The Day of Week Effect

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- Test for heteroskedasticity

To examine the seasonality, I filter the daily means and variances using the following two regressions.

- =α1Mont+α2Tuet+α3Wedt+α4Thut+z1[pic 7]

- ()^2=β1Mont+β2Tuet+β3Wedt+β4Thut+z2][pic 8]

Where Mont, Tuet, Wedt, Thur are the dummy variables for Monday, Tuesday, Wednesday, and Thursday. is the ordinary least squares(OLS) fitted value of from regression(1) at date t. [pic 9][pic 10]

Jointly test the significance of all of the lagged values in (2) equal to zero. According to the result of the above regression, we can exam whether the variance of the return has a day-of-the-week effect.

3.2 ARIMA model

We use the ARIMA model to test whether the day of the week effect exists in Chinese ETF market in daily return series for the four ETFs that didn’t present heteroskedasticity, including Shen100 ETF, XiaokangETF, ZerenETF, and ZhongpanETF. In the following model, is the return of ETF i on day t. D_Monday is a dummy variable that takes the value of 1 if the observation is from Monday and 0 otherwise. D_ Tuesday is a dummy variable that takes the value of 1 if the observation is from Tuesday and 0 otherwise. D_Wednesday is a dummy variable that takes the value of 1 if the observation is from Wednesday and 0 otherwise. D_Thursday is a dummy variable that takes the value of 1 if the observation is from Thursday and 0 otherwise. Daily returns are computed for each ETF by taking the natural logarithm of the change in daily closing price for each of the trading days in the sample, as shown in the equation:[pic 11]

[pic 12]

Where is the return on fund i during the period t, and value i,t is the value of an investment in ETF i at time t.[pic 13]

[pic 14]

3.3 AutoRegressive Conditional Heteroskedasticity(ARCH) models

For the 10 ETFs(ShenchengETF, ShenhongliETF, ZhongxiaobanETF, 50ETF, HongliETF, JiazhiETF, MinqiETF, ShangzhengETF, YangqiETF, ZhiliETF) that we proved presenting time-varing variance, we applied Auto-Regressive Conditional Heteroskedasticity(ARCH) processes using lagged disturbances and the Generalized ARCH(GARCH) model based on an infinite ARCH specification to test the day-of-week effect. Both of ARCH and GARCH models capture volatility volatility clustering and leptokurtosis. We also applied the Exponential GARCH (EARCH) model, which is a nonlinear extension of GARCH model.

Denotes is a time series of asset returns whose mean equation is given by , where refers to the information available at time t-1 and are the random innovations with E([pic 15][pic 16][pic 17][pic 18][pic 19]

The ARCH model proposed by Engle(1982) is defined as follows.

[pic 20]

Where , is i.i.d. random variable with mean zero and variance one. [pic 21][pic 22]

While ARCH model often requires many parameters and a high order q to capture the volatility process, the generalized autoregressive conditional heteroskedasticity( GARCH, Bollerslev(1986) model) enables us to reduce the number of estimated parameter by imposing nonlinear restrictions.

In the case of GARCH(p,q) model, where p is the order of the GARCH terms

+[pic 23][pic 24]

Where ,and are the control parameters of the GARCH stochastic process. In this paper, we applied the simplest GARCH process, the GARCH(1,1).[pic 25][pic 26][pic 27]

Besides GARCH(1,1), we also applied the exponential general autoregressive conditional heteroskedastic(EGARCH) model by Nelson(1991) in this study, which is another form of the GARCH model to capture the asymmetry observed in the data.

+[pic 28][pic 29]

Where g()= θ+λ(|, is the conditional variance. ,and are the parameters of the EGARCH process. may be a standard normal variable or come from a generalized error distribution. The formulation of g((allows the sign and the magnitude of to have separate effects on the volatility. [pic 30][pic 31][pic 32][pic 33][pic 34][pic 35][pic 36][pic 37][pic 38][pic 39]

- Estimation Results

As the result of the OLS estimates of the fourteen regressions shown in Table 2, 8 of the 14 ETFs(Shen100 ETF, Shencheng ETF, Shenhongli ETF, 50ETF, Jiazhi ETF, ShangzhengETF, Xiaokang ETF, Zeren ETF) have significantly negative daily means on Monday, while all of the 14examples except Shen100ETF have significantly negative daily means on Thursday. In addition, there are 10 ETFs(ShenchengETF, ShenhongliETF, ZhongxiaobanETF, 50ETF, HongliETF, JiazhiETF, MinqiETF, ShangzhengETF, YangqiETF, ZhiliETF) out of 14 ETFs presenting the problem of heteroscedasticity. As a result, we applied ARIMA model to the four ETFs(Shen100ETF, XiaokangETF, ZerenETF, ZhongpanETF) that didn’t present heteroskedasic, while applied different conditional heteroskedastic ARCH models to estimate and forecast the returns volatility of the 12 ETFs that present heteroskedasic.

Table 2. OLS Regression coefficients for day-of-the-week effect

(Note: ***,**and* indicate significance at the 1,5, and 10 percent levels, respectively.)

ETF:Shen100

Monday

Tuesday

Wednesday

Thursday

Mean(1)

-0.0103454**

-0.0019163

-0.0002079

-0.0032431

Variance(2)

0.0128267

-0.0000648

-0.0000758

-0.0001352

Prob > F = 0.3698, does not reject the hypothesis that all of the lagged values in (2) equal to zero. It does not present significantly heteroskedasic.

ETF:Shencheng

Monday

Tuesday