Error Statistics for the Survey of Professional Forecasters for Real Business Fixed Investment Release Date: 05/24/2024 Tom Stark Assistant Vice President | Assistant Director Real-Time Data Research Center Economic Research Department Federal Reserve Bank of Philadelphia Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics 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. Tables 1A-1B report 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 periods shown in Table 1A. 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. Tables 1A-1B report four such comparisons. They show 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. Table 1A reports the root-mean-square-error ratios using as many observations as possible for each model. The number of observations can differ from model to model. We first compute the RMSE for each model. We then construct the ratio. Table 1B reports RMSE ratios after we adjust the samples to include only the observations common to both models in the pair. Accordingly, the ratios reported in Table 1B may differ slightly from those of Table 1A, depending on the availability of sufficient real-time observations for estimating the benchmark models or for computing the errors of the SPF or benchmark forecasts. Table 1B also reports three two-sided p-values for each ratio. The p-values, corrected for the presence of heteroskedasticity and serial correlation in the time series of differences in squared forecast errors, are those for the test of equality of mean-square error between the SPF and the benchmark. The p-values are those for: (1) The Diebold-Mariano statistic (July 1995, Journal of Business and Economic Statistics), using a uniform lag window with the truncation lag set to the forecast horizon minus unity. When the uniform lag window produces a nonpositive standard error, the Bartlett window is used. (2) The Harvey-Leybourne-Newbold correction (1997, International Journal of Forecasting) to the Diebold-Mariano statistic. (3) The Diebold-Mariano statistic, using a Bartlett lag window with the truncation lag increased four quarters beyond that of (1) and (2). 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. In Tables 1A - 1B, 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 1A 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 Tables 1A - 1B. Any 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. They 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 postdates by one month the vintage that SPF panelists would have had at their disposal when they formed their projections. 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 in the vintage dated 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 squared root-mean-square error. The latter is estimated over the aforementioned sample, using the measure of history listed in the footnote. 4. SPECIAL TREATMENT OF COVID-19 EXTREME HISTORICAL OBSERVATIONS. Many macroeconomic variables experienced extreme values in 2020:Q2 and 2020:Q3 due to the partial shutdown of the U.S. economy at the end of March 2020. In some cases, these extreme values adversely affected the estimation of parameters in the benchmark models. The effect produced unrealistic parameter values in the models. In some cases, the models became dynamically unstable in sample periods that encompassed the 2020:Q2 and 2020:Q3 observations, leading to distorted forecast error statistics in those benchmark models for some survey variables. Comparisons with the SPF projections became hard to defend. Beginning with the error statistics published following the 2022:Q1 survey, we proceed in the following way. For some survey variables, we scale back the magnitude of the historical extreme observations for 2020:Q2 and 2020:Q3. We make these adjustment before the estimation of parameters in the benchmark models but reverse the adjustment prior to forecasting the models. The intent