\linespread{1.2}
\chapter{Spurious Inference in the GARCH(1,1) Model when Generalized ZILC Holds: Theory and Application}

\section{Introduction}

The GARCH(1,1) model is popular in modeling the changing-volatility of the time series data. It
has become one of the benchmarks in modeling time-varying volatility since introduced by Bollerslev (1986). Capturing time-varying volatility is one of the most important issues in modeling time series. The GARCH(1,1) model is originally proposed as ARCH(AutoRegressive Conditional Heteroskedasticity) in 1982 by Robert Engle. The ARCH/GARCH family has been widely used to extract a potential volatility-changing process. Bera and Higgins (1993) remarked that ``a major contribution of the ARCH literature is the finding that apparent changes in the volatility of economic time series may be predictable and result from a specific type of nonlinear dependent rather than exogenous structural changes in variables."

Zero-Information-Limit Condition (ZILC) is first proposed by Nelson and Startz (2007). Their paper has shown that when identification of the target parameter is conditional on another structural parameters in the model, the inference for the target parameters will be misleading by the realization of the structural parameters, if ZILC holds in the structure of the information matrix. In models when ZILC applies, the standard errors tend to be underestimated when the identifying structural parameters are close to some specific values, no matter how large the sample size would be. Examples include the Weak Instrument problem, ARMA(1,1) models, and some certain nonlinear regression models. Ma, Nelson, and Startz (2007) shows ZILC also holds in the GARCH(1,1) model. 


To illustrate how the ZILC applies to the ARCH-GARCH family model, I propose a simulation-based experiment as motivation. I first set up an ARCH(1) model: 


\begin{equation}
\epsilon_t \sim i.i.d.N(0, \sigma^2_t)
\end{equation}

\begin{equation}
\sigma^2_t = \omega + \alpha \cdot \epsilon^2_{t-1} 
\end{equation}


Note that $\sigma^2$ is the conditional variance and is driven by past realizations of $\epsilon$. $\alpha$ is called the ARCH coefficient. 



The typical GARCH(1,1) model is written as: 

\begin{equation}
\epsilon_t \sim i.i.d.N(0, \sigma^2_t)
\end{equation}

\begin{equation}
\sigma^2_t = \omega + \alpha \cdot \epsilon^2_{t-1} + \beta \cdot \sigma^2_{t-1} 
\end{equation}


Note that $\sigma^2$ is still the conditional variance and is driven by past realizations of $\epsilon$, but also with added persistence determined by $\beta$. Here, $\alpha$ is called ARCH coefficient, and $\beta$ is called GARCH coefficient. In the case $\beta=0$, the model reduces to the pure ARCH(1) model, and in the case $\alpha=0$, the GARCH effect $\beta$ cannot be identified. 

The GARCH coefficient $\beta$ represents the relationship between the current volatility with the past volatility. Intuitively, the GARCH model depicts the volatility clustering phenomenon, which implies ``the large changes tend to be followed by large changes, and small changes tend to be followed by small changes." This phenomenon exists among many time series data, especially for stock price, exchange rates, and oil price. That is why GARCH model is very popular and GARCH coefficient $\beta$ is on the focus of the model research. 



The purpose of the experiment is to investigate the spurious inference of GARCH estimate $\beta$ in the GARCH(1,1) model. I implement a series of Monte Carlo experiments. In the sequence of Monte Carlo experiments, I simulate the data from ARCH(1) process defined by equation (1) and (2) with three sets of parameters values: 


\[\left( \begin{array}{c}
\omega  \\
\alpha \\
\beta \\
 \end{array} \right) = 
%
  \left( \begin{array}{c}
1  \\
0.01 \\
0 \\
 \end{array} \right) 
%
\left( \begin{array}{c}
1  \\
0.05 \\
0 \\
 \end{array} \right)  
%
\left( \begin{array}{c}
1  \\
0.1 \\
0 \\
 \end{array} \right) 
%
\left( \begin{array}{c}
1  \\
0.9 \\
0 \\
 \end{array} \right)  \]  

\

Then I use the ARCH(1) simulated data and GARCH(1,1) model setting to estimate the GARCH coefficient $\beta$. The estimated variance of $\beta$ will be calculated, then the t-statistics of the $\beta$. Run the Monte Carlo simulation for 1000 times, and record the chances that the t-test of $\beta$ rejects the true null hypothesis of $\beta=0$. Table 1 gives the empirical sizes of t-test at the nominal $5\%$ level for GARCH coefficient $\beta$.



The result is very interesting. The data is generated from ARCH(1) model (which the GARCH coefficient $\beta=0$). If I estimate $\beta$ from the ARCH(1) simulated data, the probability of rejecting the null hypothesis of $\beta=0$ should be around the nominal $5\%$. However, the real Monte Carlo experiment shows, we would have around $50\%$ chance that we falsely reject the null hypothesis of $\beta=0$, and falsely reckon that the data fits GARCH(1,1) model setting. This also implies that, we might falsely reckon the data contains the significant clustering volatility phenomena. In Table 1.1, even the sample size is large as increased to 5000, the spurious inference issue still exists: the estimated variance of the GARCH coefficient $\beta$ is underestimated, and the t-statistics of $\beta$ is overestimated. The explanation of this phenomena is ZILC. 


