modelling the interaction between stocks and interest rates

MODELLING THE INTERACTION BETWEEN STOCKS AND

INTEREST RATES

KIEN TRINH

OCTOBER 2001

ABSTRACT

To analyse the intertemporal interaction between stock and interest rates, we apply the GARCH

framework to monthly U.S data and performed various diagnostic tests to assess their

performance. We find that the symmetric GARCH models reveal a common inability to explain

the asymmetry present in our sample observations. Although the EGARCH model satisfactorily

incorporates some of these leverage effects, the application of the News Impact Curve suggests

that these models tend to overfit the asymmetric feature of volatility. By including lagged

interest rate (stock) returns into our model of stocks (interest rates), we find that stock

movements respond negatively to interest rate returns while its volatility is positively affected by

interest rate shocks. Interestingly enough, interest rate volatility also influences fluctuations in

stock returns. The improvement of our univariate models by the inclusion of an exogenous

variable suggests that our previous models were mis-specified.

© 2001 All rights reserved. No part of this document may be reproduced or transmitted in any form or by

any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written

permission.

CONTENTS

1. Introduction........................................................................................................

2. Models of Volatility...........................................................................................

3. Data ..................................................................................................................

3.1 Summary Statistics..............................................................................

3.2 Tests for Stationarity...........................................................................

3.3 Mean Equations...................................................................................

3.4 Squared Returns...................................................................................

3.5 Tests for ARCH Effects......................................................................

3.6 Asymmetry Tests.................................................................................

4. Estimates of Our Parametric Volatility Models.................................................

4.1 Models of Stock Volatility...................................................................

4.2 Models of Interest Rate Volatility........................................................

4.3 Estimated News Impact Curve............................................................

5. The Relationship Between Stocks and Interest Rates........................................

5.1 Sensitivity of Interest Rate on Stock Volatility...................................

5.2 Effect of Stocks on Interest Rate Volatility.........................................

6. Summary and Conclusions.................................................................................

7. Extensions..........................................................................................................

8. References..........................................................................................................

3

5

10

12

14

16

17

18

19

21

21

25

27

30

31

34

37

38

39

1. INTRODUCTION

Modelling the conditional volatilities of asset returns, including the interaction between them, is

fundamental for the pricing of financial securities, asset allocation and risk management

strategies. While a vast amount of literature on volatility has emerged over the past decade, these

studies have often been restricted by either examining the stock or bond market separately. This

paper will extend previous investigations by modelling the volatility of stock and interest rate

markets as well as examine their co-movements. Our objective will be to determine whether

stock (interest rates) movements are sensitive to interest rate (stocks) shocks and evaluate the

implications of such sensitivities.

An analysis of volatility is fundamental in the application of asset pricing, portfolio selection,

capital structure, and valuation theory. For example, more efficient estimators can be obtained

through appropriate modelling of heteroscedasticity in the errors. In practice, it is especially

relevant for risk management, program trading and dynamic hedging strategies. By modelling

volatility, a risk manager is able to determine the likelihood that his portfolio will decline in the

future. Alternatively, an option trader desiring to hedge his/her position may need to understand

the volatility of the future life of a contract. Volatility is also applied implicitly in pricing

equities and derivatives. For instance, in the CAPM and Black-Scholes option pricing formula,

the key parameters are the beta and standard deviation of the underlying stock returns.

The correlation between the stock and interest rate market is important for a variety of

investment and risk management decisions. Portfolio managers often shift funds from stocks into

bonds when they expect stock market volatility to increase. The effectiveness of this risk

reduction strategy depends on the volatility linkages between the stock and bond markets. If the

volatilities across these markets are highly correlated than bonds may not provide the safe haven

managers are seeking. Furthermore, fund managers rely upon volatility estimates and market

correlations in determining their asset allocation through estimates of efficient portfolios.

The interaction between stock and interest rates is also important for valuing derivative securities

and hedging strategies. For instance, a derivative dealer who operates in more than one market is

vulnerable to the volatility exposure associated with cross market correlations and should

incorporate this information into his risk measurement systems. Meanwhile, a trader should

consider the correlation between stock and bond returns when taking a position in bonds to

hedge his speculative position in stocks.

Moreover, volatility linkages are also considered in setting regulatory policy. Banking regulators

are required to understand the covariance between the two markets to appropriately assess capital

adequacy. Similarly, market regulators should account for these linkages when evaluating the

effects of proposed policy changes, such as the impact of margin requirements on trading

activity.

Spiro (1990) argues that the volatility of stock prices is mainly attributable to the volatility of

interest rates. A rise in the stock market stimulates investment and increases the demand for

credit and eventually leads to higher interest rates throughout the economy. Spiro concludes that

although it is unfeasible to forecast every movement of the stock index through economic

variables such as interest rates, it is possible to obtain a reasonably good forecast of changes in

stock direction.

Accurate modelling of this behaviour can be potentially rewarding but it is surprising that

researchers have only recently begun to examine these temporal dependencies. Fischer (1981)

proposed that the variance of inflation positively responds to the volatility of interest rates. If

short-term interest rates embody expectations about inflation then it may be possible to apply

them to predict future stock volatility.

Other papers have documented a close relationship between the stock and interest rate market.

For example, Breen, Glosten and Jagannathan (1989) found a negative relationship between

short-term interest rates and stock index returns, while Schwert (1989) concluded that U.S stock

and bond volatilities commonly move together. In general, volatility is expected to change over

time and evidence of this comes from Kwan (1996). He finds that individual stocks and bonds

can either be positively or negatively correlated, depending on the information disseminated over

time. A more recent study by Fleming, Kirby and Ostdiek (1998) reveal that strong volatility

linkages between the stock and bond markets are mainly due to common information that affects

expectations in both markets and information spillover caused by cross-market hedging. Hence,

for the above reasons, it is important for us to study the behaviour of volatility and their

interactions over different markets.

2. MODELS OF VOLATILITY

The aim of any modelling process is to produce an adequate conditional characterisation of the

data. Financial time series displays strong clustering of volatility. Mandelbrot (1963) and Fama

(1965) both reported evidence that 'large changes tend to be followed by large changes of either

sign, while small changes tend to be followed by small changes'. This behaviour has been

confirmed by numerous other studies such as Chou (1988) and Bailie et al. (1996). Volatility

clustering implies that shocks today will influence the expectation of volatility shocks into the

future. It also suggests that the volatility is mean-reverting, i.e. there is a normal level to which

volatility will return.

Another common feature of financial time series is that they exhibit leptokurtosis. This means

they contain higher peaks and thicker tails relative to the normal distribution. Note that

leptokurtosis is an outcome of volatility clustering.

