Testing serial correlation in a general d-factor model with possible infinite variance

It is well-known that the presence of serial correlation may result in an inefficient or even biased estimation in time series analysis. In this paper, we consider testing serial correlation in a general d-factor model when the model errors follow the GARCH process, which is frequently used in modeling financial data. Two empirical likelihood-based testing statistics are suggested as a way to deal with problems that might come up with infinite variance. Both statistics are shown to be chi-squared distributed asymptotically under mild conditions. Simulations confirm the excellent finite-sample performance of both tests. Finally, to emphasize the importance of using our tests, we explore the impact of the exchange rate on the stock return using both monthly and daily data from eight countries.


Introduction
In economics, finance, and risk management research, it is a common interest to investigate the relationship between a dependent variable and possible important factors.For instance, the capital asset pricing model, hereafter CAPM, first introduced by [40], is widely used to explore the relationship between systemic risk and investment return.Many extended models have also been developed by noting that the CAPM cannot explain the difference in different stock returns, while company size and book-to-market ratio are also important factors in capital asset pricing [2,15] in addition to market risk.The studies include, inter alia, [16] that proposed a three-factor model to consider market capitalization and the book-to-market ratio factor.Similarly, [7,17] developed the four-factor model and fivefactor model, respectively; see [17] and references therein for details.It is widely accepted that time series data, typically financial time series data, exhibit properties such as volatility clustering over time.Therefore, since the seminal work in [12], fitting GARCH models has become a standard technique to capture such features.
In view of this, we consider a d-factor model in a more general setting as follows: where Y t may represent the excess asset return in CAPM or any other variables of interest at time t, X t = (X t,1 , . . ., X t,d ) denotes a vector of d risk factors, which may stand for, e.g. the market excess return, market capitalization, and book-to-market ratio in CAPM, and {(η t , ηt,1 , . . ., ηt,d ) } n t=1 is a sequence of random vectors with means of zero and variances of one.Note that α, β, ω, {a i }, {b j }, {μ k }, {φ i,k }, {ψ j,k }, { ωk }, {ā i,k } and { bj,k } are unknown parameters.
We assume in model ( 1) that ε t has a GARCH structure to allow for any possible conditional heteroscedasticity, and each risk factor X t,l , l ∈ {1, . . ., d} follows an ARMA-GARCH process in order to capture volatility persistence precisely.The ARMA-GARCH model is a practical and useful tool to model conditional heteroscedasticity and is widely applied in many applications [24,27,29,42,44].
Usually, the sequence {ε t } n t=1 is assumed to be serially uncorrelated.Note that when {η t } n t=1 is assumed to be independent and identically distributed (i.i.d.), {ε t } n t=1 are uncorrelated.Violation of this assumption may result in inefficient or even biased estimation.Suppose the random observations {Y t , X t } n t=1 are generated from model (1).The objective of this paper is to test the null hypothesis H 0 : γ = 0, i.e. ε t 's are serially uncorrelated, where γ = (γ 1 , γ 2 , . . ., γ q ) , with γ s = E(ε t ε t−s ) for s = 1, 2, . . ., q, t = q + 1, q + 2, . . ., n.Note that ignoring some important risk factors, such as {Z t } n t=1 , which also has an impact on Y t , may result in the sequence of residuals being serially correlated.That is, when the underlying true model is inferences on {Y t , X t } n t=1 are equivalent to considering the following model where α = α + ϕE(Z t ), and εt = ϕ(Z t − E(Z t )) + ε t .It is easy to show that {ε t } is serially correlated when {Z t } is serially correlated, since E(ε t εt−1 Such problems may appear in the social sciences, such as economics, sociology, and political science; see [4] for an example of returns to education.Hence, it is of great importance to carry out a pretest to check whether there exists any serial correlation in {ε t } n t=1 before proceeding with statistical inferences.To this end, many well-known procedures, e.g. the ACF test and the Durbin-Watson test proposed by [10], the Lagrange Multiplier test, and the portmanteau test proposed by [6], can be employed.However, these tests often involve estimating the variance of residuals {ε t } n t=1 , which may be difficult given the GARCH structure in {ε t } n t=1 , and, importantly, it can be seen from our simulations in the sequel that the portmanteau test of [6] suffers from a heavy size distortion problem because it was developed originally for white noise sequences.Therefore, we propose an empirical likelihood-based test to check the possibility of serial correlation in {ε t } n t=1 .This is motivated by the fact that empirical likelihood, firstly proposed by [34], exhibits many desirable properties.To name but a few, it usually has limited requirements for the underlying distribution and can produce data-dependent confidence intervals; no variance estimation is involved in determining its critical value.In fact, the empirical likelihood method has been considered by many authors in modeling time series data.More recently, [30] developed an efficient empirical likelihood procedure to test whether the errors have both zero mean and zero median in analyzing financial and economic data; for others, see, e.g.[20,28,43] and references therein for more details.It is shown that our proposed test follows a standard chi-squared limiting distribution under mild conditions. On the other hand, it is noted that the aforementioned empirical likelihood-based test requires ε t to have finite variance, while many time series data, such as economics and financial data, are confirmed to be heavy-tailed without having finite variances.This literature goes back to [31] who pioneered the study of heavy-tailed distributions in economics and finance (see also [14,32]) and numerous studies have subsequently contributed to this (see e.g.[9,18,21,22,39] and others).
It is known that when ε t has an infinite variance, most traditional inferential procedures are invalid.As mentioned in [33], when ε t follows an IGARCH process, the convergence rate of the ACF test is lower, and the sample autocorrelations of the squared time series have a non-degenerate limiting distribution.Consequently, the testing results are often unreliable or meaningless.Hence, it is of great interest to find valid procedures in such situations (see [8,19,36]).Motivated by [26], we further develop a weighted empirical likelihoodbased test for checking serial correlation in {ε t } when it has an infinite variance.Our weighted test statistic is also shown to be chi-squared distributed asymptotically under mild conditions.
We organize this paper as follows: In Section 2, we introduce the methodology and main results.When GRACH errors have finite variances, we propose the empirical likelihood method; then we further develop a weighted empirical likelihood method, which is robust regardless of the variance of the error term being infinite.Numerical simulations and a real data application are given in Sections 3 and 4, respectively.Section 5 concludes the paper.All proofs of the main results are given in the Supplemental Materials.