Ma, Nelson, and Startz (2007) first proves that ZILC applies in the GARCH(1,1) model. In this paper, I extend the ZILC theory to the Generalized ZILC theory, and depict the ZILC zone by constructing a complete Monte Carlo Simulation in the GARCH(1,1) model. In the ZILC zone, the estimated standard errors of $\beta$ are too small; as a result, the actual size of the t-test for the GARCH coefficient is far too large, which leads the rejection of the true null hypotheses occurring too often. Thus, researchers unaware of this spurious effect may be tempted to conclude that the persistence due to the GARCH effect is significant, while in fact the persistence of the volatility is absent. For people who care about the significance of the GARCH coefficient, this paper provides a test strategy for empirical application. In addition, for people who aim to use estimated variance of the GARCH coefficient for further empirical application, I propose a simulation-based strategy to correct the underestimated standard error, and eventually resolve the Generalized ZILC issue. This solution can be used in the Value-at-Risk analysis in the risk management. Value-at-Risk analysis uses estimated variance of the target parameter to construct the Value-at-Risk value. In my paper, I propose an investment hedging strategy with a virtual data, to show that how the spurious inference led by Generalized ZILC would influence the Value-at-Risk analysis.


This essay is organized in the following way: Section 2 introduces what is the Generalized ZILC, and how the Generalized ZILC applies in the GARCH(1,1) model theoretically. Section 3 presents evidence by Monte Carlo experiments, to elaborate what is the actual size of the t-test for the GARCH coefficient, and how the Generalized ZILC distorts the t-test results. Section 3 also illustrates the ZILC zone and safe zone of the estimated $\alpha$ and $\beta$ in the GARCH(1,1) model. Section 4 presents various tests under Generalized ZILC issue, and proposes a test strategy. This strategy can be applied for research interests on the significance of GARCH coefficient $\beta$. Section 5 design a volatility hedging investment hedging strategy, and show how the Generalized ZILC applies in the Value-at-Risk analysis in the risk management. Section 5 proposes the solution for ZILC: a simulation-based ratio to correct the underestimated variance. Section 6 concludes this essay. 



\section{The Generalized ZILC in the GARCH(1,1) Model}

\subsection{Zero-Information-Limit Condition (ZILC)}


ZILC identifies how the weak identification issue leads to the spurious inference on the target parameter estimates. In many Econometrics models, the asymptotic variance of a parameter estimate depends on the value of other structural parameters. If the data contains little information about the target parameter when the structural parameters are close to some critical values, Nelson and Startz (2007) identified this phenomena as ZILC. 



Consider an Econometrics model with parameters $\gamma$ and $\beta$. $\beta$ is of the interest for hypothesis testing. Sample size is T, and exogenous data is X. $ V_{\hat{\beta}}(\beta, \gamma, X)  $ is the asymptotic variance of $\beta$. The estimated information matrix of $\beta$ is $ I_{\hat{\beta}}(\beta, \gamma, T, X) = [V_{\hat{\beta}}(\beta, \gamma, T, X)]^{-1} \geq 0 $. 



When ZILC holds:  

\begin{equation}
\lim\limits_{\gamma \rightarrow \gamma_0} I_{\hat{\beta}} (\beta, \gamma, T, X) = 0
\end{equation}


Thus, we can get: 

\begin{equation}
\lim\limits_{\gamma \rightarrow \gamma_0} \left[ Pr \left( \frac{I_{\hat{\beta}} (\hat\beta, \hat\gamma, T, X)}{ I_{\hat{\beta}} (\beta, \gamma, T, X)} \right) > M \right] =1, \ \forall M 
\end{equation}

\

The t-statistic is  $ t^2_{\hat{\beta}} = (\hat{\beta} - \beta_0)^2 \cdot  I_{\hat{\beta}}(\hat{\beta}, \hat{\gamma}, T, X)  $. The implication of the equation is, when $\gamma$ is close to a critical value, true information of $\beta$ is close to 0. Under this scenario, the estimated information of $\beta$ will be always larger than the asymptotic true variance of $\beta$. Information is overestimated; variance would be underestimated; standard error would be underestimated; thus, t-statistics will be overestimated. Spurious inference occurs.   





\subsection{Generalized ZILC in the GARCH(1,1) Model}


Nelson and Startz (2007) illustrates that under ZILC, the standard errors tend to be underestimated; the size of the asymptotic t-test is distorted (either be oversized or undersized). Ma, Nelson, and Startz (2007) shows that ZILC applies to the GARCH(1,1) model when ARCH coefficient $\alpha$ goes to 0. The estimated standard error of GARCH coefficient $\beta$ is underestimated; the t-test is oversized. 


