Brazil: Selected Issues and Statistical Appendix

This paper analyzes several issues regarding fiscal sustainability and fiscal adjustment in Brazil during 1990 and searches for econometric evidence of a monetary dominant regime during some subperiods. The following statistical data are also presented in detail: macroeconomic flows and balances, industrial production, consumer price index, relative public sector prices and tariffs, minimum wage statistics, financial system loans, monetary aggregates, exports by principal commodity groups, direction of trade, detailed balance of payments, total external debt, central government operations, and so on.

Abstract

This paper analyzes several issues regarding fiscal sustainability and fiscal adjustment in Brazil during 1990 and searches for econometric evidence of a monetary dominant regime during some subperiods. The following statistical data are also presented in detail: macroeconomic flows and balances, industrial production, consumer price index, relative public sector prices and tariffs, minimum wage statistics, financial system loans, monetary aggregates, exports by principal commodity groups, direction of trade, detailed balance of payments, total external debt, central government operations, and so on.

VII. Forecasting Inflation in Brazil: How Useful are Time Series Techniques?1

A. Introduction

1. The numerous price indices that are produced by a variety of Brazilian institutions reflect the country’s history of high inflation, and suggest that inflation concerns still weigh heavily in Brazil’s collective memory. With cumulative inflation, as measured by one of the consumer price indices (INPC), reaching 6,550 percent during 1980–84, 573,018 percent during 1985–89, and 32,258,701 percent during 1990–94, the cumulative 49 percent that were registered during 1995–99 appeared uncharacteristically low from a historical perspective. Five years of consistently low inflation apparently were not long enough to wipe out memories of hyperinflation and of the anxieties and behavioral peculiarities that came along with it. Toward this background, it seemed almost natural that the choice of the new nominal anchor following the floating of the real in January 1999 fell on inflation itself. A few additional indicators of inflation have been added since then,2 which would seem to safeguard employment opportunities for Brazil’s inflation forecasters for the foreseeable future.

2. As is natural under inflation targeting, the Brazilian Central Bank (BCB) has been in the forefront of inflation forecasting. Since the transmission channel of monetary policy usually takes several months, if not years, to fully work its way through the economy, the monetary authorities need to be forward-looking. Sources of pressure on the price level of the economy have to be identified as early as possible, so that the monetary authorities can take action preemptively and keep inflation at targeted levels. Early identification of inflationary pressures and early corrective action usually avoid higher costs of disinflation later on.

3. Since the start of the inflation targeting regime, the BCB has been publishing in its regular Quarterly Inflation Reports (QIRs) forecasts for inflation at different horizons that represent the institution’s best estimate at the time. The main tool used to produce these forecasts has been a small-scale structural macroeconomic model, although the BCB has also developed various other tools, such as leading indicators of inflation.

4. In this section, we compare the inflation forecasting power of various models, including the BCB’s own model. In particular, we explore the inflation forecasting power of time series models, including simple univariate models, “classical” vector autoregression (VAR) models, and Bayesian vector autoregression (BVAR) models. BVARs, in particular, provide an alternative approach to inflation forecasting that has yet to be explored by the BCB. Several other central banks, such as the Federal Reserve Bank of Minneapolis, where this forecasting tool was first developed, or the Bank of Spain,3 have been using BVARs successfully as an inflation forecasting tool and as a complement to structural macroeconomic models. Considering that inflation targeting in Brazil has only been in effect for five full quarters, for the time being the application of any statistical or econometric forecasting tool to Brazilian data has to rely also on data from the period prior to the change in the monetary policy regime in January 1999. In applying these various time series techniques to Brazil, the analysis in this section focuses on the accuracy of out-of-sample forecasts, and relative performance of the models vis-à-vis the forecasting tools used by the BCB. Following the BCB’s presentation in its QIRs, we perform several out-of-sample forecasts. Special attention is given to short-term forecasts, particularly to one-month-ahead and four-month-ahead forecasts.

