# Balanced Growth Approach to Tracking Recessions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. US Postwar Recessions

## 4. Hybrid Tracking Models

#### 4.1. Key Features

#### 4.2. Implementation Details

#### 4.2.1. Core Model

#### 4.2.2. The State VAR Process

#### 4.2.3. The ECM Measurement Process

#### 4.2.4. Recursive Estimation, Calibration, and Model Validation

## 5. Pilot Application to the RBC Model

#### 5.1. Model Specification

#### 5.2. Recursive Estimation (Conditional on $\lambda $)

#### 5.3. Recursive Tracking/Forecasting (Conditional on $\lambda $)

#### 5.4. Calibration of $\lambda $

#### 5.5. Results

#### 5.6. Great Recession and FinancialSeries

#### 5.7. Policy Experiment

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Data

Fred Series Name and Identification Code | Units and Seasonal Adjustment | Frequency | Range |
---|---|---|---|

Real Personal Consumption Expenditures: Services (DSERRA3Q086SBEA) | Index 2012=100, SA | Quarterly | 1948Q1–2019Q2 |

Real Personal Consumption Expenditures: Services (PCESVC96) | Billions of chained 2012 dollars, SAAR | Quarterly | 2002Q1–2019Q2 |

Real Personal Consumption Expenditures: Nondurable Goods (DNDGRA3Q086SBEA) | Index 2012=100, SA | Quarterly | 1948Q1–2019Q2 |

Real Personal Consumption Expenditures: Nondurable Goods (PCNDGC96) | Billions of chained 2012 dollars, SAAR | Quarterly | 2002Q1–2019Q2 |

Real Gross Private Domestic Investment | Billions of chained 2012 dollars, SAAR | Quarterly | 1948Q1–2019Q2 |

(GPDIC1) | |||

Civilian Noninstitutional Population: 25 to 54 Years (LNU00000060) | Thousands of persons, NSA | Quarterly | 1948Q1–2019Q2 |

#### Appendix A.1. Consumption

#### Appendix A.2. Output

#### Appendix A.3. Fraction of Time Spent Working

## Appendix B. Pseudo Code for the RBC Application

- Set ${T}_{a}=150$.
- Start calibration loop:
- 2.1.
- Select $\lambda $.

- Start recursive loop (given $\lambda $):
- 3.1.
- Set ${T}_{\ast}={T}_{a}$.
- 3.2.
- Estimate state variables:Set ${\widehat{g}}_{t}\left({T}_{\ast}\right)$ to a principal component of $\Delta ln\left(\frac{{y}_{t}}{{n}_{t}}\right)$ and $\Delta ln\left(\frac{{c}_{t}}{{n}_{t}}\right)$, $t:1\to {T}_{\ast}$.Given ${\left\{{\widehat{g}}_{t}\left({T}_{\ast}\right)\right\}}_{t=1}^{{T}_{\ast}}$ optimize in $\left({d}_{t},{\alpha}_{t}|{\widehat{g}}_{t}\left({T}_{\ast}\right)\right)$:${\widehat{s}}_{t}\left({T}_{\ast};\lambda \right)={argmin}_{{s}_{t}}\left|\right|{\u03f5}_{t}\left({s}_{t},{\widehat{s}}_{t-1}\left({T}_{\ast};\lambda \right);\lambda \right){\left|\right|}_{2},\phantom{\rule{1.em}{0ex}}t:1\to {T}_{\ast},$where ${\u03f5}_{t}^{\prime}\left({s}_{t},{s}_{t-1};\lambda \right)=\left[{r}_{t}-{h}_{1}\left({s}_{t};\lambda \right),\Delta {x}_{t}-{h}_{2}\left({s}_{t},{s}_{t-1};\lambda \right)\right]$.
- 3.3.
- Estimate the VAR process for ${\left\{{\widehat{s}}_{t}\left({T}_{\ast};\lambda \right)\right\}}_{t=1}^{{T}_{\ast}}$:${\widehat{s}}_{t}\left({T}_{\ast};\lambda \right)={A}_{0}+{A}_{1}{\widehat{s}}_{t-1}\left({T}_{\ast};\lambda \right)+{A}_{2}{\widehat{s}}_{t-2}\left({T}_{\ast};\lambda \right)+{u}_{t},$where ${u}_{t}\sim \mathcal{IN}\left(0,{\mathrm{\Sigma}}_{A}\right)$.Store ${\left\{{\widehat{A}}_{i}\left({T}_{\ast};\lambda \right)\right\}}_{i=0}^{2}$ and ${\widehat{\mathrm{\Sigma}}}_{A}\left({T}_{\ast};\lambda \right)$.
- 3.4.
- Estimate the ECM process for ${\left\{\Delta {x}_{t}\right\}}_{t=1}^{{T}_{\ast}}$:$\Delta {x}_{t}^{o}={D}_{0}+{D}_{1}\Delta {\widehat{s}}_{t}\left({T}_{\ast};\lambda \right)+{D}_{2}\Delta {x}_{t-1}^{o}-{D}_{3}\left({r}_{t-1}-{h}_{1}\left({\widehat{s}}_{t-1}\left({T}_{\ast};\lambda \right);\lambda \right)\right)+{v}_{t},$where $\Delta {x}_{t}^{o}=\Delta {x}_{t}-{h}_{2}\left({\widehat{s}}_{t}\left({T}_{\ast};\lambda \right),{\widehat{s}}_{t-1}\left({T}_{\ast};\lambda \right);\lambda \right)$, and ${v}_{t}\sim \mathcal{IN}\left(0,{\mathrm{\Sigma}}_{D}\right).$Store ${\left\{{\widehat{D}}_{j}\left({T}_{\ast};\lambda \right)\right\}}_{j=0}^{3}$ and ${\widehat{\mathrm{\Sigma}}}_{D}\left({T}_{\ast};\lambda \right)$.
- 3.5.
- Compute fitted values ${\widehat{x}}_{{T}_{\ast}}\left({T}_{\ast};\lambda \right)$.
- 3.6.
- Conduct MC forecast simulation $\left(n:1\to N=1,000\right)$:
- 3.6.1.
- Forecast 1- to 3-step ahead from VAR: ${\left\{{\widehat{s}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right)\right\}}_{l=1}^{3}$.
- 3.6.2.
- Forecast 1- to 3-step ahead from ECM: ${\left\{\Delta {\widehat{x}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right)\right\}}_{l=1}^{3}$ given ${\left\{{\widehat{s}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right)\right\}}_{l=1}^{3}$.
- 3.6.3.
- Recover and store ${\left\{{\widehat{x}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right)\right\}}_{l=1}^{3}:={\left\{{\widehat{y}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right),{\widehat{c}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right),{\widehat{n}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right)\right\}}_{l=1}^{3}$.

