تجزیه و تحلیل مختصر از اندازه گیری تجمعی به عنوان یک جایگزین برای میانه ​​در پیش بینی های حرفه ای بررسی بانک مرکزی برزیل

This paper presents a brief analysis of representative measures of inflation expectations from Central Bank of Brazil's Survey of Professional Forecasts that are alternative to the median response. We build time series with the mode and core measures of inflation expectations from the panel of professional forecasts surveyed from January 2002 to September 2012. We compare them to the median response with respect to their predictive power in a 12-month-ahead horizon. We also compare the predictive power of the alternative measures with the realized core of consumer price inflation.

Market forecasts surveyed by the Central Bank of Brazil since 1999 are reported daily at the institution's website and published weekly at the “Focus” reports. They have become an important reference to the discussion of macroeconomic prospects in Brazil, especially with respect to inflation. The univariate measures that are usually chosen to represent the panel of forecasts both at the “Focus” reports and at the Inflation Report (Chapter 6) are the median forecasts and their standard deviations.1 Throughout the rest of the world, the use of the panel median or mean is also widely disseminated.2 Carvalho and Minella (2012) present a detailed study of the predictive power of the median forecasts surveyed by the Central Bank of Brazil for a 12-month-ahead horizon. They show that in the analyzed period, the median response does not present systematic bias, which implies a reasonable predictive power, in spite of their failure in efficiency tests. Other studies have investigated the predictive power of the median responses of Central Bank of Brazil's survey for varied forecast horizons.3 This paper assesses the predictive power of other measures representative of the panel of inflation forecasts surveyed by the Central Bank of Brazil. In particular, we build series of modes and core measures of inflation forecasts for a 12-month-ahead horizon. Except for the symmetric trimmed mean core and for one of the asymmetric core measures, all measures that we investigate are statistically different from the median response. In terms of predictive power, all measures present systematic bias in the complete sample. The evidence of bias is slightly smaller for the median. However, this conclusion does not hold for shorter sub-samples. Furthermore, for the complete sample, the investigated measures are more appropriate proxies of the smoothed trimmed core inflation