28 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

MODELING CONDITIONAL VOLATILITY IN R 
 

Omid Sabbaghi1 

 

Abstract 

 

Many economic and financial time-series exhibit time-varying volatility. In particular, volatility 

is an important input for pricing models and portfolio management decisions. This article 

demonstrates how to estimate volatility using the GARCH (1,1) model through the R analytics 

software. In addition, this study demonstrates how to employ GARCH-based estimates for the 

purposes of forecasting volatility.   

 

Key Words: financial econometrics, GARCH, volatility, forecasting 

 

JEL Classifications: C58, G12, C22 

 

Introduction 

Time-varying volatility is an important phenomenon in the economics and finance 

literature. In financial applications, the variance of the return on an asset represents the risk level 

of those returns, and it is documented that the volatility, or amplitude, of financial returns is 

time-varying as there are periods of market tranquility as well as periods of market turbulence. In 

addition to individual stocks and stock market indices, other examples of observed time-varying 

volatility in our global markets include changes in exchange rates and absolute deviations of the 

inflation rate from its mean. That is, financial markets data exhibit volatility clustering in that 

there are periods of high volatility and periods of low volatility.  

Precise estimates of volatility are an input for numerous real-world applications in 

economics and finance. For example, volatility is an important input for forecasting confidence 

intervals since time-varying volatility translates into time-varying widths for confidence 

intervals. In addition, precise estimates of volatility enable precise Value-at-Risk (VaR) 

estimates for portfolios, thereby allowing banks to better estimate the risk of loss for 

investments. Moreover, volatility is an input variable for the Black-Scholes formula, and thus 

precise estimates of volatility are important for valuing options. Importantly, the estimation of 

returns and conditional volatilities is important in identifying investment opportunities that 

increase returns since expected returns depend on volatility (French et al. 1987). Thus, for these 

reasons, among others, it is important to obtain robust estimates of volatility through GARCH 

models. 

This paper provides several important lessons in volatility for the undergraduate student 

studying economics and finance. First, this study provides an introduction to the estimation of 

time-varying volatility, through the GARCH model of Bollerslev (1986), in the R analytics 

environment. R is an open source statistical programming language in which it is estimated that 

nearly 250,000 people work with R on a regular basis, and is considered the second language for 

individuals emerging from graduate school (Vance 2009). As Li (2019) points out, R is a 

valuable aspect of an introductory research methods course when taught appropriately. This 

 
1  Associate Professor of Finance, College of Business Administration, University of Detroit Mercy, 4001 W. 

McNichols Road, Commerce and Finance 245, Detroit, MI 48221 



29 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

study demonstrates how to estimate time-varying volatility for a well-known barometer of the 

U.S. stock market, namely the S&P500 market index. In particular, this article shows how 

GARCH effects are significant and persistent throughout time.  

After estimating the GARCH model in R, this study demonstrates how to extract the 

time-series of the conditional volatilities, also referred to as the conditional standard deviations, 

using R analytics. In addition, this paper demonstrates how to construct plots that provide visual 

confirmation of time-varying volatility. Spikes in the plot convey time periods in which volatility 

is high.     

Finally, this study demonstrates how to produce time-varying confidence intervals for the 

S&P500 daily returns using the conditional volatilities obtained by the GARCH model. The R 

environment provides rich graphic plots in the construction of such time-varying intervals. 

Goodness-of-fit statistics are also discussed when interpreting the statistical output. That is, this 

study demonstrates how to gauge the adequacy of a fitted GARCH model in R.  In addition, this 

paper concludes with the interpretation of the information criteria as a means of performing 

model comparison. Figures and tables are in the appendix.  
 

R 

The R environment is open-source software, and its capabilities are extended via 

packages. The base R software, as well as additional packages, may be downloaded freely from 

the Comprehensive R Archive Network (CRAN) via the internet at https://cran.r-project.org/. 

