Bealieu maturity effect

Journal of Empirical Finance 5 1998. 177–195
Time to maturity in the basis of stock market
indices: Evidence from the SP 500 and the
MMI
Marie-Claude Beaulieu )
De´partement de finance et assurance and CRE´FA, UniÍersite´ LaÍal, Que´bec, Canada G1K 7P4
Abstract
This paper focuses on the behaviour of the basis in stock market index futures contracts
over the lifetime of futures contracts. The model in this paper relaxes the cost of carry
model assumptions of constant interest rate and known dividend yield over the lifetime of
futures contracts. This allows for a test of the presence of time to maturity in the conditional
variance of the model using GARCH. The empirical evidence reveals that, consistent with
Samuelson’s 1995. analysis, time to maturity is a determinant of the conditional variance
of the basis. Furthermore, it implies that time to maturity cannot be accounted for by
transaction costs or cost of carry. q1998 Elsevier Science B.V. All rights reserved.
JEL classification: G13
Keywords: Basis; Stock market index; Intertemporal risk; Nonsynchronous trading; Time to maturity;
One-step ahead forecasts
1. Introduction
Previous empirical studies of the basis in stock market index futures contracts
Castelino and Francis, 1982; MacKinlay and Ramaswamy, 1988; Duan and Hung,
) Tel.: q1-418-656-2926; fax: q1-418-656-2624; e-mail: marie-claude.beaulieu@fas.ulaval.ca.
0927-5398r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S0927-5398 97. 00017-0

178 M.-C. BeaulieurJournal of Empirical Finance 5 (1998) 177–195
1991. have suggested the existence of a relation between the remaining time to
maturity of the contracts and the basis which could lead to arbitrage opportunities 1
or to a predictable variance of the basis. The traditional approach to pricing futures
contracts is the cost of carry model. MacKinlay and Ramaswamy 1988. analyze
the SP 500 futures prices defining a mispricing series as the difference between
the actual futures price and an estimate of those prices constructed with the cost of
carry model. They find that the absolute value of mispricing depends on the time
to expiration. In this paper, I use a model of the intertemporal change of the basis
to gauge the presence of time to maturity in the conditional variance of the basis.
Futures prices are typically collected daily for each contract until its expiration;
data collection then continues with a new contract. The fact that futures prices are
not collected at times corresponding to constant maturity explains why remaining
time to maturity can influence the conditional variance of the time series of futures
prices. Assuming that spot prices follow a stationary autoregressive process,
Samuelson 1965. defines futures prices as the expected spot price at maturity of
the contract. He shows that the conditional variance of the futures price changes
per unit of time increases as time to maturity decreases.
Castelino and Francis 1982. build on Samuelson’s analysis of futures prices to
study the effect of time to maturity on the basis. They show that the conditional
variance of the change in the basis decreases when time to maturity decreases. As
contract maturity approaches, futures prices evolve into spot prices due to the
reduction of interest rate risk. Therefore the arrival of new information is more
likely to affect spot and futures prices in the same manner if it arrives close to
maturity, causing a reduction in the basis variance.
As pointed out by MacKinlay and Ramaswamy, the unanticipated interest
earnings arising from financing or reinvesting the marking to market cash flows to
and from the futures position may explain why the absolute value of mispricing
defined in terms of the cost of carry model diminishes with time to maturity 2. For
instance, French 1983. is critical of approaches that ignore marking to market
since he finds significant differences between futures and forward prices in copper
and silver. MacKinlay and Ramaswamy also suggest that transaction costs may
explain the presence of a maturity effect in their analysis. Indeed, Figlewski
1984. claims that large transaction costs to acquire the SP 500 stocks encour-age
the use of hedging portfolios that do not incorporate all the component stocks
in the index. In that case, the expected number of transactions is greater further
1 Even though the basis in futures contracts shrinks with approaching maturity because the futures
price must equal the spot price at maturity see Fig. 1., time to maturity should not be a characteristic
of the mean of the basis adjusted for the cost of carry. if futures and spot prices simultaneously reflect
all available information. In that case, the basis adjusted for the cost of carry. today contains all the
relevant information about the expected basis tomorrow.
2 This idea is clear in the characterization of Cox et al. 1981. of futures prices since interest rate
uncertainty decreases as the maturity date gets closer.