B. Inflation Forecasts of the Central Bank of Brazil—A Brief Review

5. Since the beginning of publication in June 1999, each QIR has provided quarterly forecasts of inflation that are based on the BCB’s small-scale structural model (SSM)—the BCB’s main tool for inflation modeling and for policy simulation and analysis. The BCB’s structural macroeconomic model, which was discussed in the QIR of March 2000, and also in Bogdanski et al. (2000), aims at capturing the main relationships between key variables in the transmission mechanism. The model includes an aggregate demand equation, an aggregate supply equation, an interest rate rule, and an exchange rate equation that are estimated using standard econometric techniques. The model’s projections are based on simulations; these can be carried out rather flexibly once the basic model structure has been estimated. Based on these simulations, the BCB has published in each of its QIR, a central quarterly projection for future inflation, as well as confidence bands for that projection which depend on an assumed probability distribution. These simulations include different assumptions about the evolution of the endogenous variables of the model (e.g., the nominal interest rate, and variables that affect the exchange rate, such as the risk premium and the U.S. Federal Funds rate), and about possible exogenous shocks (such as the evolution of oil prices, food prices, fiscal variables, minimum wages, and adjustments of administered prices).

6. The forecast horizon in the BCB’s QIR has ranged from four quarters (in the first QIR of June 1999) to ten quarters (in the QIR of June 2000). As the SSM is fairly new still, even short-term forecasts have sometimes been subject to significant errors. For instance, using a constant SELIC rate of 19 percent and data up to August 1999, the September 1999 QIR projected the 12-month rate of inflation for December 1999 at 7.4 percent, while the actual outcome was 8.9 percent, which, when the report was published, was assumed to be outside of the 50 percent confidence interval. Also, the June 2000 QIR using data through May 2000 and a constant SELIC rate of 17.5 percent, did not fully anticipate the impact on consumer prices of various adjustments in administered prices and of adverse weather conditions which jointly caused a temporary spurt in inflation in July and August of 2000. This left the 12-month inflation rate for September 2000 outside of the 40 percent confidence interval established in the June 2000 QIR.

7. In addition to the forecast for future quarters, the more recent QIRs also show a forecast for the current quarter. When the QIR is released at the end of a quarter, only the values for the first two months of the current quarter are known, but the full quarter result will only be known about two weeks after the report is released. For example, the June 2000 report included forecasts starting with Q2 2000: inflation for April and May 2000 were already known when the report was released at end-June 2000, but June 2000 inflation was not. The quarterly forecast for the current period can be derived in at least two possible ways. First, it can be done by using the small-scale structural model itself, which, as it is a quarterly model, would not take into account the information on inflation for the first two months of the quarter. Alternatively, a separate monthly forecasting tool can be used, and the projected value for the current month can be added to the known values for the two previous months.

8. The BCB’s other main approach for projecting future inflation is based on leading indicators of inflation. The general concept was presented in the March 2000 QIR, and a more detailed discussion is contained in Chauvet (2000) and Chauvet et al. (2000). A main purpose of the leading indicators approach is to project the timing of inflation turning points (“peaks and troughs”), but leading indicators can also be combined with inflation in a bivariate VAR process to yield linear forecasts of inflation. Leading indicators do not attempt to capture structural relationships between macroeconomic variables, but rather to extract information from variables that seem to have an important predictive power for future inflation. The leading indicators approach uses Kalman filtering to combine different variables into a single composite leading indicator that is meant to signal inflation turning points. With many variables potentially having some predictive power for inflation—the BCB’s March 2000 Inflation Report stated that out of over 200 possible variables, 49 were analyzed in more detail concerning their predictive power for inflation—there is a large number of possibilities for combining these variables into a single leading indicator, and the computational resources needed to identify the best leading indicator can become quite large.

9. So far, notably absent from the BCB’s presentation of its inflation forecasting tools are time series models. While these models have certainly been explored by the BCB, they have not been presented to the public, which raises the question of how useful these techniques may (or may not) be for forecasting inflation in Brazil.