index than of the actual headline consumer price inflation. Except for the mode, the investigated measures do not present systematic bias when compared to the core inflation. Notwithstanding, for subsamples beginning in January 2003 or January 2004, which are less contaminated by the confidence crisis that had hit the economy in previous years, all measures present systematic bias when compared to the inflation core.

This brief paper shows that the mode of inflation expectations for a 12-month-ahead horizon and a great number of asymmetric core measures present important differences with respect to the median inflation expectation. In the complete sample, which includes years in which the crisis of confidence in the future conduct of economic policy in Brazil affected most noticeably the predictive power of inflation expectations from professional forecasters, all analyzed measures (median, symmetric trimmed cored, asymmetric core and mode) present systematic forecast bias for the headline consumer inflation. However, in the sub-sample that begins in January 2003, the analyzed inflation forecasts cease to present bias, likely due not to an improvement in predictive capacity, but to statiscal cancelling out of positive through negative errors. In fact, the choice of subsample influences the result. Inspired in Ranchhod (2003), we carry out tests to check whether the forecasts made for the headline inflation are more appropriate representations of a less volatile measure of inflation, such as the smoothed trimmed core consumer inflation. Contrary to the unbiasedness tests for the headline inflation, there is no indication of systematic bias in the analyzed measures (with the exception of the mode using a 95% confidence interval) when we compare the inflation expectations with the core inflation. However, for subsamples beginning in January 2003 or January 2004, the tests indicate forecast bias. Appendix A. See Table A1, Table A2, Table A3, Table A4 and Table A5. Table A1. Test of statistical difference between the symmetric trimmed mean and the median 12-month-ahead inflation expectations. Dependent variable (Y): symmetric trimmed mean core of inflation Expectations Sample: January 2002 to September 2012 Number of observations: 129 Equation: Y = c(1) + c(2) × Median of infl. expectations Coefficient Std t-ratio P-value c(1) −0.037 0.020 −1.840 0.068 c(2) 1.007 0.004 273.266 0.000 R2 0.998302 Mean dependent var 5.216861 Adjusted R2 0.998289 SD dependent var 1.357727 Regression Std 0.056165 Akaike −2.90568 SSR 0.400616 Schwarz −2.861342 Log likelihood 