1.2 舉例
1.2.1 ARCH建模
以下代碼需要在 IPython notebook下運行：
In [1]:
import warnings
warnings.simplefilter('ignore')
%matplotlib inline
import seaborn
seaborn.set_style('darkgrid')
In [2]:
seaborn.mpl.rcParams['figure.figsize'] = (10.0, 6.0) seaborn.mpl.rcParams['savefig.dpi'] = 90 seaborn.mpl.rcParams['font.family'] = 'serif' seaborn.mpl.rcParams['font.size'] = 14
1.2.2 Setup
本例利用來自Yahoo金融的數據，利用pandas-datareader導入。
import datetime as dt
import pandas_datareader.data as web
st = dt.datetime(1988,1,1)
en = dt.datetime(2018,1,1)
data = web.get_data_famafrench('F-F_Research_Data_Factors_daily', start=st, end=en)
mkt_returns = data[0]['Mkt-RF'] + data[0]['RF']
returns = mkt_returns
figure = returns.plot()
1.2.3 Specifying Common Models
最簡單的模型構建方法是使用模型構建方法 arch.arch_model，它能表示多數通用模型。最簡單的arch調用將會使用均值爲常數模型的GARCH（1，1）波動過程且誤差服從正態分佈。
該模型通過調用fit方法進行估計。可選迭代參數iter控制最優化產出的頻率，disp控制是否返回收斂信息。結果類直接返回估計參數及相關數值，如果summary方法一樣。
1.2.4 GARCH (with a Constant Mean)
默認選項會產生基於均值的GARCH（1，1）條件方差以及正態分佈誤差。
from arch import arch_model
am = arch_model(returns)
res = am.fit(update_freq=5)
print(res.summary())
Iteration: 5, Func. Count: 41, Neg. LLF: 9820.63969763662
Iteration: 10, Func. Count: 74, Neg. LLF: 9817.274895653107
Optimization terminated successfully. (Exit mode 0)
Current function value: 9817.274187120329
Iterations: 13
Function evaluations: 92
Gradient evaluations: 13
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: None R-squared: -0.000
Mean Model: Constant Mean Adj. R-squared: -0.000
Vol Model: GARCH Log-Likelihood: -9817.27
Distribution: Normal AIC: 19642.5
Method: Maximum Likelihood BIC: 19670.3
No. Observations: 7561
Date: Mon, Jan 21 2019 Df Residuals: 7557
Time: 19:24:29 Df Model: 4
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0687 9.098e-03 7.555 4.176e-14 [5.091e-02,8.657e-02]
Volatility Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
omega 0.0151 4.174e-03 3.622 2.923e-04 [6.937e-03,2.330e-02]
alpha[1] 0.0836 1.231e-02 6.788 1.138e-11 [5.943e-02, 0.108]
beta[1] 0.9014 1.436e-02 62.771 0.000 [ 0.873, 0.930]
============================================================================
Covariance estimator: robust
plot() 方法可以用來快速將標準化殘差和條件波動率顯示出來。
fig = res.plot(annualize='D')
1.2.5 GJR-GARCH
其他輸入參數可以用來構建有關模型。該舉例設定o爲1，這表示該模型包括一階滯後的非對稱效應，且使GARCH模型變爲GJR-GARCH模型，其方差變化表示如下：
這裏I爲指標函數，當其參數爲真時值爲1.
