Error Statistics for the Survey of Professional Forecasters for 3-Month T-Bill Rate [QA,PPs] Release Date: 09/01/2009 Tom Stark Assistant Director and Manager Real-Time Data Research Center Economic Research Department Federal Reserve Bank of Philadelphia 1. OVERVIEW. This document reports error statistics for median projections from the Survey of Professional Forecasters (SPF), conducted since 1990 by the Federal Reserve Bank of Philadelphia. We provide the results in a series of tables and, in the PDF version of this document, a number of charts. The tables show the survey variable forecast and, importantly, the transformation of the data that we used to generate the statistics. (The transformation is usually a quarter-over-quarter growth rate, expressed in annualized percentage points. However, some variables, such as interest rates, the unemployment rate, and housing starts are untransformed and, thus, expressed in their natural units.) The paragraphs below explain the format of the tables and charts and the methods used to compute the statistics. These paragraphs are general. The same discussion applies to all variables in the survey. 2. DESCRIPTION OF TABLES. Table 1 reports error statistics for various forecast horizons, sample periods, and choices of the real-time historical value that we used to assess accuracy. In each quarterly survey, we ask our panelists for their projections for the current quarter and the next four quarters. The current quarter is defined as the quarter in which we conducted the survey. Our tables provide error statistics separately for each quarter of this five-quarter horizon, beginning with the current quarter (denoted H = 1) and ending with the quarter that is four quarters in the future (H = 5). For each horizon, we report the mean forecast error [ME(S)], the mean absolute forecast error [MAE(S)], and the root-mean-square error [RMSE(S)]. All are standard measures of accuracy, though the academic literature generally places the most weight on the latter. We define a forecast error as the difference between the historical value and the forecast. The mean error for each horizon is simply the average of the forecast errors at that horizon, constructed over the sample period shown in Table 1. Other things the same, a forecast with a mean error close to zero is better than one with a mean error far from zero. The mean absolute error is the sample average of the absolute value of the errors. Many analysts prefer this measure to the mean error because it does not allow large positive errors to offset large negative errors. In this sense, the mean absolute error gives a cleaner estimate of the size of the errors. Decision makers, however, may care not only about the average size of the errors but also about their variability, as measured by variance. Our last measure of accuracy is one that reflects the influence of the mean error and the variance of the error. The root-mean-square error for the SPF [RMSE(S)], the measure most often used by analysts and academicians, is the square root of the the average squared error. The lower the root-mean-square error, the more accurate the forecast. 2.1. Benchmark Models. The forecast error statistics from the SPF are of interest in their own right. However, it is often more interesting to compare such statistics with those of alternative, or benchmark, forecasts. Table 1 reports four such comparisons. It shows the ratio of the root-mean-square error of the SPF forecast to that of four benchmark models. The benchmark models are statistical equations that we estimate on the data. We use the equations to generate projections for the same horizons included in the survey. In effect, we imagine standing back in time at each date when a survey was conducted and generating a separate forecast with each benchmark model. We do this in the same way that a survey panelist would have done using his own model. A RMSE ratio below unity indicates that the SPF consensus (median) forecast has a root-mean-square error lower than that of the benchmark. This means the SPF is more accurate. We now describe the benchmark models. The first is perhaps the simplest of all possible benchmarks: A no-change model. In this model, the forecast for quarter T, the one-step-ahead or current-quarter forecast, is