The fGarch package deals specifically with volatility modeling and is the object of interest for 

the present study. Upon starting R, entering the following command in the R console enables the 

fGarch package, and any necessary related packages, to be installed:  

 
if(!require(fGarch)){ 

install.packages("fGarch") 

} 

 

Once the fGarch package is installed, it must be loaded upon initialization of an R working 

session to ensure that the GARCH model can be estimated in R, either via the Packages 

submenu, or through entering the following command in the R console: 
 

library(fGarch) 

 

Sample Data & Empirical Methodology 

This study applies the GARCH framework in estimating the conditional volatility for the 

S&P500 market index. Daily closing levels for the S&P500 index, adjusted for dividends and 

splits, are extracted from the Yahoo! Finance database. The data ranges from January 1950 

through December 2019. The closing price level for the S&P500 at time t is denoted tp .The 

daily rate of return for the S&P500 on day t, denoted tr , is defined as the continuously 

compounded return in Eqn. (1): 

 

𝑟𝑡 = 𝑙𝑜𝑔( 𝑝𝑡 ) − 𝑙𝑜𝑔( 𝑝𝑡−1)         (1) 
 

https://cran.r-project.org/


30 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

The GARCH model, first introduced by Bollerslev (1986), describes the conditional 

return volatility of a financial asset over time. GARCH generalizes the ARCH model which was 

proposed by Engle (1982) to model the uncertainty of inflation in the United Kingdom. 

The present study adopts a GARCH(1,1) model in understanding the time-varying 

volatility for the S&P 500 index. Research has shown that the GARCH(1,1) model is the most 

important case for practical applications. For example, Tsay (2010) and Hansen and Lunde 

(2005) argue that higher-order GARCH models fail to outperform the GARCH(1, 1) model. 

Engle (2001) states that the GARCH(1,1) model is the most robust of the family of volatility 

models. Formally the GARCH (1,1) model is given by Eqns. (2) and (3):  

 

𝑟𝑡 = √ℎ𝑡 𝑧𝑡  𝑧𝑡  iid 𝑁(0,1)         (2) 
            

ℎ𝑡 = 𝜔 + 𝛼𝑟𝑡−1
2 + 𝛽ℎ𝑡−1         (3) 

 

In the GARCH model, the conditional return variance ℎ𝑡 is itself a serially correlated time series 

process. In addition, tomorrow’s return variance ℎ𝑡 is predicted to be a weighted average of 

today’s variance forecast ℎ𝑡−1 and the new information in this period as captured by today’s 

squared return 𝑟𝑡−1
2 . Conditional return volatility, or also known as conditional standard 

deviation, is denoted √ℎ𝑡. The errors, 𝑧𝑡, is a sequence of independent and identically distributed 

(iid) random variables with zero mean and unit variance, and are assumed to be normal in Eqn. 

(2). The realized errors, also referred to as the standardized returns, are important in diagnostic 

checking since the standardized returns should no longer show significant volatility clustering.  

 The economic magnitudes of the   and   coefficient estimates are important. First, in 
the context of the GARCH(1,1) model, the long-run average variance is given by 

𝜔

1−𝛼−𝛽
 , and 

thus 𝛼 + 𝛽 < 1 as well as 𝛼 > 0 , 𝛽 > 0, and 𝜔 > 0 (Engle 2001). It is required that 𝛼 + 𝛽 < 1 
so as to ensure that the volatility is mean-reverting.  

The conditional mean represents the average return, and in the pure GARCH(1,1) model 

of Eqns. (2) and (3), conditional mean return dynamics are negligible. That is, the average daily 

returns are zero. In the event that daily average returns are statistically distinguishable from zero, 

Tsay (2010) suggests demeaning the daily returns prior to the GARCH estimation. 

 As a means of gauging the importance of conditional volatility for the S&P 500 index, 

this study estimates the GARCH(1,1) model over seven non-overlapping time periods. Since 

trading volume data for the S&P 500 begins in 1950 based on our data source, this study 

examines the following non-overlapping time periods: January 1950 – December 1959; January 

1960 – December 1969; January 1970 – December 1979; January 1980 – December 1989; 

January 1990 – December 1999; January 2000 – December 2009; and January 2010 – December 

2019. The aforementioned time periods are henceforth referred to as Periods 1, 2, 3, 4, 5, 6, and 

7, respectively.  
 

