estimating default probabilities implicit in equity prices
TRANSCRIPT
Estimating Default Probabilities Implicit in Equity Prices
Tibor Janosi* Robert Jarrow** Yildiray Yildirim***
April 9, 2001Revised September 18, 2001
*Computer Science Department, Cornell University, Ithaca, NY 14853.**Johnson Graduate School of Management, Cornell University, Ithaca, NY 14853 and KamakuraCorporation. (607-255-4729, [email protected])*** School of Management, Syracuse University, Syracuse, NY 13244.
ii
Estimating Default Probabilities Implicit in Equity Prices
Abstract
This paper uses a reduced form model to estimate default probabilities implicit in equityprices. The model implemented is a generalization of the model contained in Jarrow (2000). Thetime period covered is May 1991 – March 1997. Monthly equity prices on fifteen different firm’sdebt issues are studied. The firms are chosen to provide a stratified sample across variousindustry groupings. Four general conclusions can be drawn from this investigation. First, equityprices can be used to infer a firm’s default intensities. This is a feasibility result. Second, due tothe noise present in equity prices, the point estimates of the default intensities that are obtainedare not very precise. They have large standard errors and they exhibit time series non-stationarities. Third, equity prices do appear to have a bubble component, not explained by theFama-French (1993,1996) four-factor model and proxied by a P/E ratio. Fourth, we compare thedefault probabilities obtained from equity with those obtained implicitly from debt prices usingthe reduced form model contained in Janosi, Jarrow, Yildirim (2000). Due to the large standarderrors of the equity model’s intensity estimates, we find that one cannot reject the hypothesis thatthese default intensities are equivalent.
Estimating Default Probabilities Implicit in Equity Prices
1. Introduction
Given the recent exponential growth in the credit derivatives market (see Risk (2000)), creditrisk modeling and estimation has become a topic of current interest. The theoretical literature isquite extensive (see Bielecki and Rutkowski (2000) for a review). The empirical literatureestimating reduced form credit risk models has concentrated on using debt prices (see Duffie andSingleton (1997), Madan and Unal (1998), Duffee (1999), Duffie, Pedersen, Singleton (2000),Janosi, Jarrow, Yildirim (2000)), credit derivative prices (see Hull and White (2000, 2001)), orbankruptcy histories (see Altman (1968), Zmijewski (1984), Shumway (2001), and Chava andJarrow (2000, 2001)). Equity prices have only been used to estimate default parameters forstructural models (see Delianedis and Geske (1998)). The purpose of this paper is to use equityprices in conjunction with a reduced form modeling approach to estimate default probabilities.The approach utilized is a slight generalization of the model contained in Jarrow (2000).
The data used for this investigation are equity prices from CRSP and debt prices from theUniversity of Houston’s Fixed Income Database over the time period May 1991 – March 1997.The observation interval is one month. Debt prices consist of bids taken from Lehman Brotherstrading sheets on the last calendar day in each month, see Warga (1999) for additional details.
Fifteen different firms are included in this study, where the firms are chosen to stratifyvarious industry groupings: financial, food and beverages, petroleum, airlines, utilities,department stores, and technology. The same fifteen firms as in Janosi, Jarrow and Yildirim(2000) are included so that a comparison of the different estimation procedures can be performed.
Eight different models for equity prices are investigated herein, the simplest modelscontaining no default. Using a rolling estimation procedure, for each month during thisobservation period, the equity model’s parameters (including the bankruptcy parameters) areestimated using a linear regression. In this procedure, only information available to the market atthe time that the equations are estimated is utilized.
First, the analysis supports the feasibility of estimating default probabilities implicit inequity prices. In a relative comparison of the eight models, in-sample root mean squared errorgoodness-of-fit tests and out-of-sample generalized cross validation statistics support thenecessity of including default parameters into the equity model structure. The best performingintensity model depends on the spot rate of interest but not an equity market index, confirmingthe results of Janosi, Jarrow and Yildirim using debt prices.
Second, due to the large variability of equity prices, the point estimates of the defaultintensities obtained are not very reliable. They have large standard errors and they exhibit timeseries non-stationarities. This cast doubts on the reliability of the default probability estimatesobtained from equity prices using structural models as in Delianedis and Geske (1998),confirming the previous conclusions of Jarrow and van Deventer (1998, 1999) in this regard.
Third, we find that equity prices contain a bubble component not captured by the Fama-French (1993,1996) four-factor model for equity’s risk premium. This bubble component,proxied by the firm’s P/E ratio, is significant for many of the firms in our sample.
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Fourth, we compare the estimates of the intensity process obtained here with thoseobtained using debt prices from Janosi, Jarrow, Yildirim (2000) for the same fifteen firms overthe same time period. Due to the large standard errors of the intensity estimates in the equitymodel, the hypothesis that these two intensity functions are equivalent cannot be rejected.
The previous literature estimating reduced form credit risk models using debt pricesinclude Duffie and Singleton (1997), Madan and Unal (1998), Duffee (1999), Duffie, Pedersen,Singleton (2000), and Janosi, Jarrow, Yildirim (2000). Duffie and Singleton (1997) estimateswap spreads, Madan and Unal (1998) estimate yields on thrift institution certificates of deposit,and Duffie, Pedersen, Singleton (2000) estimate credit and liquidity spreads for Russian debt.Both Duffee (1999) and Janosi, Jarrow, Yildirim (2000) estimate default intensities using U.S.corporate debt. As mentioned earlier, we will compare our estimated default intensities withthose from Janosi, Jarrow, Yildirim (2000). Bankruptcy prediction models using historicalbankruptcy data include Altman (1968), Zmijewski (1984), Shumway (2001), and Chava andJarrow (2000, 2001), among others. Chava and Jarrow (2001) compare historically estimateddefault intensities with estimates obtained using reduced form models and debt prices. Theycannot reject the hypothesis that the explicit and implicit default intensities are identical. Astructural model for estimating default intensities is Delianedis and Geske (1998).
An outline of this paper is as follows. Section 2 presents the model structure. Section 3provides a description of the data and section 4 estimates the state variable process parameters.The equity model parameter estimation is discussed in sections 5 – 10. Section 11 compares thedefault parameter intensities estimated using the equity model to those obtained using debt pricesby Janosi, Jarrow, Yildirim (2000). A conclusion is provided in section 12.
2. The Model Structure
This section introduces the notation and provides a generalization of the reduced formcredit risk model for equity prices contained in Jarrow (2000).
Trading can take place anytime during the interval ]T,0[ . Traded are default-free zero-coupon bonds of all maturities and a firm’s common stock. Markets are assumed to befrictionless with no arbitrage opportunities, but equity prices may contain bubbles.
Let p(t,T) represent the time t price of a default-free dollar paid at time T where TTt0 ≤≤≤ .The instantaneous forward rate at time t for date T is defined by f(t,T) = T/)T,t(plog ∂∂− . Thespot rate of interest is given by r(t) = f(t,t).
Consider a firm issuing equity. This firm may default. Let τ be the random variable
representing the first time that this firm defaults (τ > T is possible if the firm does not default).We let
{ } ≤
== ≤ otherwise0
tif11)t(N t
ττ (1)
denote the point process indicating whether or not default has occurred prior to time t. )t(λdenotes its random intensity process. The time t intensity process, ∆λ )t( , gives the approximateprobability of default for this firm over the time interval ]t,t[ ∆+ .1
1 The intensity process is defined under the risk neutral probability. This statement will become clearbelow.
3
Equity pays dividends and has a liquidating payoff at time TL* for 0 ≤ TL* ≤ T . Thetime t value of these payments equals the value of the equity (per share) and is denoted by ξ(t).These promised payments are made, unless the firm defaults. If default occurs, the equity holdersget a fractional recovery payment on these promises equal to δe(τ)ξ(τ−) where δe(τ) ≡ 0.2
We need to develop some notation for these promised payments. The regular dividends
are paid at times 1, 2, ... , TD* and denoted by tD at time t. We assume that these dividends are
deterministic quantities, placed in an escrow account and paid for sure.3 The requirement thatthese dividends are deterministic implicitly determines the date TD*. For many equities TD* willbe a month or less. The liquidating dividend L(TL*) is paid at time TL* unless default occurs priorto this date where TD* ≤ TL*. The liquidating dividend consists of the time TL* (future) value ofall unannounced and random dividends paid over (TD*,TL*) plus the remaining resale value of thefirm at time TL*.
Let S(t) represent the time t present value of the liquidating dividend, conditional upon nodefault prior to time t. There is some evidence, for example the recent price growth of internetstocks4, that stock prices contain a “bubble” or “monetary value” component, see Jarrow andMadan (1999). We let 0)t( ≥θ represent this time t bubble component in the stock price.
Given this set-up, it is easy to see5 that the per share equity value at time t is given by
=
<∑++= ≥
.tif0
tif)j,t(pD)t()t(S)t( j
T
tj
*D
τ
τθξ (4)
The share price consists of the present value of the liquidating dividend (S(t)), the bubble ( )t(θ ),and the announced dividends (Dj for j = t, ... , TD*).
Under the assumption of no arbitrage, standard arbitrage pricing theory6 implies thatthere exists a probability Q (equivalent to the statistical probability) such that present values arecomputed by discounting at the spot rate of interest and then taking an expectation with respect toQ. That is,
)(du)u(r
t
T
teE)T,t(p∫−
= (5)
)()T(
du)u(r*Lt *
L
*LT
t 1e)T(LE)t(S τ<
∫−= (6)
2 This assumption is easily relaxed, see Jarrow (2000).3 One could assume that these dividends could be defaulted on as well, see Jarrow (2000).4 See Money Magazine April 1999, page 169 for Yahoo’s P/E ratio of 1176.6. 5 This is a simple no arbitrage restriction that the present value of the sum of multiple cash flows equals thesum of the present values of the cash flows. 6 See Jarrow and Turnbull (1995). No arbitrage guarantees the existence, but not the uniqueness of theprobability Q. Without any additional hypotheses on the economy, the uniqueness of Q is equivalent tomarkets being complete, see Battig and Jarrow (1999). In incomplete markets, equilibrium (additionalhypotheses) guarantees the uniqueness of Q. The uniqueness of Q is essential for estimation.