由於引進了非對稱項，對數似然率得到很大提升，估計參數高度顯著。
Constant Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable: None R-squared: -0.000
Mean Model: Constant Mean Adj. R-squared: -0.000
Vol Model: GJR-GARCH Log-Likelihood: -9713.51
Distribution: Normal AIC: 19437.0
Method: Maximum Likelihood BIC: 19471.7
No. Observations: 7561
Date: Mon, Jan 21 2019 Df Residuals: 7556
Time: 19:33:32 Df Model: 5
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0442 8.759e-03 5.051 4.398e-07 [2.707e-02,6.141e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 0.0186 4.608e-03 4.044 5.252e-05 [9.603e-03,2.766e-02]
alpha[1] 5.9351e-03 5.981e-03 0.992 0.321 [-5.788e-03,1.766e-02]
gamma[1] 0.1355 2.086e-02 6.495 8.291e-11 [9.459e-02, 0.176]
beta[1] 0.9044 1.464e-02 61.778 0.000 [ 0.876, 0.933]
=============================================================================
Covariance estimator: robust
1.2.6 TARCH/ZARCH
TARCH (也作 ZARCH)模型使用絕對值來構建波動率模型。該模型的精度參數 power=1.0。其默認精度參數爲power=2，且方差過程與平方項線性相關。
TARCH波動率過程表示如下：
更多的模型參數在精度 (κκ)下的波動率表示如下：
此處條件方差爲 
該TARCH 模型也改進了擬合度，雖然對數似然率下降。
Iteration: 5, Func. Count: 54, Neg. LLF: 9701.46915224492
Iteration: 10, Func. Count: 94, Neg. LLF: 9685.575185675976
Iteration: 15, Func. Count: 130, Neg. LLF: 9683.248100354442
Optimization terminated successfully. (Exit mode 0)
Current function value: 9683.248099008848
Iterations: 16
Function evaluations: 137
Gradient evaluations: 16
Constant Mean - TARCH/ZARCH Model Results
==============================================================================
Dep. Variable: None R-squared: -0.000
Mean Model: Constant Mean Adj. R-squared: -0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -9683.25
Distribution: Normal AIC: 19376.5
Method: Maximum Likelihood BIC: 19411.1
No. Observations: 7561
Date: Mon, Jan 21 2019 Df Residuals: 7556
Time: 19:40:04 Df Model: 5
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0386 8.899e-03 4.338 1.437e-05 [2.116e-02,5.604e-02]
Volatility Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
omega 0.0215 4.333e-03 4.961 6.998e-07 [1.301e-02,2.999e-02]
alpha[1] 0.0137 6.461e-03 2.114 3.448e-02 [9.980e-04,2.632e-02]
gamma[1] 0.1195 1.440e-02 8.301 1.031e-16 [9.130e-02, 0.148]
beta[1] 0.9213 1.010e-02 91.219 0.000 [ 0.901, 0.941]
============================================================================
Covariance estimator: robust
1.2.7 Student’s T Errors（學生T分佈）
金融收益率通常肥尾，學生T分佈是一種捕獲該特徵的簡單方法。使用arch方法可以把分佈從正態變爲學生T分佈。
標準化殘差在估計自由度10附近呈現肥尾特徵，對數似然率也有大幅增加。
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: 57, Neg. LLF: 9535.283529224867
Iteration: 10, Func. Count: 102, Neg. LLF: 9506.009581127182
Iteration: 15, Func. Count: 144, Neg. LLF: 9505.381448443233
Optimization terminated successfully. (Exit mode 0)
Current function value: 9505.38144747584
Iterations: 15
Function evaluations: 145
Gradient evaluations: 15
Constant Mean - TARCH/ZARCH Model Results
====================================================================================
Dep. Variable: None R-squared: -0.000
Mean Model: Constant Mean Adj. R-squared: -0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -9505.38
Distribution: Standardized Student's t AIC: 19022.8
Method: Maximum Likelihood BIC: 19064.3
No. Observations: 7561
Date: Mon, Jan 21 2019 Df Residuals: 7555
Time: 19:47:19 Df Model: 6
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0593 8.031e-03 7.382 1.558e-13 [4.355e-02,7.503e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 0.0153 3.142e-03 4.875 1.087e-06 [9.159e-03,2.148e-02]
alpha[1] 7.9883e-03 4.902e-03 1.630 0.103 [-1.619e-03,1.760e-02]
gamma[1] 0.1258 1.335e-02 9.420 4.523e-21 [9.959e-02, 0.152]
beta[1] 0.9291 8.309e-03 111.828 0.000 [ 0.913, 0.945]
Distribution
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
nu 6.8359 0.575 11.896 1.241e-32 [ 5.710, 7.962]
========================================================================
Covariance estimator: robust
1.2.8 Fixing Parameters
在一切情況下，固定參數而非估計參數或許更爲有趣。可以利用fix（）方法生成同模型結果類。 除了判斷信息（諸如標準誤差，t統計等等）不可用外，該類結果返回值與通常模型結果類一致.
該例中，對前一估計模型的使用對稱參數：
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: None R-squared: --
Mean Model: Constant Mean Adj. R-squared: --
Vol Model: TARCH/ZARCH Log-Likelihood: -9690.73
Distribution: Standardized Student's t AIC: 19393.5
Method: User-specified Parameters BIC: 19435.1
No. Observations: 7561
Date: Mon, Jan 21 2019
Time: 20:10:34
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.
