Asymmetric Distributions in Fan Charts: International Experience and an Application to Brazil

Transcription

1 Asymmetric Distributions in Fan Charts: International Experience and an Application to Brazil One of the operational consequences for the BCB of the adoption of an inflation-targeting regime in Brazil was the need to improve its inflation forecasting techniques, based on econometric models and leading indicators of key price dynamics. The forecasting process involves a relative weighting of the main risks to the benchmark inflation scenario, with the degree of uncertainty expressed by confidence intervals presented in the form of fan charts for inflation and GDP growth. The September 999 Inflation Report outlined the process of fan chart construction and the symmetry and variance of the projections. The inflation and GDP growth fan charts result from the Copom members collective view on the central forecast, and the range of potential divergence from that forecast, for key economic variables. With the exception of June 999 and December 00, past issues of the Inflation Report presented symmetric fan charts. This box presents the experience of other countries in describing the balance of risks facing the economy, and applies an asymmetric distribution to the fan chart for Brazilian inflation. The majority of central banks that adopt an inflationtargeting regime, including Brazil, use confidence intervals derived from historical forecast errors. The process of calculating the forecast errors to establish the variance parameters for the normal distribution is fairly standard. The cases of Sweden and the United Kingdom are benchmarks for balance of risks analysis. / Alternatives to the use of fan charts are typically based on the presentation of scenarios constructed from changes in the trajectories of certain variables relevant to the inflation dynamics. The Reserve Banks of New Zealand and Australia use this type of approach. / Alternatively, some Central Banks, like the Czech Republic, submit historical forecast errors to stochastic simulations in order to generate forecast intervals. See Coates, w., Laxton, D., and Rose, D. (003), The Czech National Bank s forecasting and policy analysis system, chapter 6, available at: 96 Inflation Report September 005

2 Since 996, the Bank of England has presented its main forecasts based on a probability distribution. 3 The Monetary Policy Committee (MPC) makes assumptions, based on current developments and the future outlook, about the central value in the distribution, the variance, and the degree of asymmetry. All of the information is weighted, such that the main risks to the baseline scenario are considered in the construction of the distribution s mode and variance. 4 The weights attributed to each scenario are derived from the discussion among the MPC members at its regular meetings. Of note, the asymmetry of the distribution is adjusted over time: phenomena that have a temporary effect on inflation could, for example, be weighted only for the estimated duration of the shock. Determining the variance of the distribution involves not only consideration of the factors inherent to economic performance, but also the operational challenges encountered in the forecasting process. For example, in its last Inflation Report, the Bank of England widened the confidence interval of its GDP forecast fan chart for the first periods of the forecast horizon. Systematic and significant revisions to GDP data means that the forecast error associated with these forecast horizons can be quite elevated. 5 The Bank of Sweden (Riksbank) adopts a very different approach from that of the Bank of England: instead of reflecting the opinions of the monetary policy committee, the uncertainty degree is the aggregate result of Bank specialists forecasts, presented to the Executive Board to serve as the starting-point in the decision-making process. 6 During the preparation of the Inflation Report, Bank specialists forecast specific variables, and note the degree of uncertainty and the possibility that the forecast will be over- or under-estimated, both measured by the deviation from the normal 3/ For more information on the use of fan charts and the two-piece normal distribution used in the U.K., see Britton, E., Fisher, P., and Whitley, J. (998) The Inflation Report projections: understanding the fan chart, Bank of England Quarterly Bulletin, February. irprobab.htm. 4/ Obviously, independent shocks can be weighted in a similar way to the different scenarios for the central value of the distribution. For non-independent shocks, some consideration over the covariance of the distribution is carried out. 5/ See Inflation Report, August 005, box on page 40. 6/ According to Blix and Sellin (999), the Swedish process takes a bottom-up approach, instead of the top-down approach of the Bank of England. See Blix, M. and Sellin, P. (999) Inflation Forecasts with Uncertainty Intervals. Sveridges Riksbank Quarterly Review, No.. Available at September 005 Inflation Report 97

3 distribution. In a final meeting, the baseline scenario is discussed, aggregating the inflation projections, the effects of uncertainty, and the asymmetry over the other relevant variables. This baseline scenario is then passed on to the Executive Board for consideration and adjustment. Figure The two-part normal distribution histogram Variance: σ Mode: 5,5 : 7,8 Variance: σ Both the Bank of England and the Riksbank use a two-part normal distribution to create their fan charts. In this distribution, identifying the mode and the two variance measures, it is possible to construct a distribution with an asymmetric confidence interval, which can, under certain conditions, converge to a normal distribution. Figure shows an asymmetric distribution on the right: the mode separates the areas of differing variance. As the variance to the right of the central value is larger, the mean of the distribution is greater than the mode, creating the asymmetry. To show that the standard normal distribution is a special case of the two-part normal distribution, we define: σ σ γ = σ + σ as a parameter of the distribution whose absolute value is less than unity, related to the asymmetry of the distribution, defined by the two variances σ and σ, in addition to: σ = σ σ σ + σ as a parameter associated with the standard deviation of the two-part normal distribution. With these definitions, the probability density function of the twopart normal distribution assumes the following formula: f ( x µ ) [ γ.sgn( ] x) =. exp ) σ π σ + γ + γ µ, σ, γ ( x µ where sgn is a function that is equal to when the difference between the arguments is greater than zero, - when the difference is less than zero, and 98 Inflation Report September 005

