Rational Speculators And Equity Volatility as a Measure of
Ex Ante Risk
Amir Kia
Emory University, Department of Economics
Atlanta, GA 30322-2240
U.S.A.
E-mail: akia@emory.edu
Tel.: (404) 727-7536
Fax: (404) 727-4639
March 2001
Rational Speculators And Equity Volatility as a Measure of
Ex Ante Risk
Abstract:
A macro-determinant model of stock return volatility using monthly data on
Canadian and U.S. markets is estimated. If volatility is a measure of risk, one should
verify whether the estimated volatility is a measure of future (ex ante) risk or realized
(ex post) risk. When speculators are rational the volatility created by their activities can
be a measure for ex ante risk. It is found that the growth of commodity price, which was
ignored in this literature, is also a factor in the equity return volatility. Furthermore, 71%
of the volatility of Canadian stock returns is explained by the behavior of rational
speculators.
Key words: stock return, forward-looking agents, superexogeneity, ex post and ex ante
risk
JEL classification = G120, G140 and G190
Rational Speculators And Equity Volatility as a Measure of
Ex Ante Risk
1. Introduction
High volatility implies higher capital costs and may also increase the value of the
“option to wait”, hence delaying investments. When making decisions, investors
incorporate information pertaining to price movements and volatility in the asset they are
trading. Since risk is directly related to volatility it is interesting to verify how much of
the estimated risk reflects the behavior of rational speculators who use, e.g., observable
fluctuations of macro-variables to forecast future volatility of dividend payments. If an
essential part of return volatility is due to activities of rational (i.e., forward-looking)
speculators then the estimated volatility can be a measure of future (ex ante) risk.
Alternatively, if speculators are backward looking then clearly the volatility created by
their activities can be a measure of the realized (ex post) risk. Unless we assume the
volatility of the return remains the same in the next holding period we cannot use the
estimated volatility as a measure of risk for the current holding of stocks or any other
assets.
Let us define rational speculators in equity markets as those investors whose
investment activities are based on changes in the fundamental economic factors (e.g.,
macro-variables) that are used in the formation of their expectations on the future stream
of dividend payments, and so, are forward looking. All other speculators in equity
markets are then mostly noise traders. Thus the volatility generated by the rational (i.e.,
forward-looking) speculative activities is a measure of the future risk of holding equities
(ex ante) and not realized risk (ex post). Realization of the ex ante risk is important to
2
both portfolio managers and policy makers. The investment decision of portfolio
managers depends on the (ex ante) expected risk-adjusted return of different assets.
If a major part of the volatility of equity return is due to rational speculative
activities then monetary policies, e.g., aimed to keep the inflation rate within a target
band, will also help to provide both stable and less volatile macroeconomic conditions as
well as asset prices. Namely, these monetary policies, if successful, can also help to
avoid, or at least to reduce, the historically relevant risk that a bubble, once out of its
norm, can easily degenerate into a panic. However, if movements in the equity return can
be explained mostly by waves of irrational optimism and pessimism, “fads”, noise trading
or feedback trading then such monetary policies help to provide stable macroeconomic
conditions, but not stable asset prices.
In a world of efficient capital market and without regulatory distortion, and in the
absence of excessive speculative activities, movements in equity prices simply reflect
changes in underlying economic fundamentals. In such world equity prices would be of
interest to central bankers to the extent that they provide useful information about the
current and future states of the economy if the estimated volatility reflects the behavior of
rational speculators. Consequently, central bankers would have no reason to concern
themselves with equity price volatility per se if a dominant part of the observed volatility
in equity return is due to rational speculative activities. However, as also mentioned by
Bernanke and Gertler (1999), matters would change if (i) “nonfundamental” factors
dominate asset market volatility and (ii) changes in asset prices unrelated to fundamental
factors have potentially significant impacts on the rest of the economy. If these two
conditions are satisfied, then equity price volatility becomes an important source of
3
economic instability of which policy makers should take account. In sum, it is very
important to verify whether the dominant part of the estimated volatility reflects the
behavior of rational speculators. Such information is useful to portfolio managers in their
estimation of risk-adjusted expected return and the policy makers in the estimation of the
future state of the economy.
To the best of my knowledge no attention has been given so far to these important
facts. For instance, Schwert (1989) and Culter, Poterba, and Summers (1989) find that
unexpected volatility in current macroeconomic variables explains only about one-fifth of
the variance in monthly stock returns. Fama (1990) and Schwert (1990) show real and
financial variables explain about half of the variance in annual returns; Gallant, Rossi and
Tauchen (1992) find a positive and non-linear relationship between volume and volatility
of the S&P 500-stock index; Kearney (1998) finds 38% of the conditional volatility of
Irish stock return can be explained by the conditional volatility of macro-variables.
Kearney and Daly (1998) find 35% of the variation in the conditional volatility of
Australian stock market is explained by the financial and business cycle variations.
Kearney (2000) examines how volatility is transmitted to national stock markets from
their international counterparts as well as from domestic business cycles variables for
Britain, France, Germany, Japan and the U.S. He finds the national stock market indices
are multivariate cointegrated with their domestic business cycle variables. None of these
studies investigates if the dominant part of the observed volatility constitutes the activity
of the forward-looking rational agents in the market, namely, if the equities volatility can
be a measure for future (ex ante) risk rather than a realized (ex post) risk.
4
Furthermore, to the best of my knowledge, not much attention was given to
resource-dependent internationally integrated stock markets volatility. For example, in
none of the existing literature, the impact of the volatility of commodity prices is
incorporated. Commodity price is an important determinant of the stock price in
resource-dependent internationally integrated markets. The purpose of this paper is,
taking into consideration the above facts, to extend the literature in the following way.
Monthly data on Canadian and U.S. stock markets for the period 1975(Jan.)1999(Dec.) are used. A macro-determinant stock price model for a small and
open-resource-oriented country is developed. The model is used to derive a conditional
stock volatility model, capable to incorporate the volatility of macro-variables, including
commodity price, which are used to forecast the volatility of future dividend payments.
The model is estimated and the superexogeneity tests are used to examine whether the
estimated volatility reflects the behavior of speculators who are forward looking and so
their expectations are formed rationally, i.e., if the estimated volatility can be a measure
of future (ex ante) risk or it is only a measure of realized (ex post) risk. It is found that the
conditional volatility of growth of commodity price index, along with other conditional
volatility of macro-determinants, is in fact an important component of the conditional
volatility of stock return determination. Furthermore, it was found that 71% of the
volatility of Canadian stock returns is explained by the behavior of forward-looking
rational speculators which, consequently, can be used to estimate ex ante risk. Namely,
noise traders’ participation in Canadian stock return volatility is only about one-fourth of
the variation in monthly stock returns.
5
The following section deals with the development of the theoretical model as well
as superexogeneity and invariance hypothesis for conditional volatility of stock returns.
Section 3 describes the data and empirical methodology and results. Section 4 is devoted
to marginal models as well as superexogeneity results. The final section provides some
concluding remarks.
2. The theoretical model
In an exchange-economy asset-pricing model, Lucas (1978) finds the equilibrium
price of an asset is the expected, discounted, present value of its real dividend stream,
conditional on current information. A close approximation of the model is
Pt = Et [(1+rt)-1(Pt+1 + Dt+1)],
(1)
where Pt is the stock price at time t, Dt+1 is the dividend paid to the shareholders between t
and t+1, 0<(1+rt)-1<1 is the discount factor and E denotes the mathematical expectation
operator for information at time t. If the transversality condition, i.e.,
limn→∞ Et[(1+rt-n)-n (Pt+n)]=0, holds then the unique solution to Equation (1) is
∞
Pt = Et [ ∑ (1 + rt+i-1 ) -i (Dt+i) ] .
i=1
(2)
Future dividends as well as discount rates are not observable. Following Lucas (1978)
and under rational expectations assumption, the market clearing price Pt is a fixed
function of the state of the economy, yt, where yt is assumed to summarize all relevant
information on the current and future physical state of the economy.
If macroeconomic data provide all relevant information on the current and future
physical state of the economy, we can assume that in a small, open and financially
6
integrated economy, the discounted expected value of future payments has the following
function:
∞
Et [ ∑ (1 + rt+i-1 ) -i (Dt+i) ] = f(yt) = F(SPt, IPt, EXt, COMPt, PLt, Rt, PREMt,
i=1
Rt-FRt, PLt/FPLt, IPt/FIPt, zt),
(3)
where SPt is the foreign country stock price, IPt is the level of total industrial production,
EXt is the level of exchange rate (domestic currency value of a unit of foreign currency),
COMPt is the commodity price index (commodities produced in the domestic country
and sold abroad), PLt is the domestic price level, Rt is the domestic overnight interest
rate, PREMt is a measure for the risk premium (corporate paper minus Treasury Bill
rates), Rt-FRt is the domestic and foreign interest rates differential, PLt/FPLt is the ratio of
domestic to foreign price levels, IPt/FIPt is the ratio of domestic to foreign levels of total
industrial production, and zt includes past values of Pt and above-mentioned
macroeconomic variables as well as current and past values of other valid conditioning
variables.1
Substituting (3) in (2) yields
Pt = F(SPt, IPt, EXt, COMPt, PLt, Rt, PREMt, Rt-FRt, PLt/FPLt, IPt/FIPt, zt).(4)
Equation (4) is an extension of the literature in the sense that it includes the
commodity price index, the domestic price level relative to the foreign price level as well
as a measure for the risk premium, which reflects, among other things, attitudes of
1
Note that, as Koutoulas and Kryzanowski (1996) also mention, there is no generally accepted theory for
linking stock returns to the economy. Consequently, general economic theory and intuition have been the
main inputs in the selection of macro-variables.
