# Wagner’s Law vs. Keynesian Hypothesis: Dynamic Impacts

^{*}

## Abstract

**:**

## 1. Introduction

- Uniform results

- Wagner’s law

- Keynesian hypothesis

- Feedback hypothesis

- Neutrality Hypothesis

- Mixed results

#### 1.1. Wagner’s Law Modeling

#### 1.1.1. Peacock and Wiseman Version

#### 1.1.2. Pryor’s Version

#### 1.1.3. Mann’s Version

#### 1.1.4. Other Versions

#### Goffman

#### Musgrave

#### Gupta and Michas

#### 1.2. Keynesian Hypothesis Modeling

#### 1.3. Theoretical Models

#### 1.3.1. Wagner’s Law

#### 1.3.2. Peacock and Wiseman

#### 1.3.3. Pryor

#### 1.3.4. Mann

#### 1.3.5. Keynesian Hypothesis

## 2. Material and Methods

#### 2.1. Materials

#### 2.2. Variables and Data

#### 2.3. Methodology

## 3. Results

#### 3.1. Seasonal Adjustment and Logarithmic Transformation

#### 3.2. Stationarity Analysis of the Series

#### 3.3. Granger Causality

#### 3.4. Cointegration and Error Correction Models

#### 3.4.1. Peacock and Wiseman’s Model and Pryor’s Model

#### 3.4.2. Keynesian Hypothesis Models according to Peacock and Wiseman’s and Pryor’s Versions

#### 3.5. Autoregressive Distributed Lag Model

#### 3.5.1. Mann’s Model

#### 3.5.2. Keynesian Hypothesis Model according Mann’s Version

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Real GDP, Real Public Consumption, and Real Public Expenditure, Peru, 1980Q1–2021Q4. * Development by authors. * Seasonal adjustment, Census ARIMA-X13.

**Figure 2.**Share of Real Public Expenditure to Real GDP. * Development by authors. * Seasonal adjustment, Census ARIMA-X13.

**Figure 3.**Inverse characteristic roots of $AR\left(3\right)$ model to $ln{\left(\frac{E}{Y}\right)}_{t}$ from Mann’s model and $AR\left(4\right)$ model to $\Delta ln{Y}_{t}$ from the Keynesian model according to Mann’s model. (Development by authors.)

Version | Function * | Elasticity |
---|---|---|

Peacock and Wiseman [27] | ${E}_{t}={f}_{1}\left({Y}_{t}\right)$ | $dln{E}_{t}/dln{Y}_{t}>1{}^{**}$ |

Pryor [9] | ${C}_{t}={f}_{2}\left({Y}_{t}\right)$ | $dln{C}_{t}/dln{Y}_{t}>1{}^{**}$ |

Mann [16] | ${E}_{t}/{Y}_{t}={f}_{3}\left({Y}_{t}\right)$ | $dln\left({E}_{t}/{Y}_{t}\right)/dln{Y}_{t}>{0}^{***}$ |

Goffman [8] | ${E}_{t}={f}_{4}\left(Y{p}_{t}\right)$ | $dln{E}_{t}/dlnY{p}_{t}>{1}^{**}$ |

Musgrave [10] | ${E}_{t}/{Y}_{t}={f}_{5}\left(Y{p}_{t}\right)$ | $dln\left({E}_{t}/{Y}_{t}\right)/dlnY{p}_{t}>1{}^{**}$ |

Gupta [7] and Michas [13] | $E{p}_{t}={f}_{6}\left(Y{p}_{t}\right)$ | $dlnE{p}_{t}/dlnY{p}_{t}>{1}^{**}$ |

${\mathit{C}}_{\mathit{t}}$ | ${\mathit{E}}_{\mathit{t}}$ | ${\mathit{Y}}_{\mathit{t}}$ | ${\mathit{E}}_{\mathit{t}}/{\mathit{Y}}_{\mathit{t}}$ | |
---|---|---|---|---|