189.4163 Hannan–Quinn −2.887664 F-statistics 74674.53 Durbin–Watson 2.187917 Prob (F-statistics) 0 Wald test: c(1) = 0, c(2) = 1 Test statistic Value dF Probability F-statistics 1.727744 (2, 127) 0.1818 Chi-square 3.455488 2 0.1777 Table options Table A2. Tests of Statistical difference between the asymmetric trimmed mean and median core and the median inflation expectations. Dependent variable (Y): asymmetric trimmed mean core of inflation Expectations (Pearson coeff.) Sample: January 2002 to September 2012 Number of observations: 129 Equation: Y = c(1) + c(2) × Median expectations Coefficient Std t-ratio P-value c(1) 0.000 0.022 −0.002 0.999 c(2) 1.000 0.004 245.890 0.000 R2 0.997904 Mean dependent var 5.216725 Adjusted R2 0.997887 SD dependent var 1.348531 Regression Std 0.061982 Akaike −2.708548 SSR 0.487912 Schwarz −2.66421 Log likelihood 176.7013 Hannan–Quinn −2.690532 F-statistics 60462.02 Durbin–Watson 1.80517 Prob (F-statistics) 0 Wald test: c(1) = 0, c(2) = 1 Test statistic Value dF Probability F-statistic 0.022819 (2, 127) 0.9774 Chi-square 0.045638 2 0.9774 Dependent variable (Y): asymmetric trimmed median core of inflation Expectations (Pearson coeff.) Sample: January 2002 to September 2012 Number of obs: 129 Equation: Y = c(1) + c(2) × Median expectations Coefficient Std t-ratio P-value c(1) 0.041 0.010 3.952 0.000 c(2) 0.992 0.002 517.150 0.000 R2 0.999525 Mean dependent var 5.215422 Adjusted R2 0.999522 SD dependent var 1.336546 Regression Std 0.029233 Akaike −4.211672 SSR 0.108528 Schwarz −4.167334 Log likelihood 273.6529 Hannan–Quinn −4.193657 F-statistics 267443.7 Durbin–Watson 1.867673 Prob (F-Statistics) 0 Wald test: c(1) = 0, c(2) = 1 Test Statistic Value dF Probability F-statistic 9.79399 (2, 127) 0.0001 Chi-square 19.58798 2 0.0001 Dependent variable (Y): asymmetric trimmed mean core of inflation Expectations (asymmetry measured by the 3rd moment) Sample: January 2002 to September 2012 Number of obs: 108 Equation: Y = c(1) + c(2) × Median Expectations Coefficient Std t-ratio P-value c(1) −0.189 0.036 −5.234 0.000 c(2) 1.041 0.007 155.116 0.000 R2 0.994749 Mean dependent var 5.242026 Adjusted R2 0.994708 SD dependent var 1.406198 Regression Std 0.102294 Akaike −1.706541 SSR 1.328946 Schwarz −1.662203 Log likelihood 112.0719 Hannan–Quinn −1.688526 F-statistics 24060.97 Durbin–Watson 1.654712 Prob(F-Statistics) 0 Wald test: c(1) = 0, c(2) = 1 Test statistic Value dF Probability F-statistic 22.17044 (2, 127) 0 Chi-square 44.34087 2 0 Dependent variable (Y): asymmetric trimmed median core of inflation Expectations (asymmetry from the 