Volatility forecasting (GARCH & ARCH)

Prerequesites This tutorial assumes basic familiarity with StatsForecast. For a minimal example visit the Quick Start

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Introduction

1. $w>0$, $\alpha_i \geq 0$ for all $i$, and $\beta_j \geq 0$ for all $j$
2. $\sum_{k=1}^{max(p,q)} \alpha_k + \beta_k < 1$. Here it is assumed that $\alpha_i=0$ for $i>p$ and $\beta_j=0$ for $j>q$.

1. Install libraries
2. Load and explore the data
3. Train models
4. Perform time series cross-validation
5. Evaluate results
6. Forecast volatility

Tip You can use Colab to run this Notebook interactively

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Install libraries

pip install statsforecast -U

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Load and explore the data

# pip install yfinance

import yfinance as yf
import pandas as pd

tickers = ['SPY', 'MSFT', 'AAPL', 'GOOG', 'AMZN', 'TSLA', 'NVDA', 'META', 'NKE', 'NFLX'] 
df = yf.download(tickers, start = '2018-01-01', end = '2022-12-31', interval='1mo', progress=False) # use monthly prices
df.head()

df = df.loc[:, (['Adj Close'], tickers)]
df.columns = df.columns.droplevel() # drop MultiIndex
df = df.reset_index()
df.head()

• unique_id: (string, int or category) A unique identifier for the series.
• ds: (datestamp or int) A datestamp in format YYYY-MM-DD or YYYY-MM-DD HH:MM:SS or an integer indexing time.
• y: (numeric) The measurement we wish to forecast.

prices = df.melt(id_vars = 'Date')
prices = prices.rename(columns={'Date': 'ds', 'Ticker': 'unique_id', 'value': 'y'})
prices = prices[['unique_id', 'ds', 'y']]
prices

from statsforecast import StatsForecast

StatsForecast.plot(prices)

import numpy as np 
prices['rt'] = prices['y'].div(prices.groupby('unique_id')['y'].shift(1))
prices['rt'] = np.log(prices['rt'])

returns = prices[['unique_id', 'ds', 'rt']]
returns = returns.rename(columns={'rt':'y'})
returns

Warning If the order of the data is very small (say $<1e-5$), scipy.optimize.minimize might not terminate successfully. In this case, rescale the data and then generate the GARCH or ARCH model.

StatsForecast.plot(returns)

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Train models

from statsforecast.models import (
    GARCH, 
    ARCH, 
    Naive
)

models = [ARCH(1), 
          ARCH(2), 
          GARCH(1,1),
          GARCH(1,2),
          GARCH(2,2),
          GARCH(2,1),
          Naive()
]

• df: The dataframe with the training data.
• models: The list of models defined in the previous step.
• freq: A string indicating the frequency of the data. Here we’ll use MS, which correspond to the start of the month. You can see the list of panda’s available frequencies here.
• n_jobs: An integer that indicates the number of jobs used in parallel processing. Use -1 to select all cores.

sf = StatsForecast(
    models = models, 
    freq = 'MS',
    n_jobs = -1
)

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Perform time series cross-validation

• df: The dataframe with the training data.
• h (int): represents the h steps into the future that will be forecasted.
• step_size (int): step size between each window, meaning how often do you want to run the forecasting process.
• n_windows (int): number of windows used for cross-validation, meaning the number of forecasting processes in the past you want to evaluate.

cv_df = sf.cross_validation(
    df = returns,
    h = 3,
    step_size = 3,
    n_windows = 4
  )

• unique_id: series identifier.
• ds: datestamp or temporal index
• cutoff: the last datestamp or temporal index for the n_windows.
• y: true value
• "model": columns with the model’s name and fitted value.

cv_df.rename(columns = {'y' : 'actual'}, inplace = True)
cv_df.head()

StatsForecast.plot(returns, cv_df.drop(['cutoff', 'actual'], axis=1))

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Evaluate results

from utilsforecast.losses import mae

models = cv_df.columns.drop(['unique_id', 'ds', 'cutoff', 'actual'])

mae_cv = mae(cv_df, models=models, target_col='actual').set_index('unique_id')
mae_cv

mae_cv.idxmin(axis=1)

unique_id
AAPL    GARCH(2,1)
AMZN    GARCH(2,2)
GOOG         Naive
META         Naive
MSFT    GARCH(1,1)
NFLX       ARCH(1)
NKE        ARCH(2)
NVDA       ARCH(1)
SPY     GARCH(2,2)
TSLA    GARCH(1,1)
dtype: object

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Forecast volatility

• h: (int) The forecasting horizon.
• level: (list[float]) The confidence levels of the prediction intervals
• fitted : (bool = False) Returns insample predictions.

levels = [80, 95] # confidence levels for the prediction intervals 

forecasts = sf.forecast(df=returns, h=3, level=levels)
forecasts.head()

• level: (list[int]) The confidence levels for the prediction intervals (this was already defined).
• unique_ids: (list[str, int or category]) The ids to plot.
• models: (list(str)). The model to plot. In this case, is the model selected by cross-validation.

StatsForecast.plot(returns, forecasts, max_insample_length=20)

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References

• Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the econometric society, 987-1007.
• Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3), 307-327.
• Hamilton, J. D. (1994). Time series analysis. Princeton university press.
• Tsay, R. S. (2005). Analysis of financial time series. John wiley & sons.