4
where Et(.) is conditional expectation with respect to Q at time t.
For simplicity, we model the bubble component as a random process that is proportionalto the present value of the liquidating dividend as in the following expression.
)( 1e)t(S)t(
t
0du)u(
−=∫ θµ
θ (7)
where 0)u( ≥θµ is the continuous return in the stock price due to the bubble component.
To obtain an empirical formulation of the above model, more structure needs to beimposed on the stochastic nature of the economy. Following Jarrow (2000) we consider aneconomy that is Markov in three state variables: the spot rate of interest, the cumulative excessreturn on an equity market index, and the liquidating dividend process. We next introduce thestochastic evolution of these state variables.
For the spot rate of interest, we use a single factor model with deterministic volatilities,sometimes called the extended Vasicek model, i.e.
(Spot Rate Evolution) [ ] )t(dWdt)t(r)t(ra)t(dr rσ+−= (8)
where a 0≠ , σr > 0 are constants, )(tr is a deterministic function of t chosen to match the initialzero-coupon bond price curve7, and W(t) is a standard Brownian motion under Q initialized atW(0) = 0. The evolution of the spot rate is given under the risk neutral probability Q.
The second state variable is related to an equity market index, denoted by M(t). Theevolution for the equity market index is assumed to satisfy a geometric Brownian motion withdrift r(t) and volatility σm. The correlation coefficient between the return on the market index andchanges in the spot rate is denoted by rmϕ .
(Market Index Evolution)( ))()()()( tdZdttrtMtdM mσ+= (9)
where σm is constant, and Z(t) is a standard Brownian motion under Q initialized at Z(0) = 0correlated with W(t) as dZ(t)dW(t) = dtrmϕ with rmϕ a constant.
Our second state variable is Z(t). Given observation dates 1, 2, 3, ..., t we can solveexpression (9) for Z(t) as a function of Z(t-1). This solution is given by
( )
.0)0(1
)()1()(log)1()( 22
1
11
=≥
+−−+−= ∫∫
−−
Zandtfor
duduurtMtMtZtZ mm
t
t
t
t
σσ (10)
7 In particular, ( )( ) aa2e1t)t,0(f)t,0(f)t(r at22
r−−+∂∂+= σ for 0a ≠ .
5
We see here that Z(t) is a measure of the cumulative excess return per unit of risk (above the spotrate of interest) on the equity market index.
The third state variable is the liquidation value of an otherwise equivalent non-defaultablefirm at time t, denoted by L(t). This liquidation value can be viewed as the market value of aportfolio consisting of the firm’s assets and liabilities, but held by a default free entity. Thisliquidation value is assumed to follow a geometric Brownian motion under the martingaleprobability Q, i.e.
)t(dw)t(Ldt)t(L)t(r)t(dL LLσ+= (11)
where 0L >σ is a constant, )t(wL is a Brownian motion under the martingale measure Q
initialized at zero, correlated with W(t) and Z(t) as dZ(t)dwL(t) = dtmLϕ and dwL(t)dW(t) =
dtrLϕ with mLϕ and rLϕ constants.
The value of this portfolio at time TL* is L(TL*).
In our model, L(t) is held by a firm that can default prior to time TL*. If the firm defaults,due to bankruptcy costs (lawyer’s fees, lost sales, etc.), the liquidation value jumps to zero. Thisimplies that the liquidation value to an equity holder in the risky firm is less than or equal to thepresent value of the underlying portfolio of assets to a default free agent, i.e. )t(L)t(S ≤ .
Given the evolutions of the state variables, we next specify the intensity process.
(Intensity a Function of the Spot Rate and the Market Index)
constants. are,,where]0),t(Z)t(rmax[)t( 210210 λλλλλλλ ++= (12)
In this formulation, the intensity function is assumed to be a linear function of the state variablesr(t) and Z(t) as long as this linear combination is non-negative, zero otherwise. Note that in thisformulation of the expected loss process, the recovery rate is allowed to be a stochastic.
For analytic tractability in the empirical implementation, we drop the maximum operatorin expression (12). In this case, as the recovery rate is non-negative, this implies that negativedefault rates (λ(t) < 0) are possible. If the likelihood of (λ(t) < 0) is small, this simplificationshould provide a reasonable approximation to expression (12). Unfortunately, when the intensityprocess is negative, the default distribution is no longer a proper probability distribution (seeBremaud [1981]). Nonetheless, given the tractability of the subsequent expressions, and thedifficulty of the numerical inversion without a closed form solution, we empirically investigatethe validity of this linear approximation.
Given these expressions, it is shown in Jarrow (2000) that the default free zero-couponbond and the present value of the liquidating dividend can be rewritten as:
2/)T,t()T,t( 211e)T,t(p σµ +−= (13)
and
6
)T,t(ve)T,t(p
)t(L)t(S *
L
2/)tT()T,t(du)T,u(b)T,t(
*L
2*LmLL2
*Lrm2
*LT
t
*LrLL1
*L
211 −−−∫−−
=ϕσληϕλϕσλσλ
(14)
where
( ) ( ) [ ] 6/tT)T,t(1)tT)(t(Z2)T,t(2)T,t()tT( 22
3rm212
21
211110e)T,t(p)T,t(v ληϕλλλσλλµλλ −+++−−++−−−=
( ) and,ae1)t,u(b
,du)T,u(b)T,t(,2du)T,u(bdu)u,t(f)T,t(
)ut(ar
2T
t
21
2T
t
T
t1
−−−=
∫=∫+∫=
σ
σµ
2r
)tT(a2r
)tT(a3r ]tT)[a2/()tT(e)a/(]e1)[a/()T,t( −+−+−−= −−−− σσση .
Expression (7) for the bubble component of the stock price and expression (14) for the presentvalue of the liquidating dividend, substituted into expression (4), gives the present value of thestock price at any date.
Unfortunately, observing values for the stock price at each date t still leaves this systemunder determined as there are more unknowns (L(t), 210 ,, λλλ ) than there are observables ( )t(ξ ).To overcome this situation, we use expression (11) in conjunction with expression (14) totransform expression (4) into a time series regression. In this regard, it is shown in the appendixthat
∆∆∆∆ξ
ξ
∆
)t(r
)j,t(pD)t(
)j,t(pD)t(
log
j
T
tj
j
T
tj
*D
*D
−−
−∑−−
∑−
−≥
≥
(15)
)t(
])2/1()t()t([
6/)t*T()tT)(T,t(b
)]t*T)(t(Z)t*T)(t(Z[
)T,t(p
)T,t(plog
2
)T,t(b
22LLL
222
*L
*Lrm21
2
*L
*L
2*L
10
∆ε
∆σ∆µσ∆Θσ
∆λ∆ϕλλ
∆∆λ
∆∆
∆λ∆λ
ξθ
−+
+−+−−+
−−−−
−−−−−−
−−
−−−≈
where
))t(w)t(w()t( LLL ∆σ∆ε −−≡− and )t(LΘ is the liquidation value’s risk premium.
Expression (15) gives a time series expression for the capital gains component of the log stockprice return over the time period ]t,t[ ∆− . This expression forms the basis for our empiricalestimation in the subsequent sections.
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One can think of this estimation equation in three different, but related ways. The firstinterpretation of expression (15) is that it is equivalent to a reduced form model for the firm’sequity given an evolution for the liquidation value of the firm and an intensity process for default.This interpretation follows the method of derivation. The second interpretation of expression (15)is that it is a type of structural model for the firm’s equity where the firm’s net value (assets lessliabilities) is exogenously given and default occurs according to an intensity process correlatedwith the randomness inherent in the firm’s net value fluctuations. The third interpretation ofexpression (15) is that it is a generalized asset pricing model with an explicit representation of
bankruptcy where the equity’s excess return is linearly related to a risk premium )t(LL ∆Θσ −plus a bubble component )t( ∆µθ − . Given this perspective, expression (15) makes explicit thebond market factors contained in Fama and French (1993). 3. Description of the Data
The time period covered in this study is May 1991 – March 1997. All individual firmequity data (including earnings, dividends, etc.) are obtained from CRSP. For the equity marketindex, we used the S&P 500 index. For estimating an equity risk premium, we will employ theFama-French benchmark portfolios (book-to-market factor (HML), small firm factor (SMB)), anda momentum factor (UMD). These monthly portfolio returns were obtained from Ken French’swebpage.8
The U.S. Treasuries prices used for this investigation were obtained from the Universityof Houston’s Fixed Income Database. This data consists of monthly bid prices of all outstandingbills, notes and bonds taken from Lehman Brothers trading sheets on the last calendar day in eachmonth, see Warga (1999) for additional details. Being such a large database (containing over 2million entries), the potential for data errors is quite large. Indeed, a careful examination of thedata confirmed this suspicion. Hence, we filtered the data to remove obvious data errors. Weexcluded Treasury bonds with inconsistent or suspicious issue/dated/maturity dates and matrixprices. Lastly, using a median yield filter of 2.5%, we also removed U.S. Treasury debt listingswhose yields exceeded the median yield by this percent. After filtering, there are approximately29,100 U.S. Treasury prices left in the sample set. For parameter estimation of the state variableprocesses a daily spot rate is needed. Since the fixed income data provides only monthlyobservations, we use daily observations of the 3-month T-bill yield available from CRSP as well.
