Error Statistics for the Survey of Professional Forecasters for Real Residential Investment Release Date: 08/29/2021 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. _________________________________________________________________________________________________ Table 1A. Forecast Error Summary Statistics for SPF Variable: RRESINV (Real Residential 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: 08/29/2021 15:46 _________________________________________________________________________________________________ H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Initial Release 1985:01-2019:01 1 0.49 6.58 8.64 0.65 0.70 0.70 NA 136 135 2 -0.23 8.28 10.65 0.72 0.80 0.84 NA 136 135 3 -0.87 9.30 11.71 0.81 0.91 0.92 NA 136 135 4 -1.19 9.84 12.29 0.79 0.91 0.88 NA 136 135 5 -1.71 9.49 12.48 0.80 0.90 0.91 NA 136 135 1985:01-1996:04 1 0.75 6.63 8.56 0.64 0.71 0.71 NA 47 46 2 -0.01 7.39 9.59 0.60 0.74 0.78 NA 47 46 3 0.24 8.78 10.92 0.67 0.90 0.89 NA 47 46 4 0.57 9.23 11.32 0.65 0.93 0.94 NA 47 46 5 0.26 8.96 11.62 0.72 0.98 0.97 NA 47 47 1997:01-2019:01 1 0.35 6.55 8.68 0.66 0.69 0.69 NA 89 89 2 -0.35 8.75 11.16 0.79 0.83 0.87 NA 89 89 3 -1.46 9.57 12.10 0.90 0.91 0.93 NA 89 89 4 -2.11 10.16 12.77 0.87 0.90 0.86 NA 89 89 5 -2.75 9.77 12.91 0.85 0.87 0.89 NA 89 88 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 1985:01-2019:01 1 0.54 6.49 8.40 0.65 0.69 0.69 NA 137 136 2 -0.16 8.17 10.41 0.72 0.80 0.85 NA 137 136 3 -0.77 9.15 11.41 0.81 0.90 0.91 NA 137 136 4 -1.05 9.54 11.96 0.78 0.90 0.87 NA 137 136 5 -1.59 9.40 12.12 0.78 0.89 0.90 NA 137 136 1985:01-1996:04 1 0.54 6.93 8.37 0.67 0.72 0.72 NA 48 47 2 -0.14 7.69 9.60 0.63 0.76 0.80 NA 48 47 3 0.18 8.82 10.71 0.67 0.88 0.88 NA 48 47 4 0.58 9.02 11.11 0.65 0.91 0.91 NA 48 47 5 0.22 9.24 11.48 0.70 0.96 0.96 NA 48 48 1997:01-2019:01 1 0.53 6.26 8.42 0.64 0.67 0.67 NA 89 89 2 -0.17 8.42 10.82 0.78 0.82 0.87 NA 89 89 3 -1.28 9.33 11.76 0.91 0.91 0.93 NA 89 89 4 -1.93 9.82 12.39 0.86 0.90 0.85 NA 89 89 5 -2.57 9.49 12.45 0.83 0.86 0.87 NA 89 88 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 1985:01-2019:01 1 0.94 6.85 8.64 0.66 0.70 0.70 NA 137 136 2 0.24 8.33 10.57 0.74 0.81 0.86 NA 137 136 3 -0.37 9.22 11.53 0.84 0.91 0.92 NA 137 136 4 -0.65 9.47 12.09 0.78 0.90 0.87 NA 137 136 5 -1.19 9.44 12.30 0.78 0.89 0.90 NA 137 136 1985:01-1996:04 1 1.60 7.32 8.98 0.69 0.74 0.74 NA 48 47 2 0.92 8.09 10.17 0.66 0.78 0.82 NA 48 47 3 1.23 9.24 11.18 0.71 0.90 0.89 NA 48 47 4 1.64 9.18 11.58 0.66 0.93 0.93 NA 48 47 5 1.28 9.59 11.98 0.68 0.97 0.97 NA 48 48 1997:01-2019:01 1 0.58 6.60 8.45 0.64 0.67 0.67 NA 89 89 2 -0.13 8.46 10.77 0.79 0.83 0.88 NA 89 89 3 -1.24 9.21 11.71 0.93 0.92 0.93 NA 89 89 4 -1.89 9.62 12.35 0.87 0.89 0.84 NA 89 89 5 -2.53 9.36 12.47 0.84 0.86 0.87 NA 89 88 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 1985:01-2019:01 1 0.32 7.04 8.81 0.68 0.69 0.69 NA 137 136 2 -0.38 8.43 10.75 0.75 0.80 