M.-C. BeaulieurJournal of Empirical Finance 5 (1998) 177–195 179
away from maturity because the incomplete hedge portfolio will need to be
adjusted over time.
The model presented in this paper relaxes the assumption of constant cost of
carry 3 and allows me to infer whether a stochastic interest rate eliminates the
time to maturity effect in the variance of the basis. Furthermore, I compare results
from the Standard and Poor’s 500 SP 500. index basis and the Major Market
Index MMI. basis. This comparison is important because of possible differences
in the extent of nonsynchronous trading in the index. From the observed rate of
change on the SP 500 index and the MMI, Stoll and Whaley 1990. and Chan
1992. find evidence of a higher degree of serial correlation in the return on the
SP 500 index than in the return on the MMI. They interpret this result as
evidence that the SP 500 index is more subject to nonsynchronous trading than
the MMI, since the MMI is a subset of twenty blue-chip stocks more actively
traded than those typical of the SP 500. from the SP 500 index 4. A
comparison of the results for the basis in the SP 500 and in the MMI allows me
to gauge whether transaction costs can explain the relevance of time to maturity in
the conditional variance. Suppose hedgers hold an approximate instead of an exact
replica of the index. Then, ex ante, the expected cost of revising the portfolio will
be higher for the SP 500 because of its greater diversification.
The paper is organized as follows. Section 2 derives the model of the
intertemporal change in stock market index. Section 3 presents data sources and
related descriptive statistics. Section 4 reports empirical results of estimation of the
model of the intertemporal change in the basis. The one-step-ahead forecasting
properties of two different time to maturity specifications in the conditional
variance of the univariate estimation are also investigated at various horizons.
Section 5 concludes.
2. Exposition of the model
The model of the intertemporal change in the basis developed in this paper is
based on the equilibrium valuation of the basis in foreign exchange futures
3 Constant cost of carry refers to two important assumptions limiting the explanatory power of the
cost of carry model. First, the interest rate is assumed constant and second the dividend yield on the
stock is assumed known over the lifetime of the futures contract.
4 Kleidon 1992. distinguishes between nonsynchronous trading or nontrading and stale pricing.
According to his definitions, nonsynchronous trading is the event where the recorded price of a stock is
for the last trade which occurred previously, while stale pricing occurs when a trade is executed at a
price set by a limit order issued much earlier than the moment it arrives at the market and so does not
incorporate current information. With the exception of October 19, 1987, nonsynchronous trading will
be the dominant factor in the analysis and a comparison between the basis in the SP 500 index and
the MMI should reveal the effects of different degrees of nonsynchronous trading in the two indices
and not of stale pricing.

contracts, first presented by McCurdy and Morgan 1993.. The intertemporal
change in the basis is defined as the basis today minus the basis yesterday adjusted
for the net cost of carry. It represents the return on a position in the basis between
ty1 and t.
Let F be the price at time t of a futures contract to deliver one unit of the t
index at time T, S be the spot price at time t of one unit of the index, R be t ty1
one plus the U.S. riskless rate of interest from time ty1 to time t, D be the t
dividend, announced at time ty1 and received at time t from one unit of the
index, d be the dividend yield on the index from time ty1 to time t or ty1
D rS , M be the nominal benchmark variable at time t and E be the t ty1 t ty1
expectation operator conditional on the information set available at time ty1.
To adapt McCurdy and Morgan’s model for stock index data, one has to take
into account the return obtained from a long position in the stock index net of the
borrowing cost and the payoff from a short position in futures contracts. The
strategy is to borrow at ty1 to buy one unit of the index on the spot market,
obtaining a capital gain or loss plus the dividend yield d at t. At the same time, ty1
one goes short one futures contract on the index. This strategy requires no net
investment at ty1. At t, the holder of the unit of index will have StqDtsStq
d S and will owe S R on it. At t, to eliminate the short position in ty1 ty1 ty1 ty1
futures contracts, the holder will go long one futures contract. The resulting payoff
of this strategy is
S qd S yS R y FyF . , 1. t ty1 ty1 ty1 ty1 t ty1
which has a net present value of zero. Using the intertemporal valuation operator
Richard and Sundaresan, 1981., M , and the definition of covariance, the t
valuation model can be written as
FyS F FyS t t ty1 t t E y yR qd sycov M R , . ty1 ty1 ty1 ty1 t ty1 S S S ty1 ty1 ty1
2.
The model of the intertemporal change in the basis leads to better properties of the
time series than the cost of carry model. As described by MacKinlay and
Ramaswamy 1988., the time series of the level of the basis is highly serially
correlated creating inefficient estimates. To get around that problem, Miller et al.
1994. use first differences in the basis. By eliminating overnight price changes,
they can ignore the opportunity cost of holding the basis position from one period
to the next, as well as any systematic risk in holding the basis position. The model
of the intertemporal change in the basis provides a theoretical framework for the
intertemporal valuation of the basis that is consistent with cash dividend payments,
the interest cost of carrying the index, and the systematic risk in the position.

