using stock index futures., while restricting the hedge ratio to be constant ... currency futures. .... ^The PPT is the Phillips-Perron i statistic in an estimated model.
Applied Financial Economics, 1995, 5, 131-137
Time-varying distributions and the optimal hedge ratios for stock index futures TAE H. PARK and LORNE N. SWITZER Faculty of Commerce, Concordia University, Montreal, Canada
This paper estimates the risk-minimizing futures hedge ratios for three types of stock index futures: S&P 500 index futures, major market index (MMI) futures and Toronto 35 index futures. Spot and futures prices are first analysed to adjust for non-stationarity and cointegration. Using a bivariate cointegration model with a generalized ARCH error structure, we estimate the optimal hedge ratio as a ratio of the conditional covariance between spot and futures to the conditional variance of futures. Both within-sample comparisons and out-of-sample comparisons reveal that the dynamic hedging strategy based on the bivariate GARCH estimation improves the hedging performance over the conventional constant hedging strategy.
I. I N T R O D U C T I O N Since the introduction of stock index futures markets in the early 1980s, numerous studies have addressed the efficiency of these markets in risk management (i.e. hedging). Hedging with stock index futures applies directly to the management of stock portfolios. Stock index futures can be used to hedge market risk caused by spot price fluctuations. Several studies (Figlewski, 1984; Lee et al., 1987; Ghosh, 1993) have investigated the optimal hedge for stock market portfolios using stock index futures., while restricting the hedge ratio to be constant over time. However, if the joint distribution of stock index and futures prices is changing through time, estimating a constant hedge ratio may not be appropriate. In this paper, optimal hedge ratios are estimated by modelling the distribution of stock index and futures price changes within the generalized autoregressive conditional heteroscedastic (GARCH) framework of Engle (1982) and Bollerslev (1986). The advantage of the GARCH specification is that very convenient assumptions about the conditional density of commodity prices changes, such as the normal or t distributions, can lead to a rich model that allows for time-dependent conditional variances in the unconditional distribution of price changes. This is achieved through relaxing the assumption that successive price changes are independent. Price changes may still be uncorrelated, but the GARCH model allows for weak dependence in the form of interactions at higher moments. The GARCH 0960-3107
© 1995 Chapman & Hail
models have already proved useful in explaining stock returns by Bollerslev (1987), French et ai (1987), and Baillie and DeGennaro (1990). Estimation of optimal or minimum risk hedge with futures contracts has also benefited from the time-dependent conditional variance models. Cecchetti et al. (1988) apply ARCH in estimating an optimal futures hedge with Treasury bonds. Baillie and Myers (1991) and Myers (1991) examine commodity futures and report improvements in hedging performance over the constant hedge approach by following a dynamic strategy based on the GARCH framework. Kroner and Suhan (1991, 1993) find similar results with currency futures. With the exception of Gagnon and Lypny (1994), no study has shown the hedging performance of stock index futures by applying the GARCH framework. Based on the several results on stock-return distribution, the GARCH hedge ratio is expected to provide greater reduction of risk with stock index futures. We examine three of the most heavily traded stock index futures contracts in North America: the S&P 500 index futures, the major market index (MMI) futures, and the Toronto 35 index futures. The S&P 500 index futures contract trades on the Chicago Mercantile Exchange. The MMI contract, until recently, traded on the Chicago Board of Trade. The last contract is traded on the Toronto Futures Exchange and is the most heavily traded index futures in Canada. The New York Stock Exchange composite index futures, another popular index futures, which is similar in 131
132
T. H. Park and L. N. Switzer
structure with the S&P 500 index futures, is excluded in this study because the stocks covered by the NYSE composite index is too large in number (around 2000) to be held in a portfolio, making the hedge only approximate.
