#
Simultaneous Indirect Inference, Impulse Responses and ARMA Models^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**ARMA**) models and production of simultaneous confidence bands from its impulse-response functions.2

**AR**) unit roots cause the so-called non-uniform convergence problem, which means that methods requiring stationarity break down at, or close to the unit boundary, whereas local-to-unity motivated methods perform poorly away from unity; see Hansen (1999); Mikusheva (2007); Andrews and Guggenberger (2010); Gorodnichenko et al. (2012); Mikusheva (2012, 2014) and Phillips (2014). Second, identification failure related to the invertibility of Moving Average (

**MA**) components causes the so-called pile up effect, since estimating functions, including likelihoods, can have a local maximum at the invertibility boundary even when the true process is invertible; see Sargan and Bhargava (1983); Davis and Dunsmuir (1996); Gospodinov (2002); Genton and Ronchetti (2003); Billio and Monfort (2003); Davis and Song (2011); and Gospodinov and Ng (2015).

## 2. General Framework

#### 2.1. Nuisance Parameters

**Assumption**

**1.**

**1**to the calibration in (6), we have:

**Assumption 1**yields the following,

## 3. Inference via Test Inversion

#### 3.1. Exact p-Values

## 4. ARMA Special Case

#### 4.1. Impulse-Response Confidence Bands

## 5. Simulation Study

#### 5.1. Main Results

#### 5.2. Robustness Checks

## 6. Empirical Application

#### 6.1. Data

#### 6.2. Impulse-Response Confidence Bands for Oil Series

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | See, for example, Robins et al. (2000); Ronchetti and Trojani (2001); Calzolari et al. (2004); Dridi et al. (2007); Gouriéroux et al. (2010); Li (2010); Dominicy and Veredas (2013); Fuleky and Zivot (2014); Calvet and Czellar (2015); Chaudhuri et al. (2018) and Forneron and Ng (2018). |

2. | On simultaneous inference in related contexts see e.g., Jorda (2009); Jorda and Marcellino (2010); Jorda et al. (2013) on simultaneous path forecasts, and recently Montiel et al. (2019). |

3. | Throughout this document, size and coverage are used interchangeably; see Andrews and Cheng (2012). |

4. | In this regard, our work relates to Dufour and Valéry (2006, 2009) in stochastic volatility models, Dufour and Kurz-Kim (2010) and Beaulieu et al. (2007, 2013, 2014) in models with fat-tailed fundamentals. |

5. | |

6. | Alternatively, the data can be burned in to induce stationarity. Extensions to the mean-non stationarity framework as in Khalaf and Saunders (2019) is a worthy research direction. |

7. | Useful insight beyond the ARMA (1,1) on minimum dimension is found in the literature, in particular, Galbraith and Zinde-Walsh (1997) and more recently Gospodinov and Ng (2015). |

8. | This is noteworthy in view of the above cited problems discussed in the literature. |

9. | |

10. | Lütkepohl et al. (2015) provide an overview and evaluation in terms of coverage of the available literature dedicated to impulse-responses for VAR models and propose a new approach based on adjustments to the Bonferroni bands. |

11. | Alquist et al. (2013) explain that after mid-1980, due to the U.S. deregulation, the co-movement between the U.S. oil price series, including WTI, refiners acquisition cost for domestically produced oil and for imported crude oil, became stronger. This means that we can take the WIT, since results obtained with other series are expected to be similar due to the high correlation between the various U.S. oil price series. |

12. | At the time the series is retrieved, the latest available monthly value corresponds to May, 2019. |

13. | The U.S. subprime mortgage crisis triggered an international financial crisis in 2008. According to the U.S. National Bureau of Economics Research (NBER), the U.S. recession is identified in terms of peaks and troughs from the last quarter of 2007 to the second quarter of 2009. |

**Figure 1.**West Texas Intermediate (WTI).

**Note**: WTI prices escalated up to the highest peak during the month of July in 2008, when the price reached US$145 per barrel. Then, the effect of the financial crisis and Great Recession caused a price drop to US$30.28 in December of the same year.13 After 2014, WTI prices plummeted during the following three years, down to US$26.68 in January, 2016, mainly due to the effect of supply strategies, the surprising growth of U.S. shale oil production and by a slower-than-expected global growth, among others. The behaviour of WTI prices afterwards is associated with geopolitical risks and growth expectations for the U.S. and Chinese economies.

