# Forecasting US Inflation in Real Time

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

^{,}4 Moreover, while attention has recently shifted to the PCE price index, its construction by the BEA largely relies on source data from disaggregate CPI series collected by the BLS. This fact has implications for our forecasting exercise, because it implies that monthly CPI releases provide information about quarterly PCE price inflation that can be exploited in forecasting.5

^{,}6

## 3. Forecasting Methodology

#### 3.1. Model-Based Forecasts

#### 3.1.1. Autoregressive Model (AR)

#### 3.1.2. Inflation Gap Model (AR-Gap)

#### 3.1.3. Phillips Curve Models

**unifying conceptual framework**to think about how the different forecasting models use additional information to forecast inflation, one can consider the following extended version of the Phillips curve model that nests the AR, AR-Gap, and standard Phillips curve models described in the paragraphs above:

#### 3.1.4. Vector Autoregressive Model (VAR)

#### 3.2. Aggregating Forecasts of Disaggregate Inflation

#### 3.3. Judgmental Forecasts

#### 3.3.1. Survey of Professional Forecasters (SPF)

#### 3.3.2. Tealbook Forecasts of Federal Reserve Board Staff

#### 3.4. Forecast Combination

## 4. Results: Forecasting US PCE Inflation in Real Time

#### 4.1. Real-Time Analysis Including Public Tealbook Forecasts

#### 4.2. Pre- vs. Post-Crisis Analysis Including Public Tealbooks

#### 4.3. Full Sample through 2019

#### 4.4. Summary of Results

## 5. Remarks

- (1)
**Forecast encompassing, forecast combination and forecast accuracy tests:**Having the smallest RMSE comparisons of a set of forecasts is a necessary but not sufficient condition for forecast encompassing (see Ericsson (1992)). The concept of forecast encompassing has been proposed by Chong and Hendry (1986) and can be tested by investigating whether the forecast of the alternative forecast model can explain the forecast error of a benchmark forecast model of interest. We explore forecast combination as one possible forecast method. Forecast combination is closely related to the concept of forecast encompassing. Evidence that forecast combination of two forecasting models provides smaller RMSE than the benchmark model implies that the benchmark forecast does not encompass the alternative model forecast. Our result that forecast combination does improve over simple benchmark models and also over Phillips curve models does suggest that some of the alternative models contain additional predictive content. This is confirmed by our result of the Diebold and Mariano test that forecast combination significantly outperforms the simple benchmark model for horizons of one quarter and two years across all samples and most pre- and post-crisis subsamples. One extension for further research would be to apply the test suggested by Hubrich and West (2010) to compare small nested model sets via adjust MSFEs relevant to some of the comparisons, that can be viewed as a forecast encompassing test for small nested model sets.- (2)
**Other models**We have included (but do not present) in our forecast comparison a random walk model that has often been used as a benchmark model in the literature. We also considered forecasts based on an AR(1) model estimated with a rolling estimation window instead of a recursively expanding estimation window. We find that the rolling window AR model performs slightly better than the benchmark and all the other models for a one year horizon for the post-crisis period that includes the published Tealbook as well as the full post-crisis period, and performs better than the benchmark for most horizons for the pre-crisis period. Other than these few instances, neither of these models does outperform our benchmark inflation gap model in RMSE terms except at very few horizons and in those cases the improvement was negligible and in any case clearly outperformed by our best forecasting methods.- (3)
**RMSEs comparisons:**We compare the different forecast models and methods in terms of RMSE. As Clements and Hendry (1993) have pointed out, RMSE are not invariant to certain transformations. For example, different transformations (first differences or annual differences) might affect the RMSE ranking of the forecast models. We have focused on the forecast performance for quarterly inflation, and note that the RMSE based forecast comparison might be different for annual inflation. However, we choose out-of-sample RMSE comparisons because parameter estimation uncertainty and structural breaks often imply that good in-sample fit does not translate into out-of-sample forecasting (see, e.g., Clements and Hendry (1998); Giacomini and Rossi (2009)).- (4)
**SPF**It should be noted that the SPF is itself an average (or a median) and so may already benefit from any aggregation effects due to differentially misspecified models or methods by forecasters in the sample.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | |

2 | |

3 | While the CPI and PCE price index share a similar low-frequency evolution, differences in formula, weight, and scope—as discussed in, for example, McCully et al. (2007)—can result in persistent differences in measured inflation. One commonly noted implication of the formula effect is that the CPI—which employs a Laspeyres index concept—is slower to accommodate consumer substitution between goods, and so tends to increase at a faster pace than the PCE price index. |