II. E S T I M A T I O N O F O P T I M A L HEDGE RATIOS Hedging models can be constructed using a two-period investment decision based on utility maximization, first introduced by Markowitz (1952). Assuming that the only hedging instrument available to the investor is the futures contract, a hedge portfolio consisting of spot and futures is constructed. Let us consider the following model, which allows for time-varying variances of the spot and futures prices. Let/ 41 and s, +1 be the changes in the prices of the futures and the spot between time t and r -I- I, respectively, and define b, as the holding of futures at time t. Then
moments replace the time invariant unconditional moments.' Because conditional moments change as new information arrives to the market, the risk-minimizing hedge ratio will change through time. The optimal hedge ratio changes through time accordingly. This conditional model is the same as the conventional model if the joint distribution of spot and futures is constant through time. The GARCH specification requires modelling the first two conditional moments ofthe bivariate distributions of s, and/;. In order to account for potential cointegration, the first moment can be modelled with a bivariate error correction model as proposed in Engle and Granger (1987). In order to account for the time-varying variances and covariances, the second moment can be parameterized with a bivariate constant correlation GARCH(1,1) model, as originally proposed in Bollerslev (1986) and applied in Baillie and Bollerslev (1990) and Kroner and Sultan (1991. 1993). This model parameterizes the eonditional variances of two variables as ARMA models in squared residuals, while assuming constant correlations between the two. The bivariate distributions of spot and futures are assumed be as follows:
is the pay off at time t + \ to purchasing one unit ofthe spot and going short in b, units of the futures at time t. At each time (, the investor with a mean-variance expected utility function will maximize:
where the constant term, r, denotes the level of risk aversion. The expectation and variance operators are subscripted with t to denote that they are calculated conditional on all information available at time t. By definition, the predictable component of volatility in the return is the conditional variance, and thus, risk is measured by conditional variance. The utility maximizing hedge ratio at time f is (3) Assuming that investment in futures contracts is a zero-sum game, or futures prices are martingale (i.e. £,(F, + i) = F,), Equation 3 simplifies to
(5)
A/(0,//,) hf.,
h,,
0
I
P
K.,
0
0
hf
P
I
0
/if
which is similar to the conventional ordinary least squares (OLS) hedge ratio except that time-varying conditional
(7)
(8)
hi =
.,., +
where T,_i is the information set at time t - I. The constant >• that appears Equation 5 is called the cointegrating parameter that links the spot and futures prices together. If the spot and futures prices show a long-run or equilibrium relationship, the error term below should be stationary: yF,
(4)
(6)
(9)
even when the two time series S, and F, are non-stationary/ Thus, the term (5,_i -yF,-i) in Equation 5 is the error correction term, which imposes the long-run relationship
' In the case of stoek index futures, the portfolio to \x hedged is the stock market portfolio. Thus, the spot price corresponds to the price of a stock portfolio which moves closely with the stoek index in price. In the following analysis, the stock index replaces the value of stock market portfolio which cannot be directly traded in the open market. Recently, however, basket instruments that track the movement of stock indices and are traded like shares of common stock have begun trading in the American Stock Exchange for S&P 500 index (S&P 500 depositary receipts) and in the Toronto Stock Exchange (Toronto 35 index participation units). It should be of future interest to examine the hedging performance of stock index futures on these stock index participation units. Mt is generally true that any linear combination of two non-stationary time series is non-stationary. It is a common presumption (that we test) that spot and futures prices are non-stationary.
133
Stock index futures (cointegration) between the spot and futures prices into this short-run model.^ The parameter p in Equation 7 is the constant correlation between the conditional spot and futures variances. Given the bivariate model of spot and futures prices developed above, the time-varying hedge ratio can be expressed as
IV. DISTRIBUTION OF SPOT AND FUTURES PRICES Several tests on the distributional properties of the data were performed and the results are reported in Tables 1-3. Table 1 shows the stationarity test results for the spot and futures data using the Phillips and Perron (1988) test for
(10) Off I
To estimate b*, we first estimate the parameters in Equations 5, 7 and 8. The conditional variances and covariances can be forecasted at each time and provide the optimal hedge ratio as given above. In addition to the bivariate GARCH estimation, the conventional OLS hedge ratios can be obtained by setting as = bs = a^ = bf = oii = pi = 0. A variation of the conventional OLS hedge that accounts for cointegration between the spot and futures prices can be obtained by setting as = b^ = Uf = b^ = 0.