C. Time Series Techniques For Projecting Inflation: An Overview

10. In this section, three different time series techniques for projecting inflation are presented and evaluated with regard to their forecasting performance: univariate regression, unrestricted VAR, and Bayesian VAR. The inflation projections derived from these models are compared with the BCB’s own SSM-based forecasts. The first of these techniques, univariate regression, only employs information available on past values of inflation, and is meant as a benchmark naïve forecast to help assess the possible forecast improvement that the other two methods, VARs and BVARs, may bring about.4 Both VAR and BVAR forecasts are derived from a system of several variables.

11. Unrestricted (or nonstructural) VARs consist of a system of variables, which is modeled with little economic theory, in the sense that no exclusion restrictions are imposed. It is assumed that every endogenous variable in the system is a linear function of the lagged values of all endogenous variables in the system. The unrestricted VAR approach has become popular since it requires a minimum of assumptions, and can be estimated efficiently using OLS. Once estimated, the VAR can also be used for forecasting purposes, given a set of initial values for the endogenous variables. It can also be used to analyze the dynamic impact of random disturbances on the system of variables, and for variance decomposition exercises. Here, we focus on the VAR as a forecasting tool.

12. While unrestricted VAR models are a useful and powerful tool to analyze statistical and structural relationships between economic variables, their forecasting power has often been poor.5 One reason for this is that forecasts made using unrestricted VAR models often suffer from overparametrization. Usually, the number of observations available is fairly small, so that the resulting degrees of freedom may be inadequate to produce precise coefficient estimates, which may then result in large out-of-sample forecast errors. Another possible consequence of generously parametrizing VAR models is that the resulting estimates may reflect random rather than systematic empirical variability (also called “overfitting”). Overfitting becomes particularly problematic when using a VAR for forecasting. The usual approach to degrees of freedom issues is to reduce the number of regressors, which in a VAR translates into reducing the number of lags, based on statistical criteria.6 By doing so, a coefficient of zero is implicitly imposed for all lags that are dropped.

13. While in VAR models the problem of overfitting is addressed by imposing exclusion restrictions (i.e., dropping lags), the Bayesian VAR approach offers an alternative approach that consists of using prior statistical and economic knowledge for an initial guess of the values of all coefficients. The Bayesian procedure suggests to specify “fuzzy” restrictions, where information is processed based on assumed probability distributions for the model’s various coefficients. Therefore, the BVAR approach complements the autoregressive representation of the model with a prior distribution of the coefficients.7 In general, the prior distribution should be selected so as to offer a reasonably large range of uncertainty, and to be modified by the sample distribution if both distributions were to differ substantially.

14. Selecting the prior distribution is probably the most distinctive aspect of BVAR modeling. In principle, this may take different forms and rely on information from a variety of sources, which makes it a rather flexible tool. Exact restrictions (including exclusion restrictions similar to those used in VARs) can be viewed as a special case that increases the “tightness” of the restrictions. The prior information used in the BVAR framework is usually of statistical-empirical origin, and lacks economic content. This economic “neutrality” often makes the resulting specification more broadly acceptable, as it does not require an agreement on the “true” structure of the economy.

15. The essential part of the prior information consists of three empirical regularities that are characteristic of time series analysis. First, the best forecast of the future value of a series is its current value (the “random walk” assumption); this satisfactorily approximates the behavior of many macroeconomic series, as suggested by Nelson and Plosser (1982). Second, recent lagged values of a series usually contain more information on its current value than the more distant lagged values. Third, lagged values of a series contain more information on that series’ current value than the lagged values of other variables.

16. The actual data can override these assumptions if there is strong evidence about a coefficient. When formulating the prior distribution in a BVAR model it is common to assume a Normal distribution. However, the complete specification of a Normal distribution of the prior on a VAR would be intractable because of the large number of parameters involved, instead, a general form for the prior involving a few hyperparameters is chosen.8

D. Methodology and Results

17. This section presents the estimates derived from the three time series techniques (univariate model, classical VAR, and Bayesian VAR), and compares them with those of the BCB’s SSM model. There are three main results: first, even naïve models for predicting inflation do not necessarily fare much worse than more sophisticated ones; second, the various models usually overpredict and underpredict actual inflation in the same direction; and third, forecasts beyond the short term (i.e., more than four months ahead) tend to be fairly inaccurate under any model.