3rd moment) Sample: January 2002 to September 2012 Number of obs: 129 Equation: Y = c(1) + c(2) × Median expectations Coefficient Std t-ratio P-value c(1) −0.036 0.010 −3.603 0.000 c(2) 1.008 0.002 541.864 0.000 R2 0.999568 Mean dependent var 5.22563 Adjusted R2 0.999564 SD dependent var 1.35903 Regression Std 0.028369 Akaike −4.271629 SSR 0.102212 Schwarz −4.227291 Log likelihood 277.5201 Hannan–Quinn −4.253614 F-statistics 293616.4 Durbin–Watson 1.593859 Prob(F-Statistics) 0 Wald test: c(1) = 0, c(2) = 1 Test statistic Value dF Probability F-statistic 15.06299 (2, 127) 0 Chi-square 30.12599 2 0 Table options Table A3. Test of statistical difference between the mode and the median inflation expectations. Dependent variable (Y): mode of inflation expectations Sample: January 2002 to September 2012 Number of obs: 129 Equation: Y = c(1) + c(2) × Median expectations Coefficient Std t-ratio P-value c(1) 0.367 0.069 5.346 0.000 c(2) 0.925 0.013 72.629 0.000 R2 0.97649 Mean dependent var 5.195851 Adjusted R2 0.976305 SD dependent var 1.261863 Regression Std 0.194242 Akaike −0.424042 SSR 4.791699 Schwarz −0.379704 Log likelihood 29.35074 Hannan–Quinn −0.406027 F-Statistics 5274.929 Durbin–Watson 1.99016 Prob (F-statistics) 0 Wald test: c(1) = 0, c(2) = 1 Test statistic Value dF Probability F-statistic 17.94627 (2, 127) 0 Chi-square 35.89254 2 0 Table options Table A4. P-value of the unbiasedness tests of aggregate measures of inflation expectations as predictions for the symmetric trimmed mean core of realized consumer inflation (IPCA). a January 2002 to September 2012 January 2004 to September 2012 Median 0.2184 0.0001 Mean 0.0975 0.0000 Mode 0.0433 0.0002 Symmetric core of expectations 0.1138 0.0000 a Test equation: Forecast error of the measure representative of expectations = c(1) + noise (MA(12)). The p-values shown refer to the test with H0: c(1) = 0. Table options Table A5. Series of measures representative of inflation expectations for the 12-month-ahead consumer inflation (IPCA). Mode of 12-month-ahead inflation expectations Jan-01 5.0 Jun-06 3.7 Feb-01 4.8 Jul-06 3.9 Mar-01 4.6 Aug-06 3.7 Apr-01 4.5 Sep-06 4.0 May-01 4.1 Oct-06 4.1 Jun-01 4.6 Nov-06 3.8 Jul-Ol 4.6 Dec-06 4.2 Aug-01 5.7 Jan-07 4.2 Sep-01 5.5 Feb-07 3.9 Oct-01 8.0 Mar-07 4.2 Nov-01 8.8 Apr-07 4.6 Dec-01 11.0 May-07 4.7 Jan-02 10.3 Jun-07 5.2 Feb-02 9.8 Jul-07 5.3 Mar-02 8.4 Aug-07 5.4 Apr-02 8.1 Sep-07 5.2 May-02 8.4 Oct-07 5.1 Jun-02 7.3 Nov-07 5.6 Jul-02 6.5 Dec-07 5.0 Aug-02 6.5 Jan-08 4.7 Sep-02 6.3 Feb-08 4.5 Oct-02 6.2 Mar-08 4.6 Nov-02 6.1 Apr-08 4.5 Dec-02 6.0 May-08 4.0 