4 | Indeed, the CPI was the only measure of inflation explicitly included in the projections of Federal Reserve Banks and Board members produced as part of the semi-annual Monetary Policy Report (MPR) to Congress during the period 1992–1999. We thank Neil Ericsson for pointing this out to us. |

5 | While our econometric forecasting models are specified at the quarterly frequency and so do not take this higher-frequency information into account, we include judgmental forecasts from the Survey of Professional Forecasters and Federal Reserve Board Tealbooks that do incorporate this information. Recent work on mixed frequency econometric models that can be used for the purpose of “nowcasting” inflation includes Modugno (2013) and Knotek and Zaman (2017). |

6 | It is worth noting that the raw price data that underlies both of these measures is known to be subject to measurement error, as documented in Shoemaker (2011) and Eichenbaum et al. (2014). While we are not able to correct for these errors, they primarily affect the most disaggregate inflation series, and are less of a concern for the high-level aggregates that we use for forecasting. |

7 | Indeed, the MPR (see Note 6) replaced overall PCE prices with core PCE prices in 2004, and the FOMC Summary of Economic Projections, introduced in 2007, includes both measures. |

8 | Croushore and Stark (2019) provide a recent overview of the details of this survey. |

9 | For example, if the estimate of the history of the unemployment gap was revised during a given quarter, then the specific timing of the forecast could have a small effect on models that include that variable. |

10 | Measurement errors to a particular variable might be systematic, and one line of research has distinguished between “news” and “noise” in the revision process of data. In practice, data revisions are difficult to model. |

11 | Alternatively, we could have considered to start from a general unrestricted model using a general-to-specific model selection strategy involving multiple path searches, encompassing tests and a set of diagnostic tests, as has been advocated by David Hendry and is implemented in Autometrics (see, e.g., Doornik (2009)). More generally, model selection can be considered as a strategy where smaller models are tested against more general model. Our comparison of forecasting models and methods using a smaller information set with models and methods using larger information sets is in that spirit. See Castle et al. (2021). |

12 | We also examined both the Bayes Information Criteria and the Akaike Information Criteria in real time. Our selected lag order $p=1$ is competitive across most of the sample period for both criteria, and it is the model most preferred by the Bayes criteria since about 2009. |

13 | We also examined other common parsimonious univariate models, including the random walk forecast and the model of Atkeson and Ohanian (2001). We report results for the AR(p) model since it exhibited better forecasting performance in our sample. |

14 | There are a huge number of empirical Phillips curve specifications that have been considered in the literature, and although we report results for only one specification, we considered many alternatives. For example, we considered models with the inflation trend derived from different survey measures or from the Federal Reserve Board staff forecasts and for economic slack we considered various measures of both the unemployment gap and output gap. |

15 | In the aggregated model presented below we model energy inflation as a function of oil prices to capture a different dimension of international influences on inflation. |

16 | The good performance of this interpolation approach noted by Faust and Wright (2013) suggests that it would also be interesting to apply it using our model-based forecasts in place of the SPF forecasts, although we leave this for future work. |

17 | This feature is discussed in Note 6 and can be seen in Figure 2. |

18 | Note that combining forecasts with overlapping information content can also lead to improved MSFE due to differentially mis-specified forecasts. Comparing in-sample to out-of-sample weighting in terms of equal weights versus MSE weights would be an interesting extension |

19 | Note that this sample period describes the included forecast periods. While the final public Tealbook is from December 2014, its $h=8$ forecast corresponds to 2016Q3. |

20 | Note that the ranking between the different forecasting models and methods might differ based on a different transformation, such as annual inflation. |

21 | Statements in the text refer to the 10 percent significance level. |

22 | Indeed, these were the horizons emphasized as most important by Faust and Wright (2013). |

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**Figure 2.**Annualized quarterly inflation rates for various price indexes using data from the 2020Q3 vintage.