SIZE | ||||
---|---|---|---|---|

SIM Method | Gaussian MA | Student-t (df = 5) MA | ||

OLS-Long AR | Simplified | OLS-Long AR | Simplified | |

T = 50 | 0.046 | 0.040 | 0.043 | 0.038 |

T = 100 | 0.054 | 0.055 | 0.047 | 0.042 |

T = 200 | 0.046 | 0.049 | 0.053 | 0.039 |

MLE | Gaussian MA | Student-t (df = 5) MA | ||

T = 50 | 0.096 | 0.099 | ||

T = 100 | 0.054 | 0.062 | ||

T = 200 | 0.057 | 0.049 | ||

POWER [SIM Method] | ||||

T = 50 | Gaussian MA | Student-t (df = 5) MA | ||

Tested Value$\mathbf{\left\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{\right\}}$ | OLS-Long AR | Simplified | OLS-Long AR | Simplified |

{0.00} | 0.717 | 0.861 | 0.720 | 0.865 |

{0.30} | 0.322 | 0.364 | 0.302 | 0.370 |

{0.85} | 0.276 | 0.225 | 0.266 | 0.211 |

{0.90} | 0.368 | 0.286 | 0.371 | 0.267 |

{0.96} | 0.436 | 0.324 | 0.430 | 0.302 |

{0.99} | 0.442 | 0.334 | 0.442 | 0.309 |

T = 100 | Gaussian MA | Student-t (df = 5) MA | ||

Tested Value$\mathit{\left\{}{\mathit{\theta}}_{\mathbf{0}}\mathit{\right\}}$ | OLS-Long AR | Simplified | OLS-Long AR | Simplified |

{0.00} | 0.988 | 0.994 | 0.989 | 0.995 |

{0.30} | 0.628 | 0.699 | 0.634 | 0.703 |

{0.85} | 0.611 | 0.476 | 0.623 | 0.470 |

{0.90} | 0.721 | 0.569 | 0.729 | 0.558 |

{0.96} | 0.785 | 0.634 | 0.796 | 0.610 |

{0.99} | 0.790 | 0.640 | 0.806 | 0.618 |

T = 200 | Gaussian MA | Student-t (df = 5) MA | ||

Tested Value$\mathbf{\left\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{\right\}}$ | OLS-Long AR | Simplified | OLS-Long AR | Simplified |

{0.00} | 1.000 | 1.000 | 1.000 | 1.000 |

{0.30} | 0.926 | 0.935 | 0.934 | 0.942 |

{0.85} | 0.910 | 0.786 | 0.910 | 0.779 |

{0.90} | 0.960 | 0.869 | 0.967 | 0.862 |

{0.96} | 0.978 | 0.914 | 0.983 | 0.908 |

{0.99} | 0.980 | 0.919 | 0.984 | 0.912 |

**Note**: Numbers reported are empirical rejections for the hypothesis ${H}_{0}:\theta ={\theta}_{0}$. Size is measured with ${\theta}_{0}={\theta}_{DGP}$ and other choices for ${\theta}_{0}$ are reported as tested values in the power tables.

SIZE | ||||
---|---|---|---|---|

SIM Method | Gaussian MA | Student-t (df = 5) MA | ||

OLS-Long AR | Simplified | OLS-Long AR | Simplified | |

T = 50 | 0.052 | 0.045 | 0.048 | 0.041 |

T = 100 | 0.054 | 0.053 | 0.045 | 0.038 |

T = 200 | 0.039 | 0.042 | 0.039 | 0.042 |

MLE | Gaussian MA | Student-t (df = 5) MA | ||

T = 50 | 0.167 | 0.156 | ||

T = 100 | 0.294 | 0.229 | ||

T = 200 | 0.429 | 0.347 | ||

POWER [SIM Method] | ||||

T = 50 | Gaussian MA | Student-t (df = 5) MA | ||

Tested Value$\mathbf{\left\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{\right\}}$ | OLS-Long AR | Simplified | OLS-Long AR | Simplified |