${C}_{t}$ | 1 | 0.98244 | 0.93961 | 0.23206 |

${E}_{t}$ | 0.98244 | 1 | 0.91774 | 0.31138 |

${Y}_{t}$ | 0.93961 | 0.91774 | 1 | −0.04498 |

${E}_{t}/{Y}_{t}$ | 0.23206 | 0.31138 | −0.04498 | 1 |

Series | Auxiliary Model | Criteria * | Lag | $\mathit{\tau}$ | Probability | Integration ** |
---|---|---|---|---|---|---|

$ln{C}_{t}$ | Constant and Trend | SIC | 0 | −3.27388 | 0.07420 | $I\left(1\right)$ |

$\Delta ln{C}_{t}$ | Constant and Trend | SIC | 3 | −9.37206 | 0.00000 | $I\left(0\right)$ |

$ln{E}_{t}$ | Constant and Trend | SIC | 0 | −2.87296 | 0.17410 | $I\left(1\right)$ |

$\Delta ln{E}_{t}$ | None | SIC | 0 | −18.01177 | 0.00000 | $I\left(0\right)$ |

$ln{Y}_{t}$ | Constant and Trend | SIC | 0 | −2.21610 | 0.47720 | $I\left(1\right)$ |

$\Delta ln{Y}_{t}$ | Constant | SIC | 0 | −13.24513 | 0.00000 | $I\left(0\right)$ |

$ln\left({E}_{t}/{Y}_{t}\right)$ | Constant | SIC | 1 | −2.92010 | 0.04520 | $I\left(0\right)$ |

Version | Function | Causality | Criteria | Lag | $\mathit{F}$ | $\mathit{p}$ |
---|---|---|---|---|---|---|

Peacock and W. (P&W) | ${E}_{t}={f}_{1}\left({Y}_{t}\right)$ | $\Delta ln{Y}_{t}\to \Delta ln{E}_{t}$ | SIC * | 1 | 12.78720 | 0.00050 |

Keynes (to P&W) | ${Y}_{t}={g}_{1}\left({E}_{t}\right)$ | $\Delta ln{E}_{t}\nrightarrow \Delta ln{Y}_{t}$ | SIC * | 1 | 0.02791 | 0.86750 |

Pryor (P) | ${C}_{t}={f}_{2}\left({Y}_{t}\right)$ | $\Delta ln{Y}_{t}\to \Delta ln{C}_{t}$ | SIC * | 1 | 16.08020 | 0.00009 |

Keynes (to P) | ${Y}_{t}={g}_{2}\left({C}_{t}\right)$ | $\Delta ln{C}_{t}\nrightarrow \Delta ln{Y}_{t}$ | SIC * | 1 | 1.01699 | 0.31470 |

Mann (M) | ${E}_{t}/{Y}_{t}={f}_{3}\left({Y}_{t}\right)$ | $\Delta ln{Y}_{t}\to \Delta ln\left({E}_{t}/{Y}_{t}\right)$ | SIC * | 1 | 2.88881 | 0.00270 |

Keynes (to M) | ${Y}_{t}={g}_{3}\left({E}_{t}/{Y}_{t}\right)$ | $\Delta ln\left({E}_{t}/{Y}_{t}\right)\nrightarrow \Delta ln{Y}_{t}$ | SIC * | 10 | 0.38062 | 0.53810 |

Version | Function | Coefficients * | Engle-Granger Cointegration Test | |||||
---|---|---|---|---|---|---|---|---|

Constant | Trend | Long-Run Elasticity | $\mathit{\tau}$ | Probab. | Criteria | Lag | ||

Peacock and W. (P&W) | ${Y}_{t}={f}_{1}\left({E}_{t}\right)$ | −8.27917 | −0.00628 | 1.62964 | −4.07915 | 0.02676 | SIC ** | 1 |