is to prevent adverse, unrealistic effects on the parameter estimates while allowing the extreme historical observations to condition the projections. These adjustments to extreme historical observations do not change the historical realizations used to compute forecast errors or forecast error statistics: We continue to use unadjusted, official U.S. government data for these purposes. The survey variables for which we make the adjustments are: Nominal GDP, Unemployment, Employment, Industrial Production, Real GDP, and Real Personal Consumption Expenditures. We do not adjust the historical values for the remaining variables. _________________________________________________________________________________________________ Table 1A. Forecast Error Summary Statistics for SPF Variable: RNRESIN (Real Business Fixed Investment) _________________________________________________________________________________________________ Computed Over Various Sample Periods Various Measures of Realizations Transformation: Q/Q Growth Rate Lag Length for IAR(p), DAR(p), and DARM(p) Models: AIC Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 05/24/2024 12:50 _________________________________________________________________________________________________ H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Initial Release 1985:01-2021:04 1 -0.09 5.11 6.68 0.64 0.73 0.73 NA 147 146 2 -0.48 6.07 8.13 0.74 0.85 0.87 NA 147 146 3 -0.99 6.43 8.61 0.75 0.90 0.91 NA 147 146 4 -0.80 6.70 8.93 0.75 0.90 0.91 NA 147 146 5 -0.72 6.99 9.15 0.68 0.90 0.93 NA 147 146 1985:01-1996:04 1 1.24 5.52 7.07 0.58 0.70 0.70 NA 47 46 2 1.36 6.45 8.10 0.75 0.84 0.83 NA 47 46 3 1.24 6.83 8.36 0.73 0.86 0.86 NA 47 46 4 1.42 6.89 8.58 0.79 0.84 0.88 NA 47 46 5 1.53 7.54 8.85 0.63 0.85 0.91 NA 47 47 1997:01-2021:04 1 -0.71 4.92 6.49 0.68 0.74 0.74 NA 100 100 2 -1.34 5.90 8.14 0.74 0.86 0.89 NA 100 100 3 -2.04 6.24 8.73 0.76 0.92 0.93 NA 100 100 4 -1.84 6.62 9.09 0.73 0.93 0.93 NA 100 100 5 -1.77 6.73 9.29 0.71 0.93 0.94 NA 100 99 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 1985:01-2021:04 1 0.66 5.10 6.86 0.66 0.73 0.73 NA 148 147 2 0.29 6.03 8.24 0.74 0.84 0.85 NA 148 147 3 -0.21 6.36 8.71 0.74 0.88 0.88 NA 148 147 4 -0.03 6.54 8.96 0.73 0.87 0.87 NA 148 147 5 0.05 6.91 9.25 0.68 0.88 0.89 NA 148 147 1985:01-1996:04 1 1.91 5.87 7.52 0.65 0.76 0.76 NA 48 47 2 2.05 6.59 8.26 0.71 0.80 0.80 NA 48 47 3 1.97 6.74 8.59 0.73 0.87 0.84 NA 48 47 4 2.13 6.58 8.63 0.78 0.82 0.83 NA 48 47 5 2.23 7.24 8.85 0.63 0.82 0.84 NA 48 48 1997:01-2021:04 1 0.07 4.73 6.51 0.66 0.72 0.72 NA 100 100 2 -0.56 5.76 8.23 0.76 0.86 0.89 NA 100 100 3 -1.26 6.17 8.77 0.74 0.89 0.90 NA 100 100 4 -1.06 6.51 9.11 0.71 0.90 0.89 NA 100 100 5 -1.00 6.74 9.43 0.70 0.91 0.92 NA 100 99 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 1985:01-2021:04 1 0.64 5.13 6.64 0.64 0.72 0.72 NA 148 147 2 0.26 6.12 8.14 0.72 0.84 0.85 NA 148 147 3 -0.24 6.33 8.51 0.73 0.88 0.88 NA 148 147 4 -0.05 6.57 8.72 0.71 0.87 0.86 NA 148 147 5 0.03 6.98 9.08 0.67 0.88 0.89 NA 148 147 1985:01-1996:04 1 1.62 6.18 7.62 0.64 0.74 0.74 NA 48 47 2 1.76 7.15 8.63 0.72 0.83 0.82 NA 48 47 3 1.68 7.07 8.74 0.72 0.87 0.84 NA 48 47 4 1.84 7.04 8.75 0.75 0.83 0.83 NA 48 47 5 1.94 7.79 9.18 0.64 0.86 0.87 NA 48 48 1997:01-2021:04 1 0.17 4.63 6.12 0.65 0.70 0.70 NA 100 100 2 -0.46 5.63 7.89 0.73 0.84 0.87 NA 100 100 3 -1.16 5.98 8.40 0.73 0.88 0.89 NA 100 100 4 -0.96 6.35 8.71 0.69 0.89 0.88 NA 100 100 5 -0.89 6.60 9.03 0.68 0.90 0.90 NA 100 99 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 1985:01-2021:04 1 -0.37 4.82 6.34 0.63 0.71 0.71 NA 148 147 2 -0.75 5.84 7.96 0.73 0.84 0.85 NA 148 147 3 -1.25 6.22 8.40 0.75 0.89 0.90 NA 148 147 4 -1.06 6.36 8.66 0.73 0.89 0.89 NA 148 147 5 -0.98 6.66 9.03 0.68 0.90 0.92 NA 148 147 1985:01-1996:04 1 0.70 5.90 7.28 0.62 0.73 0.73 NA 48 47 2 0.84 6.80 8.44 0.71 0.84 0.83 NA 48 47 3 0.76 7.15 8.60 0.73 