- If ${T}_{\ast}<T$, then ${T}_{\ast}={T}_{\ast}+1$ and go to 3.2. Else, end recursive loop.
- Evaluate recursive performance. For ${T}_{\ast}:{T}_{a}\to T$:
- 5.1.
- Graph ${\left\{{\widehat{A}}_{i}\left({T}_{\ast};\lambda \right)\right\}}_{i=0}^{2}$ and ${\left\{{\widehat{D}}_{j}\left({T}_{\ast};\lambda \right)\right\}}_{j=0}^{3}$.
- 5.2.
- Compute mean 1- to 3-step ahead forecasts: ${\widehat{x}}_{{T}_{\ast}+l}\left({T}_{\ast};\lambda \right)=\frac{1}{N}{\sum}_{n=1}^{N}{\widehat{x}}_{{T}_{\ast}+l}^{n}\left({T}_{\ast};\lambda \right)$, $l=1,2,3$.
- 5.3.
- Graph fitted values ${\widehat{x}}_{{T}_{\ast}}\left({T}_{\ast};\lambda \right)$ and 1- to 3-step ahead mean forecasts ${\left\{{\widehat{x}}_{{T}_{\ast}+l}\left({T}_{\ast};\lambda \right)\right\}}_{l=1}^{3}$.
- 5.4.
- Graph MC 95 percent confidence intervals for ${\left\{{\widehat{x}}_{{T}_{\ast}+l}\left({T}_{\ast};\lambda \right)\right\}}_{l=1}^{3}$.
- 5.5.
- Compute the MAE and RMSE for ${\left\{{\widehat{x}}_{{T}_{\ast}+l}\left({T}_{\ast};\lambda \right)\right\}}_{l=0}^{3}$, for ${T}_{\ast}\in {W}_{j}$, $j=1,2,3$.

- As needed, return to 2.1 and select a different value of $\lambda $.

## Appendix C. Additional Figures

**Figure A1.**Recession of 2007–2009: Out-of-sample 0- to 3-step ahead recursive forecasts for real output per capita. In the top left figure, the solid thin lines correspond to fitted values. In the remaining figures, the solid thin lines denote the mean forecasts calculated over 1000 MC repetitions and the dashed lines the corresponding 95 percent confidence intervals. In all figures the solid thick line denotes the Fred data. Shaded regions correspond to NBER recession dates.