The same twenty firms as in Janosi, Jarrow, Yildirim (2000) were initially selected foranalysis. These firms were selected to stratify various industry groupings: financial, food andbeverages, petroleum, airlines, utilities, department stores, and technology. Due to unavailabilityof balance sheet data or stock prices, five of these companies were eliminated. The remaining 15firms included in this study and the industry represented by each firm are provided in Table 1.The Moodies and S&P’s ratings for each company’s debt issues at the start of our sample period(May 24, 1991) are also included.
The observation interval for the equity estimation was selected to be one month. Themonthly observation interval was chosen for two reasons. First, the default parameter estimationusing debt prices in Janosi, Jarrow, Yildirim (2000) was based on monthly data, so that a monthlyobservation interval for the equity estimation will provide a more equal comparison. Second, andmore importantly, it is believed that the use of monthly data for equities eliminates market-
8 The address is: http://web.mit.edu/kfrenc/www/data_library.html.
8
microstructure noise more prevalent in smaller observation intervals (daily or weekly), see Smith(1978), Schwartz and Whitcomb (1977a, 1977b) and Dimson (1979).
4. Estimation of the State Variable Process Parameters
To implement the estimation of the default and liquidity discount parameters, we firstneed to estimate the parameters for the state variable processes (r(t),Z(t)).
a. Spot Rate Process Parameter Estimation
The inputs to the spot rate process evolution are the forward rate curves over an extendedobservation period (f(t,T) for all months t ∈ January 1975 – March 1997) and the spot rateparameters ( r,a σ ). We discuss the estimation of these inputs in this section.
For the estimation of the forward rate curves, a two-step procedure is utilized. First, for agiven time t, the discount bond prices (p(t,T) for various T) are estimated by solving the followingminimization problem:
choose (p(t,T) for all relevant }Ii:Tmax{T ti ∈≤ )
to minimize [ ] 2
Ii
bidiiii
t
)T,t(B)T,t(B∑ −∈
(16)
where )t,t(pC)T,t(B j
n
1jtii
i
j∑==
is a U.S. Treasury security with coupons of jC dollars at times jt for j = 1, ...,ni where in Tti
=
is the maturity date, tI is an index set containing the various U.S. Treasury bonds, notes and bills
available at time t, and bidii )T,t(B is the market bid price for the thi bond with maturity iT .
The discount bond prices’ maturity dates T coincide with the maturities of the Treasury bills, andthe coupon payment and principal repayment dates for the Treasury notes and bonds.
Step 2 is to fit a continuous forward rate curve to the estimated zero-coupon bond prices(p(t,T) for all }Ii:Tmax{T i ∈≤ ). We use the maximum smoothness forward rate curve asdeveloped by Adams and van Deventer (1994) and refined by Janosi (2000). Briefly, we choosethe unique piecewise, 4th degree polynomial with the left and right end points left “dangling” that
minimizes ∫ ∂∂∈ }Ii:Tmax{
t
22ti
dss/)s,t(f . A time series graph of these curves is contained in
Figure 1.
For the spot rate parameters ( r,a σ ) estimation, the procedure follows that used in Janosi,Jarrow, Yildirim (2000). A rolling estimation of the parameters using only information availableat the time of the estimation is performed, making the parameter estimates ( rtt ,a σ ) dependent ontime t as well.
9
The procedure is based on an explicit formula for the variance of the default-free zero-coupon bond prices derived using expression (8), see Heath, Jarrow and Morton (1992). For ∆ =1/12 (a month), the expression is:
( ) ∆σ∆∆
−=−+ −− 2t
2)tT(a2rtt a/1e])t(r))T,t(P/)T,t(P[log(var t . (17)
First we fix a time to maturity T− t ∈ {3 months, 6 months, 1 year, 5 years, 10 years, thelongest time to maturity of an outstanding Treasury bond closest to 30 years}. Then, we fix acurrent date t ∈ {May 1991 – March 1997}. Going backwards in time 60 months (5 years), wecompute the sample variance, denoted tTv , using the smoothed forward rate curves previouslygenerated. Note that the sample variance depends on both the date of estimation and the bond’smaturity.
Then, for each date t ∈ {May 1991 – March 1997}, to estimate the parameters ( trt a,σ )we run a nonlinear regression
( ) tT2
t
2)tT(a2rttT ea/1ev t +
−= −− ∆σ (18)
across the bond time to maturities T− t ∈ { ¼, 1/2, 1, 5, 10, longest time to maturity closest to 30} where tTe is the error term.
The parameter estimates and the 90 percent confidence bands are graphed in Figures 2.aand 2.b. The 90 percent confidence bands are computed point-wise, using the standard errorsfrom the regression coefficients at each date. The R^2 for each of these monthly non-linearregressions (not reported) exceeded .99 in all cases. As illustrated in Figure 2.a, the spot ratevolatility ( rtσ ) is nearly constant over this period. For most dates, the mean spot rate volatility is
within the 90 percent confidence band. In contrast, the mean reverting parameter ( ta ) appears tobe more volatile with an increasing trend.
To test for the time series stability of these parameter estimates, a unit root test wasperformed.9 These tests are contained in Table 2. For the volatility rtσ , the test rejects a unitroot, implying the time series is stationary. In contrast, one cannot reject a unit root for the meanreverting parameter ta .
b. Market Index Parameter Estimation
Using the daily S&P 500 index price data and the 3-month T-bill spot rate data, we needto estimate the parameters of the market index process ),( rmm ϕσ as given in expression (9) and
9 We perform a Dickey-Fuller (DF) test. The DF test statistic is the t-statistic for the ρ coefficient in the
regression: t1t1tt yyy ερµ ++=− −− where ρµ , are constants and tε is an error term. The null
hypothesis of a unit root for ty is 0=ρ . A rejection of the null hypothesis implies that there is not a unitroot.
10
the cumulative excess return on the market index as given by Z(t) in expression (10). This sectiondiscusses this estimation.
This estimation is based on daily data (∆ = 1/365). As before, the procedure involves arolling estimation of the parameters using only information available at the time of the estimation.For a given date t ∈ {May 24,1990 – March 31,1997}, we go back in time 365 business days andestimate the time dependent sample variance and correlation coefficients ),( rmtmt ϕσ using thesample moments, i.e.
∆∆∆σ 1
)t(M
)t(M)t(Mvart
2mt
−
−−=
and
−−
−−−
= )t(r)t(r,)t(M
)t(M)t(Mcorrtrmt ∆
∆∆ϕ . (19)
These estimates and their 90 percent confidence bands are graphed in Figures 3a and 3b. Thestandard deviation of the time series values for ( rmm ,ϕσ ) over the observation period is used togenerate the confidence bands.
The market volatility is relatively constant between .1 and .2 over this observation period.For most dates, the average market volatility over this sample period lies within the 90 percentconfidence band. Although the correlation coefficient appears to be more variable, for most datesthe average correlation coefficient over this sample period lies within the 90 percent confidenceband. As before, to test for the stability of the parameters a unit root test was performed, seeTable 2. The results show that a unit root can be rejected at the 90 percent confidence level forthe market volatility but not for the correlation coefficient.
Given the parameter estimates for the market volatility )( mtσ and the daily 3-monthTreasury bill yields, the Z(t) process is computed using expression (10), starting the series onMay 24, 1991.10 The Z(t) process is graphed in Figure 4. For comparison purposes, the 3-monthT-bill yield is also graphed in Figure 4. As seen, Z(t) is steadily increasing over our observationperiod. The spot rate process exhibits significant variation, first declining and then increasingover the sample period.
5. Equity Model Parameter Estimation
Given the state variables (r(t),Z(t)) parameters as estimated in the previous sections, thissection presents the default parameter estimation. The basis for this estimation is expression (15).To empirically implement expression (15) we need to specify models for both the risk premiumand the equity price bubble.
Following Fama and French (1993, 1996) we use a four-factor asset pricing model withthe factors being the excess return on a market portfolio, SMB(t), HML(t) and UMD(t) , i.e.
10 In this estimation, ∫ −=−
tt )t(rdu)u(r∆ ∆∆ and ∫ =−
tt
2m
2m du∆ ∆σσ .
11
∆∆∆Θσ ))t(X,t(LL −− = (20)
[ ] [ ] [ ])t(UMD)t(HML)t(SMB)t(r)t(M
)t(M)t(M3210 βββ∆∆
∆∆β +++
−−
−−−
where SMB(t) is the difference between the return on a portfolio of small stocks and the return ona portfolio of large stocks at time t, HML(t) is the difference between the return on a portfolio ofhigh-book-to-market stocks and the return on a portfolio of low-book-to-market stocks at time t,and UMD(t) is a momentum factor.
The equity price bubble and the volatility of L(t), )t(ξ are proxied by the volatility of thestock and a price earnings ratio,
∆∆∆µσ θ ))]t(X,t([ 2L −−+− + ∆σξ )t()2/1( 2 = [ ]
+ )t(
Earnings
icePr)t( 5
24 β∆σβ ξ (21)
where ∆∆ξ
∆ξξσξ1
)t(
)t()t(var)t(2
−
−−≡ .
There is some concern that the SMB(t), HML(t) and UMD(t) factors may already include anadjustment for bubbles. For this reason, the subsequent regressions are run both with and withoutthe P/E ratio included.
For the estimation we fix 20)tT( *L =− years and we set tT *
D = . The first restrictionmakes the firm’s valuation horizon 20 years, making equity comparable with long term debt. Thesecond restriction implies that all future dividends are viewed as random. Consequently, we onlyneed to make an adjustment for dividends in the pay out month.