18. Using the various estimated models, we computed out-of-sample one-month, four-months (one quarter), and seven-months (two quarters) ahead forecasts for inflation. For the three time series techniques, we used the specification that delivered the best estimates in terms of minimizing the mean squared forecast error. For the univariate model, we estimated a simple AR(1) process for inflation. For the multivariate models, the system that was estimated consisted of seven endogenous variables: the SELIC rate, the average bank lending rate,9 the consumer price index (IPCA), the wholesale price index (IPA-DI), the nominal exchange rate of the real against the U.S. dollar, and M1.10 The VAR was estimated in first differences, using four lags of every endogenous variable in the system; using a larger number of lags generally worsened the accuracy of the results. The Bayesian VAR was estimated in levels, using six lags of every endogenous variable.11

19. In addition to these three estimates, we also present a fourth forecast, which is obtained by using the BVAR1 model and assuming that the SELIC rate will be left constant during the forecast period at the level of the last observation. This is done to match the main scenario that the BCB presents in its QIRs.12 In the BVAR context, this fourth estimate assumes that the SELIC rate is not necessarily a systematic function of past values of other endogenous variables.

20. The selection and ordering of the endogenous variables in the VARs and BVARs reflects our general views on the transmission mechanism of monetary policy in Brazil, and the passthrough of exchange rates on domestic inflation.13 The variables considered here (i.e., money, interest rates, exchange rates, wholesale prices) could potentially have some effects on future consumer price inflation, and, a priori, it seems reasonable to think that their contribution to the forecasting of consumer price inflation should be statistically significant. Initially, we also tried several other indicators that, a priori, may be considered good candidates for inflation forecasting, including world oil prices, domestic fuel and energy prices, and nominal wages. However, adding these variables to the system did not improve the forecasts, and they were excluded.14

21. Table 7.1 presents the one month, four-months (next quarter), and seven-months (two quarters) ahead forecast for inflation for the different models that were estimated; Table 7.2 presents the deviations of predicted from actual inflation. The various time series models were all estimated with monthly data from January 1995 onward.15

Table 7.1.

Actual Inflation and Inflation Forecasts

(12-Month Rate, in Percent)

article image
Sources: BCB; and authors’ estimates.

Reflects BCB forecasts from the quarterly SSM model with constant nominal interest rates, as presented in the QIRs for June 1999, September 1999, December 1999, March 2000, June 2000, and September 2000, respectively.

Reflects change of the projected 12-month rate for a given quarter vis-à-vis the actual (for the one-month forecast) or the projected 12-month rate for the previous quarter.

Table 7.2.

Deviation of Forecast from Actual Inflation

(in Percentage Points)

article image
Sources: BCB; and authors’ estimates.

Reflects BCB forecasts from the quarterly SSM model with constant nominal interest rates, as presented in QIRs for June 1999, September 1999, December 1999, March 2000, and June 2000, respectively.

Mean squared errors of the forecast, multiplied by 100.

Mean squared errors of the forecast, multiplied by 100, excluding all projections made with data up to May 1999.

Mean squared errors of the forecast, multiplied by 100, excluding all projections made with data up to May 1999, and excluding projections for September 2000.

22. Table 7.1 suggests that, in general, the various models forecasted inflation as going in the same direction, that is either up or down. For example, using data to either November 1999 or February 2000, all models predicted the 12-month rate of inflation for March 2000 to decrease relative to December 1999. There were only few exceptions: for example, using data up to either February 2000 or May 2000, the BCB’s SSM model was the only one that (correctly) predicted September 2000 inflation to go up relative to June 2000. Similarly, using data to February 2000, the BVAR1 model was the only one that (correctly) predicted June 2000 inflation to go down relative to March 2000.