0.88 0.88 NA 48 47 4 0.92 6.93 8.54 0.74 0.84 0.86 NA 48 47 5 1.02 7.38 8.93 0.62 0.86 0.91 NA 48 48 1997:01-2021:04 1 -0.88 4.31 5.84 0.64 0.69 0.69 NA 100 100 2 -1.51 5.39 7.72 0.74 0.84 0.87 NA 100 100 3 -2.21 5.77 8.30 0.76 0.90 0.91 NA 100 100 4 -2.01 6.09 8.72 0.73 0.91 0.90 NA 100 100 5 -1.94 6.31 9.07 0.71 0.92 0.92 NA 100 99 H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Latest Vintage 1985:01-2021:04 1 0.17 4.15 5.29 0.57 0.67 0.67 NA 148 147 2 -0.21 4.97 6.72 0.67 0.81 0.83 NA 148 147 3 -0.71 5.21 7.08 0.67 0.85 0.86 NA 148 147 4 -0.52 5.36 7.37 0.67 0.86 0.85 NA 148 147 5 -0.44 5.70 7.73 0.64 0.87 0.87 NA 148 147 1985:01-1996:04 1 0.20 4.14 5.04 0.51 0.69 0.69 NA 48 47 2 0.35 4.71 5.70 0.59 0.75 0.76 NA 48 47 3 0.27 4.81 5.76 0.57 0.79 0.78 NA 48 47 4 0.43 4.70 5.84 0.63 0.76 0.77 NA 48 47 5 0.53 5.20 6.25 0.54 0.76 0.79 NA 48 48 1997:01-2021:04 1 0.15 4.15 5.41 0.60 0.67 0.67 NA 100 100 2 -0.47 5.09 7.16 0.70 0.83 0.86 NA 100 100 3 -1.18 5.40 7.63 0.71 0.87 0.89 NA 100 100 4 -0.98 5.68 8.00 0.69 0.90 0.87 NA 100 100 5 -0.91 5.94 8.35 0.68 0.90 0.90 NA 100 99 Notes for Table 1A. (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 Philadelphia 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 1B. Ratios of Root-Mean-Square Errors for SPF Variable: RNRESIN (Real Business Fixed Investment) Alternative P-Values in Parentheses ________________________________________________________________________________________________________ Computed Over Various Sample Periods Various Measures of Realizations Transformation: Q/Q Growth Rate Lag Length for IAR(p), DAR(p), and DARM(p) Models: AIC Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 05/24/2024 12:50 ________________________________________________________________________________________________________ History: Initial Release 1985:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.641 0.725 0.725 NA 146 146 146 NA (0.000) (0.001) (0.001) ( NA ) (0.000) (0.001) (0.001) ( NA ) (0.000) (0.003) (0.003) ( NA ) 2 0.745 0.855 0.869 NA 146 146 146 NA (0.003) (0.018) (0.019) ( NA ) (0.004) (0.021) (0.022) ( NA ) (0.002) (0.025) (0.021) ( NA ) 3 0.751 0.897 0.909 NA 146 146 146 NA (0.006) (0.033) (0.040) ( NA ) (0.007) (0.038) (0.045) ( NA ) (0.002) (0.040) (0.042) ( NA ) 4 0.749 0.905 0.916 NA 146 146 146 NA (0.006) (0.045) (0.046) ( NA ) (0.008) (0.053) (0.054) ( NA ) (0.004) (0.049) (0.054) ( NA ) 5 0.681 0.904 0.931 NA 146 146 146 NA (0.001) (0.041) (0.102) ( NA ) (0.001) (0.050) (0.115) ( NA ) (0.001) (0.039) (0.135) ( NA ) History: Initial Release 1985:01-1996:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.581 0.697 0.697 NA 46 46 46 NA (0.000) (0.000) (0.000) ( NA ) (0.000) (0.001) (0.001) ( NA ) (0.000) (0.003) (0.003) ( NA ) 2 0.756 0.844 0.833 NA 46 46 46 NA (0.065) (0.125) (0.068) ( NA ) (0.081) (0.145) (0.084) ( NA ) (0.048) (0.080) (0.025) ( NA ) 3 0.731 0.854 0.859 NA 46 46 46 NA (0.000) (0.097) (0.109) ( NA ) (0.000) (0.124) (0.137) ( NA ) (0.000) (0.100) (0.093) ( NA ) 4 0.803 0.853 0.887 NA 46 46 46 NA (0.064) (0.138) (0.218) ( NA ) (0.094) (0.178) (0.261) ( NA ) (0.054) (0.131) (0.212) ( NA ) 5 0.631 0.855 0.913 NA 47 47 47 NA (0.002) (0.166) (0.361) ( NA ) (0.006) (0.217) (0.413) ( NA ) (0.001) (0.149) (0.399) ( NA ) History: Initial Release 1997:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.682 0.741 0.741 NA 100 100 100 NA (0.021) (0.030) (0.030) ( NA ) (0.023) (0.033) (0.033) ( NA ) (0.017) (0.049) (0.049) ( NA ) 2 0.740 0.861 0.888 NA 100 100 100 NA (0.016) (0.067) (0.107) ( NA ) (0.020) (0.075) (0.116) ( NA ) (0.014) (0.103) (0.141) ( NA ) 3 0.761 0.917 0.933 NA 100 100 100 NA (0.059) (0.147) (0.173) ( NA ) (0.069) (0.160) (0.187) ( NA ) (0.037) (0.166) (0.192) ( NA ) 4 0.729 0.930 0.928 NA 100 100 100 NA (0.020) (0.162) (0.113) ( NA ) (0.028) (0.180) (0.129) ( NA ) (0.015) (0.187) (0.140) ( NA ) 5 0.707 0.928 0.939 NA 99 99 99 NA (0.018) (0.119) (0.162) ( NA ) (0.027) (0.140) (0.185) ( NA ) (0.020) (0.126) (0.206) ( NA ) History: One Qtr After Initial Release 1985:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.657 0.732 0.732 NA 147 147 147 NA (0.000) (0.001) (0.001) ( NA ) (0.000) (0.001) (0.001) ( NA ) (0.000) (0.003) (0.003) ( NA ) 2 0.743 0.844 0.857 NA 147 147 147 NA (0.001) (0.007) (0.007) ( NA ) (0.001) (0.008) (0.008) ( NA ) (0.001) (0.013) (0.010) ( NA ) 3 0.736 0.875 0.879 NA 147 147 147 NA (0.004) (0.013) (0.008) ( NA ) (0.005) (0.016) (0.010) ( NA ) (0.002) (0.016) (0.007) ( NA ) 4 0.731 0.876 0.877 NA 147 147 147 NA (0.004) (0.012) (0.007) ( NA ) (0.006) (0.016) (0.009) ( NA ) (0.002) (0.013) (0.009) ( NA ) 5 0.681 0.884 0.897 NA 147 147 147 NA (0.001) (0.011) (0.008) ( NA ) (0.001) (0.015) (0.011) ( NA ) (0.001) (0.010) (0.014) ( NA ) History: One Qtr After Initial Release 1985:01-1996:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.647 0.762 0.762 NA 47 47 47 NA (0.000) (0.002) (0.002) ( NA ) (0.000) (0.004) (0.004) ( NA ) (0.000) (0.005) (0.005) ( NA ) 2 0.720 0.809 0.805 NA 47 47 47 NA (0.019) (0.025) (0.012) ( NA ) (0.028) (0.036) (0.019) ( NA ) (0.047) (0.046) (0.014) ( NA ) 3 0.717 0.853 0.832 NA 47 47 47 NA (0.000) (0.115) (0.070) ( NA ) (0.000) (0.143) (0.093) ( NA ) (0.001) (0.125) (0.051) ( NA ) 4 0.784 0.830 0.839 NA 47 47 47 NA (0.013) (0.119) (0.107) ( NA ) (0.026) (0.156) (0.143) ( NA ) (0.010) (0.109) (0.108) ( NA ) 5 0.634 0.823 0.839 NA 48 48 48 NA (0.008) (0.084) (0.028) ( NA ) (0.019) (0.124) (0.053) ( NA ) (0.004) (0.073) (0.050) ( NA ) History: One Qtr After Initial Release 1997:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.664 0.716 0.716 NA 100 100 100 NA (0.015) (0.018) (0.018) ( NA ) (0.017) (0.020) (0.020) ( NA ) (0.012) (0.029) (0.029) ( NA ) 2 0.755 0.863 0.886 NA 100 100 100 NA (0.014) (0.068) (0.100) ( NA ) (0.017) (0.075) (0.109) ( NA ) (0.012) (0.090) (0.118) ( NA ) 3 0.744 0.885 0.901 NA 100 100 100 NA (0.050) (0.055) (0.049) ( NA ) (0.059) (0.064) (0.058) ( NA ) (0.030) (0.062) (0.056) ( NA ) 4 0.711 0.898 0.895 NA 100 100 100 NA (0.019) (0.040) (0.025) ( NA ) (0.026) (0.050) (0.033) ( NA ) (0.013) (0.053) (0.033) ( NA ) 5 0.704 0.914 0.925 NA 99 99 99 NA (0.016) (0.043) (0.080) ( NA ) (0.024) (0.056) (0.098) ( NA ) (0.017) (0.044) (0.102) ( NA ) History: Five Qtrs After Initial Release 1985:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.643 0.718 0.718 NA 147 147 147 NA (0.000) (0.000) (0.000) ( NA ) (0.000) (0.001) (0.001) ( NA ) (0.000) (0.003) (0.003) ( NA ) 2 0.723 0.837 0.849 NA 147 147 147 NA (0.000) (0.002) (0.002) ( NA ) (0.001) (0.003) (0.002) ( NA ) (0.001) (0.006) (0.004) ( NA ) 3 0.727 0.876 0.873 NA 147 147 147 NA (0.001) (0.005) (0.004) ( NA ) (0.002) (0.006) (0.005) ( NA ) (0.001) (0.005) (0.002) ( NA ) 4 0.715 0.871 0.866 NA 147 147 147 NA (0.002) (0.005) (0.002) ( NA ) (0.003) (0.007) (0.003) ( NA ) (0.001) (0.005) (0.003) ( NA ) 5 0.668 0.886 0.896 NA 147 147 147 NA (0.001) (0.008) (0.009) ( NA ) (0.001) (0.011) (0.012) ( NA ) (0.001) (0.008) (0.017) ( NA ) History: Five Qtrs After Initial Release 1985:01-1996:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.640 0.742 0.742 NA 47 47 47 NA (0.000) (0.000) (0.000) ( NA ) (0.000) (0.001) (0.001) ( NA ) (0.000) (0.004) (0.004) ( NA ) 2 0.716 0.834 0.822 NA 47 47 47 NA (0.016) (0.019) (0.006) ( NA ) (0.025) (0.027) (0.010) ( NA ) (0.030) (0.027) (0.004) ( NA ) 3 0.712 0.861 0.833 NA 47 47 47 NA (0.000) (0.056) (0.056) ( NA ) (0.000) (0.076) (0.076) ( NA ) (0.001) (0.063) (0.033) ( NA ) 4 0.762 0.841 0.834 NA 47 47 47 NA (0.001) (0.080) (0.057) ( NA ) (0.004) (0.112) (0.085) ( NA ) (0.001) (0.072) (0.063) ( NA ) 5 0.643 0.860 0.874 NA 48 48 48 NA (0.008) (0.088) (0.073) ( NA ) (0.020) (0.129) (0.111) ( NA ) (0.005) (0.083) (0.116) ( NA ) History: Five Qtrs After Initial Release 1997:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.646 