**Figure A2.**Recession of 2007–09: Out-of-sample 0- to 3-step ahead recursive forecasts for real consumption per capita. In the top left figure, the solid thin lines correspond to fitted values. In the remaining figures, the solid thin lines denote the mean forecasts calculated over 1,000 MC repetitions and the dashed lines the corresponding 95 percent confidence intervals. In all figures the solid thick line denotes the Fred data. Shaded regions correspond to NBER recession dates.

**Figure A3.**Recession of 2007–09: Out-of-sample 0- to 3-step ahead recursive forecasts for the fraction of time spent working. In the top left figure, the solid thin lines correspond to fitted values. In the remaining figures, the solid thin lines denote the mean forecasts calculated over 1000 MC repetitions and the dashed lines the corresponding 95 percent confidence intervals. In all figures the solid thick line denotes the Fred data. Shaded regions correspond to NBER recession dates.

**Figure A4.**Hedgehog graphs for 0- to 3-step ahead forecasts for y, c, and n around the 1990–91 and 2001 recessions. Shaded regions correspond to NBER recession dates. Filled circles denote tracked values and empty circles 1- to 3-step ahead forecasts.

**Figure A5.**Hedgehog graphs for 0- to 3-step ahead forecasts for y, c, and n around the 2007-09 Great Recession. Shaded regions correspond to NBER recession dates. Filled circles denote tracked values and empty circles 1- to 3-step ahead forecasts.

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1. | |

2. | It follows that the ECM parsimounsly encompasses the initial VAR model. See Hendry and Richard (1982, 1989); Mizon and Richard (1986); Mizon (1984) for a discussion of the concept of encompassing and its relevance for econometric models. |

3. | See also An and Schorfheide (2007) for a survey of Bayesian methods used to evaluate DSGE models and an extensive list of related references. |

4. | In the present paper, we follow Pagan (2003) by using an unrestricted VAR as a standard benchmark to assess the empirical relevance of our proposed model. Potential extensions to Bayesian VARs belong to future research (though imposing a DSGE-type prior density on VAR in order to improve its theoretical relevance could negatively impact its empirical performance). |

5. | A similar message was delivered by Jerome Powell in his swearing-in ceremony as the new Chair of the Federal Reserve: “The success of our institution is really the result of the way all of us carry out our responsibilities. We approach every issue through a rigorous evaluation of the facts, theory, empirical analysis and relevant research. We consider a range of external and internal views; our unique institutional structure, with a Board of Governors in Washington and 12 Reserve Banks around the country, ensures that we will have a diversity of perspectives at all times. We explain our actions to the public. We listen to feedback and give serious consideration to the possibility that we might be getting something wrong. There is great value in having thoughtful, well-informed critics”. (See https://www.federalreserve.gov/newsevents/speech/powell20180213a.htm for the complete speech given during the ceremonial swearing-in on February 13, 2018). |

6. | For more details see https://www.youtube.com/watch?v=lyzS7Vp5vaY (Stiglitz’s interview posted on May 6, 2019) and https://www.youtube.com/watch?v=rUYk2DA8PH8 (Schiller’s interview posted on April 1, 2019). |

7. | For the full article see https://www.theguardian.com/business/2019/feb/05/financial-crisis-us-uk-crash. February, 2019. |

8. | By doing so, we avoid producing “series with spurious dynamic relations that have no basis in the underlying data-generating process” (Hamilton 2018) as well as “mistaken influences about the strength and dynamic patterns of relationships” (Wallis 1974). |

9. | There is no evidence that seasonality plays a determinant role in recessions and recoveries. Therefore, without loss of generality we rely upon seasonally adjusted data, instead of substantially increasing the number of model parameters by inserting quarterly dummies, potentially in every equation of the state VAR and/or ECM processes. |

10. | NBER recession dating is based upon GDP growth, not per capita GDP growth. However, our objective is not that of dating recessions, for which there exists an extensive and expanding literature. Instead, our objective is that of tracking macroeconomic aggregates at times of rapid changes, and for that purpose per capita data can be used without loss of generality. Note that if needed per capita projections can be ex-post back-transformed into global projections. |

11. | Since we rely upon real data, it is apparent that the great ratios vary considerably over time. Most importantly, their long term dynamics appear to be largely synchronized with business cycles providing a solid basis for our main objective of tracking recessions. |