3. Data and descriptive statistics
3.1. Data
The data used in this paper come principally from two sources: the Chicago
Mercantile Exchange CME. for the SP 500 and the Chicago Board of Trade
CBOT. for the MMI. The SP 500 data set contains daily futures and spot prices
from September 30, 1985 to December 31, 1991. The SP 500 index is a
value-weighted index composed of 500 widely held stocks. The index is updated
continuously using the most recent prices as reported of the component 500 stocks.
The index series contains some prices based on nonsynchronous trading, especially
for the thinly traded stocks. The daily futures prices database consists of transac-tion
data for the SP 500 futures contracts 5. Up to March 1987, the futures
contract’s expiration was the third Friday of the delivery month March, June,
September and December.. In order to avoid problems due to the triple witching
hour, the CME then changed the last day of trading of the contract to the last
business day prior to the third Friday of the delivery month of the contract.
The MMI database contains daily futures and spot prices over the same time
period. It is a price-weighted index for which the price of each stock at each point
in time is adjusted for stock splits and stock dividends. The last day of trading of
these futures contracts is the third Friday of the delivery month. In the case of the
MMI, the futures contract delivery months are the first three consecutive months
e.g. October, November and December. plus the next three months in the March,
June, September and December cycle. In order to make results comparable for the
SP 500 and the MMI indices, I consider the contracts on a three-month to
maturity cycle only March, June, September and December.. Furthermore, be-cause
the basis is deterministic on the expiration day of futures contracts, the last
observation of the basis in the series for each contract is dropped.
The dividends for the stocks in the SP 500 index are estimated by the
realized daily dividend yield of the value-weighted index of all NYSE stocks
supplied by the Center for Research in Security Prices CRSP.. To get the
dividend yield, I subtracted VWRET from VWRETD in the CRSP file. The
dividends for the MMI are the actual dividends paid on the 20 component stocks
of the index obtained from CBOT. To construct the dividend yield, I divided the
5 Because the model uses a modified lagged endogenous variable, it is important to construct the
series consistently when a contract change occurs. In this event, I have to compare the value of the
basis for the new contract with the lagged value of the basis of the same contract. In other words, when
a contract change has occurred, the data for the basis and its lagged value must both refer to the new
contract.

Table 1
Estimated serial correlation coefficients. Observed rates of return of the SP 500 index Rs ., the MMI
Rm., and rates of change of the SP 500 index futures contract Rsf ., and the MMI futures contract
Rmf .
Lag Rs Rsf Rm Rmf
rk t rk . rk t rk . rk t rk . rk t rk .
1 0.0561 2.24 y0.0146 y0.58 0.0192 0.77 0.0273 1.09
2 y0.0267 y1.06 y0.0404 y1.62 y0.0471 y1.88 y0.0579 y2.32
3 y0.1390 y5.56 y0.1810 y7.24 y0.1270 y5.08 y0.1380 y5.52
4 y0.0327 y1.31 y0.0464 y1.86 y0.0194 0.78 y0.0515 y2.06
5 y0.0205 y0.82 y0.0046 y0.18 y0.0091 0.36 y0.0057 y0.23
6 0.0213 0.85 0.0488 1.95 0.0133 0.53 0.0379 1.52
7 y0.0191 y0.76 0.0210 0.84 y0.0158 y0.63 y0.0015 y0.06
8 0.0360 1.44 0.0181 0.72 0.0286 1.14 0.0201 0.80
9 y0.0003 y0.01 y0.0192 y0.77 y0.0209 y0.84 y0.0140 y0.56
10 y0.0179 y0.72 y0.0346 y1.38 y0.0186 y0.74 y0.0241 y0.96
The sample includes data from September 30, 1985 to December 31, 1991. The estimated serial
correlation coefficients are for the residuals from the regression of the specified rate of return or rate of
change on a constant and a dummy variable which takes the value of one for Mondays, holidays and
two other dummy variables that take the value of one for the market crashes of October 19, 1987 and
of October 13, 1989 and zero otherwise. The t-ratio corresponds to the null hypothesis of rk the
estimated serial correlation coefficient. equals zero, where k is the number of lags.
sum of those dividends by the index price the day before the ex-dividend date and
adjusted this ratio with the MMI divisor 6.
The interest rates are the overnight effective Federal funds rate, adjusted for
weekends and holidays, so that the rate for a normal weekend is approximately
three times that of a typical weekday. When any interest rate was missing because
of holidays, I used the previous day rate. Finally, the benchmark portfolio is the
daily excess return on the Morgan Stanley Capital International MSCI. world
index in U.S. dollars over the Federal funds rate for a maturity of one day. MSCI
calculates their index from closing values of the 19 country component indices.
3.2. DescriptiÍe statistics
Table 1 shows the autocorrelation function of the rate of change of the spot and
futures prices for the SP 500 and MMI indices 7. Daily rates of change are not
6 I am thankful to Barry Schachter from the Comptroller of the Currency for making the MMI
dividends available to me and to Louis Gagnon from Queen’s University for providing the MMI
divisor.
7 Each index price series is converted into a rate of return and each futures price series is converted
into a rate of change. Then all rates are prewhitened by a regression on a constant, a dummy variable
that takes the value of one on the first trading day of the week, a dummy for the market crash of
October 19, 1987 and a dummy variable for the market crash of October 13, 1989.