{0.00} | 0.998 | 0.993 | 0.998 | 0.993 |

{0.30} | 0.948 | 0.854 | 0.961 | 0.842 |

{0.60} | 0.594 | 0.363 | 0.573 | 0.353 |

{0.85} | 0.095 | 0.073 | 0.091 | 0.061 |

{0.90} | 0.061 | 0.054 | 0.059 | 0.049 |

{0.96} | 0.057 | 0.045 | 0.051 | 0.041 |

T = 100 | Gaussian MA | Student-t (df = 5) MA | ||

Tested Value$\mathbf{\left\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{\right\}}$ | OLS-Long AR | Simplified | OLS-Long AR | Simplified |

{0.00} | 1.000 | 1.000 | 1.000 | 1.000 |

{0.30} | 1.000 | 0.990 | 1.000 | 0.992 |

{0.60} | 0.918 | 0.676 | 0.916 | 0.690 |

{0.85} | 0.141 | 0.094 | 0.130 | 0.078 |

{0.90} | 0.087 | 0.060 | 0.071 | 0.059 |

{0.96} | 0.059 | 0.053 | 0.045 | 0.040 |

T = 200 | Gaussian MA | Student-t (df = 5) MA | ||

Tested Value$\mathbf{\left\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{\right\}}$ | OLS-Long AR | Simplified | OLS-Long AR | Simplified |

{0.00} | 1.000 | 1.000 | 1.000 | 1.000 |

{0.30} | 1.000 | 1.000 | 1.000 | 1.000 |

{0.60} | 0.996 | 0.922 | 0.996 | 0.928 |

{0.85} | 0.204 | 0.099 | 0.196 | 0.124 |

{0.90} | 0.084 | 0.050 | 0.079 | 0.063 |

{0.96} | 0.042 | 0.039 | 0.043 | 0.040 |

**Note**: Numbers reported are empirical rejections for the hypothesis ${H}_{0}:\theta ={\theta}_{0}$. Size is measured with ${\theta}_{0}={\theta}_{DGP}$ and other choices for ${\theta}_{0}$ are reported as tested values in the power tables.

SIZE | ||||||
---|---|---|---|---|---|---|

SIM Method | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr | |

T = 50 | 0.056 | 0.053 | 0.038 | 0.041 | 0.049 | 0.033 |

T = 100 | 0.053 | 0.054 | 0.058 | 0.041 | 0.048 | 0.051 |

T = 200 | 0.049 | 0.045 | 0.054 | 0.049 | 0.043 | 0.061 |

MLE | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

MA | AR | Joint | MA | AR | Joint | |

T = 50 | 0.470 | 0.454 | 0.523 | 0.446 | 0.401 | 0.500 |

T = 100 | 0.432 | 0.417 | 0.482 | 0.404 | 0.392 | 0.457 |

T = 200 | 0.414 | 0.414 | 0.470 | 0.370 | 0.371 | 0.445 |

POWER [SIM Method] | ||||||

T = 50 | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

Tested Pair$\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}},{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 0.430 | 0.225 | 0.181 | 0.418 | 0.229 | 0.174 |

{0.3, 0.2} | 0.806 | 0.728 | 0.858 | 0.796 | 0.740 | 0.866 |

{0.3, 0.85} | 0.923 | 0.699 | 0.453 | 0.932 | 0.702 | 0.428 |

{0.5, 0.5} | 0.956 | 0.866 | 0.981 | 0.958 | 0.858 | 0.986 |

{0.6, 0} | 0.836 | 0.436 | 0.180 | 0.828 | 0.431 | 0.170 |

{0.6, 0.85} | 0.985 | 0.878 | 0.880 | 0.992 | 0.884 | 0.880 |

{0.85, 0.2} | 0.999 | 0.916 | 0.307 | 0.999 | 0.904 | 0.271 |

{0.85, 0.6} | 1.000 | 0.955 | 0.568 | 1.000 | 0.961 | 0.557 |

{0.96, 0.5} | 1.000 | 0.961 | 0.481 | 1.000 | 0.963 | 0.471 |

{0.99, 0.99} | 1.000 | 0.980 | 0.997 | 1.000 | 0.979 | 0.998 |

T = 100 | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

Tested Pair$\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{,}{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 0.797 | 0.695 | 0.500 | 0.793 | 0.700 | 0.493 |