simply the historical value for the prior quarter (T - 1). There is, in other words, no change in the forecast compared with the historical value. Moreover, the forecast for the remaining quarters of the horizon is the same as the forecast for the current quarter. We denote the relative RMSE ratio for this benchmark as RMSE(S/NC), using NC to indicate no change. The second and third benchmark models generate projections using one or more historical observations of the the variable forecast, weighted by coefficients estimated from the data. Such autoregressive (AR) models can be formulated in two ways. We can estimate one model to generate the forecasts at all horizons, using an iteration method to generate the projections beyond the current quarter (IAR), or we can directly estimate a new model for each forecast horizon (DAR). The latter formulation has been shown to reduce the bias in a forecast when the underlying model is characterized by certain types of misspecification. The root-mean-square error ratios are denoted RMSE(S/IAR) and RMSE(S/DAR), respectively. The one- through five-step-ahead projections of the benchmark models use information on the quarterly average of the variable forecast. The latest historical observation is for the quarter that is one quarter before the quarter of the first projection in the horizon. In contrast, the panelists generate their projections with the help of additional information. They submit their projections near the middle of each quarter and hence have access to some monthly indicators for the first month of each quarter, when those data are released before the survey deadline. This puts the projections of panelists for some variables at an advantage relative to the corresponding benchmark projections. Moreover, the panelists may also examine the very recent historical values of such monthly indicators in forming their projections for quarterly averages. Such monthly statistical momentum represents an advantage not shared by the benchmark models, which use only quarterly averages. For survey variables whose observations are reported at a monthly frequency, such as interest rates, industrial production, housing starts, and unemployment, we estimate and forecast a fourth benchmark model, the DARM. This model adds recent monthly historical values to the specification of the DAR model. For the projections for unemployment, nonfarm payroll employment, and interest rates, we add the values of monthly observations, beginning with that for the first month of the first quarter of the forecast horizon. These values should be in the information set of the survey panelists at the time they formed their projections. In contrast, for variables such as housing starts and industrial production, we include only lagged values of monthly observations. For such variables, the panelists would not have known the monthly observation for the first month of the first quarter of the forecast horizon. In general, we find that adding monthly observations to the benchmark DAR models improves accuracy. Indeed, for the projections for interest rates and the unemployment rate, the accuracy of the benchmark DARM projections rivals that of the SPF projections. 2.2. Real-Time Data. All benchmark models are estimated on a rolling, fixed window of 60 real-time quarterly observations. Lag lengths, based on either the Akaike information criterion (AIC) or the Schwarz information criterion (SIC), are re-estimated each period. The tables below indicate whether the lag length was was chosen by the AIC or SIC. We would like to make the comparison between the SPF forecast and the forecasts of each benchmark as fair as possible. Therefore, we must subject the benchmark models to the same data environment the survey panelists faced when they made their projections. This is important because macroeconomic data are revised often, and we do not want the benchmark models to use a data set that differs from the one our panelists would have used. We estimate and forecast the benchmark models with real-time data from the Philadelphia Fed real-time data set, using the vintage of data that the survey panelists would have had at the time they generated their own projections. (For more information on the Philadelphia Fed real-time data set, go to