ti) (al: econometrics_module_5_lesson_1 November 28, 2024 0.1 FINANCIAL ECONOMETRICS MODULE 5 | LESSON 1 ee 1 ARCH MODEL Reading Time 60 minutes Prior Knowledge Basics of time series, ARIMA Keywords Conditional Means, Conditional Variance, Volatility Clustering, ARCH(1) Model In the last module, we used the ARIMA model to analyze Google stock price. One key issue we observed was that Google stock price does not exhibit constant variance throughout our modeling time period. This issue does not just happen to Google’s stock price. It is very common among many financial asset returns. In this module, we are going to address this phenomenon and find a model to analyze it. We will introduce the ARCH model and GARCH model to handle this issue. They are popular methods in finance to model volatility for many asset returns. In this lesson, we will start with learning the ARCH model and its properties. We will simulate the ARCH process to give @ visual example and learn how to use ACF and PACF plots to identify the ARCH process. import matplotlib.pyplot as plt import numpy as np import pandas as pd import statsmodels.api as sm from arch.univariate import ARCH, ConstantMean, Normal from scipy import stats plt.rcParams["figure.figsize"] = (16, 9) # Figure size and width # Download the dataset mS data = pd.read_cev("../M5. goog_eur_10.csv") 4 Convert date variable to date format and set index m§_data["Date2"] = pd.to_datetime(m5_data["Date"], format="Zm/%d/2Y") goog = m§_data.loc[:, ["Date2", "GOOGLE"]].set_index("Date2")(3) 1.1 1. Common Features of Financial Asset Returns There are several special features from financial asset returns. The existence of these features motivates us to use ARCH and GARCH models to analyze financial asset returns. In this section, We are going to go through these features first so that you will have a better understanding of why we need to learn the ARCH and GARCH models. The first feature we'll talk about is the reason to use asset returns to conduct a research project instead of asset prices. 11.1 11 Asset Return Time Series Data is More Stable Than Asset Price Time Series Data In a lot of financial research, we usually use asset returns instead of asset prices to conduct analysis for a financial asset. Here are some reasons to use asset returns: a. An asset return provides same-scale time series data for all financial assets. It is easier for comparison among several financial assets with different price levels. b. Asset return time series is usually more stable than asset price time series. Let's use Google's stock price and return as an example, Figure 1: Google Daily Stock Price and Retum Historical Chart 2016 — 2021 # Plot Google Daily stock price and return # Google Daily stock price plt plot (goog. GOOGLE) Plt.ylabel ("Google Stock Price") plt.show() # Google Daily stock price return goog("GOUGLE_R"] = goog.GO0GLE.pct_change() Plt plot (goog. GOOGLE_R) Plt. ylabel ("Google Stock Return") plt.show()In figure 1, the upper chart is Google’s daily stock price chart. The bottom chart is Google's stock daily return chart. We can see that Google's price plot has an upward trend whereas Google's return plot oscillates around 0 and is stable. We can say the return on average is 0. c. If the asset retum is small, we can approximate the return by using the difference of the natural log of price. Here is the formula:x log(p:) — log(pr-1) Pt 11.2 1.2 The Volatility of Asset Returns Can Vary During Different Time Periods A Portfolio management and risk management, managers control risk by monitoring the volatility of underlying assets. Variance and standard deviations are common metrics to measure volatility We can see from figure 1 that the local variance of Google stock returns changes substantially across the time period. This time series data exhibits heteroskedasticity. In the return plot, we can see there are pockets of time in the whole period where the movement of returns are more volatile than others, These high volatility pockets do not spread randomly during the whole time period. We call these volatility clustering. We notice that there is autocorrelation of asset returns. A highly volatile return one day would be followed by another highly volatile return the next day. It means that the past volatilities will still have an impact on today’s volatility. The highly volatile periods tend to group together. This is a key property that later we would like to model. 11.3 1.3 Asset Return Distribution Has Heavier Tails Than Normal Distribution We have discussed many times in the previous lessons that financial asset returns often show a distribution with heavier tails than normal distribution. Let’s take a look at Google's stock retum, distribution to see if the same feature shows up. All rights reserved WQU WorldQuant University QQQQ Figure 2: Histogram and Normal QQ Plot for Google Stock Returns # Histogram and Normal QQ Plot for Google Stock Returns fig, (axl, ax2) = plt.subplots(1, 2, figsize=(16, 5)) goog_r = goog.GOOGLE_R.dropna() # Histogram with density x = mp.linspace(min(goog_r), max(goog_r), len (goog_r)) (gu, sigma) = stats.norm.fit (goog_r) values, bins, _ = ax1.hist(goog_r, bins=25) # Histogram axi.plot(x, stats.norm.pdf(x, mu, sigma) * sum(values * np.diff(bins)), "r") 4 —Density axi.set(title="Google Stock Return Histogram") # Normal 9Q plot sm.qqplot (goog_r, stats.norm, fit=True, lin