0.84 NA 137 136 3 -0.99 9.11 11.74 0.84 0.89 0.90 NA 137 136 4 -1.27 9.57 12.34 0.76 0.88 0.85 NA 137 136 5 -1.81 9.59 12.58 0.77 0.88 0.89 NA 137 136 1985:01-1996:04 1 0.70 7.61 9.17 0.74 0.75 0.75 NA 48 47 2 0.02 8.40 10.41 0.69 0.77 0.81 NA 48 47 3 0.33 9.40 11.41 0.70 0.86 0.86 NA 48 47 4 0.74 9.53 11.97 0.65 0.90 0.90 NA 48 47 5 0.37 10.12 12.50 0.67 0.95 0.96 NA 48 48 1997:01-2019:01 1 0.11 6.73 8.61 0.65 0.66 0.66 NA 89 89 2 -0.59 8.44 10.92 0.79 0.82 0.86 NA 89 89 3 -1.70 8.96 11.92 0.94 0.91 0.93 NA 89 89 4 -2.35 9.59 12.53 0.85 0.88 0.83 NA 89 89 5 -2.99 9.31 12.62 0.85 0.85 0.86 NA 89 88 H ME(S) MAE(S) RMSE(S) RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) Nspf N History: Latest Vintage 1985:01-2019:01 1 0.60 6.50 8.54 0.64 0.67 0.67 NA 137 136 2 -0.10 7.86 10.49 0.76 0.81 0.86 NA 137 136 3 -0.71 8.77 11.40 0.81 0.89 0.90 NA 137 136 4 -0.99 9.16 12.02 0.74 0.89 0.86 NA 137 136 5 -1.53 9.10 12.23 0.79 0.89 0.90 NA 137 136 1985:01-1996:04 1 0.94 6.61 8.35 0.63 0.68 0.68 NA 48 47 2 0.26 7.36 9.61 0.67 0.81 0.84 NA 48 47 3 0.57 8.53 10.46 0.68 0.89 0.89 NA 48 47 4 0.98 8.49 11.02 0.61 0.93 0.94 NA 48 47 5 0.62 8.82 11.33 0.66 0.98 0.98 NA 48 48 1997:01-2019:01 1 0.41 6.45 8.65 0.65 0.67 0.67 NA 89 89 2 -0.29 8.14 10.94 0.81 0.82 0.87 NA 89 89 3 -1.40 8.91 11.88 0.89 0.89 0.90 NA 89 89 4 -2.05 9.52 12.52 0.83 0.87 0.83 NA 89 89 5 -2.69 9.25 12.68 0.88 0.86 0.86 NA 89 88 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: RRESINV (Real Residential 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: 08/29/2021 15:46 ________________________________________________________________________________________________________ History: Initial Release 1985:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.655 0.697 0.697 NA 135 135 135 NA (0.000) (0.005) (0.005) ( NA ) (0.000) (0.006) (0.006) ( NA ) (0.008) (0.019) (0.019) ( NA ) 2 0.717 0.798 0.837 NA 135 135 135 NA (0.006) (0.016) (0.013) ( NA ) (0.007) (0.018) (0.015) ( NA ) (0.003) (0.026) (0.019) ( NA ) 3 0.813 0.909 0.920 NA 135 135 135 NA (0.053) (0.186) (0.188) ( NA ) (0.060) (0.197) (0.199) ( NA ) (0.036) (0.160) (0.144) ( NA ) 4 0.790 0.915 0.883 NA 135 135 135 NA (0.139) (0.333) (0.203) ( NA ) (0.152) (0.348) (0.217) ( NA ) (0.104) (0.251) (0.152) ( NA ) 5 0.807 0.905 0.916 NA 135 135 135 NA (0.074) (0.077) (0.157) ( NA ) (0.086) (0.090) (0.174) ( NA ) (0.074) (0.158) (0.188) ( 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 0.645 0.716 0.716 NA 46 46 46 NA (0.001) (0.003) (0.003) ( NA ) (0.002) (0.005) (0.005) ( NA ) (0.000) (0.001) (0.001) ( NA ) 2 0.594 0.734 0.771 NA 46 46 46 NA (0.003) (0.001) (0.003) ( NA ) (0.006) (0.003) (0.005) ( NA ) (0.002) (0.002) (0.004) ( NA ) 3 0.676 0.906 0.897 NA 46 46 46 NA (0.018) (0.125) (0.107) ( NA ) (0.030) (0.153) (0.135) ( NA ) (0.003) (0.154) (0.155) ( NA ) 4 0.659 0.941 0.945 NA 46 46 46 NA (0.044) (0.470) (0.508) ( NA ) (0.069) (0.508) (0.544) ( NA ) (0.023) (0.478) (0.516) ( NA ) 5 0.716 0.984 0.975 NA 47 47 47 NA (0.071) (0.841) (0.745) ( NA ) (0.109) (0.857) (0.770) ( NA ) (0.053) (0.832) (0.731) ( NA ) History: Initial Release 1997:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.660 0.688 0.688 NA 89 89 89 NA (0.012) (0.042) (0.042) ( NA ) (0.015) (0.047) (0.047) ( NA ) (0.082) (0.096) (0.096) ( NA ) 2 0.787 0.826 0.866 NA 89 89 89 NA (0.126) (0.132) (0.130) ( NA ) (0.136) (0.142) (0.140) ( NA ) (0.084) (0.165) (0.156) ( NA ) 3 0.902 0.911 0.930 NA 89 89 89 NA (0.438) (0.345) (0.401) ( NA ) (0.453) (0.361) (0.416) ( NA ) (0.407) (0.308) (0.332) ( NA ) 4 0.873 0.905 0.860 NA 89 89 89 NA (0.532) (0.419) (0.247) ( NA ) (0.550) (0.440) (0.269) ( NA ) (0.493) (0.327) (0.186) ( NA ) 5 0.858 0.876 0.894 NA 88 88 88 NA (0.314) (0.054) (0.156) ( NA ) (0.342) (0.071) (0.182) ( NA ) (0.326) (0.146) (0.195) ( NA ) History: One Qtr After Initial Release 1985:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.652 0.689 0.689 NA 136 136 136 NA (0.000) (0.004) (0.004) ( NA ) (0.000) (0.004) (0.004) ( NA ) (0.008) (0.016) (0.016) ( NA ) 2 0.722 0.799 0.842 NA 136 136 136 NA (0.005) (0.015) (0.014) ( NA ) (0.006) (0.017) (0.017) ( NA ) (0.003) (0.025) (0.023) ( NA ) 3 0.812 0.904 0.914 NA 136 136 136 NA (0.049) (0.158) (0.185) ( NA ) (0.055) (0.168) (0.195) ( NA ) (0.033) (0.137) (0.151) ( NA ) 4 0.782 0.905 0.872 NA 136 136 136 NA (0.109) (0.267) (0.160) ( NA ) (0.121) (0.281) (0.174) ( NA ) (0.081) (0.195) (0.117) ( NA ) 5 0.784 0.890 0.899 NA 136 136 136 NA (0.050) (0.059) (0.118) ( NA ) (0.061) (0.070) (0.133) ( NA ) (0.043) (0.107) (0.123) ( 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 0.675 0.723 0.723 NA 47 47 47 NA (0.004) (0.005) (0.005) ( NA ) (0.006) (0.007) (0.007) ( NA ) (0.000) (0.003) (0.003) ( NA ) 2 0.618 0.745 0.786 NA 47 47 47 NA (0.002) (0.005) (0.008) ( NA ) (0.004) (0.009) (0.014) ( NA ) (0.002) (0.006) (0.007) ( NA ) 3 0.673 0.887 0.880 NA 47 47 47 NA (0.012) (0.116) (0.102) ( NA ) (0.021) (0.143) (0.128) ( NA ) (0.002) (0.132) (0.131) ( NA ) 4 0.659 0.921 0.920 NA 47 47 47 NA (0.036) (0.349) (0.345) ( NA ) (0.058) (0.390) (0.387) ( NA ) (0.018) (0.357) (0.350) ( NA ) 5 0.696 0.964 0.956 NA 48 48 48 NA (0.059) (0.532) (0.461) ( NA ) (0.094) (0.574) (0.508) ( NA ) (0.038) (0.553) (0.481) ( NA ) History: One Qtr After Initial Release 1997:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.641 0.673 0.673 NA 89 89 89 NA (0.006) (0.029) (0.029) ( NA ) (0.007) (0.033) (0.033) ( NA ) (0.059) (0.077) (0.077) ( NA ) 2 0.783 0.824 0.869 NA 89 89 89 NA (0.117) (0.121) (0.131) ( NA ) (0.127) (0.131) (0.141) ( NA ) (0.082) (0.158) (0.167) ( NA ) 3 0.908 0.911 0.931 NA 89 89 89 NA (0.466) (0.345) (0.435) ( NA ) (0.480) (0.362) (0.450) ( NA ) (0.436) (0.313) (0.383) ( NA ) 4 0.862 0.898 0.853 NA 89 89 89 NA (0.480) (0.379) (0.227) ( NA ) (0.499) (0.400) (0.249) ( NA ) (0.441) (0.294) (0.171) ( NA ) 5 0.837 0.862 0.876 NA 88 88 88 NA (0.261) (0.064) (0.148) ( NA ) (0.289) (0.082) (0.173) ( NA ) (0.256) (0.119) (0.152) ( NA ) History: Five Qtrs After Initial Release 1985:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.658 0.696 0.696 NA 136 136 136 NA (0.000) (0.003) (0.003) ( NA ) (0.000) (0.003) (0.003) ( NA ) (0.005) (0.012) (0.012) ( NA ) 2 0.736 0.808 0.853 NA 136 136 136 NA (0.004) (0.005) (0.004) ( NA ) (0.005) (0.006) (0.005) ( NA ) (0.002) (0.012) (0.013) ( NA ) 3 0.840 0.913 0.917 NA 136 136 136 NA (0.087) (0.127) (0.179) ( NA ) (0.095) (0.137) (0.189) ( NA ) (0.061) (0.121) (0.162) ( NA ) 4 0.786 0.906 0.870 NA 136 136 136 NA (0.104) (0.221) (0.124) ( NA ) (0.115) (0.235) (0.137) ( NA ) (0.079) (0.161) (0.089) ( NA ) 5 0.778 0.896 0.901 NA 136 136 136 NA (0.051) (0.067) (0.157) ( NA ) (0.061) (0.079) (0.173) ( NA ) (0.038) (0.100) (0.139) ( 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 0.689 0.743 0.743 NA 47 47 47 NA (0.004) (0.002) (0.002) ( NA ) (0.006) (0.004) (0.004) ( NA ) (0.000) (0.001) (0.001) ( NA ) 2 0.648 0.765 0.806 NA 47 47 47 NA (0.006) (0.002) (0.005) ( NA ) (0.011) (0.005) (0.009) ( NA ) (0.005) (0.007) (0.010) ( NA ) 3 0.715 0.905 0.895 NA 47 47 47 NA (0.038) (0.147) (0.120) ( NA ) (0.056) (0.177) (0.147) ( NA ) (0.009) (0.196) (0.182) ( NA ) 4 0.668 0.934 0.934 NA 47 47 47 NA (0.054) (0.529) (0.506) ( NA ) (0.081) (0.563) (0.541) ( NA ) (0.029) (0.520) (0.494) ( NA ) 5 0.679 0.971 0.973 NA 48 48 48 NA (0.041) (0.702) (0.719) ( NA ) (0.070) (0.730) (0.746) ( NA ) (0.024) (0.702) (0.722) ( NA ) History: Five Qtrs After Initial Release 1997:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.640 0.672 0.672 NA 89 89 89 NA (0.003) (0.026) (0.026) ( NA ) (0.004) (0.029) (0.029) ( NA ) (0.048) (0.065) (0.065) ( NA ) 2 0.791 0.830 0.877 NA 89 89 89 NA (0.086) (0.074) (0.076) ( NA ) (0.095) (0.083) (0.085) ( NA ) (0.059) (0.112) (0.127) ( NA ) 3 0.930 0.916 0.928 NA 89 89 89 NA (0.555) (0.292) (0.407) ( NA ) (0.568) (0.309) (0.423) ( NA ) (0.531) (0.271) (0.371) ( NA ) 4 0.869 0.893 0.844 NA 89 89 89 NA (0.476) (0.288) (0.159) ( NA ) (0.495) (0.311) (0.179) ( NA ) (0.447) (0.210) (0.112) ( NA ) 5 0.847 0.865 0.871 NA 88 88 88 NA (0.313) (0.057) (0.160) ( NA ) (0.341) (0.075) (0.186) ( NA ) (0.290) (0.096) (0.139) ( NA ) History: Nine Qtrs After Initial Release 1985:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.679 0.694 0.694 NA 136 136 136 NA (0.001) (0.002) (0.002) ( NA ) (0.001) (0.003) (0.003) ( NA ) (0.014) (0.015) (0.015) ( NA ) 2 0.745 0.795 0.838 NA 136 136 136 NA (0.007) (0.003) (0.003) ( NA ) (0.008) (0.004) (0.004) ( NA ) (0.003) (0.008) (0.008) ( NA ) 3 0.840 0.897 0.905 NA 136 136 136 NA (0.113) (0.070) (0.122) ( NA ) (0.122) (0.078) (0.131) ( NA ) (0.079) (0.061) (0.108) ( NA ) 4 0.767 0.885 0.856 NA 136 136 136 NA (0.088) (0.139) (0.086) ( NA ) (0.098) (0.152) (0.097) ( NA ) (0.065) (0.088) (0.061) ( NA ) 