1.2.9 構建模型
整體上，模型可以從三個方面構建：
- 平均模型(
arch.mean)- 0均值 (
ZeroMean) - 可用，如果單獨使用模型殘差。 - 常數均值(
ConstantMean) - 適用於多數流動金融資產。 - 可選外生變量自迴歸 (
ARX) - 可選外生變量異方差自迴歸
- 僅僅外生迴歸 (
LS)
- 0均值 (
- 波動率過程 (
arch.volatility)- ARCH (
ARCH) - GARCH (
GARCH) - GJR-GARCH (
GARCHusingoargument) - TARCH/ZARCH (
GARCHusingpowerargument set to1) - Power GARCH and Asymmetric Power GARCH (
GARCHusingpower) - 使用估計係數的指數加權平均移動方差 (
EWMAVariance) - 異方差 ARCH (
HARCH) - 參數模型
- 指數加權平均移動方差,即爲風險指標 RiskMetrics (
EWMAVariance) - EWMAs的加權平均數,即爲 RiskMetrics 2006 方法風險指標 (
RiskMetrics2006)
- 指數加權平均移動方差,即爲風險指標 RiskMetrics (
- ARCH (
- 分佈 (
arch.distribution)- Normal (
Normal) - Standardized Students’s T (
StudentsT)
- Normal (
1.2.10 均值模型
第一個選擇爲均值模型。對於很多流動性金融資產而言，常數均值或0均值模型已經足夠。對於其他時間序列數據，比如通脹，則可能需要更爲複雜的模型。這些例子使用了美聯儲數據庫的核心CPI數據Federal Reserve Economic Data .
core_cpi = web.DataReader("CPILFESL", "fred", dt.datetime(1957,1,1), dt.datetime(2014,1,1))
ann_inflation = 100 * core_cpi.CPILFESL.pct_change(12).dropna()
fig = ann_inflation.plot()
所有均值模型均由常數方差和正態分佈進行初始化.對於 ARX 類模型，模型中的滯後階數由lags表示。
from arch.univariate import ARX
ar = ARX(ann_inflation, lags = [1, 3, 12])
print(ar.fit().summary())
AR - Constant Variance Model Results
==============================================================================
Dep. Variable: CPILFESL R-squared: 0.991
Mean Model: AR Adj. R-squared: 0.991
Vol Model: Constant Variance Log-Likelihood: -13.7178
Distribution: Normal AIC: 37.4356
Method: Maximum Likelihood BIC: 59.9043
No. Observations: 661
Date: Mon, Jan 21 2019 Df Residuals: 656
Time: 21:23:29 Df Model: 5
Mean Model
===============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-------------------------------------------------------------------------------
Const 0.0424 2.197e-02 1.929 5.372e-02 [-6.781e-04,8.542e-02]
CPILFESL[1] 1.1927 3.513e-02 33.950 1.229e-252 [ 1.124, 1.262]
CPILFESL[3] -0.1803 4.123e-02 -4.374 1.218e-05 [ -0.261,-9.954e-02]
CPILFESL[12] -0.0235 1.384e-02 -1.695 9.001e-02 [-5.060e-02,3.663e-03]
Volatility Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
sigma2 0.0610 6.992e-03 8.728 2.590e-18 [4.733e-02,7.474e-02]
============================================================================
Covariance estimator: White's Heteroskedasticity Consistent Estimator
1.2.11 波動率過程
波動率過程可以由volatility屬性添加到均值模型中去。該舉例使用volatility參數給均值模型添加了一個ARCH（5）的波動率過程。參數iter和disp則在擬合過程fit()中時可用來描述估計結果。
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: CPILFESL R-squared: 0.991
Mean Model: AR Adj. R-squared: 0.991
Vol Model: ARCH Log-Likelihood: 83.8447
Distribution: Normal AIC: -147.689
Method: Maximum Likelihood BIC: -102.752
No. Observations: 661
Date: Mon, Jan 21 2019 Df Residuals: 651
Time: 21:27:32 Df Model: 10
Mean Model
===============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-------------------------------------------------------------------------------
Const 0.0268 2.207e-02 1.212 0.225 [-1.651e-02,7.002e-02]
CPILFESL[1] 1.0854 3.840e-02 28.266 9.155e-176 [ 1.010, 1.161]
CPILFESL[3] -0.0755 4.148e-02 -1.821 6.854e-02 [ -0.157,5.747e-03]
CPILFESL[12] -0.0210 1.195e-02 -1.761 7.829e-02 [-4.446e-02,2.381e-03]
Volatility Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
omega 9.9111e-03 2.195e-03 4.515 6.342e-06 [5.608e-03,1.421e-02]
alpha[1] 0.1291 4.090e-02 3.157 1.594e-03 [4.895e-02, 0.209]
alpha[2] 0.2279 6.468e-02 3.524 4.258e-04 [ 0.101, 0.355]