7
investors toward risk. In a small, open and financially integrated economy one would
expect stock markets to rally as stock markets in the rest of the world or its largest trading
partner rallies. For example, Canadian equity prices may move with U.S. equity prices as
economic news in the U.S. may also affect Canadian equity markets.2
Furthermore, market psychology and speculative activities could lead stock
markets in Canada to move with the stock markets in the U.S. For example, a rally in the
U.S. stock markets and so higher equity prices will increase expectations of a higher
demand for Canadian exports through the wealth effect in the U.S. Expectations of higher
demand for Canadian exports lead to expectations of higher corporate profits in Canada.
This will increase demand for equity in Canada and result in higher equity prices.
Consequently, one would expect theoretically the sign of SPt in Equation (4) to be
positive. It should be noted that it is more appropriate for the foreign stock price to be
adjusted for the exchange rate. However, to avoid the introduction of an extra stochastic
variable in the system, I follow Kearney (1998) and Koutoulas and Kryzanowski (1996)
and allow the exchange rate to appear as a separate variable in the equation.3
Moreover, in a small open economy with high capital mobility, cash flows, Dt+i,
as mentioned and shown by Kearney (1998) and others (see reference list given in
Kearney, 1998), are influenced by developments in the domestic macroeconomic
variables like the total level of industrial production, the price level, the exchange rate
and interest rates. For instance, a higher level of industrial production is associated with
2
Note that the U.S. is Canada’s major trading partner. For example, in 1998, about 82% of Canadian
exports were sent to the U.S. while about 78% of Canadian imports originated from the U.S.
3
This argument is also applicable for variables Rt-FRt, PLt/FPLt and IPt/FIPt.
8
higher corporate profits and cash flows. Consequently, it is expected for the industrial
production theoretically to have a positive relationship with the price of equity.
Furthermore, a higher exchange rate, which is associated with a lower foreign
currency price of the exports, results in an appreciation of the balance of trade, which
leads to higher corporate profits and cash flows. Consequently, a higher exchange rate is
expected to lead to a higher equity price. It should, of course, be mentioned that exchange
rate appreciation could result in an improvement of the balance of trade if the MarshallLerner condition, a long-run condition, is satisfied. However, in the short term, investors
associate the deterioration of the domestic currency with a higher likelihood that the
central bank would raise the interest rate to defend the currency. Consequently, the short
run impact of the exchange rate appreciation on the equity price may be negative.
The weakness in the commodity price does hurt stock markets of a highly open
economy like Canada.4 Such a weakness can reduce corporate profits and lower
expectations of future cash flows. Consequently, one would expect a positive relationship
between the commodity and the stock prices. To the best of my knowledge, no study so
far incorporated the commodity price index in the determination of stock price or return.
4
Note that Canada’s reliance on resource-based exports has declined from 80% to 40% of total exports
during the past quarter century, but its dependence remains high compared with most advanced countries.
Furthermore, about one-fifth of the Toronto Stock Exchange 300 Composite Index is made up of
commodity related companies and as much as 35% of Canada’s exports are raw materials. It should, of
course, be mentioned that global commodity price weakness is offset by Canadian dollar weakness, but the
offset is only partial. However, in this paper the impact of exchange rate on the commodity price is already
incorporated, see the description of the commodity price in Table 1.
9
Moreover, while a higher price level increases expectations of a future hike in
interest rates, it may also lead to higher offsetting cash flows. However, cash flows may
not rise at the same rate as the inflation rate; see Mukherjee and Naka (1995) and
references therein. Consequently, one would expect a negative relationship between the
level of price and the stock price. Furthermore, if stocks hedge against inflation, at least
in the long run, as it was shown by Ely and Robinson (1997), Kaul (1986) and Kia
(1997a, 1997b), then a positive relationship between stock price and the level of domestic
price can be expected.
A higher domestic interest rate as well as the level of interest rate relative to
foreign interest rate is expected to have a positive impact on the investors’ subjective
discount rate. Furthermore, a higher interest rate is associated with expectations of lower
corporate profits. Both a higher subjective discount rate and expectations of lower
corporate profits result in a fall of the equity price. Furthermore, the more risk averse
investors are the higher risk premium they will require and, therefore, the higher their
subjective discount rate will be. Consequently, we would expect the domestic level of
interest rate, and the domestic rate relative to foreign rate (interest rate differential) as
well as risk premium to have negative relationships with the stock price.
Cash flows are also influenced by the above macro-variables relative to their
foreign counterparts. For example, a higher domestic price relative to its international
level in a small open economy results in a deterioration of balance of trade and
expectations of a lower cash flow and, therefore, a lower stock price. Finally, if the
industrial production in the domestic country continuously outgrows its foreign
10
counterpart, rational investors expect a higher cash flow relative to the cash flow of
foreign stocks and a higher stock price.
Let us define gt the actual equity return (excluding dividends) between time t-1
and t, and let get = Et [gt│It-1] denote its expected return conditional on the available
information set at time t-1, i.e., It-1. Furthermore, define vgt (= Et [gt - get]2) conditional
volatility of the equity return. According to Equation (4), the conditionally expected
equity return is a function, g, of the conditionally expected macro-variables used to
forecast the future stream of dividend payments. Namely, we will have
get = Et [gt│It-1] = g {Et [F(spt, indpt, ext, compt, cpict, onrt, premt, ronfft,
rindpcust, rcpt, zzt) │It-1]}.
(5)
The variable spt is the monthly return of S&P 500-stock index, indpt is the
monthly growth of total industrial production, ext is the monthly growth of the exchange
rate, compt is the monthly growth of commodity price index, cpict is the monthly growth
of the Consumer Price Index in Canada, onrt is the Canadian overnight rate, premt is the
monthly corporate rate less T-bill rate, ronfft is the difference between the Canadian
overnight and Fed Fund rates, rindpcust is the monthly growth of the ratio of industrial
production in Canada and U.S., rcpt is the monthly change of the log of the ratio of
Canada to U.S. consumer price indexes, and zzt includes past values of gt and
above-mentioned macroeconomic variables as well as current and past values of other
valid conditioning variables.
Following Kearney (1998), Kearney and Daly (1998) and Kearney (2000) and
using Equation (5), the standard asymptotic approximation of the conditional variance of
11
the stock return, vgt (= Et [gt - get]2), depends on the volatility of macro-variables included
in yt (Equation 3), assuming these macro-variables are independent. This implies that
vgt = vtset = vg(vspt, vindpt, vext, vcompt, vcpict, vonrt, vpremt, vronfft,
vrindpcust,
vrcpt, vzt),
(6)
where vtset is the conditional volatility of monthly return of TSE 300, vspt is the
conditional volatility of monthly return of S&P 500-stock index, vindpt is the conditional
volatility of the monthly growth of total industrial production, vext is the conditional
volatility of the monthly growth of the exchange rate, vcompt is the conditional volatility
of the monthly growth of commodity price index, vcpict is the conditional volatility of the
monthly growth of the Consumer Price Index in Canada, vonrt is the conditional volatility
of the Canadian overnight rate, vpremt is the conditional volatility of the monthly
corporate rate less T-bill rate, vronfft is the conditional volatility of the difference
between the Canadian overnight and Fed Fund rates, vrindpcust is the conditional
volatility of the monthly growth of the ratio of industrial production in Canada and U.S,
vrcpt is the conditional volatility of the monthly growth of the ratio of Canada to U.S.
consumer price indexes and vzt includes past values of vtset and above-mentioned
volatilities of macroeconomic variables as well as current and past values of other valid
conditioning volatilities.
It is extremely important to mention that Equation (6) reflects the theoretical
behavioral Equation (2) if and only if it is a forward-looking relationship. Equation (6)
represents agents’ expectations on volatility of future dividend payments. Namely, it
estimates an ex ante measure of risk. Consequently, its parameters are no longer invariant
12
to the process of forcing variables as was mentioned by Lucas (1976). Namely, at least
one of the parameters varies with changes in the expectation process. This requires that at
least one of the variables in (6) fails to be superexogeneous in the sense of Engle, Hendry
and Richard (1983) and Engle and Hendry (1993). This important fact, to the best of my
knowledge, was ignored in this literature.
To formulate superexogeneity and invariance hypothesis associated with
conditional model (6), assume a linear approximation relationship of Equation (6) is:
vtset = Γ1 vspt + Γ2 vindpt + Γ3 vext + Γ4 vcompt + Γ5 vcpict + Γ6 vonrt
+ Γ7 vpremt + Γ8 vronfft + Γ9 vrindpcust + Γ10 vrcpt + vz t’Γ + εt,
(6)’
where Γ1,...,Γ10 and Γ are constant parameters. The disturbance term εt is assumed to be
normally, identically and independently distributed. Let the set of variables Zt include all
contemporaneous macroeconomic volatilities in the conditional model and assume the
information set It includes the past values of vtset and Zt as well as the current and past
values of other valid conditioning variables (volatilities) included in vzt. Define,
respectively, the conditional moments of vtset and Zt as ηPt=E(vtset │It), ηZt=E(Zt│It),
σtPP=E[(vtset – ηPt)2│It] and σtZZ=E[(Zt – ηZt )2│It], and let σtPZ=E[(vtset – ηPt)
(Zt - ηZt )│It]. Consider the joint distribution of vtset and Zt conditional on information set
It to be normally distributed with mean ηt=[ηPt, ηZt] and a non-constant error covariance
 PP PZ 
σ σ  . Then following Engle, Hendry and Richard (1983), Engle and
matrix ∑ = 
 ZP ZZ 
σ σ

Hendry (1993) and Psaradakis and Sola (1996) we can write the relationship between
vtset and Zt as:
13
vtset = α0 + ψ0 Zt + (δ0 - ψ0) (Zt - ηZt) + δ1 σtZZ (Zt - ηZt) + ψ1 (ηZt)2 +
ψ2 σtZZ + ψ3 σtZZ ηZt + ψ4 σtZZ (ηZt)2 + vz’tγ + ut,
(7)
where α0, ψ0, ψ1, ψ2, ψ3, ψ4, δ0 and δ1 are regression coefficients of vtset on Zt conditional
on z’tγ, and term ut is assumed to be, as before, white noise, normally, identically and
independently distributed.