Methodology and main results
In this section, we provide two empirical likelihood-based tests for checking the possible serial correlation in model (1).The first one is mainly developed for the case that ε t has a finite variance, while the second test has no such constraint condition.

Finite variance case
Since γ s 's involve the expectation, i.e. γ s = E(ε t ε t−s ), one may apply the empirical likelihood method to test H 0 along the same line as [34] based on {ε t ε t−s } n t=q+1 , s = 1, 2, . . ., q.However, since ε t = Y t − α − β X t are unobserved and contain the unknown α and β, we propose to use the profile empirical likelihood method suggested originally by [38], and treat α and β as redundant parameters.
= E{Z t (α 0 , β 0 , γ )Z t (α 0 , β 0 , γ ) } is positive definite when γ = 0. C4.There exists δ > 0 such that Condition C1 is quite mild (see, e.g.conditions in Theorem 3.1 of [3]).Condition C2 is a technical assumption to ensure the consistency of least squares estimation, while C3 and C4 are mainly assumed to the validity of Wilks' theorem for the log-profile empirical likelihood function defined above.Based on these conditions, we can obtain the following theorem.
Theorem 2.1: Suppose the sequence of {η t } n t=1 is i.i.d., implying γ = 0.Then, under conditions C1-C4, we have as n → ∞, where χ 2 q denotes a chi-squared distributed variable with q degrees of freedom.
Although L(α, β, γ ) above takes the same form as the empirical likelihood function proposed by [38], they are different in the following two ways: First, {p t } in [38] are essentially generated from replacing the distribution, say F, of the i.i.d.observations with the general empirical distribution F = t p t δ t with δ t being a point mass at the tth observation, while {p t } are just some weights in L(α, β, γ ).Second, as seen from the Supplemental Materials, owning to the dependence between {Z t (α, β, γ )}, the proof of Theorem 2.1 depends on the central limit theorem for martingale difference sequences, whereas that of [38] is based on the conventional central limit theorem for i.i.d.observations.Theorem 2.1 indicates that the limit distribution is standard and no variance estimation is involved.Hence, using the Theorem 2.1 is convenient.For any given data, we may reject the null hypothesis