4 zero when the values are equal. As in the case of the symmetric distribution, we have σ = σ, the parameter g becomes equal to zero, making a standard normal distribution. In addition to the definition of the density function, the moments of the two-part normal distribution can be written in terms of the two parameters above. 7 Also of note, the positive values of g imply an asymmetric distribution to the right.the variance will be higher than σ as the higher is the difference between σ and σ, regardless of the direction of the asymmetry. Table Central Value and means of alternative scenarios Period Central Value Mode Pessimistic Scenario Optimistic Scenario γ = 0 / γ = +0,70 γ = -0,70 Q Q Q Q Q Q Q / This is the central value of the distribution presented in the benchmark scenario of the June 004 Inflation Report. The following section aims to compare the normal distribution, typically used by the BCB, with the implications of the use of possible asymmetries in the two-part normal distribution. To this end, we present two extreme conditions cited in the June 004 Inflation Report. At the time, the Report pointed out, in section 6., that the main risks to inflation convergence to the target path were the pace of the adjustment in U.S. monetary policy and the future path of oil prices. Given the high volatility of oil prices, the use of asymmetric fan charts could reflect the balance of risks that included a higher probability of inflation being above the central forecast. On the other hand, the same document pointed out that the scenarios for lower inflation, despite being of diminishing importance in the period, could be taken into consideration in the construction of a positive scenario for the probability distributions for inflation. Using the benchmark scenario at the time as the mode of the inflation distribution expected by the Copom members, and the forecast error parameters used at the time, we present in Table what would be the mean of the distribution, supposing two different degrees of asymmetry. As explained, in the pessimistic scenario, the mean would be systematically greater than the mode, leading to the asymmetry to the right of the distribution of the fan chart. Figure, shown earlier, depicts the distribution 7/ Additional details on the two-part normal distribution can be found in: Blix, M., and Sellin, P. (998) Uncertainty Bands for Inflation Forecasts. Sveridges Riksbank Working Paper Series, No. 65. Available at Britton, E., Fisher, P., and Whitley, J. (998) The Inflation Report projections: understanding the fan chart. Bank of England Quarterly Bulletin, February. and Riella, G., Staub, R.B. and Rodrigues Neto, J.A., A Distribuição Normal de Duas Partes: Algumas Extensões, internal working paper of the Brazilian Central Bank Research Department. September 005 Inflation Report 99

5 for the forecast from the first quarter of 005. It bears emphasizing, nonetheless, that this example is merely illustrative as the value of γ was arbitrarily selected. Graph June 004 Inflation Report Standard and asymmetric fan chart (%) Q003 Q004 4Q004 Q005 4Q005 Table Central value and means of scenarios with time-varying γ Period Central Value - IR Central value Mode Pessimistic scenario γ = 0 / γ = +0,70 50% CI - IR 50% CI - Pessimistic 50% CI - Optimistic Alternative scenario Decreasing γ along time Q Q Q Q Q Q Q / This is the central value of the distribution presented in the benchmark scenario of the June 004 Inflation Report. Graph June 004 Inflation Report Time-varying asymmetric fan charts (%) Q003 Q004 4Q004 Q005 4Q005 Central Value - IR 50% CI - IR 50% CI - Pessimistic 50% CI - Alternative Graph shows the 50% uncertainty interval for each scenario of the above-mentioned asymmetric distribution, assuming that the central value of the distribution always corresponds to the mode shown in the Inflation Report. As emphasized above, in both asymmetric approaches, the variance of the distribution is larger than the variance of the standard normal distribution. In any case, with the use of asymmetric fan charts, the monetary authority obtains an additional instrument for signaling the risks surrounding future inflation. The scenarios constructed consider only shocks perceived as permanent by the monetary authority, at least over the forecast horizon. An alternative approach, for example, would consider that the volatility then present in the oil market was a temporary phenomenon. Table shows the benchmark scenario (again taken as the mode of the distribution), the mean of the pessimistic scenario with the parameter γ = +0.7, and the mean of a scenario assuming that the parameter g is initially equal to +0.7, falling in a linear path to zero in the last point of the forecast horizon (4Q005). This depicts a scenario where the risks associated with the benchmark scenario would be gradually eliminated over time. The scenario of declining risk underscores the linearity of the relationship between the mean, mode, and the defined asymmetry parameter, taking into account the convergence of the inflation estimates in the two extremes of the forecast horizon. Graph shows the distribution around the mode of these three scenarios. Note that the variance also possesses a linear relationship with the variance of the distribution; therefore, the initial values at the extremes of the confidence interval are equal to the pessimistic scenario, converging over the forecast horizon to the values of the scenario presented in June 004. In sum, the use of confidence intervals in inflation projections always illustrates some combination between the appropriate statistical technique and the judgment of the monetary authority. Any forecast necessarily implies some degree of uncertainty with 00 Inflation Report September 005

6 regard to its outcome, which should be measured in line with the subjective opinions of the monetary authority. Therefore, the purpose of using fan charts is to convey to economic agents not only the main value expected for the variables, but also to clarify the degree of uncertainty inherent in the forecasts and the associated risks. September 005 Inflation Report 0