Note that Zt includes volatilities of some control/target variables (i.e., the
volatility of interest, exchange and inflation rates) that are subject to policy interventions.
Under the null of weak exogeneity, δ0-ψ0=0. Under the null of invariance, ψ1=ψ2=ψ3= ψ4
=0 in order to have ψ0 = ψ. Finally, if we assume that σtZZ has distinct values over
different, but clearly defined regimes, then under the null of constancy of δ, we need
δ1=0. If these entire hypotheses are accepted the equation will be reduced to Equation
(6)’. However, in such a case Equation (6)’ will no longer reflect a volatility model
associated with Equation (2). Namely, the estimated volatility is a measure for a realized
(ex post) risk and it is not a measure for risk (ex ante) for currently holding equity. It
must, of course, be emphasized that in this case Equation (6)’ can be used for future
prediction of the volatility of stock returns conditional on expected future values of Z and
vz. Furthermore, “superexogeneity is sufficient but not a necessary condition for valid
inference under intervention” (Engle, Hendry and Richard, 1983, p. 284). This is due to
the fact that estimable models with invariant parameters, but with no weakly exogenous
variables are easily formulated.
14
3. Data, empirical methodology and results
I will test the model on monthly Canadian and U.S. data. All observations are for
the last day of the month. The sample period is January 1975-December 1999. All data
are obtained from Statistics Canada CANSIM database. Domestic stock markets are
represented by all stocks traded on the Toronto Stock Exchange, and so the TSE 300
Composite Index is considered to be the price of domestic stock markets. The Standard
and Poor 500 Index is a representative of the price of foreign stock markets.
To empirically implement the model, we must obtain monthly estimates of the
standard deviations of the relevant variables. Following, among many, Schwert (1989),
Koutoulas and Kryzanowski (1996), Kearney (1998), Kearney and Daly (1998) as well as
Kearney (2000) the methodology developed by Davidian and Carroll (1978) was used.
Let vector Xt = [gt, spt, indpt, ext, compt, cpict, onrt, premt, ronfft, rindpcust, rcpt], and
estimate equations (8a) for gt and spt:
Xt =
12
12
i =1
i =1
∑α ix X t-i + ∑ µ ix M it + β1 Oct87 + β2 AS97 + uxt,
(8a)
and Equation (8b) for the rest of the variables in vector X.
Xt =
12
12
i =1
i =1
∑α ix X t-i + ∑ µ ix M it + uxt.
uxt ~niid(0, Σ), t = 1, …,N
(8b)
M’s are monthly seasonal dummy variables, Oct87 is a dummy variable used to
capture the impact of the October 87 stock market crisis. It is equal to one in October 87
and zero otherwise. AS97 is a dummy variable, which is equal to one for October and
15
November 1997 observations, and is zero, otherwise.5 The parameters αx’s and µx’s are
assumed to be constant. The monthly dummy variables capture seasonal variations in the
means and standard deviations of the variables while Oct87t, and AS97t dummy variables
capture the shock on stock prices during the October 87 and the recent Asian crises.
Furthermore, a 12th-order autoregression for the absolute values of errors from
equations (8a) and (8b), including monthly dummy variables to allow for different
monthly standard deviations, should be estimated:
|ûxt |= σ tx =
12
12
i =1
i =1
∑ δ ixσ tx-i + ∑ηix M it + vt
(9)
where δ ix and η ix , for i = 1 to 12 are constant parameters. The absolute value of the fitted
value of uxt (i.e., |ûxt |) is the standard deviation of X. However, as it was mentioned by
Schwert (1989), since the expected error is less than the standard deviation from a normal
distribution, following Schwert (1989), all absolute errors are multiplied by the constant
1.2533.
5
The TSE 300 Composite Index hit a record high on October 7, 1997, just 10 days before the crisis. Up to
January 12, 1998 the index fell 13.45%. However, the index hit a record high on March 9, 1998 and by the
end of March 1998 it hit 10 record highs. Consequently, if the Asian crisis had any impact on Canadian
stock markets, the impact would have been completely dissipated by the end of February 1998. However,
from the end of September 1997 till the end of November 1997 the TSE 300 Composite Index fell by
7.79% and rebounded after. This implies that for our monthly observations the appropriate dummy variable,
which may reflect the Asian Crisis, is a variable that is one for the October-November period and zero,
otherwise. This dummy variable is also relevant for the S&P500 index. The S&P500 index closed at 947.28
on the last day of September 1997 and fell by 3.45% and closed at 914.62 on the last day of October 1997.
But the index rebounded and increased by 4.46% and closed at 955.40 on the last day of November 1997.
16
As it was also mentioned by Kearney and Daly (1998), the conditional volatility in
Equation (9) represents a generalization of the 12-month rolling standard estimator used
by Officer (1973), Fama (1976) and Merton (1980). This is due to the fact that the
conditional volatility estimated by Equation (9) allows the conditional mean to vary over
time in equations (8a) and (8b), while it also allows different weights to be applied to the
lagged absolute unpredicted changes in stock market returns and macroeconomic
variables in equations (8a) and (8b), respectively.
Note that here I also allow the conditional mean of stock returns to vary with the
stock crisis of 1987 and the Asian crisis of 1997. Furthermore, the merit of this measure
of volatility is reviewed, among others, by Engle (1993) and Diebold and Lopez (1995).
This measure of volatility, as also indicated by Kearney and Daly (1998), is similar to the
autoregressive conditional heteroskedasticity (ARCH) model of Engle (1982), which, in
its various forms, has been widely used in the finance literature. Davidian and Carroll
(1987) argue that the specification in Equation (9) based on the absolute value of the
prediction errors is more robust than those on the squared residuals in equations (8a) and
(8b).
However, in estimation equations (6)’, (9) as well as superexogeneity
Equation (7), the dependent variables are generated regressor and their t-statistic should
be interpreted with caution (Pagan (1984) and (1986)). To cope with this problem,
following Pagan (1984 and 1986), Kearney and Daly (1998) as well as Kearney (2000),
the equation for the conditional volatility (i.e., Equation (6)’) is estimated jointly with the
equations determining the conditional volatilities of all variables included in the model
using the generalized least squares (GLS) estimation procedure.
17
In our GLS system, two equations are generated by Equation (8a), nine equations
are generated by Equation (8b), eleven equations are generated by Equation (9), and
including Equation (6)’ a system of 23 equations with 300 observations will be estimated.
The GLS estimator incorporates the possibility of cross-equation correlation among the
error terms. As it was also mentioned by Kearney and Daly (1998), in the current context
of estimating volatilities of financial and economic variables, the potential for efficiency
gain by estimating the system using GLS is considerable.
Table 1 about here
Table 1 reports the description of variables and data sources. Tables 2 and 3
reports the OLS result of the estimation of equations (8a) and (8b) and diagnosis tests on
these estimations, respectively. Tables 4 and 5 report the OLS estimation results on
Equation (9) and diagnosis test on the estimation, respectively. In the estimation process
whenever the errors exhibit heteroskedasticity I used the robust-error regression
estimation technique (White (1980) and Hansen (1982)) to correct for heteroskedasticity.
In tables 3 and 5, White is the White’s (1980) general test for heteroskedasticity, ARCH
is five-order Engle’s (1982) test, Godfrey is five-order Godfrey’s (1978) test, RESET is
the Ramsey (1969) misspecification test.
According to Godfrey’s test result in Table (3) none of the equations suffers from
autocorrelation while according to White and ARCH test results, TSE return, the growth
of industrial product, U.S.-Canada industrial production growth rates differential, the
Canadian overnight rate, Canada-U.S. overnight rates differential, premium rate and
Canada-U.S. inflation rates differential suffer from heteroskedasticity. The robust-error
regression estimation technique (White (1980) and Hansen (1982)) was, therefore, used to
18
correct the standard errors. According to R2 test result reported in Table 2 the equations
explain between 9% (for the growth of the commodity price) and 98% (for the Canadian
overnight interest rate) of the variation in the dependent variables. The lagged dependent
variables, according to the F-test result (column 2), are jointly significant for all equations
except for the S&P 500-stock index and the growth of the commodity price index.
Furthermore, the F-test for joint exclusion of monthly seasonal dummy variables is
significant only for the rate of return on the TSE 300 index, the growth of industrial
production, premium, inflation rate and Canada-U.S. inflation rates differential. Since
there is no study of this sort for Canada it is not possible to confirm these results with
other studies. Studies on other countries did ignore commodity price and premium
variables, but as far as other variables are concerned, the result on the lagged dependent
variable on the Canadian stock market confirms, e.g., the result of Kearney and Daly
(1998) for the Australian stock market.
