# Improved Average Estimation in Seemingly Unrelated Regressions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model and Notation

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

## 3. Estimators

#### 3.1. Unrestricted Estimator

#### 3.2. Restricted Estimator

#### 3.3. Average Estimator

## 4. Large-Sample Approximate Bias and MSE

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 5. Monte Carlo Simulation

## 6. Application: Returns to Scale in US Banking Industry

#### 6.1. The Model

#### 6.2. The Data

#### 6.3. Estimation

- Ignore the extra observations in estimating $\mathbf{\Omega}$;
- Use the extra observations to estimate variances. This procedure has the disadvantage of producing estimates of $\mathbf{\Omega}$ that are not positive definite;
- Use the extra observations to estimate variances, and modifying the estimates of covariances using the method of (Srivastava and Zaatar 1973);
- Use all observations in estimation, following the method of (Hocking and Smith 1968).

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Lemma**

**A1.**

**Proof.**

**Proof.**

**Proof.**

**Proof.**

## References

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1. | Note that we do not assume that ${X}_{i}$’s are the same, nor do we assume they are different across equations. In other words, our model supports complete heterogeneity, partial heterogeneity, and complete homogeneity of regressors. |

2. | The second equality holds by using Equation (A.15). |

3. | The weight can be replaced by a positive part weight, that is, when $\tau /D<0$, we assign a zero weight to the restricted estimator. It is easy to verify that the MSE of the positive part is smaller. However, it will not change the approximations, so for simplicity we do not impose it at this stage. Nevertheless, the Monte Carlo and the application results are reported using the positive part weights. |

4. | See Equation (A.19) in Appendix A. |

5. | The data from 2000–2010 is downloaded from the Federal Reserve Bank of Chicago website, and the rest of the data from 2011–2018 is downloaded from the FFIEC Central Data Repository’s Public Data Distribution website. |

6. | The summary of the data for other years are not reported to save the space, but it is available upon request. |

7. | We only report the results for these three years to save the space. However, the results for other years are available upon request. |

8. | ${(I+A)}^{-1}=I-A+{A}^{2}-{A}^{3}+\dots \phantom{\rule{0.166667em}{0ex}}.$ |

**Figure 5.**United States Commercial Banks, Number and Average Assets (source: Federal Reserve Bank of St. Louis FRED database).

Variable | Min | Max | ||
---|---|---|---|---|

2000 | 2018 | 2000 | 2018 | |

C | 136.49 | 151.78 | 18,369,359.04 | 26,757,643.39 |

${w}_{1}$ | 7015.14 | 15,321.43 | 163,358.47 | 261,240.11 |

${w}_{2}$ | 6.75 | 9.61 | 12375.00 | 43,469.79 |

${w}_{3}$ | 3.20 | 0.02 | 206.53 | 48.82 |

${w}_{4}$ | 0.26 | 0.02 | 265.36 | 153.96 |

${w}_{5}$ | 0.46 | 0.06 | 49.21 | 16.82 |

${y}_{1}$ | 1.25 | 1.00 | 42,638,250.00 | 173,922,000.00 |

${y}_{2}$ | 0.25 | 170.50 | 168,465,250.00 | 686,161,250.00 |

${y}_{3}$ | 45.75 | 125.50 | 178,056,500.00 | 457,517,750.00 |

${y}_{4}$ | 89.50 | 0.25 | 144,188,250.00 | 703,099,250.00 |

${y}_{5}$ | 39.00 | 42.50 | 86,346,000.00 | 704,384,250.00 |

Variable | Mean | STD | ||

2000 | 2018 | 2000 | 2018 | |

C | 24,454.49 | 43,803.47 | 331,091.51 | 643,872.50 |

${w}_{1}$ | 26,175.32 | 49,386.16 | 7084.42 | 15,332.39 |

${w}_{2}$ | 230.31 | 265.78 | 351.89 | 920.70 |

${w}_{3}$ | 33.22 | 3.51 | 6.38 | 2.75 |

${w}_{4}$ | 14.79 | 2.43 | 9.01 | 3.87 |

${w}_{5}$ | 22.87 | 3.46 | 4.79 | 1.90 |

${y}_{1}$ | 57,014.23 | 268,717.11 | 754,295.15 | 4,722,816.78 |

${y}_{2}$ | 175,843.57 | 881,915.93 | 2,961,928.59 | 15,632,429.24 |

${y}_{3}$ | 206,836.22 | 929,043.01 | 2,423,963.16 | 10,526,805.47 |

${y}_{4}$ | 192,820.92 | 862,701.61 | 2,985,314.66 | 16,392,980.08 |

${y}_{5}$ | 85,019.93 | 842,793.39 | 1,554,035.20 | 16,868,329.90 |

Bank Groups | Asset Size (in Millions of Dollars of the Year) | Number of Banks | Share of Banks |
---|---|---|---|