Substitution of the above into expression (15) yields:
−∈−−
−−−
−−−
−
]t,t[xatdividendif)t(r)x,t(pD)t(
)t(log
]t,t[overdividendnoif)t(r)t(
)t(log
x
∆∆∆∆∆ξ
ξ
∆∆∆∆ξ
ξ
[ ]))t(T)(t(Z)tT)(t(Z2
)T,t(b
)T,t(p
)T,t(plog *
L*L2
2*L
*L
*L
10 ∆∆Λ∆∆
∆ΛΛ −−−−−+
−−
−+≈
[ ] [ ] [ ]
[ ]
++
+++
−−
−−−+
)t(Earnings
icePr)t(
)t(UMD)t(HML)t(SMB)t(r)t(M
)t(M)t(M
52
4
3210
β∆σβ
βββ∆∆∆
∆β
ξ
(22)
where
12
.
6/)tT()tT)(T,t(b
22
11
2*L
22
*L
*Lrm2100
λΛλΛ
∆λ∆ϕλλ∆λΛ
−==
−−−−−=
To insure that the intensity process is non-negative when both 21 ,λλ are zeros, we
impose the constraint that 00 ≤Λ in the estimation. The final computations for the defaultparameters are:
6/)tT()t*T)(T,t(b)/( 2*L
22
*Lrm2100 −−−+−= ΛϕΛΛ∆Λλ
11 Λλ =
22 Λλ −= .
The time period covered is May 1991 – March 1997. The estimating regression is runusing 48 months of historical data. Thus, the first regression estimation occurs four years into thedata set on May 31, 1995. For each subsequent month, until March 1997, the regression is re-estimated and parameter estimates obtained. This generates 23 regressions. As before, onlyinformation available to the market at the time of the estimation is utilized. This rollingestimation procedure gives a time series of parameter estimates( t2t1t0 ,, λλλ , t5t4t3t2t1t0 ,,,,, ββββββ ) based on 46 (48 – 2) months of overlapping data. Thechoice of a four-year estimation period was based on trading off the stability of the estimatesversus larger standard errors. Although longer estimation periods are likely to make the standarderrors smaller, they also imply that structural shifts are more likely to occur, making the estimatedparameters less stable.
Eight different models for equity prices are estimated. The models differ with respect tothe number of independent variables included in the regression. Models 1 and 2 have no default( 0210 ≡≡≡ λλλ ). They differ only in the inclusion of a price/earnings ratio ( 05 ≡β or not).Models 3 – 8 include default, and they differ with respect to the default intensity investigated andthe inclusion of the P/E ratio or not. Models 3 and 4 have only 0λ non-zero. Models 5 and 6
have both 0λ and 1λ non-zero. Models 7 and 8 have all default parameters non-zero. Theseeight models are nested and a relative comparison of model performance is subsequentlyprovided.
To summarize the time series estimates across all models and across all times, Table 3provides the average values for the point estimates of the parameters. The average adjusted R^2 isalso included. The values in Table 3 are averages over the number of months in the observationperiod (May 1995 – March 1997) for which the linear regression estimates of the parameters arecomputed.11 A typical time series graph of the default intensities for Eastman Kodak (ek) usingmodels 3 – 8 is contained in Figure 5. As depicted, models 3 and 4 have the default intensityidentically zero. Models 5 and 6 have similar intensity estimates, greater than zero, but quitesmall. Models 7 and 8 exhibit larger estimates for the default intensity with significantly greatertime series variation.
11 This is not to be confused with the number of observations used in the time t regression for a particularfirm. At the time t regression, we use 48 months of data.
13
Table 4 provides the t-scores12 for the averages of the parameter estimates as contained inTable 3 as well as the average P-scores for the coefficients (across the number of regressions).The P-score is the probability of rejecting the null hypothesis (that the coefficient is zero), when itis true. Summary statistics for various F-tests are also provided. The first F-test has as its nullhypothesis ( 0t4t3t2t1t0 ===== βββββ ). Given are the average P-scores of the F-tests(across the number of regressions). The F-tests for models 3, 5 and 7 test for the joint hypothesisthat all default parameters are zero, i.e. 0t0 =λ , 0t1t0 == λλ , and 0t2t1t0 === λλλrespectively. The F-tests from models 2,4,6 and 8 test the hypothesis that 0t5 =β . The nextsections discuss these statistics and various tests for the relative performance of the differentequity models.
Table 7 provides some summary statistics (both in- and out-of-sample) regarding thequality of the models fit to the data. Included are the average root mean squared error, theaverage Generalized Cross Validation statistic, the average default intensity, the average standarderror of the default intensity, and the average one-year default probability.
6. Analysis of the Time Series Properties of the Parameters
Under the assumed model structure, the default and equity risk premium parameters( t2t1t0 ,, λλλ , t3t2t1t0 ,,, ββββ ) should be constant across time. The four-year estimation periodwas selected to better insure this hypothesis, by minimizing the structural shifts in the economythat would more likely occur using a longer horizon interval. Given measurement error in theinput data (equity prices and the state variable parameters) and its effect on the parameterestimates, we test the hypothesis that the time series variation in these parameters is solely due torandom (white) noise. Alternatively stated, we test to see if the parameter estimates follow arandom walk around a given mean. A unit root test is used in this regard.
Table 6 contains these unit root test statistics. The last panel in Table 6 contains asummary of the unit root rejections across model types. As seen, we accept a unit root (non-stationarity) for almost all the parameters in all the models. This includes both the defaultparameters and the risk premium coefficients. The best-performing models in terms of stabilityare models 5 – 8 that include default.
This stability analysis suggests that lengthening the estimation period will only worsenthe results. Perhaps a smaller observation period (daily or weekly) is needed instead. However,this extension of our analysis awaits subsequent research.
7. Analysis of Fama-French Four-Factor Model with No Default
Before analyzing the default parameters, it is important to document the performance of thesimple Fama-French four-factor model with no default. Table 3 contains the estimates for thecoefficients of the Fama-French four-factor model with and without a P/E ratio (models 1 and 2,respectively), and Table 4 contains their t-scores and p-values. The F-test for model 1 in Table 4tests the hypothesis that the model is significant (i.e., 0t4t3t2t1t0 ===== βββββ ). Theaverage p-value for this F-test for every firm is .0000, strongly rejecting the hypothesis of nosignificance. The average R2 for model 1 is .5018.
12 The t-score is adjusted to reflect the fact that the regressions contain overlapping time intervals, seeJanosi, Jarrow, Yildirim (2000) for more details on the adjustment.
14
8. Analysis of a Bubble Component (P/E ratio) in Stock Prices
This section tests for the significance of a bubble component in the equity price model bytesting the null hypothesis that the P/E ratio is insignifcant, i.e. 0t5 =β . The F- test for model 2in Table 4 tests this hypothesis. For models 1 and 2 (not including default) the average p-valuefor 3 firms (mer, amr, ek) is significantly different from zero. The individual t-scores and averagep-values for t5β show significance for six firms (mer, amr, txu, wmt, ek, txn), confirming thisconclusion.
For models 3 – 8 (including default), the average F-test gives significance for the samethree firms (mer, amr, ek). The average p-values and t-scores for t5β confirm this significancefor four firms (amr, txu, wmt, ek).
It appears that the P/E ratio does proxy for a bubble component in stock prices notcontained in the four-factors of Fama-French.
9. Analysis of the Default Intensity
As mentioned earlier, the average default intensity parameters are contained in Table 3with t-scores and average values for the P-scores provided in Table 4. The firms’ estimates arepresented in industry groupings for easy comparison.
First to be noticed in Table 3 is that the fit of the linear regressions are quite good. TheR^2 varies between .1675 (for cce, model 1) to .8459 (for mer, model 8). The average R^2 acrossall models varies from .5018 for model 1 to .5548 for model 8. R^2 uniformly increases acrossfirms with increasing model complexity up to model 6. Model 7’s R^2 often is less than model 6,before increasing again for model 8.
Second, it is interesting to examine the signs of the coefficients for the default intensityparameters. The signs of 1λ and 2λ indicate the sensitivity of the firm’s default likelihood tochanges in the spot rate and the equity market index, respectively. For example, for bt (BankersTrust) the sign of 1λ is positive indicating that as interest rates rise, the likelihood of default
increases. Continuing, the sign of 2λ is negative, indicating that as the market index rises, thelikelihood of default declines. The signs of these coefficients differ across firms and betweenfirms within an industry. An example of different signs within an industry is for the departmentstores grouping, where Sears Roebuck and Company (s) and Wal-Mart Stores, Inc. (wmt) havecontrasting signs for both the interest rate and market index variables.
Glancing now at Table 4, we discuss the statistical significance of these point estimates.First, in discussing these estimates, it is important to emphasize the fact that the coefficientsshould (in theory) be small numbers and there are only 48 observations per regression. For 0λ , itis insignificantly different from zero for all companies and all models except amr. But this in notunexpected, since the magnitude of average historical default intensities observed frombankruptcy data is less than .01 (see Chava and Jarrow (2000)), which is much less than thestandard error of this coefficient. The F-test for model 3 tests the hypothesis that 0t0 =λ . TheF-test average p-score provided confirms that this hypothesis cannot be rejected for all firms inthe data set.
15
With respect to the spot rate coefficient in the default intensity, 1λ , the significance of itst-scores varies across firms. For 8 out of the 15 firms, the average p-score is less than .15 for thiscoefficient for either model 5 or 6. This observation is supported by the lower values for the F-test average p-score for model 5 that tests the joint hypothesis 0t1t0 == λλ .