www.philadelphiafed.org/econ/forecast/real-time-data/.) An open question in the literature on forecasting is: What version or vintage of the data should we use to compute the errors? A closely related question is: What version of the data are professional forecasters trying to predict? Our computations take no strong position on these questions. Rather, in Table 1, we evaluate the projections (SPF and benchmark) with five alternative measures of the historical values, all from the Philadelphia Fed real-time data set. These measures range from the initial-release values to the values as we know them today. All together, we compute the forecast error statistics using the following five alternative measures of historical values: (1) The initial or first-release value; (2) The revised value as it appears one quarter after the initial release; (3) The revised value as it appears five quarters after the initial release; (4) The revised value as it appears nine quarters after the initial release; (5) The revised value as it appears today. Each measure of the historical value has advantages and disadvantages. The initial-release value is the first measure released by government statistical agencies. A forecaster might be very interested in this measure because it enables him to evaluate his latest forecast soon after he generated it. However, early releases of the data are often subject to large measurement error. Subsequent releases [(2) - (5)] are more accurate, but they are available much later than the initial release. As we go from the first measure to the fifth, we get more reliability, at the cost of higher delays in availability. The last two columns in Table 1 report the number of observations that we used to compute the error statistics. Some observations are omitted because the data are missing in the real-time data set, such as occurred when federal government statistical agencies closed in late 1995. 2.3. Recent Projections and Realizations. Tables 2 to 7 provide information on recent projections and realizations. They show how we align the data prior to computing the forecast errors that form the backbone of the computations in Table 1. Any forecast error can be written as the equation given by error = realization - forecast. For our computations, we must be more precise because, for each projection (SPF and benchmarks), we have different periods forecast (T) different forecast horizons (h), and several measures of the realization (m). Thus, we can define the forecast error more precisely as error( T, h, m ) = realization( T, m ) - forecast( T, h ). Tables 2 to 7 are organized along these lines. Table 2 shows recent forecasts from the SPF. Each column gives the projection for a different horizon or forecast step (h), beginning with that for the current quarter, defined as the quarter in which we conducted the survey. The dates (T) given in the rows show the periods forecast. These also correspond to the dates that we conducted the survey. Tables 3 to 6 report the recent projections of the four benchmark models. These are organized in the same way as Table 2. Table 7 reports recent values of the five alternative realizations (m) we use to compute the error statistics. 2.4. Qualifications. We note two minor qualifications to the methods discussed above. The first concerns the vintage of data that we used to estimate and forecast the benchmark models for CPI inflation. The second concerns the five measures of realizations used for the unemployment rate, nonfarm payroll employment, and CPI inflation. To estimate and forecast the benchmark models for CPI inflation, we use the vintage of data that would have been available in the middle of each quarter. This predates by one month the vintage that SPF panelists would have had at their disposal when they formed their projections. The effect is likely small because revisions to the CPI are generally small. To compute the realizations for unemployment, nonfarm payroll employment, and CPI inflation, we use the vintages associated with the middle of each quarter. The measure that we call initial comes from this vintage, even though the initial estimate was available one month earlier. Thus, for these variables, our initial estimate reflects some revision by government statistical agencies. The effect for unemployment and CPI inflation is likely small. The effect could be somewhat larger for nonfarm payroll employment. 