The autoregressive conditional heteroscedasticity (ARCH) model was introduced by Engle

(1982) to capture the volatility clustering as well as leptokurtosis inherent in many financial

times series. It is unique in that it specifies the variance of the error term in a regressive equation

as conditional on the squared of past errors.

To understand the advantage of conditioning on past information, let us first consider a simple

stationary AR(1) model without drift:

�� = �� + �� where |�| < 1 ��~�(0, ��) The unconditional mean is �(��) = �(�� + ��)

= ��(�� + ��) + ��) = �(�� + �� + �� +⋯) = 0

The unconditional variance is �(��) = �(�� + ��) = �(�� + �� + �� +⋯) = ��(1 + �� + �� +⋯) = ��/(1 − ��)

We can calculate the mean and variance of �� by conditioning on all available information up to

time t -1. The conditional mean and variance are respectively:

�(��|��) = �� and �( �|��) = �� where �� represents the information set at time t

These values provide better forecasts because the conditional mean is time-varying and the

conditional variance is smaller than the unconditional variance eg. �� < ��/(1 − ��) Let us now return to our ARCH model.

The ARCH(p) model involves a joint estimation of a conditional mean and variance equation:

!� = "#$� + ��%ℎ'(')��|��*~�(0, ��) �� = + + �� + �� +⋯+ �,��,� (1.0)

The mean is expressed as a function of exogenous variables $�with "# coefficients and an error term ��. For example, we can represent the mean by an autoregressive AR(p) equation:

AR(p) !� = "�!�� + "�!�� +⋯+ ",!, + �� The distribution of the stochastic error term �� is conditional on the information set �� and is assumed to follow a normal distribution with mean zero and variance ��. The conditional variance comprises a constant ω and past news on volatility reflected by the squared residuals

from (t-p) to (t-1). We assume that the error terms are uncorrelated and the conditional variance

is time-varying. This gives rise to the name heteroscedasticity. Note that in practice the residuals

in many time series regressions are correlated.

Due to difficulties in attaining precision by selecting the optimal lag p, we can replace many of

the lagged values of �� by one or two lagged values of ��. This leads us to the generalised autoregressive conditional heteroscedasticity (GARCH) model proposed by Bollerslev (1986).

The GARCH(p,q) model has a conditional variance containing lagged variances as well as past

squared error terms.

�� = + + ∑ �#��#� + ∑ .#��#�/#0�,#0� (1.1)

where +, �# , .# are constant parameters.

This model allows a more parsimonious representation because it corresponds to an infinite order

ARCH model. To ensure a positive conditional variance, we place non-negativity constraints on

the coefficients whereby +# > 0 and �#, .# ≥ 0. Such restrictions are more easily satisfied for the

GARCH(p,q) since there are fewer parameters to be estimated. Note that changes in news, either

positive or negative, leads to increases in the variance with more emphasis being place on the

most recent innovations. Therefore, the model effectively captures the volatility-clustering

phenomenon.

Since GARCH models are a function of lagged squared errors and not their signs, they are

severely limited by their inability to capture the asymmetric volatility frequently inherent in

stock market data. The sign asymmetry of stocks suggests that there is a tendency for negative

shocks to elicit a larger response than positive shocks of equal magnitude. Furthermore, large

shocks have been found to produce a proportionately larger response when compared to smaller

shocks. This has been known as size asymmetry.

Asymmetry was first noted by Black (1976) and is commonly ascribed to the leverage effect.

Black (1976), Christie (1982) and Nelson (1991) argued that an increase in stock volatility

associated with 'bad news' lowers the stock price and increases the debt-equity ratio of a

company. As a result, the financial leverage of the firm rises, increasing the risk of holding stock

and reinforcing further volatility and price declines. On the other hand, an increase in stock price

volatility associated with the arrival of 'good news' decreases the firm's corporate leverage and

reduces the risk. Thus the increased price is effectively offset by the reduced corporate risk,

producing a lower impact on volatility.

Black (1976) noted that leverage alone does not fully explain this asymmetry in equity volatility.

Several recent papers have proposed alternative explanations. One approach is that of French,

Schwert and Stambaugh (1987) and Campbell and Hentschel (1992) with their volatility

feedback hypothesis. Arrival of good news signals a positive change in market volatility. This

direct positive effect is partially offset by an increase in risk premium causing only a minor

impact. Alternatively, when bad news arrives, both the direct effect and risk premium effect

move in the same direction so the impact of news is amplified. Another explanation was

developed by Veronosi (1999), who attributed it to a following-the-herd-effect. Here, investors

imitate one another in selling stocks when bad news arrives and this induces relatively higher

volatility. Whatever the reasons, we acknowledge that volatility asymmetry exists and will now

focus on the exciting part – modelling.

The presence of asymmetric responses can be parsimoniously accounted for by explicitly

incorporating an indicator variable, d, into the conditional variance equation. This leads us to the

Threshold (TARCH) model introduced independently by Zakoian (1994) or otherwise known as

the GJR-GARCH model, named after Glosten, Jagannathan and Runkle (1993).

�� = + + �� + 3�� + .�� or more generally,

�� = + + ∑ �#��#� + 3�� + ∑ .#��#�/#0�,#0� (1.2)

Where �� = 145�� < 0, �� = 0 otherwise +, �#, .#, 3 are constant parameters

The TARCH(p,q) model implies that a positive innovation at time 6(�� > 0) has an impact on

volatility at time t + 1 equal to �# times the residual squared, while a negative innovation

(�# < 0) has an impact equal to (�# + 3) times the squared residual. The presence of leverage

effects suggests that the coefficient γ is positive ie. negative innovations have a greater impact

on volatility.

Another historically successful asymmetric specification is the Exponential GARCH (EGARCH)

proposed by Nelson (1991). Here, the parameters are created to be non-linear in a logarithmic

transformation of the conditional variance:

log(��) = + + .log(�� ) + � :;<=>?<=> − @�A: + 3 ;<=>

?<=> (1.3)

This model implies that the leverage effect is exponential rather than quadratic. Hence, it is not

necessary to enforce non-negative constraints on the coefficients because the conditional

variance is guaranteed to be positive. The presence of leverage effects can be tested by the

hypothesis that 3 > 0, and the impact is asymmetric if 3 ≠ 0 Other models such as the GARCH-in-Mean (GARCH-M) have been developed by Engle,

Lillien, and Robins (1987), to incorporate a direct relation between the conditional variance and

mean equation. This accounts for the fact that risk-averse investors trade off higher risk for

higher expected return. It is worth noting that other models like these exist even though they will

not be applied in this study.

The models explained above have been quite successful in characterising volatility. However, we

must be aware of their limitations such as the assumption of normally distributed errors.