Jan-03 6.0 Jun-08 4.0 Feb-03 5.3 Jul-08 4.0 Mar-03 5.4 Aug-08 4.3 Apr-03 5.5 Sep-08 4.4 May-03 5.6 Oct-08 4.4 Jun-03 5.8 Nov-08 4.4 Jul-03 5.9 Dec-08 4.5 Aug-03 5.8 Jan-09 4.6 Sep-03 6.1 Feb-09 4.3 Oct-03 6.2 Mar-09 4.5 Nov-03 5.9 Apr-09 4.6 Dec-03 5.8 May-09 5.1 Jan-04 5.7 Jun-09 4.8 Feb-04 5.4 Jul-09 4.8 Mar-04 5.3 Aug-09 4.9 Apr-04 5.3 Sep-09 5.1 May-04 5.2 Oct-09 5.0 Jun-04 5.1 Nov-09 5.0 Jul-04 5.0 Dec-09 5.5 Aug-04 4.9 Jan-10 5.5 Sep-04 4.7 Feb-10 5.2 Oct-04 4.5 Mar-10 5.2 Nov-04 4.7 Apr-10 5.5 Dec-04 4.5 May-10 5.0 Jan-05 4.7 Jun-10 5.1 Feb-05 4.5 Jul-10 5.5 Mar-05 4.3 Aug-10 5.5 Apr-05 4.2 Sep-10 5.7 May-05 4.2 Oct-10 5.8 Jun-05 4.6 Nov-10 5.5 Jul-05 4.2 Dec-10 5.2 Aug-05 4.2 Jan-11 5.3 Sep-05 4.3 Feb-11 5.3 Oct-05 4.0 Mar-11 5.4 Nov-05 4.2 Apr-11 5.5 Dec-05 4.2 May-11 5.4 Asymmetric trimmed median core of inflation expectations (asymmetry from the 3rd moment) Jan-01 4.9 Jun-06 3.5 Feb-01 4.6 Jul-06 3.6 Mar-01 4.8 Aug-06 3.7 Apr-01 4.4 Sep-06 3.8 May-01 4.3 Oct-06 3.8 Jun-01 4.5 Nov-06 3.9 Jul-Ol 4.5 Dec-06 4.2 Aug-01 5.1 Jan-07 4.2 Sep-01 5.5 Feb-07 4.2 Oct-Ol 7.9 Mar-07 4.4 Nov-01 10.2 Apr-07 4.3 Dec-01 11.6 May-07 4.7 Jan-02 10.4 Jun-07 5.1 Feb-02 9.8 Jul-07 5.3 Mar-02 9.3 Aug-07 5.3 Apr-02 8.6 Sep-07 5.2 May-02 8.1 Oct-07 5.3 Jun-02 7.5 Nov-07 5.4 Jul-02 7.0 Dec-07 5.1 Aug-02 6.4 Jan-08 4.8 Sep-02 6.3 Feb-08 4.5 Oct-02 6.1 Mar-08 4.2 Nov-02 6.0 Apr-08 4.1 Dec-02 6.0 May-08 4.0 Jan-03 5.8 Jun-08 4.1 Feb-03 5.3 Jul-08 4.1 Mar-03 5.4 Aug-08 4.1 Apr-03 5.5 Sep-08 4.2 May-03 5.8 Oct-08 4.3 Jun-03 6.0 Nov-08 4.4 Jul-03 6.0 Dec-08 4.5 Aug-03 6.1 Jan-09 4.5 Sep-03 6.2 Feb-09 4.4 Oct-03 6.2 Mar-09 4.6 Nov-03 6.0 Apr-09 4.7 Dec-03 5.8 May-09 4.9 Jan-04 5.6 Jun-09 4.9 Feb-04 5.4 Jul-09 5.1 Mar-04 5.4 Aug-09 5.1 Apr-04 5.5 Sep-09 5.1 May-04 5.2 Oct-09 5.1 Jun-04 5.1 Nov-09 5.2 Jul-04 5.1 Dec-09 5.3 Aug-04 5.0 Jan-10 5.4 Sep-04 4.7 Feb-10 5.3 Oct-04 4.6 Mar-10 5.4 Nov-04 4.6 Apr-10 5.2 Dec-04 4.5 May-10 5.0 Jan-05 4.5 Jun-10 5.3 Feb-05 4.4 Jul-10 5.4 Mar-05 4.3 Aug-10 5.5 Apr-05 4.2 Sep-10 5.6 May-05 4.2 Oct-10 5.6 Jun-05 4.3 Nov-10 5.5 Jul-05 4.5 Dec-10 5.3 Aug-05 4.5 Jan-11 5.3 Sep-05 4.3 Feb-11 5.4 Oct-05 4.0 Mar-11 5.4 Nov-05 4.2 Apr-11 5.5 Dec-05 4.0 May-11 5.5 Asymmetric trimmed median core of inflation expectations (Pearson's coeff.) Jan-01 4.9 Jun-06 3.6 Feb-01 4.6 Jul-06 3.7 Mar-01 4.7 Aug-06 3.7 Apr-01 4.5 Sep-06 3.9 May-01 4.3 Oct-06 3.9 Jun-01 4.4 Nov-06 4.0 Jul-Ol 4.5 Dec-06 4.2 Aug-01 5.1 Jan-07 4.2 Sep-01 5.5 Feb-07 4.2 Oct-01 7.9 Mar-07 4.3 Nov-01 10.1 Apr-07 4.4 Dec-01 11.6 May-07 4.7 Jan-02 10.3 Jun-07 5.1 Feb-02 9.8 Jul-07 5.3 Mar-02 