12. | It is sometimes argued that in order to be interpreted as structural and/or to be instrumental for policy analysis, a parameter needs to be time invariant. We find such a narrow definition to be unnecessarily restrictive and often counterproductive. The very fact that some key structural parameters are found to vary over time in ways that are linked to the business cycles and can be inferred from a state VAR process paves the way for policy interventions on these variables, which might not be available under the more restricted interpretation of structural parameters. An example is provided in Section 5.7. |

13. | Potential exogenous variables are omitted for the ease of notation. |

14. | It is also meant to be parsimonious in the sense that the number of state variables in ${s}_{t}$ has to be less that the number of equations. |

15. | The benchmark VAR process for $\Delta {x}_{t}$ is given by $\Delta {x}_{t}={Q}_{0}+{Q}_{1}\Delta {x}_{t-1}+{Q}_{2}{x}_{t-2}+{w}_{t}$. |

16. | See Appendix A for the full description of the data. |

17. | See also DeJong and Dave (2011, sct. 5.1.2). |

18. | The risk aversion parameter $\varphi $ could also be considered, except for the fact that it is loosely identified to the extent that letting $\varphi $ vary over time serves no useful purpose, and worse, can negatively impact the subsequent recursive invariance of the model. |

19. | For the ease of interpretation, the second component of ${r}_{t}^{\ast}$ is redefined as the sum of the original great ratios in Equation (11). |

20. | It follows that standard cointegration rank tests are not applicable in this context. Bierens and Martins (2010) propose a vector ECM likelihood ratio test for time-invariant cointegration against time-varying cointegration. However, it is not applicable as such to our two stage model and, foremost, Figure 2 offers clear empirical evidence in favor of time-varying cointegration. |

21. | Individual elimination would be undermined by the fact that the estimated residual covariance matrix ${\widehat{\mathrm{\Sigma}}}_{D}$ is ill-conditioned with condition numbers of the order of $2.4\times {10}^{-5}$, which raises concerns about the validity of asymptotic critical values for system test statistics. One advantage of the sequential system elimination is that we can rely upon standard single equation t- and F-test statistics. |

22. | Both eliminations appear to be meaningful. First, equilibrium adjustments in ${n}_{t}$ are undoubtedly impeded by factors beyond agents control. Second, the elimination of $\Delta {\widehat{d}}_{t}$ is likely driven by the fact that the quarterly variations of ${\widehat{d}}_{t}$ are too small to have a significant impact on $\Delta {x}_{t}^{o}\left(\lambda \right)$. |

23. | |

24. | We investigated a number of alternative time windows and arrived at similar qualitative results. |

25. | Depending upon an eventual decision context, alternative metrics could be used (see Elliott and Timmermann 2016). |

26. | It is important to note that the MAE and RMSE have inherent shortcomings because they measure a single variable’s forecast properties at a single horizon (see Clements and Hendry 1993). While measures do exist for assessing forecast accuracy for multiple series across multiple horizons, we believe that they would not impact our conclusions in view of the evidence provided further below (tables, figures, and hedgehog graphs). |

27. | Analogous figures for all other coefficients of the VAR-ECM model and of VAR benchmark are presented in Figures S2–S4 of the Online Supplementary Material and confirm the overall recursive invariance of our estimates and those of the VAR benchmark. |

28. | Analogous figures for the other two recessions are presented in Figures S17–S22 of the Online Supplementary Material. |

29. | Note that the 95 percent confidence intervals are those of the 1000 individual MC draws. The mean forecasts are much more accurate with standard deviations divided by the square root of 1000. |

30. | The average CRPS is given by $\mathrm{CRPS}\left(j,i;\widehat{\lambda}\right)=\frac{1}{{N}_{j}}{\sum}_{{T}_{\ast}\in {W}_{j}}{\int}_{\mathbb{R}}{\left[{\widehat{F}}_{m}\left({\widehat{x}}_{{T}_{\ast}+i}\right)-\mathbf{1}\left({\widehat{x}}_{{T}_{\ast}+i}\ge {x}_{{T}_{\ast}+i}\right)\right]}^{2}\mathrm{d}{\widehat{x}}_{{T}_{\ast}+i}$, where ${\widehat{x}}_{{T}_{\ast}+i}$ stands for ${\widehat{x}}_{{T}_{\ast}+i}\left({T}_{\ast};\widehat{\lambda}\right)$ and ${\widehat{F}}_{m}$ denotes the predictive CDF. See Grimit et al. (2006, Formula (3)) for the discrete version of the CRPS. |

31. | The CRPS accounts for the full predictive CDF and as such was not used as one of the calibration criteria for $\widehat{\lambda}$ since our objective is that of producing mean rather than point forecasts. |