Furthermore, although these models may appear correctly specified for one frequency of data,

they could be incorrectly specified for data with different time scales.

3. DATA

The data comprised of 378 monthly observations from the period 01/1970 to 06/2000 on the

Standard and Poor's (S&P) 500 Index and the Three Month Treasury Bill in the secondary

market (TB3MS). Data were obtained from the Federal Reserve Bank of St Louis website under

FRED - an economic time series database which provides historical U.S. economic and financial

data1. All data was transformed to continuously compounded returns over the period (t -1) to t,

and is calculated as !� = log(��) − log(��), where �� represents the price for the S&P index or

the yield for the TB3MS.

Figures 1.1 presents the monthly time series plots for our stock data. The S&P 500 index exhibits

a general positive exponential trend until year 2000. It is clearly non-stationary. Figure 1.2

represents continuously compounded returns calculated by taking the difference of logarithms.

Note that the mean return on the S&P500 index is slightly above zero, indicating a drift

component in the time series. The peak in volatility observed in 1974 coincides with the

dramatic rise in world oil prices associated with OPEC, while the downward spike in late 1987

corresponds to the October 1987 stock market crash. Apart from other lesser peaks, the

fluctuations appear fairly evenly distributed.

Figure 1.3 displays the 3-Month Treasury-bill yield over time. Observe the interesting variation

as rates fluctuated upwards until early 1980, and then oscillated back down. Figure 1.4 represents

the continuously compounded return on the T-Bill. Note the sharp downward spike in rates in

1980. There is a high risk and return period around 1979-1983 with excess fluctuations and may

be attributed to a period of monetary change in the Federal Reserve Bank discussed by Hamilton

(1988). In contrast to Figure 1.2, the series appears to exhibit no drift component with the mean

very close to zero.

3.1 SUMMARY STATISTICS

The tables below exhibit descriptive statistics for the S&P500 index and T-Bill respectively. The

probability values are displayed in parenthesis. Jarque-Bera is a statistic to test whether the series

is normally distributed. The test measures the difference in the data's skewness and kurtosis from

those of the normal distribution. The formula is:

C (DE' − F'( = G�HI JK� + (L�M)N� O~P�� (1.4)

where S represents the skewness, K is kurtosis and k represents the number of estimated

coefficients used to create the series. A small probability value leads to a rejection of the null

hypothesis of normal distribution.

The Ljung-Box Q-statistic at lag k tests for the null hypothesis that there is no autocorrelation up

to order k. A lag k of 12 months is applied here.

QRS = ( + 2)∑ UVNW�X

HY0� (1.5)

Where ZY is the j-th autocorrelation and T is the number of observations.

Table 1.0 Descriptive Statistics for the S&P500 Index

Table 1.1 Descriptive Statistics for the Three-Month Treasury Bill

As suspected, stocks have higher average returns than bonds with a positive mean significantly

different from zero. A comparison of the variances reveals that interest rate returns are much

more volatile over the same period. Both sets of data exhibit negative skewness such that large

negative returns are more common than large positive ones. Note that the return on the T-Bill is

much more negatively skewed. This can be attributable to the negative spikes over the period

1980-1983.

Both stock and interest rate returns are strongly leptokurtic with a kurtosis significantly greater

than 3. This indicates that both series have an excessive number of observations in the tails

compared to the normal distribution (which has a kurtosis equal to 3). As expected, the Jarque-

Bera test for normality reveals that both data sets are non-normal.

Numerous studies including Mckenzie et al. (1999), and Brailsford and Faff (1993) have shown

that stock data exhibits volatility persistence and serial correlation between consecutive returns.

Hence it was surprising to see that our Q(12) statistic showed significant autocorrelation for

interest rate returns but rejected the autocorrelation between stock returns.

3.2 TESTS FOR STATIONARITY

A series is stationary if both the mean and variance of a process is constant over time. Hamilton

(1994) previously noted that financial time series data are typically non-stationary and contain a

unit root. However our returns series will need to be stationary in order for our model

assumptions to be applicable. We can establish whether a series is stationary by using two

common root tests to our return series.

The Augmented Dickey-Fuller (ADF) Test is based on the regression equation:

∆�� = \ + ] �� + ∑ ^#∆��# + _6 + ��,#0� (1.6)

If the coefficient p is significantly different from zero, than the null hypothesis that Y is non-

stationary is rejected. The test statistic is 6 = ]̀ a'̂(])⁄ where ]̀ is the Ordinary Least Squares (OLS) Estimator of p and a'̂(]) is its standard error. The criterion is to reject non-stationarity if 6defd < 6dg#�#def. Second is the Phillips-Perron (PP) Test, which uses the equation:

∆�� = \ + ]�� + _6 + �� (1.7)

Similar to 1.6, this test takes the difference of the series and identifies the statistical significance

of ]. The primary difference between the ADF and PP tests is the way each test deals with serial

correlation. The ADF includes lagged values of ∆��# while the PP test applies a Newey-West

Correction Procedure which is used to generate a consistent heteroscedasticity autocorrelation

estimate. For more information, refer to Phillips and Perron (1988).

The table below presents the test results for both returns. The 5% critical values are in

parenthesis. The tests included a trend and a constant with lag of 4.

Table 1.3 Unit Root Test for Stationarity

Note that the calculated t-statistics are substantially less than the critical values for both stock

and interest rates. Hence both tests clearly reject the null hypothesis of a unit root and indicate

that our returns are stationary.

3.3 MEAN EQUATIONS

Optimal inference requires the correct specification of the conditional mean and variance

equations. On the basis of the significance of the parameters using the t-statistic, we model the

S&P500 index return by a simple mean equation with no autoregressive terms. By doing this, we

illustrate that most of the movements in stock returns are primarily due to disturbances in the

residual rather than by past period returns. The Q-statistics reveal that the mean equation has

adequately modelled the serial correlation in the residuals. The Jarque-Bera statistic is highly

significant, indicating that the distribution of the residuals are non-normal - a fact which is

largely explained by strong leptokurtosis (kurtosis = 5.96).

Interest rates present a much more interesting analysis. The mean equation shows that interest

rates are affected by a negative trend and approximately 36% of last period's return. The constant

in the equation is insignificant and confirms our previous observation that the trend is very close

to zero. The Q-statistic of up to 4th lag indicates a marginal acceptance of no correlation while

the Q(12)-statistic demonstrates that there is a significant amount of correlation in the residuals

that are unaccounted for. We must note here that the Ljung-Box test loses power for higher order

lags. In addition, the statistics indicate that the residual distribution is clearly non-normal with an

extreme leptokurtosis (8.24) and negative skewness.