5 0.773 0.885 0.892 NA 136 136 136 NA (0.056) (0.031) (0.099) ( NA ) (0.067) (0.039) (0.113) ( NA ) (0.040) (0.054) (0.093) ( 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 0.745 0.752 0.752 NA 47 47 47 NA (0.020) (0.004) (0.004) ( NA ) (0.026) (0.007) (0.007) ( NA ) (0.005) (0.010) (0.010) ( NA ) 2 0.673 0.755 0.791 NA 47 47 47 NA (0.022) (0.002) (0.005) ( NA ) (0.031) (0.005) (0.010) ( NA ) (0.010) (0.008) (0.014) ( NA ) 3 0.711 0.869 0.867 NA 47 47 47 NA (0.049) (0.058) (0.056) ( NA ) (0.069) (0.079) (0.077) ( NA ) (0.017) (0.085) (0.094) ( NA ) 4 0.652 0.905 0.912 NA 47 47 47 NA (0.037) (0.296) (0.329) ( NA ) (0.060) (0.339) (0.371) ( NA ) (0.016) (0.284) (0.308) ( NA ) 5 0.666 0.950 0.959 NA 48 48 48 NA (0.037) (0.301) (0.444) ( NA ) (0.065) (0.354) (0.491) ( NA ) (0.019) (0.332) (0.467) ( NA ) History: Nine Qtrs After Initial Release 1997:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.647 0.664 0.664 NA 89 89 89 NA (0.005) (0.020) (0.020) ( NA ) (0.007) (0.023) (0.023) ( NA ) (0.060) (0.066) (0.066) ( NA ) 2 0.786 0.816 0.863 NA 89 89 89 NA (0.084) (0.058) (0.066) ( NA ) (0.093) (0.066) (0.074) ( NA ) (0.053) (0.087) (0.095) ( NA ) 3 0.937 0.911 0.925 NA 89 89 89 NA (0.624) (0.262) (0.388) ( NA ) (0.635) (0.279) (0.404) ( NA ) (0.589) (0.227) (0.348) ( NA ) 4 0.851 0.876 0.832 NA 89 89 89 NA (0.451) (0.238) (0.135) ( NA ) (0.471) (0.260) (0.155) ( NA ) (0.422) (0.163) (0.099) ( NA ) 5 0.857 0.856 0.862 NA 88 88 88 NA (0.371) (0.044) (0.124) ( NA ) (0.398) (0.060) (0.148) ( NA ) (0.345) (0.075) (0.114) ( NA ) History: Latest Vintage 1985:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.640 0.671 0.671 NA 136 136 136 NA (0.000) (0.002) (0.002) ( NA ) (0.000) (0.002) (0.002) ( NA ) (0.009) (0.014) (0.014) ( NA ) 2 0.756 0.807 0.851 NA 136 136 136 NA (0.019) (0.014) (0.018) ( NA ) (0.021) (0.017) (0.021) ( NA ) (0.007) (0.022) (0.024) ( NA ) 3 0.814 0.890 0.901 NA 136 136 136 NA (0.064) (0.082) (0.129) ( NA ) (0.072) (0.090) (0.139) ( NA ) (0.048) (0.062) (0.095) ( NA ) 4 0.746 0.888 0.859 NA 136 136 136 NA (0.064) (0.174) (0.123) ( NA ) (0.073) (0.187) (0.135) ( NA ) (0.046) (0.117) (0.094) ( NA ) 5 0.797 0.896 0.899 NA 136 136 136 NA (0.092) (0.074) (0.163) ( NA ) (0.106) (0.087) (0.180) ( NA ) (0.071) (0.109) (0.154) ( 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 0.624 0.683 0.683 NA 47 47 47 NA (0.004) (0.000) (0.000) ( NA ) (0.006) (0.000) (0.000) ( NA ) (0.002) (0.004) (0.004) ( NA ) 2 0.657 0.787 0.819 NA 47 47 47 NA (0.021) (0.019) (0.028) ( NA ) (0.031) (0.028) (0.039) ( NA ) (0.006) (0.016) (0.031) ( NA ) 3 0.684 0.900 0.898 NA 47 47 47 NA (0.023) (0.202) (0.194) ( NA ) (0.037) (0.233) (0.225) ( NA ) (0.011) (0.216) (0.217) ( NA ) 4 0.615 0.936 0.951 NA 47 47 47 NA (0.033) (0.567) (0.664) ( NA ) (0.054) (0.599) (0.689) ( NA ) (0.015) (0.559) (0.653) ( NA ) 5 0.662 0.978 0.979 NA 48 48 48 NA (0.040) (0.757) (0.793) ( NA ) (0.069) (0.781) (0.813) ( NA ) (0.021) (0.747) (0.780) ( NA ) History: Latest Vintage 1997:01-2019:01 H RMSE(S/NC) RMSE(S/IAR) RMSE(S/DAR) RMSE(S/DARM) N1 N2 N3 N4 1 0.648 0.665 0.665 NA 89 89 89 NA (0.009) (0.026) (0.026) ( NA ) (0.011) (0.030) (0.030) ( NA ) (0.076) (0.076) (0.076) ( NA ) 2 0.809 0.816 0.865 NA 89 89 89 NA (0.164) (0.079) (0.101) ( NA ) (0.175) (0.087) (0.110) ( NA ) (0.113) (0.103) (0.121) ( NA ) 3 0.894 0.886 0.902 NA 89 89 89 NA (0.431) (0.169) (0.258) ( NA ) (0.446) (0.185) (0.275) ( NA ) (0.395) (0.132) (0.202) ( NA ) 4 0.832 0.870 0.828 NA 89 89 89 NA (0.377) (0.212) (0.130) ( NA ) (0.398) (0.234) (0.150) ( NA ) (0.345) (0.138) (0.096) ( NA ) 5 0.888 0.867 0.870 NA 88 88 88 NA (0.485) (0.062) (0.156) ( NA ) (0.510) (0.080) (0.182) ( NA ) (0.460) (0.102) (0.148) ( 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: RRESINV (Real Residential Investment) By Forecast Step (1 to 5) Transformation: Q/Q Growth Rate Last Updated: 08/29/2021 15:46 ______________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2014:04 6.750 7.018 9.747 14.713 9.054 2015:01 7.081 8.352 10.726 10.401 14.031 2015:02 8.750 8.890 9.092 8.680 10.984 2015:03 8.000 8.126 10.500 8.462 12.568 2015:04 8.175 8.659 7.843 10.535 7.905 2016:01 7.000 7.916 8.725 10.196 10.000 2016:02 7.641 8.392 7.585 5.873 7.532 2016:03 4.980 6.788 8.112 6.486 6.681 2016:04 4.000 6.761 7.772 6.000 7.644 2017:01 5.198 5.553 7.840 5.553 7.337 2017:02 4.897 4.789 6.073 6.688 7.246 2017:03 3.544 4.779 5.108 5.780 5.058 2017:04 3.000 2.581 4.884 4.589 5.632 2018:01 4.050 4.744 3.879 5.090 5.316 2018:02 3.500 3.175 4.015 3.955 5.062 2018:03 1.000 3.000 3.105 4.472 3.522 2018:04 1.000 2.862 3.000 3.124 2.892 2019:01 -0.500 2.000 2.124 3.345 3.199 2019:02 0.537 0.297 2.000 2.015 3.756 2019:03 1.248 1.398 1.105 1.593 3.367 2019:04 2.995 1.227 0.660 1.103 2.762 2020:01 3.978 2.501 1.175 0.699 1.566 2020:02 -44.622 2.032 1.507 1.532 1.805 2020:03 14.404 0.532 2.155 1.102 1.646 2020:04 6.901 7.509 3.507 2.112 2.311 2021:01 11.643 5.618 5.076 6.683 2.834 2021:02 6.763 6.809 4.286 2.426 7.212 2021:03 4.298 7.762 9.691 2.721 4.356 2021:04 NA 4.986 7.378 2.508 2.563 2022:01 NA NA 5.245 7.989 0.223 2022:02 NA NA NA 3.700 2.018 2022:03 NA NA NA NA 3.276 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: RRESINV (Real Residential 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: 08/29/2021 15:46 ______________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2014:04 1.227 -0.086 -3.304 -0.078 7.659 2015:01 4.008 2.845 1.232 -2.705 -0.733 2015:02 1.990 2.849 1.424 -0.303 -2.915 2015:03 6.989 1.697 2.706 1.268 -0.017 2015:04 6.843 6.243 1.175 2.367 1.279 2016:01 6.980 6.025 5.290 1.131 1.955 2016:02 9.269 6.359 5.205 4.917 0.919 2016:03 2.863 9.415 5.873 4.946 4.390 2016:04 -4.994 1.070 9.147 5.289 4.466 2017:01 0.515 -4.015 -0.216 7.742 4.850 2017:02 8.710 1.933 -3.228 0.931 7.310 2017:03 2.360 8.021 3.257 -2.603 0.536 