32. | A four quarter lag allows us to produce 4-step ahead forecasts, without ex-ante forecasting any of the auxiliary series added into the VAR-ECM baseline model. 4-step ahead forecasts are available upon request and were not included in the paper as they only confirm further the ex-ante forecasting delays already illustrated in Figure A1, Figure A2 and Figure A3. |

33. | The history of earlier postwar recessions unambiguously suggest that even if such series were available for the entire postwar period, they would likely fail to explain earlier recessions and would, therefore, be irrelevant at those time. Hence, we believe that any potential bias resulting from the missing data would also be insignificant. This is confirmed further by the fact that the auxiliary series incorporated into the VAR component of the VAR-ECM model turn out to be largely insignificant for the Great Recession, even though they are directly related to its cause. |

**Figure 1.**Laws of motion for individual variables. Shaded regions correspond to NBER recession dates.

**Figure 2.**Balanced growth ratios. The dotted line’s vertical axis is on the left and that of the solid line’s on the right. Shaded regions correspond to NBER recession dates.

**Figure 3.**Estimated trajectory of state variable ${\phi}_{t}={\left(exp\left({d}_{t}\right)+1\right)}^{-1}$. Fitted values result from a single equation OLS estimation of the state VAR model. Shaded regions correspond to NBER recession dates.

**Figure 4.**Estimated trajectory of state variable ${g}_{t}$. Fitted values result from a single equation OLS estimation of the state VAR model. Shaded regions correspond to NBER recession dates.

**Figure 5.**Estimated trajectory of state variable ${\alpha}_{t}$. Fitted values result from a single equation OLS estimation of the state VAR model. Shaded regions correspond to NBER recession dates.

**Figure 6.**Recursive equilibrium correction coefficients in the hybrid Real Business Cycle (RBC) model. The solid lines represent the recursive parameter estimates and dashed lines the corresponding 95 percent confidence intervals. Vertical shaded regions correspond to NBER recession dates.

**Figure 7.**Effects of policy interventions for $\widehat{\alpha}$ and $\widehat{d}$ designed to mitigate the impact of the Great Recession on output and consumption. Policies 1a and 1b pertain to interventions for $\widehat{\alpha}$. Policies 2a and 2b pertain to interventions for $\widehat{d}$. Shaded regions correspond to NBER recession dates.

Regressor | Dependent Variable | |||||
---|---|---|---|---|---|---|

${\widehat{\mathit{s}}}_{\mathit{t},1}$ | ${\widehat{\mathit{s}}}_{\mathit{t},2}$ | ${\widehat{\mathit{s}}}_{\mathit{t},3}$ | ||||

$const$ | 0.014 | (2.66) | −0.002 | (−0.36) | 0.031 | (2.60) |

${\widehat{s}}_{t-1,1}$ | 2.422 | (3.94) | 2.715 | (3.87) | 3.768 | (2.79) |

${\widehat{s}}_{t-1,2}$ | 0.133 | (3.10) | 1.314 | (26.81) | 0.281 | (2.97) |

${\widehat{s}}_{t-1,3}$ | −1.019 | (−3.68) | −1.287 | (−4.07) | −1.565 | (−2.57) |

${\widehat{s}}_{t-2,1}$ | −2.162 | (−3.41) | −2.388 | (−3.30) | −5.470 | (−3.92) |

${\widehat{s}}_{t-2,2}$ | −0.139 | (−3.21) | −0.324 | (−6.56) | −0.294 | (−3.09) |

${\widehat{s}}_{t-2,3}$ | 0.994 | (3.58) | 1.300 | (4.10) | 2.517 | (4.11) |

Regressor | Dependent Variable | |||||
---|---|---|---|---|---|---|

$\mathbf{\Delta}{\mathit{x}}_{\mathit{t},1}^{\mathit{o}}$ | $\mathbf{\Delta}{\mathit{x}}_{\mathit{t},2}^{\mathit{o}}$ | $\mathbf{\Delta}{\mathit{x}}_{\mathit{t},3}^{\mathit{o}}$ | ||||

$const$ | 0.002 | (4.41) | −0.002 | (−3.98) | −0.002 | (−4.18) |

${r}_{t-1,1}^{o}$ | −0.078 | (−5.78) | 0.079 | (4.68) | 0.078 | (5.16) |

$\Delta {x}_{t-1,1}^{o}$ | −0.258 | (−1.92) | 1.115 | (6.70) | 0.677 | (4.53) |

$\Delta {x}_{t-1,2}^{o}$ | −0.224 | (−1.83) | 1.004 | (6.62) | 0.605 | (4.44) |

$\Delta {x}_{t-1,3}^{o}$ | −0.053 | (−1.66) | 0.026 | (0.67) | 0.540 | (15.14) |

$\Delta {\widehat{s}}_{t,1}^{o}$ | 0.169 | (0.40) | 3.865 | (7.39) | 1.941 | (4.13) |