R and the GARCH(1,1) Model 

 The first step in our analysis consists of inputting the daily returns data. To illustrate the 

estimation of the GARCH(1,1) model in R, this study considers the S&P 500 returns data for the 

most recent decade, spanning from January 2010 through December 2019, namely Period 7. In 

R, we define the Period 7 data with the following commands in the R console:  



31 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

 

period7data <- read.delim("C:\\data\\period7data.txt",header=T) 

period7return <- period7data[,1] 

 

In the first command above, note that we are reading in the tab-delimited text data file from its 

corresponding location via the read.delim command. The header=T argument is allowing 

R to read the first row of the data matrix as column names. By applying the read.delim 

command, we are defining the data matrix period7data which contains the returns time-

series data. The second command above is defining period7return as the first column of the 

period7data matrix, namely the returns. Figure 1, Panel A presents a time-series plot of the 

daily squared returns over the January 2010 – December 2019 time period.2 Such a plot is 

constructed in the R environment via the following command in the R console: 
 

ts.plot(period7return^2, main="Panel A: S&P500 Daily Squared 
Returns: 2010-2019", ylab="Daily Squared Return"))  

 

In the R command above, the ts.plot command is being applied to the daily squared returns 

for Period 7. The main= subargument is allowing for us to provide a title for the plot, while the 

ylab= argument allows for the labeling of the y-axis. Figure 1 Panel A presents evidence of 

clustering in return volatility over time. That is, large changes in the daily squared returns are 

followed by large changes, and small changes in the daily squared returns are followed by small 

changes. Panel B plots the sample autocorrelation function for the S&P500 daily squared returns 

over the identical time period, and visually confirms the presence of volatility clustering. The 

autocorrelation function for the Period 7 squared returns data is constructed in the R environment 

via the following acf command in the R console: 
 

acf(period7return^2,lag.max=40,main="Panel B: ACF of S&P500 

Daily Squared Returns: 2010-2019")  

 

In the above command, the number of lags utilized for the autocorrelation plot is provided by the 

lag.max= subargument. Similarly, the main= subargument allows for the title of the plot. 

Statistically, the return autocorrelations through the 40th lag are statistically distinguishable from 

zero since the autocorrelations lie outside the dotted lines which represent the significance 

bounds. That is, the statistically significant autocorrelations suggest that the S&P500 return 

volatility is serially correlated throughout time.3 

 The fGarch package in R estimates a GARCH(1,1) model for the S&P 500 daily returns. 

For classroom illustrative purposes, we assume that the conditional mean dynamics for returns 

are negligible and that the standardized returns t

t

r

h
 are normally distributed, thereby justifying 

the estimation of a pure GARCH(1,1) model with normal innovations zt. The conditional 

normality assumption may sometimes be violated. However, Bollerslev and Woolridge (1992) 

show that GARCH parameters are consistently estimated by Gaussian maximum likelihood even 

when the normality assumption is incorrect.  

 
2 Daily squared returns for Periods 1 through 6 are similar in appearance. 
3 Autocorrelation functions for the S&P500 daily squared returns in Periods 1 through 6 are similar in appearance. 



32 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

 We estimate the GARCH(1,1) model for the January 2010 through December 2019 time 

period. Such a model is defined and estimated in R using the following garchFit command in 

the R console: 
 

garch11.model <- garchFit(~garch(1,1),data=period7return, 
cond.dist = c("norm"), include.mean=FALSE) 

 

In the above syntax, the R environment is defining garch11.model as an object in its 

working memory. However, the garchFit command requires several inputs. First and 

foremost, the order of the GARCH model is specified: the ~garch(1,1) substring conveys the 

estimation of a GARCH(1,1) model. Higher-order GARCH models may be estimated by 

inputting different values for this substring. Second, the garchFit command requires the 

vector of daily returns via the data= substring. In the above example command, 

period7return is referring to the vector of S&P500 daily returns that reside in Period 7, 

namely the January 2010 – December 2019 time period. The cond.dist substring is referring 

to the conditional distribution of the standardized returns. In the present study, we assume 

conditional normality for the standardized returns. In the event that we wish to impose a Student-

t conditional distribution, we would alter the substring to cond.dist = c("std"). This is 

especially important for the case of high-frequency intra-day data, in that the conditional density 

is leptokurtic. In Bollerslev’s (1987) conditionally Student’s t-GARCH model, the conditional 

density is assumed to be Student’s t, with the degrees of freedom treated as another parameter to 

be estimated. Finally, the include.mean binary substring is set to FALSE so that the R 

environment ignores the estimation of conditional mean dynamics, in particular ARIMA effects.  