3. DESCRIPTION OF GRAPHS. 3.1. Root-Mean-Square Errors. For each sample period shown in Table 1, we provide graphs of the root-mean-square error for the SPF forecast. There is one page for each sample period. On each page, we plot (for each forecast horizon) the RMSE on the y-axis. The x-axis shows the measure of the historical value that we used to compute the RMSE. These range from the value on its initial release to the value one quarter later to the value as we know it now (at the time we made the computation). The graphs provide a tremendous amount of information. If we focus on a particular graph, we can see how a change in the measure of the realization (x-axis) affects the root-mean-square-error measure of accuracy. The effect is pronounced for some variables, such as real GDP and some of its components. For others, there is little or no effect. For example, because the historical data on interest rates are not revised, the estimated RMSE is the same in each case. If we compare a particular point on one graph with the same point on another, we see how the forecast horizon affects accuracy. In general, the RMSE rises (accuracy falls) as the forecast horizon lengthens. Finally, if we compare a graph on one page with the corresponding graph on another page, we see how our estimates of accuracy in the SPF change with the sample period. Periods characterized by a high degree of economic turbulence will generally produce large RMSEs. 3.2. Fan Charts. The last chart plots recent historical values and the latest SPF forecast. It also shows confidence intervals for the forecast, based on back-of-the-envelope calculations. The historical values and the SPF forecast are those associated with the latest vintage of data and survey, respectively, available at the time we ran our computer programs. The confidence intervals are constructed under the assumption that the historical forecast errors over the sample (shown in the footnote) follow a normal distribution with a mean of zero and a variance given by the root-mean-square error. The latter is estimated over the aforementioned sample, using the measure of history listed in the footnote. ---------------------------------------------------------------------------------- Table 1. Forecast Error Statistics for SPF Variable: TBILL (3-Month T-Bill Rate [QA,PPs]) ---------------------------------------------------------------------------------- Computed Over Various Sample Periods Various Measures of Realizations Transformation: Level Lag Length for IAR(p), DAR(p), and DARM(p) Models: AIC Last Updated: 09/01/2009 10:22 ---------------------------------------------------------------------------------- H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Initial Release 1985:01-2007:01 1 -0.04 0.08 0.10 0.24 0.25 0.25 0.55 89 89 2 -0.15 0.32 0.45 0.54 0.55 0.52 0.79 89 89 3 -0.30 0.65 0.83 0.72 0.77 0.72 0.87 89 89 4 -0.47 0.94 1.20 0.83 0.88 0.81 0.95 89 89 5 -0.60 1.18 1.50 0.89 0.92 0.85 0.99 89 89 1985:01-1996:04 1 -0.02 0.08 0.10 0.23 0.22 0.22 0.48 48 48 2 -0.15 0.38 0.48 0.55 0.51 0.48 0.82 48 48 3 -0.33 0.76 0.92 0.76 0.76 0.69 0.87 48 48 4 -0.53 1.07 1.31 0.88 0.89 0.82 1.00 48 48 5 -0.67 1.29 1.59 0.93 0.92 0.85 1.04 48 48 1997:01-2007:01 1 -0.05 0.08 0.10 0.24 0.32 0.32 0.68 41 41 2 -0.16 0.25 0.40 0.52 0.67 0.62 0.74 41 41 3 -0.27 0.52 0.72 0.65 0.80 0.79 0.86 41 41 4 -0.40 0.79 1.05 0.75 0.87 0.81 0.88 41 41 5 -0.52 1.05 1.39 0.84 0.93 0.86 0.92 41 41 H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: One Qtr After Initial Release 1985:01-2007:01 1 -0.04 0.08 0.10 0.24 0.25 0.25 0.55 89 89 2 -0.15 0.32 0.45 0.54 0.55 0.52 0.79 89 89 3 -0.30 0.65 0.83 0.72 0.77 0.72 0.87 89 89 4 -0.47 0.94 1.20 0.83 0.88 0.81 0.95 89 89 5 -0.60 1.18 1.50 0.89 0.92 0.85 0.99 89 89 1985:01-1996:04 1 -0.02 0.08 0.10 0.23 0.22 0.22 0.48 48 48 2 -0.15 0.38 0.48 0.55 0.51 0.48 0.82 48 48 3 -0.33 0.76 0.92 0.76 0.76 0.69 0.87 48 48 4 -0.53 1.07 1.31 0.88 0.89 0.82 1.00 48 48 5 -0.67 1.29 1.59 0.93 0.92 0.85 1.04 48 48 1997:01-2007:01 1 -0.05 0.08 0.10 0.24 0.32 0.32 0.68 41 41 2 -0.16 0.25 0.40 0.52 0.67 0.62 0.74 41 41 3 -0.27 0.52 0.72 0.65 0.80 0.79 0.86 41 41 4 -0.40 0.79 1.05 0.75 0.87 0.81 