$\Delta {\widehat{s}}_{t,3}^{o}$ | −0.991 | (−5.15) | −0.461 | (−1.93) | 0.224 | (1.05) |

F-statistic | 1.64 | 2.30 | 1.92 |

Mean Absolute Error | Root Mean Square Error | Continuous Rank Probability Score | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

VAR-ECM | Benchmark VAR | VAR-ECM | Benchmark VAR | VAR-ECM | Benchmark VAR | ||||||||||||||||||

$\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | $\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | $\mathit{y}$ | $\mathit{c}$ | n | y | $\mathit{c}$ | $\mathit{n}$ | $\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | $\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | ||||||

Recession 1990–1991 | |||||||||||||||||||||||

$h=0$ | 50 | 11 | 0.44 | 104 | 73 | 0.99 | 59 | 13 | 0.51 | 137 | 88 | 1.20 | - | - | - | - | - | - | |||||

$h=1$ | 98 | 67 | 1.10 | 108 | 74 | 1.04 | 135 | 86 | 1.38 | 140 | 89 | 1.24 | 334 | 436 | 6.32 | 210 | 345 | 6.33 | |||||

$h=2$ | 141 | 97 | 1.87 | 193 | 111 | 2.01 | 222 | 116 | 2.36 | 271 | 141 | 2.45 | 1155 | 719 | 4.51 | 1377 | 846 | 4.50 | |||||

$h=3$ | 207 | 89 | 3.17 | 275 | 124 | 3.45 | 277 | 117 | 3.77 | 359 | 167 | 4.04 | 5731 | 3803 | 5.13 | 5857 | 3828 | 5.31 | |||||

Recession 2001 | |||||||||||||||||||||||

$h=0$ | 62 | 16 | 0.43 | 138 | 38 | 1.52 | 79 | 22 | 0.56 | 167 | 49 | 1.96 | - | - | - | - | - | - | |||||

$h=1$ | 106 | 45 | 1.44 | 144 | 39 | 1.60 | 134 | 57 | 1.74 | 173 | 51 | 2.05 | 4736 | 3140 | 3.79 | 4586 | 3063 | 3.72 | |||||

$h=2$ | 164 | 60 | 2.45 | 268 | 75 | 3.05 | 237 | 90 | 2.71 | 319 | 98 | 3.53 | 1867 | 1400 | 5.33 | 1514 | 1306 | 6.75 | |||||

$h=3$ | 211 | 68 | 3.07 | 376 | 107 | 4.46 | 292 | 112 | 3.33 | 455 | 138 | 5.36 | 2501 | 1768 | 3.35 | 2736 | 1849 | 3.30 | |||||

Recession 2007–2009 | |||||||||||||||||||||||

$h=0$ | 117 | 27 | 0.72 | 145 | 49 | 1.42 | 161 | 38 | 1.00 | 189 | 64 | 1.62 | - | - | - | - | - | - | |||||

$h=1$ | 139 | 65 | 1.65 | 153 | 51 | 1.47 | 217 | 85 | 1.87 | 197 | 66 | 1.66 | 7103 | 5246 | 6.40 | 7074 | 5199 | 7.37 | |||||

$h=2$ | 268 | 103 | 3.29 | 335 | 84 | 2.89 | 431 | 126 | 3.91 | 433 | 114 | 3.58 | 2442 | 2522 | 19.4 | 2116 | 2444 | 22.0 | |||||

$h=3$ | 410 | 130 | 5.05 | 528 | 138 | 4.57 | 625 | 162 | 6.20 | 676 | 175 | 6.05 | 514 | 129 | 9.64 | 576 | 165 | 9.84 |

Housing Variables | Financial Variables | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean Absolute Error | Root Mean Square Error | Mean Absolute Error | Root Mean Square Error | |||||||||

$\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | $\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | $\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | $\mathit{y}$ | $\mathit{c}$ | $\mathit{n}$ | |

Housing Starts: Total: New Privately Owned Housing Units Started (1959Q1) | Chicago Fed National Financial Conditions Index (1971Q1) | |||||||||||

h = 1 | −3.8 | −18.6 | −5.5 | −3.7 | −19.8 | 0.3 | −6.6 | −12.4 | −4.9 | −5.2 | −10.3 | −3.1 |

h = 2 | −1.2 | −28.5 | 3.0 | −6.1 | −26.4 | −3.6 | −4.2 | −19.2 | −4.2 | −5.9 | −15.9 | −5.8 |

h = 3 | −5.5 | −26.2 | 5.9 | −5.9 | −24.4 | −4.8 | −5.0 | −21.5 | −3.6 | −5.5 | −17.6 | −5.7 |

