Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations
This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" displa...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/8/816 |
| _version_ | 1797538285382270976 |
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| author | Eunju Hwang |
| author_facet | Eunju Hwang |
| author_sort | Eunju Hwang |
| collection | DOAJ |
| description | This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> is derived uniformly over stationary values in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, focusing on <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></mrow></semantics></math></inline-formula> as sample size <i>n</i> tends to infinity. For tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, but no condition on the rate 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> is required. It is shown that, for the tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the LSE is inconsistent, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mi>n</mi><mo>/</mo><mo>(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow></semantics></math></inline-formula>-consistent, and for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>n</mi><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mo>/</mo><mi>α</mi></mrow></msup><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>; and no restriction on the rate 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> is necessary. |
| first_indexed | 2024-03-10T12:29:20Z |
| format | Article |
| id | doaj.art-d2720711277346e892faeb8e5866197f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T12:29:20Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d2720711277346e892faeb8e5866197f2023-11-21T14:49:31ZengMDPI AGMathematics2227-73902021-04-019881610.3390/math9080816Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH InnovationsEunju Hwang0Department of Applied Statistics, Gachon University, Seongnam 13120, KoreaThis paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> is derived uniformly over stationary values in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, focusing on <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></mrow></semantics></math></inline-formula> as sample size <i>n</i> tends to infinity. For tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, but no condition on the rate 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> is required. It is shown that, for the tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the LSE is inconsistent, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mi>n</mi><mo>/</mo><mo>(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow></semantics></math></inline-formula>-consistent, and for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>n</mi><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mo>/</mo><mi>α</mi></mrow></msup><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>; and no restriction on the rate 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> is necessary.https://www.mdpi.com/2227-7390/9/8/816autoregressionaugmented GARCHheavy-tailedlimit theory |
| spellingShingle | Eunju Hwang Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations Mathematics autoregression augmented GARCH heavy-tailed limit theory |
| title | Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations |
| title_full | Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations |
| title_fullStr | Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations |
| title_full_unstemmed | Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations |
| title_short | Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations |
| title_sort | limit theory for stationary autoregression with heavy tailed augmented garch innovations |
| topic | autoregression augmented GARCH heavy-tailed limit theory |
| url | https://www.mdpi.com/2227-7390/9/8/816 |
| work_keys_str_mv | AT eunjuhwang limittheoryforstationaryautoregressionwithheavytailedaugmentedgarchinnovations |