Financial time series analysis poses significant challenges due to the presence of non-stationarity and long-range dependence. These two characteristics are often observed in financial data and need to be accounted for in any statistical analysis to avoid spurious results.
Non-stationarity refers to the fact that key statistical properties of a time series, such as mean and variance, change over time, making it difficult to model the series using traditional stationary models. This concept is particularly relevant in the financial world, where market conditions are continuously evolving, leading to non-stationary returns. To address non-stationarity, one can either transform a non-stationary time series into a stationary form through differencing, or use models specifically designed to handle non-stationary data, such as autoregressive integrated moving average (ARIMA) or autoregressive conditional heteroskedasticity (ARCH) models.
Long-range dependence refers to the persistence of shocks or volatility in a time series beyond what is expected in stationary processes. This means that a shock or volatility event can have a lasting impact on future observations. In finance, the long-range dependence is often generated by the high degree of interdependence among financial markets due to global factors such as political and economic events. To address this challenge, one can use models that account for long-range dependence, such as the autoregressive fractionally integrated moving average (ARFIMA) or fractional stochastic volatility (FSV) models.
Here is an example of how to address non-stationarity and long-range dependence in financial time series analysis using Python:
import pandas as pd
import numpy as np
from statsmodels.tsa.arima_process import ArmaProcess
from statsmodels.tsa.arima_model import ARIMA
# generate non-stationary time series
np.random.seed(123)
ar3 = np.array([1, -0.8, 0.2, -0.1])
ma3 = np.array([1, 0.5, 0.2, 0.1])
ar3ma3 = ArmaProcess(ar3, ma3)
nonstationary_series = ar3ma3.generate_sample(nsample=100)
# transform non-stationary series into stationary form
stationary_series = np.diff(nonstationary_series)
# fit ARIMA model to stationary series
model = ARIMA(stationary_series, order=(1, 1, 1))
results = model.fit()
# generate long-range dependent time series using FSV
from arch import arch_model
np.random.seed(123)
sigma = 0.1 + 0.8*np.abs(np.random.randn(1000))
epsilon = sigma*np.random.randn(1000)
y = np.cumsum(epsilon)
# fit FSV model to long-range dependent series
model = arch_model(y, vol='fsv', p=1, o=1, q=1, dist='normal')
results = model.fit()In this example, we first generated a non-stationary time series using an ARMA process with non-zero mean. We then transformed this series into a stationary form using differencing and fitted an ARIMA model to the stationary series. This approach enables us to analyze the stationary part of the series without being affected by non-stationarity. We then generated a long-range dependent time series using a fractional stochastic volatility (FSV) model and fitted an FSV model to the data. This approach accounts for long-range dependence, which allows us to capture the volatility clustering and persistence observed in the financial data.