9.0 Aug-07 5.3 Apr-02 8.5 Sep-07 5.2 May-02 8.1 Oct-07 5.3 Jun-02 7.5 Nov-07 5.4 Jul-02 6.8 Dec-07 5.1 Aug-02 6.4 Jan-08 4.7 Sep-02 6.2 Feb-08 4.5 Oct-02 6.1 Mar-08 4.3 Nov-02 6.0 Apr-08 4.1 Dec-02 6.0 May-08 4.0 Jan-03 5.8 Jun-08 4.1 Feb-03 5.3 Jul-08 4.1 Mar-03 5.4 Aug-08 4.1 Apr-03 5.5 Sep-08 4.3 May-03 5.7 Oct-08 4.3 Jun-03 5.9 Nov-08 4.4 Jul-03 5.9 Dec-08 4.5 Aug-03 6.1 Jan-09 4.5 Sep-03 6.1 Feb-09 4.4 Oct-03 6.2 Mar-09 4.6 Nov-03 6.0 Apr-09 4.7 Dec-03 5.8 May-09 4.9 Jan-04 5.6 Jun-09 4.9 Feb-04 5.4 Jul-09 5.0 Mar-04 5.3 Aug-09 5.0 Apr-04 5.4 Sep-09 5.1 May-04 5.2 Oct-09 5.1 Jun-04 5.1 Nov-09 5.1 Jul-04 5.1 Dec-09 5.3 Aug-04 5.0 Jan-10 5.4 Sep-04 4.7 Feb-10 5.3 Oct-04 4.6 Mar-10 5.3 Nov-04 4.6 Apr-10 5.3 Dec-04 4.5 May-10 5.0 Jan-05 4.5 Jun-10 5.2 Feb-05 4.4 Jul-10 5.4 Mar-05 4.3 Aug-10 5.5 Apr-05 4.2 Sep-10 5.7 May-05 4.2 Oct-10 5.7 Jun-05 4.4 Nov-10 5.5 Jul-05 4.5 Dec-10 5.3 Aug-05 4.5 Jan-11 5.3 Sep-05 4.3 Feb-11 5.3 Oct-05 4.1 Mar-11 5.4 Nov-05 4.2 Apr-11 5.5 Dec-05 4.1 May-11 5.5 Symmetric core of inflation expectations Jan-01 4.8 Jun-06 3.5 Feb-01 4.6 Jul-06 3.6 Mar-01 4.7 Aug-06 3.7 Apr-01 4.4 Sep-06 3.8 May-01 4.4 Oct-06 3.9 Jun-01 4.4 Nov-06 4.0 Jul-01 4.6 Dec-06 4.2 Aug-01 5.1 Jan-07 4.2 Sep-01 5.5 Feb-07 4.2 Oct-01 7.7 Mar-07 4.4 Nov-01 10.4 Apr-07 4.4 Dec-01 11.3 May-07 4.7 Jan-02 10.5 Jun-07 5.0 Feb-02 9.9 Jul-07 5.3 Mar-02 9.4 Aug-07 5.3 Apr-02 8.6 Sep-07 5.1 May-02 8.2 Oct-07 5.2 Jun-02 7.7 Nov-07 5.4 Jul-02 6.9 Dec-07 5.1 Aug-02 6.4 Jan-08 4.8 Sep-02 6.2 Feb-08 4.5 Oct-02 6.1 Mar-08 4.2 Nov-02 6.0 Apr-08 4.1 Dec-02 5.9 May-08 4.0 Jan-03 5.7 Jun-08 4.1 Feb-03 5.4 Jul-08 4.1 Mar-03 5.4 Aug-08 4.1 Apr-03 5.5 Sep-08 4.2 May-03 5.8 Oct-08 4.3 Jun-03 6.0 Nov-08 4.4 Jul-03 6.0 Dec-08 4.5 Aug-03 6.1 Jan-09 4.5 Sep-03 6.1 Feb-09 4.5 Oct-03 6.1 Mar-09 4.6 Nov-03 6.0 Apr-09 4.7 Dec-03 5.8 May-09 4.8 Jan-04 5.5 Jun-09 4.9 Feb-04 5.4 Jul-09 5.1 Mar-04 5.4 Aug-09 5.1 Apr-04 5.5 Sep-09 5.1 May-04 5.2 Oct-09 5.1 Jun-04 5.1 Nov-09 5.2 Jul-04 5.1 Dec-09 5.3 Aug-04 5.0 Jan-10 5.4 Sep-04 4.8 Feb-10 5.3 Oct-04 4.6 Mar-10 5.4 Nov-04 4.5 Apr-10 5.2 Dec-04 4.5 May-10 5.1 Jan-05 4.5 Jun-10 5.2 Feb-05 4.4 Jul-10 5.4 Mar-05 4.2 Aug-10 5.4 Apr-05 4.1 Sep-10 5.6 May-05 4.2 Oct-10 5.6 Jun-05 4.4 Nov-10 5.6 Jul-05 4.5 Dec-10 5.3 Aug-05 4.6 Jan-11 5.3 Sep-05 4.3 Feb-11 5.4 Oct-05 4.1 Mar-11 5.5 Nov-05 4.2 Apr-11 5.5 Dec-05 4.0 May-11 5.5 Asymmetric trimmed mean core of inflation expectations (asymmetry from the 3rd moment) Jan-01 4.9 Jun-06 3.5 Feb-01 4.6 Jul-06 3.6 Mar-01 4.7 Aug-06 3.6 Apr-01 4.3 Sep-06 3.7 May-01 4.3 Oct-06 3.8 Jun-01 4.6 Nov-06 3.9 Jul-Ol 