Infinite variance case
As mentioned in Section 1, the error term ε t may possibly have an infinite variance, especially when daily data are present.That is, the condition E(|σ t σt,k | 2+δ ) < ∞ in C4 is invalid.Under this situation, Wilks' theorem obtained in Theorem 2.1 no longer holds.Note that when there is no serial correlation in {ε t }, γ = 0 is equivalent to γ = 0 under the same conditions of Theorem 2.1, where γ = ( γ1 , γ2 , . . ., γq ) , and Motivated by [26], we propose to construct the empirical likelihood function based on the weighted score vectors Similar to Section 2.1, we here still handle α and β as redundant parameters.As did in [26], we here use some proper weights w t−1,X and w t−1,XY to control the moment effects of σ t on ε t , and ( σt,1 , σt,2 , . . ., σt,d ) on X t , respectively, for t = 1, 2, . . ., n. Remarkably, the reason for using two weights, i.e. w −1 t−1,X and w −1 t−1,XY , in Zt,1 (α, β, γ ) and Zt,2 (α, β, γ ) lies in the following fact: Zt,1 (α, β, γ ) and Zt,2 (α, β, γ ) are obtained similarly from the partial derivatives of the weighted sum of lease squares as follows: In this paper, we define and M Y are the 90% sample quantiles of {|X t,k |} and {|Y t |}, respectively.Similar weighting strategies can also be found in [30].
The weighted empirical likelihood function based on the weighted score vectors is as follows: Define ˜l(α, β, γ ) := −2 log L(α, β, γ ), and the profile empirical likelihood function as ˜ ( γ ) := min α,β ˜l(α, β, γ ).To address the case of possible infinite variance, we assume instead the following conditions: It is noted that, in comparison to the finite variance case, we remove the assumptions on σ t and σ t,k , k = 1, 2, . . ., d, and add an assumption on the weights.That is, the conditional variance similarly, the conditional variance of εt,k , can be infinite, where F t is the σ -field generated by {(η s , ηs,1 , . . ., ηs,d ) : s ≤ t} for any t = 1, 2, . . ., n.Furthermore, some simple derivations can show that the weights defined above satisfy C5.
On the weighted empirical likelihood function, we have the following result.

Simulation results
In this section, we carry out simulation experiments to illustrate the finite sample properties of the proposed empirical likelihood and weighted empirical likelihood methods when the variance of ε t is finite and infinite.
The simulated data {Y t , X t } n t=1 are generated from the following model: We set α = 0.5, β = (0.5, 0.5) and ω = 0.2.For the GARCH process of ε t , we choose (a, b) = (0.1, 0.3) to represent the case that the variance of ε t is finite, while (a, b) = (0.111, 0.888), implying that the conditional variance of ε t may tend to infinity as t → ∞.It is noted that only when the value of a + b is very close to 1, it shows heavy-tailed.
We replicate the experiments 2000 times for the sample size n ∈ {200, 400, 800, 2000}, respectively.Three different testing methods are investigated, i.e. the weighted empirical likelihood method (WEL), the empirical likelihood method (EL), and the Ljung-Box test (LB).Here, we employ directly the build-in R function Box.test to carry out the Ljung-Box test and emplik package to carry out the WEL and EL tests.The R function nlm is used in the optimization step when computing the profile empirical likelihood function.The R code of this paper is provided in the Supplemental Materials.The empirical sizes and powers at significance levels τ = 0.1 and τ = 0.05 are reported in Tables 1-6.
The results in these Tables show that: (i) For (a, b) = (0.1, 0.3), i.e. when the variance of ε t is finite, the sizes of both EL and WEL are very close to the nominal levels under all three different settings of X t,k , while LB test has size distortion; (ii) For the case with an approximately infinite variance of ε t , i.e. (a, b) = (0.111, 0.888), as expected, the WEL performs the best among these three methods while LB is highly over-sized and such size distortion deteriorates with the sample size n increases.Furthermore, EL exhibits significant oversize patterns in particular when X t,k ∼ AR (1).Given the robustness of WEL in the case of the infinite variance, WEL is recommended to use if the infinite variance is hard to judge, despite the cost of a slight power loss.