To illustrate how ZILC applies to the GARCH(1,1) Model, I rewrite the archetypal GARCH(1,1) model as: 

\begin{equation}
\epsilon_t = \sqrt{\sigma^2_t} \cdot \phi_t 
\end{equation}

\begin{equation}
\sigma^2_t = \omega + \alpha \cdot \epsilon^2_{t-1} + \beta \cdot \sigma^2_{t-1} 
\end{equation}


Following the previous discussion, $\sigma^2$ is the conditional variance and is driven by past realizations of $\epsilon$, with added persistence determined by $\beta$. Here, the mean of equation (7) is set to be zero without loss of generality (see Bollerslev (1986)). $\phi_t$ is independently and identically distributed with zero mean and unit variance, i.e., $i.i.d.(0,1)$, with finite higher moments (see Lumsdaine(1996) for details). 



Write up the log-likelihood function: 

\begin{equation}
L_{{T}}(\theta) = T^{-1} \cdot \sum_{t=1}^{T} l_{t} (\theta)
\end{equation}

\begin{equation}
 \sum_{t=1}^{T} l_{t} (\theta) = - \frac{1}{2} \  \log 2 \pi - - \frac{1}{2} \ \log h_{t} - \frac{1}{2} \frac{\epsilon^2_{t}}{h_{t}}
\end{equation}


Where, $ \theta = (\omega, \alpha, \beta)'$ and $\hat{\theta_{T}}$ maximizes the quasi log-likelihood function for a given sample data {$\epsilon_{1}$,$\epsilon_{2}$, $\epsilon_{3}$, ... $\epsilon_{T}$}, and therefore is the QMLE. Fiorentini, Calzolari, and Panattoni (1996) derives the first and second derivatives in GARCH(1,1) models. Ma (2008) takes a local approximation of each element in the neighborhood of small $\alpha$ to avoid taking the expectation of a non-linear form. Based upon Ma (2008)'s results, it is straight forward to show that the inverse of the asymptotic variance of $\hat{\beta}$, the information measure of Nelson and Startz (2007), goes to 0 as $\alpha$ approaches 0, i.e., ZILC holds:

\begin{equation}
\lim \limits_{\alpha \rightarrow 0} I_{\hat{\beta}} (\omega, \alpha, \beta) = 0
\end{equation}


Appendix gives a formal proof of (1.11) based upon Ma (2008)'s analytic results.




Ma, Nelson, and Startz (2007) proves that when $\alpha$ is close to 0, information of $\beta$ is close to 0. In this paper, as proved in Appendix, when $\beta$ goes to 0, the information of $\beta$ will also go to 0, which is presented as: 


\begin{equation}
\lim \limits_{\beta \rightarrow 0} I_{\hat{\beta}} (\omega, \alpha, \beta) = 0
\end{equation}



Appendix gives a formal proof of (1.12). Furthermore, the following Monte Carlo experiments investigate that the spurious inference occurs when both $\alpha$ and $\beta$ go $0$. That is how this paper extends ZILC to the Generalized ZILC in the GARCH(1,1) model. 




In the GARCH(1,1) model, the estimated variance of $\hat{\beta}$ is $V_{\hat{\beta}} (\omega, \alpha, \beta, T, X)$. The inverse of the variance is a natural measure of information associated with $\hat{\beta}$, so the estimated information of $\hat{\beta}$ is $ I_{\hat{\beta}}(\omega, \alpha, \beta, T, X) = V^{-1}_{\hat{\beta}} (\omega, \alpha, \beta, T, X)$. Under the Generalized ZILC issue, either $\alpha$ or $\beta$ close to 0 would both lead the information measure of $\beta$ close to 0. The estimated information measure will be always larger than the true information measure. Proved by the series of Monte Carlo experiments, the estimated variance of the GARCH estimate $\beta$ is too small under both of the scenarios; t-statistics of $\beta$ is overestimated.

Generally speaking, consider a model with scalar parameters $\gamma$. $\beta$ is the parameter of interest for hypothesis testing. The asymptotic variance of estimator $\hat{\beta}$ is assumed to have a representation as a function of $\beta$, other parameters $\gamma$, and exogenous data X. The Generalized ZILC issue can be illustrated as:




\textbf{Definition:} The Generalized Zero-Information-Limit Condition (Generalized ZILC) holds for estimator $\hat{\beta}$ if any value in the domain $\Theta$ makes the following equation: \footnote{The domain $\Theta$ represents any values to make $I_{\hat{\beta}}(\beta, \gamma, X)$ goes to 0, even including the condition of $\beta$ itself.}
 

\begin{equation}
\lim \limits_{\Theta} I_{\hat{\beta}} (\beta, \gamma, X) = 0
\end{equation}

So 

\begin{equation}
\lim\limits_{\Theta} \left[ Pr \left( \frac{I_{\hat{\beta}} (\hat\beta, \hat\gamma, X)}{ I_{\hat{\beta}} (\beta, \gamma, X)} \right) > M \right] =1, \ \forall M 
\end{equation}