4.6 Dec-06 4.1 Aug-01 5.2 Jan-07 4.3 Sep-01 5.8 Feb-07 4.2 Oct-Ol 8.2 Mar-07 4.4 Nov-01 10.4 Apr-07 4.3 Dec-01 11.2 May-07 4.8 Jan-02 10.8 Jun-07 5.1 Feb-02 10.1 Jul-07 5.3 Mar-02 9.6 Aug-07 5.2 Apr-02 8.9 Sep-07 5.1 May-02 8.2 Oct-07 5.2 Jun-02 7.7 Nov-07 5.4 Jul-02 6.9 Dec-07 5.1 Aug-02 6.6 Jan-08 4.9 Sep-02 6.3 Feb-08 4.5 Oct-02 6.1 Mar-08 4.2 Nov-02 6.0 Apr-08 4.1 Dec-02 5.9 May-08 4.0 Jan-03 5.7 Jun-08 4.0 Feb-03 5.4 Jul-08 4.1 Mar-03 5.4 Aug-08 4.1 Apr-03 5.5 Sep-08 4.2 May-03 5.8 Oct-08 4.3 Jun-03 6.1 Nov-08 4.4 Jul-03 6.1 Dec-08 4.5 Aug-03 6.2 Jan-09 4.5 Sep-03 6.2 Feb-09 4.5 Oct-03 6.2 Mar-09 4.5 Nov-03 6.1 Apr-09 4.7 Dec-03 5.9 May-09 4.9 Jan-04 5.7 Jun-09 4.9 Feb-04 5.5 Jul-09 5.1 Mar-04 5.4 Aug-09 5.2 Apr-04 5.5 Sep-09 5.2 May-04 5.2 Oct-09 5.2 Jun-04 5.1 Nov-09 5.3 Jul-04 5.1 Dec-09 5.4 Aug-04 4.9 Jan-10 5.5 Sep-04 4.8 Feb-10 5.4 Oct-04 4.6 Mar-10 5.4 Nov-04 4.5 Apr-10 5.2 Dec-04 4.5 May-10 5.0 Jan-05 4.5 Jun-10 5.3 Feb-05 4.4 Jul-10 5.5 Mar-05 4.2 Aug-10 5.5 Apr-05 4.1 Sep-10 5.6 May-05 4.1 Oct-10 5.6 Jun-05 4.3 Nov-10 5.5 Jul-05 4.5 Dec-10 5.3 Aug-05 4.5 Jan-11 5.3 Sep-05 4.2 Feb-11 5.4 Oct-05 4.0 Mar-11 5.5 Nov-05 4.1 Apr-11 5.5 Dec-05 4.0 May-11 5.5 Asymmetric trimmed mean core of inflation expectations (Pearson coeff.) Jan-01 4.9 Jun-06 3.5 Feb-01 4.7 Jul-06 3.6 Mar-01 4.7 Aug-06 3.7 Apr-01 4.4 Sep-06 3.8 May-01 4.3 Oct-06 3.9 Jun-01 4.5 Nov-06 3.9 Jul-Ol 4.6 Dec-06 4.2 Aug-01 5.2 Jan-07 4.3 Sep-01 5.6 Feb-07 4.2 Oct-01 7.9 Mar-07 4.3 Nov-01 10.1 Apr-07 4.4 Dec-01 11.2 May-07 4.8 Jan-02 10.6 Jun-07 5.1 Feb-02 9.9 Jul-07 5.3 Mar-02 9.3 Aug-07 5.3 Apr-02 8.5 Sep-07 5.2 May-02 8.2 Oct-07 5.2 Jun-02 7.5 Nov-07 5.4 Jul-02 6.8 Dec-07 5.1 Aug-02 6.4 Jan-08 4.8 Sep-02 6.2 Feb-08 4.5 Oct-02 6.1 Mar-08 4.3 Nov-02 6.0 Apr-08 4.2 Dec-02 5.9 May-08 4.0 Jan-03 5.8 Jun-08 4.1 Feb-03 5.4 Jul-08 4.1 Mar-03 5.4 Aug-08 4.1 Apr-03 5.5 Sep-08 4.2 May-03 5.7 Oct-08 4.3 Jun-03 5.9 Nov-08 4.4 Jul-03 6.0 Dec-08 4.5 Aug-03 6.1 Jan-09 4.5 Sep-03 6.1 Feb-09 4.4 Oct-03 6.1 Mar-09 4.6 Nov-03 6.0 Apr-09 4.7 Dec-03 5.8 May-09 4.9 Jan-04 5.6 Jun-09 4.9 Feb-04 5.4 Jul-09 5.0 Mar-04 5.4 Aug-09 5.1 Apr-04 5.4 Sep-09 5.2 May-04 5.2 Oct-09 5.1 Jun-04 5.1 Nov-09 5.2 Jul-04 5.1 Dec-09 5.4 Aug-04 5.0 Jan-10 5.5 Sep-04 4.8 Feb-10 5.3 Oct-04 4.5 Mar-10 5.3 Nov-04 4.6 Apr-10 5.3 Dec-04 4.5 May-10 5.0 Jan-05 4.5 Jun-10 5.2 Feb-05 4.4 Jul-10 5.5 Mar-05 4.2 Aug-10 5.5 Apr-05 4.1 Sep-10 5.7 May-05 4.2 Oct-10 5.7 Jun-05 4.4 Nov-10 5.5 Jul-05 4.5 Dec-10 5.3 Aug-05 4.5 Jan-11 5.3 Sep-05 4.3 Feb-11 5.3 Oct-05 4.0 Mar-11 5.4 Nov-05 4.2 Apr-11 5.5 Dec-05 4.1 May-11 5.5 Table options