Applications
In this section, we do a modeling-based empirical analysis of how the exchange rate affects the stock return.In financial economics studies, the link and interaction between the exchange rate and the stock return have received a great deal of attention since both variables are regarded as leading economic and financial variables.Any dynamic relationships between them may have an impact on the implementation of monetary and fiscal policy.As in the study of [37], the 'flow-oriented' models of exchange rate determination affirm that currency movements affect international competitiveness and the balance of trade position, and consequently, the real output of the country in turn affects current and future cash flows of companies and their stock prices.[11,35,41] supported that depreciation of the real exchange rate increases the values and stock prices of the firms.During a financial crisis in particular, any large fluctuations in the exchange rate could have significant consequences for international trade, foreign investment, and asset prices.Hence, an accurate estimate of the association between financial return and exchange rate is of great importance.However, before verifying the existence of a significant exchange rate impact on stock index return, it is vital to ensure no serial correlation exists in the residuals, as we discussed in Section 1, and this is the motivation for our empirical analysis.
For the dataset, we collected both monthly and daily exchange rate and stock price data from eight countries, including emerging and developed countries.The currencies of emerging countries that we use are the Indian rupee (INR), Malaysian ringgit (MYR), South Korean Won (KRW), and Thai baht (THB); the currencies in developed countries include the Canadian dollar (CAD), British sterling (GBP), Euro (EUR) and Japanese yen (JPY).The stock indices are: S&P/TSX (Canada), DAX (Germany), Nifty 50 (India), Nikkei 225 (Japan), FTSE KLCI (Malaysia), KOSPI Composite Index (South Korea), SET 50 (Thailand) and FTSE 100 (UK).All data are downloaded from investing.comand Yahoo Finance.We then transform all data by using log(P t /P t−1 ), where P t is the exchange rate or stock price at time t.Therefore, Y t and X t represent the stock return and the exchange rate return respectively in our model.

Monthly data
We first consider the monthly data.The time spans of the data sets for these eight countries are summarized in Table 7.The monthly data is fitted to our model (1) where the exchange rate return is allowed to follow an ARMA-GARCH structure.Summarized in Table 8, the estimated coefficients in the GARCH model in all countries are significant, which shows the rationality of fitting the GARCH model.Here, we use ArchTest function to check whether the data has ARCH effect, and we use the garchFit function to fit the GARCH model and estimate the corresponding parameters.
To ensure that we use the appropriate test, it is important to check if there is any heavytailed in the residuals.As pointed out in [23], the heavy-tailed feature is of key interest to risk managers, financial regulators, financial stability analysts, and policymakers.Several recent studies have suggested that many financial variables may be driven by innovations with infinite variance.For example, studies by [1,5,13,23,25,31] provide evidence for infinite variance behavior in the exchange rate returns.A simple way to detect the possible  heavy tail is based on a Q-Q plot.Note that in the literature there is another tool, i.e. the Hill plot, which can serve the same purpose, and both the Q-Q plot and Hill plot are only descriptive tools.The corresponding Q-Q plots of the model residuals are provided in Figure 1.It seems that there is no strong evidence supporting a significant heavy-tailed trend in residuals as the sample quantiles are very close to those of normal distribution.All the test results for WEL, EL, and LB are summarized in Table 7.The p-values of WEL and EL in all countries are greater than the nominal significance level, which implies nonrejection of the null hypothesis, hence no serial correlation is found in the residuals.For the results from LB, except that the null hypothesis is only marginally rejected at 0.1 significance level for Canada, the same conclusions can be reached for other countries in this case.This is not surprising given that the size distortion in our simulations is not severe for LB when there is no infinite variance in residuals.However, WEL and EL tests are always more reliable to use in the current model setting.