as highly autocorrelated as returns over five-minute intervals Stoll and Whaley,
1990. but the return on the SP 500 index is positively and significantly
autocorrelated at its first lag. This is not the case for the return on the MMI; the
t-statistic on the first lag is not significantly different from zero at a five percent
level of significance. This supports Lo and MacKinlay’s 1990. model according
Fig. 1. Basis defined as in Table 2 in both indices over remaining days to maturity of the futures
contract. The sample includes data from September 30, 1985 to December 31, 1991.

to which nonsynchronous trading can be best represented by an autoregressive
process of order one.
Fig. 2 plots the intertemporal change in the basis against time to maturity for
the two indices. Comparing Fig. 2 with Fig. 1, one can see that the former series is
Fig. 2. Intertemporal change in the basis defined as in Table 3 in both indices over remaining days to
maturity of the futures contract. The sample includes data from September 30, 1985 to December 31,
1991.

more evenly spread around zero. Therefore, the intertemporal change in the basis
produces a series which does not depend as strongly on time to maturity as does
the basis itself.
Table 2 presents summary statistics of the basis and Table 3 of the intertempo-ral
change in the basis. Comparing the two, it appears that the series in the latter
table are not as autocorrelated as those of the basis, probably because the
intertemporal change in the basis approximates a first difference. The autocorrela-tion
function of the intertemporal change shows a significant coefficient at its first
two lags for the SP 500 index but not for the MMI, for which the autocorrela-tion
function has only one significant coefficient at its first lag. This is an
indication that the SP 500 basis has a more complicated structure than that in
the MMI.
Furthermore, consistent with Miller et al. 1994., Beaulieu and Morgan 1996.
show that the first difference of the return on the basis depends on an autoregres-
sive term f. that captures the nonsynchronous trading resulting from the basis
position if futures and spot prices are perfectly correlated and of equal variance.
Table 2
Descriptive statistics of the basis
Descriptive statistics
SP 500 index MMI
Sample size 1582 1582
Mean 3.89=10y3 3.19=10y3
Std. deviation 5.56=10y3 3.66=10y3
Minimum y9.49=10y2 y3.56=10y2
Maximum 2.95=10y2 1.51=10y2
Estimated serial correlation coefficients
Lag SP 500 index MMI
rk t rk . rk t rk .
1 0.607 24.18 0.631 25.10
2 0.403 12.18 0.535 15.88
3 0.401 11.12 0.459 11.86
4 0.376 9.69 0.414 9.86
5 0.317 7.73 0.379 8.52
6 0.271 6.36 0.340 7.31
7 0.230 5.28 0.290 6.03
8 0.116 2.61 0.234 4.76
9 0.103 2.31 0.218 4.37
10 0.149 3.33 0.215 4.26
The sample includes data from September 30, 1985 to December 31, 1991. The basis is defined as
FtySt .rSty1 for each index. The estimated serial correlation coefficients are for the basis of both
indices. The t-ratio corresponds to the null hypothesis of rk the estimated serial coefficient. equals
zero, where k is the number of lags.

Table 3
Descriptive statistics of the intertemporal change in the basis
Descriptive statistics
SP 500 index MMI
Sample size 1582 1582
Mean 2.31=10y5 3.07=10y5
Std. deviation 4.82=10y3 3.01=10y3
Minimum y8.66=10y2 y2.49=10y2
Maximum 9.49=10y2 2.73=10y2
Estimated serial correlation coefficients
Lag SP 500 index MMI
rk t rk . rk t rk .
1 y0.257 y10.23 y0.426 y16.97
2 y0.270 y10.27 y0.028 y0.98
3 0.032 1.12 y0.035 y1.21
4 0.039 1.40 y0.002 y0.07
5 0.002 0.07 0.002 0.06
6 y0.012 y0.44 0.026 0.87
7 0.106 3.72 y0.001 y0.04
8 y0.111 y3.87 y0.037 y1.27
9 0.091 y3.15 y0.037 y1.24
10 0.101 3.46 0.105 3.56
The sample includes data from September 30, 1985 to December 31, 1991. The intertemporal change in
the basis is defined as wFtySt .rSty1 xywFty1.rSty1.yRty1qdty1 x for each index. The esti-mated
serial correlation coefficients are for the intertemporal change in the basis of both indices. The
t-ratio corresponds to the null hypothesis of rk the estimated serial coefficient. equals zero, where k is
the number of lags.
These assumptions also imply that futures and spot prices are cointegrated. In this
case, the first-order serial correlation of the intertemporal change in the basis is
1yf
r 1.sy . 3.
2
Again in Table 3, the coefficients on the first-order serial correlation of the
intertemporal change in the basis is closer y1r2 for the MMI than for the SP
500 index. One can therefore interpret the smaller first-order autocorrelation
coefficient in absolute value. on the SP 500 index basis as evidence of larger
problems due to nonsynchronous trading in this index than in the case of the MMI.
4. Empirical results
This paper utilizes GARCH Engle, 1982; Bollerslev, 1986. to estimate the
model of the basis. The basis, like many financial series, is subject to the presence