0.88 41 41 5 -0.52 1.05 1.39 0.84 0.93 0.86 0.92 41 41 H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Five Qtrs After Initial Release 1985:01-2007:01 1 -0.04 0.08 0.10 0.24 0.25 0.25 0.55 89 89 2 -0.15 0.32 0.45 0.54 0.55 0.52 0.79 89 89 3 -0.30 0.65 0.83 0.72 0.77 0.72 0.87 89 89 4 -0.47 0.94 1.20 0.83 0.88 0.81 0.95 89 89 5 -0.60 1.18 1.50 0.89 0.92 0.85 0.99 89 89 1985:01-1996:04 1 -0.02 0.08 0.10 0.23 0.22 0.22 0.48 48 48 2 -0.15 0.38 0.48 0.55 0.51 0.48 0.82 48 48 3 -0.33 0.76 0.92 0.76 0.76 0.69 0.87 48 48 4 -0.53 1.07 1.31 0.88 0.89 0.82 1.00 48 48 5 -0.67 1.29 1.59 0.93 0.92 0.85 1.04 48 48 1997:01-2007:01 1 -0.05 0.08 0.10 0.24 0.32 0.32 0.68 41 41 2 -0.16 0.25 0.40 0.52 0.67 0.62 0.74 41 41 3 -0.27 0.52 0.72 0.65 0.80 0.79 0.86 41 41 4 -0.40 0.79 1.05 0.75 0.87 0.81 0.88 41 41 5 -0.52 1.05 1.39 0.84 0.93 0.86 0.92 41 41 H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Nine Qtrs After Initial Release 1985:01-2007:01 1 -0.04 0.08 0.10 0.24 0.25 0.25 0.55 89 89 2 -0.15 0.32 0.45 0.54 0.55 0.52 0.79 89 89 3 -0.30 0.65 0.83 0.72 0.77 0.72 0.87 89 89 4 -0.47 0.94 1.20 0.83 0.88 0.81 0.95 89 89 5 -0.60 1.18 1.50 0.89 0.92 0.85 0.99 89 89 1985:01-1996:04 1 -0.02 0.08 0.10 0.23 0.22 0.22 0.48 48 48 2 -0.15 0.38 0.48 0.55 0.51 0.48 0.82 48 48 3 -0.33 0.76 0.92 0.76 0.76 0.69 0.87 48 48 4 -0.53 1.07 1.31 0.88 0.89 0.82 1.00 48 48 5 -0.67 1.29 1.59 0.93 0.92 0.85 1.04 48 48 1997:01-2007:01 1 -0.05 0.08 0.10 0.24 0.32 0.32 0.68 41 41 2 -0.16 0.25 0.40 0.52 0.67 0.62 0.74 41 41 3 -0.27 0.52 0.72 0.65 0.80 0.79 0.86 41 41 4 -0.40 0.79 1.05 0.75 0.87 0.81 0.88 41 41 5 -0.52 1.05 1.39 0.84 0.93 0.86 0.92 41 41 H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Latest Vintage 1985:01-2007:01 1 -0.04 0.08 0.10 0.24 0.25 0.25 0.55 89 89 2 -0.15 0.32 0.45 0.54 0.55 0.52 0.79 89 89 3 -0.30 0.65 0.83 0.72 0.77 0.72 0.87 89 89 4 -0.47 0.94 1.20 0.83 0.88 0.81 0.95 89 89 5 -0.60 1.18 1.50 0.89 0.92 0.85 0.99 89 89 1985:01-1996:04 1 -0.02 0.08 0.10 0.23 0.22 0.22 0.48 48 48 2 -0.15 0.38 0.48 0.55 0.51 0.48 0.82 48 48 3 -0.33 0.76 0.92 0.76 0.76 0.69 0.87 48 48 4 -0.53 1.07 1.31 0.88 0.89 0.82 1.00 48 48 5 -0.67 1.29 1.59 0.93 0.92 0.85 1.04 48 48 1997:01-2007:01 1 -0.05 0.08 0.10 0.24 0.32 0.32 0.68 41 41 2 -0.16 0.25 0.40 0.52 0.67 0.62 0.74 41 41 3 -0.27 0.52 0.72 0.65 0.80 0.79 0.86 41 41 4 -0.40 0.79 1.05 0.75 0.87 0.81 0.88 41 41 5 -0.52 1.05 1.39 0.84 0.93 0.86 0.92 41 41 Table 1 notes. (1) The forecast horizon is given by H, where H = 1 is the SPF forecast for the current quarter. (2) The headers ME(S), MAE(S), and RMSE(S) are mean error, mean absolute error, and root-mean-square error for the SPF. (3) The header RMSE(S/NC) is the ratio of the SPF RMSE to that of the no-change (NC) model. (4) The headers RMSE(S/IAR), RMSE(S/DAR) and RMSE(S/DARM) are the ratios of the SPF RMSE to the RMSE of the iterated and direct autoregressive models and the direct autoregressive model augmented with monthly observations, respectively. All models are estimated on a rolling window of 60 observations from the Phila Fed real-time data set. (5) The headers Nspf and N are the number of observations analyzed for the SPF and benchmark models. (6) When the variable forecast is a growth rate or an interest rate, it is expressed in annualized percentage points. When the variable forecast is the unemployment rate, it is expressed in percentage points. (7) Sample periods refer to the dates forecast, not the dates when the forecasts were made. Source: Tom Stark, Research Department, FRB Philadelphia. -------------------------------------------------------------------- Table 2. Recent SPF Forecasts - Dated at the Quarter Forecast -------------------------------------------------------------------- Variable: TBILL (3-Month T-Bill Rate [QA,PPs]) By Forecast Step (1 to 5) Transformation: Level Last Updated: 09/01/2009 10:22 -------------------------------------------------------------------- Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2002:04 1.400 1.700 2.325 2.450 3.105 2003:01 1.200 1.300 1.900 2.730 2.955 2003:02 1.170 1.250 1.410 2.280 3.300 2003:03 0.950 1.200 1.450 1.790 2.700 2003:04 0.960 1.020 1.280 1.800 2.100 2004:01 0.910 1.025 1.120 1.430 2.300 2004:02 1.000 1.000 1.120 1.310 1.770 2004:03 1.445 1.300 1.200 1.250 1.600 2004:04 1.910 1.765 1.574 