2017:04 -1.215 1.499 6.192 1.800 -2.103 2018:01 0.488 -3.660 -0.742 5.250 2.009 2018:02 1.732 1.628 -2.936 0.597 4.303 2018:03 1.458 3.657 3.478 -2.222 0.219 2018:04 -2.817 -0.836 1.426 1.501 -2.392 2019:01 -2.763 -2.639 -0.287 1.642 1.633 2019:02 -3.340 -3.063 -2.919 -0.078 1.674 2019:03 -2.277 -3.269 -2.957 -2.683 -0.449 2019:04 0.074 -1.837 -2.945 -2.826 -2.642 2020:01 1.922 0.308 -2.002 -2.954 -2.835 2020:02 8.982 2.715 0.813 -2.093 -2.858 2020:03 -2.795 9.502 2.210 -0.058 -2.066 2020:04 -13.458 -8.420 9.278 1.437 -0.126 2021:01 11.108 18.118 -11.637 7.021 1.254 2021:02 34.226 32.854 9.231 -6.401 6.372 2021:03 2.098 21.494 19.607 4.308 -7.344 2021:04 NA -2.118 17.023 16.405 8.993 2022:01 NA NA 0.514 21.073 19.544 2022:02 NA NA NA -0.145 16.036 2022:03 NA NA NA NA 0.507 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: RRESINV (Real Residential 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: 08/29/2021 15:46 _________________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2014:04 1.869 7.540 -5.761 -9.794 14.617 2015:01 4.064 1.869 7.540 -5.761 -9.794 2015:02 1.275 4.064 1.869 7.540 -5.761 2015:03 6.638 1.275 4.064 1.869 7.540 2015:04 6.088 6.638 1.275 4.064 1.869 2016:01 8.173 6.088 6.638 1.275 4.064 2016:02 14.867 8.173 6.088 6.638 1.275 2016:03 -6.114 14.867 8.173 6.088 6.638 2016:04 -6.236 -6.114 14.867 8.173 6.088 2017:01 10.187 -6.236 -6.114 14.867 8.173 2017:02 13.670 10.187 -6.236 -6.114 14.867 2017:03 -6.821 13.670 10.187 -6.236 -6.114 2017:04 -5.988 -6.821 13.670 10.187 -6.236 2018:01 11.653 -5.988 -6.821 13.670 10.187 2018:02 0.066 11.653 -5.988 -6.821 13.670 2018:03 -1.036 0.066 11.653 -5.988 -6.821 2018:04 -3.983 -1.036 0.066 11.653 -5.988 2019:01 -3.507 -3.983 -1.036 0.066 11.653 2019:02 -2.834 -3.507 -3.983 -1.036 0.066 2019:03 -1.547 -2.834 -3.507 -3.983 -1.036 2019:04 5.068 -1.547 -2.834 -3.507 -3.983 2020:01 5.781 5.068 -1.547 -2.834 -3.507 2020:02 21.049 5.781 5.068 -1.547 -2.834 2020:03 -38.732 21.049 5.781 5.068 -1.547 2020:04 59.273 -38.732 21.049 5.781 5.068 2021:01 33.537 59.273 -38.732 21.049 5.781 2021:02 10.785 33.537 59.273 -38.732 21.049 2021:03 -9.750 10.785 33.537 59.273 -38.732 2021:04 NA -9.750 10.785 33.537 59.273 2022:01 NA NA -9.750 10.785 33.537 2022:02 NA NA NA -9.750 10.785 2022:03 NA NA NA NA -9.750 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: RRESINV (Real Residential 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: 08/29/2021 15:46 ______________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2014:04 1.227 0.497 -3.561 1.521 7.259 2015:01 4.008 3.464 3.577 -4.146 -4.719 2015:02 1.990 2.412 0.731 -0.943 -2.975 2015:03 6.989 1.594 1.974 2.921 2.973 2015:04 6.843 5.926 0.473 1.531 0.399 2016:01 6.980 5.663 3.613 1.340 1.433 2016:02 9.269 6.090 3.533 4.747 0.306 2016:03 2.863 9.384 4.763 4.681 2.984 2016:04 -4.994 0.474 8.613 4.829 3.000 2017:01 0.515 -4.813 -2.933 7.034 4.116 2017:02 8.710 