{0.3, 0.2} | 0.980 | 0.998 | 0.991 | 0.981 | 0.999 | 0.990 |

{0.3, 0.85} | 0.996 | 0.995 | 0.875 | 0.996 | 0.994 | 0.870 |

{0.5, 0.5} | 0.999 | 1.000 | 0.999 | 1.000 | 1.000 | 0.999 |

{0.6, 0} | 0.983 | 0.901 | 0.509 | 0.989 | 0.899 | 0.502 |

{0.6, 0.85} | 1.000 | 1.000 | 0.992 | 1.000 | 1.000 | 0.991 |

{0.85, 0.2} | 1.000 | 1.000 | 0.727 | 1.000 | 0.999 | 0.727 |

{0.85, 0.6} | 1.000 | 1.000 | 0.933 | 1.000 | 1.000 | 0.932 |

{0.96, 0.5} | 1.000 | 1.000 | 0.901 | 1.000 | 1.000 | 0.894 |

{0.99, 0.99} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

T = 200 | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

Tested Pair$\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{,}{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 0.984 | 0.979 | 0.887 | 0.987 | 0.983 | 0.894 |

{0.3, 0.2} | 0.999 | 1.000 | 1.000 | 0.999 | 1.000 | 1.000 |

{0.3, 0.85} | 1.000 | 1.000 | 0.998 | 1.000 | 1.000 | 0.996 |

{0.5, 0.5} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

{0.6, 0} | 0.999 | 0.999 | 0.911 | 1.000 | 0.999 | 0.918 |

{0.6, 0.85} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

{0.85, 0.2} | 1.000 | 1.000 | 0.994 | 1.000 | 1.000 | 0.992 |

{0.85, 0.6} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.999 |

{0.96, 0.5} | 1.000 | 1.000 | 0.998 | 1.000 | 1.000 | 0.998 |

{0.99, 0.99} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

**Note**: Numbers reported are empirical rejections for the hypothesis ${H}_{0}:\theta ={\theta}_{0}$ and $\psi ={\psi}_{0}$. Size is measured with ${\theta}_{0}={\theta}_{DGP}$ and ${\psi}_{0}={\psi}_{DGP}$. Other choices for ${\theta}_{0}$ and ${\psi}_{0}$ are reported as tested values in the power tables.

SIZE | ||||||
---|---|---|---|---|---|---|

SIM Method | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr | |

T = 50 | 0.044 | 0.041 | 0.052 | 0.049 | 0.038 | 0.049 |

T = 100 | 0.049 | 0.041 | 0.053 | 0.049 | 0.035 | 0.048 |

T = 200 | 0.052 | 0.058 | 0.051 | 0.048 | 0.046 | 0.043 |

MLE | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

MA | AR | Joint | MA | AR | Joint | |

T = 50 | 0.140 | 0.060 | 0.171 | 0.119 | 0.060 | 0.153 |

T = 100 | 0.281 | 0.068 | 0.259 | 0.223 | 0.049 | 0.227 |

T = 200 | 0.390 | 0.074 | 0.349 | 0.317 | 0.049 | 0.281 |

POWER [SIM Method] | ||||||

T = 50 | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

Tested Pair$\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{,}{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 0.922 | 1.000 | 0.915 | 0.921 | 1.000 | 0.914 |