 Following the estimation of the GARCH(1,1) model, a concise summary output of the 

coefficient estimates and their corresponding significance levels is made possible through the 

summary command. The following command line illustrates the application of the summary 

command to the garch11.model object defined within R:  
 

summary(garch11.model) 
 

Following the application of the summary command in the R console, the R environment will 

produce summary output comprising of coefficient estimates, statistical significance levels, 

model information criteria, and goodness-of-fit statistics. For the Period 7 data, the summary 

command outputs the following information: 
 

Error Analysis: 

        Estimate  Std. Error  t value Pr(>|t|)     

omega  3.661e-06   5.491e-07    6.667 2.61e-11 *** 

alpha1 1.585e-01   1.704e-02    9.303  < 2e-16 *** 

beta1  8.005e-01   1.824e-02   43.886  < 2e-16 *** 

--- 

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

 

Log Likelihood: 

 8560.509    normalized:  3.403781  

 



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Standardised Residuals Tests: 

                                Statistic p-Value   

 Jarque-Bera Test   R    Chi^2  616.843   0         

 Shapiro-Wilk Test  R    W      0.9720951 0         

 Ljung-Box Test     R    Q(10)  11.2828   0.3359141 

 Ljung-Box Test     R    Q(15)  18.31018  0.24671   

 Ljung-Box Test     R    Q(20)  26.78719  0.1413581 

 Ljung-Box Test     R^2  Q(10)  9.867153  0.4522248 

 Ljung-Box Test     R^2  Q(15)  16.51333  0.3487793 

 Ljung-Box Test     R^2  Q(20)  17.58753  0.6145591 

 LM Arch Test       R    TR^2   11.76193  0.4649839 

 

Information Criterion Statistics: 

      AIC       BIC       SIC      HQIC  

-6.805176 -6.798222 -6.805179 -6.802652  
 

In the R output above, the GARCH(1,1) results are given. The three coefficients in the 

variance equation are listed as the intercept omega (𝜔); the first lag of the squared return 
alpha1 (𝛼); and the first lag of the conditional variance beta1 (𝛽). In particular, the 
coefficient estimates are listed underneath the Estimate column, and we observe that the 

coefficient estimates are ω =3.661e-06, α = 0.1585, and β = 0.8005. Thus, the garchFit 

command has estimated the following GARCH(1,1) model: 
 

 

ℎ𝑡 = 3.66𝑒 − 06 + 0.1585𝑟𝑡−1
2 + 0.8005ℎ𝑡−1      (4) 

 

In Eqn. (4), we discern that the process is mean-reverting since the sum of the coefficients is less 

than 1. As Engle and Patton (2001) point out, mean reversion in volatility conveys that there is a 

normal level of volatility to which the volatility process will return.  

 Having estimated the intercept , 𝛼 , and 𝛽, we can employ the coefficient estimates for 
our GARCH(1,1) model to  construct volatility forecasts, since the forecast for tomorrow’s 

volatility √ℎ𝑡+1, can be calculated using Eqn. (5): 

 

ℎ𝑡+1 = 𝜔 + 𝛼𝑟𝑡
2 + 𝛽ℎ𝑡         (5) 

 

 To illustrate, consider a trader who thinks that the long-run average daily return standard 

deviation √
𝜔

1−𝛼−𝛽
 for the S&P 500 is 1 percent. In addition, suppose today’s observed 

unexpected return (𝑟𝑡 ) is 4 percent, and that yesterday, the trader made a forecast of 3 percent for 

today’s volatility (√ℎ𝑡 ). Thus, based on the estimated GARCH(1,1) model in Eqn. (4), our 
forecast for tomorrow’s volatility is: 

 
√ℎ𝑡+1 = √𝜔 + 𝛼𝑟𝑡

2 + 𝛽ℎ𝑡 = √[(1 − 𝛼 − 𝛽) ∗ 1] + (𝛼 ∗ 4
2) + (𝛽 ∗ 32) = √[(1 − 0.1585 − 0.8005) ∗ 1] + (0.1585 ∗ 42) + (0.8005 ∗ 32)  

 

√ℎ𝑡+1= 3.1275 percent 
 



34 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

Hence, R is useful for estimating the GARCH model and predicting future volatility based on the 

above output. The R output from above also presents standard errors under the Std. Error 

column, t-values under the t value column, and corresponding p-values under the 

Pr(>|t|)column. The t-values are the ratio of the estimated coefficients and standard errors, 

and convey the degree of statistical significance for the coefficient estimates. Based on the 

summary output, we find that all of the coefficient estimates are highly significant at the 1 

percent level.  