Table 1.4 Mean Equation Model Summary

3.4 SQUARED RETURNS

Before estimating our ARCH model, it is first appropriate to establish whether ARCH effects are

present in our data. Through visual inspection of the squared return in Figure 1.5 and Figure 1.6,

we find presence of volatility clustering, that is, shocks of similar magnitude of either sign are

followed by similar degree of shocks. For instance, large returns are clustered together around

the period 1975 while small movements in returns are evident around 1995.

3.5 TESTS FOR ARCH EFFECTS

A formal ARCH test is the Lagrange Multiplier (LM) test for autoregressive conditional

heteroscedasticity (ARCH) in the residual suggested by Engle (1982). This is a regression of

squared residuals on a constant and lagged squared residuals up to order q:

�� = �h + i∑ �Y��Y�/Y0� j + E� (1.8)

where E� is a white noise disturbance term. The null hypothesis is that there is no ARCH up to

lag q in the residuals. Table 1.5 presents the F-statistic for the joint significance of all lagged

residuals.

Table 1.5 Arch LM Test on the Mean Equations

The test reveals strong presence of up to the 12th order ARCH for our interest rate data. For

stock returns there is reasonable evidence of 1st order ARCH but less evidence of 4th and 12th

order ARCH. This suggests that our mean equation may have reasonably modelled the serial

correlation within the residuals.

3.6 ASYMMETRY TESTS

As noted in §2.0, GARCH models are symmetric in their response to news. Aside from

examining the skewness of the standardised residuals, we can conduct formal diagnostic tests

developed by Engle and Ng (I993), which focus on assessing the asymmetry of volatility in

response to past innovations. The test is based on dummy variables that identifies negative

innovations and includes (a) a sign bias test; (b) a negative size bias test; and (c) a positive size

bias test.

The tests are based on the OLS regression:

k�� = ∅h + ∅�K�� + ∅�K�� + ∅MK�m�� + E� (1.9)

where k� are the standardised residuals from the GARCH model and K��(K�m) is a dummy

variable that takes a value of one when �� is negative (positive) and zero otherwise. Here,E� is a disturbance term.

The sign bias test refers to the statistical significance of ∅�. If ∅� is found to be statistically non-zero then positive and negative shocks have different impacts on volatility. Similarly, the size

bias tests relate to the significance of ∅�(∅M) and if these are non-zero then large and small

negative (positive) innovations impact differently on volatility. The standard symmetric GARCH

models require ∅�, ∅�, and∅M to be jointly equal to zero. This hypothesis can be tested based on the Lagrange Multiplier Principle and is performed as . !�~PMfrom the estimation of Equation

1.9. The results of these diagnostic tests for our ARCH and GARCH models are presented in

Table 1.6. Note the null hypothesis .# equals zero so that our series is symmetric.

Table 1.6. Diagnostic Tests of Asymmetric Responses in Volatility

According to our tests, size bias is strongly insignificant for both sets of data. This implies that

shocks in the residuals of either sign have a symmetric impact on volatility. The test strongly

rejects positive size bias for stock returns but has marginally accepted negative size bias at

around the 10% critical level. The estimated coefficient of the variable that captures the negative

shock for ARCH(1) is equal to -0.017563, which means that large negative innovations causes

more volatility than can be explained by the above models. Existing literature like Glosten,

Jagannathan and Runkle (1993), and Engle and Ng (1993) all claimed significant asymmetry

effects in the stock indices, so it is quite surprising to see that leverage effects are only

marginally present here.

Strong size bias is evident for our Treasury-bill data with all size bias p-values (in parenthesis)

close to zero. Given that ARCH(3) and GARCH(1,1) models incorporate a symmetric response

to past innovations, it is not surprising that they both fail the positive and negative size bias test.

For our interest rate data, we can interpret that large innovations, whether positive or negative,

causes more volatility than small innovations. Note that the leverage explanation of Black (1976)

cannot be the only valid argument for this effect since it is based on a debt-to-equity ratio which

only holds true for stocks. The asymmetric feature of volatility is reinforced by the highly

significant joint statistic, indicating a rejection of the joint hypothesis of the absence of sign and

size biases.

The Engle and Ng (1993) test demonstrates that we should be able to estimate asymmetric

models for the data. We must note however that this test has low power and results should be

interpreted indicatively rather than conclusively.

The program Eviews applies the Berndt, Hall, Hall and Hausman (BHHH) algorithm of

maximum likelihood estimation to calculate the parameters. For details, refer to Berndt, Hall,

Hall and Hausman (1974). Given the inherent non-normality in the returns series, we will use

Bollerslev and Wooldridge’s (1992) robust standard errors to perform inference on the model

estimates. In a sense, our approach is "Quasi Maximum Likelihood".

4.0 ESTIMATES OF PARAMETRIC VOLATILITY MODELS

4.1 MODELS OF STOCK VOLATILTY

Various models were estimated with different lags and the four most appropriate for our S&P500

data were presented in Table 1.7. We restrict our estimations to a few lags since it is well known

that parsimonious models typically outperform more complicated models {see Hansen (2001)}.

Note that a double line separates the conditional mean and variance equations while the

parameters used are analogous to the equations displayed in §2.0. To determine the adequacy of

these models we initially check for positive and significant ARCH and GARCH coefficients in

the variance. Clearly, the ARCH(1) and TARCH(1,1) models have failed this test with an

insignificant �� value (p-value =0.2606) and a negative �� parameter (-0.0885) respectively.

The coefficients in our GARCH(1, I) model reveal that approximately 87% of the previous

period's volatility and 5% of past residuals affect the current period’s conditional variance. There

is significant evidence of asymmetry in the estimates of our EGARCH(2,1). This, coupled with

the previous evidence from §3.5, suggests that the GARCH(1,I) model is mis-specified. Another

constraint is for the sum of the coefficients to be less than unity. This restriction allows the

GARCH processes to be stationary so that shocks to the system are temporary rather than

permanent. All four models passed this test.

For the GARCH(1, 1) model, �� appears to be marginally insignificantly different from zero at

the 5% level. The ARCH term also presents problems for our asymmetric EGARCH(2,1) model

(with a p-value of 0.1165). The insignificance of the ARCH term in modelling stock index

returns has not been documented in history. This may be attributed to the fact that most of past

literature has concentrated on modelling daily and weekly returns which seem to exhibit stronger

ARCH effects. Despite this, the GARCH(1,1) and EGARCH(2,1) model appear to be the most

feasible set of symmetric and asymmetric models respectively for our monthly data. This is

consistent with evidence from Bollerslev, Chou and Kroner (1992) who indicate that the

majority of U.S studies found the GARCH(1,1) specification to be preferred.