of heteroscedasticity and leptokurtosis. By using conditional second moments,
GARCH helps taking these factors into account Bollerslev et al., 1992..
4.1. Estimation of the model
The bivariate estimation of the model consists of a joint estimation of the basis
equation, a benchmark equation and a conditional covariance matrix. The positive
definite parameterization follows that proposed by Engle and Kroner 1995.
BEKK..
The intertemporal capital asset pricing model CAPM., which I use for the
valuation of the intertemporal change in the basis, involves the intertemporal
marginal rate of substitution between periods as the benchmark for asset pricing. I
replace the benchmark variable M McCurdy and Morgan, 1991. with a bench- t
mark portfolio the return on a worldwide index. hypothesized to be conditionally
mean-variance efficient Hansen and Richards, 1984.. This is an attractive alterna-tive
to the benchmark based on consumption because of measurement problems in
consumption data Breeden et al., 1989.. This approach has been found empiri-
cally useful Mark, 1988. but is, of course, subject to the critique of Roll 1977..
Two instrumental variables are used to predict market excess return. An
indicator variable for Mondays is used for the weekend effect French, 1980..
Second, the excess market return is assumed to follow an autoregressive process
Lo and MacKinlay, 1990.. The conditional variance of the market excess return
follows a GARCH process with an asymmetric term Glosten et al., 1993..
The basis equation is as specified in Eq. 2. after invoking rational expectations
and replacing the expectations in Eq. 2. by the ex post values of the variables at
time t. Note that a moving average MA. term might be necessary if the level of
the basis is stationary, that is if futures and spot prices are cointegrated Beaulieu
and Morgan, 1996.. The conditional variance of the basis follows a GARCH
process with asymmetric matrices A, E and B and is a function of time to
maturity. The model also includes dummy variables for the market crash of
October 1987 for the SP 500 basis and the MMI basis in the conditional
variance of the basis and that of the market portfolio’s excess return.
Let R be the rate of return on the benchmark portfolio from time ty1 to mt
time t, r be the riskless rate of return from time ty1 to time t, m be an f t t
indicator variable that takes the value of one when t is the first trading day of the
week and zero otherwise, X be a set of instruments known at time ty1, m,ty1
Sy be an indicator variable in the conditional variance of the market excess ty1
return equation that takes the value of one when « is negative and zero mt
otherwise, « be an error term at time t for the market excess return equation, mt
« be an error term at time t for the intertemporal change in the basis, u be a bt t
vector with components « ,«
X . , m be a coefficient on basis risk which takes mt bt
the value of one if the conditional CAPM model is appropriate, H be the t
variance–covariance matrix of the system of mean equations with asymmetric

components, hmt be the variance of the benchmark return on the Ht matrix, hbmt
be the covariance between the benchmark return and the intertemporal change in
the basis in the H matrix, d be a dummy variable taking the value of one on t mt
October 19 and 21, 1987 and zero otherwise, d be a dummy variable that takes cc t
the value of one at the beginning of each contract and zero otherwise, l be a t
vector with components 0,d X, f be a vector with components d ,d X . . , K cct t mt mt t
be a matrix equal to L L with L sL sL s0 and L sTyt ., L be i j i jt 11t 12 t 21t 22t t
a function of remaining days to maturity in spike form such as introduced by
Baillie and Bollerslev 1989. and extended to the multivariate BEKK model by
Morgan and Neave 1992.. The equivalent formulation for the time to maturity
specification is L sK y AXK AqBXK BqEXK E., C be an upper t t ty1 ty1 ty1
triangular matrix for the constants in the variance, A be an asymmetric matrix for
the ARCH terms, E be an asymmetric matrix for the asymmetric ARCH terms
and B be an asymmetric matrix for the GARCH terms.
The estimated system is
R yr sg qhm qg R yr .q« 4. mt f t 0m t 1m mty1 fty1 mt
FyS F h t t ty1 bmt X y yR qd sg qm g X .qC« ty1 ty1 0b m mty1 bty1 S S h ty1 ty1 mt
q« 5. bt
with either
H sCXCqAXu uX AqBXHX BqEXSXyu uX Sy EqDXl lX D t ty1 ty1 ty1 ty1 ty1 ty1 ty1 t t
qFXf fXF 6. t t
or
H sCXCqAXu uX AqBXHX BqEXSXyu uX Sy E t ty1 ty1 ty1 ty1 ty1 ty1 ty1
qL qFXf fXF. 6X . t t t
In Eq. 6., the maturity effect in the conditional variance of the basis is
parameterized with a dummy variable that takes the value of one at the beginning
of new contracts and zero otherwise. Through the recursive form of the GARCH
variance, the time to maturity effect will gradually become smaller. Eq. 6X ., as in
Morgan 1995., uses a spike function for time to maturity. In this form, the
remaining days to maturity variable is introduced at time t in the conditional
variance in such a way that the conditional variance at time tq1 does not depend
on the variable introduced at time t. This insures that this source of change in
variance does not influence the conditional variance in following periods.
Table 4 presents the principal coefficient estimates of the basis with both
indices. The estimates reveal that all coefficients are positive and significantly
different from zero, except for the constant in the MMI basis equation. With
respect to the time to maturity coefficient, results imply that the conditional