Finally, with respect to the market index coefficient in the default intensity, 2λ , theaverage p-scores indicate that it is insignificant from zero at the 15 percent level for all firmsexcept three (amr, bt, cce). Unfortunately, the introduction of the independent variable
( ))t(T)(t(Z)tT)(t(Z *L
*L ∆∆ −−−−− ) into this regression causes a severe multi-collinearity
problem with another independent variable, the excess return on the market portfolio. As seen in
models 7 and 8, for all firms the introduction of ( ))t(T)(t(Z)tT)(t(Z *L
*L ∆∆ −−−−− ) causes
the point estimates of 0β to change dramatically from its values in models 1 – 6 (see Table 3)
and 0β becomes insignificantly different from zero (see Table 4). The correlation matrix inTable 5 confirms this multi-collinearity problem. The correlation between these two variables is.9934. This implies that the coefficient 2λ and 0β cannot be separated using this regressionmodel.
The impact of these different default intensity models on the point estimates of thedefault intensities can be dramatic. This can be seen from Table 7. Included in Table 7 are theaverage default intensities, the average standard error for the default intensities, and the averageone-year default probabilities. Although the differences are dramatic, the average standard errorsexceed the differences in these default intensity estimates across the models. Note also that theaverage default intensity is sometimes negative, especially for models 7 and 8 that include themarket index in the default intensity. Negative default intensities should be interpreted as being apoint estimate of zero. These occur due to the linear intensity approximation to expression (12)and the large standard errors for the estimates.
Also to be noticed at this juncture are the magnitudes of the RMSE of the regressionmodel in comparison to the average magnitude of the dependent variable. The RMSE is anestimate of the standard error of the unpredictable component of the equity returns. The averagemagnitude of the dependent variable is the average equity return over this period. As indicated inTable 7, the standard error of the unpredictable component of the equity’s return is over 10 timesthe magnitude of its average value. This is true for all firms in our sample. This illustrates themagnitude of the “noise” in equity prices relative to our predictive ability using the Fama-Frenchfour-factor model. This noise inhibits our ability to estimate the default intensities with a greatdeal of precision. It provides an alternative perspective on the reason that the default parameterestimates are often insignificantly different from zero.
10. Relative Performance of the Equity Models
This section studies the relative performance of the eight equity pricing models. For eachfirm and for each model’s regression, both a root mean squared error statistic (RMSE) and ageneralized cross validation statistic (GCV) are computed. The RMSE statistic measures the“average” pricing error between the model and the market bid. It is an in-sample goodness of fitmeasure. As with all in-sample goodness of fit measures, a potential problem with RMSE is thatit may provide a biased picture of the quality of model performance due to a model over-fittingthe noise in the data. The second GCV test statistic is designed to partially overcome this
16
problem, as an out-of-sample goodness of fit measure that is predictive in nature.13 The lower theGCV statistic, the better the out-of-sample model fit.
The average RMSE and GCV statistics for each firm and model are contained in Table 7.For 6 of the 15 firms (amr, cce, cpl, ibm, luv, txu) the RMSE statistic is smallest (or within .0001of the smallest value) for models 3 – 6. For 7 of the 15 firms (amr, bt, cpl, ibm, mob, txu, wmt)the GCV statistic is smallest (or within .0001 of the smallest value) for models 3 – 6. This isstrong evidence consistent with the importance of including the default parameters into the equityregression equation and the feasibility of using equity prices to infer default intensity estimates.
Surprisingly, for no firms do models 7 – 8 have the smallest RMSE or GCV statistics.Despite the multi-collinearity problem present when including Z(t) into the default intensityprocess, this is strong evidence consistent with the insignificance of the 2λ coefficient. This
relative performance analysis confirms the insignificance of the 2λ coefficient documented inJanosi, Jarrow, Yildirim (2000) for the same firms, but using debt prices.
11. Comparison of Default Intensities based on Debt versus Equity
This section investigates the equality between the default intensities estimated using theequity price model with the default intensities estimated implicitly using a reduced form creditrisk model and debt prices as contained in Janosi, Jarrow, Yildirim (2000).
11.a The Reduced Form Model
We briefly review the reduced form credit risk model and the estimation procedures ofJanosi, Jarrow, Yildirim (2000). First some notation. Let v(t,T) represent the time t price of apromised dollar to be paid by a risky firm at time T where Tt0 ≤≤ . The debt is risky because ifthe firm defaults prior to time T, then the promised dollar may not be paid. If default occurs, therisky zero-coupon bond receives a fractional recovery of )T,(v −τδ dollars where δ≤0 is aconstant and −τ represents an instant before default.
The risky debt is assumed to trade in potentially illiquid markets with a liquidity premium
)t(γ interpreted as a convenience yield. Under this structure and the assumption of no arbitrage,Janosi, Jarrow and Yildirim (2000) show that the risky debt’s value can be written as:
)1(du)]1)(u()u(r[
t)t(
T
teEe)T,t(vδλ
γ−+∫−
− ⋅= (23)
where )(Et ⋅ denotes conditional expectation with respect to the martingale measure Q . Expression (23) represents a liquidity discount times the expected value of a dollar received attime T discounted at the spot rate r(u) augmented by the expected loss per unit time )1)(t( δλ − .This expectation representation is a standard result in the literature. It is important to note that theintensity process and recovery rate always appear as a product in expression (23).
13 Roughly speaking, The GCV statistics measures the average predictive error obtained by systematicallyeliminating each data point from the time series regression, predicting that point’s value with theregression, and then measuring the “average” predictive error that results, after adjusting for degrees offreedom (see Wahba (1985)).
17
To estimate the intensity process, an additional assumption is imposed: an approximate
linear intensity process, i.e.]0),t(Z)t(rmax[)t( 210 λλλλ ++= (24)
where 210 ,, λλλ are constants. In the empirical implementation, as an approximation tosimplify the numerical inversion, the max operator is removed.
Janosi, Jarrow and Yildirim also assume that the spot rate r(t) evolves according to anextended Vasicek model and that Z(t) evolves as a standard Brownian motion.
Matching coupon-bearing bond prices, Janosi, Jarrow, Yildirim (2000) implicitlyestimate the expected loss per unit time )1)(t( δλ − in expression (23). The bond prices areobtained from the University of Houston’s Fixed Income Database, see Warga (1999). The spotrate is the 3-month Treasury bill yield obtained from the CRSP database. The time period coveredis May 1991 – March 1997. Z(0) is initialized at zero on May 24, 1991, the start of the estimationperiod.
They selected twenty different firms, fifteen overlap with this study, see Table 1. Janosi,Jarrow, Yildirim estimated five different liquidity premium models. We use only the best fittingmodel, the constant liquidity premium. To match the best fitting equity pricing model (model 6)above, we use the intensity function from Janosi, Jarrow, Yildirim (2000) without the marketindex included. For this credit risk model, we have the time series of the estimated expected lossper unit time )1)(t( δλ − for each of the fifteen companies from Janosi, Jarrow, Yildirim (2000)and their standard errors.
11.b A Comparison Test
This section provides the statistical tests for equality of the explicit hazard rates estimatedusing the equity price model with the implicit hazard rates estimated in Janosi, Jarrow, Yildirim(2000) using a reduced form model. Due to non-overlapping periods of observations in the twostudies, only 10 firms are included in this comparison. The five firms omitted are: amr, bt, cce,cpl, ek.
Given are estimated intensities )t(ieλ from the equity price model and estimated
expected losses (per unit time) =)t(aid )1)(t( ii
d δλ − from expression (23) for ten firms. Thenull hypothesis to be tested is:
(debt) )t()1/()t(a ie
iid λδ =− (equity) for all firms i and all times t.
For the equity price model we use model 6 where the default intensity depends only on the spotrate of interest and the equity premium includes the P/E ratio. To be consistent, for the debtbased estimates, we drop the Z(t) term for the default intensity process. In Jarrow, Janosi,Yildirim (2000) this term is never significant, so this omission is consistent with the evidenceobtained therein.
A Test for Equality of the Implicit and Explicit Estimates
18
This statistical test provides a direct test for the equality of the two intensity estimates.
Given an assumed value for the recovery rate iδ , we can do a pair-wise t-test (using the standard
errors of the estimates) of the difference [ )t(aid )1/( iδ− ]– )t(i
eλ for a given firm i for eachdate t. Under the null hypothesis this difference is zero. Unfortunately, this is a joint test of the
null hypothesis and the assumed value of iδ . To eliminate this joint hypothesis, we test for
equality of these differences over a range of different values for iδ from 0 to .9. If there is a
value for iδ where the null hypothesis [ )t(a~ id )1/( iδ− ]– )t(i
eλ = 0 is not rejected, then this
value for iδ is an estimate of the recovery rate. The relevant summary statistics for these testsare contained in Tables 8.
For each company, Table 8 provides the average default intensities over the sampleperiod, the average standard errors of these estimates, and the average t-score of their differences.For the debt based estimates, we only report values for a recovery rate of zero. As seen, the theaverage differences are much less than the average standard error of the difference. In fact, for all
firms, for all time periods, and for all recovery rates iδ between 0 and .9, one cannot reject thenull hypothesis that the two intensity estimates are equivalent.
Although this evidence is consistent with the equivalence of the two default intensities,the large standard errors for the equity model estimates indicates that this is a weak test of the nullhypothesis. In fact, the evidence is consistent with any recovery rate between 0 and less than one.The large standard errors for the equity model estimates generate these results. Indeed, the debtbased estimates standard errors are of the same order of magnitude as the intensity values, whilethe standard errors for the equity based estimates are approximately ten times the size of theintensity values.
A Test for Constant Recovery Rates
This section provides an independent test for the time series stationarity of the recovery
rate, estimated as the ratio of the implicit and explicit estimates, i.e. )1()t(/)t(a iie
id δλ −= .
