10.3 Arch/Garch 模型 · 如何使用優礦進行 GARCH 模型分析 · Python 量化交易教程

# 10.3 Arch/Garch 模型 · 如何使用優礦進行 GARCH 模型分析 > 來源：https://uqer.io/community/share/56209478f9f06c4c5e2fb5f1 ## ARCH 建模示例小弟近來學習波動率建模相關知識，正巧發現優礦中有ARCH包，所以通過翻譯ARCH包文檔中的示例，來學習相關函數用法。翻譯有不少不準確的地方，請大家指出，我會及時改進。 ```py import numpy as np import matplotlib.mlab as mlab import matplotlib.pyplot as plt try: import seaborn except ImportError: pass ``` ## 一、準備工作這個例子中使用通聯DataAPI提供的行情數據。以平安銀行為例。下面畫出了平安銀行從1990年到2015年收益率情況。 ```py import datetime as dt st = dt.datetime(1990,1,1) en = dt.datetime(2015,6,30) data = DataAPI.MktEqudGet(secID=u"",ticker=u"000001",tradeDate=u"",beginDate=u"",endDate=u"",field=u"ticker,secShortName,tradeDate,closePrice,PE",pandas="1") # data.index = data.tradeDate returns = 100 * data['closePrice'].pct_change().dropna() figure = returns.plot(figsize=(20,6)) ``` ![](https://box.kancloud.cn/2016-07-30_579cbdb48037a.png) ## 二、實現常見模型描述一個模型最簡單的方法就是使用Python的類庫`arch.arch_model`。使用這個類庫可以實現大多數常見的模型。簡單的調用`arch `類庫會得到一個均值恒定，誤差符合正態分布，符合`GARCH(1,1)`波動率過程的模型。 ![](https://box.kancloud.cn/2016-07-30_579cbdb49b92e.jpg) 通過調用`fit`方法，可以對這個模型進行估計。可選輸入項 `update_freq`控制優化器輸出結果的頻率，`disp`控制是否返回收斂相關信息。返回結果直接提供了估計的參數值和相關數量，同時包含估計結果的摘要信息。 ## GARCH (均值恒定) 使用默認選項，可以生成一個均值恒定，誤差符合正態分布，同時符合`GARCH(1,1)`條件方差的模型。通過擬合獲得模型的參數，下面展示擬合結果的摘要。 ```py from arch import arch_model am = arch_model(returns) res = am.fit(update_freq=5) print(res.summary()) Iteration: 5, Func. Count: 38, Neg. LLF: 5787.7752693 Iteration: 10, Func. Count: 75, Neg. LLF: 5785.39088499 Optimization terminated successfully. (Exit mode 0) Current function value: 5785.0196556 Iterations: 14 Function evaluations: 101 Gradient evaluations: 14 Constant Mean - GARCH Model Results ============================================================================== Dep. Variable: closePrice R-squared: -0.000 Mean Model: Constant Mean Adj. R-squared: -0.000 Vol Model: GARCH Log-Likelihood: -5785.02 Distribution: Normal AIC: 11578.0 Method: Maximum Likelihood BIC: 11601.1 No. Observations: 2373 Date: Fri, Oct 16 2015 Df Residuals: 2369 Time: 11:08:36 Df Model: 4 Mean Model ============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------ mu 0.0349 2.825e-03 12.348 4.985e-35 [2.935e-02,4.042e-02] Volatility Model ============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------ omega 0.0383 2.763e-03 13.861 1.085e-43 [3.288e-02,4.371e-02] alpha[1] 0.0287 2.439e-04 117.761 0.000 [2.825e-02,2.920e-02] beta[1] 0.9694 4.516e-04 2146.801 0.000 [ 0.969, 0.970] ============================================================================== Covariance estimator: robust ``` `plot()` 函數可以快速展示標的的標準偏差和條件波動率。 ```py fig = res.plot(annualize='D') ``` ![](https://box.kancloud.cn/2016-07-30_579cbdb4b06be.png) ## GJR-GARCH `arch_model` 在構建模型時，還可以添加附加參數。在這個例子中，設置`o`為1，即包含了非對稱沖擊的一階滯后項，從而將原GARCH模型轉換為一個GJR-GARCH模型。新的模型具有動態方差，由下面公式給出： ![](https://box.kancloud.cn/2016-07-30_579cbdb4c948e.jpg) 其中，`I`是一個指標函數，當它的輸入參數為真時，返回為1. 對數似然函數值改進了非對稱項引入的平穩性問題，從而使參數估計具有很高的顯著性。 ```py am = arch_model(returns, p=1, o=1, q=1) res = am.fit(update_freq=5, disp='off') print(res.summary()) ``` ## TARCH/ZARCH TARCH模型 (又稱為 ZARCH模型) 是對波動率的絕對值進行建模. 使用該模型時，需要在`arch_model`建構函數中，設置`power=1.0`。因為默認的階數為2，對應的是用平方項表示的方差變化過程。 TARCH model的波動率過程由下面公式給出： ![](https://box.kancloud.cn/2016-07-30_579cbdb4dd0d9.jpg) 更一般的情況，模型的動態波動率代碼`κ`階。