Tables 2 and 3 about here
According to Godfrey’s test results in Table 5, none of the error term in volatility
equations suffers from autocorrelation. The error terms in volatility equation of stock
return, growth of industrial production, U.S.-Canada industrial production growth rate
differential, Canadian overnight rate, Canada-U.S. overnight rates differential, premium
and Canada-U.S. inflation differential, however, suffers from heteroskedasticity. As
before, the robust-error regression estimation technique (White (1980) and Hansen
(1982)) was, therefore, used to correct the standard errors. According to Ramsey’s (1969)
RESET test result, the conditional volatility equation for S&P 500-stock index return,
U.S.-Canada industrial production growth rates differential, growth of exchange rate,
19
premium, Canadian inflation rate and Canada-U.S. inflation rates differential are not
misspecified.
According to the F-test result reported in Table 4 the lagged values of conditional
volatilities (column 2 of Table 4), except the growth of commodity price, U.S.-Canada
industrial production growth rate differential and growth of the exchange rate, should be
included in all models. This result confirms, among others, findings of Kearney and Daly
(1998) for the Australian stock market, growth of Australian industrial production,
interest rate, inflation and exchange rates, namely, all our common macro-variables.
According to the F-test result, column 3 of Table 4, the seasonal dummy variables
of the conditional volatility of domestic stock return, as well as the growth of industrial
production, Canadian overnight rate and premium are jointly significant, indicating that
these conditional volatilities are time varying. This result confirms earlier findings of
Koutoulas and Kryzanowski (1996), who also used Canadian data, on macro-variables
like the growth of industrial production and exchange rate. Furthermore, the result also
confirms other work on other countries. For instance, Kearney (1998) finds a strong
monthly seasonality effect for British stock markets as well as Irish interest rate,
Irish-U.K. exchange rate, Irish industrial production and inflation rate. Furthermore, our
result also confirms Kearney and Daly’s (1998) findings for seasonality effect on the
conditional volatility of Australian stock return and the growth of industrial production.
Tables 4 and 5 about here
Table 6 reports the summary statistics from the OLS estimates of Equation (9).
All of the conditional volatility variables have low standard deviation relative to their
means. Except for the conditional volatility of the return of the S&P 500-stock index, of
20
the differential between the growth of the Canadian and U.S. industrial production and of
the growth of the exchange rate, there is evidence of non-normality in the conditional
volatility variables. This result is consistent with the existing literature, e.g., Koutoulas
and Kryzanowski (1996), Kearney (1998) and Kearney and Daly (1998).
Tables 6 and 7 about here
As stationary test results reported in Table 7 indicate all conditional volatility
variables are integrated of degree zero (stationary). Given all conditional volatility
variables are stationary, assuming a lag length of six, to capture two-quarter impact, a
parsimonious GLS was obtained by engaging in general-to-specific-modeling procedure.
The general-to-specific-modeling procedure is common in this literature, e.g., Koutoulas
and Kryzanowski (1996), Kearney (1998) and Kearney and Daly (1998). Table 8 reports
GLS estimation result of conditional volatility model (6)’. In Table 5, the R 2, σ and DW,
respectively, denote the adjusted squared multiple correlation coefficient, the residual
standard deviation and the Durbin-Watson statistic. Hansen’s (1992) stability L test for
the null hypothesis that the estimated coefficient is stable (5% critical value=0.47,
Table 1, Hansen (1992)) denotes all of the coefficients are stable.
Table 8 about here
Furthermore, the joint Hansen’s (1992) stability Lc test result for the null
hypothesis that the estimated coefficients as well as the error variance are jointly constant
is 3.39 (<4.14 for 18 degrees of freedom), which indicates we can not reject the null of
joint stability of the coefficients together with the estimated variance. I also estimated
recursively computed ‘break point’ Chow test (not reported, but available upon request)
which also indicated all coefficients are stable. Consequently, the conditional volatility
21
Equation (6)’appears to be constant over the sample period. In Table 8 White is the
White’s (1980) general test for heteroskedasticity, ARCH is five-order Engle’s (1982)
test, Godfrey is five-order Godfrey’s (1978) test, RESET is the Ramsey (1969)
misspecification test, Normality is Jarque-Bera (1987) normality statistic. None of these
diagnostic checks is significant.
According to the R 2 statistics the model explains 71% of the variation in the
conditional variance of the Canadian stock market with the standard error of the estimate
being one-fifth of the mean of the dependent variable. This result compares extremely
well with related existing work of, for instance, Schwert (1989) on U.S. data (i.e.,
R2 = 0.208 for the 1900-1987 period, the standard error of the estimate not reported),
Kearney (1998) on Irish stock market (i.e., R2 = 0.38, with the standard error of the
estimate being 0.011, the mean of the dependent variable not reported), Kearney and Daly
(1998) on Australian stock market (i.e., R2 = 0.35, with the standard error of the estimate
being one-third of the mean of the dependent variable) and Kearney (2000) on British
stock (i.e., R2 = 0.46, with the standard error of the estimate being one-fifth of the mean
of the dependent variable), on French stock (i.e., R2 = 0.48, with the standard error of the
estimate being 18% of the mean of the dependent variable), on German stock (i.e.,
R2 = 0.43, with the standard error of the estimate being 31% of the mean of the dependent
variable) and on Japanese stock (i.e., R2 = 0.63, with the standard error of the estimate
being 35% of the mean of the dependent variable) .
As the results in Table 8 indicate, while the intercept is zero in nine months of the
year, it is positive in February and April and negative in December indicating no January
effect on the conditional volatility of the stock return. December’s tax loss selling could
22
be a factor in a lower volatility in December. As it would be expected for a small open
economy under perfect capital mobility the conditional volatility of stock return is
instantaneously affected by the conditional volatility of the exchange rate and
Canada-U.S. interest rates differential. The estimated coefficient of the conditional
volatility of the exchange rate is 0.52 and statistically significant (t-statistic of 3.06),
indicating that, everything else being constant, more than half of the exchange rate
volatility is instantaneously transmitted to the stock market volatility. The instantaneous
and positive sign of the exchange volatility on the conditional volatility of stock return
confirms the finding of Kearney and Daly (1998) and Kearney (2000) for French stock
market. The coefficient of the volatility of interest rate differential is also positive and
statistically significant (t-statistic = 3.37) implying an instantaneous impact on the
conditional volatility of the stock return.6 Furthermore, the impact lasts for four months
with an overall impact (the sum of the coefficients) of 0.51% for a one-percent increase
of the conditional volatility of the interest rate differential.
The conditional volatility of the other macro-variables influences the conditional
volatility of stock return with lags. The volatility of the growth of the industrial
production influences positively the conditional volatility of stock return after a
one-month lag length and it lasts for two months. The overall impact is 1.24, indicating
that an increase of one percent in the volatility of the growth of industrial production
results in an increase of 1.24% in the volatility of stock return within two months.7 This
result is consistent with the finding of Schwert (1989) on U.S. data for the period
6
To the best of my knowledge no study of this sort incorporated interest rate differential.
7
Note that standard deviation has the same unit as the variable.
23
1953-1987 with statistically significant coefficient of 0.18, Kearney (1998) on the Irish
stock market with a statistically significant coefficient of 0.458.
The conditional volatility of the commodity price index influences the stock
volatility after a two-month lag length and it lasts for three months though the overall
impact is a negative 0.08. Namely, a 1% increase in the volatility of the growth of
commodity price index results in an increase of 0.21% in the conditional volatility of
stock return after a lag length of two months, but after a three-month lag length, such an
increase will reduce the volatility of stock return by 0.29%. To the best of my knowledge
no study has included the commodity price in the analysis.
The impact of the conditional volatility of interest rate on the conditional volatility
of stock return is negative with a lag length of two months. This result is in contrast with
Kearney and Daly (1998) who find the impact of the conditional volatility of interest rate
on the conditional volatility of Australian stock return to be positive with a lag length of
one month. Interestingly, while the conditional volatility of interest rate differential is
found to influence instantaneously and positively the conditional volatility of stock return
it takes two months for the volatility of interest rate to influence negatively the volatility
of stock return. One possible explanation is that, if a higher volatility of interest rate
differential is due to a higher volatility of Canadian overnight rate, given Fed Fund rate,
then stock volatility will increase instantaneously as interest rate differential varies, but
after two months when the market adjusts to the variations of interest rate differential the
stock volatility will fall.
24
The estimated coefficient of the conditional volatility of the premium is positive
while the estimated coefficient of conditional volatility of inflation rate is negative; both
impacts take a lag length of one month. The estimated coefficient of the lagged dependent
variable is statistically significant up to a six-month lag length indicating that the
conditional volatility of stock return in Canada responds dynamically to the conditional
volatility of the related macro-variables. In sum, this section adds to the existing literature
by showing that in a small open economy with no restriction on the mobility of capital
both volatility of commodity price and interest rate differential are important
determinants, among the volatility of other macro variables, in the determination of the
volatility of equity return.
4. Marginal models: Superexogeneity test result
Having established in the previous section that a dynamic relationship to describe
conditional volatility of stock return and conditional volatility of related macro-variables
exists, we need to verify if our model is also forward looking, namely, if the estimated
conditional volatility represents future volatility of dividend payments. This requires a
superexogeneity test on the variables. For this test we need to establish the marginal
models.
25
4.1. Marginal models
There have been several potential regime changes over the sample period as
follows:
(i)
The introduction of SPRA (the Special Purchase and Resale Agreements)
and SRA (Sales Repurchase Agreements) in June 1985.8
(ii)
The change of the Bank of Canada’s policy management approach (tight
monetary policy) under Governor Crow, February 1987-February 1994.9
(iii)
The implementation of a free trade agreement between Canada and the
United States in January 1991.
(iv)
The implementation of NAFTA (North American Free Trade Agreement
between Canada, the United States and Mexico in January 1994).