The adequacy of a fitted GARCH model warrants attention. For the case of the estimated 

GARCH(1,1) model above, the output also presents examination of the standardized residuals for 

serial correlation. If the evidence suggests that serial correlation, or autocorrelation, exists for the 

standardized residuals, then the GARCH(1,1) model has not captured all of the return volatility 

dynamics. From the GARCH summary output above, we find that the statistics for the Ljung-

Box test for the squared standardized residuals are presented in the rows beginning with 

Ljung-Box Test. Evidently, the Ljung-Box test of interest has a p-value near 0.61 when 

testing correlation up through the 20th lag. Thus, the estimated GARCH model captures volatility 

in the returns and there are no more GARCH effects in the residuals since we fail to reject the 

null hypothesis of no serial correlation for the squared standardized residuals.  

 The summary output provides several goodness-of-fit metrics. For example, the 

summary output presents a log-likelihood value of 8560.5 for our estimated GARCH(1,1) 

model. The log-likelihood is a goodness-of-fit metric since GARCH models that achieve higher 

log-likelihood values are preferred in practice. In addition, the R summary output also presents 

the normalized log-likelihood value of 3.40, which is the log-likelihood divided by n. The R 

summary output also provides the Akaike information criterion (AIC), the Bayesian information 

criterion (BIC), the Schwarz information criterion (SIC), and the Hannan-Quinn information 

criterion value (HQIC). The aforementioned information criteria values are helpful for model 

selection purposes in assessing higher-order GARCH models, such as GARCH(2,2) models. In 

practice, smaller values of the information criteria reflect preferred models.   

As Engle and Patton (2001) point out, persistence in volatility reflects the extent to which 

today’s return exhibits a large impact on future volatility forecasts, and furthermore, present a 

measure of persistence known as the ‘half-life’ of volatility: the ‘half-life’ of volatility is defined 

as the time taken for the volatility to move halfway back towards its unconditional mean 

following a deviation from it. Based on data from 2010-2019, this study finds that the half-life of 

volatility for the S&P500 index is 16.56 days.  

 A time series plot of the S&P500’s in-sample sequence of conditional volatilities may be 

extracted within the R environment. In particular, the plot command provides a rich array of 

graphical options for the Rmetrics user. For the Period 7 data, the associated GARCH(1,1) 

conditional volatilities are plotted using the following R command: 

 
plot(garch11.model,which=2) 

 

In the above example command, the which substring is directing the R environment to the 

relevant plot of interest. For a list of the possible different plots that can be generated by the 

fGarch package, one would enter the following command in the R console: 
 

plot(garch11.model) 
 



35 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

The upper panel in Figure 2 plots the resulting time-series of the S&P500 conditional 

volatility for the January 2010 through December 2019 time period when adopting a 

GARCH(1,1) model with normal innovations. The conditional volatilities in Figure 2 are 

important when constructing dynamic confidence intervals for the S&P500 daily returns. For a 

given day, a 95% confidence interval for the S&P500 daily return on day t is given by Eq. (5): 
 

2
t t

r h            (5) 

 

Since conditional mean dynamics are negligible in the pure GARCH(1,1) model of Eqs. (2) and 

(3), the daily forecasted return for day t, tr , is zero. Thus, the 95% confidence interval for the 

S&P500 daily return on day t is subsequently given by Eq. (6): 
 

0 2
t

h            (6) 

 

In Eq. (6), it is evident that the conditional volatilities determine the confidence interval over 

time. In the R environment, the 95% confidence interval for the S&P500 daily returns is plotted 

via the appropriate plot command. Specifically, the which substring must be set to 3: 
 

plot(garch11.model,which=3) 

 

The lower panel in Figure 2 plots 95% confidence intervals for the S&P500 daily returns, 

superimposed on the daily realized returns for the January 2010 through December 2019 time 

period. This figure highlights the importance of GARCH volatilities in predicting future returns 

since the confidence interval captures, to a great extent, the dynamic behavior of the daily 

S&P500 returns. That is, nearly 95 percent of the realized S&P500 daily returns lie within the 

upper and lower limits of the confidence bands.  

 Table 1 presents the results of the estimated GARCH(1,1) model across the different time 

periods. In Table 1, we observe that all of the GARCH coefficient estimates are highly 

significant at the 1 percent level across time, further underlining the robustness of the 

GARCH(1,1) model. In addition, Table 1 reports the long-run average daily return standard 

deviation as well as the half-life of the S&P500 index over different time periods. In Table 1, we 

find that the 2000 – 2009 time period exhibits the highest long-run average daily return standard 

deviation while the 1990 – 1999 time period exhibits the highest half-life of volatility. 