III. DATA Our spot market data consist of daily closing prices for S&P 500 index, major market index and the Toronto 35 index. The futures data correspond to daily settlement prices for the three futures contracts. The US data were obtained from Commodity Systems Inc. and the Toronto 35 data were provided by the Toronto Stock Exchange. The first step is to construct series of weekly prices, including dividends, from every Wednesday.* To construct a dividend inclusive series for the US indices, we take the daily dividend yield for the New York Stock Exchange available from the Center for Research in Security Prices tapes and compute the nominal amount of daily dividends. For the Toronto 35 index, the dividend inclusive index (total return index value) is available from the TSE and is used without any adjustment. The price data begin on 8 June 1988 and end on 18 December 1991. Ofthe several futures contracts that are outstanding at any given time, we use the price of the nearest contract. To avoid thin markets and expiration effects, however, we roll over to the next nearest contract one week prior to expiration ofthe current contract. This provides 185 weekly prices for the S&P 500 and MMI, and 184 prices for Toronto 35.^
Table 1. Unit root tests {for log prices) 8 June 1988-18 December Differences
Prices pp
PPT S&P 500 spot S&P 500 futures MMI spot MMI futures TSE35 spot TSE35 futures 95% critical value
- 2.247 - 2.265 - 2.625 - 2.591 - 1.994 - 1.986 -3.80
- 1.339 - 1.350 - 1.132 -1.131 - 2.084 - 2.084 -3.37
PPT -
14.410 14.275 15.098 14.647 13.388 12.580 - 3.80
PF - 14.403 - 14.270 - 15.088 - 14.639 - 13.347 - 12.541 -3.37
^The PPT is the Phillips-Perron i statistic in an estimated model with a time trend. The PP is the corresponding statistic In the estimated model without a time trend. Critical values can be found in Engle and Granger (1987) and Phillips and Ouliaris (1990).
Table 2. Cointegration tests {for log of prices): estimation of Equation 9, with spot prices adjusted for dividends ilog{S, — DIV,) ^S + y\ogF, + £,) 8 June 1988-18 December 1991'' DW
ADF (4)
PP
S&P 500
- 5.920
- 7.096 0.846
MMI
- 5.554
-9.102
1.230
TSE35
- 8.566
- 11.451
1.706
95% CV.
-3.17
-3.37
0.386
y 1.003 (0.002) 1,00! (0.001) 0.996 (0.007)
6 - 0,025 (0.014) -0.013 (0.007) 0.015 (0,040)
'The PPT is the Phillips-Perron f statistic in an estimated model with a time trend. The PP is the corresponding statistic in the estimated model without a time trend. The ADF(4) statistic is a fourth order augmented Dickey-Fuller test for a unit root. The DW statistic is the Durbin-Watson statistic of the cointegration equation between cash and futures prices. Critical values can be found in Engle and Granger (1987) and Phillips and Ouliaris (1990). The OLS estimates of y and 6 are shown with standard errors in parentheses.
^Earlier studies (Kawaller et ai, 1987; Ng, 1987; Stoll and Whaley, 1990) have documented the lead-lag relationships between stock index and futures prices. Recent studies (Wahab and Lashgari, 1993; Ghosh, 1993) document that the stock index and futures prices are cointegrated and show that an error correction representation for each series is appropriate. For other contracts, the cointegration between spot and futures was found in currencies by Kroner and Sultan (1993) but not found in commodities by Baillie and Myers (1991). ^We chose Wednesday-to-Wednesday sequence because Wednesday data have the largest number of observations and do not coincide with regular announcement of public information. logarithms of the prices, multiplied by 100, are used in this study.