(v)
The introduction of inflation rate target band by the Department of
Finance and the Bank of Canada in February 1991.10
8
With SPRA instrument, Bank of Canada is involved in the purchase of short-term Government of Canada
securities under an agreement to sell them back on the following day. This temporary supply of funds can
ease the market. SRA is the reverse of SPRA (Bank of Canada, 1989).
9
In this period Canada-U.S. overnight interest rate differential (overnight rate minus Fed Fund rate) went
up from an average of 1.41% (standard deviation=2.39) during the 1975:01-1987:01 period to an average of
2.64% (standard deviation=1.33) during 1987:02 to 1994:02 period.
10
“… from 1982 to 1991, monetary policy in Canada was carried out with price stability as the longer-term
goal and inflation containment as the shorter-term goal, but without intermediate targets or a specified path
to the longer-term objective. In February 1991, explicit targets for reducing inflation were introduced
through joint announcements by the Bank and the federal government. These announcements confirmed
26
(vi)
The introduction of term deposit auction in April 1986.11
Dummy variables were created for step changes, i.e., (i) spra = 1 for June 1985
and after, zero otherwise, (ii) crow = 1 for February 1987-February 1994, zero otherwise,
(iii) free = 1 for January 1991 and after, zero otherwise, (iv) nafta = 1 for January 1994
price stability as the appropriate long-term objective for monetary policy in Canada and specified a target
path to low inflation….”(Thiessen, 1998-1999, p. 91).
11
In Canada, the day-to-day operations of monetary policy seek to influence the overnight financing rate,
primarily through the management of the supply of settlement balances provided to the direct clearers. The
direct clearers use their accounts at the Bank of Canada only to settle transactions between themselves or
with the government. Currently, Bank of Canada influences the daily level of these balances, retroactively,
through the drawdown/redeposit (D/R) mechanism.
Even though the bank’s actions as fiscal agent of the government are not directly linked to the
implementation of monetary policy, the two functions are related. Interactions occur principally on two
major fronts. The use of the Receiver General deposits in the D/R mechanism changes the overall level of
government balances every day. Consequently, treasury management decisions must take into account
actual and potential monetary operations. The net investment of government balances in the overnight
market can then be an important net source or use of funds for the market, influencing the evolution of the
overnight financing rate and should, therefore, be taken into account by the Bank when determining the
supply of settlement balances (Montador, 1995).
A portion of Receiver General deposits has been auctioned among direct clearers since April 1986,
and has become the largest component of the government’s cash balances. Furthermore, the auction for term
deposits is now an important daily event for the overnight money market in Canada. Indeed, the yields on
these deposits are one of the key indicators for the evolution of overnight rates during the course of a daily
overnight funds cycle. The one-day funds won at the term deposit auctions are part of the pool of one-day
resources available to the financial institutions to lend or meet their financing needs (Kia, 1996).
27
and after, zero otherwise, (v) inftar = 1 for February 1991 and after, zero otherwise, and
(vi) term = 1 for April 1996 and after, zero otherwise. 12
Level and interactive combinations of these dummy variables were tried for the
impact of these potential shift events in the marginal models for lspt, lext, lcompt and onrt
and any first round significant effects were retained. The resulting marginal models took
the form:
12
Dummies for other potential regimes were also created and used as regressors in marginal equations.
However, none of these dummies was found to be significant in any of the marginal equations. These
potential regime changes include: the revision to the reserve requirement in Canada in August 1983 (Bank
of Canada, 1983) and the introduction of zero reserve requirements and operating band in July 1994. In
compliance with the Bank Act, the statutory requirement on chartered banks to hold reserves against certain
of their deposit liabilities was reduced to zero in July 1994, Bank of Canada, (1994, p. 80, Footnote 1).
28
vext = 0.07 - 0.18 vext-1 - 0.50 vext-2 - 0.42 vex t-3 - 0.53 vex t-4
[SE]→[0.005] [0.05]
[0.05]
[0.05]
[0.06]
- 0.45 vext-5 - 0.38 vext-6 - 0.48 vex t-7 - 0.15 vex t-8
[SE]→ [0.06]
[0.05]
[0.06]
[0.06]
- 0.52 vext-9 - 0.16 vext-10 - 0.22 vex t-11 - 0.12 (vex t-7)(crowt-7)
[SE]→ [0.04]
[0.05]
[0.05]
[0.04]
+ 0.73 vrcpt-2 - 0.81 vrcpt-4 - 0.51 (vrcpt-9)(naftat-9)
[SE]→ [0.14]
[0.13]
[0.15]
- 0.07 vronfft-5 + 0.11 (vronfft-3)(freet-3)
[SE]→ [0.02]
[0.03]
- 0.08 (vronfft-9)(crowt-9) + 0.10 (vronff t-12)(crowt-12) + 0.06 vonrt-4
[SE]→ [0.02]
[0.03]
[0.02]
+ 0.09 vonrt-11 - 0.18 (vonrt-2)(freet-2) -0.13 (vonrt-9)(freet-9)
[SE]→ [0.02]
[0.04]
[0.04]
+ 0.20 (vonrt-2)(crowt-2) + 0.12 (vonrt-11)(crowt-11)
[SE]→ [0.05]
[0.04]
- 0.19 vindpt-3 - 0.28 vindpt-4-0.17 vindpt-5
[SE]→ [0.06]
[0.06]
[0.06]
+ 0.28 (vindpt-1)(sprat-1) +0.22 (vindpt-6)(zerot-6)
[SE]→ [0.07]
[0.06]
+ 0.15 (vindpt-2)(naftat-2) -0.27 (vindpt-1)(inftart-1)
[SE]→ [0.05]
[0.07]
+ 0.21 (vindpt-3)(inftart-3) + 0.14 (vindpt-4)(inftart-4)
[SE]→ [0.06]
[0.08]
- 0.18 (vindpt-8)(inftart-8) + 0.13 (vindpt-11)(inftart-11)
[SE]→ [0.07]
[0.06]
+ 0.11 (vindpt-8)(crowt-8) - 0.004 sprat – 0.003 AS97t
[SE]→ [0.05]
[0.001]
[0.001]
(10)
29
2
R =0.73, σ=0.002, DW=1.93, Godfrey(5)=0.53 (significance level=0.78), White=234
(significance level=1.00), ARCH(5)=22.17 (significance level=0.04), RESET=0.83
(significance level=0.47), Normality(χ2=2)=0.07 (significance level=0.97),
and
vronfft = 0.36 vronfft-1 + 0.23 vronfft-5 + 0.48 vronfft-7- 0.15 vronff t-11
[SE]
[0.07]
[0.07]
[0.08]
[0.07]
- 0.25 (vronfft-2)(sprat-2) - 0.33 (vronfft-7)(sprat-7) - 0.18 (vronfft-8)(sprat-8)
[SE]→ [0.07]
[0.09]
[0.06]
+ 0.18 (vronfft-9)(sprat-9) + 0.47 vex t-1 + 0.23 (vex t-8)(spra t-8)
[SE]→ [0.06]
[0.07]
[0.08]
- 0.30 vindpt-12 - 0.43 (vindpt-10)(sprat-10) + 0.77 vrcp t-1 - 1.28 vrcp t-7
[SE]→ [0.10]
[0.10]
[0.36]
[0.32]
+ 0.77 (vrcpt-6)(sprat-6) + 0.81 (vrcp t-11)(sprat-11)
[SE]→ [0.22]
[0.25]
(11)
2
R =0.71, σ=0.004, DW=1.85, Godfrey(5)=1.67 (significance level=0.13), White=168.68
(significance level=0.51), ARCH(5)=77.53 (significance level=0.00), RESET=0.02
(significance level=0.99), Normality(χ2=2)=145.88 (significance level=0.00),
Equations (10) and (11) pass the diagnostic checks for residual autocorrelation
test and the RESET test. However, both equations’ residual suffers from
heteroskedasticity in ARCH sense. To cope with heteroskedasticity problem the robusterror regression estimation technique (White (1980) and Hansen (1982)) was used to
correct the standard errors. Furthermore, Equation (11) fails for the normality of the
residual. The failure of the residual normality is common in the estimation of marginal
equations, e.g., Hurn and Muscatelli (1992) and Metin (1998).
Overall, equations (10) and (11) seem reasonable marginal models for the
analogues of ηz, especially since the standard errors are very small, i.e., σ respectively is
30
0.002 and 0.004. Clearly there is evidence of the structural break in all these equations,
i.e., possible break points are due to the introduction of Free Trade, NAFTA, overnight
repoes (spra and sra), Bank of Canada’s auctions of the Government of Canada funds
(term), the introduction of inflation target (inftar), the appointment of Governor Crow
(crow), the elimination of reserve requirements and the Asian Stock Market Crisis. Note
that non-constancy of the marginal models is related to the concept of superexogeneity,
which implies that the parameters of conditional model remain constant if agents are not
forward looking.
In Equation (10) the ‘crow’,‘free’, ‘nafta’, ‘spra’, ‘zero’, ‘inftar’ and ‘AS97’
dummy variables are significant. Dummy variable ‘AS97’affects only the intercept while
dummy variable ‘spra’ affects the intercept and the slope. Dummy variables ‘crow’,‘free’,
‘nafta’, ‘zero’, and ‘inftar’ influence only the slope. According to the estimated results the
conditional volatility of growth of the exchange rate fell during the Asian crisis. After the
introduction of NAFTA a higher conditional volatility of Canada-U.S. inflation
differential, after a lag length of nine months, reduces the conditional volatility of the
exchange rate while an increase in the conditional volatility of the growth of the industrial
production, after a lag length of two months, increases the volatility of the growth of the
exchange rate.