Under the null hypothesis, the difference between adjacent time series estimates of the ratio is
zero for any firm i, i.e. 0)1t(
)1t(a
)t(
)t(aie
id
ie
id =
−−
−λλ
. Given noise in the estimated hazard rates, we
test to see if this difference follows a random walk. That is, we run the following regression:
tie
id
ie
id
ie
id
ie
id
t
ta
t
ta
t
ta
t
ta ελλ
ηηλλ
+
−−
−−−
+=
−−
−)2(
)2(
)1(
)1(
)1(
)1(
)(
)(10 where tε is an error term.
We test the hypotheses 00 =η and 01 =η . Accepting the first hypothesis 00 =η wouldprovide evidence consistent with a constant recovery rate. Accepting the second hypothesis
01 =η would imply that the differences in the ratios are uncorrelated over time. Thisuncorrelated hypothesis may be rejected, however, because of the nature of the estimation
procedures for )t(aid and )t(i
eλ as described above. Both of these estimation procedures useoverlapping data in their computations, which could induce a correlation in the time series of thedifferences of the ratios of the estimates.
19
Tables 9 contain the results of these regressions. As shown, the first hypothesis 00 =η is
accepted in all cases. The second hypothesis 01 =η is rejected at the 10 percent level for five ofthe ten firms. This is strong evidence consistent with a constant recovery rate.
12. Conclusion
This paper uses a reduced form model to estimate default probabilities implicit in equityprices. The model implemented is a generalization of the model contained in Jarrow (2000). Thetime period covered is May 1991 – March 1997. Monthly equity prices on 15 different firms’debt issues are studied. The firms are chosen to provide a stratified sample across variousindustry groupings.
Four general conclusions can be drawn from this investigation. First, equity prices can beused to infer a firm’s default intensities. This is a feasibility result. Second, due to the noisepresent in equity prices, the point estimates of the default intensities that are obtained are not veryprecise. They have large standard errors and they exhibit time series non-stationarities. Thisconclusion casts doubts upon the reliability of the default probability estimates obtained fromstructural models as in Delianedis and Geske (1998) (confirming the previous conclusions ofJarrow and van Deventer (1998, 1999) in this regard). Third, equity prices appear to contain abubble component, as proxied by the firm’s P/E ratio. Bubbles in equity returns are notcompletely captured by the four factor model of Fama and French (1993, 1996). Fourth, wecompare the default probabilities obtained from equity with those obtained implicitly from debtprices using the reduced form model contained in Janosi, Jarrow, Yildirim (2000). We find thatdue to the large standard errors of the equity price estimates, one cannot reject the hypothesis thatthese default intensities are equivalent. This re-confirms both the feasibility of using equity datato estimate default probabilities and the imprecision of the estimates obtained.
20
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22
Appendix
From the appendix in Jarrow (2000) we have that:
.)tT(
)T,t(b3/)tT(2/)T,t(b
)]tT)(t(Z)tT)(t(Z[
]2/)T,t(b)T,t(p
)T,t(plog[
du)u(L
Llog
)e:j,t(vD
)e:j,t(vD
log
*LmLL2
*LrLL1
2*L
22
2*L
21
*L
*L2
2*L*
L
*L
1
0
t
tt
t
T
tjjt
T
tjjt
*D
*D
∆ϕσλ
∆∆ϕσλ∆λ∆∆λ
∆∆λ
∆∆∆
λ
∆λµ∆ξ
ξ
∆θ
∆
∆∆
−−
−−−+−+
−−−−−−
−+
−−−
−∫+
=
∑ −−
∑−
−−
−≥−
≥
In the previous expression the following quantities are unobservable: mLrL ,ϕϕ . To eliminatethese quantities from this expression, we compute the variance of the preceding expression.
≡∆σξ )t(2
−∑−−
∑−
−≥
≥
)e:j,t(vD)t(
)e:j,t(vD)t(
logvar
j
*T
tj
j
*T
tjt
D
D
∆∆ξ
ξ
∆
=
])]tT)(t(Z)tT)(t(Z[
]2/)T,t(b)T,t(p
)T,t(plog[
L
Llogvar
*L
*L2
2*L*
L
*L
1t
tt
∆∆λ
∆∆∆
λ∆
−−−−−−
−+
−−−
−
But, we have that:
DET)T,t()T,t()T,t(p
)T,t(plog
2
)T,t(b *L1
*L1*
L
*L
2*L +−−=
−−
−∆µµ
∆∆
∆
DET/)]t(r)t(r)[T,t(b
DET/)]t(r)T,t(b)t(r)T,t(b[
r*L
r*L
*L
+−−=
+−−−=
σ∆
σ∆∆
where DET indicates non-random terms. Also note that
DET)v(dWe)t(r)t(rt
t
)tv(ar +∫=−−
−
−−
∆σ∆ . Substitution gives
=
−−
−)T,t(p
)T,t(plog
2
)T,t(b*L
*L
2*L
∆∆
∆
∫
−
−−t
t
)tv(a*L )v(dWe)T,t(b
∆+ DET.
Combined we get that:
23
]DET)]t(Z)t(Z)[tT(
)v(dWe)T,t(b)]t(w)t(w[var)t(
*L2
t
t
)tv(a*L1LLLt
2
+−−−−
∫−−−=
−
−−
∆λ
λ∆σ∆σ∆
ξ
Computing this variance yields:
∆ϕλλ∆ϕσλ∆ϕσλ
∆σ∆λ∆λ∆σξ
)t*T)(T,t(b2)t*T(2)T,t(b2
)tT()T,t(b)t(
*Lrm21mLL2rLL
*L1
2L
2*L
22
2*L
21
2
−+−−−
+−+≈
Where we have used the facts that:
( ) ∆∆
∆∆≈−=∫=∫
−
−−
−
−− a2/e1dve))v(dWe(var a2t
t
)tv(a2t
t
)tv(at and
( ) ∆∆
∆≈−=∫
−
−− a/e1dve at
t
)tv(a .
Rearranging terms gives:
.)t*T()T,t(b3/)tT(2/)T,t(b
6/)tT()t*T)(T,t(b2/2/)t(
mLL2rLL*L1
2*L
22
2*L
21
2*L
22
*Lrm21
2L
2
∆ϕσλ∆ϕσλ∆λ∆λ
∆λ∆ϕλλ∆σ∆σξ
−−−−+≈
−−−−−
Substitution gives the result:
6/)tT()t*T)(T,t(b2/2/)t(
)]tT)(t(Z)tT)(t(Z[
]2/)T,t(b)T,t(p
)T,t(plog[
du)u(L
Llog
)e:j,t(vD
)e:j,t(vD
log
2*L
22
*Lrm21
2L
2
*L
*L2
2*L*
L
*L
1
0
t
tt
t
T
tjjt
T
tjjt
*D
*D
∆λ∆ϕλλ∆σ∆σ
∆∆λ
∆∆∆
λ
∆λµ∆ξ
ξ
ξ
∆θ
∆
∆∆
−−−−−+
−−−−−−
−+
−−−
−∫+
=
∑ −−
∑−
−−
−≥−
≥
Using Girsanov’s theorem, we have that the original Brownian motion
∫+=t
0LLL
du)u()t(w)t(w Θ where )t(wL is a Brownian motion under the statistical measure and
)u(L
Θ is the liquidation value’s risk premium. Hence,
)].t(w)t(w[)2/1()t()t(r
)L/Llog(
LLL2LLL
tt
∆σ∆σ∆∆Θσ∆∆∆
−−+−−+−
=−
Here is the final result:
24
2/)t()t()t(
)t(r
)e:j,t(vD
)e:j,t(vD
log
22LLL
T
tjjt
T
tjjt
*D
*D
∆σ∆∆µ∆σ∆∆Θσ
∆∆∆ξ
ξ
ξθ
∆∆
+−+−−+
=−−
∑ −−
∑−
−≥−
≥
)].t(w)t(w[
6/)tT()t*T)(T,t(b
)]tT)(t(Z)tT)(t(Z[
]2/)T,t(b)T,t(p
)T,t(plog[
LLL
2*L
22
*Lrm21
*L
*L2
2*L*
L
*L
10
∆σ∆λ∆ϕλλ
∆∆λ
∆∆∆
λ∆λ
−−+−−−−
−−−−−−
−+
−−−−
25
Figure 1: Treasury Forward Rate Curves from January 1990 to March 1997.
Maximum smoothness forward rate curves (piecewise-joined 4th degree polynomials) fitted to stripped U.S.Treasury zero-coupon bond prices. The stripping procedure chooses the zero-coupon bond prices tominimize the sum of squared differences between the market price of the coupon bearing U.S. Treasurydebt and the price of a portfolio of zero-coupon bonds generating the same cash flows.
26
Figure 2: Time Series Estimates of the Spot Rate Parameters (a, rσ ) with 90 percentConfidence Bands from May 1991 – March 1997.
The spot rate model is )t(dWdt)]t(r)t(r[a)t(dr rσ+−= where r(t) is the spot rate of interest at time t,a is the mean reversion parameter, r(t) is a deterministic function of t chosen to match the initial zero-coupon bond price curve, rσ is the spot rate volatility per year, and W(t) is a standard Brownian motion.
The parameters (a, rσ ) are estimated using a non-linear regression matching the model’s implied variancesfor zero-coupon bond prices to the observed sample variances of stripped U.S. Treasury zero-coupon bondprices. These estimates are based on monthly observations. A rolling estimation procedure is utilized sothat the time t estimated parameters depend only on information available at time t. At each date, theregression generates a standard error for the estimates (a, rσ ). The 90 percent confidence band and the
average values for (a, rσ ) over the observation period are depicted.