Specifically, the half-life of volatility for the S&P500 index was 125.68 days for the 1990 – 1999 

time period, underlining high persistence during this time period. Thus, in the most recent decade 

spanning 2010 - 2019, volatility was less persistent relative to earlier decades. In addition, when 

examining the Ljung-Box test statistics, we find that the GARCH(1,1) model is adequate in 

capturing the volatility dynamics for all of the remaining time periods.  

  

Concluding Remarks 

This study illustrates how to estimate GARCH models for the purpose of modeling 

volatility using the R analytics platform. In particular, the present study demonstrates the 

necessary commands and use of the fGarch package for R in estimating and forecasting 

conditional volatility. The R environment is open source software that is flexible, user-friendly, 

and allows for the estimation of higher order GARCH models with different distributional 

assumptions. 



36 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

 In an application to the S&P500 returns, this study estimates a GARCH(1,1) model and 

provides interpretations of the summary output provided by R. In addition, this study 

demonstrates how to forecast future volatility based on the estimated parameters and, moreover, 

demonstrates how to extract the time-series of conditional volatility through R’s rich graphical 

capabilities. The present study also underlines the importance of the GARCH(1,1) model in 

constructing dynamic confidence intervals, and how to assess the adequacy of a fitted GARCH 

model. Finally, this article presents results on the half-life of volatility for the S&P500 index as a 

measure of volatility persistence.  

 Several directions lie ahead for future research in the economics and financial education 

literature. Firstly, the GARCH model estimates provided by R are useful for conducting dynamic 

portfolio allocation exercises in university trading rooms and classrooms. For example, a mean-

variance investment strategy could be specified for an investor exhibiting quadratic utility 

preferences. That is, conditional utility would depend on the time-varying return variances. For 

specific risk aversion levels, the student can then solve for optimal portfolio weights that vary 

over time on the basis of optimizing the investor’s conditional utility. The actual utility obtained 

by the dynamic portfolio strategy could then be compared to the actual utility of an investor who 

selects fixed portfolio weights that do not vary over time. Such an exercise allows the student to 

conduct market timing strategies on the basis of the observed time-varying volatility. 

Another avenue for future research surrounds forecasting in the context of the GARCH 

model. For example, the student may be interested in a 2-day ahead forecast given today’s 

realization of conditional volatility. Thus, a detailed examination of conducting multi-day ahead 

volatility forecasts is essential.   

Third, it is of interest to relate GARCH volatility forecasts to how banks assess future 

portfolio losses through volatility-based Value-at-Risk (VaR) measures.  In particular, out-of-

sample forecasts for the conditional volatility, in conjunction with the probability distribution of 

returns, provide a higher degree of accuracy when estimating potential portfolio losses. In 

addition, it would be interesting to compare VaR forecasts across different distributional 

assumptions of the standardized returns.  

 Finally, the asymmetric class of GARCH models, proposed by Engle and Ng (1993) and 

Nelson (1991), suggest that volatility tends to be higher on days in which negative returns are 

realized. That is, asymmetric volatility models allow for good news and bad news to exhibit 

different predictability for realized volatility. Given the downside risks observed in 2008 through 

2010, and in 2018, it would be interesting to explore the estimation of such models in the R 

environment. These topics are left for future research.  

 

References 

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of 

Econometrics, 31(3): 307-327.  

Bollerslev, T. 1987. “A Conditional Heteroskedastic Time Series Model for Speculative Prices  

and Rates of Return.” Review of Economics and Statistics, 69(3): 542-547. 

Bollerslev, T., and J.M. Woolridge. 1992. “Quasi-Maximum Likelihood Estimation and  

Inference in Dynamics Models with Time-Varying Covariances.” Econometric Reviews, 

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Engle, R.F. 1982. “Autoregressive Conditional Heteroskedasticity with Estimates of the  

Variance of U.K. Inflation.” Econometrica, 50(4): 987-1008.  

Engle, R.F. 2001. “GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics.”  



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Journal of Economic Perspectives, 15(4): 157 – 168.  

Engle, R.F., and A.J. Patton. 2001. “What good is a volatility model?” Quantitative Finance, 1:  

237-245. 