Before the introduction of Free Trade Agreement between U.S. and Canada, the
conditional volatility of Canadian-U.S. interest rate differentials had a negative impact on
the conditional volatility of the growth of the exchange rate while the conditional
volatility of overnight interest rate had a positive impact on the volatility of the growth of
the exchange rate. The introduction of free trade reversed the impact of these variables.
31
Furthermore, during the period of tight monetary policy of John Crow the positive impact
of the conditional volatility of overnight rate on the conditional volatility of the growth of
the exchange rate was overemphasized. During this period the negative impact of the
conditional volatility of interest rate differential on the growth of the exchange rate was
reduced after a lag length of twelve months.
The negative impact of the conditional volatility of industrial production on the
conditional volatility of the growth of the exchange rate was reduced after the
introduction of SPRA, zero reserve requirement and inflation target policy. Such an
observation is also evidenced during the period of tight monetary policy of John Crow. It
should, of course, be mentioned that according to the estimation result there are different
lag lengths in the above procedure.
In Equation (11), as it would be expected, the ‘spra’ dummy variable is only
significant. This dummy variable influences only the slope. The equation suggests that
while a higher conditional volatility of the growth of the exchange rate and Canada-U.S.
inflation rates differential increases the conditional volatility of Canada-U.S. interest rates
differential after a one-month lag length, the conditional volatility of the growth rate of
the industrial production reduces the interest rates differential after a 12-month lag length.
The introduction of SPRA emphasized these impacts.
4.2. Superexogeneity test results
From marginal models 10 and 11, estimates of ηZ, and σtZZ, for Z = vex and vronff
were calculated. As for σtZZ, since the error for vex and vronff variables at the
conventional level is heteroskedastic according to ARCH test, a five-period ARCH error,
therefore, was estimated. I also constructed DevZ (for Z=vex and vronff) as differences
32
between the variance of the error term in (10) and (11) and the variance constructed by
ARCH estimation. All of these constructed variables were then included in Equation (6)’
reported in Table 8. The estimation results on these constructed variables are given in
Table 9.
Table 9 about here
The individual F-test is on the null hypothesis that each variable is zero. The
F-test on the null hypothesis that all constructed variables are jointly zero is given in the
last row of the table. The F-test, given in the third to last row in Table 9, is on the
hypothesis that only the constructed variables related to each contemporaneous variable
while all other constructed variables are included, are jointly zero. I also included only the
constructed variables related to each contemporaneous variable and tested whether these
constructed variables are jointly zero. The F-test result is given in the second to last row
in Table 9.
As the estimation result in Table 9 shows the joint F-test on the null hypothesis
that coefficients of these constructed variables are jointly zero is significant, indicating
that these variables together should be included. This result immediately implies that the
contemporaneous variables in the conditional model, reported in Table 8, are not jointly
superexogenous, i.e., agents are forward looking and Equation (6)’ reflects a volatility
model associated with Equation (2) and, consequently, the estimated volatility can be a
measure of future risk for holding equity in Canada. Since the coefficient of (Z-ηZ) of
both contemporaneous variables vex and vronff is statistically insignificant, both of these
variables are weakly exogenous. However, the coefficient of σZZ(Z-ηZ) of vronff variable
is significant implying that the null of constancy is rejected for this variable.
33
The coefficient of σZZ ηZ and σZZ (ηZ)2 of both variables vex and vronff is
significant. Furthermore, the coefficient of σZZ of vex is significant. These results together
imply that the null of invariance with respect to policy changes is rejected for both of
these variables. Consequently, we reject the null of constancy and invariance, while
accepting weak exogeneity conditions for vronff variable, and reject the null of invariance
for the contemporaneous variable vex while accepting weak exogeneity condition. Note
that constancy and invariance are different concepts. Parameters could vary over time, but
be invariant with respect to policy changes.
However, as it was mentioned by Engle and Hendry (1993), we need all three
conditions to be satisfied in order to ensure superexogeneity. The failure of the constancy
and invariance conditions, therefore, justifies the result of the joint F-test on the null
hypothesis that all coefficients of the constructed variables are jointly zero. Namely, in
general, we reject the null hypothesis that variables vex and vronff are superexogenous.
Note that Fed Fund rate is exogenous to Canadian financial markets. Hence, the above
result means that, for a given Fed Fund rate, the monetary policy which alters the
conditional volatility of interest rate differential and/or the growth of the exchange rate
will affect investment decisions made by the economic agents in Canadian stock markets.
Namely the agents in equity markets are forward looking.
It should be noted that, instead of marginal models 10 and 11, one could simply
estimate the DGP (data generating process) of each contemporaneous variable in the
volatility model (6)’, while allowing level and interactive combinations of the above
dummy variables to reflect potential shifts events. The estimated equations, of course,
would not have any economic interpretation, but would be valid marginal equations for
34
the superexogeneity test, see Engle and Hendry (1993). In fact, I also tried this approach
and found no materially different result with what is reported. For the sake of brevity this
result is not reported, but available upon request.
It should be mentioned that since both contemporaneous variables vex and vronff
in the agents’ model (Equation (6)’) are weakly exogenous the inferences on the
parameters of agents’ model are efficient. In sum, it was found in this section that the
conditional volatility of the stock return represented by Equation (6)’can be a measure of
the ex ante risk for holding equities in Canada. The agents in the stock markets in Canada
are forward looking. Furthermore, agents form expectations by using non-constant
parameter models. This result also implies that Canadian investors in equity markets form
rational (or near rational) expectations. Namely, they incorporate expectations of future
economic changes in determining their investment decisions in equity markets.
5. Concluding remarks
This paper extends the existing literature by developing a model of stock return
volatility capable to capture whether the observed estimated equity return volatility can be
a measure of the future (ex ante) risk or realized (ex post) risk. The monthly data on
Canadian and U.S. stock markets for the period 1975-1999 are used. A macrodeterminant stock price model for a small and open-resource-oriented country is
developed. The model is used to derive a conditional stock volatility model capable to
incorporate the volatility of macro-variables, including commodity price, which are used
to forecast the volatility of future dividend payments. The model is estimated and the
superexogeneity tests are used to examine whether the estimated volatility reflects the
behavior of agents who are forward looking and their expectations are formed rationally,
35
i.e., if the estimated volatility can be a measure of future (ex ante) risk or it is only a
measure of realized (ex post) risk.
It is found that the conditional volatility of both the growth of commodity price
index and Canada-U.S. interest rate differentials, along with other conditional volatility of
macro-determinants, are in fact important component of the conditional volatility of stock
return determination. Furthermore, it was found that 71% of the volatility of the Canadian
stock return is explained by the behavior of forward-looking rational speculators which,
consequently, can be used to estimate ex ante risk. Namely, noise traders’ participation in
Canadian stock return volatility is only about one-fourth of the variation in monthly stock
returns.
36
References
Bank of Canada, 1983. Overnight Financing in Canada: Special Call Loans, Bank of
Canada Review, May, 4-13.
Bank of Canada, 1989. Buy-back Techniques in the Conduct of Monetary Policy, Bank of
Canada Review, July, 3-17.
Bank of Canada, 1994. Bank of Canada Review, Autumn.
Bernanke, Ben and Mark Gertler, 1999, Monetary Policy and Asset Price Volatility,
Federal Reserve Bank of Kansas City, Fourth Quarter, 17-51.
Culter, David M., James M. Poterba, and Lawrence H. Summers, 1989. What Moves
Stock Prices? Journal of Portfolio Management 15, 4-12.
Davidian, M. and R. J. Carroll, 1978. Variance Function Estimation Journal of the
American Statistical Association 82, 1079-1091.
Dickey, D. A. and W. A. Fuller, 1981. Likelihood Ratio Statistics for Autoregressive
Time Series with a Unit Root, Econometrica 49, 1057-1072.
Diebold, F. X. and J. A. Lopez, 1995. Modeling Volatility Dynamics, in
Macroeconometrics; Developments, Tensions and Prospects, Ed. K. D. (Kluwer
Academic Publisher Hoover, Boston).
Ely, David P. and Kenneth J Robinson, 1997. Are Stocks a Hedge Against Inflation?
International Evidence Using a Long-run Approach, Journal of International
Money and Finance 16, 141-167.
Engle, Robert F., 1982. Autoregressive Conditional Heteroskedasticity With Estimates of
the Variance of United Kingdom Inflation, Econometrica 50, 987-1007.
37
Engle, Robert F., 1993. Statistical Models for Financial Volatility, Financial Analysts
Journal 49, 72-78.
Engle, Robert F., David F. Hendry and Jean-Francois Richard, 1983. Exogeneity,
Econometrica 51, 277-304.
Engle, Robert F. and David F. Hendry, 1993. Testing Superexogeneity and Invariance in
Regression Models, Journal of Econometrics 56, 119-139.
Fama, Eugene F., 1976. Forward Rate as Predictors of Future Rates, Journal of Financial
Economics 3, 351-377.
Fama, Eugene F., 1990. Stock Returns, Expected Returns, and Real Activity, Journal of
Finance 45, 1089-1108.
Gallant, A. Ronald, Peter E. Rossi and George Tauchen, 1992. Stock Prices and Volume,
Review of Financial Studies 5, 199-242.
Godfrey, Les G., 1978. Testing Against General Autoregressive And Moving Average
Error Models When the Regressors Include Lagged Dependent Variables,
Econometrica 46, 1293-1301.
Hansen, Bruce E., 1992. Testing for Parameter Instability in Linear Models, Journal of
Political Modeling 14, 517-533.
Hansen, Lars P., 1982. Large Sample Properties of Generalized Method of Moments
Estimators, Econometrica 50, 1029-1054.