May91 Mar92 Jan93 Nov93 Sep94 Jul95 May96 Mar970
0.01
0.02
0.03
0.04
0.05Figure 2a: Spot Rate Parameter (a)
acimean(a)
May91 Mar92 Jan93 Nov93 Sep94 Jul95 May96 Mar970.009
0.01
0.011
0.012
0.013Figure 2b: Spot Rate Parameter (σr)
σr
cimean(σr)
27
Figure 3: Time Series Estimates of the Market Index Parameters ( rmm ,ϕσ ) with 90percent Confidence Bands from May 1991 – March 1997.
The market index model is )]t(dZdt)t()[t(M)t(dM mm σµ += where M(t) is the S&P 500 index value
at time t, mµ is the expected return per year on the market index, mσ is the market volatility per year,
rmϕ is the correlation coefficient between dr(t) and dM(t)/M(t) where r(t) is the spot interest rate at time t
(the 3 month Treasury bill yield), and Z(t) is a standard Brownian motion. The parameters (rmm ,ϕσ ) are
estimated using the sample moments based on the previous 365 business days. These estimates are basedon daily observations starting May 24, 1991. A rolling estimation procedure is utilized so that the time testimated parameters depend only on information available at time t. The standard deviation of the timeseries values for (
rmm ,ϕσ ) over the entire observation period were used to generate the 90 percent
confidence bands. The average values for (rmm ,ϕσ ) over the observation period are also depicted.
May91 Apr92 Feb93 Dec93 Oct94 Aug95 Jun96 Mar970.05
0.1
0.15
0.2
0.25
Figure 3a: Market Volatility σm
σmcimean(σm)
May91 Apr92 Feb93 Dec93 Oct94 Aug95 Jun96 Mar97-0.6
-0.4
-0.2
0
0.2
0.4Figure 3b: Market and Interest Rate Correlation φrm
φrm
cimean(φrm)
28
Figure 4: Time Series Graph of the Spot Interest Rate (r(t)) and the CumulativeExcess Return per Unit of Risk on the Market Index (Z(t)) over the time period May1991 to March 1997.
Given are daily observations of the 3-month Treasury Bill yield (r(t)) and the cumulative excess return perunit of risk on the S&P 500 (Z(t)). The cumulative excess return per unit of risk is computed as
365/1/)]365/1()2/1()365/1)(t(r))1t(M/)t(M[log()1t(Z)t(Z m2m σσ+−−+−= with Z(0)
= 0 on May 24, 1991, where M(t) is the market index at time t, r(t) is the spot rate of interest at time t, and
mσ is the market index’s volatility per year.
May91 Apr92 Feb93 Dec93 Oct94 Aug95 Jun96 Mar97-5
0
5
10Evolution of r(t) and Z(t)
May91 Apr92 Feb93 Dec93 Oct94 Aug95 Jun96 Mar970.02
0.04
0.06
0.08
r(t)
Z(t)
29
Figure 5: Time Series Estimates of Eastman Kodak Company's Intensity Function
May95 Aug95 Oct95 Dec95 Feb96 Apr96 Jun96 Aug96 Oct96 Dec96 Mar97-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
λ(t)=λ0+λ1 r(t)+λ2 z(t)
Model 3Model 4Model 5Model 6Model 7Model 8
30
TickerSymbol
SICCode
First Dateused in theEstimation
Last Dateused in theEstimation
Numberof
Bonds
Moodies S&P
Financials
BANKERS TRUST NY bt 6022 01/31/1994 04/30/1994 3 A1 AAMERRILL LYNCH & CO mer 6211 12/31/1991 03/31/1997 14 A2 A
Food & BeveragesPEPSICO INC pep 2086 12/31/1991 03/31/1997 8 A1 ACOCA - COLA ENT. INC cce 2086 12/31/1991 06/30/1994 3 A2 AA-
AirlinesAMR CORPORATION amr 4512 02/29/1992 08/31/1994 2 Baa1 BBB+SOUTHWEST AIRLINES luv 4512 05/31/1992 03/31/1997 3 Baa1 A-
UtilitiesCAROLINA POWER LIGHT cpl 4911 08/31/1992 01/31/1993 3 A2 ATEXAS UTILITIES ELE CO txu 4911 04/30/1994 03/31/1997 4 Baa2 BBB
PetroleumMOBIL CORP mob 2911 12/31/1991 02/29/1996 3 Aa2 AA
Department StoresSEARS ROEBUCK + CO s 5311 12/31/1991 08/31/1996 7 A2 AWAL-MART STORES, INC wmt 5331 12/31/1991 03/31/1997 3 Aa3 AA
TechnologyEASTMAN KODAK COMPANY ek 3861 01/31/1992 09/30/1994 3 A2 A-XEROX CORP xrx 3861 12/31/1991 03/31/1997 4 A2 ATEXAS INSTRUMENTS txn 3674 10/31/1992 03/31/1997 3 A3 AINTL BUS MACHINES ibm 3570 01/31/1994 03/31/1997 3 A1 AA-
Table 1: Details of the Firms Included in the Empirical Investigation.
Ticker Symbol is the firm’s ticker symbol. SIC is the Standard Industry Code. Number of bondscorresponds to the number of distinct bond issues used in the estimation. Moodies refers to Moodies’ debtrating for the company’s senior debt on the first date used in the estimation. S&P refers to S&P’s debtrating for the company’s debt on the first date used in the estimation.
31
Unit Root TestStatistics
σr -2.63480.1000
a -1.16320.9999
σm -3.94070.0100
rmσ -1.34790.9999
Table 2: Unit Root Test for the Spot Rate of Interest and Market Index Parameters
There are two entries in each cell. The first entry is the modified Dickey-Fuller (DF) test statistic for a unitroot test. The modified DF test statistic is given as the t-statistic of the 1β coefficient in the linear
regression: t1t10t yy εββ∆ ++= − where yt represents the time t value of each parameter,
1ttt yyy −−=∆ , and tε is an error term. The null hypothesis for a unit root is 01 =β . The second entryis the P-score for the DF test statistic, i.e. it is the probability of rejecting the unit root when it is true.
32
FINANCIALS1-BANKERS TRUST NY
bt O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 1.7055 0.0071 0.0102 -0.0018 -2.3971 0.4873Model 2 1.7051 0.0068 0.0097 -0.0024 -2.8753 0.0034 0.4901Model 3 0.0000 1.7055 0.0071 0.0102 -0.0018 -2.3971 0.4873Model 4 0.0000 1.7051 0.0068 0.0097 -0.0024 -2.8753 0.0034 0.4901Model 5 0.0000 0.1342 1.5888 0.0075 0.0094 -0.0025 -2.3947 0.4933Model 6 0.0000 0.1188 1.6023 0.0072 0.0091 -0.0030 -2.8431 0.0031 0.4954Model 7 0.0000 0.1113 -0.0146 -0.9716 0.0071 0.0089 -0.0020 -2.4422 0.5101Model 8 0.0000 0.0940 -0.0151 -1.0620 0.0068 0.0086 -0.0025 -2.9128 0.0033 0.51192-MERRILL LYNCH & CO
mer O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 2.0601 0.0046 0.0055 0.0063 -1.7785 0.8292Model 2 1.9509 0.0033 0.0035 0.0041 -1.9064 0.0089 0.8379Model 3 0.0000 2.0601 0.0046 0.0055 0.0063 -1.7785 0.8292Model 4 0.0000 1.9509 0.0033 0.0035 0.0041 -1.9064 0.0089 0.8379Model 5 0.0000 0.0543 2.0238 0.0048 0.0055 0.0059 -1.7770 0.8319Model 6 0.0000 0.0891 1.8840 0.0036 0.0033 0.0035 -1.9029 0.0089 0.8406Model 7 0.0000 0.0224 -0.0094 0.4801 0.0044 0.0052 0.0065 -1.7676 0.8377Model 8 0.0000 0.0596 -0.0115 -0.0478 0.0032 0.0029 0.0041 -1.8902 0.0089 0.8459
FOOD & BEVERAGES3-PEPSICO INC
pep O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 1.1323 -0.0058 -0.0046 0.0053 -0.8301 0.5877Model 2 1.1177 -0.0057 -0.0046 0.0049 -0.9515 0.0053 0.5889Model 3 0.0000 1.1323 -0.0058 -0.0046 0.0053 -0.8301 0.5877Model 4 0.0000 1.1177 -0.0057 -0.0046 0.0049 -0.9515 0.0053 0.5889Model 5 0.0000 -0.0025 1.1326 -0.0058 -0.0046 0.0053 -0.8257 0.5884Model 6 0.0000 -0.0031 1.1190 -0.0057 -0.0046 0.0049 -0.9485 0.0052 0.5895Model 7 0.0000 -0.0147 -0.0031 0.6321 -0.0062 -0.0048 0.0056 -0.8734 0.5944Model 8 0.0000 -0.0155 -0.0029 0.6606 -0.0061 -0.0048 0.0053 -0.9483 0.0046 0.59594-COCA-COLA