Table 4
Selected coefficient estimates
SP 500 MMI
contract change spike contract change spike
Parameters for the market equation
g0m 0.009 0.002. 0.009 0.002. 0.009 0.002. 0.009 0.002.
h y0.014 0.005. y0.014 0.005. y0.017 0.005. y0.016 0.005.
g1m 0.176 0.030. 0.183 0.036. 0.196 0.030. 0.193 0.027.
Parameters for the basis equation
g0b y0.005 0.002. y0.004 0.001. 0.001 0.001. 0.001 0.001.
Tyt 0.052 0.014. 0.072 0.032. 0.058 0.015. 0.054 0.016.
C y0.790 0.021. y0.790 0.022. y0.786 0.021. y0.784 0.018.
The sample includes data from September 30, 1985 to December 31, 1991. Contract change refers to
the maturity effect in the conditional variance of the basis parameterized with a dummy variable that
takes the value of one at the beginning of new contracts and zero otherwise. Spike refers to the
remaining days to maturity variable introduced at time t in the conditional variance of the intertempo-ral
change in the basis so that the conditional variance at time tq1 is made independent of the variable
introduced at time t. The selected coefficient estimates come from systems of Eqs. 4.–6. and Eqs.
4.–6X .. g0m is the constant in the market portfolio equation. h is the coefficient for the indicator
variable for the first trading day of the week and g1m is the coefficient on the first lag of the market
portfolio excess return over the riskfree rate. g0b is the constant in the intertemporal change in the
basis equation, Tyt refers to coefficient estimates of the different time to maturity specifications
presented for the conditional variance in Eq. 6. and Eq. 6X . andC is the coefficient on the first-order
moving average term. Robust standard errors Bollerslev and Wooldridge, 1992. are in parentheses.
variance of the basis decreases as the futures contract approaches expiration.
Furthermore, there is not much difference across indices and time to maturity
specifications. Overall, the size of the maturity effect is comparable in both bases
and since the cost of approximating the MMI is almost certainly lower than that
for the SP 500 index, this is an indication that transaction costs are not a valid
explanation of the presence of a maturity effect in the conditional variance of the
basis.
Table 5 presents results of tests for the SP 500 index and the MMI basis. The
benchmark portfolio test is a joint test of the conditional CAPM and of the
conditional efficiency of the benchmark portfolio Mark, 1988.. As described by
McCurdy and Morgan 1993., one can treat M , the true benchmark portfolio, as a t
latent variable and project it onto a constant, R) sR yr and DB swFy mt mt ft t t
S .rS xywF rS .yR qd x. Let c be the coefficient for the former t ty1 ty1 ty1 ty1 ty1 1
and c the coefficient for the latter. The test equation for the benchmark portfolio 2
according to those authors is
c rc .Var DB .qCov DB ,R) . E 2 1 ty1 t ty1 t mt ) DB s E R 7. ty1 t c rc .Cov DB ,R) .qVar R) . ty1 mt 2 1 ty1 t mt ty1 mt