1.500 1.500 2005:01 2.500 2.260 2.150 2.000 1.830 2005:02 3.000 2.890 2.520 2.500 2.364 2005:03 3.395 3.432 3.200 2.870 2.900 2005:04 3.900 3.852 3.775 3.500 3.253 2006:01 4.364 4.294 4.087 3.950 3.700 2006:02 4.800 4.600 4.500 4.275 4.060 2006:03 5.075 5.000 4.650 4.560 4.305 2006:04 4.962 5.150 4.981 4.650 4.560 2007:01 5.000 4.962 5.160 4.960 4.635 2007:02 4.985 4.995 4.900 5.065 4.850 2007:03 4.830 4.953 4.950 4.830 4.953 2007:04 3.950 4.850 4.920 4.900 4.750 2008:01 2.407 3.945 4.900 4.900 4.900 2008:02 1.510 2.225 3.910 4.840 4.900 2008:03 1.785 1.683 2.200 3.925 4.845 2008:04 0.650 1.850 1.750 2.300 4.100 2009:01 0.200 0.697 2.060 1.920 2.580 2009:02 0.200 0.276 0.800 2.300 2.200 2009:03 0.200 0.250 0.300 1.000 2.640 2009:04 NA 0.236 0.304 0.400 1.180 2010:01 NA NA 0.280 0.400 0.575 2010:02 NA NA NA 0.416 0.500 2010:03 NA NA NA NA 0.710 Table 2 notes. (1) Each column gives the sequence of SPF projections for a given forecast step. The forecast steps range from one (the forecast for the quarter in which the survey was conducted) to four quarters in the future (step 5). (2) The dates listed in the rows are the dates forecast, not the dates when the forecasts were made, with the exception of the forecast at step one, for which the two dates coincide. Source: Tom Stark, Research Department, FRB Philadelphia. ------------------------------------------------------------------------------------ Table 3. Recent Benchmark Model 1 IAR Forecasts - Dated at the Quarter Forecast ------------------------------------------------------------------------------------ Variable: TBILL (3-Month T-Bill Rate [QA,PPs]) By Forecast Step (1 to 5) Transformation: Level Lag Length for IAR(p): AIC Last Updated: 09/01/2009 10:22 ------------------------------------------------------------------------------------ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2002:04 1.767 2.131 2.438 1.014 3.728 2003:01 1.277 2.015 2.601 2.875 1.367 2003:02 1.186 1.392 2.329 3.021 3.110 2003:03 1.103 1.351 1.615 2.669 3.503 2003:04 0.988 1.283 1.599 1.898 3.007 2004:01 1.035 1.150 1.533 1.890 2.209 2004:02 1.039 1.213 1.373 1.818 2.197 2004:03 1.294 1.240 1.439 1.624 2.114 2004:04 1.823 1.517 1.459 1.692 1.882 2005:01 2.448 2.146 1.770 1.700 1.936 2005:02 3.016 2.850 2.477 2.011 1.936 2005:03 3.179 3.403 3.186 2.737 2.226 2005:04 3.699 3.423 3.685 3.431 2.956 2006:01 4.086 3.893 3.555 3.867 3.613 2006:02 4.736 4.310 4.072 3.658 3.962 2006:03 4.860 4.914 4.429 4.133 3.710 2006:04 4.975 4.904 4.975 4.428 4.123 2007:01 4.847 4.947 4.870 4.956 4.395 2007:02 4.983 4.760 4.855 4.785 4.888 2007:03 4.519 4.930 4.655 4.725 4.671 2007:04 3.990 4.335 4.845 4.546 4.577 2008:01 2.746 3.776 4.183 4.743 4.440 2008:02 1.077 2.337 3.643 4.063 4.635 2008:03 1.458 0.472 2.124 3.572 3.971 2008:04 1.542 1.480 0.188 2.067 3.547 2009:01 -0.077 1.721 1.642 0.175 2.126 2009:02 0.386 -0.058 1.982 1.894 0.377 2009:03 -0.090 0.243 -0.260 2.285 2.195 2009:04 NA 0.261 0.685 0.000 2.600 2010:01 NA NA 0.611 1.205 0.468 2010:02 NA NA NA 0.893 1.564 2010:03 NA NA NA NA 1.414 Table 3 notes. (1) Each column gives the sequence of benchmark IAR projections for a given forecast step. The forecast steps range from one to five. The first step corresponds to the forecast that SPF panelists make for the quarter in which the survey is conducted. (2) The dates listed in the rows are the dates forecast, not the dates when the forecasts were made, with the exception of the forecast at step one, for which the two dates coincide. (3) The IAR benchmark model is estimated on a fixed 60-quarter rolling window. Its forecasts are computed with the indirect method. Estimation uses data from the Philadelphia Fed real-time data set. Source: Tom Stark, Research Department, FRB Philadelphia. --------------------------------------------------------------------------------------- Table 4. Recent Benchmark Model 2 No-Change Forecasts - Dated at the Quarter Forecast --------------------------------------------------------------------------------------- Variable: TBILL (3-Month T-Bill Rate [QA,PPs]) By Forecast Step (1 to 5) Transformation: Level Last Updated: 09/01/2009 10:22 --------------------------------------------------------------------------------------- Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2002:04 1.643 1.717 1.723 