\


\section{Evidence of Spurious Inference from Monte Carlo Experiments}


In the GARCH(1,1) model, the problem of Generalized ZILC can be represented as: 


\begin{equation}
\lim \limits_{\Theta} \left[ Pr \left( \frac{I_{\hat{\beta}} (\hat\omega, \hat\alpha, \hat{\beta}, T, X)}{ I_{\hat{\beta}} (\hat\omega, \hat\alpha, \hat{\beta}, T, X)} \right) > M \right] = 1, \forall M 
\end{equation}


Estimated information of $\beta$ will be always larger than the true information of $\beta$ under Generalized ZILC. Since $V^{-1}_{\hat{\beta}} = I_{\hat{\beta}}$, estimated variance will be underestimated, and $ t^2_{\hat{\beta}} = (\hat{\beta} - \beta_0)^2 \cdot  V^{-1}_{\hat{\beta}}  $ will be oversized. 








A series of Monte Carlo experiments is implemented to investigate how worse the spurious inference occurs when the GARCH(1,1) model is under the Generalized ZILC issue. 






\subsection{Inference when there is no GARCH effect}






In this sequence of Monte Carlo experiments, data is simulated from the GARCH(1,1) defined by equation (1.7) and (1.8) with the following sets of parameter values: 



\[  \left( \begin{array}{c}
\omega  \\
\alpha \\
\beta \\
 \end{array} \right) = 
%
  \left( \begin{array}{c}
1  \\
0.01 \\
0 \\
 \end{array} \right),
%
  \left( \begin{array}{c}
1  \\
0.02 \\
0 \\
 \end{array} \right) ...
%
\left( \begin{array}{c}
1  \\
0.1 \\
0 \\
 \end{array} \right) ,  
%
\left( \begin{array}{c}
1  \\
0.2 \\
0 \\
 \end{array} \right) ...  
%
\left( \begin{array}{c}
1  \\
0.9 \\
0 \\
 \end{array} \right)  \]  


\



Ma (2008) sets the local approximation for each element in the neighborhood of small $\alpha$ to avoid taking the expectation of non-linear form. This paper extends the Monte Carlo experiment range to a complete area of $\alpha$ and $\beta$. Note that in the GARCH(1,1) model $\alpha + \beta$ should be smaller than 1. 





Since $\beta$ is 0, there is no GARCH effect and the process is actually an ARCH(1) process. The scale parameter $\omega$ is normalized to be unity. For each set of parameter values, I set up the sample size $T=1000$ to be consistent with the experiments in Nelson and Startz (2007), and Ma, Nelson, and Startz (2007). For all experiments, 1000 simulated paths of sample data of length T are generated. Figure 1.1 gives the empirical size of t-test at the nominal $5\%$ level for estimated parameter $\hat{\beta}$.






Ma, Nelson, and Startz (2007) shows the similar results that when ARCH coefficient $\alpha$ is close to 0, the actual size of t-test for $\beta$ is around $50\%$ even for a large sample size as shown in Table 1. Their paper also mentions that, for sufficiently large $\alpha$, and for sufficiently large sample size, the size distortion of $\beta$ is greatly reduced. As Figure 1.1 shows, the size distortion of $\beta$ is reduced as $\alpha$ increases at the beginning. However, when $\alpha$ continues to become larger, the actual size of $\beta$ goes worse. This is due to the Generalized ZILC proved in Section 2: in the GARCH(1,1) model, when $\beta$ is small, the estimated information of $\beta$ is always close to 0, no matter how the $\alpha$ changes. 








\subsection{Inference when there is moderate GARCH effect}


Ma, Nelson, and Startz (2007) also proposes the moderate GARCH effect Monte Carlo experiment. It is important to note that ZILC holds whenever the ARCH coefficient $\alpha$ is small, regardless of the magnitude of true $\beta$. In the moderate GARCH model, the $\beta$ is set as 0.5. Here I extend the Monte Carlo experiment to the complete rang of $\alpha$ and $\beta$, which $\beta$ is set as 0.5, and $\alpha$ is increased from 0.01 to 0.49 ($\alpha + \beta \textless 1 $). Data is simulated from the GARCH(1,1) process defined by the following equation: 


\[  \left( \begin{array}{c}
\omega  \\
\alpha \\
\beta \\
 \end{array} \right) = 
%
  \left( \begin{array}{c}
1  \\
0.01 \\
0.5 \\
 \end{array} \right) ...
%
\left( \begin{array}{c}
1  \\
0.1 \\
0.5 \\
 \end{array} \right) ,  
%
\left( \begin{array}{c}
1  \\
0.2 \\
0.5 \\
 \end{array} \right) ...  
%
\left( \begin{array}{c}
1  \\
0.49 \\
0.5 \\
 \end{array} \right)  \]  


\

The sample size is fixed at 1000 and the number of simulation is also 1000. Graph 2 presents the empirical size of t-test of $\beta$ at the nominal $5\%$ level. 