134
. H. Park and L. N. Switzer
Table 3. Descriptive statistics of spot and futures {based on logdifferenced data) 8 June 1988-18 December 1991
S&P 500 spot S&P 500 futures MMI spot MMI futures TSE35 spot TSE35 futures 95% critical value
Skewness
Kurtosis
B-J"
e(24)
-0.413 - 0.392 - 0.341 - 0.294 0.112 0.288
3,996 3,969 4,068 3,942 3.940 3.790
12.906 11.976 12.378 9.505 7.003 7.169 5.99
22.252 24.519 20.407 18.537 16.102 \1.161 36.42
"B-J is the Bera-Jarque test for normality. It is distributed under the null hypothesis of normality.
unit roots. If the Phillips and Perron f-statistic is below the critical value, the null hypothesis ofthe presence of unit root cannot be rejected. The results show the significance of unit roots and thus the nonstationarity of the price series, suggesting that St and F, should be differenced to induce stationarity. To test for cointegration between spot and futures in Equation 9, the spot price is adjusted for dividends. Substituting the dividend-adjusted prices for S, in Equation 9 corresponds to the cost-of-carry model offutures prices, assuming that the dividend yield remains constant until the time of maturity." If the spot and futures prices show a long-run or "equilibrium' relationship, such a cointegration measure needs to be added to Equation 4 to account for the long-run behaviour of spot and futures price changes. Table 2 shows that the OLS estimation of Equation 9 yields estimates of y that are not significantly different from 1. Table 2 also shows the significance of cointegration by using the augmented Dickey-Fuller test, the Phillips and Perron test and the Durbin-Watson test. The results indicate that for the stock indices considered in this paper, the spot and futures prices are cointegrated at y = 1. Because this estimate is consistent, we impose y = 1 in Equation 5 throughout the following analysis.^ Table 3 shows the higher moments of percentage changes in price (log-diiTerenced S, and F,), The unconditional distributions of the spot and futures price changes are nonnormal, as evidenced by high skewness and high kurtosis. The US indices show high skewness but the Toronto 35 index does not exhibit high skewness. Kurtosis values suggest that the distribution of the stock indices are leptokurtic for ail three indices. Based on the Bera-Jarque statistic we reject normality for these data. Tests for autocorrelation, conducted by estimating the Ljung-Box Q statistics on the
first 24 lags of the autocorrelation, reveal that the null hypothesis of no autocorrelation cannot be rejected for all three types of spot and futures price changes. Overall, the results are consistent with time-varying conditional heteroscedasticity models of Engle (1982) and Boilerslev (1986).
V. C O M P A R I S O N A M O N G H E D G I N G TECHNIQUES In this section, four types of hedging models are compared in their hedging effectiveness. The first one is the naive hedging model where the hedge ratio is 1 over time. The second model is the OLS hedge, and the third one is the OLS with cointegration (OLS-CI) between spot and futures. The last one is the bivariate GARCH model. The maximum likelihood estimation is used to estimate the parameters in Equations 5, 7 and 8 for the GARCH model and the parameters in Equation 5 for the OLS and the OLS-CI models. Table 4 shows the likelihood ratio tests that compare the three hedging models. The likelihood ratio test statistics of the validity of restrictions on the parameters {a^ = b, = Qf = b{ = oLi = pj = 0 for the O L S model and
as = bs = af = bf = 0 for the OLS-CI model) are reported with their respective critical values. Each of these estimated statistics exceeds its critical value at the 95% level of significance. The OLS-CI model shows a better fit than the conventional OLS model. However, the GARCH model describes the distribution of spot and futures price changes better than both of the constant hedge ratio models. The parameter estimates for the bivariate GARCH model are shown in Table 5. Most of the parameters are significant in explaining the time-varying distribution of spot and
Table 4. Likelihood ratio tests of the models Data Series Test
S&P 500
MMI
TSE 35
DF
95% CV.
A B C
36.778 43.716 6.938
43.024 48,406 7,622
12.024 22.968 10.944
2 4 2
5.99 9.49 5.99
"Test A compares the OLS-CI hedge model with the conventional OLS hedge model. Test B compares the GARCH hedge model with the conventional OLS model. Test C compares the GARCH hedge model with the OLS-CI hedge model, DF is the degree of freedom for each test that accounts for the difference in the number of parameters used in each estimation.
^An ideal representation ofthe cointegration model would use the spot price minus the average amount of dividends to be paid out until the maturity of the futures contract. Since the pay out pattern of the market dividends is very lumpy within a quarter, the assumption of constant dividend flow may not be appropriate in the long-run relationship between the spot and futures prices. In this study, however, the constant dividend flow is assumed for expositionai purposes. ''For the error-correction term in Equation 5, the spot price is the dividend adjusted spot price, S,- DIV,. For the price change series, however, the dividend inclusive spot price is used.
