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ritvikmath
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12742924c6
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"from random import gauss\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"from arch import arch_model\n",
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"from statsmodels.graphics.tsaplots import plot_acf, plot_pacf"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# GARCH(2,2) Model\n",
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"\n",
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"$$\n",
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"a_t = \\varepsilon_t \\sqrt{\\omega + \\alpha_1 a_{t-1}^2 + \\alpha_2 a_{t-2}^2 + \\beta_1 \\sigma_{t-1}^2 + \\beta_2 \\sigma_{t-2}^2}\n",
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"$$\n",
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"\n",
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"$$\n",
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"a_0, a_1 \\sim \\mathcal{N}(0,1)\n",
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"$$\n",
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"\n",
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"$$\n",
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"\\sigma_0 =1, \\sigma_1 = 1\n",
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"$$\n",
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"\n",
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"$$\n",
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"\\varepsilon_t \\sim \\mathcal{N}(0,1)\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# create dataset\n",
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"n = 1000\n",
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"omega = 0.5\n",
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"\n",
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"alpha_1 = 0.1\n",
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"alpha_2 = 0.2\n",
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"\n",
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"beta_1 = 0.3\n",
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"beta_2 = 0.4\n",
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"\n",
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"test_size = int(n*0.1)\n",
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"\n",
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"series = [gauss(0,1), gauss(0,1)]\n",
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"vols = [1, 1]\n",
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"\n",
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"for _ in range(n):\n",
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" new_vol = np.sqrt(omega + alpha_1*series[-1]**2 + alpha_2*series[-2]**2 + beta_1*vols[-1]**2 + beta_2*vols[-2]**2)\n",
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" new_val = gauss(0,1) * new_vol\n",
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" \n",
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" vols.append(new_vol)\n",
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" series.append(new_val)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"plt.figure(figsize=(10,4))\n",
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"plt.plot(series)\n",
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"plt.title('Simulated GARCH(2,2) Data', fontsize=20)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"plt.figure(figsize=(10,4))\n",
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"plt.plot(vols)\n",
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"plt.title('Data Volatility', fontsize=20)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"plt.figure(figsize=(10,4))\n",
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"plt.plot(series)\n",
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"plt.plot(vols, color='red')\n",
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"plt.title('Data and Volatility', fontsize=20)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# PACF Plot"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"plot_pacf(np.array(series)**2)\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Fit the GARCH Model"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"train, test = series[:-test_size], series[-test_size:]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"model = arch_model(train, p=2, q=2)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"model_fit = model.fit()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"model_fit.summary()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Predict"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"predictions = model_fit.forecast(horizon=test_size)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"plt.figure(figsize=(10,4))\n",
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"true, = plt.plot(vols[-test_size:])\n",
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"preds, = plt.plot(np.sqrt(predictions.variance.values[-1, :]))\n",
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"plt.title('Volatility Prediction', fontsize=20)\n",
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"plt.legend(['True Volatility', 'Predicted Volatility'], fontsize=16)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"predictions_long_term = model_fit.forecast(horizon=1000)\n",
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"plt.figure(figsize=(10,4))\n",
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"true, = plt.plot(vols[-test_size:])\n",
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"preds, = plt.plot(np.sqrt(predictions_long_term.variance.values[-1, :]))\n",
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"plt.title('Long Term Volatility Prediction', fontsize=20)\n",
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"plt.legend(['True Volatility', 'Predicted Volatility'], fontsize=16)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Rolling Forecast Origin"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"rolling_predictions = []\n",
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"for i in range(test_size):\n",
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" train = series[:-(test_size-i)]\n",
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" model = arch_model(train, p=2, q=2)\n",
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" model_fit = model.fit(disp='off')\n",
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" pred = model_fit.forecast(horizon=1)\n",
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" rolling_predictions.append(np.sqrt(pred.variance.values[-1,:][0]))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"plt.figure(figsize=(10,4))\n",
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"true, = plt.plot(vols[-test_size:])\n",
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"preds, = plt.plot(rolling_predictions)\n",
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"plt.title('Volatility Prediction - Rolling Forecast', fontsize=20)\n",
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"plt.legend(['True Volatility', 'Predicted Volatility'], fontsize=16)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.4"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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