ax2.set (title="Normal QQ Plot for Google Stock Return") aq", ax-ax2) plt.show()[Sr St atu stagram os HOT: tf Geng Rtn In figure 2, we show Gi istogram re 2, ‘oogle’s stock ret Google's stock return exhibits fat tail Sina ma a ene a 1.2 2. Conditional Means and Conditional Variances pone ated so aca features for financial asset returns time series. Before we dive into ‘mally analyze those features, let’s start by learning some concepts that will pre us for explaining the models later. a * ” i We talked about conditional probability in the last module. Now let's learn how to obtain conditional means and conditional variances. Assume X and Y are two discrete random variables. ‘The conditional mean of ¥ given X is as follows: Hix = OF LOY = EX) where f(Y|X) is the conditional probability of Y given X. The conditional variance of Y given X is as follows: Ome = YIN) ~ thay = BVA) ~ 0g ‘We can see that the calculation for conditional means and conditional variances are very similar to calculation for regular means and variances. The difference is we use conditional probability to calculate conditional means and conditional variances. Why do we need to know conditional means and conditional variances in the time series context? The central idea is how we can incorporate the concept of using past value information to predict today’s value in a model. We actually did that for the ARMA model in the last module. We used the concept of conditional mean to run an ARMA model. Let’s use the following example to illustrate the idea. Consider an AR(p) model with a constant conditional variance Var(X¢|Xe-1, Xe-24-+> 5 Xt) ‘We can write the AR(p) model as follows:Xe = F(X Xia Xeep) +e ; iance 2, f() is the conditional Serie ole ey has mean = 0 and constant conditional variance 0, /0 ; ane es amy eee +++, Xt-p and a? is the constant conditional variance for X; given § X_ {UI}, X_e3} x CE pys. is od data in the ow if we want to introduce non-constant variance like we observed from the Google data last section to the above AR(p) model, we can rewrite the model X= FX, Xtoay ++ Xtp) + 0( Kins Xian, Xp) ee Where white noise e has conditional mean = 0 and conditional pyariance | 4 2(Xi-1, Xe_a,-++ ,X¢_p)e is the standard deviation for X; given X11, Xi-2,-+-,Xi-p. This stan- dard deviation will change when Xt1,Xea-++ ,Xep change. Since o(Xt-1, Xray." ,Xe-p)e is legative. Later, we will provide regularity conditions to make Cat we just described id a general vatiance function ‘model for conditional variance. ARCH and GARCH are all types of vatiance function sees 2h {the following sections, we are going to focus our analysis on asset returns. ‘The common obser Tation is that the conditional mean of asset neta is 0, s0 we will drop the conditional mean part from the above model £0 focus on the conditional variance part in the following sections. We will start with a simple ARCH(1) model in the is lesson. Then, we will introduce a, general ARCH(m) ‘model. In the next lesson, we will expand the ARCH model and introduce the GARCH model, 1.3 3. ARCH(1) Model Now let’s formally introduce our first conditional Variance model, ARCH(1) model. ARCH stands for autoregressive conditional hteroskedastcity. If we have a stationary asset roten time series ry, ‘we can define ARCH(1) model for ry as follows r= oer Where e; ~ iidN(0,1) and the 1 in ARCH(1) refers to the lag 1 asset return in the variance equation, ‘Now let's review the properties of ARCH(1) model/process.13.1 3.1 ARCH(1) is Strictly Stationary From r= oye; and o? = ap +anr?_y, we know 12 = aoe? + axe?r?_, By iterating the above definition, we can show that ‘We will leave the proof as one open-ended question. | 13.2 3.2 1 is Conditionally Normally Distributed The conditional mean of r is given by E(rilrs,s 0 Heence, the unconditional distribution of ry can be written as follows: ‘We can see from the ab is stationary (0 < a < 134° 3.41% Is Not iid. Although r, has 0 Covariance to its past: values, 1? does depend on the squared past returns. It can be shown as follows: PUrélre-1) = Blage? + axe?r? Ie-1) a9 + rrp) So we can see ARCH(1) is a case where the clements in the process have 0 covariance, but the Clements are not independent. This coneltuion also confirms our previous knowledge that two variables with 0 covariance do not mean they are independent. 1.3.5 3.5 7? IS A non-Gaussian AR(1) Process From the ARCH(1) definition, we can take the Square of rz and rewrite the equations as follows: 2,2 te R= 0 + aur? =o? ‘We then subtract the second equation from the first equation to get the following equation: 72 — (a0 + arr? 1) = 0}(€2 ~ 1) P= ontarts +n where 1; = 7 (e? ~ 1) is non-normal white noise. (¢?°1) follows a shifted Chi-Square distribution since ¢? is the square of a N(0,1) random variable,Also if SE(e_{t}°{4}) < 008, thenr? has the same ACF as AR(1) process, ‘The ACF for 7? is as follows : for all h and ay <1 wary AR(1) process, we can use look for an ARCH(1) model. If m and the squared asset return PACF (1) model for modeling variance of the Tetum and squared asset return to I the asset return time series plot exhibits a white noise patter Plot shows cut off pattern after lag 1, we can use an ARCH| asset return. We will talk more about this in later sections. 18-6 3.6 r; Has Heavier Tails Than Standard Normal Distribution If ay < 3 When 3a; <1 and SE(r —{t}°(4})