Stock index futures
135
Table 5, Ma.ximum likelihood estimates ofthe GARCH model" Coefficient
S&P 500
So
0,114 (0.770) -0.117 (1.293) 0.414 (2.831) 0.285 (3.143) 3.118 (10.102) 3.073 (7,816) - 0.034 (1,417) -0.051 (3.391) 0.239 (2.309) 0.303 (3.917) 0.987 (500.977)
«!
fh> ih C» Cf
^^
Of
b. hf
P
MMI 0.040 (0.474) - 0.474 (2.713) 0.220 (2.854) 0.006 (0.034) 2.983 (29.734) 2.422 (29.731) 0.016 (0.431) -0.009 (0.427) 0.234 (8.181) 0.378 (18.660) 0.993 (1094.742)
TSE 35 0.477 (2.927) 0,720 (4.099) 0.263 (1.731) 0.395 (2.394) 2.535 (6.099) 2.966 (11,514) - 0.034 (0.645) - 0.056 (1.431) 0.197 (0.900) 0.071 (0.570) 0,884 (52.789)
^Absolute (-statistics are in parentheses.
ratios given by Equation 10 are also changing through time. For S&P 500 index futures, the OLS-CI hedge ratio is higher than the OLS hedge ratio, but the reverse is true for the other two index futures. The constant hedge ratios are clearly unable to recognize the trend in the spot and futures price changes. Note that in some cases the GARCH hedge ratio changes abruptly over the sample period, implying the usefulness of the time-varying hedge method. For the Toronto 35 index futures, it takes longer for the GARCH hedge ratio to stabilize after each jump. In order to compare the performances of each type of hedge, we construct the portfolios implied by the computed hedge ratios each week and calculate the variance of the returns to the constructed portfolios over the sample. Thus, we evaluate a^{s, — bff), where bf is the computed hedge ratio for each hedge method (the subscript t would not appear for the OLS and the OLS-CI models). The portfolio variances are reported in Table 6. All four types of hedging reduce the variance of the spot portfolio significantly. The variance reduction is much greater for the US index futures. Table 6 also reports the percentage variance reductions of the GARCH hedge over other positions. The variance reduction is calculated as _2 "Others
futures. The high significance of the constant correlation coefficient, p. supports the initial hypothesis that the instantaneous correlation between spot and futures price changes is constant over time. Next, the different hedging methods are compared within the estimation (within-sample) period. For the entire observation period (185 weeks for S&P 500 and MMI, 184 weeks for TSE 35), the above parameters are estimated for each model and used to compute the hedge ratios. For the bivariate GARCH method, a hedge ratio for each week is computed. For the OLS and the OLS-CI model, the hedge ratios are estimated and they remain constant over the testing period. In Fig. 1. three types of hedge ratios are plotted over the sample period. Compared to the OLS and The OLS-CI hedge ratios, the GARCH hedge ratios show considerable variation over time for all three types of index futures.'* This variation occurs even though the stock markets were not particularly volatile over the period. This result is expected given the significance of the GARCH model parameters in Table 5. In the GARCH model, the variances and covariances are constantly changing through time, hence the hedge
"•GARCH
The percentage variance reduction of the GARCH hedge over the three alternative hedges shows noticeable improvement of hedging effectiveness through GARCH over the conventional methods, especially for the S&P 500 and MMI futures contracts. The more reliable measure of hedging effectiveness is the hedging performance of different hedging methods for the out-of-sample periods. To compare the out-of-sample hedging performance of the four strategies, we perform the following calculations for the OLS, the OLS-CI and the GARCH methods. One half of the sample (the first 93 observations) is used to estimate the parameters for each method. The same parameters are used to forecast the conditional variance and covariance of the following week in the testing period. The forecasted hedge ratio will be the one-period forecast of the covariance divided by the oneperiod forecast ofthe variance. Such forecasts are conducted for each week for the following 92(91) observations for S&P 500 and MMI (for TSE 35). For the OLS and the OLS-CI models, the hedge ratios are estimated in the first half period and the same ratios are used for the second half.