1.907 3.170 2003:01 1.333 1.643 1.717 1.723 1.907 2003:02 1.157 1.333 1.643 1.717 1.723 2003:03 1.040 1.157 1.333 1.643 1.717 2003:04 0.930 1.040 1.157 1.333 1.643 2004:01 0.917 0.930 1.040 1.157 1.333 2004:02 0.917 0.917 0.930 1.040 1.157 2004:03 1.077 0.917 0.917 0.930 1.040 2004:04 1.487 1.077 0.917 0.917 0.930 2005:01 2.007 1.487 1.077 0.917 0.917 2005:02 2.537 2.007 1.487 1.077 0.917 2005:03 2.863 2.537 2.007 1.487 1.077 2005:04 3.360 2.863 2.537 2.007 1.487 2006:01 3.827 3.360 2.863 2.537 2.007 2006:02 4.393 3.827 3.360 2.863 2.537 2006:03 4.703 4.393 3.827 3.360 2.863 2006:04 4.907 4.703 4.393 3.827 3.360 2007:01 4.903 4.907 4.703 4.393 3.827 2007:02 4.983 4.903 4.907 4.703 4.393 2007:03 4.737 4.983 4.903 4.907 4.703 2007:04 4.303 4.737 4.983 4.903 4.907 2008:01 3.390 4.303 4.737 4.983 4.903 2008:02 2.043 3.390 4.303 4.737 4.983 2008:03 1.627 2.043 3.390 4.303 4.737 2008:04 1.493 1.627 2.043 3.390 4.303 2009:01 0.297 1.493 1.627 2.043 3.390 2009:02 0.213 0.297 1.493 1.627 2.043 2009:03 0.173 0.213 0.297 1.493 1.627 2009:04 NA 0.173 0.213 0.297 1.493 2010:01 NA NA 0.173 0.213 0.297 2010:02 NA NA NA 0.173 0.213 2010:03 NA NA NA NA 0.173 Table 4 notes. (1) Each column gives the sequence of benchmark no-change projections for a given forecast step. The forecast steps range from one to five. The first step corresponds to the forecast that SPF panelists make for the quarter in which the survey is conducted. (2) The dates listed in the rows are the dates forecast, not the dates when the forecasts were made, with the exception of the forecast at step one, for which the two dates coincide. (3) The projections use data from the Philadelphia Fed real-time data set. Source: Tom Stark, Research Department, FRB Philadelphia. ------------------------------------------------------------------------------------ Table 5. Recent Benchmark Model 3 DAR Forecasts - Dated at the Quarter Forecast ------------------------------------------------------------------------------------ Variable: TBILL (3-Month T-Bill Rate [QA,PPs]) By Forecast Step (1 to 5) Transformation: Level Lag Length for DAR(p): AIC Last Updated: 09/01/2009 10:22 ------------------------------------------------------------------------------------ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2002:04 1.767 2.281 2.466 1.922 3.733 2003:01 1.277 2.188 2.628 2.961 2.409 2003:02 1.186 1.584 2.531 3.169 3.464 2003:03 1.103 1.373 1.843 2.913 3.623 2003:04 0.988 1.309 1.759 2.366 3.474 2004:01 1.035 1.230 1.630 2.289 2.794 2004:02 1.039 1.295 1.503 2.101 2.896 2004:03 1.294 1.313 1.577 1.869 2.685 2004:04 1.823 1.570 1.534 1.933 2.372 2005:01 2.448 2.187 1.838 1.878 2.306 2005:02 3.016 2.833 2.473 2.141 2.241 2005:03 3.179 3.396 3.075 2.738 2.493 2005:04 3.699 3.424 3.561 3.250 3.018 2006:01 4.086 3.889 3.597 3.674 3.445 2006:02 4.736 4.312 4.219 3.662 3.764 2006:03 4.860 4.955 4.506 4.298 3.702 2006:04 4.975 4.942 5.097 4.537 4.243 2007:01 4.847 4.978 4.955 5.043 4.504 2007:02 4.983 4.730 4.947 4.850 4.930 2007:03 4.519 4.927 4.599 4.809 4.694 2007:04 3.990 4.323 4.802 4.483 4.637 2008:01 2.746 3.723 4.143 4.653 4.328 2008:02 1.077 2.359 3.534 3.980 4.525 2008:03 1.458 0.536 2.105 3.460 3.865 2008:04 1.542 1.394 0.407 2.117 3.403 2009:01 -0.077 1.717 1.532 0.622 2.277 2009:02 0.386 -0.351 1.928 1.893 0.981 2009:03 -0.090 0.155 -0.754 2.258 2.250 2009:04 NA 0.694 1.056 -0.496 2.638 2010:01 NA NA 0.957 1.639 0.221 2010:02 NA NA NA 1.572 2.157 2010:03 NA NA NA NA 2.184 Table 5 notes. (1) Each column gives the sequence of benchmark DAR projections for a given forecast step. The forecast steps range from one to five. The first step corresponds to the forecast that SPF panelists make for the quarter in which the survey is conducted. (2) The dates listed in the rows are the dates forecast, not the dates when the forecasts were made, with the exception of the forecast at step one, for which the two dates coincide. (3) The DAR benchmark model is estimated on a fixed 60-quarter rolling window. Its forecasts are computed with the direct method. Estimation uses data from the Philadelphia Fed real-time data set. Source: Tom Stark, Research Department, FRB Philadelphia. ------------------------------------------------------------------------------------ Table 6. Recent Benchmark Model 4 DARM Forecasts - Dated