**Figure 3.**Example of data release and forecast timing for the period November 2011 through March 2012.

Horizon | AR(1), Gap | AR(1) | Phillips Curve | VAR(1) | Aggregation | SPF | Tealbook | Combination | |
---|---|---|---|---|---|---|---|---|---|

Simple | MSE | ||||||||

Full sample with public Tealbooks (1999Q3–2016Q3) | |||||||||

1 | 1.65 | 1.73 | 1.65 | 1.68 | 1.46 | 1.05 | 0.47 | 1.49 | 1.43 |

– | −1.75 | 0.23 | −0.71 | 1.81 * | 1.57 | 2.03 * | 1.68 * | 1.73 * | |

– | [0.086] | [0.388] | [0.310] | [0.078] | [0.117] | [0.051] | [0.097] | [0.090] | |

2 | 1.73 | 1.86 | 1.73 | 1.81 | 1.75 | 1.67 | 1.35 | 1.73 | 1.74 |

– | −1.86 | −0.08 | −1.66 | −0.63 | 0.90 | 1.49 | −0.23 | −0.39 | |

– | [0.070] | [0.398] | [0.100] | [0.326] | [0.265] | [0.132] | [0.388] | [0.370] | |

4 | 1.67 | 1.77 | 1.69 | 1.73 | 1.68 | 1.70 | 1.77 | 1.68 | 1.68 |

– | −1.72 | −0.44 | −1.68 | −0.31 | −1.01 | −1.34 | −0.73 | −0.48 | |

– | [0.092] | [0.362] | [0.097] | [0.380] | [0.240] | [0.163] | [0.306] | [0.356] | |

8 | 1.71 | 1.85 | 1.67 | 1.75 | 1.67 | 1.51 | 1.79 | 1.70 | 1.68 |

– | −2.33 | 1.03 | −0.51 | 1.40 | 0.96 | −0.81 | 0.96 | 2.00 * | |

– | [0.026] | [0.235] | [0.350] | [0.150] | [0.251] | [0.287] | [0.252] | [0.054] | |

Post-crisis with public Tealbooks (2010Q2–2016Q3) | |||||||||

1 | 1.11 | 1.17 | 1.07 | 1.17 | 1.05 | 0.91 | 0.32 | 1.05 | 1.02 |

– | −1.23 | 0.46 | −0.49 | 1.41 | 2.77 ** | 4.51 ** | 1.61 | 1.95 * | |

– | [0.187] | [0.359] | [0.353] | [0.147] | [0.009] | [0.000] | [0.109] | [0.059] | |

2 | 1.43 | 1.59 | 1.37 | 1.43 | 1.47 | 1.40 | 1.05 | 1.42 | 1.42 |

– | −2.07 | 0.54 | −0.00 | −0.53 | 0.39 | 1.25 | 0.22 | 0.19 | |

– | [0.047] | [0.345] | [0.399] | [0.347] | [0.370] | [0.183] | [0.389] | [0.391] | |

4 | 1.35 | 1.54 | 1.33 | 1.36 | 1.29 | 1.34 | 1.34 | 1.33 | 1.34 |

– | −2.36 | 0.16 | −0.10 | 0.94 | 0.13 | 0.05 | 0.33 | 0.13 | |

– | [0.025] | [0.394] | [0.397] | [0.257] | [0.396] | [0.399] | [0.378] | [0.396] | |

8 | 1.40 | 1.66 | 1.25 | 1.45 | 1.26 | 1.38 | 1.31 | 1.37 | 1.34 |

– | −4.15 | 2.01 * | −0.36 | 3.55 ** | 0.57 | 0.62 | 1.43 | 1.86 * | |

– | [0.000] | [0.053] | [0.374] | [0.001] | [0.339] | [0.328] | [0.143] | [0.070] | |

Pre-crisis (1999Q3–2008Q2) | |||||||||

1 | 1.30 | 1.39 | 1.36 | 1.31 | 1.13 | 1.00 | 0.56 | 1.20 | 1.16 |

– | −1.50 | −1.95 | −0.09 | 2.54 ** | 1.96 * | 3.28 ** | 2.44 ** | 2.54 ** | |

– | [0.129] | [0.060] | [0.397] | [0.016] | [0.058] | [0.002] | [0.021] | [0.016] | |

2 | 1.31 | 1.37 | 1.39 | 1.37 | 1.35 | 1.30 | 1.26 | 1.32 | 1.31 |

– | −0.58 | −1.65 | −1.72 | −0.64 | 0.06 | 0.38 | −0.56 | 0.05 | |

– | [0.338] | [0.102] | [0.091] | [0.325] | [0.398] | [0.371] | [0.342] | [0.398] | |

4 | 1.31 | 1.31 | 1.39 | 1.38 | 1.35 | 1.34 | 1.51 | 1.32 | 1.30 |

– | −0.02 | −1.53 | −1.72 | −0.72 | −0.53 | −1.53 | −0.49 | 0.44 | |

– | [0.399] | [0.124] | [0.091] | [0.307] | [0.346] | [0.123] | [0.353] | [0.362] | |