estimated OLS constant hedge ratios and OLS-CI hedge ratios are shown below: OLS OLS-CI
S&P 500 0.961 0,967
MMI 1.008 1.009
T35 0.890 0.880
The GARCH hedge ratios vary considerably for each contract as illustrated in Fig, 1, The hedge ratio ranges from 0.926 to 1.234 for the S&P 500 index. The range for the MMI is 0.989 to 1,156, while for the T35 it is 0.710 to 0,998.
136
T. H. Park and L. N. Switzer S&P 500 index
Table 6. Within-sample comparisons of hedging effectiveness Hedge type Unhedged position Naive hedge {h = I) OLS conventional hedge OLS CI hedge GARCH hedge
S&P 500 3.9760 0.1-100 0.1239 0.1239 0.1205
MMI
TSE 35
4.0621 0.0715 0.0713 0.0713 0.0692
3.3874 0.7617 0.7296 0.7305 0.7212
Percentage variance reductions of GARCH hedge over Unhedged position Naive hedge OLS conventional hedge OLS CI hedge 1988
1991
96.969 7.310 2.742 2.741
98.295 3.225 2.951 2.951
78.708 5.329 1.154 1.277
"The estimation and forecast period is from 8 June 1988 to 18 December 1991.
Major market index ( MM )
1.16
Table 7. Out-of sample comparisons of hedge effectiveness Hedge type Unhedged position Naive hedge (h = \) OLS conventional hedge OLS CI hedge GARCH hedge
S&P 500 5.0322 0.1089 0.1151 0.1070 0.1048
MMI 4.7915 0.0571 0.0592 0.0585 0.0551
TSE 35 4.3500 1.1610 1.0360 0.9970 0.9800
Percentage variance reductions of GARCH hedge over 1.00 0.98 4, 1988
1991
97.916 3.762 8.953 2.059
98.843 3.558 6.932 5.814
77.471 15-591 5.414 1.715
"The estimation period is from 8 June 1988 to 14 March 1990. The forecast period is from 21 March 1990 to 18 December 1991.
Toronto 35 index 1.00 0.95-
0.70 1988
Unhedged position Naive hedge OLS conventional hedge OLS CI hedge
1991
Fig. 1. Hedge ratios from the OLS, the OLS-cointegrated, and the GARCH estimations
Table 7 reports the out-of-sample hedging effectiveness of the four strategies. Again, the bivariate GARCH method outperforms the other hedges. The variance reduction is more effective with GARCH for S&P 500 index and MMI futures and less effective for Toronto 35 index futures. In fact, for TSE 35 futures, none of the hedging methods reduces the variance ofthe unhedged position by more than 80%., whereas almost 97-98% variance reduction is achieved for the US counterparts. This poor performance for Toronto 35 futures could be caused by a few reasons. First, the Toronto 35 index experienced several large price drops in the 1988-90 period, combined with large price swings. Several price limit calls made to the futures at the time of severe bear market could have caused futures prices to deviate from the index prices. The Toronto 35 futures' lower liquidity and the resulting inaccurate prices could have also made the futures to be a poor hedging tool for the index.
Stock index futures Overall, the bivariate GARCH hedge outperforms all the constant hedge methods in both within-sample and out-ofsample periods. Another interesting finding is that the naive hedge performs as weil-as the OLS hedges, especially in the out-of-sample period.
VI.
