazilianquarterly GDP ("+") in the period from 1996 to 2017, at 1995 prices. Forecast (blueline) for the horizon of 12 quarters ahead with its interval of 95% credibility (red line),superimposed on the values observed in this period (circles).It is observed that the model was able to fit the data reasonably. The fit also occurredsignificantly in periods of strong recession, such as the international financial crisis of 2008and the period of recession in the Brazilian economy between the second quarter of 2014and the fourth quarter of 2017.
To apply SARIMA model, the behavior of autocorrelation (ACF) and partial autocorrela-tion functions (PACF) were verified. In Figure 2 (a), it is possible to see a slow decay rateof the autocorrelation function to zero. This behavior indicates the non-stationarity of theseries, which needs to be differentiated in order to make it stationary.9 − . . . . (a) Autocorrelation function A C F − . . . . . . . (b) Partial autocorrelation function PA C F Figure 2: Autocorrelation function (a) and partial autocorrelation function (b) to theobserved Brazilian quarterly GDP in the period from 1996 to 2019, at 1995 prices.Figure 3 (a) shows the autocorrelation function of the differentiated series with an ex-ponential decay in the lags multiples of 4, indicating a possible series stationarity. Throughthe Phillips-Perron test (Dickey-Fuller Z α = -62.816; p-value = 0.01), the alternativehypothesis of stationarity of the differentiated series was accepted at a significance level of1%. − . . . . (a) Autocorrelation function A C F − . − . . . . . (b) Partial autocorrelation function PA C F Figure 3: Autocorrelation function (a) and partial autocorrelation function (b) to thedifferentiated series of Brazilian quarterly GDP in the period from 1996 to 2019, at 1995prices.We used an algorithm to generate sixteen SARIMA models following the principle10f parsimony. From the generated models, the structure with the best results was theSARIMA (0 , , × (0 , , , with metrics: Akaike information criterion (-438.58); thesum of squared error (0.01686); and the Ljung-Box test (p-value = 0.96).Figure 4 shows the model fitted to the Brazilian quarterly GDP data for the period1996 to 2016 and the forecast between the years 2017 and 2019. It is observed that themodel also fits the data reasonably. It includes periods when economic shocks occurred,such as the mentioned crises. G D P l l l l l l l l l l l l Figure 4: SARIMA (0 , , × (0 , , fitted (solid line) to the observed Brazilian quarterlyGDP ("+") in the period from 1996 to 2017, at 1995 prices. Forecast (blue line) for thehorizon of 12 quarters ahead with its interval of 95% credibility (red line), superimposedon the values observed in this period (circles). In this work, the dynamic regression matrix F t and the evolution matrix G t of the modelare F t = h i and G t = − − −
10 0 1 0 00 0 0 1 0 . (9)For the study, it was assumed the observational variance V t = σ , and the covariance matrixof the system W t is a diagonal matrix introduced by W t = diag ( σ µ , σ β , σ γ , , a, a θ , a θ , a θ and variances b, b θ , b θ , b θ , respectively, fixed in known values.Therefore, by using the unobservable states as latent variables, a Gibbs sampler canbe run on the basis of the following full conditional densities: σ ∼ IG (cid:16) a b + n , ab + SS y (cid:17) ,σ µ ∼ IG (cid:18) a θ b θ + n , a θ b θ + SS θ (cid:19) ,σ β ∼ IG (cid:18) a θ b θ + n , a θ b θ + SS θ (cid:19) ,σ γ ∼ IG (cid:18) a θ, b θ + n , a θ b θ + SS θ (cid:19) , (10)with SS y = P nt =1 ( y t − F t θ t ) and SS θ i = P Tt =1 ( θ t,i − ( G t θ t − ) i ) , for i = 1 , ,
3. The fullconditional density of the states is a normal distribution and it is covered in the used dlmpackage (Petris, 2010).From the Gibbs sampler, 5000 iterations were generated for each parameter, modelvariances, out of which the 1000 initial iterations were considered as burn-in period anddiscarded. Hence, the remaining iterations were used to compose the posterior samplesof the estimated variances. Posterior estimates of the four unknown variances, from theGibbs sampler output, can be seen in Figure 5. s . + . + . + . + s m s b s g . . . . . . Figure 5: Trajectory of simulated variances (top) and the ergodic means (bottom).Figure 6 shows the adjustment of the dynamic linear model to the observed data andthe quarterly forecast between the years 2017 and 2019 with a confidence interval of 95%.12t is possible to observe that the fit of the dynamic linear model to the data was alsosignificant. This means that the model was probably able to capture the data structureand generate forecasts effectively. G D P l l l l l l l l l l l l Figure 6: Dynamic linear model fitted (solid line) to the observed Brazilian quarterly GDP("+") in the period from 1996 to 2017, at 1995 prices. Forecast (blue line) for the horizonof 12 quarters ahead with its interval of 95% credibility (red line), superimposed on thevalues observed in this period (circles).