Pryor (P) | ${C}_{t}={f}_{2}\left({Y}_{t}\right)$ | −6.85207 | −0.00481 | 1.45999 | −8.52144 | 0.00000 | SIC ** | 0 |

Keynes (to P&W) | ${Y}_{t}={g}_{1}\left({E}_{t}\right)$ | 5.87281 | 0.00456 | 0.52178 | −3.99765 | 0.03329 | SIC ** | 1 |

Keynes (to P) | ${Y}_{t}={g}_{2}\left({C}_{t}\right)$ | 5.68261 | 0.00421 | 0.56565 | −7.81920 | 0.00000 | SIC ** | 0 |

ECM | ||||
---|---|---|---|---|

Wagner’s Law | Keynesian Hypothesis | |||

Peacock and W. (P&W) | Pryor (P) | Keynes (to P&W) | Keynes (to P) | |

Function | ${E}_{t}={f}_{1}\left({Y}_{t}\right)$ | ${C}_{t}={f}_{2}\left({Y}_{t}\right)$ | ${Y}_{t}={g}_{1}\left({E}_{t}\right)$ | ${Y}_{t}={g}_{2}\left({C}_{t}\right)$ |

Regressand variable | $\Delta ln{E}_{t}$ | $\Delta ln{C}_{t}$ | $\Delta ln{Y}_{t}$ | $\Delta ln{Y}_{t}$ |

Error correction coefficient | −0.24360 * | −0.21145 * | −0.10835 * | −0.14636 * |

Short-run elasticity | 1.01455 * | 0.57024 * | 0.16841 * | 0.05366 * |

Observations | 159 | 159 | 159 | 159 |

Number of coefficients ** | 18 | 17 | 24 | 22 |

${R}^{2}$-adjusted | 0.73536 | 0.82233 | 0.85929 | 0.85057 |

$\mathrm{Probability}\text{}F$-statistic | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

$\mathrm{Durbin}\u2013\mathrm{Watson}\text{}d$-statistic | 1.70714 | 1.81202 | 1.98001 | 2.04776 |

$\mathrm{Breusch}\u2013\mathrm{Godfrey}\text{}\mathrm{Prob}.\text{}LM$-stat. *** | 0.06697 | 0.11470 | 0.86588 | 0.66910 |

$\mathrm{Breusch}\u2013\mathrm{Pagan}\u2013\mathrm{Godfrey}\text{}\mathrm{Prob}.\text{}LM$-stat. | 0.97857 | 0.60050 | 0.25184 | 0.86180 |

$\mathrm{White}\text{}\mathrm{cross}\text{}\mathrm{terms}\text{}\mathrm{Prob}.\text{}LM$-statistic | – | 0.07840 | – | – |

$\mathrm{White}\text{}\mathrm{non}-\mathrm{cross}\text{}\mathrm{terms}\text{}\mathrm{Prob}.\text{}LM$-stat. | 0.41175 | 0.35220 | 0.42939 | 0.90960 |

$\mathrm{Glejser}\text{}\mathrm{Prob}.\text{}LM$-statistic | 0.19224 | 0.28020 | 0.05265 | 0.20890 |

$\mathrm{Jarque}\u2013\mathrm{Bera}\text{}\mathrm{Prob}.\text{}JB$-statistic | 0.27556 | 0.80323 | 0.17638 | 0.05644 |

$\mathrm{Dickey}\u2013\mathrm{Fuller}\text{}\mathrm{Prob}.\text{}\tau $-statistic **** | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

**Table 7.**Estimation of Mann’s model and estimation of the Keynesian hypothesis model according Mann’s version.

ARDL models | ||||

Wagner’s Law | Keynesian Hypothesis | |||

Version | Mann (M) | Version | Keynes (to M) | |

Function | ${E}_{t}/{Y}_{t}={f}_{3}\left({Y}_{t}\right)$ | Function | ${Y}_{t}={g}_{3}\left({E}_{t}/{Y}_{t}\right)$ | |