2000 | |||

Large Banks | Assets ≥ 500 | 739 | 8.9 |

Medium Banks | 100 ≤ Assets < 500 | 2946 | 35.5 |

Small Banks | Assets < 100 | 4620 | 55.6 |

2018 | |||

Large Banks | Assets ≥ 729 | 988 | 19.8 |

Medium Banks | 146≤ Assets <729 | 2298 | 46.1 |

Small Banks | Assets < 146 | 1704 | 34.1 |

Bank Size | Estimator | Estimates of RTS | |||||
---|---|---|---|---|---|---|---|

${\mathit{D}}_{10}$ | ${\mathit{Q}}_{25}$ | ${\mathit{Q}}_{50}$ | ${\mathit{Q}}_{75}$ | ${\mathit{D}}_{90}$ | Mean | ||

2000 | |||||||

Restricted | 0.979 | 0.988 | 0.997 | 1.008 | 1.019 | 0.998 | |

Large Banks | Unrestricted | 0.971 | 0.985 | 0.998 | 1.009 | 1.021 | 0.996 |

Average Estimator | 0.980 | 0.989 | 0.998 | 1.007 | 1.018 | 0.998 | |

Medium Banks | Unrestricted | 0.969 | 0.992 | 1.017 | 1.043 | 1.071 | 1.019 |

Average Estimator | 0.995 | 1.002 | 1.011 | 1.020 | 1.028 | 1.011 | |

Small Banks | Unrestricted | 0.997 | 1.014 | 1.034 | 1.055 | 1.075 | 1.035 |

Average Estimator | 1.006 | 1.014 | 1.024 | 1.034 | 1.044 | 1.025 | |

2009 | |||||||

Restricted | 1.007 | 1.016 | 1.026 | 1.038 | 1.051 | 1.028 | |

Large Banks | Unrestricted | 1.001 | 1.025 | 1.053 | 1.080 | 1.108 | 1.055 |

Average Estimator | 1.004 | 1.026 | 1.051 | 1.075 | 1.101 | 1.052 | |

Medium Banks | Unrestricted | 0.956 | 0.990 | 1.032 | 1.079 | 1.125 | 1.038 |

Average Estimator | 0.961 | 0.993 | 1.031 | 1.076 | 1.118 | 1.037 | |

Small Banks | Unrestricted | 0.979 | 1.013 | 1.049 | 1.085 | 1.124 | 1.050 |

Average Estimator | 0.983 | 1.015 | 1.048 | 1.082 | 1.118 | 1.049 | |

2018 | |||||||

Restricted | 1.009 | 1.021 | 1.034 | 1.049 | 1.066 | 1.037 | |

Large Banks | Unrestricted | 1.009 | 1.052 | 1.100 | 1.140 | 1.184 | 1.098 |

Average Estimator | 1.010 | 1.052 | 1.094 | 1.132 | 1.172 | 1.093 | |

Medium Banks | Unrestricted | 0.915 | 0.950 | 0.988 | 1.030 | 1.070 | 0.989 |

Average Estimator | 0.919 | 0.953 | 0.989 | 1.029 | 1.068 | 0.990 | |

Small Banks | Unrestricted | 0.960 | 0.997 | 1.042 | 1.104 | 1.166 | 1.054 |

Average Estimator | 0.962 | 0.998 | 1.040 | 1.098 | 1.158 | 1.051 |

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**MDPI and ACS Style**

Mehrabani, A.; Ullah, A.
Improved Average Estimation in Seemingly Unrelated Regressions. *Econometrics* **2020**, *8*, 15.
https://doi.org/10.3390/econometrics8020015

**AMA Style**

Mehrabani A, Ullah A.
Improved Average Estimation in Seemingly Unrelated Regressions. *Econometrics*. 2020; 8(2):15.
https://doi.org/10.3390/econometrics8020015

**Chicago/Turabian Style**

Mehrabani, Ali, and Aman Ullah.
2020. "Improved Average Estimation in Seemingly Unrelated Regressions" *Econometrics* 8, no. 2: 15.
https://doi.org/10.3390/econometrics8020015