23. The direction of change that the various projections pointed to was not always correct, suggesting that all models have some difficulty in predicting inflation turning points. For example, using data to November 1999, the various models wrongly predicted inflation to rise in the second quarter of 2000; the same also holds true for the BCB’s model, and the AR1, VAR, and BVAR2 models, with data up to February 2000. Hence, with the models generally projecting inflation to increase through June 2000, the further decline in inflation that actually occurred through June may have come as a surprise also to the BCB. This may have led the BCB to initiate its 200 basis point reduction in the SELIC rate during June-July 2000. The subsequent upward revision of the December 2000 inflation forecast by 1.1 percentage points that occurred from the June 2000 QIR to the September 2000 QIR suggests that the BCB was equally surprised by the high July-August inflation outcomes as it had been by the fairly low inflation that prevailed through June. While the planned increases in administered prices in July 2000 and continued relatively high international oil prices were taken into account in the inflation projections that were presented in the June 2000 QIR, it was probably the unexpected increase in the IPCA food-price component,16 that, in retrospect, seems to make the June-July reductions in the overnight interest rate appear as a departure from a cautious monetary policy stance.

24. Table 7.2 suggests that the BCB’s SSM model generally outperforms the various time series models. However, this seems less related to the model itself, but to the fact that the SSM forecasts make use of exogenous information, e.g., on planned price increases.

25. Still, while September 2000 inflation was underestimated by all models, only the BCB’s SSM model with data up to May 2000 correctly predicted an upturn in inflation for September 2000. Even though the BCB’s SSM model underpredicted the upturn by about 1.1 percentage points, the various time series models underpredicted by more, reflecting the fact that, in contrast to the SSM model, they were not fed with any exogenous information on the increases in administered prices that were planned for July. Hence, the relevant question would not be why the time series models fare worse than the SSM in predicting September inflation, but why the SSM model did not fare better.

26. In general, Table 7.2 indicates that, at the four-month and seven-month horizons, all models fared far worse than at the one-month horizon. Still, the one-month-ahead forecasts presented in the QIRs are likely to have been generated largely outside of the SSM model, also since the SSM uses quarterly rather than monthly information.17 Hence, a direct comparison between the performance of the quarterly SSM and the monthly time series models would not be strictly valid at the one-month horizon. It is interesting to note, though, that among the time series models, the BVARs easily outperformed the AR1 and VAR models at the one-month horizon, as indicated by the mean squared errors (MSEs) of forecasts.

27. As expected, the models generally tended to either overpredict or underpredict jointly. For example, using data to August 1999, all models underpredicted December 1999 inflation; the same happened again with September 2000 inflation, which was underpredicted by all models with data to either February 2000 or May 2000. Similarly, using data to November 1999 that included the sharp increase in inflation which had occurred during July-November 1999, all models overpredicted inflation for March 2000 and June 2000. The same holds again with data up to February 2000, where all models continued to overpredict June 2000 inflation.

28. The results in Table 7.2 also suggest that having more data did not always make for a better forecast, except for the very short run, one-month ahead projection. For example, the inflation outcome for March 2000 was first generally underpredicted based on data to August 1999, and then overpredicted based on data to November 1999; in two of the five models (AR1 and VAR) having observations for three more months actually worsened the absolute prediction error. Also, having data up to May 2000 instead of only February 2000 did not help to improve the forecast error for September 2000 inflation in any of the models. On the other hand, having data up to February 2000 instead of only November 1999 significantly improved the forecast for June 2000 in all models. This again suggests that all models have problems to predict inflation turning points, reflecting the fact that most recent information weighs heavily in the various forecasts.

29. Interestingly, Table 7.2 indicates that the BCB’s SSM model does a better job forecasting seven months ahead than it does forecasting four months ahead. This is evidenced by a comparison of the MSE for the two forecast horizons but also by the fact that in three out of four occasions, the absolute forecast error for the seven-month forecast was lower than for the four-month forecast. This is not the case for the various time series models, which seem to do relatively well for the four-months ahead forecasts, but are significantly less accurate for the seven-months ahead forecasts. This may suggest that the various time series models may work best for fairly short-term forecasts of inflation.