Note that the log likelihood has moderately increased from the simple ARCH(1) model to the

more sophisticated asymmetric EGARCH(2,1) model. Although this suggests that the parameters

in the EGARCH(2,1) more closely resemble the data than the other models, a direct comparison

on the basis of Likelihood values is not possible given the difference in functional forms.

Two statistics often used for model selection are the Akaike Information Criterion (AIC) and

Schwarz Bayesian Criterion (SBC). They are computed as follows:

r�s = −2 tuv)wx* + 2D KFs = −2 tuv)wx* + Dtuv)�* (2.0)

where ML is the maximum likelihood value of the model, q is the number of independently

estimated parameters, and n is the sample size. The model selection criterion is to choose the

model with the lowest AIC and SBC value. By examining these statistics, we see that the

EGARCH(2,1) model generated the lowest AIC (-3.4502) and SBC (-3.3772) values and thus is

preferred.

However, Pagan and Schwert (1990) argued that the use of the AIC and SBC may be

inappropriate because these statistics are concerned with the first moment. Correct specification

of the GARCH and EGARCH models implies that the moment condition �(��) = �� be satisfied. This naturally lends itself to the diagnostic _h = 0 and _� = 1 in the following regression:

��̂� = _h + _��̀�� + '((u( (2.1) The test is wald-type from the estimation of Equation 2.1 and is distributed as P��. In Table 1.7, this test strongly accepts the joint hypothesis that _h = 0 and _� = 1 for all models with a

higher preference for the EGARCH(2,1) specification.

The Durbin Watson (DW) statistic tests for first order serial correlation within the residuals.

There is no serial correlation when DW = 2, however there is evidence of positive serial

correlation when 0 < DW < 2 and negative correlation when 2 < DW < 4. The DW statistic for

all models is marginally below 2, suggesting that serially correlation is not evidently present in

the residuals.

Recall from §2.0 that our assumption inferred that the standardised residuals are distributed

normally with zero mean and unit variance, ie k� = ��̂ �̀�⁄ ~�(0,1). We can examine descriptive

statistics of the standardised residuals to assess the adequacy of this assumption. The results of

this test favoured the EGARCH(2,1) model because it showed the least negative skewness and

kurtosis. The mean and variance of all standardised residuals are close to zero and one

respectively. Nevertheless, they are strongly non-normal (as shown by the Jarque-Bera test) with

extreme negative skewness, and leptokurtosis (a minimum kurtosis = 4.306).

A satisfactory model must be able to account for most of the serial correlation in the residuals.

The Q-statistics and Q2-statistics clearly accept the hypothesis of no serial correlation at all

levels of confidence. The ARCH tests strongly rejects the presence of further ARCH effects.

Hence we can safely conclude that our models have adequately accounted for any serial

correlation among the residuals. Note that when our mean equations were modelled in §3.3, the

Q-statistics had already indicated no evidence of serial correlation. Therefore, the conditional

variance may have played only a minor role in the improvement of our characterisation.

Table 1.7. Estimated volatility models for the Return on the SP500 Index

4.2 MODELS OF INTEREST RATE VOLATILITY

Various models of different lags were fitted to the return on the 3-month Treasury Bill and those

with the closest fit were presented in Table 1.8. First, note that all parameters in the conditional

variance equations are significant apart from the 3 coefficient in the TARCH(1,l) (with p-value = 0.2386) and EGARCH(1,l) (p-value= 0.1448) models. Although evidence of asymmetry was

present in our data, these effects were rejected by the t-statistics. In time, further investigations

led to an improved asymmetric model - the EGARCH(1,2). Here the 3 coefficient is -0.157144, indicating a leverage effect on the conditional variance of approximately 0.85('�h.�yz��) times.

The negative sign suggests that a larger volatility response is generated from a negative shock

than a positive shock. Notice that this value is significant at the 10% critical level (p-value =

0.0821). Another alternative is the GARCH(1,l) with all parameters significant. It implies that

this period's fluctuations are influenced by 73% of last period's volatility and 26% of last

period’s squared errors.

The Log Likelihood and AIC statistics further reinforces the superiority of the EGARCH(1,2)

model. Note that the Wald Test rejected our assumption of �(��) = �� for all models but has

given preference to the EGARCH(1,2) specification. On the other hand, the SBC statistic

suggests a conflicting opinion by indicating a lower SBC for the GARCH(1,1). Due to this, we

will now need to examine the residuals to improve the closeness of fit with the models.

Despite failing the normality tests with insignificant Jarque-Bera statistics, all models behaved

reasonably well with the mean of the standardised residuals not significantly different from zero

while the standard deviations were very close to unity. However, the skewness and excess

kurtosis should have been smaller in absolute size. The preferred model for this test is the

EGARCH(I,2) which has the smallest negative skewness (-0.008) and kurtosis (3.685).

Furthermore, these models show a stronger conformity to our normality assumptions than the

stock models in Table 1.7.

The Q-statistics rejects the presence of serially correlation among the residuals and marginally

rejects the 12th lag for most models which is a cause for concern. To eliminate this problem,

perhaps we could include moving average terms. Engle's (1982) ARCH tests of the 4th and 12th

order clearly show that most of the ARCH effects have been accounted for. We are unable to

differentiate between the GARCH(I,I) and EGARCH(1,2) models from these tests because they

contain approximately the same results. Nevertheless, on an overall basis, the EGARCH(1,2) is

favoured because it accounts for the asymmetry within the volatility of the residuals.

Table 1.8 Volatility models for the Return on 3 Month Treasury-Bill

4.3 ESTIMATED NEWS IMPACT CURVE

One can plot �� against �� on any volatility model to illustrate how the volatility reacts to the

difference between the realised and expected return. This plot allows a visual inspection to the

extent to which models facilitate the asymmetry response to news. It also characterises the

differences among the various specifications of volatility. This method was introduced by Pagan

and Schwert (1990), and later named the News Impact Curve (NIC) by Engle and Ng (1993).

Suppose that the information at and before time (t-1) is held constant.

The NIC of the GARCH(l,l) model can be graphed as:

�� = s + �� where C is a constant and s = + + .��

For the EGARCH models, we will plot the variance �� against k� = �� ⁄ for simplicity.

The equations are as follows:

EGARCH(2,1) : tuv�� = s + ��|k��| + ��|k��| + 3�k�� + 3�k�� where s = + + .�tuv��

EGARCH(1,2): tuv�� = s + ��|k��| + 3�k�� where s = + + .�tuv�� + .�tuv��

Table 1.9 presents the news impact curves calculated for our selected four models. Figure 1.5

demonstrates that the GARCH(1,1) NIC is symmetric and centred at �� = 0. Comparing

Figure 1.5 and 1.7 we see that the GARCH(1,1) specification on the Treasury Bill has a much

larger response to volatility (about five-fold) from changes in the previous residuals than the

GARCH(1,1) specification for stocks.

