Seemingly Unrelated Regression (SUR)

In econometrics, seemingly unrelated regression (SUR), model developed by Arnold Zellner and first published in Zellner (1962), is a technique for analyzing a system of multiple equations with cross-equation parameter restrictions and correlated error terms. An economic model may contain multiple equations which are independent of each other on the surface:

they are not estimating the same dependent variable, they have different independent variables, etc. However, if the equations are using the same data, the errors may be correlated across the equations. SUR is an extension of the linear regression model which allows correlated errors between equations. Suppose that the Gauss-Markov assumptions hold for all the equations.

Then the OLS estimates are BLUE. However, by using the SUR method to estimate the equations jointly, efficiency is improved. The mathematics is very similar to computing Huber-White standard errors. Suppose we have a series of equations

i i i i

y =xβ ε+ where :

x, β, and ε are vectors and i = 1, ..., M where M is the number of equations.

Assume each equation has N observations. Let Σ be an M × M matrix representing the covariance of residuals between the equations. Even though each equation satisfies the OLS assumptions, the joint model exhibits serial correlation due to the correlation of the error terms. Standard OLS estimation, then, will be inefficient (unless all the equations have the identical explanatory variables). Thus, SUR uses generalized least squares to estimate β:

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(

)

SUR X V X X V Y

β^∧ = ^{− −}

where

( ) _N

V Y = Σ ⊗I

where is the Kronecker product and V(Y) is an M × N matrix. Once SUR model estimates are obtained, inferences are mainly about testing the validity of cross-equation parameter restrictions.

We realize the economic events happened to affect price for three stock market of U.S. may be relative to each other. In order to solve problem, the last robustness test which we used is SUR model. We adopt the SUR model to reexamine the results of simple time-series regression. Furthermore, we added some variables which can have influence for volatility spread. Expectantly, we fully describe the relationship between option market and spot market momentum. In addition, we lengthen the past returns periods to expectantly observe how long the past ETF returns affect option price. The following is the SUR model of our study.

, , 1, 0 , , 1 , 1, 1 , 1, , , 1, , , ,

p c t i i R_τt i R_τt i p c t i s_τ t i skewnesst i pcratiot i j i

σ ₋ =α +γ +γ ₋ +θ σ _{− −} +ησ ₋ +ω +λ +ε

where:

p c t i, ,

σ ₋ = The volatility spread in the current day for ith market and i = 1, 2, 3.

, 1,t i

R_{τ −} = The returns on the ETF in the preceding period.

, ,t i

R_τ = The return in the subsequent period (revision).

, 1, p c t i

σ _{− −} = The lagged values of the volatility spread.

τ = The past return period (10, 20, 30, 40, 60, 80, 100, 120, 150, 200, 300, 400, 500, 600 days).

, , 1, sτ t i

σ ₋ = the historical volatility of τ days in the preceding period.

t i,

skewness = the skewness coefficient for pastτ days.

,

pcratiot i= the ratio of puts volume divided by calls volume.

Using the SUR model need to have the same researching period for three markets, because the SPY option is listed on 2005/01/10, the beginning of sample period on other markets are identical. The following five reults are raised from the SUR model. Firstly of all, when adding the one-lagged volatility spread , we observe the Durbin-Watson statistics are close to 2 , abhering to no autocorrelation assumption.

Second, the results in table 4-9, 4-10, and 4-11 indicate that past stock returns continue to show up with negative coefficient against the volatility spread as in table 4-5, 4-6, and 4-7.

Hence, including higher moments of stock return distributions does not eliminate the negative relation between past stock returns and volatility spread (γ₁<0). This finding suggests that investors’ expectations about future returns directly affect their valuations of ETF options, independent of other channels of influence. This result suggests that (1) past returns do not act as a proxy variable for higher moments of stock return, such as increased volatility, and (2) the market momentum hypothesis is not rejected even when we control for other factors. The results present the volatility spreads of QQQQ are affected significantly by past 120 days return and the volatility spreads of SPY and DIA are affected significantly by past 400 days return. We supposed that the U.S. stock market from 911 Terrorist Attacks to Subprime Mortgage Crisis is the bull market, causing momentum can affect option markets longer.

Hence, in this interval, given positive past market returns, investors expect the positive returns

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to continue, and bid up the prices of call options. Given negative past market returns, investors expect the negative returns to continue, and bid up the prices of put options. We call this the market momentum hypothesis. This hypothesis predicts that past stock returns exhibit an independent positive influence on the volatility spread. Besides, the finance literature documents a negative relation between stock returns and volatility. When stock prices fall, volatility increases. When stock prices rise, volatility decreases. We replicate this finding in table 4-1 to table 4-4. Implied volatilities for puts increase more than implied volatilities for calls, so it cause the negative relationship between volatility spread and past spot returns.

Notice that the components of QQQQ are more active, so the momentum trading is not significant for short-term period (120 days). Additionally, the historical volatility and volatility spread turn into the negative relation. It shows price reversal for short-term period on QQQQ and long-term period presents price reversal more easier than other ETF market . Third, investors’ supply and demand for options is not only affected by their return expectations from market momentum, but also by portfolio insurance considerations and both effects are present. When the volatility of stock returns increases, a greater number of investors seek reduced exposure to the stock market and bid up the prices of put options.

When the volatility of stock returns decreases, a greater number of investors seek increased exposure to the stock market and bid up the prices of call options. This idea suggests that, if a separate estimate of the volatility is included as a regressor in table 4-9 to table 4-11, it would show up with a positive coefficient (η >0) but would not necessarily drive away the significance of the past stock returns. Both past stock returns and volatility would show up with significant influences.

Fourth, Studies in the stock market have found that both stock returns are right skewed and investors have a preference for (systematic) right skewness. As hypothesized before, we find a positive relation (ω>0) between skewness expectations and the volatility spread. An

expectation of increased skewness leads to increased volatility spread. This finding is consistent with a scenario where investors prefer skewness in ETF returns and bid up the call prices when they expect higher skewness. Also, inclusion of the skewness variable does not drive away the effects of past ETF returns or volatility

Fifth, we added the put-call ratio regressor to examine how the volume affected the volatility spread. The results found that only QQQQ has significantly positive relation (λ>0).

It presents that when put volume increase more than call volume, causing the put price raise, the call price decline, and the implied volatility spread increase. This result may be due to the QQQQ is quite active, investors prefer the option market to hedge. When the past ETF returns are positive, some investors may establish contrary position which causing the put volume raise and volatility spread increase.

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TABLE 4-9 SUR Regression of Volatility Spread on Past ETF Returns, Historical Volatility, Skewness, and Put-Call Ratio for SPY

Estimates are from SUR. The p-values for the estimated coefficients are reported in parentheses.

Volatility spread is computed from vega-weighted options for each day.

, , 1, 0 , , 1 , 1, 1 , 1, , , 1, , , ,

TABLE 4-10 SUR Regression of Volatility Spread on Past ETF Returns, Historical Volatility, Skewness, and Put-Call Ratio for QQQQ

Estimates are from SUR. The p-values for the estimated coefficients are reported in parentheses.

Volatility spread is computed from vega-weighted options for each day.

, , 1, 0 , , 1 , 1, 1 , 1, , , 1, , , ,

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TABLE 4-11 SUR Regression of Volatility Spread on Past ETF Returns, Historical Volatility, Skewness, and Put-Call Ratio for DIA

Estimates are from SUR. The p-values for the estimated coefficients are reported in parentheses.

Volatility spread is computed from vega-weighted options for each day.

, , 1, 0 , , 1 , 1, 1 , 1, , , 1, , , ,