0.701 0.701 NA 100 100 100 NA (0.011) (0.015) (0.015) ( NA ) (0.013) (0.018) (0.018) ( NA ) (0.013) (0.037) (0.037) ( NA ) 2 0.727 0.838 0.865 NA 100 100 100 NA (0.009) (0.027) (0.042) ( NA ) (0.012) (0.031) (0.047) ( NA ) (0.007) (0.052) (0.065) ( NA ) 3 0.734 0.883 0.895 NA 100 100 100 NA (0.036) (0.034) (0.028) ( NA ) (0.044) (0.041) (0.034) ( NA ) (0.020) (0.033) (0.026) ( NA ) 4 0.695 0.886 0.883 NA 100 100 100 NA (0.015) (0.024) (0.014) ( NA ) (0.021) (0.032) (0.020) ( NA ) (0.008) (0.030) (0.020) ( NA ) 5 0.682 0.900 0.907 NA 99 99 99 NA (0.013) (0.035) (0.054) ( NA ) (0.020) (0.047) (0.069) ( NA ) (0.014) (0.037) (0.069) ( NA ) History: Nine Qtrs After Initial Release 1985:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.630 0.707 0.707 NA 147 147 147 NA (0.000) (0.001) (0.001) ( NA ) (0.000) (0.001) (0.001) ( NA ) (0.000) (0.004) (0.004) ( NA ) 2 0.726 0.844 0.855 NA 147 147 147 NA (0.001) (0.006) (0.004) ( NA ) (0.001) (0.008) (0.005) ( NA ) (0.001) (0.013) (0.007) ( NA ) 3 0.751 0.892 0.900 NA 147 147 147 NA (0.003) (0.028) (0.025) ( NA ) (0.004) (0.032) (0.029) ( NA ) (0.001) (0.026) (0.021) ( NA ) 4 0.732 0.889 0.891 NA 147 147 147 NA (0.002) (0.018) (0.012) ( NA ) (0.003) (0.022) (0.015) ( NA ) (0.001) (0.015) (0.014) ( NA ) 5 0.680 0.898 0.921 NA 147 147 147 NA (0.000) (0.017) (0.042) ( NA ) (0.001) (0.022) (0.051) ( NA ) (0.000) (0.015) (0.062) ( NA ) History: Nine Qtrs After Initial Release 1985:01-1996:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.617 0.727 0.727 NA 47 47 47 NA (0.000) (0.001) (0.001) ( NA ) (0.000) (0.002) (0.002) ( NA ) (0.000) (0.006) (0.006) ( NA ) 2 0.709 0.848 0.837 NA 47 47 47 NA (0.012) (0.034) (0.009) ( NA ) (0.018) (0.046) (0.015) ( NA ) (0.018) (0.034) (0.006) ( NA ) 3 0.731 0.881 0.873 NA 47 47 47 NA (0.000) (0.106) (0.119) ( NA ) (0.000) (0.132) (0.146) ( NA ) (0.001) (0.101) (0.085) ( NA ) 4 0.747 0.850 0.870 NA 47 47 47 NA (0.001) (0.077) (0.131) ( NA ) (0.004) (0.109) (0.169) ( NA ) (0.001) (0.057) (0.139) ( NA ) 5 0.618 0.859 0.911 NA 48 48 48 NA (0.004) (0.080) (0.277) ( NA ) (0.012) (0.120) (0.330) ( NA ) (0.002) (0.076) (0.329) ( NA ) History: Nine Qtrs After Initial Release 1997:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.640 0.694 0.694 NA 100 100 100 NA (0.014) (0.023) (0.023) ( NA ) (0.016) (0.025) (0.025) ( NA ) (0.021) (0.049) (0.049) ( NA ) 2 0.736 0.842 0.865 NA 100 100 100 NA (0.015) (0.047) (0.060) ( NA ) (0.018) (0.053) (0.067) ( NA ) (0.010) (0.076) (0.084) ( NA ) 3 0.761 0.898 0.914 NA 100 100 100 NA (0.055) (0.109) (0.102) ( NA ) (0.064) (0.121) (0.114) ( NA ) (0.031) (0.106) (0.106) ( NA ) 4 0.726 0.909 0.901 NA 100 100 100 NA (0.023) (0.098) (0.044) ( NA ) (0.031) (0.114) (0.054) ( NA ) (0.015) (0.097) (0.049) ( NA ) 5 0.717 0.919 0.926 NA 99 99 99 NA (0.010) (0.089) (0.078) ( NA ) (0.016) (0.107) (0.096) ( NA ) (0.013) (0.081) (0.097) ( NA ) History: Latest Vintage 1985:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.568 0.673 0.673 NA 147 147 147 NA (0.000) (0.002) (0.002) ( NA ) (0.000) (0.002) (0.002) ( NA ) (0.000) (0.007) (0.007) ( NA ) 2 0.667 0.806 0.830 NA 147 147 147 NA (0.000) (0.002) (0.003) ( NA ) (0.000) (0.003) (0.004) ( NA ) (0.000) (0.006) (0.006) ( NA ) 3 0.670 0.853 0.857 NA 147 147 147 NA (0.001) (0.011) (0.003) ( NA ) (0.002) (0.014) (0.004) ( NA ) (0.000) (0.012) (0.004) ( NA ) 4 0.673 0.861 0.849 NA 147 147 147 NA (0.003) (0.005) (0.001) ( NA ) (0.004) (0.007) (0.002) ( NA ) (0.002) (0.006) (0.002) ( NA ) 5 0.643 0.869 0.874 NA 147 147 147 NA (0.000) (0.008) (0.002) ( NA ) (0.001) (0.011) (0.004) ( NA ) (0.000) (0.008) (0.005) ( NA ) History: Latest Vintage 1985:01-1996:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.506 0.681 0.681 NA 47 47 47 NA (0.000) (0.004) (0.004) ( NA ) (0.000) (0.007) (0.007) ( NA ) (0.000) (0.011) (0.011) ( NA ) 2 0.585 0.744 0.760 NA 47 47 47 NA (0.001) (0.013) (0.014) ( NA ) (0.003) (0.020) (0.021) ( NA ) (0.009) (0.033) (0.021) ( NA ) 3 0.564 0.783 0.766 NA 47 47 47 NA (0.000) (0.044) (0.008) ( NA ) (0.000) (0.063) (0.016) ( NA ) (0.000) (0.040) (0.010) ( NA ) 4 0.623 0.752 0.765 NA 47 47 47 NA (0.002) (0.048) (0.024) ( NA ) (0.007) (0.074) (0.042) ( NA ) (0.002) (0.041) (0.015) ( NA ) 5 0.539 0.764 0.795 NA 48 48 48 NA (0.001) (0.083) (0.040) ( NA ) (0.005) (0.122) (0.069) ( NA ) (0.001) (0.072) (0.054) ( NA ) History: Latest Vintage 1997:01-2021:04 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.601 0.669 0.669 NA 100 100 100 NA (0.007) (0.018) (0.018) ( NA ) (0.008) (0.020) (0.020) ( NA ) (0.010) (0.039) (0.039) ( NA ) 2 0.699 0.828 0.855 NA 100 100 100 NA (0.008) (0.025) (0.034) ( NA ) (0.011) (0.029) (0.039) ( NA ) (0.005) (0.044) (0.050) ( NA ) 3 0.709 0.874 0.886 NA 100 100 100 NA (0.038) (0.064) (0.045) ( NA ) (0.046) (0.074) (0.053) ( NA ) (0.021) (0.070) (0.049) ( NA ) 4 0.687 0.897 0.875 NA 100 100 100 NA (0.022) (0.041) (0.016) ( NA ) (0.029) (0.051) (0.023) ( NA ) (0.015) (0.051) (0.023) ( NA ) 5 0.684 0.906 0.899 NA 99 99 99 NA (0.013) (0.035) (0.021) ( NA ) (0.019) (0.047) (0.030) ( NA ) (0.014) (0.043) (0.038) ( NA ) Notes for Table 1B. (1) The forecast horizon is given by H, where H = 1 is the SPF forecast for the current quarter. (2) The headers RMSE(S/NC), RMSE(S/IAR), RMSE(S/DAR), and RMSE(S/DARM) are the ratios of the SPF root-mean-square error to that of the benchmark models: No-change (NC), indirect autoregression (IAR), direct autoregession (DAR), and direct autoregression augmented with monthly information (DARM). These statistics may differ slightly from those reported in Table 1A because they incorporate only those observations common to both the SPF and the benchmark model. The previous statistics make use of all available observations for each model. (3) All models are estimated on a rolling window of 60 observations from the Philadelphia Fed real-time data set. (4) A set of three two-sided p-values (in parentheses) accompanies each statistic. These are the p-values for the test of the equality of mean-square-error. The first is for the Diebold-Mariano (1995, JBES) statistic, using a uniform lag window with the trunction lag set to the forecast horizon minus one. (The tables report the p-values using a Bartlett window when the uniform window produces a negative standard error.) The second is for the Harvey-Leybourne-Newbold (1997, IJF) correction to the Diebold-Mariano statistic. The third is for the Diebold-Mariano statistic, using a Bartlett lag window with the truncation lag increased four quarters. (5) The headers N1, N2, N3, and N4 show the number of observations used in constructing each ratio of root-mean-square errors. (6) 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: RNRESIN (Real Business Fixed Investment) By Forecast Step (1 to 5) Transformation: Q/Q Growth Rate Last Updated: 05/24/2024 12:50 ______________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2017:03 3.787 3.991 3.586 3.448 2.773 2017:04 4.432 4.134 4.321 4.458 3.256 2018:01 5.338 3.934 3.280 4.300 4.079 2018:02 5.531 5.276 4.092 4.000 4.355 2018:03 5.689 5.439 6.433 4.161 3.728 2018:04 5.949 4.986 5.559 4.880 3.792 2019:01 4.065 5.350 5.021 4.412 5.038 2019:02 3.960 4.259 4.421 4.323 4.759 2019:03 2.394 3.260 3.382 4.048 5.232 2019:04 1.824 2.252 3.631 3.881 4.099 2020:01 0.675 1.860 2.376 3.489 3.816 2020:02 -34.900 2.046 2.347 2.177 3.049 2020:03 -0.600 -7.856 2.730 1.006 2.350 2020:04 4.874 8.349 5.076 2.066 1.648 2021:01 6.775 2.329 8.067 7.713 2.724 2021:02 8.160 5.422 4.079 -1.552 2.599 2021:03 7.463 7.657 7.504 5.517 2.837 2021:04 7.507 7.169 9.395 5.646 4.483 2022:01 7.643 7.436 6.238 4.195 5.365 2022:02 7.021 5.999 5.909 5.521 5.563 2022:03 3.792 4.877 5.531 4.199 4.223 2022:04 1.683 3.366 4.188 5.037 3.329 2023:01 1.400 1.037 2.056 3.119 5.139 2023:02 0.300 1.458 0.197 