Median Number of Months on Sales Market for Newly Completed Homes (1975Q1) | Delinquency Rate on Commercial and Industrial Loans at All Commercial Banks (1987Q1) | |||||||||||

h = 1 | 2.6 | −17.2 | 0.2 | −2.0 | −10.5 | 0.6 | 3.3 | 1.3 | 0.3 | 2.3 | 2.8 | 0.1 |

h = 2 | 11.4 | −14.8 | −0.1 | −1.7 | −10.2 | −0.2 | 8.2 | 7.6 | −0.4 | 2.5 | 8.2 | 1.5 |

h = 3 | 7.6 | −5.0 | 2.7 | −1.1 | −8.7 | 0.4 | 7.5 | 14.8 | 1.2 | 4.0 | 13.0 | 2.6 |

Median Sales Price of Houses Sold (1963Q1) | Delinquency Rate on Consumer Loans at All Commercial Banks (1987Q1) | |||||||||||

h = 1 | 7.9 | 9.8 | −7.0 | −2.9 | 17.7 | −7.4 | −1.0 | −3.9 | −0.4 | −0.7 | −2.0 | 0.1 |

h = 2 | 17.5 | 13.4 | −18.0 | −2.4 | 24.3 | −9.5 | −1.0 | −2.9 | 0.1 | −1.6 | −2.9 | 0.4 |

h = 3 | 13.2 | 14.6 | −14.2 | −0.3 | 16.1 | −7.9 | −1.9 | 1.6 | 1.4 | −1.6 | −1.6 | 1.0 |

Monthly Supply of Houses (1963Q1) | Delinquency Rate on Loans Secured by Real Estate at All Commercial Banks (1987Q1) | |||||||||||

h = 1 | −4.9 | −9.2 | 0.9 | −3.9 | −5.4 | 1.3 | 4.2 | −11.9 | 0.3 | −1.5 | −9.7 | 0.4 |

h = 2 | −1.9 | −10.5 | 2.2 | −3.7 | −8.0 | 2.0 | 9.4 | −15.8 | 0.3 | −1.6 | −10.8 | −0.4 |

h = 3 | −4.0 | −11.5 | 2.9 | −4.0 | −10.4 | 2.7 | 4.5 | −12.8 | 3.1 | −1.0 | −10.5 | 0.2 |

New One Family Homes for Sale (1963Q1) | Household Financial Obligations as a Percent of Disposable Personal Income (1980Q1) | |||||||||||

h = 1 | 2.5 | 0.3 | −1.7 | 2.2 | −0.1 | −0.4 | 4.0 | 3.3 | 0.5 | 2.9 | 3.8 | 0.8 |

h = 2 | 0.1 | 1.6 | −1.0 | 0.2 | −0.5 | −2.1 | 6.4 | 19.3 | −0.5 | 3.2 | 15.1 | 3.7 |

h = 3 | 0.9 | 8.9 | 1.9 | 1.3 | 5.3 | −2.2 | 7.0 | 32.1 | −1.6 | 4.8 | 24.3 | 4.6 |

New One Family Houses Sold (1963Q1) | Mortgage Debt Service Payments as a Percent of Disposable Personal Income (1980Q1) | |||||||||||

h = 1 | 8.0 | −0.5 | 20.9 | −14.4 | −9.8 | 28.1 | 2.7 | 1.9 | −0.3 | 2.0 | 2.1 | 0.4 |

h = 2 | 11.0 | −9.9 | 26.8 | −18.1 | −11.3 | 26.2 | 3.4 | 13.5 | 0.2 | 1.9 | 9.9 | 2.2 |

h = 3 | −0.8 | −13.2 | 30.5 | −22.2 | −17.1 | 21.2 | 4.1 | 21.9 | −1.0 | 3.1 | 16.7 | 2.6 |

New Private Housing Units Authorized by Building Permits (1960Q1) | Mortgage Real Estate Investment Trusts: Liability Level of Debt Securities (1969Q2) | |||||||||||

h = 1 | −6.6 | −21.7 | −6.9 | −8.4 | −26.7 | 1.5 | 3.0 | 6.5 | −0.9 | 0.7 | 7.8 | −2.4 |

h = 2 | 1.0 | −38.4 | 3.5 | −11.1 | −37.6 | −2.7 | 12.1 | 25.1 | −6.0 | 1.8 | 23.3 | 2.5 |

h = 3 | −8.6 | −39.0 | 8.3 | −12.0 | −40.4 | −4.3 | 12.0 | 43.6 | −4.8 | 3.5 | 31.7 | 5.7 |