The news impact curves for the EGARCH models are also centred at �� = 0 but contain a steeper slope when �� < 0. The EGARCH(2,1) NIC indicates that a negative shock will have an impact on stock volatility roughly equal to 2.6 times the impact of a positive shock. This

appears reasonably consistent with the leverage theory presented in §2.0. On the other hand, the

NIC of the EGARCH(1,2) model seems extreme in that positive values of �� have only a minute effect on the variance relative to negative innovations. This suggests that the

EGARCH(1,2) model is heavy in the tails and may be an inadequate characterisation of interest

rates. This fact was also described by Henry (1998) who stated that the EGARCH model can be

overly sensitive to extremely large positive and negative shocks.

Note that the NIC for the EGARCH is plotted in different units to the GARCH specification.

However, by comparing them we can decipher that the GARCH(1,1) would underestimate large

negative shocks and overestimate large positive shocks. Meanwhile, the EGARCH(1,2) model

may reflect excessive fluctuations in volatility for large negative innovations.

Table 1.9 Estimated News Impact curves

Further Asymmetric tests shown in Table 2.0 suggests that the EGARCH(2,1) specification has

successfully modelled the asymmetry of stock volatility with tests adequately accepting the

hypothesis that the coefficients are indifferent from zero. On the contrary, a substantial amount

of size asymmetry remains with our EGARCH(1,2) model, indicating that large and small

negative (positive) innovations continue to impact differently on volatility. We can account for

this by observing Figure 1.8. Notice that large positive innovations have a minor influence on

volatility compared with the excessive impact of large negative innovations. This extreme

imbalance between positive and negative innovations on volatility may be impractical for our

interest rate data. Nevertheless, our EGARCH(1,2) proved significant in modelling part of the

asymmetry as shown by the increased p-values relative to Table 1.6.

Table 2.0 Final Tests for Further Asymmetric Responses

5.0 THE RELATIONSHIP BETWEEN STOCKS & INTEREST RATES

In §1.0 numerous studies confirmed that strong correlations exists between stocks and interest

rates. In this section we will attempt to statistically detect whether there is causality between our

data, that is, whether one market causes the other to change. By combining the returns of our two

series, we can visually interpret these findings. Figure 1.9 displays a moderate trend between

stock and interest rates. Fluctuations from both the S&P500 and 3-month Treasury bill returns lie

relatively in line with one another. Notice also that there are many extreme outliers, which may

be the result of various macroeconomic shocks such as changes in monetary policy, recessions,

speculative bubbles and oil price shocks.

Figure 1.9 Volatility of T-Bill returns relative to S&P500 returns

5.1 SENSITIVITY OF INTEREST RATES ON STOCK VOLATILITY

To formally assess whether there is any effect of interest rates on stock volatility we place the

lagged level of the 3-month Treasury-Bill return as an exogenous regressor within the

conditional mean and variance of our final stock models. The T-Bill rate is correlated with the

cost of borrowing to firms and thus may carry information that is relevant to the volatility of the

S&P 500 index. Note that an absolute value of the T-bill regressor is applied in the conditional

variance of the GARCH(1, 1) model to avoid a negative volatility.

The results from Table 2.1 reveal the significance of the T-Bill regressor in the mean equation of

both the GARCH(1,1) and EGARCH(2,1) with coefficients of -0.ll04 and -0.0717 respectively.

This indicates that information driving individual stock and bond prices tend to be dominated by

the information about the mean value of the underlying asset rather than by information about

their variances. The value of the coefficients suggests that stock returns are negatively affected

by interest rates from the previous period by around 10%. A possible reason for this is that as

interest rates rise, consumer spending drops and retail sales begin to slow, leading to decreased

corporate profits and a declining stock market. The effect of this decline is compounded by the

fact that investors often borrow to finance their position. Higher interest rates reduce borrowing

and makes interest-bearing investments more attractive, causing an exodus of money from the

stockmarket and a decline in prices. This is also consistent with the study by Breen, Glosten and

Jagannathan (1989) who demonstrated a negative relation between short-term interest rates and

stock index returns.

The impact of the T-bill rate on the volatility process of the S&P500 return is small, but quite

significant. Although the t-statistic has rejected the variance coefficient in the EGARCH(2,1)

specification (p-value = 0.1645), the positivity of this parameter in both models indicate that

high interest rates are generally associated with high levels of equity return volatility. This result

confirms a variety of studies such as Campbell (1987) and Glosten, Jagannathan, and Runkle

(1993) who documented a positive relation between stock volatility and the level of short-term

interest rates. Fleming et al. (1998) suggested an explanation whereby volatility linkages exist

between these two markets due to information flow that affects expectations in both markets,

such as news about inflation, which is often proxied by the short-term interest rate. The second

source of volatility linkage comes from information spillover caused by cross-market hedging.

When information alters expectations in one market, traders will adjust their holdings across all

markets causing the higher volatility.

Results from the Log Likelihood, Wald and AIC tests suggest an improvement in our

characterisation relative to the models presented in Table 1.7. The DW test is closer to 2, while

the AIC and SBC criterions are lower than previously revealed. Although the normality test once

again failed, serial correlation in the residuals has been adequately accounted for. These results

suggest that our previous models in Table 1.7 were mis-specified.

Table 2.1 Effect of Interest rates on the SP500 Index

5.2 EFFECT OF STOCKS ON INTEREST RATE VOLATILITY

Various regressions were performed with the lagged return of the S&P 500 index applied to both

the conditional mean and variance of the interest rate models. We find that stocks returns do not

directly affect the T-Bill rate and therefore, is insignificant in the mean equation. Despite this,

we discover that stocks have a minor influence on the volatility of our EGARCH models. Table

2.2 present the results.

Firstly, observe that the coefficient for the absolute value of the S&P 500 index return in both the

GARCH and TARCH models are insignificant with values close to zero. This finding is

consistent with results from Scruggs and Glabadanidis (2001) who reported that bond market

variance is relatively unresponsive to stock return shocks. On the other hand, our EGARCH

representations indicate that stocks have a minor effect on interest rate volatility equal to a factor

of approximately 7% (calculated as '��.Iz). This is a relatively new discovery and may be

attributed to the fact that negative stock return shocks typically lead to higher demands for bonds

and thus raises bond prices and volatility. Another explanation can be attributed to the effect of

similar expectations in both markets from common information, indicated by Fleming et al.

(1998).