8 | 1.45 | 1.50 | 1.50 | 1.52 | 1.47 | 1.51 | 1.74 | 1.46 | 1.43 |

– | −0.33 | −0.63 | −0.85 | −0.24 | −2.42 | −1.75 | −0.33 | 1.81 * | |

– | [0.378] | [0.326] | [0.279] | [0.388] | [0.022] | [0.086] | [0.378] | [0.077] |

Horizon | AR(1), Gap | AR(1) | Phillips Curve | VAR(1) | Aggregation | SPF | Tealbook ^{†} | Combination | |
---|---|---|---|---|---|---|---|---|---|

Simple | MSE | ||||||||

Full sample (1999Q3–2019Q4) | |||||||||

1 | 1.53 | 1.61 | 1.52 | 1.56 | 1.34 | 0.95 | – | 1.38 | 1.31 |

– | −2.40 | 0.39 | −0.76 | 2.16 ** | 1.81 * | – | 1.91 * | 1.98 * | |

– | [0.023] | [0.369] | [0.298] | [0.039] | [0.078] | – | [0.065] | [0.056] | |

2 | 1.56 | 1.68 | 1.56 | 1.62 | 1.58 | 1.51 | – | 1.56 | 1.56 |

– | −2.34 | 0.02 | −1.57 | −0.75 | 0.77 | – | −0.32 | −0.49 | |

– | [0.026] | [0.399] | [0.117] | [0.301] | [0.297] | – | [0.379] | [0.354] | |

4 | 1.52 | 1.63 | 1.53 | 1.57 | 1.54 | 1.56 | – | 1.53 | 1.53 |

– | −2.13 | −0.39 | −1.51 | −0.50 | −1.18 | – | −0.91 | −0.66 | |

– | [0.042] | [0.369] | [0.127] | [0.352] | [0.199] | – | [0.263] | [0.322] | |

8 | 1.57 | 1.70 | 1.53 | 1.60 | 1.53 | 1.40 | – | 1.56 | 1.55 |

– | −2.46 | 0.99 | −0.52 | 1.29 | 0.93 | – | 0.81 | 1.67 * | |

– | [0.019] | [0.244] | [0.349] | [0.173] | [0.258] | – | [0.288] | [0.100] | |

Post-crisis (2010Q2–2019Q4) | |||||||||

1 | 1.10 | 1.17 | 1.06 | 1.13 | 0.96 | 0.75 | – | 1.00 | 0.95 |

– | −2.36 | 0.78 | −0.60 | 2.26 ** | 2.49 ** | – | 2.65 ** | 3.06 ** | |

– | [0.025] | [0.294] | [0.334] | [0.031] | [0.018] | – | [0.012] | [0.004] | |

2 | 1.16 | 1.31 | 1.11 | 1.15 | 1.19 | 1.15 | – | 1.16 | 1.16 |

– | −3.10 | 0.71 | 0.12 | −0.76 | 0.15 | – | 0.14 | 0.09 | |

– | [0.003] | [0.310] | [0.396] | [0.299] | [0.395] | – | [0.395] | [0.397] | |

4 | 1.13 | 1.31 | 1.11 | 1.13 | 1.11 | 1.14 | – | 1.13 | 1.13 |

– | −3.03 | 0.22 | −0.04 | 0.52 | −0.25 | – | 0.08 | −0.10 | |

– | [0.004] | [0.389] | [0.399] | [0.348] | [0.386] | – | [0.398] | [0.397] | |

8 | 1.22 | 1.44 | 1.11 | 1.26 | 1.11 | 1.21 | – | 1.19 | 1.17 |

– | −3.80 | 1.79 * | −0.37 | 2.74 ** | 0.08 | – | 1.13 | 1.42 | |

– | [0.000] | [0.080] | [0.373] | [0.009] | [0.398] | – | [0.210] | [0.145] |

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## Share and Cite

**MDPI and ACS Style**

Fulton, C.; Hubrich, K.
Forecasting US Inflation in Real Time. *Econometrics* **2021**, *9*, 36.
https://doi.org/10.3390/econometrics9040036

**AMA Style**

Fulton C, Hubrich K.
Forecasting US Inflation in Real Time. *Econometrics*. 2021; 9(4):36.
https://doi.org/10.3390/econometrics9040036

**Chicago/Turabian Style**

Fulton, Chad, and Kirstin Hubrich.
2021. "Forecasting US Inflation in Real Time" *Econometrics* 9, no. 4: 36.
https://doi.org/10.3390/econometrics9040036