New Privately-Owned Housing Units Completed (1968Q1) | Mortgage Real Estate Investment Trusts: Liability Level of Mortgage-Backed Bonds (1984Q1) | |||||||||||

h = 1 | 23.5 | 30.0 | −0.5 | 15.2 | 23.9 | −0.7 | 1.3 | 5.1 | −1.8 | 0.3 | 6.0 | −3.3 |

h = 2 | 18.9 | 54.0 | 0.8 | 13.3 | 44.0 | 1.8 | 8.7 | 17.7 | −8.7 | 0.7 | 16.7 | −0.2 |

h = 3 | 21.0 | 80.2 | 3.2 | 17.7 | 65.7 | 2.8 | 7.6 | 31.9 | −6.4 | 1.8 | 22.0 | 2.5 |

New Privately-Owned Housing Units Under Construction (1970Q1) | 10-Year Treasury Constant Maturity Minus 3-Month Treasury Constant Maturity (1982Q1) | |||||||||||

h = 1 | 14.7 | 17.6 | −0.6 | 8.4 | 11.9 | −0.4 | −0.7 | −1.9 | 0.4 | 0.3 | −2.3 | 1.4 |

h = 2 | 9.2 | 32.6 | 0.8 | 6.7 | 24.8 | 0.8 | −0.5 | 0.3 | 6.0 | −0.2 | −0.8 | 3.3 |

h = 3 | 10.3 | 46.5 | 2.5 | 9.3 | 38.6 | 1.3 | −0.6 | 5.4 | 7.4 | 0.1 | 1.5 | 4.7 |

S&P/Case-Shiller U.S. National Home Price Index (1987Q1) | 10-Year Treasury Constant Maturity Minus 2-Year Treasury Constant Maturity (1976Q3) | |||||||||||

h = 1 | 8.0 | 4.8 | −0.3 | 3.2 | 5.5 | −1.2 | 5.8 | 17.8 | 0.9 | 3.6 | 9.6 | 2.5 |

h = 2 | 15.6 | 19.3 | −3.3 | 3.8 | 17.5 | 0.1 | 5.4 | 21.0 | 8.5 | 3.3 | 16.0 | 5.8 |

h = 3 | 16.7 | 37.6 | −2.0 | 6.4 | 26.5 | 1.2 | 5.1 | 28.8 | 8.0 | 4.2 | 21.6 | 7.8 |

^{−3}. Both metrics are computed based on a time window covering 2 quarters before and 6 quarters after the 2007–2009 Great Recession. All auxiliary variables are introduced one at a time as a fourth lag into the state VAR equation. In parenthesis we indicate each series starting date.

Year | Quarter | Policy 1a | Policy 1b | Policy 2a | Policy 2b |
---|---|---|---|---|---|

2007 | Q4 | 0.0005 | - | −0.0030 | - |

2008 | Q1 | 0.0005 | - | −0.0030 | - |

2008 | Q2 | 0.0020 | - | −0.0100 | - |

2008 | Q3 | 0.0030 | - | −0.0200 | - |

2008 | Q4 | 0.0040 | 0.0060 | −0.0200 | −0.0200 |

2009 | Q1 | 0.0040 | 0.0060 | −0.0200 | −0.0200 |

2009 | Q2 | 0.0040 | 0.0060 | −0.0200 | −0.0200 |

2009 | Q3 | 0.0030 | 0.0030 | −0.0100 | −0.0100 |

2009 | Q4 | 0.0020 | 0.0030 | −0.0100 | −0.0100 |

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**MDPI and ACS Style**

Boczoń, M.; Richard, J.-F. Balanced Growth Approach to Tracking Recessions. *Econometrics* **2020**, *8*, 14.
https://doi.org/10.3390/econometrics8020014

**AMA Style**

Boczoń M, Richard J-F. Balanced Growth Approach to Tracking Recessions. *Econometrics*. 2020; 8(2):14.
https://doi.org/10.3390/econometrics8020014

**Chicago/Turabian Style**

Boczoń, Marta, and Jean-François Richard. 2020. "Balanced Growth Approach to Tracking Recessions" *Econometrics* 8, no. 2: 14.
https://doi.org/10.3390/econometrics8020014