Although it appears that this result contradicts Scruggs and Glabadanidis (2001), we must

understand that covariances between assets changes are higher (lower) at times of high (low)

volatility (This fact was indicated by Goeij and Leuven (2001)). Given that our data type and

time frame are different from Scruggs and Glabadanidis, we would expect to obtain varied

results.

Table 2.2 Effect of stock on interest rate volatility

The EGARCH models have reasonably improved with the input of lagged stock returns into the

variance specification. Notice that both 3 coefficients, especially for the EGARCH(1,1), have become increasingly significant relative to Table 1.8. This suggests that interest rates have a

minor asymmetric effect on fluctuations in stocks which cannot be accounted for by single

GARCH(1,1) specifications, and may be due to similar investor expectations from common

information revealed in both markets.

The suitability of these EGARCH models is reinforced by a higher Log Likelihood value and

lower AIC criteria than previously stated. The only drawback comes from information revealed

by statistics of the standardised residuals for our EGARCH(1,2) model. Although the mean and

variance are reasonable close to zero and one respectively, the addition of the lagged stock

parameter caused an increase in negative skewness (-0.028) and excess kurtosis (3.76) by a

fraction. This resulted in stronger acceptance of non-normality. Note that our inference is robust

to this departure from conditional normality, due to the "Quasi-Maximum Likelihood" correction

to the variance-covariance matrix of the parameters. Nonetheless, the presence of serial

correlation within the residuals is similar to Table 1.8, with Q-statistics remaining fairly stable,

marginally accepting no correlation at the 5% level.

This approach in assessing the significance of stock (interest rate) returns in the conditional

mean and volatility of interest rates (stocks) is limited by the fact that it does not provide

information on the magnitude of the correlation. Nor does it offer any information regarding

whether the covariances exhibit asymmetric effects. More sophisticated modelling techniques

have been recently proposed to capture these traits, and will be discussed in §7.0.

6.0 SUMMARY AND CONCLUSIONS

Historically, the majority of U.S studies, as outlined by Bollerslev, Chou and Kroner (1992) have

favoured the GARCH(1,1) specification. While this model parsimoniously characterises both

stock and interest rate data, it neglects to capture an important feature of volatility - the

asymmetric responses to past innovations. Our application of the news impact curve reveal that

the GARCH(1, 1) specification tends to underestimate (overestimate) the conditional variance

for large negative (positive) shocks. We discover that the EGARCH specifications were able to

adequately characterise a portion of this leverage effect, but similarly to Henry (1998), we also

find that the EGARCH model is restricted in that it is overly sensitive to extreme negative

shocks.

Our investigation into the relation between stock and interest rates supports the result of Breen,

Glosten and Jagannathan (1989). We find a negative relation between short-term interest rates

and stock returns. In other words, rises in interest rates tend to be associated with reductions in

stock returns, assuming all other variables are constant. This information is reflected in the

determination of security prices as indicated by the Efficient Market Hypothesis. In addition, we

find that interest rates shocks have a positive impact (to a lesser extent) on conditional stock

volatility. Hence by examining the volatility of interest rates, it is possible to forecast future

stock return volatility to a limited degree. As indicated in §1.0, these results have important

implications for asset allocation and risk management strategies.

Perhaps more interesting is our analysis on whether stock return shocks have an impact on

interest rate volatility. Unlike Scruggs and Glabadanidis (2001), we find that movements in

interest rates have an asymmetric response to stock return shocks. This is a relatively new result

and can be attributed to the relation between volatility and common information flows by Ross

(1989).

Furthermore, the improvement of our univariate models by the inclusion of an exogenous

variable suggests that our original models were mis-specified. In order for us to obtain

reasonably accurate forecasts of volatility changes, we may need to account for intertemporal

dependencies and causality across markets.

7.0 EXTENSIONS

Although ARCH models have been quite successful in capturing the volatility clustering and

heteroscedasticity characteristic exhibited in financial time series data, they do not accurately

describe the leptokurtosis element very well, as indicated in this paper with most of our models

rejecting the normality assumption. We can extend the GARCH models by specifying

conditionally t-distributed errors as done by Bollerslev (1987). The t-distributed series contains a

larger proportion of outliers (fat-tails), and hence is able to characterise the data more

effectively.

Another extension is to allow for seasonal effects by incorporating dummy variables in our mean

equation. This accounts for any predictable influences such as holidays and non-synchronous

trading in the return series.

Finally, the interactions between stock and interest rate volatility can be more thoroughly

captured by Multivariate GARCH models. These models specify the error term as a column

vector and describe the relationship between the variances by a conditional covariance matrix. In

doing so, they are able to portray the way in which correlation varies, as well as whether

asymmetric effects are present in covariances. For further details, refer to the recent study by

Goeij and Leuven (2001).

8.0 REFERENCES

Baillie, R.T., T. Bollerslev, (1998), "Answering the Skeptics: Yes, Standard Volatility Models

Do Provide Accurate Forecasts", International Economic Review, 39, pp. 885- 905

Berndt, E.K, B.H. Hall, R.E. Hall, J.A. Hausman, (1974), "Estimation and Inference in Nonlinear

Structural Models," Annals of Economic and Society Measurement 3, pp. 653-665.

Black, F. (1976), "Studies of Stock Price Volatility Changes: Proceedings of the 1976 Business

Meeting of the Business and Economics Statistics Section", American Statistical Association, pp.

177-181

Bollerslev, T., (1986), "Generalized Autoregressive Conditional Heteroscedasticity", Journal of

Econometrics, 31, pp. 307-327.

Bollerslev, T., R.Y. Chou, K. Kroner, (1992), "ARCH Modelling in Finance: A Review of the

theory & Empirical Evidence", Journal of Econometrics, 52, pp. 61-90

Bollerslev, T., J. Wooldridge, (1992), "Quasi-maximum Likelihood Estimation and Inference in

Dynamic Models with Time Varying Covariances", Econometric Reviews, 11, pp. 143-172

Brailsford, T.J., R.W. Faff, (1993) "Modelling Australian Stock Market Volatility", Australian

Journal of Management, 18,2, pp. 109-132

Breen, W., L.R. Glosten and R. Jagannathan (1989), "Predictable Variations on Stock Index

Returns", Journal of Financial Economics, 31, pp. 281-318

Brenner, R.I, R.H Harjes, K.F Kroner, (1996), "Another Look at Models of the 8hortTerm

Interest Rate", Journal of Financial and Quantitative Analysis, 31, no.1, pp. 85- 107

Campbell, J., (1987), "Stock Returns and the Term Structure", Journal of Financial Economics,

18, pp. 373-399

Campbell, J., L. Hentschel, (1992), "No News is Good News: An Asymmetric Model of

Changing Volatility in Stock Returns", Journal of Financial Economics, 31, pp. 218- 318

Chou, R., Yeutien, (1988), "Volatility Persistence and Stock Valuations: Some Empirical

Evidence using GARCH", Journal of Applied Econometrics, 3, pp. 279-294

Christie, A.A (1982), "The Stochastic Behaviour of Common Stock Variances: Value, Leverage

and Interest Rate Effects", Journal of Financial Economics, 10, pp. 407-432

Dickey, D., W. Fuller, (1981), "Likelihood Ratio Statistics for Autoregressive Time Series with a

Unit Root", Econometrica, 49, pp. 1057-1072

Enders, W., (1995), "Testing for Trends and Unit Roots", Chapter 4, Applied Econometric Time

Series, Wiley Publishing.