Daily data
In this subsection, we consider daily data of stock prices and exchange rates.This is motivated by the fact that in this big data era, there is an upsurge of interest in studying higher frequency data, in particular in finance, due to the data availability.The description of the data is provided in Table 9.Like in monthly data, it is shown in Table 10 that using the daily data, the estimated coefficients in GARCH model in all countries are significant which confirms the suitability of fitting GARCH model.Different from the monthly data case, Q-Q plots provided in Figure 2 on daily data clearly show that there may exist a weak heavy-tailed trend in the residuals.From our previous discussion, WEL will be the only one among the three tests to be used to test for the possible serial correlation.
We then examine the performance of three tests reported in Table 9.For the country of India, three tests lead to different conclusions, i.e.WEL indicates a strong rejection of the null hypothesis of no serial correlation at 0.01 significance level, whereas LB and EL imply non-rejection of the null hypothesis.Similarly, for the series of South Korea, LB suggests the rejection of the null hypothesis.However, WEL indicates the opposite result, i.e. the serial correlation is not present.In terms of Germany, the null hypothesis is marginally rejected by WEL at 0.1 significance level while it is not rejected by either LB or EL.Finally, WEL results in UK suggest the rejection of the null hypothesis while EL indicates the non-rejection at 0.05 significance level.Hence, it is evident that the conclusions can be misleading if an inappropriate test is used.In conclusion, the results from WEL confirm the existence of serial correlation in most countries except South Korea, Malaysia, and Thailand.
Our empirical results above imply that when we try to examine the relationship between the exchange rate and stock return, traditional econometric and statistical methods should be applied with care.This is especially important when the residuals have infinite variance.

Conclusions
We proposed two empirical likelihood-based methods, that is, an empirical likelihood (EL) and a weighted empirical likelihood (WEL) test, for serial correlation of a general dfactor model with GARCH errors.The asymptotic distribution of both statistics is proved to be chi-squared.To accommodate possible heavy-tailed characteristics of the error term, we have further shown that the WEL is relatively robust while the traditional test such as Ljung-Box (LB) test will be invalid.Our simulation results show that when there is finite variance in residuals, both WEL and EL have better finite sample performance than LB; EL test has more accurate size and higher power than WEL test.On the other hand, when ε t has an infinite variance, WEL has the best performance for all settings; however, LB suffers from significant size distortion in all settings.In the empirical application, we examine the serial correlation issue when modeling the impact of exchange rate on stock return for eight countries.Both monthly and daily data are analyzed.There is no evidence of heavytailed found using monthly data and both WEL and EL tests indicate no serial correlation is present.For the daily data, given that errors contain infinite variance, WEL results are used which confirm the strong serial correlation for most countries.This study intends to provide practical guidance as our conclusion implies that traditional econometric and statistical methods to detect the serial correlation must be used with caution especially when the series has an infinite variance.

Figure 1 .
Figure 1.The Q-Q plots of residuals of monthly data with eight countries.

Figure 2 .
Figure 2. The Q-Q plots of residuals of daily data with eight countries.

Table 7 .
The p-values of different tests with monthly data.

Table 10 .
The coefficients of different countries with daily data.