SUMMARY
Based on the substantial evidence of time-varying distributions of stock index levels and index futures, it is only natural to consider a time-varying distribution to estimate the optimal hedge ratios for index futures. In this paper, we use the bivariate GARCH model to capture the time-varying distributions of spot and futures price changes for three types of stoek indices. Since the spot and futures prices exhibit a long-term cointegration relationship, such cointegration is taken into account in the GARCH estimation. The parameter estimation ofthe GARCH method shows significance of several parameters and high likelihood ratios, supporting the modelling framework. The GARCH-based hedge ratios show considerable variations between June 1988 and December 1991. Such variation in the hedge ratio indicates the unreliability ofthe constant hedge ratio based on the conventional risk-minimizing estimation methods. Both within-sample and out-of-sample evidence presented in this paper indicate that the hedging strategy using the bivariate GARCH method is potentially superior to the other conventional methods, including the constant hedge with cointegration. Of course, the only shortfall of the bivariate GARCH method is the frequent rebalancing ofthe hedge portfolio to follow the changing optimal hedge ratio. In the examples given above, the hedge ratio for the GARCH method was changed every week, which is not considered too frequent for a short-term hedging strategy. The trade off between the risk reduction and transactions cost will determine the practicality of the GARCH hedging method. Perhaps an alternative strategy which involves less frequent rebalancing, such as rebalancing only when the optimal hedge ratio changes by some fixed amount, may prove to be more ideal. The benefits of this alternative approach as opposed to that shown here remains a topic for future research.
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137 exchange rates markets. Journal of International Money and Finance, 9, 309-24. Baillie, R. T. and DeGennaro, R. P. (1990) Stock returns and volatility. Journal of Financial and Quantitative Analysis, 25. 203-14. Baillie, R. T. and Myers, R. J. (1991) Bivariate GARCH estimation of the optimal commodity futures hedge, Journal of Apptied Econometrics, 6, 109-24. Bhatt, S. and Cakici, N. (1990) Premiums on stock index futures - some evidence, Journal of Futures Markets, 10, 367-75. Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity, Journat of Econometrics. 31, 307-27. Bollerslev, T. (1987) A conditional heteroscedastic times series model for speculative prices and rates of return. Review of Economics and Statistics, 31, 307-27. Cecchetti, S. G., Cumby, R. E. and Figlewski, S. (1988) Estimation of optimal futures hedge. Review of Economics and Statistics, 70, 623-30. Engle, R. F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987-1007. Engle, R. F. and Granger, C. W. J. (1987) cointegration and error correction: representation, estimation and testing, Econometrica, 55, 251-76. Figlewski, S. (1984) Hedging performance and basis risk in stock index futures, Journat of Finance, 39, 657-69. French. K., Schwert, W. and Stambaugh, R. (1987) Expected stock returns and volatility. Journal of Financial Economics, 19, 137-55. Gagnon, L. and Lypny, G. (1994) The Benefits of Dynamically Hedging Toronto 35 Stock Index, H'orking Paper, Queens University. Ghosh, A. (1993) Hedging with stock index futures: estimation and forecasting with error correction model. Journal of Futures Markets, 13, 743-52. Kawaller, I. G., Koch, P. D. and Kock, T. W. (1987) The temporal price relationship between S&P 500 futures and the S&P 500 index. Journal of Finance, 42. 1309-30. Kroner, K. F. and Sultan, J. (1993) Time varying distribution and dynamic hedging with foreign currency futures. Journal of Financial and Quantitative Analysis, 28, 535-51. Lee, C. F., Bubnys, E. L. and Lin, Y. (1987) Stock index futures hedge ratios: tests on horizon effects and functional form. Advances in Futures and Options Research. 2, 291-311. Markowitz, H. M. (1952) Portfolio selection. Journal of Finance, 7, 77-91. Myers, R. J. (1991) Estimating time varying optimal hedge ratios on future markets. Journal of Futures Markets, II, 39-53. Ng, N. (1987) Detecting spot price forecasts in futures prices using causality tests. The Review of Futures Markets. 3, 15-41. Phillips, P, C. B. and Ouliaris, S. (1990) Asymptotic properties of residual based tests for cointegration, Econometrica, 58, 165-93. Phillips, P. C. B. and Perron. P. (1988) Testing for a unit root in time series regression, Biometrika, 75, 335-46. Stoll, R. H. and Whaley, R. E. (1990) The dynamics of stock index and stock index futures returns. Journal of Financial and Quantitative Analysis, 25, 441-68. Wahab, M. and Lashgari, M. (1993) Price dynamics and error correction in stock index and stock index futures markets: a cointegration approach. Journal of Futures Markets, 13, 711-42.