EngleR.F, D. Lillien and R. Robins (1987), "Estimating Time Varying Risk Premia in the Term

Structure: The ARCH-M model, Econometrica, 55, pp. 391-408

Engle, R.F., V.K. Ng, (1993), "Measuring and Testing the Impact of News on Volatility",

Journal of Finance, 48, pp. 1749-1778

Engle, R.F., A.J. Patton., (2001), "What Good is a Volatility Model?", Manuscript at Stem,

NYU, http://weber.ucsd.edul-mbacci/engle/403.PDF

Engle, Robert F., (1982), "Autoregressive Conditional Heteroscedasticity with Estimates of the

Variance of the United Kingdom Inflation", Econometrica, 50, pp. 987 -1007.

Eviews 4.0 Users Guide, (2000), Quantitative Micro Software, LLC, USA

Fama, F. Eugene, (1965), "The Behaviour of Stock-Market Prices", Journal of Business, 38(1),

pp. 34-105

French, K., G. Schwert, R. Stambaugh, (1987), "Expected Stock Returns and Volatility", Journal

of Financial Economics, 19, pp. 3-29

Fischer, S., (1981), "Relative Shocks, Relative Price Variability, and Inflation", Brookings

Papers on economic Activity, 2, pp. 381-431.

Fleming, J., C. Kirby, B. Ostdiek, (1998), "Information & Volatility Linkages in the Stock,

Bond, and Money Markets", Journal of Financial Economics, 49, pp. 111-137

Glosten, 1., R. Jaganathan, and D. Runkle, (1993), "On the Relation Between the Expected

Value and Volatility of the Nominal Excess Return on Stocks", Journal of Finance, 48, pp. 1779-

1801.

Goeij, P., K.U. Leuven, (2001)" Modelling the Conditional Covariance between Stock and Bond

Returns: A Multivariate GARCH Approach", Draft Version,

http://www.econ.kuleuven.ac.be/ew/academic/econrnetrlPaperslDownload.htm

Hamilton, J. D., (1988), "Rational Expectations Econometric Analysis of Changes in Regime;

An Investigation of the Term Structure of Interest Rates", Journal of Economic Dynamics and

Control, 12, pp. 231-54

Hansen, P.R, (2001), "A Forecast Comparison of Volatility Models: Does Anything Beat a

GARCH(I,I)?", Working Paper No. 01-04, http://www.cls.dklcaflasger.pdf

Harris, L., (1989), "S&P 500 Cash Stock Price Volatilities", Journal of Finance, Vol. XLIV,

No.5, pp. 1155-1157

Henry, O., (1998), "Modelling the Asymmetry of Stock Market Volatility, Applied Financial

Economics", 8, pp. 145-53

Henry, O., (1999), "The Volatility of US Term Structure Term Premia 1952-1991", Applied

Financial Economics, 9, pp. 263-271

Henry, O., (1999), "Univariate Arch Models: Modelling Asset Return Volatility", Lecture Notes.

Keirn, D.B, R.F. Stambaugh, (1986), "Predicting Returns in the Stock and Bond Markets",

Journal of Financial Economics, 17, pp. 357-390

Kwan, S.H., (1996) "Firm-specific Information and the Correlation Between Individual Stock

and Bonds", Journal of Financial Economics, 40, pp. 63-80

Lee, W., (1997), "Market Timing and Short-Term Interest Rates", Journal of Portfolio

Management, 23, no. 3, pp. 35-46

Mandelbrot, B. (1963), "The Variation of Certain Speculative Prices", Journal of Business, 36,

pp. 394-419

McKenzie, M.D., R.D. Brooks, (1999), "Research Designs Issues in Time Series Modelling of

Financial Market Volatility", IrwinlMcGraw-Hill, Sydney NSW

Nelson, D.B, (1991), "Conditional Heteroscedasticity III Asset Returns: A New Approach",

Econometrica, 59, pp. 347-370

Pagan, A.R., G.W. Schwert, (1990), "Alternative Models for Conditional Stock Volatility",

Journal of Econometrics, 45, pp. 267-290

Park, H.S., (1999), "ForecastingThree-Month Treasury Bills using ARIMA and GARCH

Models", Term Paper 1999-11, Kansas State University http://www.personal.ksu.edul-

hsp3704/wpI99911.pdf Phillips, P., P. Perron, (1988), "Testing for a Unit Root in Time Series

Regression",

Biometrika, 75, pp. 335-346 Pindyck, R.S., D.L. Rubinfield, (1998), "Econometric Models and

Economic Forecasts", Published: New York, Irwin, McGraw-Hill

Ross, S.A (1989), " Information and Volatility: The No-Arbitrage Martingale Approach to

Timing and Resolution Irrelevancy, Journal of Finance, 44, pp. 1-17

Rossi, Peter E., Editor, (1996), "Modelling Stock Market Volatility - Bridging the Gap to

Continuous Time", Academic Press, Chicago, Various contributors.

Schwert, G.W., (1989), "Why does Stock Market Volatility Change over TimeT', Journal of

Finance, 44, pp. 1115-1153

Schott, L.O., (1997), "Pricing Stock Options in a Jurnp Diffusion Model with Stochastic

volatility and Interest rates", Journal of Mathematical Finance, 7, no. 4, pp. 413-424

Scruggs, J.T., P. Gabadanidis, (2001), "Risk Premia and the Dynamic Covariance Between Stock

and Bond Returns" Draft Version, www.olin.wustl.edulfaculty /scruggs/papers.htrn

Spiro, P.S., (1990), "The Impact ofInterest Rate Changes on Stock Price Volatility", Journal of

Portfolio Management, 16, no.2, pp. 63-68

Veronosi, P., (1999), "Stock Market Overreaction to Bad News in Good Times: A Rational

Expectations Equilibrium Model", Review of Financial Studies, 12, pp. 975- 1007

Zakoian, J. M., (1994), "Threshold Heteroskedastic Models", Journal of Economic Dynamics

and Control, 18, pp. 931-955

modelling the interaction between stocks and interest rates

Documents