As proved in Section 2, for the moderate GARCH(1,1) model, when $\alpha$ is small, ZILC applies to the GARCH(1,1) model as the estimated information matrix of $\beta$ is close to 0. So the estimated standard error of $\beta$ is underestimated, t-statistics is overestimated. Figure 1.2 shows the actual size of t-test for $\beta$ is around $40\%$ when $\alpha$ is small in the moderate GARCH(1,1) model, while the t-test goes back to the normal size level when $\alpha$ increases, which implies the ZILC issue goes away.  




\subsection{ZILC Zone and Safe Zone in the GARCH(1,1) Model}


As Section 2 proves, and Figure 1.1 and Figure 1.2 show, ZILC issue suffers either $\alpha$ is small or $\beta$ is small, which is the Generalized ZILC. Thus, information of $\beta$ close to 0 leads the estimated information of $\beta$ would be always larger than the true information of $\beta$. Therefore, estimated standard error of $\beta$ is underestimated, t-statistics of $\beta$ is overestimated, actual size of the t-test for $\beta$ is too large. When $\alpha$ and $\beta$ increase, estimated information of $\beta$ does not suffer the ZILC issue. It would be interesting to know that, regarding the estimated $\alpha$ and $\beta$, under what range, the t-test of $\beta$ is not distorted. Figure 1.3 shows the ZILC zone and safe zone of the GARCH(1,1) model. To get Figure 1.3, I simulate the data from the GARCH(1,1) process starting from $\alpha = 0.01$ and $\beta = 0$, run the Monte Carlo simulation for 1000 times to calculate the actual size of t-test of $\beta$. Then I increase $\alpha$ and $\beta$ each with 0.01 value, run the Monte Carlo simulation again, and record the actual size of t-test of $\beta$. When the actual size of the GARCH coefficient $\beta$ falls into the nominal $5\%$, the estimated $\alpha$ and $\beta$ combine a boundary, which forms the ZILC zone and safe zone, as shown in Figure 1.3.      




Figure 1.3 provides a useful way for future empirical work. If a series of data is estimated in GARCH(1,1) model, and the estimated $\alpha$ and $\beta$ fall into the safe zone, then we can trust the t-test for $\beta$. However, if the estimated $\alpha$ and $\beta$ fall into the ZILC zone, we should be careful of using t-statistics and the estimated standard error of $\beta$, since the estimated standard error of $\beta$ might be underestimated and therefore the t-statistics might be oversized.   








\section{Application: Distorted t-statistics}



\subsection{Different Tests Compare}


Assume research focus of empirical work is on the significance of $\beta$, t-test is distorted and the standard error is underestimated due to the overestimated information of $\beta$ under Generalized ZILC. Although t-test is distorted, Likelihood Ratio(LR) test and lagrange Multiplier(LM) test do not suffer from ZILC. This will be proved by the Monte Carlo experiment with the same modeling setting in Section 3.1 and Section 3.2. Table 1.2 gives the empirical sizes of LR test and LM test when the model is setting as no GARCH effect. 




It's good for practitioners that LR test and LM test perform much better than the t-test for model with no GARCH effect. The better performance of the LR test and LM test can be traced to the fact that both of these two tests are calculated without the restriction on the weakly identified parameters; see Zivot, Startz, and Nelson (1998) for discussion of this in the weak instrument case. 



Table 1.3 also proves the LR and LM tests perform better than t-test when there is moderate GARCH effect. I run the complete Monte Carlo experiment under the domain with $\alpha$ \textgreater 0, $\beta$ \textgreater 0, and $\alpha + \beta \textless 1$. The conclusion is, LR test and LM test perform the actual size of the test, which proves LR test and LM test are valid no matter the ARCH and GARCH estimates fall into ZILC zone or not. 







\subsection{Strategy when the Research Focuses on the Significance of $\beta$}



Based on the analysis in Section 1.4.1, an empirical strategy could be proposed when the application focuses on the significance of $\beta$:




\begin{itemize}

\item ZILC holds in GARCH(1,1) model when $\alpha \rightarrow 0$ and $\beta \rightarrow 0$. t-test is distorted (t-statistics is overestimated) proved by a series of Monte Carlo experiments. So when we use GARCH(1,1) to model the data, we should be careful about the t-test of $\beta$. 


\item LR test and LM test are not distorted, so a practice strategy is proposed as:


\begin{itemize}

\item If t-test rejects the null, but LR test and LM test don't reject null, we go for LR and LM test and don't reject null. 

\item If t-test, LR test, and LM test all reject the null, then we reject the null.  