這種情況由下面的公式給出 ![](https://box.kancloud.cn/2016-07-30_579cbdb4f26ab.jpg) 其中，條件方差為![](https://box.kancloud.cn/2016-07-30_579cbdb51609d.jpg) 盡管對數似然函數值變動范圍更小，TARCH 模型還是對擬合過程做了改進。 ```py am = arch_model(returns, p=1, o=1, q=1, power=1.0) res = am.fit(update_freq=5) print(res.summary()) Iteration: 5, Func. Count: 45, Neg. LLF: 5765.36462439 Iteration: 10, Func. Count: 84, Neg. LLF: 5758.70411096 Iteration: 15, Func. Count: 121, Neg. LLF: 5758.63601597 Optimization terminated successfully. (Exit mode 0) Current function value: 5758.6360268 Iterations: 15 Function evaluations: 121 Gradient evaluations: 15 Constant Mean - TARCH/ZARCH Model Results ============================================================================== Dep. Variable: closePrice R-squared: -0.000 Mean Model: Constant Mean Adj. R-squared: -0.000 Vol Model: TARCH/ZARCH Log-Likelihood: -5758.64 Distribution: Normal AIC: 11527.3 Method: Maximum Likelihood BIC: 11556.1 No. Observations: 2373 Date: Fri, Oct 16 2015 Df Residuals: 2368 Time: 12:50:24 Df Model: 5 Mean Model ============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------ mu 0.0625 4.323e-03 14.456 2.296e-47 [5.402e-02,7.096e-02] Volatility Model ============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------ omega 0.0457 4.933e-03 9.257 2.107e-20 [3.599e-02,5.533e-02] alpha[1] 0.0594 1.012e-03 58.701 0.000 [5.742e-02,6.139e-02] gamma[1] -0.0184 5.771e-04 -31.882 4.768e-223 [-1.953e-02,-1.727e-02] beta[1] 0.9498 2.510e-03 378.466 0.000 [ 0.945, 0.955] ============================================================================== Covariance estimator: robust ``` ## 學生T分布誤差金融資產回報率的分布往往體現出肥尾現象，學生`T`分布是一種簡單的方法，可以用來捕捉這種特性。在調用`arch_model` 構建函數時，可以將概率分布從正態分布轉換為學生T分布。標準化的新息展示出，分布函數具有一個將近10個估計自由度的肥尾。對數似然函數值同樣有大的改善。 ```py am = arch_model(returns, p=1, o=1, q=1, power=1.0, dist='StudentsT') res = am.fit(update_freq=5) print(res.summary()) Iteration: 5, Func. Count: 48, Neg. LLF: 5522.16193119 Iteration: 10, Func. Count: 93, Neg. LLF: 5475.09377571 Iteration: 15, Func. Count: 138, Neg. LLF: 5451.04968458 Iteration: 20, Func. Count: 179, Neg. LLF: 5435.39156625 Iteration: 25, Func. Count: 223, Neg. LLF: 5434.83467797 Iteration: 30, Func. Count: 269, Neg. LLF: 5434.83086355 Optimization terminated successfully. (Exit mode 0) Current function value: 5434.83085822 Iterations: 33 Function evaluations: 304 Gradient evaluations: 33 Constant Mean - TARCH/ZARCH Model Results ==================================================================================== Dep. Variable: closePrice R-squared: -0.001 Mean Model: Constant Mean Adj. R-squared: -0.001 Vol Model: TARCH/ZARCH Log-Likelihood: -5434.83 Distribution: Standardized Student's t AIC: 10881.7 Method: Maximum Likelihood BIC: 10916.3 No. Observations: 2373 Date: Fri, Oct 16 2015 Df Residuals: 2367 Time: 13:04:54 Df Model: 6 Mean Model =============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------- mu -1.3518e-08 1.493e-10 -90.539 0.000 [-1.381e-08,-1.323e-08] Volatility Model =============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------- omega 0.0894 0.304 0.294 0.769 [ -0.507, 0.686] alpha[1] 0.1185 9.143e-03 12.962 2.004e-38 [ 0.101, 0.136] gamma[1] -2.9293e-03 4.117e-03 -0.712 0.477 [-1.100e-02,5.139e-03] beta[1] 0.8829 9.652e-02 9.148 5.803e-20 [ 0.694, 1.072] Distribution ============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------ nu 3.6773 0.142 25.919 4.047e-148 [ 3.399, 3.955] ============================================================================== Covariance estimator: robust ``` ## 使用固定參數在一些場景下，相比估計出來的參數，使用固定參數可能更讓人感興趣。