alpha[3] 0.1698 7.121e-02 2.385 1.709e-02 [3.026e-02, 0.309]
alpha[4] 0.2643 8.318e-02 3.177 1.489e-03 [ 0.101, 0.427]
alpha[5] 0.1686 7.426e-02 2.271 2.316e-02 [2.309e-02, 0.314]
============================================================================
Covariance estimator: robust
即使經過了標準化處理，標準化的殘差和條件波動率仍然顯示出更大的誤差。
fig = res.plot()
1.2.12 分佈
最後，分佈可以通過distribution屬性，把默認的正態分佈修改爲標準的學生T分佈。
學生T分佈改進了模型，自由度大約在8個附近。
from arch.univariate import StudentsT
ar.distribution = StudentsT()
res = ar.fit(update_freq=0, disp='off')
print(res.summary())
AR - ARCH Model Results
====================================================================================
Dep. Variable: CPILFESL R-squared: 0.991
Mean Model: AR Adj. R-squared: 0.991
Vol Model: ARCH Log-Likelihood: 89.4699
Distribution: Standardized Student's t AIC: -156.940
Method: Maximum Likelihood BIC: -107.509
No. Observations: 661
Date: Mon, Jan 21 2019 Df Residuals: 650
Time: 21:33:12 Df Model: 11
Mean Model
===============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-------------------------------------------------------------------------------
Const 0.0281 2.252e-02 1.249 0.212 [-1.600e-02,7.227e-02]
CPILFESL[1] 1.0851 3.920e-02 27.683 1.133e-168 [ 1.008, 1.162]
CPILFESL[3] -0.0697 4.277e-02 -1.629 0.103 [ -0.154,1.415e-02]
CPILFESL[12] -0.0266 1.525e-02 -1.742 8.157e-02 [-5.643e-02,3.328e-03]
Volatility Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
omega 0.0117 3.142e-03 3.710 2.071e-04 [5.499e-03,1.782e-02]
alpha[1] 0.1669 5.233e-02 3.189 1.426e-03 [6.433e-02, 0.269]
alpha[2] 0.2185 6.648e-02 3.287 1.012e-03 [8.823e-02, 0.349]
alpha[3] 0.1370 6.928e-02 1.977 4.804e-02 [1.179e-03, 0.273]
alpha[4] 0.2186 7.738e-02 2.825 4.734e-03 [6.691e-02, 0.370]
alpha[5] 0.1596 8.579e-02 1.861 6.275e-02 [-8.495e-03, 0.328]
Distribution
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
nu 9.1015 3.834 2.374 1.760e-02 [ 1.587, 16.616]
========================================================================
Covariance estimator: robust
1.2.13 WTI 原油
下一個例子使用了FRED的西得克薩斯中級原油數據.這些模型使用替代分佈假定進行擬合。打印出的結果顯示，正態分佈比標準學生T分佈或標準偏態學生T分佈的對數似然率更低。然而，後兩者比較接近。T和偏態T分佈的密切性表明收益率並沒有嚴重的偏態。
from collections import OrderedDict
crude=web.get_data_fred('DCOILWTICO',dt.datetime(2000, 1, 1),dt.datetime(2015, 1, 1))
crude_ret = 100 * crude.dropna().pct_change().dropna()
res_normal = arch_model(crude_ret).fit(disp='off')
res_t = arch_model(crude_ret, dist='t').fit(disp='off')
res_skewt = arch_model(crude_ret, dist='skewt').fit(disp='off')
lls = pd.Series(OrderedDict((('normal', res_normal.loglikelihood),
('t', res_t.loglikelihood),
('skewt', res_skewt.loglikelihood))))
print(lls)
params = pd.DataFrame(OrderedDict((('normal', res_normal.params),
('t', res_t.params),
('skewt', res_skewt.params))))
print(params)
normal -8227.359031
t -8128.534732
skewt -8126.303934
dtype: float64
normal t skewt
alpha[1] 0.054488 0.046069 0.045908
beta[1] 0.940953 0.949954 0.950364
lambda NaN NaN -0.048593
mu 0.065643 0.076392 0.057599
nu NaN 6.841493 6.889696
omega 0.034733 0.026497 0.025239
標準化的殘差可以把殘差除以條件波動率後計算得出。上述均同殘差一併畫出（非標準化，但時間長度不一）。非標準化的殘差中間的尖峯更爲突出，表明其分佈比標準化殘差更具有厚尾特徵。
std_resid = res_normal.resid / res_normal.conditional_volatility
unit_var_resid = res_normal.resid / res_normal.resid.std()
df = pd.concat([std_resid, unit_var_resid],1)
df.columns = ['Std Resids', 'Unit Variance Resids']
subplot = df.plot(kind='kde', xlim=(-4,4))