1.192 3.997 2023:03 0.992 -0.704 1.235 1.910 2.832 2023:04 1.766 2.035 0.202 1.286 1.795 2024:01 2.225 1.167 0.685 1.933 2.287 2024:02 2.074 1.776 1.384 1.626 2.626 2024:03 NA 2.579 2.176 1.886 1.013 2024:04 NA NA 2.237 1.505 1.528 2025:01 NA NA NA 2.946 2.225 2025:02 NA NA NA NA 2.401 Notes for Table 2. (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: RNRESIN (Real Business Fixed Investment) By Forecast Step (1 to 5) Transformation: Q/Q Growth Rate Lag Length for IAR(p): AIC Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 05/24/2024 12:50 ______________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2017:03 4.618 6.187 2.920 2.327 1.735 2017:04 3.765 4.242 5.318 2.998 2.439 2018:01 5.816 3.677 3.994 4.735 3.050 2018:02 5.287 5.146 3.619 3.830 4.342 2018:03 6.284 4.771 4.707 3.581 3.722 2018:04 1.892 5.574 4.429 4.420 3.556 2019:01 5.519 2.593 5.103 4.203 4.232 2019:02 3.349 5.100 3.054 4.790 4.053 2019:03 0.679 3.757 4.823 3.358 4.583 2019:04 -1.032 1.580 4.028 4.639 3.559 2020:01 0.343 0.839 2.211 4.208 4.518 2020:02 -5.750 1.712 2.051 2.653 4.328 2020:03 -23.225 -2.497 2.538 2.738 2.963 2020:04 11.657 -15.334 -0.242 2.996 3.104 2021:01 8.481 7.449 -8.801 1.139 3.239 2021:02 6.477 5.949 5.394 -4.408 1.948 2021:03 5.759 4.835 4.739 4.391 -1.714 2021:04 2.524 4.611 4.048 4.161 3.901 2022:01 2.651 2.896 4.032 3.670 3.885 2022:02 6.314 2.964 3.083 3.739 3.489 2022:03 1.544 4.853 3.122 3.176 3.591 2022:04 3.342 2.329 4.122 3.200 3.223 2023:01 1.858 3.154 2.711 3.758 3.240 2023:02 1.924 2.409 3.065 2.898 3.575 2023:03 5.299 2.494 2.666 3.023 2.988 2023:04 1.927 4.193 2.762 2.786 3.003 2024:01 3.323 2.892 3.677 2.888 2.842 2024:02 4.247 3.927 3.357 3.436 2.947 2024:03 NA 4.657 4.190 3.582 3.324 2024:04 NA NA 4.781 4.304 3.690 2025:01 NA NA NA 4.818 4.354 2025:02 NA NA NA NA 4.830 Notes for Table 3. (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: RNRESIN (Real Business Fixed Investment) By Forecast Step (1 to 5) Transformation: Q/Q Growth Rate Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 05/24/2024 12:50 _________________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2017:03 5.187 9.398 2.393 1.140 -2.256 2017:04 3.898 5.187 9.398 2.393 1.140 2018:01 6.839 3.898 5.187 9.398 2.393 2018:02 6.068 6.839 3.898 5.187 9.398 2018:03 7.354 6.068 6.839 3.898 5.187 2018:04 0.829 7.354 6.068 6.839 3.898 2019:01 6.151 0.829 7.354 6.068 6.839 2019:02 2.735 6.151 0.829 7.354 6.068 2019:03 -0.606 2.735 6.151 0.829 7.354 2019:04 -2.968 -0.606 2.735 6.151 0.829 2020:01 -1.494 -2.968 -0.606 2.735 6.151 2020:02 -8.628 -1.494 -2.968 -0.606 2.735 2020:03 -27.029 -8.628 -1.494 -2.968 -0.606 2020:04 20.277 -27.029 -8.628 -1.494 -2.968 2021:01 13.777 20.277 -27.029 -8.628 -1.494 2021:02 9.903 13.777 20.277 -27.029 -8.628 2021:03 8.031 9.903 13.777 20.277 -27.029 2021:04 1.780 8.031 9.903 13.777 20.277 2022:01 2.026 1.780 8.031 9.903 13.777 2022:02 9.239 2.026 1.780 8.031 9.903 2022:03 -0.067 9.239 2.026 1.780 8.031 2022:04 3.742 -0.067 9.239 2.026 1.780 2023:01 0.677 3.742 -0.067 9.239 2.026 2023:02 0.711 0.677 3.742 -0.067 9.239 2023:03 7.668 0.711 0.677 3.742 -0.067 2023:04 -0.073 7.668 0.711 0.677 3.742 2024:01 1.938 -0.073 7.668 0.711 0.677 2024:02 2.891 1.938 -0.073 7.668 0.711 2024:03 NA 2.891 1.938 -0.073 7.668 2024:04 NA NA 2.891 1.938 -0.073 2025:01 NA NA NA 2.891 1.938 2025:02 NA NA NA NA 2.891 Notes for Table 4. (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: RNRESIN (Real Business Fixed Investment) By Forecast Step (1 to 5) Transformation: Q/Q Growth Rate Lag Length for DAR(p): AIC Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 05/24/2024 12:50 ______________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2017:03 4.618 5.973 2.745 2.280 2.368 2017:04 3.765 4.182 4.907 2.714 2.449 2018:01 5.816 3.670 3.870 3.872 2.685 2018:02 5.287 4.964 3.599 3.479 2.933 2018:03 6.284 4.695 4.445 3.440 3.128 2018:04 