使用`arch_model`的`fix()`可以同樣生成一個模型。除了沒有與推斷相關的值(標準差，t統計量等)的差別之外，新的模型和通常的模型沒有什么差別。在這個例子中，將參數固定設置為之前估計模型的對稱版本。 ```py fixed_res = am.fix([0.0235, 0.01, 0.06, 0.0, 0.9382, 8.0]) print(fixed_res.summary()) Constant Mean - TARCH/ZARCH Model Results ===================================================================================== Dep. Variable: closePrice R-squared: -- Mean Model: Constant Mean Adj. R-squared: -- Vol Model: TARCH/ZARCH Log-Likelihood: -5579.95 Distribution: Standardized Student's t AIC: 11171.9 Method: User-specified Parameters BIC: 11206.5 No. Observations: 2373 Date: Fri, Oct 16 2015 Time: 13:04:57 Mean Model ===================== coef --------------------- mu 0.0235 Volatility Model ===================== coef --------------------- omega 0.0100 alpha[1] 0.0600 gamma[1] 0.0000 beta[1] 0.9382 Distribution ===================== coef --------------------- nu 8.0000 ===================== Results generated with user-specified parameters. Since the model was not estimated, there are no std. errors. ``` ```py import pandas as pd df = pd.concat([res.conditional_volatility,fixed_res.conditional_volatility],1) df.columns = ['Estimated', 'Fixed'] df.plot() <matplotlib.axes.AxesSubplot at 0x6b5cd90> ``` ![](img/Hw+FwOBwOh8PhcDgcDofD4XA4HA6Hw+H0Jf8PCkWn9o7i7uoAAAAASUVORK5CYII=.png) 三、通過多個組件模塊創建模型模型同樣可以使用代表三類模型的`arch`模塊進行系統性的構建。 + A mean model (arch.mean) + Zero mean (ZeroMean) - useful if using residuals from a model estimated separately + Constant mean (ConstantMean) - common for most liquid financial assets + Autoregressive (ARX) with optional exogenous regressors + Heterogeneous (HARX) autoregression with optional exogenous regressors + Exogenous regressors only (LS) + A volatility process (arch.volatility) + ARCH (ARCH) + GARCH (GARCH) + GJR-GARCH (GARCH using o argument) + TARCH/ZARCH (GARCH using power argument set to 1) + Power GARCH and Asymmetric Power GARCH (GARCH using power) + Heterogeneous ARCH (HARCH) + Parameterless Models + Exponentially Weighted Moving Average Variance, known as RiskMetrics (EWMAVariance) + Weighted averages of EWMAs, known as the RiskMetrics 2006 methodology (RiskMetrics2006) + A distribution (arch.distribution) + Normal (Normal) + Standardized Students's T (StudentsT) ## Mean Models 均值模型第一種選擇是使用均值模型。對于很多流動性充足的金融資產來說，恒定均值(甚至是0均值)的模型就足夠了。對于其他一些時間序列，若通貨膨脹率數據，可能需要更加復雜的模型。下面的例子使用了中國居民消費價格指數(CPI)數據。這些數據由通聯DataAPI提供。 ```py core_cpi = DataAPI.ChinaDataCPIGet(indicID=u"M030000003",indicName=u"",beginDate=u"20050101",endDate=u"",field=u"",pandas="1") ann_inflation = 100 * core_cpi.sort(columns='periodDate').dataValue.pct_change(12).dropna() fig = ann_inflation.plot() fig <matplotlib.axes.AxesSubplot at 0x6e6a090> ``` ![](https://box.kancloud.cn/2016-07-30_579cbdb52cb3b.png) 所有的均值模型都派生自恒定方差、正態分布誤差的基礎模型。對于`ARX`模型，`lags`輸入參數制定了模型需要包括的滯后項階數。 ```py from arch.univariate import ARX ar = ARX(ann_inflation, lags = [1, 3, 12]) print(ar.fit().summary()) AR - Constant Variance Model Results ============================================================================== Dep. Variable: dataValue R-squared: 0.264 Mean Model: AR Adj. R-squared: 0.242 Vol Model: Constant Variance Log-Likelihood: -685.867 Distribution: Normal AIC: 1381.73 Method: Maximum Likelihood BIC: 1395.00 No. Observations: 105 Date: Fri, Oct 16 2015 Df Residuals: 100 Time: 13:21:29 Df Model: 5 Mean Model ================================================================================= coef std err t P>|t| 95.0% Conf. Int. --------------------------------------------------------------------------------- Const -1.4076 254.081 -5.540e-03 0.996 [-4.994e+02,4.966e+02] dataValue[1] 0.4233 0.112 3.777 1.590e-04 [ 0.204, 0.643] dataValue[3] 0.1590 1.423e-02 11.173 5.523e-29 [ 0.131, 0.187] dataValue[12] -0.0117 1.050e-03 -11.133 8.695e-29 [-1.374e-02,-9.628e-03] Volatility Model ============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------ sigma2 2.7619e+04 2.259e+08 1.223e-04 1.000 [-4.427e+08,4.427e+08] ============================================================================== Covariance estimator: White's Heteroskedasticity Consistent Estimator ``` ## Volatility Processes 波動率過程波動率過程可以通過在均值模型中添加`volatility`屬性來實現。下面的例子中將模型的波動率設置為`ARCH(5)`，`update_freq`和`disp`用來約束`fit()`進行估計時候的輸出內容。 ```py from arch.univariate import ARCH, GARCH ar.volatility = ARCH(p=5) res = ar.fit(update_freq=0, disp='off') print(res.summary()) AR - ARCH Model Results ============================================================================== Dep. Variable: dataValue R-squared: 0.104 Mean Model: AR Adj. R-squared: 0.077 Vol Model: ARCH Log-Likelihood: -576.003 Distribution: Normal AIC: 1172.01 Method: Maximum Likelihood BIC: 1198.55 No. Observations: 105 Date: Fri, Oct 16 2015 Df Residuals: 95 Time: 13:25:52 Df Model: 10 Mean Model ================================================================================= coef std err t P>|t| 95.0% Conf. Int. --------------------------------------------------------------------------------- Const -11.9096 29.775 -0.400 0.689 [-70.268, 46.449] dataValue[1] 0.8647 2.093e-02 41.313 0.000 [ 0.824, 0.906] dataValue[3] -0.0296 9.925e-03 -2.978 2.904e-03 [-4.901e-02,-1.010e-02] dataValue[12] -0.0154 2.936e-04 -52.419 0.000 [-1.597e-02,-1.482e-02] Volatility Model =============================================================================== coef std err t P>|t| 95.0% Conf. Int. ------------------------------------------------------------------------------- omega 404.9890 1.164e+05 3.480e-03 0.997 [-2.277e+05,2.285e+05] alpha[1] 0.6348 5.176e-02 12.266 1.383e-34 [ 0.533, 0.736] alpha[2] 0.2690 1.767e-02 15.221 2.552e-52 [ 0.234, 0.304] alpha[3] 0.0962 1.161e-02 8.283 1.196e-16 [7.341e-02, 0.119] alpha[4] 2.5205e-10 3.554e-04 7.093e-07 1.000 [-6.965e-04,6.965e-04] alpha[5] -4.6303e-10 1.178e-05 -3.929e-05 1.000 [-2.310e-05,2.309e-05] =============================================================================== Covariance estimator: robust ``` 下面的圖標，展示了標準化后的新息和條件波動率的情況。可以看出，雖然進行了標準化處理，一些位置還是出現了較大的誤差(通過振幅來看) ```py fig = res.plot() ``` ![](https://box.kancloud.cn/2016-07-30_579cbdb542518.png) ## Distributions 概率分布情況最后提一下，模型的概率分布可以通過設置`distribution`屬性來修改，將默認的正態分布修改為學生T分布。學生T分布改進了模型，模型的估計自由度大約在8左右。備注 1 本文章是將Python ARCH類庫的幫助文檔進行翻譯產生的。原文地址： http://nbviewer.ipython.org/github/bashtage/arch/blob/master/examples/univariate_volatility_modeling.ipynb 本文將原文中的數據源替換為優礦/通聯DataAPI的數據，方便優礦用戶進行參照和分析。 2 文中，多次提到`iter`，但這個疑為原作者筆誤，故都改為`update_freq` 3 原文最后一章‘WTI Crude’由于給出的例子和當前的`arch`包版本不兼容，無法正確運行，故沒有引入