\end{itemize}



\end{itemize}








\section{Application: Distorted Variance}


\begin{flushleft}
If the empirical work is emphasized on the estimated variance of $\beta$, I propose a Monte Carlo based experiment to correct the underestimated variance. The methodology is Parametric Bootstrapping. This methodology is followed by an empirical application of Value-at-Risk analysis in the risk management. 
\end{flushleft}


\subsection{Value-at-Risk Analysis in GARCH(1,1) Model}



Value-at-Risk (VaR) is a statistical technique term in risk management. It is used to measure and quantify the level of financial risk within a firm or investment portfolio over a specific time frame. Value-at-Risk is measured through three fundamental variables: the amount of potential loss, the probability of that amount of loss, and the time frame. For example, a financial firm may determine that it has a 1\% one month value at risk of \$100 million. This means there is a 1\% chance that the firm could lose more than \$100 million in any given month. So the Value-at-Risk for 1\% and one month is \$100 million. 


Assume an investment strategy to hedge the persistence volatility index $\beta$ constructed by the GARCH(1,1) model. The benchmark model is MA(1)-GARCH(1,1), and estimated daily Value-at-Risk is at 99\% confidence level. $\hat{\omega} = 1.061$, $\hat{\alpha} = 0.077$ and $\hat{\beta} = 0.773$. Generalized ZILC applies in this case as estimated $\alpha$ is close to 0. The estimated variance of $\beta$ is 0.05246. If the capital exposure for this investment strategy is \$10 million, within one day, there is 1\% chance that the bank would lose more than 0.05246 $\times$ \$10m $\times$ 2.2 = \$1.16m. Here, 2.2 is calculated by the Quantile Loss Function proposed by Angelidis, Benos, and Degiannakis(2004). Because of Generalized ZILC, estimated variance of $beta$ is underestimated; the Value-at-Risk is underestimated. The risk of this strategy tends to be underestimated. 


\subsection{Parametric Bootstrapping to Calculate the Corrected Ratio for Generalized ZILC}



The following steps propose how a ratio is calculated to correct the underestimated variance based on the Parametric Bootstrapping methodology:


\begin{itemize}
\item Set a sequence of Monte Carlo experiment that $\omega = 1.061$, $\alpha = 0.077$ and $\beta = 0.773$. Also set the sample size T = 1000, and 1000 simulated paths of sample data of length T are generated. 

\item For each of the simulated sample data, use GARCH(1,1) model to estimate and record the 1000 of the estimated variance of $\beta$. 

\item The asymptotic variance of $\beta$ can be evaluated numerically through the equation in Appendix.

\item Take the average of the 1000 estimated variance of $\beta$, and divide the value of the numerical asymptotic variance of $\beta$. The ratio is 7.1. 

\end{itemize}


This ratio is the corrected ratio for the GARCH(1,1) model when $\omega = 1.061$, $\alpha = 0.077$ and $\beta = 0.773$. The underestimated variance of $\beta$ is 0.05246. By multiplying the corrected ratio 7.1, the corrected estimated variance of $\hat{\beta}$ is 0.05246 $\times$ 7.1 = 0.372. Therefore, the corrected Value-at-Risk is 0.372 $\times$ \$10m $\times$ 2.2 = \$8.19m. Previously, the Value-at-Risk is \$1.16m, which means there is a 1\% that this investment strategy could lose \$1.16m in any given business day. However, because of the Generalized ZILC, the variance is underestimated, thus the risk is underestimated. By conducting the parametric bootstrapping, the Monte Carlo based corrected ratio is generated. By multiplying the corrected variance ratio, the amount that under 1\% this investment strategy might lose in any given business day could increase from \$1.16m to \$8.19m. We might underestimate the risk if we fail to consider Generalized ZILC in the GARCH(1,1) model in the application of the Value-at-Risk analysis. 



\section{Conclusion}


This paper shows that Generalized ZILC holds in the GARCH(1,1) model. ZILC is formulated by Nelson and Startz(2007). Ma, Nelson, and Startz (2007) proves when $\alpha$ is close to 0, ZILC holds in the GARCH(1,1) model. This paper proves when $\beta$ is close to 0, ZILC still applies in the GARCH(1,1) model. Therefore, the Generalized ZILC theory is proposed. This paper depicts the ZILC zone and safe zone for detecting ZILC issue, and illustrates two empirical strategies to deal with the Generalized ZILC issue. One strategy is the test choice when we focus on the significance of $\beta$. The other strategy is to generate a Monte Carlo based ratio by parametric bootstrapping, to correct the underestimated variance of $\beta$. This strategy proves its value in the Value-at-Risk analysis, since the risk would be underestimated if we failed to consider the Generalized ZILC issue in the GARCH(1,1) model application.  