Table 5
Specification tests for the bivariate analysis
SP 500 MMI
BT 3.61 0.01. y1.08 0.14. 1.01 0.16. y1.11 0.13.
ms0 1.90 0.17. 2.03 0.15. 0.31 0.58. 0.01 0.99.
ms1 22.45 0.00. 18.97 0.00. 0.29 0.53. 0.01 0.99.
remaining days to maturity variable introduced at time t in the conditional variance of the intertemporal
introduced at time t. Tests on BT, the benchmark test statistic, adapted from Mark 1988.; p-values for
this test, in parentheses, are for the unit normal distribution. Tests on m are for the coefficient in
systems of equations Eqs. 4.–6. and Eqs. 4.–6X .. ms0 tests the null hypothesis that the basis
systematic risk is zero, ms1 tests the null hypothesis the CAPM restriction that m is not significantly
different from 1 p-values in both cases are for the x 2 1. distribution..
For this equation to be consistent with the conditional CAPM, c rc has to equal 2 1
zero 8. The benchmark test statistics presented in Table 5 test whether that
constraint holds. A rejection can imply that the benchmark used in the estimation
R) . is not conditionally mean variance efficient or that the conditional CAPM mt
does not hold. The results of the benchmark test reveal that for the SP 500 basis,
only in the case where the dummy variable for the contract change is used can one
reject the hypothesis that c rc is equal to zero. In no case is the benchmark 2 1
rejected for the MMI.
The evidence on the SP 500 and the MMI basis risk provides an interesting
contrast. For the SP 500 basis, regardless of the specifications used for the
maturity effect and of the choice of conditional distribution, m which is equal to
one if the conditional CAPM holds., the coefficient on basis risk, is never
significantly different from zero. When m is constrained to one, the OPG–LM
statistic has a p-value of 0.00, clearly rejecting the hypothesis of m equal to one.
Similar evidence is not found in the MMI basis since neither hypothesis, m equal
to one or m equal to zero, is rejected. Therefore, the risk premium in the SP 500
basis is not consistent with the conditional CAPM while the conditional CAPM
cannot clearly be rejected using the MMI. However, it is possible that basis risk is
a component of the SP 500 basis. Yet some feature of the data set, possibly
8 Note that according to Mark 1988., the benchmark portfolio test is for the hypothesis of c not 2
significantly different from zero. In order to make the estimation procedure more parsimonious and
reduce the estimation by one coefficient, using simple algebra one can simplify the original benchmark
portfolio test to the hypothesis of c2 rc1 not significantly different from zero, where c2 rc1 is
estimated as a single coefficient.

Table 6
Conditional moment tests and likelihood ratio tests
SP 500 MMI
Conditional moment tests
Tyt in mean 0.57 0.45. 0.40 0.53. 1.45 0.23. 0.45 0.50.
Tyt in covariance 0.76 0.38. 0.29 0.59. 0.01 0.96. 0.49 0.48.
Likelihood ratio tests
Tyt in variance 77.55 0.00. 99.73 0.00. 54.60 0.00. 73.22 0.00.
remaining days to maturity variable introduced at time t in the conditional variance of the intertempo-ral
introduced at time t. Conditional moment tests evaluate whether Tyt. belongs in the mean of the
intertemporal change in the basis or in the covariance with the market portfolio; while the likelihood
ratio test statistic evaluates how much explanatory power is lost in the model of the intertemporal
change in the basis when time to maturity is excluded from the conditional variance of the basis
equation. P-values, in parentheses, are for the x 2 1. distribution.
nonsynchronous trading, is interfering with the estimation causing m to be
significantly different from one.
Finally, Table 6 presents results of the likelihood ratio LR. tests and condi-
tional moment CM. tests when m is constrained to one. Test statistics for time to
maturity in the mean of the basis equation and in the conditional covariance of the
basis are insignificant. For the mean, this implies that the intertemporal change in
the basis is consistent with market efficiency and that it is not possible to use time
to maturity to take advantage of potential arbitrage opportunities. In the case of the
covariance, it means that the systematic risk of the basis position is also indepen-dent
of remaining days to maturity and that the risk premium is unpredictable in
that respect. Table 6 also reports LR tests for the influence of time to maturity on
the conditional variance of both indices. The results indicate that the conditional
variance of the intertemporal change in the basis in both indices is clearly a
function of time to maturity. That is, a model that incorporates marking to market
does not seem to resolve the question of why time to maturity influences the
conditional variance of the basis.
4.2. Out-of-sample forecasts
In order to assess the importance of time to maturity in the variance of the basis
in stock market index futures contracts, I present out-of-sample forecasts at
various horizons. Since basis risk does not appear to be a very large component of
the basis, and to facilitate calculations, I assume it to be equal to zero. This