1.892 5.448 3.985 3.813 3.254 2019:01 5.519 2.943 4.989 3.814 3.299 2019:02 3.349 4.904 3.448 3.209 3.386 2019:03 0.679 3.786 4.643 2.138 4.014 2019:04 -1.032 2.289 3.962 5.159 4.437 2020:01 0.343 0.880 3.042 3.539 4.093 2020:02 -5.750 1.403 1.596 2.962 4.490 2020:03 -23.225 -1.969 2.330 3.579 4.553 2020:04 11.657 -10.827 -0.277 4.129 4.686 2021:01 8.481 9.569 -5.023 1.962 4.367 2021:02 6.477 5.799 7.045 -3.827 4.969 2021:03 5.759 4.842 5.383 20.401 6.256 2021:04 2.524 4.504 4.037 -0.376 1.076 2022:01 2.651 2.988 3.639 1.723 2.031 2022:02 6.314 3.001 3.283 3.094 2.355 2022:03 1.544 4.652 3.262 3.481 2.814 2022:04 3.342 2.479 3.628 3.435 3.490 2023:01 1.858 3.135 3.072 2.897 3.442 2023:02 1.924 2.465 3.061 3.526 2.879 2023:03 5.299 2.519 2.815 2.983 3.577 2023:04 1.927 4.112 2.850 3.202 2.994 2024:01 3.323 2.931 3.465 3.246 3.184 2024:02 4.247 3.774 3.399 2.695 3.219 2024:03 NA 4.515 3.988 3.919 2.790 2024:04 NA NA 4.586 4.166 3.739 2025:01 NA NA NA 4.700 4.094 2025:02 NA NA NA NA 4.610 Notes for Table 5. (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: RNRESIN (Real Business Fixed Investment) By Forecast Step (1 to 5) Transformation: Q/Q Growth Rate Lag Length for DARM(p): AIC Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 05/24/2024 12:50 ______________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2017:03 NA NA NA NA NA 2017:04 NA NA NA NA NA 2018:01 NA NA NA NA NA 2018:02 NA NA NA NA NA 2018:03 NA NA NA NA NA 2018:04 NA NA NA NA NA 2019:01 NA NA NA NA NA 2019:02 NA NA NA NA NA 2019:03 NA NA NA NA NA 2019:04 NA NA NA NA NA 2020:01 NA NA NA NA NA 2020:02 NA NA NA NA NA 2020:03 NA NA NA NA NA 2020:04 NA NA NA NA NA 2021:01 NA NA NA NA NA 2021:02 NA NA NA NA NA 2021:03 NA NA NA NA NA 2021:04 NA NA NA NA NA 2022:01 NA NA NA NA NA 2022:02 NA NA NA NA NA 2022:03 NA NA NA NA NA 2022:04 NA NA NA NA NA 2023:01 NA NA NA NA NA 2023:02 NA NA NA NA NA 2023:03 NA NA NA NA NA 2023:04 NA NA NA NA NA 2024:01 NA NA NA NA NA 2024:02 NA NA NA NA NA 2024:03 NA NA NA NA NA 2024:04 NA NA NA NA NA 2025:01 NA NA NA NA NA 2025:02 NA NA NA NA NA Notes for Table 6. (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) Philadelphia Fed Real-Time Data Set ______________________________________________________________________ Variable: RNRESIN (Real Business Fixed Investment) Transformation: Q/Q Growth Rate Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 05/24/2024 12:50 Column (1): Initial Release Column (2): One Qtr After Initial Release Column (3): Five Qtrs After Initial Release Column (4): Nine Qtrs After Initial Release Column (5): Latest Vintage _______________________________________________________________________ Obs. Date (1) (2) (3) (4) (5) 2017:03 3.898 4.652 3.442 2.437 3.905 2017:04 6.839 6.839 4.850 8.402 9.908 2018:01 6.068 11.508 8.838 12.179 11.678 2018:02 7.354 8.727 7.859 6.235 4.612 2018:03 0.829 2.518 2.098 2.641 2.490 2018:04 6.151 5.432 4.803 4.977 3.665 2019:01 2.735 4.429 4.177 4.693 2.128 2019:02 -0.606 -1.023 0.029 6.677 7.919 2019:03 -2.968 -2.271 1.875 2.933 4.324 2019:04 -1.494 -2.456 -0.344 -1.648 -1.573 2020:01 -8.628 -6.685 -8.096 -8.096 -7.698 2020:02 -27.029 -27.168 -30.333 -29.388 -28.589 2020:03 20.277 22.881 18.730 20.233 18.273 2020:04 13.777 13.083 12.533 11.499 10.475 2021:01 9.903 12.938 12.938 8.859 8.936 2021:02 8.031 9.227 9.881 9.706 9.706 2021:03 1.780 1.639 0.634 -1.277 -1.277 2021:04 2.026 2.901 1.127 2.695 2.695 2022:01 9.239 9.976 7.887 NA 10.702 2022:02 -0.067 0.069 5.297 NA 5.297 2022:03 3.742 6.203 4.736 NA 4.736 2022:04 0.677 3.991 1.748 NA 1.748 2023:01 0.711 0.604 NA NA 5.734 2023:02 7.668 7.441 NA NA 7.441 2023:03 -0.073 1.450 NA NA 1.450 2023:04 1.938 3.754 NA NA 3.754 2024:01 2.891 NA NA NA 2.891 2024:02 NA NA NA NA NA 2024:03 NA NA NA NA NA 2024:04 NA NA NA NA NA 2025:01 NA NA NA NA NA 2025:02 NA NA NA NA NA Notes for Table 7. (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.