AMRA Model and Kalman Filter (01)

Autocorrelation

We often need to characterize time series autocorrelation, , which is defined as

When the sequence is weakly stationary, i.e.



where , , and are constant, we write

Now, let’s work on more technical stuffs. If are independent and identically distributed sequence, then the sample autocorrelation

is asympotically normal with mean zero (trivially) and variance . For weakly stationary sequence , is asympotitically normal with mean zero and variance

if we assume for . Therefore, a t-ratio test statistic can be derived such that

for the test

: :

Q Test

When testing on multiple autocorrelations being zero, Box and Pierce (1970) and Ljung and Box (1978) classic work provided test on the hypothesis

: : $\rho_i \neq 0$ for some

The Q test defined as

which follows a chi-squared distribution with $n$ degree of freedom. With Python `statsmodels` module, we can easily check the test results of some financial data.

S&P 500 Data

[sourcecode language=”python” light=”true” wraplines=”false” collapse=”false”]
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import pandas.io.data as web
import statsmodels.tsa.api as sma

from statsmodels.tsa.arima_model import ARMA
%matplotlib inline

start = ‘2006-01-01’
end = ‘2010-12-31’
df_data = web.DataReader(‘GSPC’, ‘yahoo’, start, end)
df_data.tail()
[/sourcecode]

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S&P 500 Daily Log Returns Analysis

[sourcecode language=”python” light=”true” wraplines=”false” collapse=”false”]
array_adjclose = df_data[‘Adj Close’].values
array_logret_adjclose = np.log(array_adjclose)[1:] – np.log(array_adjclose)[:-1]
array_logret_adjclose = np.insert(array_logret_adjclose, 0, 0)
df_data[‘Log Return’] = array_logret_adjclose

ax = df_data[‘Log Return’].plot()
ax.set_title(‘Daily Log Return’)
[/sourcecode]

—

[sourcecode language=”python” light=”true” wraplines=”false” collapse=”false”]
fig, ax = plt.subplots(2, 1)
ax[0].plot(sma.stattools.acf(array_logret_adjclose))
ax[1].plot(sma.stattools.pacf(array_logret_adjclose))

fig, ax = plt.subplots(3, 1)
acf, confint, qstat, pvalues = sma.stattools.acf(array_logret_adjclose, alpha=0.05, qstat=True)
ax[0].plot(acf)

# Q Stats
ax[1].plot(qstat)

# P Values
ax[2].plot(pvalues)
[/sourcecode]

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What’s Next?

We will examine how to choose and calibrate an ARMA model.