Hurn, A.S. and V.A. Muscatelli, 1992. Testing Superexogeneity: The Demand for Broad
Money in the UK, Oxford Bulletin of Economics and Statistics 54, 543-556.
Jarque, Carlos M. and Anil K. Bera, 1987. A Test for Normality of Observations and
Regression Residuals, International Statistical Review 55, August, 163-172.
38
Kaul, Gautam, 1986. Stock Returns and Inflation: the Role of the Monetary Sector,
Journal of Financial Economics 18, 253-276.
Kearney, Colm, 1998. The Causes of Volatility in a Small, Internationally Integrated
Stock Market: Ireland, July 1975-June 1994, The Journal of Financial Research
XXI, 85-104.
Kearney, Colm and Kevin Daly, 1998. The Causes of Stock Market Volatility in
Australia, Applied Financial Economics 8, 597-605.
Kearney, Colm, 2000. The determination and international transmission of stock market
volatility, Global Finance Journal 11, 31-52.
Kia, Amir, 1996. Day-of-the-Week Effect and the Overnight Money Market Efficiency,
Bank of Canada mimeograph. Paper presented at the annual meeting of the
Canadian Economics Association, May 1996.
Kia, Amir, 1997a. Inflation, Stock Returns and the Relative Performance of Equities and
Bonds: A Brief Survey of Empirical Studies, Bank of Canada, Financial Markets
Department Working Paper, FMD 97-110.
Kia, Amir, 1997b. Hedging Against Inflation: The Best Investment Strategy in a Low
Inflation Environment, Bank of Canada, Financial Markets Department Working
Paper, FMD 97-195.
Korkie, Bob, 1990. A Note on the TSE/Western Treasury Bill Returns, Canadian Journal
of Administrative Sciences 7, 26-28.
Koutoulas, George and Lawrence Kryzanowski, 1996. Microfactor conditional
volatilities, time-varying risk premia and stock return behavior, The Financial
Review 31, 169-227.
39
Lucas Jr., Robert E., 1976. Econometric Policy Evaluation: A critique in K. Brunner and
A.H. Meltzer, eds., The Phillips Curve and Labor Markets (North-Holland,
Amsterdam), 19-46.
Lucas Jr., Robert E., 1978. Asset Prices in an Exchange Economy, Econometrica 46,
1429-1445.
Merton, R.C., 1980. On Estimating the Expected Return on the Market: an Explanatory
Investigation, Journal of Financial Economics 8, 323-361.
Metin, Kivilcim, 1998. The Relationship Between Inflation and the Budget Deficit in
Turkey, Journal of Business & Economic Statistics 16, No. 4, 412-422.
Montador, Bruce, 1995. The Implementation of Monetary Policy in Canada, Canadian
Public Policy XXX:1, 107-120.
Mukherjee, Tarun K. and Atsuyuki Naka, 1995. Dynamic Relations Between
Macroeconomic Variables and the Japanese Stock Market: An Application of A
Vector Error Correction Model, The Journal of Financial Research XVIII,
223-237.
Officer, R., R., 1973. The Variability of the Market Factor of the New York Stock
Exchange, Journal of Business 46, 434-453.
Pagan, Adrian, 1984. Econometric Issues in the Analysis of Regressions with Generated
Regressors, International Economic Review 25, 221-247.
Pagan, Adrian, 1986. Two Stage and Related Estimators and Their Applications, Review
of Economic Studies 53, 517-538.
Psaradakis, Zacharias and Martin Sola, 1996. On the Power of Tests for Superexogeneity
and Structural Invariance, Journal of Econometrics 72, 151-175.
40
Ramsey, J.B., 1969. Tests for Specification Errors in Classical Linear Least-squares
Regression Analysis, Journal of Royal Statistical Society 31 Series B, 350-371.
Schwert, G. William, 1989. Why Does Stock Market Volatility Change Over Time?, The
Journal of Finance XLIV, 1115-1153.
Schwert, G. William, 1990. Stock Returns and Real Activity: A Century of Evidence,
Journal of Finance 45, 1237-1257.
Thiessen, Gordon, 1998-1999. The Canadian Experience with Targets for Inflation
Control, Bank of Canada Review, winter, 89-107.
White, Halbert, 1980. A Heteroskedasticity-Consistent Covariance Matrix Estimator and
a Direct Test for Heteroskedasticity, Econometrica 48, 817-837.
41
Table 1
Description of Variables Of Which The Conditional Volatilities Are Used
in the Analysis: CANSIM Number
a. The commodity price index is a fixed-weight index of the spot or transaction prices of 23
commodities produced in Canada and sold in world markets. The weight of each commodity in the
total index is based on the average value of the Canadian production over the 1982-90 period (Bank
of Canada Review (1994)). Consequently, the data were converted to the Canadian dollar by
multiplying it by the exchange rate and dividing it by the average of exchange rate for the period
1982-90 (i.e., 1.269193474994).
b. Since corresponding data were not available the ratio was constructed by using the ratio of the log
of real indexes plus the log of the ratio of Canadian and U.S. consumer price indexes: D360048
converted to 1992=100, D360061, I56010, P100000 and D139105.
c. The overnight financing rate is the call loan rate. Call loans are money market instruments
designed to finance the acquisition or holding of securities by investment dealers for short periods of
time. These loans are callable and a wide range of collateral is accepted by their suppliers (Bank of
Canada, 1983).
d. Note that monthly T-bill rates are only available from 1989. We, consequently, followed Korkie
(1990) and Koutoulas and Kryzanowski (1996) and calculated these rates for the 1975:1-1979:12
period as tbr1t=log(1+kt), where
kt=1200[(1+(91/365)*tbr3t-1/100)30.4/91-1] and tbr3 is the three-month T-bill rate (B14060).
g = Monthly return of the TSE 300 Composite Index, closing quotations at month-end: B4237
sp = Monthly return of the S&P 500-stock index, quotations at month-end: B4291
indp = Monthly growth of nominal (unadjusted) total industrial production= change of the log of
real total industrial production: I57001 plus change of the log of Consumer Price (unadjusted)
Index: P100000
comp = Monthly growth of commodity price index in Canadian dollars: B3300a
rindpcus = Canadian (SA) and U.S. (SA) industrial production monthly growth rates differentialsb
ex = Monthly growth of the exchange rate ($Canadian of one unit $US), the last day of the month:
B3414
onr = Canadian overnight financing ratec: B14044
ronff = Canadian overnight rate less Fed Fund rate: B14044 and B54408
cpi = Monthly Canadian inflation rate (Consumer Price Index: P100000)
rcp = Monthly Canadian-U.S. inflation rate differentials (Consumer Price indexes: P100000 and
D139105)
prem = Premium=Differences between monthly Canadian corporate paper and T-bill ratesd:
B14039 and B14059
42
Table 2
The ARCH Models of Stock Market Returns and Macro-variables
Sample Period:1975:1-1999:12 (Note the first two years were reserved for lags)
*The robust-error regression estimation technique (White, 1980; and
Hansen, 1982) was used to correct for heteroskedasticity.
12
Xt =
∑α ix X t-i
12
+
i =1
∑α ix X t-i
x
i
M t -i + β1 Oct87 + β2 AS97 + uxt
(8a)
x
i
M t -i
(8b)
i =1
12
Xt =
∑µ
12
+
i =1
X
Return TSE: Eq(8a)*
∑µ
+ uxt
i =1
F-test:
(p-value) Null:
t-stat.
t-stat.
α1x=...=α12x=0
F-test:
(p-value)
Null:
µ1=...=µ12=0
on
on
β1
β2
R2
0.01
0.03
-22
-4
0.21
Return S&P: Eq(8a)
0.83
0.57
-5
-
0.21
Growth rate: total
indus. prod.: Eq(8b)*
Growth rate:
commodity:Eq(8b)
U.S.-Canada indus.
prod. growth rate diff.:
Eq(8b)*
Growth of exchange
rate: Eq(8b)
Canadian overnight
rate: Eq(8b)*
Canada-U.S. overnight
rate diff.: Eq(8b)*
Premium: Eq(8b)*
0.00
0.00
-
-
0.96
0.49
0.46
-
-
0.09
0.00
0.05
-
-
0.17
0.03
0.76
-
-
0.10
0.00
0.39
-
-
0.98
0.00
0.39
-
-
0.74
0.00
0.00
-
-
0.66
Inflation: Eq(8b)
0.00
0.00
-
-
0.76
Canada-U.S. inflation
diff.: Eq(8b)*
0.00
0.00
-
-
0.25
43
Table 3
Diagnosis tests of the estimation of equations (8a) and (8b)
Godfrey
White
ARCH
RESET
X
(p-value)
(p-value)
(p-value)
(p-value)
Return TSE: Eq(8a)
0.54
0.02
0.00
0.98
Return S&P: Eq(8a)
0.92
0.47
0.89
0.01
Growth rate: total
indus. prod.: Eq(8b)
Growth rate:
commodity: Eq(8b)
U.S.-Canada indus.
prod. growth rate diff.:
Eq(8b)
Growth of
exchange rate: Eq(8b)
Canadian overnight
rate: Eq(8b)*
Canada-U.S. overnight
rate diff.: Eq(8b)
Premium: Eq(8b)
0.09
0.00
0.00
0.00
0.89
0.41
0.13
0.96
0.99
0.07
0.01
0.39
0.99
0.80
0.77
0.87
0.99
0.00
0.00
0.32
0.96
0.00
0.00
0.00
0.21
0.00
0.10
0.40
Inflation: Eq(8b)
0.99
0.27
0.27
0.70
Canada-U.S. inflation
diff.: Eq(8b)
0.81
0.00
0.69
0.73
44
Table 4
The ARCH Models of Volatility of Stock Market Returns and Macro-variables
Sample Period: 1975:1-1999:12 (Note the first two years were reserved for lags).