{0.3, 0.2} | 0.831 | 1.000 | 0.992 | 0.830 | 1.000 | 0.991 |

{0.3, 0.85} | 0.609 | 0.979 | 0.331 | 0.615 | 0.985 | 0.332 |

{0.5, 0.5} | 0.295 | 0.968 | 0.904 | 0.305 | 0.980 | 0.906 |

{0.5, 0.96} | 0.285 | 0.778 | 0.046 | 0.290 | 0.794 | 0.046 |

{0.6, 0} | 0.427 | 0.998 | 0.998 | 0.442 | 0.999 | 0.992 |

{0.6, 0.85} | 0.131 | 0.653 | 0.317 | 0.131 | 0.663 | 0.308 |

{0.85, 0.2} | 0.017 | 0.763 | 0.980 | 0.020 | 0.772 | 0.986 |

{0.85, 0.6} | 0.023 | 0.304 | 0.815 | 0.024 | 0.293 | 0.825 |

{0.96, 0.5} | 0.040 | 0.257 | 0.891 | 0.038 | 0.241 | 0.893 |

T = 100 | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

Tested Pair$\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{,}{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 0.997 | 1.000 | 0.994 | 0.997 | 1.000 | 0.995 |

{0.3, 0.2} | 0.980 | 1.000 | 1.000 | 0.984 | 1.000 | 1.000 |

{0.3, 0.85} | 0.908 | 1.000 | 0.635 | 0.914 | 1.000 | 0.648 |

{0.5, 0.5} | 0.690 | 1.000 | 0.993 | 0.701 | 1.000 | 0.994 |

{0.5, 0.96} | 0.662 | 0.984 | 0.074 | 0.671 | 0.988 | 0.070 |

{0.6, 0} | 0.826 | 1.000 | 1.000 | 0.824 | 1.000 | 1.000 |

{0.6, 0.85} | 0.418 | 0.939 | 0.610 | 0.435 | 0.947 | 0.626 |

{0.85, 0.2} | 0.036 | 0.983 | 1.000 | 0.042 | 0.984 | 1.000 |

{0.85, 0.6} | 0.035 | 0.508 | 0.976 | 0.033 | 0.531 | 0.973 |

{0.96, 0.5} | 0.039 | 0.453 | 0.991 | 0.048 | 0.444 | 0.994 |

T = 200 | Gaussian ARMA | Student-t (df = 5) ARMA | ||||

Tested Pair$\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{,}{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

{0.3, 0.2} | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

{0.3, 0.85} | 0.991 | 1.000 | 0.922 | 0.992 | 1.000 | 0.921 |

{0.5, 0.5} | 0.943 | 1.000 | 1.000 | 0.943 | 1.000 | 1.000 |

{0.5, 0.96} | 0.932 | 1.000 | 0.157 | 0.935 | 1.000 | 0.163 |

{0.6, 0} | 0.984 | 1.000 | 1.000 | 0.984 | 1.000 | 1.000 |

{0.6, 0.85} | 0.778 | 0.998 | 0.911 | 0.780 | 0.999 | 0.910 |

{0.85, 0.2} | 0.121 | 1.000 | 1.000 | 0.100 | 1.000 | 1.000 |

{0.85, 0.6} | 0.052 | 0.840 | 1.000 | 0.034 | 0.833 | 0.999 |

{0.96, 0.5} | 0.067 | 0.767 | 1.000 | 0.067 | 0.763 | 1.000 |

**Note**: Numbers reported are empirical rejections for the hypothesis ${H}_{0}:\theta ={\theta}_{0}$ and $\psi ={\psi}_{0}$. Size is measured with ${\theta}_{0}={\theta}_{DGP}$ and ${\psi}_{0}={\psi}_{DGP}$. Other choices for ${\theta}_{0}$ and ${\psi}_{0}$ are reported as tested values in the power tables.

**Table 5.**Data generated process (DGP) with $\{{\theta}_{1,DGP},{\theta}_{2,DGP},{\psi}_{DGP}\}=\{0.30,0.85,0.60\}$.

POWER [SIM Method] | |||
---|---|---|---|

T = 100 | Gaussian ARMA | ||

Tested Pair $\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{,}{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 0.903 | 0.998 | 0.326 |