The RMSE, the MAPE, the MAE, and the U-Theil were calculated for the fitted valuesof models, and their results are shown in Table 1. The RMSE and MAPE metrics wereinvestigated for the forecast values and the results are shown in Table 2.It is observed that the better results were given through the models SARIMA (0 , , × (0 , , and dynamic linear, the latter being one that best fits the series of Brazilian GDP,at 1995 prices, for having achieved the lowest values in all metrics for fitted and forecastvalues. 13able 1: Comparison between Holt-Winters additive method, SARIMA (0 , , × (0 , , and dynamic linear in relation to the fitted of models to Brazilian quarterly GDP databetween the years 1996 and 2016, at 1995 prices. MetricModel RMSE MAE MAPE U-TheilHolt-Winters Additive Method 5924.012 5189.844 2.168 0.047SARIMA (0 , , × (0 , , SARIM A (0 , , × (0 , , and dynamic linear in relation to the forecast of models to Brazilian quarterly GDP databetween the years 1996 to 2016, at 1995 prices. MetricModel RMSE MAPEHolt-Winters Additive Method 3661.262 1.026SARIMA (0 , , × (0 , , As the dynamic linear model was chosen the best model, in the fit and prediction criteria,it is shown in Table 3 the comparison between the growth rate for the Brazilian GDPaccording to the observed values and the growth rate of values predicted by the dynamiclinear model. 14able 3: Comparison between observed values and forecast results in dynamic linear model.Quarter DLM Growth Rate Observed Growth Rate2017.1 0,0084 0,00012017.2 -0,0211 0,01662017.3 -0,0042 0,01882017.4 0,0041 -0,01152018.1 0,0123 -0,00902018.2 -0,0208 0,01262018.3 -0,0041 0,02362018.4 -0,0041 -0,01462019.1 0,0121 -0,01522019.2 -0,0205 0,01742019.3 -0,0040 0,02492019.4 -0,0039 -0,0100Figure 7 shows that the proposed model in this study obtained satisfactory resultswhen the observed and predicted growth rates are compared. The projection data alsomaintained the tendency of Brazilian economic growth to stagnate in the analyzed period.Therefore, economic policies were not effective for a consistent recovery in the short andmedium term. 15 . − . − . . . . . l l l l l l l l l l l ll l l l l l l l l l l l Figure 7: Dynamic linear model growth rate forecast (blue line) and observed valuesgrowth rate (black line) to the Brazilian quarterly GDP in the period from 2017 to 2019,at 1995 prices.
Understanding the GDP behavior is a topic of study and discussion by society and theacademic community. In the present work, we proposed the application of the Holt-Wintersadditive method, SARIMA and dynamic linear model with interest in forecast the behaviorof Brazilian quarterly GDP, at 1995 prices. The data comprise the period between thefirst quarter of 1996 and the fourth quarter of 2019.Theil’s inequality coefficient (U-Theil) shows that the models used in the study arebetter than the naive prediction, i.e, when the forecast at time t is the value observed in t − .Both the analyzed series and the models’ forecast show the necessity for sustained growthin a market economy. By the metrics RMSE, MAE, MAPE and U-Theil, it appears thatthe dynamic linear model presented the best fit to data and efficient forecast performance,with MAPE of 0.839.We find evidence in this study that corroborates with the observed results of stagnationin the Brazilian economy after a crisis period started in the second quarter of 2014.Therefore, the dynamic linear model proved to be efficient for forecasting and fit to GDPdata even with economic shocks.From the pandemic caused by Covid-19 in 2020 and your economic and humankindconsequences, the time series forecasting models must be adjusted so that they can adaptto a significant exogenous and structure economic shock.16 eferences M. R. Abonazel and A. I. Abd-Elftah. Forecasting egyptian gdp using arima models.
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