\newpage


\section{Appendix}

\subsection{The Closed Form Information Matrix of the GARCH(1,1) Model}

\begin{flushleft}
The information matrix of the GARCH(1,1) model given by Ma (2008):
\end{flushleft}
 \[ I = \left( \begin{array}{ccc}
A & B & B  \\
B & C & D \\
B & D & E \\
 \end{array} \right)\] 

\begin{flushleft}
Where A, B, C, D, E are all the functions of estimates ${\omega}$, ${\alpha}$, and ${\beta}$.  
\end{flushleft}

$ A = \frac{(1 - \alpha - \beta)^2}{ 2 \omega^2 (1-\beta)^2}$,


$ B = \frac{1 - \alpha - \beta}{2 \omega (1-\beta)^2} $,


$ C = \frac{1 - \alpha - \beta}{ 2 (1 - 2 \alpha \beta -\beta^2) } \cdot [\frac{3(1 + \alpha + \beta)}{1 - 3\alpha^2 - 2 \alpha \beta - \beta^2} +  \frac{2 \beta}{(1 - \beta)^2}]$

$ D = \frac{(1 + \alpha + \beta) \cdot (1 - \alpha - \beta)}{2 (1 - 3 \alpha^2 - 2 \alpha \beta - \beta^2) \cdot (1 - \beta^2) } (\frac{1}{1 - \alpha \beta - \beta^2} + \frac{3 \alpha \beta}{1 -2 \alpha \beta - \beta^2}) + \frac{\beta}{2 (1 - \beta^2)} ( \frac{2}{1 - \beta} - \frac{\alpha + \beta}{ 1 - \alpha \beta - \beta^2 } - \frac{\alpha}{ 1 - 2 \alpha \beta - \beta^2}  )$

$ E = \frac{1 - \alpha - \beta}{ 2 (1 - \beta^2) \cdot (1 - \alpha \beta - \beta^2)} [ \frac{ (1 + \alpha \beta + \beta^2) \cdot (1 + \alpha + \beta)}{ 1 - 3\alpha^2 - \alpha \beta - \beta^2} + \frac{2 \beta}{ 1 - \beta}   ]  $

\begin{flushleft}
The information measure for $\hat{\beta}$ is the inverse of $\hat{\beta}$'s variance, which is:
\end{flushleft}
$$ I_{\hat{\beta}} = \frac{T}{I^{-1}(3,3)} = T \cdot \frac{[1-(\alpha+\beta)]^2}{2\omega^2} \cdot \frac{B^2 (2 D - C - E) + A (CE - D^2)}{AC - B^2}  $$

\begin{flushleft}
Ma, Nelson, and Startz (2007) proves that, 
\end{flushleft}

$T \cdot \frac{(1 - \alpha - \beta)^2}{2 \omega^2} \ \neq \ 0 \ \ as \ \ \alpha \rightarrow 0$

$  AC - B^2 \ \neq \ 0 \ \ as \ \ \alpha \rightarrow 0$


$B^2 (2 D - C - E) + A (CE - D^2) \ \rightarrow \ 0 \ \ as \ \ \alpha \rightarrow 0$

\begin{flushleft}
This completes the equation (1.11):
\end{flushleft}
$$\lim \limits_{\alpha \rightarrow 0} I_{\hat{\beta}} (\omega, \alpha, \beta) = 0 $$

\begin{flushleft}
In addition, 
\end{flushleft}
$B^2 (2 D - C - E) + A (CE - D^2) \ \rightarrow \ 0 \ \ as \ \ \beta \rightarrow 0$


\begin{flushleft}
Which leads, 
\end{flushleft}
$$\lim \limits_{\beta \rightarrow 0} I_{\hat{\beta}} (\omega, \alpha, \beta) = 0 $$


\begin{flushleft}
Equation (1.12) is proved, which is the Generalized ZILC case. 
\end{flushleft}



\subsection{The Asymptotic Variance of $\beta$}

\begin{flushleft}
The information matrix of parameter $(\omega, \alpha, \beta)'$ is
\end{flushleft}
\[ I = \left( \begin{array}{ccc}
A & B & B  \\
-- & C & D \\
-- & -- & E \\
 \end{array} \right)\] 

\begin{flushleft}
The information measure of $\hat{\beta}$ is: $ I_{\hat{\beta}} = \frac{T}{I^{-1}(3,3)} $ 
\end{flushleft}

\begin{flushleft}
The asymptotic variance of $\beta$  is $V_{\hat{\beta}} = I_{\hat{\beta}}^{-1} $
\end{flushleft}
















































\newpage

\
\
\
\

\begin{table}[h!]



\begin{center}


\caption {Size of t-test for $\beta$ at $5\%$ Level} \label{tab:title} 
\begin{tabular}{@{}lllll@{}}

\toprule[1pt]
  & T=500  &  T=1000 & T=5000  & \\
   


 $\alpha=0.01$ & 55.2\% & 51.7\% & 48.9\% &  \\
 
 $\alpha=0.05$ & 41.5\% & 36.9\% & 25.9\% &  \\

 $\alpha=0.1$ & 30.5\% & 21.9\% & 10.8\% &  \\

 $\alpha=0.9$ & 36.7\% & 24.8\% & 21.7\% &  \\


\bottomrule[1pt]
\end{tabular}
\end{center}
\scriptsize Notes: Each row represents a different parameter setting of the GARCH(1,1) Monte Carlo simulation. The sample size increase as 500, 1000, and 5000. The repeated time of the Monte Carlo simulation is 1000. 
\end{table}