30. Clearly, as presented here, a main shortcoming of the various time series techniques is that they are purely backward looking in the sense that forecasts are based exclusively on past values of the endogenous variables. Hence, information on anticipated shocks, such as future adjustments of administered prices, did not enter the projection. Of course, the models could easily be extended to take such information into account. An interesting extension in this regard would be to feed the various time series models the same information on exogenous shocks used for the SSM estimates that are presented in the QIRs.

31. Still, as already indicated, it is interesting to note that, while the BCB’s SSM model did take into account information on anticipated shocks, the better time series models, generally the BVARs, came fairly close to the performance of the SSM. For example, while, for the overall sample, the MSE of the SSM was lower than the MSE for the BVARs, this no longer holds when excluding the June 1999 inflation report (the very first QIR) and projections made for September 2000 (where the SSM forecast clearly benefited from exogenous information), as shown in Table 7.2.

32. For any forecasting exercise it is desirable to take into account all known information on future events, so as to avoid that the model is determined only by “history.” For example, a main reason why, using data to November 1999, inflation for the first quarter of 2000 was significantly overpredicted seems to have been that inflation for the most recent actual observations (i.e., October and November 1999) were particularly high. For the BVAR2 model, the March 2000 forecast with data for November 1999 was 7.48 percent and resulted in an absolute forecast error of 0.56 percentage points. Already with one more month of data, for December 1999, which had a much lower inflation reading than October-November 1999, the forecast error would have been roughly halved. In an environment where even one more data point may significantly affect the forecast, it becomes even more important to take into account information on likely future shocks.18

33. Although the “backward-looking” time series estimates presented here, particularly of the BVARs, compare relatively well to the BCB’s SSM model, at least for the four-month forecasting horizon, it remains to be shown whether or not a BVAR that would make use of forward looking information could fare better than the SSM model. Still, the initial estimates presented here would suggest that exploring the BVAR models more fully would be a promising area for future research.

E. Conclusions

34. Having good forecasts of inflation is important for any monetary authority, but particularly for one that operates with an explicit inflation target, such as in Brazil. With the transmission channels of monetary policy operating with a lag, the central bank needs good inflation forecasts to be able to act preemptively, if needed, so as to keep the inflation outcome close to the inflation target.

35. This section has compared the inflation forecasting performance of three time series techniques—univariate autoregressive estimation, multivariate “classical” VAR, and multivariate Bayesian VAR—with that of the small-scale macroeconomic model of the Brazilian Central Bank. Bayesian VAR, in particular, provides a possible solution to the problem of classical VAR forecasting where, often, too many parameters are estimated with too few observations, resulting in large out-of-sample forecasting errors. This problem is judged to be particularly relevant for Brazil, where, given the many structural breaks and the hyperinflation that prevailed during much of the 1980s and early 1990s, reliable time series data are often fairly short. Bayesian VAR modeling calls for introducing prior statistical information on the endogenous variables, and allows the forecaster to decide how much weight to put on the prior information and how much on the sample data.

36. The results presented in this section suggested that a Bayesian VAR can, generally, be expected to do a better job in forecasting inflation than either a classical VAR model or a univariate model. Although the Bayesian VAR forecasts presented here scored worse than the BCB’s small-scale macroeconomic model, this seems to have resulted largely from the fact that, contrary to the small-scale macroeconomic model, the Bayesian VAR forecasts did not take into account information on likely or expected future shocks (e.g., planned adjustments in administered prices). Still, the Bayesian VAR model could potentially take such forward-looking information into account. Similarly, it would also be possible to incorporate into the Bayesian VAR model information on leading indicators or market expectations of future inflation. Another issue that would seem to warrant further exploration is the forecasting horizon. It is unclear whether the observed relative deterioration of the forecasting performance of the Bayesian VAR model beyond the four-month horizon reflects on the technique itself, or whether improved models would also produce better longer-term forecasts. These and other issues remain to be explored in a more comprehensive assessment. Nevertheless, the results presented in this section suggest that Bayesian VAR modeling presents a promising tool for forecasting inflation in Brazil that would seem to warrant further exploration.