{0.3, 0.2} | 0.978 | 0.999 | 0.974 |

{0.3, 0.85} | 0.994 | 0.999 | 0.310 |

{0.5, 0.5} | 0.998 | 1.000 | 0.275 |

{0.5, 0.96} | 1.000 | 1.000 | 0.918 |

{0.6, 0} | 0.998 | 0.999 | 0.993 |

{0.6, 0.85} | 1.000 | 1.000 | 0.327 |

{0.85, 0.2} | 1.000 | 1.000 | 0.862 |

{0.85, 0.6} | 1.000 | 1.000 | 0.105 |

{0.96, 0.5} | 1.000 | 1.000 | 0.231 |

{0.99, 0.99} | 1.000 | 1.000 | 0.987 |

**Table 6.**Data generated process (DGP) with $\{{\theta}_{DGP},{\psi}_{1,DGP},{\psi}_{2,DGP}\}=\{0.30,0.60,0.85\}$.

POWER [SIM Method] | |||
---|---|---|---|

T = 100 | Gaussian ARMA | ||

Tested Pair $\mathbf{\{}{\mathit{\theta}}_{\mathbf{0}}\mathbf{,}{\mathit{\psi}}_{\mathbf{0}}\mathbf{\}}$ | OLS-FBLC | OLS-Long AR | AutoCorr |

{0, 0.6} | 0.635 | 0.931 | 0.962 |

{0.3, 0.2} | 0.403 | 0.957 | 1.000 |

{0.3, 0.85} | 0.171 | 0.198 | 0.233 |

{0.5, 0.5} | 0.066 | 0.130 | 0.959 |

{0.5, 0.96} | 0.105 | 0.117 | 0.103 |

{0.6, 0} | 0.056 | 0.733 | 1.000 |

{0.6, 0.85} | 0.289 | 0.227 | 0.208 |

{0.85, 0.2} | 0.858 | 0.562 | 0.997 |

{0.85, 0.6} | 0.930 | 0.814 | 0.898 |

{0.96, 0.5} | 0.974 | 0.860 | 0.956 |

{0.99, 0.99} | 0.981 | 0.969 | 0.282 |

**Note**: Numbers reported are empirical rejections for the hypothesis ${H}_{0}:\theta ={\theta}_{0}$ and $\psi ={\psi}_{0}$. Various choices for ${\theta}_{0}$ and ${\psi}_{0}$ are reported as tested values in the power tables.

Time Period | Sample | Obs. | Mean | Std. Dev. | Skewness | Kurtosis |
---|---|---|---|---|---|---|

Jan. 1986–June 2019 | Log of Weekly Nominal Prices | 1732 | −2.8881 × ${10}^{-15}$ | 0.6582 | 0.6582 | 1.6844 |

Jan.1986–June 2019 | Log of Monthly Nominal Prices | 402 | 5.0485 × ${10}^{-16}$ | 0.6566 | 0.2541 | 1.6844 |

Jan. 1986–May 2019 | Log of Monthly Real Prices | 401 | 9.5241 × ${10}^{-16}$ | 0.8719 | 0.1165 | 1.5428 |

Jan. 1986–June 2019 | Weekly Nominal Returns | 1731 | 4.7302 × ${10}^{-19}$ | 0.05128 | −0.1449 | 5.9181 |

Jan. 1986–June 2019 | Monthly Nominal Returns | 401 | 6.9216 × ${10}^{-20}$ | 0.10253 | −0.35965 | 4.9748 |

Jan. 1986–May 2019 | Monthly Real Returns | 400 | 2.2898 × ${10}^{-18}$ | 0.10341 | −0.37958 | 5.0612 |

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## Share and Cite

**MDPI and ACS Style**

Khalaf, L.; Peraza López, B.
Simultaneous Indirect Inference, Impulse Responses and ARMA Models. *Econometrics* **2020**, *8*, 12.
https://doi.org/10.3390/econometrics8020012

**AMA Style**

Khalaf L, Peraza López B.
Simultaneous Indirect Inference, Impulse Responses and ARMA Models. *Econometrics*. 2020; 8(2):12.
https://doi.org/10.3390/econometrics8020012

**Chicago/Turabian Style**

Khalaf, Lynda, and Beatriz Peraza López.
2020. "Simultaneous Indirect Inference, Impulse Responses and ARMA Models" *Econometrics* 8, no. 2: 12.
https://doi.org/10.3390/econometrics8020012