Regressand variable | $ln{\left(E/Y\right)}_{t}$ | Regressand variable | $\Delta ln{Y}_{t}$ | |

Constant | −0.11957 * | Constant | −0.03732 * | |

$ln{\left(E/Y\right)}_{t-1}$ | 0.41981 * | $\Delta ln{Y}_{t-1}$ | 0.09558 * | |

$ln{\left(E/Y\right)}_{t-2}$ | 0.29535 * | $\Delta ln{Y}_{t-3}$ | −0.10972 * | |

$ln{\left(E/Y\right)}_{t-3}$ | 0.22443 * | $\Delta ln{Y}_{t-4}$ | −0.11666 * | |

$\Delta ln{Y}_{t}$ | 0.19863 * | $ln{\left(E/Y\right)}_{t}$ | −0.05259 * | |

$\Delta ln{Y}_{t-1}$ | 0.16100 | $ln{\left(E/Y\right)}_{t-4}$ | 0.02546 * | |

$\Delta ln{Y}_{t-2}$ | −0.15970 | |||

$\Delta ln{Y}_{t-6}$ | 0.06057 | |||

$\Delta ln{Y}_{t-7}$ | 0.72731 * | |||

Observations | 160 | 163 | ||

Number of coefficients ** | 18 | 16 | ||

${R}^{2}$-adjusted | 0.85976 | 0.81571 | ||

$\mathrm{Probability}\text{}F$-statistic | 0.00000 | 0.00000 | ||

$\mathrm{Durbin}\u2013\mathrm{Watson}\text{}d$-statistic | 1.80746 | 1.77090 | ||

$\mathrm{Breusch}\u2013\mathrm{Godfrey}\text{}\mathrm{Prob}.\text{}LM$-statistic *** | 0.09960 | 0.18260 | ||

$\mathrm{Breusch}\u2013\mathrm{Pagan}\u2013\mathrm{Godfrey}\text{}\mathrm{Prob}.\text{}LM$-statistic | 0.25573 | 0.64210 | ||

$\mathrm{White}\text{}\mathrm{cross}\text{}\mathrm{terms}\text{}\mathrm{Prob}.\text{}LM$-statistic | – | 0.36510 | ||

$\mathrm{White}\text{}\mathrm{non}-\mathrm{cross}\text{}\mathrm{terms}\text{}\mathrm{Prob}.\text{}LM$-statistic | 0.36501 | 0.81110 | ||

$\mathrm{Glejser}\text{}\mathrm{Prob}.\text{}LM$-statistic | 0.12381 | 0.10520 | ||

$\mathrm{Jarque}\u2013\mathrm{Bera}\text{}\mathrm{Prob}.\text{}JB$-statistic | 0.76809 | 0.13538 | ||

$\mathrm{Dickey}\u2013\mathrm{Fuller}\text{}\mathrm{Prob}.\text{}\tau $-statistic **** | 0.00000 | 0.00000 |

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**MDPI and ACS Style**

Bazán, C.; Álvarez-Quiroz, V.J.; Morales Olivares, Y.
Wagner’s Law vs. Keynesian Hypothesis: Dynamic Impacts. *Sustainability* **2022**, *14*, 10431.
https://doi.org/10.3390/su141610431

**AMA Style**

Bazán C, Álvarez-Quiroz VJ, Morales Olivares Y.
Wagner’s Law vs. Keynesian Hypothesis: Dynamic Impacts. *Sustainability*. 2022; 14(16):10431.
https://doi.org/10.3390/su141610431

**Chicago/Turabian Style**

Bazán, Ciro, Víctor Josué Álvarez-Quiroz, and Yennyfer Morales Olivares.
2022. "Wagner’s Law vs. Keynesian Hypothesis: Dynamic Impacts" *Sustainability* 14, no. 16: 10431.
https://doi.org/10.3390/su141610431