Mildly Explosive Autoregression with Strong Mixing Errors
In this paper, we consider the mildly explosive autoregression <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>=</mo><...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-11-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/12/1730 |
| _version_ | 1797459310106640384 |
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| author | Xian Liu Xiaoqin Li Min Gao Wenzhi Yang |
| author_facet | Xian Liu Xiaoqin Li Min Gao Wenzhi Yang |
| author_sort | Xian Liu |
| collection | DOAJ |
| description | In this paper, we consider the mildly explosive autoregression <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>=</mo><msub><mi>ρ</mi><mi>n</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mi>t</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>c</mi><mo>/</mo><msup><mi>n</mi><mi>ν</mi></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>u</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are arithmetically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing errors. Under some weak conditions, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>E</mi><mo>|</mo></mrow><msub><mi>u</mi><mn>1</mn></msub><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn><mo>+</mo><mi>δ</mi></mrow></msup><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and mixing coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mo>(</mo><mn>2</mn><mo>+</mo><mn>8</mn><mo>/</mo><mi>δ</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the Cauchy limiting distribution is established for the least squares (LS) estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>n</mi></msub></semantics></math></inline-formula>, which extends the cases of independent errors and geometrically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing errors. Some simulations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>n</mi></msub></semantics></math></inline-formula>, such as the empirical probability of the confidence interval and the empirical density, are presented to illustrate the Cauchy limiting distribution, which have good finite sample performances. In addition, we use the Cauchy limiting distribution of the LS estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> to illustrate real data from the NASDAQ composite index from April 2011 to April 2021. |
| first_indexed | 2024-03-09T16:49:37Z |
| format | Article |
| id | doaj.art-2cef0f328bf645efa64550437a948b81 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T16:49:37Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-2cef0f328bf645efa64550437a948b812023-11-24T14:41:57ZengMDPI AGEntropy1099-43002022-11-012412173010.3390/e24121730Mildly Explosive Autoregression with Strong Mixing ErrorsXian Liu0Xiaoqin Li1Min Gao2Wenzhi Yang3School of Big Data and Statistics, Anhui University, Hefei 230039, ChinaSchool of Big Data and Statistics, Anhui University, Hefei 230039, ChinaSchool of Big Data and Statistics, Anhui University, Hefei 230039, ChinaSchool of Big Data and Statistics, Anhui University, Hefei 230039, ChinaIn this paper, we consider the mildly explosive autoregression <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>=</mo><msub><mi>ρ</mi><mi>n</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mi>t</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>c</mi><mo>/</mo><msup><mi>n</mi><mi>ν</mi></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>u</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are arithmetically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing errors. Under some weak conditions, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>E</mi><mo>|</mo></mrow><msub><mi>u</mi><mn>1</mn></msub><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn><mo>+</mo><mi>δ</mi></mrow></msup><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and mixing coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mo>(</mo><mn>2</mn><mo>+</mo><mn>8</mn><mo>/</mo><mi>δ</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the Cauchy limiting distribution is established for the least squares (LS) estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>n</mi></msub></semantics></math></inline-formula>, which extends the cases of independent errors and geometrically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing errors. Some simulations for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>n</mi></msub></semantics></math></inline-formula>, such as the empirical probability of the confidence interval and the empirical density, are presented to illustrate the Cauchy limiting distribution, which have good finite sample performances. In addition, we use the Cauchy limiting distribution of the LS estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> to illustrate real data from the NASDAQ composite index from April 2011 to April 2021.https://www.mdpi.com/1099-4300/24/12/1730mildly explosive autoregressionleast squares estimatorCauchy distributionstrong mixing sequences |
| spellingShingle | Xian Liu Xiaoqin Li Min Gao Wenzhi Yang Mildly Explosive Autoregression with Strong Mixing Errors Entropy mildly explosive autoregression least squares estimator Cauchy distribution strong mixing sequences |
| title | Mildly Explosive Autoregression with Strong Mixing Errors |
| title_full | Mildly Explosive Autoregression with Strong Mixing Errors |
| title_fullStr | Mildly Explosive Autoregression with Strong Mixing Errors |
| title_full_unstemmed | Mildly Explosive Autoregression with Strong Mixing Errors |
| title_short | Mildly Explosive Autoregression with Strong Mixing Errors |
| title_sort | mildly explosive autoregression with strong mixing errors |
| topic | mildly explosive autoregression least squares estimator Cauchy distribution strong mixing sequences |
| url | https://www.mdpi.com/1099-4300/24/12/1730 |
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