simplification allows me to produce forecasts from a univariate model of the
intertemporal change in the basis with conditional variance h in which L t t
sTyt .. The model is
FyS F t t ty1 y yR qd sg qC« q« 8. ty1 ty1 0b bty1 bt S S ty1 ty1
h scqa« 2 qbh qDd qff 9. t bty1 ty1 t t
or
h scqa« 2 qbh qg L y aqb. L .qff 9X . t bty1 ty1 t ty1 t
The parameterizations for time to maturity correspond to those in the bivariate
analysis. Eq. 9. corresponds to Eq. 6.; d is an indicator variable for a contract t
change but D is now a scalar coefficient. Given the recursive form of the GARCH
conditional variance, the influence of time to maturity gradually becomes smaller
over the life of the futures contract. Eq. 9X . corresponds to Eq. 6X . and uses a
spike function of time to maturity. In this form, the remaining days to maturity
variable is introduced at time t in the conditional variance in such a way that the
conditional variance at time tq1 is made independent of the variable at time t.
Table 7
One-step ahead out-of-sample variance forecasts
Forecasting horizon SP 500 MMI
LR test MAE LR test MAE
Contract change
May 22, 1990 to December 19, 1991 10.75 0.00. 0.69 0.25. 17.58 0.00. 0.18 0.43.
May 22, 1990 to March 7, 1991 3.34 0.07. 0.15 0.44. 17.10 0.00. 0.28 0.39.
March 8, 1991 to December 19, 1991 7.69 0.01. 0.19 0.42. 0.48 0.51. y0.72 0.24.
Spike
May 22, 1990 to December 19, 1991 11.81 0.00. 1.13 0.13. 21.09 0.00. 0.72 0.23.
May 22, 1990 to March 7, 1991 5.51 0.02. 0.05 0.48. 16.98 0.00. 0.13 0.45.
March 8, 1991 to December 19, 1991 6.30 0.01. y1.02 0.15. 4.11 0.04. y0.32 0.37.
Contract change refers to the maturity effect in the conditional variance of the basis parameterized with
a dummy variable that takes the value of one at the beginning of new contracts and zero otherwise.
Spike refers to the remaining days to maturity variable introduced at time t in the conditional variance
of the intertemporal change in the basis so that the conditional variance at time tq1 is made
independent of the variable introduced at time t. The LR test statistic evaluates how much explanatory
power is lost in the model of the intertemporal change in the basis when time to maturity is excluded
from the conditional variance of the basis equation. MAE is a t-test statistic on the difference between
the mean absolute error of a model that includes time to maturity in its conditional variance and that for
a model that does not include it. The time period from May 22, 1990 to December 19, 1991 contains
400 one-step-ahead out-of-sample forecasts. Time periods from May 22, 1990 to March 7, 1991 and
from March 8, 1991 to December 19, 1991 each contain 200 one-step-ahead out-of-sample forecasts.
P-values are in parentheses.

In order to forecast the conditional variance for the error term at tq1, I use
observations 1 through t. Then, I add the observation for tq1, update the model
parameter estimates and forecast the variance for tq2. I follow the same
procedure until I have reached the last observation of my forecasting horizon.
I have chosen two criteria for comparison of the results of the out-of-sample
forecasts of the models including and excluding time to maturity. The first
criterion is the difference in mean absolute error MAE. between the two models.
The second criterion is a likelihood ratio test evaluating the gain in explanatory
power resulting from the inclusion of time to maturity in forecasts of the
conditional variance of the basis. The results of the LR tests, reported on Table 7,
show that overall for sets of 200 and 400 of one-step ahead forecasts, the inclusion
of time to maturity adds to the explanatory power of forecasts of the variance of
the intertemporal change in the basis in both indices, independently of the
parameterization used. Furthermore, time to maturity specifications in the variance
have more explanatory power over longer time periods. Nonetheless, over shorter
subperiods, only in two cases out of eight the explanatory power gained is not
significantly different from zero at a five percent level of significance. The
evidence with MAE is weaker.
5. Summary and conclusions
The model of the intertemporal change in the basis relaxes the cost of carry
model assumptions of constant interest rate and known dividend yield over the
lifetime of the futures contract. Using this model, in two different parameteriza-tions,
I present evidence that the conditional variance of the basis decreases as the
maturity date gets closer. This result is consistent with Castelino and Francis’
1982. modelling of the conditional variance of the basis and, ultimately, with
Samuelson’s 1965. hypothesis. Furthermore, the evidence also suggests that in
out-of-sample forecasts, the model of the conditional variance of the basis with
time to maturity is a better predictor of the conditional variance of the basis than
the model that does not incorporate time to maturity. This evidence is consistent
with the fact that interest rate risk cannot explain the presence of a maturity effect
in the conditional variance of the basis in stock market index futures contracts.
The comparison of the SP 500 and the MMI basis reveals interesting features
of the data. First, it appears that time to maturity in the conditional variance of the
basis is of similar importance in both indices. This implies that the presence of
time to maturity in the conditional variance of the basis cannot be explained by
transaction costs. Furthermore, the analysis finds no evidence supporting the
presence of basis risk in the SP 500 basis. This contrasts with the MMI basis
model in which case I cannot reject the CAPM and positive basis risk. Since the
MMI is a subset of the SP 500 index and that the analysis is the same for both
bases, nonsynchronous trading problems in the SP 500 index is the most likely
explanation for this phenomenon.

Acknowledgements
This paper is based on my doctoral dissertation presented at Queen’s Univer-sity,
March 1994. My thanks to I.G. Morgan who has directed this research. I also
wish to thank A.C. MacKinlay, T.H. McCurdy and E.H. Neave and an anonymous
referee. A previous version of this paper was presented at the 1993 Northern
Finance Association conference. Funding from Queen’s University School of
Business and School of Graduate Studies as well as the Fonds pour la formation
de jeunes chercheurs et l’aide a` la recherche FCAR. is gratefully acknowledged.
The usual disclaimer applies.
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