at the Quarter Forecast ------------------------------------------------------------------------------------ Variable: TBILL (3-Month T-Bill Rate [QA,PPs]) By Forecast Step (1 to 5) Transformation: Level Lag Length for DARM(p): AIC Last Updated: 09/01/2009 10:22 ------------------------------------------------------------------------------------ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2002:04 1.699 2.115 2.444 2.579 2.454 2003:01 1.227 2.042 2.483 3.009 2.849 2003:02 1.192 1.475 2.348 2.867 3.447 2003:03 0.939 1.394 1.677 2.759 3.367 2003:04 0.997 1.112 1.726 2.177 3.214 2004:01 0.946 1.203 1.341 2.227 2.667 2004:02 1.028 1.171 1.455 1.752 2.841 2004:03 1.451 1.231 1.405 1.799 2.342 2004:04 1.896 1.715 1.479 1.718 2.289 2005:01 2.506 2.236 2.021 1.784 2.086 2005:02 2.958 2.900 2.539 2.347 2.145 2005:03 3.392 3.298 3.164 2.791 2.538 2005:04 3.833 3.700 3.542 3.352 2.813 2006:01 4.399 4.076 3.855 3.408 3.261 2006:02 4.700 4.713 4.397 3.987 3.470 2006:03 5.068 4.915 4.831 4.519 3.940 2006:04 4.948 5.243 5.068 4.806 4.317 2007:01 5.013 4.907 5.195 5.019 4.734 2007:02 4.872 4.944 4.897 5.134 4.596 2007:03 4.832 4.808 4.823 4.783 4.884 2007:04 3.813 4.776 4.665 4.727 4.478 2008:01 2.552 3.426 4.588 4.493 4.654 2008:02 0.975 2.109 3.286 4.513 4.246 2008:03 1.648 0.381 1.833 3.213 4.596 2008:04 0.603 1.624 0.214 1.789 3.253 2009:01 0.136 0.569 1.774 0.431 2.224 2009:02 0.198 0.046 0.637 2.149 1.223 2009:03 0.227 0.152 -0.175 0.778 2.745 2009:04 NA 0.572 0.863 0.165 1.358 2010:01 NA NA 0.900 1.416 0.783 2010:02 NA NA NA 1.408 1.944 2010:03 NA NA NA NA 2.036 Table 6 notes. (1) Each column gives the sequence of benchmark DARM projections for a given forecast step. The forecast steps range from one to five. The first step corresponds to the forecast that SPF panelists make for the quarter in which the survey is conducted. (2) The dates listed in the rows are the dates forecast, not the dates when the forecasts were made, with the exception of the forecast at step one, for which the two dates coincide. (3) The DARM benchmark model is estimated on a fixed 60-quarter rolling window. Its forecasts are computed with the direct method and incorporate recent monthly values of the dependent variable. Estimation uses data from the Philadelphia Fed real-time data set. Source: Tom Stark, Research Department, FRB Philadelphia. ------------------------------------------------------------ Table 7. Recent Realizations (Various Measures) Source: Philadelphia Fed Real-Time Data Set ------------------------------------------------------------ Variable: TBILL (3-Month T-Bill Rate [QA,PPs]) Transformation: Level Last Updated: 09/01/2009 10:22 1- Initial Release 2- One Qtr After Initial Release 3- Five Qtrs After Initial Release 4- Nine Qtrs After Initial Release 5- Latest Vintage ------------------------------------------------------------- Obs. Date (1) (2) (3) (4) (5) 2002:04 1.333 1.333 1.333 1.333 1.333 2003:01 1.157 1.157 1.157 1.157 1.157 2003:02 1.040 1.040 1.040 1.040 1.040 2003:03 0.930 0.930 0.930 0.930 0.930 2003:04 0.917 0.917 0.917 0.917 0.917 2004:01 0.917 0.917 0.917 0.917 0.917 2004:02 1.077 1.077 1.077 1.077 1.077 2004:03 1.487 1.487 1.487 1.487 1.487 2004:04 2.007 2.007 2.007 2.007 2.007 2005:01 2.537 2.537 2.537 2.537 2.537 2005:02 2.863 2.863 2.863 2.863 2.863 2005:03 3.360 3.360 3.360 3.360 3.360 2005:04 3.827 3.827 3.827 3.827 3.827 2006:01 4.393 4.393 4.393 4.393 4.393 2006:02 4.703 4.703 4.703 4.703 4.703 2006:03 4.907 4.907 4.907 4.907 4.907 2006:04 4.903 4.903 4.903 4.903 4.903 2007:01 4.983 4.983 4.983 4.983 4.983 2007:02 4.737 4.737 4.737 NA 4.737 2007:03 4.303 4.303 4.303 NA 4.303 2007:04 3.390 3.390 3.390 NA 3.390 2008:01 2.043 2.043 2.043 NA 2.043 2008:02 1.627 1.627 NA NA 1.627 2008:03 1.493 1.493 NA NA 1.493 2008:04 0.297 0.297 NA NA 0.297 2009:01 0.213 0.213 NA NA 0.213 2009:02 0.173 NA NA NA 0.173 2009:03 NA NA NA NA NA 2009:04 NA NA NA NA NA 2010:01 NA NA NA NA NA 2010:02 NA NA NA NA NA 2010:03 NA NA NA NA NA Table 7 notes. (1) Each column reports a sequence of realizations from the Philadelphia Fed real-time data set. (2) The date listed in each row is the observation date. (3) Moving across a particular row shows how the observation is revised in subsequent releases. Source: Tom Stark, Research Department, FRB Philadelphia.