APPENDIX I Selecting the Prior Distribution of a Bayesian VAR Model

37. Selecting the prior distribution of the coefficients is a key aspect of BVAR modeling. Accordingly, both the mean vector and the variance covariance matrix of the multivariate prior distribution have to be specified. Nelson and Plosser (1982) suggested that most macroeconomic series can be characterized by a random walk process with drift. The vector of means of the prior distribution sets the value of all coefficients to zero. The only exception is the mean of the prior distribution for the first own lag of each variable in the system, which is set to one.19

38. To specify the structure of the standard deviations of the prior distribution, BVAR modeling uses so-called “hyperparameters.”20 The specification of a complete Normal distribution of the priors of a VAR would be intractable because of the size of the variance-covariance matrix. Instead, a general form for the prior involving a few hyperparameters is chosen, where, for each equation i, for each variable j in the system, and for every lag l, the structure of the standard deviation is:

(A1)S(i,j,l)=w*g(l)*f(i,j)*stSj
A07app01

In this equation, st is the standard deviation of a univariate autoregression on equation i;21 w is an overall “tightness”22 parameter, which is also the standard deviation of the first own lag: g(l)=l-d is the tightness of lag l relative to lag 1, with g(1)=1, d>1; and f(i, j) is the tightness on variable j for equation i relative to variable i. It is assumed that f(i, j)=1 when i=j and k otherwise (0<k<1).

39. Therefore, all relevant information about the prior distribution is contained in the hyperparameters w, d and k. Note that a smaller value of w places a smaller weight on all standard deviations, making the prior distribution “tighter.” Also, the relative standard deviation of lags of all endogenous variables decays geometrically at a rate d. Similarly, the higher d, the smaller is the weight placed on lags of all endogenous variables. Finally, the smaller k, the smaller is the importance placed on other endogenous variables when forecasting a given variable.

40. The BVAR methodology comprises the other forecasting models used in this paper: the other time series models explored here may be viewed as special cases of a BVAR model with extreme values for the hyperparameters. For instance, an extremely tight prior distribution (e.g., w=0.0001) implies the univariate model. An extremely loose prior distribution (e.g., w=5) implies an unrestricted VAR model.

41. Once the hyperparameters are selected, the BVAR model is estimated using Theil’s (1971) mixed estimation technique. This technique provides a suitable framework for obtaining the posterior distribution of the coefficient vector by allowing to combine the different sources of information available (prior and sample). In a more technical way, Theil’s mixed estimation technique involves supplementing data with prior information on the distributions of the coefficients. For each restriction on the parameter estimates, the number of observations and degrees of freedom increase by one in an artificial way; hence, the loss of degrees of freedom due to overparametrization in a VAR model disappears in the BVAR model.23

42. The BVAR model presented in this section was estimated using the RATS econometric package. To estimate the model, the following hyperparameters were chosen: d=1, w=0.2 and k=0.7. This compares to the RATS manual,24 which, following Litterman (1986), suggests to use the following values: d=1, w=0.2 and k=0.5. We found that, for Brazil, assigning a slightly less tight prior distribution increased the accuracy of the forecasts. To select the optimal hyperparameters, we tried out various combinations of the hyperparameters, where the range of combinations that was considered was d=1 or 2, w=0.1, 0.15 or 0.2, k=0.3, 0.5 or 0.7. Our final choice consisted in the combination that delivered the overall best out-of-sample forecasts, as measured by the MSE criterion. In future research along these lines it would seem helpful, once the available time series allow for more in-sample forecasts, to explore the stability of the hyperparameters that produce the best forecasts.

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1

Prepared by Pau Rabanal and Gerd Schwartz.

2

This includes various preview indicators, such as the mid-month indicator of the index used for inflation targeting (IPCA), and various newly-developed indicators of core inflation.

4

This benchmark naïve forecast could be improved upon, even in the univariate context, by using more elaborate ARIMA techniques. This option was not explored.