Engle, R.F., and V.K. Ng. 1993. “Measuring and Testing the Impact of News on Volatility.” 

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French, K., Schwert, G., and R. Stambaugh. 1987. “Expected stock returns and volatility.” 

Journal of Financial Economics, 19(1): 3-29. 

Hansen, P. R., and A. Lunde. 2005. “A forecast comparison of volatility models: Does anything  

beat a GARCH(1,1)?” Journal of Applied Econometrics, 20(7): 873-889.   

Li, R. 2019. “Teaching Undergraduates R in an Introductory Research Methods Course: A Step- 

by-Step Approach.” Journal of Political Science Education.  

Nelson, D.B. 1991. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” 

Econometrica, 59(2): 347-370.  

Tsay, R.S. 2010. Analysis of Financial Time Series. 3rd edition. New York: John Wiley & Sons.  

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38 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

APPENDIX 

Figure 1 

Panel A plots the time-series of daily squared returns for the S&P500 index over the January 

2010 – December 2019 time period. Panel B plots the autocorrelation function (ACF) for the 

S&P500 daily squared returns over the identical time period. 

 

 

 

 

 



39 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

Figure 2 

The upper panel plots the conditional volatilities for the S&P500 index over the January 2010 

through December 2019 time period. The lower panel plots the realized S&P500 daily returns 

and the associated 95% confidence bands over the identical time period. The conditional 

volatilities are estimated using a GARCH(1,1) model with Normal innovations. 

 
 

 

 

 

 



40 |JOURNAL FOR ECONOMIC EDUCATORS, 20 (1), 2020 

 

Table 1 

This table reports the results of fitting a GARCH(1,1) model with Normal innovations for the S&P500 daily returns using R’s fGarch package. Quasi-

maximum likelihood parameter estimates, the associated t-ratios, log likelihood, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) 

are reported.  This table reports the long-run average daily return standard deviation (SD), persistence, and half-life of volatility for the S&P500 index based on 

the estimated GARCH(1,1) models across the different time periods. The Ljung-Box test for autocorrelation through the 20th lag for the squared standardized 

residuals, denoted by Q(20), is presented.  The sample period is January 1950 through December 2019. Seven non-overlapping time-periods are examined. 

Period 1 extends from January 1950 through December 1959. Period 2 extends from January 1960 through December 1969. Period 3 extends from January 1970 

through December 1979. Period 4 extends from January 1980 through December 1989. Period 5 extends from January 1990 through December 1999. Period 6 

extends from January 2000 through December 2009. Period 7 extends from January 2010 through December 2019. Adjusted closing p rice data is obtained from 

Yahoo! Finance. The GARCH(1,1) model is of the form: 
2

1 1
, ~ (0,1)

iid

t t t t t t t
r h z h r h z N  

− −
= = + +  

           
  

Period 1 2 3 4 5 

 

6 

 

7 

Coefficient      
  

𝜔 1.668e-06 1.596e-06 7.515e-07 4.886e-06 4.855e-07 1.065e-06 3.661e-06 

 (2.42)** (4.06)*** (3.23)*** (4.20)*** (2.75)*** (3.83)*** (6.67)*** 

𝛼 0.0594 0.1649 0.0631 0.0927 0.0481 0.0737 0.1585 

 (3.91)*** (7.90)*** (3.23)*** (8.20)*** (5.85)*** (8.18)*** (9.30)*** 

𝛽 0.9103 0.8012 0.9260 0.8632 0.9464 0.9196 0.8005 

 (35.05)*** (30.96)*** (94.18)*** (44.41)*** (103.20)*** (99.42)*** (43.89)*** 

Long-run 

average daily 

return SD 0.12 0.11 0.13 0.17 0.15 

 

 

0.20 

 

 

0.15 

Persistence 0.91 0.80 0.93 0.86 0.95 
0.92 0.80 

Half-life 22.53 20.10 63.24 15.37 125.68 103.11 16.56 

Log-Likelihood 8914.1 9382.1 8705.9 8170.4 8597.6 7789.4 8560.5 

AIC -7.10 -7.54 -6.89 -6.46 -6.80 -6.19 -6.81 

BIC -7.09 -7.53 -6.89 -6.46 -6.80 -6.19 -6.80 

Q(20) 5.33 21.38 14.99 7.30 10.87 19.18 17.59 

      ***,** indicate statistical significance at the 0.01 and 0.05 level, respectively