2.516 -3.192 2.187 7.420 2017:03 2.360 8.269 5.702 -3.475 -2.124 2017:04 -1.215 1.182 7.472 0.923 -2.379 2018:01 0.488 -4.703 -3.518 6.203 5.054 2018:02 1.732 2.259 -3.267 2.241 6.598 2018:03 1.458 4.365 6.113 -3.782 -3.076 2018:04 -2.817 -1.695 0.057 1.046 -2.759 2019:01 -2.763 -2.420 -0.646 4.014 5.232 2019:02 -3.340 -3.104 -2.664 -1.600 -0.071 2019:03 -2.277 -3.210 -2.566 -1.847 -0.672 2019:04 0.074 -1.574 -2.299 -2.702 -2.326 2020:01 1.922 0.360 -1.648 -3.079 -2.357 2020:02 8.982 3.095 0.604 -1.807 -2.213 2020:03 -2.795 9.662 1.103 -0.989 -1.850 2020:04 -13.458 -9.562 8.528 1.841 1.079 2021:01 11.108 17.491 -10.913 5.670 1.623 2021:02 34.226 36.442 30.038 -2.365 8.307 2021:03 2.098 19.134 18.960 3.091 -17.823 2021:04 NA -3.445 6.326 27.698 24.752 2022:01 NA NA 2.648 14.584 15.019 2022:02 NA NA NA 3.041 5.507 2022:03 NA NA NA NA -3.587 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: RRESINV (Real Residential 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: 08/29/2021 15:46 ______________________________________________________________________________________________ Qtr Forecast Step 1 Step 2 Step 3 Step 4 Step 5 2014:04 NA NA NA NA NA 2015:01 NA NA NA NA NA 2015:02 NA NA NA NA NA 2015:03 NA NA NA NA NA 2015:04 NA NA NA NA NA 2016:01 NA NA NA NA NA 2016:02 NA NA NA NA NA 2016:03 NA NA NA NA NA 2016:04 NA NA NA NA NA 2017:01 NA NA NA NA NA 2017:02 NA NA NA NA NA 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 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: RRESINV (Real Residential Investment) Transformation: Q/Q Growth Rate Source for Historical Realizations: Bureau of Economic Analysis via Haver Analytics Last Updated: 08/29/2021 15:46 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) 2014:04 4.064 3.816 9.930 11.326 15.060 2015:01 1.275 10.119 13.356 11.421 8.078 2015:02 6.638 9.368 14.810 11.670 10.391 2015:03 6.088 8.178 12.624 10.634 11.229 2015:04 8.173 10.091 11.480 7.342 7.212 2016:01 14.867 7.819 13.420 13.648 12.850 2016:02 -6.114 -7.754 -4.763 -1.012 0.542 2016:03 -6.236 -4.147 -4.492 -1.686 -2.078 2016:04 10.187 9.598 7.079 7.710 5.122 2017:01 13.670 11.080 11.069 11.844 10.810 2017:02 -6.821 -7.321 -5.488 -2.267 2.574 2017:03 -5.988 -4.695 -0.462 -1.956 -1.935 2017:04 11.653 12.767 11.158 9.922 7.033 2018:01 0.066 -3.435 -5.245 -3.284 -4.232 2018:02 -1.036 -1.358 -3.737 -1.730 3.284 2018:03 -3.983 -3.540 -4.028 -5.393 -5.853 2018:04 -3.507 -4.719 -4.649 -5.214 -8.350 2019:01 -2.834 -1.075 -1.719 0.067 0.067 2019:02 -1.547 -2.943 -2.122 NA 4.135 2019:03 5.068 4.644 4.579 NA 3.617 2019:04 5.781 6.485 5.839 NA 1.052 2020:01 21.049 18.974 20.337 NA 20.337 2020:02 -38.732 -35.544 NA NA -30.761 2020:03 59.273 62.978 NA NA 59.974 2020:04 33.537 36.643 NA NA 34.353 2021:01 10.785 13.265 NA NA 13.265 2021:02 -9.750 NA NA NA -9.750 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 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.