*The robust-error regression estimation technique (White, 1980; and Hansen,1982) was used to
correct for heteroskedasticity.
σ tx
12
=
∑ δ ixσ tx-i
12
+
i =1
∑η
x
i
M t -i
+ vt
(9)
i =1
X
F-test: (p-value)
Null:δ1x=...=δ12x=0
F-test: ( p-value)
Null:η1=...=η12
R2
Return TSE*
0.00
0.00
0.63
Return S&P
0.03
0.33
0.21
Growth rate: total indus. prod.*
0.00
0.00
0.61
Growth rate: commodity price
0.48
0.29
0.63
U.S.-Canada indus. prod.
growth rate diff.*
0.33
0.39
0.61
Growth of exchange rate
0.74
0.27
0.59
Canadian overnight rate*
0.00
0.02
0.57
Canada-U.S. overnight rate
diff.*
0.00
0.12
0.62
Premium*
0.00
0.00
0.58
Inflation
0.00
0.10
0.59
Canada-U.S. inflation diff.*
0.01
0.09
0.58
45
Table 5
Diagnosis tests of the estimation of Equation (9)
Godfrey
White
ARCH
RESET
X
(p-value)
(p-value)
(p-value)
(p-value )
Return TSE
0.99
0.04
0.39
0.02
Return S&P
0.99
0.39
0.55
0.72
Growth rate: total indus. prod.
0.99
0.00
0.00
0.00
Growth rate: commodity
0.99
0.83
0.68
0.04
U.S.-Canada indus. prod.
growth rate diff.
0.99
0.06
0.03
0.75
Growth of exchange rate
0.99
0.97
0.91
0.98
Canadian overnight rate
0.99
0.00
0.14
0.00
Canada-U.S. overnight rate diff.
0.99
0.00
0.00
0.00
Premium
0.99
0.01
0.99
0.18
Inflation
0.99
0.93
0.99
0.22
Canada-U.S. inflation diff.
0.99
0.00
0.73
0.08
46
Table 6
Summary of statistics of conditional volatility of the stock returns and macro-variables
a. SK is skewness, KU is kurtosis and JB is Jarque-Bera normality test. Sample Period: 1975:11999:12 (Note the first two years were reserved for lags).
** vtse is the conditional volatility of monthly return of TSE 300, vsp is the conditional volatility of
monthly return of S&P 500-stock index, vindp is the conditional volatility of the monthly growth of
total industrial production, vcomp is the conditional volatility of the monthly growth of commodity
price index, vrindpcus is the conditional volatility of the monthly growth of the ratio of industrial
production in Canada and U.S., vex is the conditional volatility of the monthly growth of the exchange
rate, vonr is the conditional volatility of the Canadian overnight rate, vronff is the conditional volatility
of the difference between the Canadian overnight and Fed Fund rates, vcpic is the conditional volatility
of the monthly change of the log of Consumer Price Index in Canada, vrcp is the conditional volatility
of the monthly change of the log of the ratio of Canada to U.S. consumer price indexes and vprem is
the conditional volatility of the monthly corporate rate less T-bill rate.
Variable
Mean
Standard
SKa
KUa
JBa
Deviation
(p-value for
(p-value for
(sig. lev.for
SK=0)
KU=0)
JB=0)
vtse
0.04
0.01
0.48
-0.11
11.13
(0.00)
(0.71)
(0.00)
.
vsp
0.04
0.01
vindp
0.01
0.00
vcomp
0.02
0.00
vrindpcus
0.01
0.00
vex
0.01
0.00
vonr
0.01
0.00
vronff
0.01
0.00
vcpic
0.002
0.00
vrcp
0.003
0.00
vprem
0.002
0.00
0.33
(0.02)
1.46
(0.00)
0.42
(0.00)
0.18
(0.23)
0.28
(0.06)
1.68
(0.00)
1.91
(0.00)
0.95
(0.00)
0.72
(0.00)
1.11
(0.00)
0.21
(0.47)
4.43
(0.00)
-0.07
(0.83)
0.004
(0.99)
-0.18
(0.55)
3.86
(0.00)
4.69
(0.00)
1.33
(0.00)
0.47
(0.11)
2.92
(0.00)
5.58
(0.05)
324
(0.00)
8.49
(0.01)
1.46
(0.48)
4.04
(0.13)
300
(0.00)
420
(0.00)
62.16
(0.00)
26.3
(0.00)
155
(0.00)
47
Table 7*
Stationary Tests: 1975 (Jan.) - 1999 (Dec.)
* All tests include constant and trend. The critical value for Augmented Dickey-Fuller τ test
(lag-length = 5) and for Phillips-Perron non-parametric Z test (window size = 4) is 3.42 at 5%
and 3.98 at 1%.
** vtse is the conditional volatility of monthly return of TSE 300, vsp is the
conditional volatility of monthly return of S&P 500-stock index, vindp is the
conditional volatility of the monthly growth of total industrial production, vcomp is
the conditional volatility of the monthly growth of commodity price index, vrindpcus
is the conditional volatility of the monthly growth of the ratio of industrial
production in Canada and U.S., vex is the conditional volatility of the monthly
growth of the exchange rate, vonr is the conditional volatility of the Canadian
overnight rate, vronff is the conditional volatility of the difference between the
Canadian overnight and Fed Fund rates, vcpic is the conditional volatility of the
monthly change of the log of Consumer Price Index in Canada, vrcp is the
conditional volatility of the monthly change of the log of the ratio of Canada to U.S.
consumer price indexes and vprem is the conditional volatility of the monthly
corporate rate less T-bill rate.
Absolute Values
Variables
Augmented Dickey-Fuller
τ-stat.
Phillips-Perron
Z-stat.
4.39
7.54
8.74
6.91
6.39
11.26
3.78
3.50
6.03
4.36
6.30
17.68
18.78
9.77
16.06
17.18
12.85
10.17
8.44
12.82
16.10
13.75
Volatilities:**
vtse
vsp
vindp
vcomp
vrindpcus
vex
vonr
vronff
vcpic
vrcp
vprem
48
Table 8*
GLS Estimation Results of Conditional Volatility Model
* Period=1975(Jan.)-1999(Dec.), Mean of Dep. Variable=0.04, R 2=0.71, σ=0.008, DW=1.99,
Godfrey(5)=1.25 (significance level=0.28), White=140 (significance level=0.09), ARCH(5)=15.27
(significance level=0.23), RESET=1.56 (significance level=0.20), and Normality(χ2=2)=7.00
(significance level=0.03).
** Feb, Apr and Dec are dummy variables for the month of February, April and December,
respectively. For the description of the remaining mnemonics see notes of Table 7.
Dependent Variable = vtset
Variable**
Coefficient
Standard Error
Hansen’s (1992) Stability Li test (5%
critical value = 0.47)
vindpt-1
0.68
0.21
0.09
vindpt-2
0.56
0.22
0.11
vext
0.52
0.17
0.09
vcompt-2
0.21
0.10
0.11
vcompt -3
-0.29
0.09
0.12
vonrt-2
-0.52
0.10
0.10
vpremt-1
2.18
0.54
0.09
vronfft
0.27
0.08
0.05
vronfft-4
0.24
0.08
0.02
vcpict-1
-1.75
0.56
0.04
vtset-2
0.41
0.05
0.11
vtset-3
0.20
0.04
0.06
vtset-4
0.18
0.05
0.04
vtset-6
-0.23
0.04
0.05
Feb
0.01
0.002
0.29
Apr
0.02
0.002
0.20
Dec
-0.01
0.002
Hansen’s (1992) Stability Li test on Variance of the
Equation
Joint (coefficients and the error variance) Hansen’s
(1992) Lc test (5% critical value=4.14 for degrees of
freedom=18)
0.06
0.24
3.39
49
Table 9*
GLS Estimation Results of Superexogeneity Tests
* vex is the conditional volatility of the monthly growth of the exchange rate, vronff is the conditional
volatility of the difference between the Canadian overnight and Fed Fund rates, ηZ is the conditional
mean of Z, σZZ is the conditional variance of Z, and DevZ is the deviation of variance of the error term
from a five-period ARCH error of Z.
** vtset = α0 + ψ0 Zt + (δ0 - ψ0) (Zt - ηZt) + δ1 σtZZ (Zt - ηZt) + ψ1 (ηZt)2 + ψ2 σtZZ + ψ3 σtZZ ηZt + ψ4 σtZZ
(ηZt)2 +z’tγ + ut
F-statistics
(p-value)
Variable (Z)**
vex
vronff
Z- ηZ
0.60
2.53
(0.43)
(0.11)
σZZ (Z - ηZ)
0.19
333.31
(0.66)
(0.00)
(ηZ)2
0.09
0.78
(0.79)
(0.38)
σZZ
10.14
2.67
(0.00)
(0.10)
σZZ ηZ
13.62
12.51
(0.00)
(0.00)
σZZ (ηZ)2
15.55
15.17
(0.00)
(0.00)
DevZ
1.30
0.07
(0.26)
(0.79)
F-statistics on null hypothesis that coefficients of all
constructed variables in this column are jointly zero.
The equation includes the constructed variables
related to other contemporaneous variables.
29.88
356.28
(0.00)
(0.00)
F-statistics on null hypothesis that coefficients of all
constructed variables in this column are jointly zero.
The equation excludes the constructed variables
related to other contemporaneous variables.
14.72
326.43
(0.04)
(0.00)
F statistics on coefficients of all variables in rows and 386.27
columns
(0.00)