\newpage

\begin{table}[h!]
\begin{center}


\caption {Size of Various Tests for $\beta$ at $5\%$ in GARCH(1,1) Model when $\beta$ = 0} \label{tab:title} 

\

\begin{tabular}{@{}lllll@{}}


\toprule[1pt]
  & T=500  &  T=1000 & T=5000  & \\
   


\cline{2-4}

& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.01$, $\beta=0$}       \\


t-test & 55.2\% & 51.7\% & 48.9\% &  \\
 
LR test & 13.3\% & 10.7\% & 8.3\% &  \\

LM test & 4.7\% & 5.2\% & 4.6\% &  \\

\midrule[0.5pt]


& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.05$, $\beta=0$}       \\


t-test & 41.5\% & 36.9\% & 25.9\% &  \\
 
LR test & 11.0\% & 9.9\% & 7.3\% &  \\

LM test & 4.7\% & 6.2\% & 4.2\% &  \\


\midrule[0.5pt]


& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.10$, $\beta=0$}       \\


t-test & 30.5\% & 21.9\% & 10.8\% &  \\
 
LR test & 4.5\% & 6.1\% & 4.5\% &  \\

LM test & 3.7\% & 6.0\% & 4.0\% &  \\

\midrule[0.5pt]
& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.9$, $\beta=0$}       \\


t-test & 36.7\% & 24.8\% & 21.7\% &  \\
 
LR test & 5.5\% & 5.2\% & 4.5\% &  \\

LM test & 3.9\% & 4.0\% & 3.7\% &  \\



\bottomrule[1pt]
\end{tabular}
\end{center}
\scriptsize Notes: Each column represents the size of various tests when $\alpha$ changes and samples size is the same. $\beta=0$ implies no GARCH effect. Each row represents, for the same $\alpha$, when sample size increases, whether the t-test, LR test, and LM test become more valid. The repeated time of each Monte Carlo simulation is 1000. The table results is consistent with Ma, Nelson, and Startz (2007). 


\end{table}































\newpage 

\begin{landscape}
\begin{figure}[h]

\begin{center}
\caption{Frequency to Reject t-test for $\beta$: no GARCH effect ($\beta=0$)}
\includegraphics[scale=0.5]{plot_no_garch.png}


\end{center}
\end{figure}
\end{landscape}









\newpage 

\begin{landscape}
\begin{figure}[h]

\begin{center}
\caption{Frequency to Reject t-test for $\beta$: moderate GARCH effect ($\beta=0.5$)}

\includegraphics[scale=0.5]{plot_moderate_garch.png}

\end{center}
\end{figure}
\end{landscape}








\newpage 

\begin{landscape}
\begin{figure}[h]

\begin{center}
\caption{The Actual Size of the t-test ZILC Zone and Safe Zone in the GARCH(1,1) Model}

\includegraphics[scale=0.6]{plot_zone.png}

\end{center}
\end{figure}
\end{landscape}




\newpage


\begin{table}[h!]
\begin{center}
\caption {Size of Various Tests for $\beta$ at $5\%$ in GARCH(1,1) Model when $\beta$ = 0.5} \label{tab:title} 

\

\begin{tabular}{@{}lllll@{}}


\toprule[1pt]
  & T=500  &  T=1000 & T=5000  & \\
   


\cline{2-4}

& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.01$, $\beta=0.5$}       \\


t-test & 43.5\% & 40.6\% & 37.9\% &  \\
 
LR test & 10.0\% & 8.8\% & 9.1\% &  \\

LM test & 5.4\% & 4.9\% & 4.1\% &  \\

\midrule[0.5pt]


& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.05$, $\beta=0.5$}       \\


t-test & 30.1\% & 27.0\% & 19.1\% &  \\
 
LR test & 7.1\% & 9.8\% & 6.5\% &  \\

LM test & 3.6\% & 3.2\% & 5.2\% &  \\


\midrule[0.5pt]


& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.10$, $\beta=0.5$}       \\


t-test & 19.9\% & 16.3\% & 9.1\% &  \\
 
LR test & 4.1\% & 5.1\% & 4.2\% &  \\

LM test & 3.3\% & 4.3\% & 3.8\% &  \\

\midrule[0.5pt]
& \multicolumn{3}{c}{$\omega=1$, $\alpha=0.49$, $\beta=0.5$}       \\


t-test & 6.3\% & 4.5\% & 4.9\% &  \\
 
LR test & 5.4\% & 6.2\% & 4.7\% &  \\

LM test & 3.7\% & 4.0\% & 3.0\% &  \\



\bottomrule[1pt]
\end{tabular}
\end{center}

\scriptsize Notes: Each column represents the size of various tests when $\alpha$ changes and samples size is the same. $\beta=0.5$ implies the moderate GARCH effect. Each row represents, for the same $\alpha$, when sample size increases, whether the t-test, LR test, and LM test become more valid. The repeated time of each Monte Carlo simulation is 1000. The table results is consistent with Ma, Nelson, and Startz (2007). 



\end{table}