5

For instance, Robertson and Tallman (1999) show that the information contained in futures contracts is a better predictor of the U.S. Federal Funds rate than VAR-based forecasts.

6

In general, these criteria, like the Akaike or Schwarz criteria, select an optimal lag length by weighing the gains from using more lags against the related loss of degrees of freedom.

7

Traditional econometric techniques (including VARs) view model coefficients as parameters, whereas Bayesian econometrics considers model coefficients to be random variables, and, hence, to have a distribution function. Through the prior distribution that is imposed, it is possible to control how much weight is given to each source of information. By using a combination of the prior information on the coefficents’ distribution and the sample information, Bayesian estimation techniques produce a posterior distribution of coefficients.

8

See Appendix I for a detailed explanation of the hyperparameters and the associated prior distribution. The term “hyperparameter” in Bayesian VAR modeling was adopted to avoid confusion with the term “parameter” used in classical econometrics.

9

This is a market interest rate that averages bank lending rates for individuals and corporate entities, as reported in the monthly BCB press releases available on the BCB’s website (http://www.bcb.gov.br). Also see BCB (1999) for a detailed description on how this variable was derived.

10

The IPCA, IPA-DI, M1, and the nominal exchange rate are all expressed in natural logarithms.

11

See Appendix on how hyperparameters for the prior distribution were chosen.

12

The QIRs of September 1999, December 1999, and March 2000 assumed a constant path for the annualized SELIC rate of 19 percent. The June 2000 and September 2000 QIRs assumed a constant path for the annualized SELIC rate of 17.5 percent and 16.5 percent, respectively.

13

See Rabanal and Schwartz (2000a and 2000b), and also McCarthy (1999).

14

Since domestic ex-refinery fuel prices are determined by the government, world oil price developments do not automatically impact on domestic inflation.

15

The comparisons presented in Tables 7.1 and 7.2 should be interpreted with caution. Strictly speaking, it would not be valid to compare a model projection made with one SELIC rate assumption with the actual inflation outcome that occurred with another SELIC rate. In practice, however, the sensitivity of the one-quarter or two-quarters ahead inflation forecasts to fairly moderate changes in the SELIC rate is not very large. As a result, most of the deviations between actual inflation and model predictions that are shown in Tables 7.1 and 7.2 may safely be attributed to forecast errors, rather than different prevailing SELIC rates.

16

Food and drink account for about 22 percent of the IPCA basket. While food and drink prices experienced deflation of 0.8 percent during January-June 2000, they experienced inflation of 3.9 percent in July-August 2000, thereby pushing up the IPCA index by a cumulative 0.9 percent during these two months (i.e., over 50 percent of total inflation).

17

Also, note, that the QIRs for June 1999 and September 1999 did not include one-month ahead forecasts.

18

In Brazil, administered prices have a strong influence on the evolution of the overall CPI. For example, the December 1999 QIR states that “the impact of this shock on inflation in 1999 has been far from negligible. Out of the 8.29 percent IPCA inflation between January and November, 3.5 percentage points are directly due to government managed prices.” Forecasts that do not take into account likely future adjustments in these prices are likely to be significantly more inaccurate.

19

This vector of means for the prior distribution is commonly known as the “Minnesota prior,” since it was developed by researchers at the University of Minnesota and the Federal Reserve Bank of Minneapolis. See, for instance, Litterman (1986).

20

Also see Doan (1996) for a good description of Bayesian VAR forecasting.

21

The ratio Si/Sj scales the variables to account for differences in units of measurement. Thus, it enables specification of the prior without consideration of the magnitudes of the variables.

22

The terms “tightness” and “looseness” reflect how much weight is placed on the prior distribution relative to the sample information in formulating the BVAR model. For instance, a “tight” prior involves setting relatively small standard deviations in the prior distribution. Therefore, with tight priors there is less room for the data to override the priors, implying that a relatively low weight is put on the information contained in the data.

23

For a detailed explanation on Theil’s estimation technique, see Ballabriga et al. (2000).