cce O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 0.9856 -0.0028 0.0001 0.0009 0.5367 0.1675Model 2 0.9706 -0.0025 -0.0002 0.0007 0.5427 0.0243 0.1764Model 3 0.0000 0.9867 -0.0027 0.0001 0.0010 0.8118 0.1714Model 4 0.0000 0.9711 -0.0024 -0.0002 0.0008 0.8050 0.0237 0.1800Model 5 0.0000 -0.4262 1.3757 -0.0041 0.0029 0.0033 0.6184 0.2154Model 6 0.0000 -0.4209 1.3553 -0.0038 0.0026 0.0031 0.6119 0.0226 0.2227Model 7 0.0000 -0.4112 0.0246 5.8176 -0.0038 0.0038 0.0031 0.5164 0.2470Model 8 0.0000 -0.4050 0.0250 5.8588 -0.0035 0.0034 0.0029 0.5195 0.0240 0.2556
AIRLINES5-AMR CORPORATION
amr O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 1.7603 0.0065 0.0008 -0.0028 -0.2679 0.3543Model 2 1.9060 0.0060 0.0012 -0.0027 -0.8753 -0.0100 0.4100Model 3 0.3620 1.8585 0.0087 0.0008 -0.0027 2.3496 0.3920Model 4 0.3237 1.9864 0.0079 0.0012 -0.0026 1.4977 -0.0094 0.4407Model 5 0.3400 -0.1733 2.0157 0.0080 0.0021 -0.0018 2.1111 0.3999Model 6 0.2846 -0.2695 2.2471 0.0068 0.0032 -0.0011 1.0456 -0.0104 0.4569Model 7 0.3220 -0.2093 -0.0252 -2.4435 0.0068 0.0010 -0.0012 1.9903 0.4350Model 8 0.2645 -0.3157 -0.0279 -2.6644 0.0055 0.0021 -0.0004 0.8783 -0.0110 0.4979
33
6-SOUTWEST AIRLINES COluv O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 2.0166 0.0034 0.0113 0.0014 -1.6570 0.4663Model 2 1.9524 0.0032 0.0109 0.0001 -1.7540 0.0410 0.4796Model 3 0.0000 2.0166 0.0034 0.0113 0.0014 -1.6570 0.4663Model 4 0.0000 1.9524 0.0032 0.0109 0.0001 -1.7540 0.0410 0.4796Model 5 0.0003 -0.7625 2.6835 0.0010 0.0163 0.0049 -1.6201 0.5093Model 6 0.0301 -0.7950 2.6472 0.0007 0.0160 0.0037 -1.7129 0.0512 0.5244Model 7 0.0000 -0.7994 -0.0103 0.9490 0.0003 0.0158 0.0053 -1.5969 0.5143Model 8 0.0303 -0.8231 -0.0077 1.3494 -0.0001 0.0155 0.0040 -1.7036 0.0508 0.5286
UTILITIES7-CAROLINA POWER + LIGHT
cpl O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 0.8464 -0.0031 0.0081 0.0060 -1.8684 0.8275Model 2 0.8692 -0.0029 0.0084 0.0064 -1.6545 -0.0039 0.8294Model 3 0.0381 0.8704 -0.0030 0.0084 0.0064 -1.6212 0.8281Model 4 0.0311 0.8752 -0.0029 0.0085 0.0065 -1.6170 -0.0017 0.8292Model 5 0.0390 0.2121 0.6874 -0.0024 0.0072 0.0052 -1.6262 0.8348Model 6 0.0182 0.2127 0.6921 -0.0022 0.0073 0.0054 -1.6213 -0.0032 0.8357Model 7 0.0334 0.1983 -0.0099 -1.0674 -0.0029 0.0066 0.0055 -1.6736 0.8401Model 8 0.0181 0.1991 -0.0096 -1.0114 -0.0027 0.0068 0.0056 -1.6657 -0.0026 0.84058-TEXAS UTILITIES ELE CO
txu O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 0.2394 -0.0040 0.0032 0.0035 -3.0857 0.2101Model 2 0.2289 -0.0041 0.0033 0.0034 -3.0423 -0.0003 0.2278Model 3 0.0000 0.2394 -0.0040 0.0032 0.0035 -3.0857 0.2101Model 4 0.0000 0.2289 -0.0041 0.0033 0.0034 -3.0423 -0.0003 0.2278Model 5 0.0000 0.3676 -0.0850 -0.0028 0.0010 0.0015 -2.8828 0.2988Model 6 0.0000 0.3553 -0.0853 -0.0030 0.0011 0.0015 -2.9138 0.0003 0.3100Model 7 0.0000 0.3697 0.0034 0.5108 -0.0030 0.0010 0.0015 -2.8770 0.3169Model 8 0.0000 0.3588 0.0034 0.5161 -0.0032 0.0011 0.0015 -2.9263 0.0005 0.3257
PETROLEUM9-MOBIL CORP
mob O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 0.6879 -0.0040 0.0031 0.0031 0.8424 0.3943Model 2 0.6894 -0.0040 0.0029 0.0031 1.5378 -0.0028 0.4077Model 3 0.0192 0.6922 -0.0040 0.0031 0.0031 1.1650 0.4018Model 4 0.0109 0.6909 -0.0040 0.0029 0.0031 1.6001 -0.0023 0.4094Model 5 0.0401 0.1431 0.5607 -0.0036 0.0021 0.0024 1.7330 0.4170Model 6 0.0613 0.1502 0.5530 -0.0035 0.0020 0.0025 2.0864 0.0002 0.4234Model 7 0.0278 0.1308 -0.0094 -1.1221 -0.0039 0.0017 0.0027 1.5653 0.4408Model 8 0.0525 0.1382 -0.0096 -1.1741 -0.0038 0.0015 0.0027 1.9800 0.0002 0.4478
DEPARTMENT STORES10-SEARS ROEBUCK + CO
s O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 1.7989 0.0076 0.0091 0.0002 -1.6245 0.7250Model 2 1.8000 0.0077 0.0091 0.0002 -1.6207 0.0004 0.7251Model 3 0.0000 1.7989 0.0076 0.0091 0.0002 -1.6245 0.7250Model 4 0.0000 1.8000 0.0077 0.0091 0.0002 -1.6207 0.0004 0.7251Model 5 0.0000 -0.0489 1.8525 0.0074 0.0097 0.0004 -1.6342 0.7289Model 6 0.0000 -0.0481 1.8531 0.0075 0.0097 0.0003 -1.6305 0.0002 0.7289Model 7 0.0000 -0.0737 -0.0107 0.0078 0.0071 0.0094 0.0008 -1.6406 0.7341Model 8 0.0000 -0.0718 -0.0108 -0.0013 0.0071 0.0094 0.0007 -1.6364 0.0004 0.7342
34
11-WAL-MART STORES, INCwmt O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 1.1142 0.0061 -0.0023 -0.0104 -1.6027 0.7309Model 2 0.9708 0.0048 -0.0042 -0.0125 -1.6049 0.0481 0.7770Model 3 0.0000 1.1142 0.0061 -0.0023 -0.0104 -1.6027 0.7309Model 4 0.0000 0.9580 0.0044 -0.0046 -0.0124 -1.5771 0.0879 0.7761Model 5 0.0000 0.1456 0.9904 0.0065 -0.0031 -0.0113 -1.5993 0.7342Model 6 0.0000 0.1169 0.8604 0.0048 -0.0052 -0.0130 -1.5737 0.0847 0.7779Model 7 0.0000 0.1612 0.0025 1.3806 0.0067 -0.0029 -0.0118 -1.5961 0.7368Model 8 0.0000 0.1327 0.0014 1.0377 0.0050 -0.0050 -0.0135 -1.5752 0.0817 0.7805
TECHNOLOGY12-EASTMAN KODAK COMPANY
ek O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 0.8666 0.0011 0.0058 0.0041 -2.5096 0.2610Model 2 0.8786 0.0024 0.0058 0.0048 -3.5253 0.0305 0.3036Model 3 0.0000 0.8666 0.0011 0.0058 0.0041 -2.5096 0.2610Model 4 0.0000 0.8786 0.0024 0.0058 0.0048 -3.5253 0.0305 0.3036Model 5 0.0000 0.1988 0.7020 0.0017 0.0049 0.0030 -2.5636 0.0000 0.2732Model 6 0.0000 0.2603 0.6611 0.0033 0.0045 0.0035 -3.6315 0.0319 0.3204Model 7 0.0000 0.2050 0.0188 4.1316 0.0018 0.0053 0.0029 -2.5050 0.2892Model 8 0.0000 0.2587 0.0159 3.5874 0.0031 0.0047 0.0035 -3.5886 0.0320 0.335513-XEROX CORP
xrx O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 1.2132 0.0103 0.0018 -0.0045 -0.0673 0.6048Model 2 1.2573 0.0105 0.0024 -0.0041 -0.1420 -0.0019 0.6125Model 3 0.0000 1.2132 0.0103 0.0018 -0.0045 -0.0673 0.6048Model 4 0.0000 1.2573 0.0105 0.0024 -0.0041 -0.1420 -0.0019 0.6125Model 5 0.0000 -0.1778 1.3697 0.0097 0.0029 -0.0035 -0.0905 0.6135Model 6 0.0000 -0.2029 1.4490 0.0099 0.0038 -0.0030 -0.1817 -0.0023 0.6230Model 7 0.0000 -0.1744 -0.0040 0.6035 0.0097 0.0028 -0.0036 -0.0833 0.6167Model 8 0.0000 -0.1977 -0.0034 0.8002 0.0099 0.0037 -0.0031 -0.1731 -0.0022 0.625914-TEXAS INSTRUMENTS
txn O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 1.8865 0.0121 0.0043 0.0057 -1.2969 0.5997Model 2 1.7710 0.0128 0.0051 0.0044 -1.3995 0.0111 0.6083Model 3 0.0000 1.8865 0.0121 0.0043 0.0057 -1.2969 0.5997Model 4 0.0000 1.7710 0.0128 0.0051 0.0044 -1.3995 0.0111 0.6083Model 5 0.0000 -0.3016 2.1478 0.0110 0.0061 0.0072 -1.2859 0.6056Model 6 0.0000 -0.2451 1.9907 0.0119 0.0065 0.0058 -1.3867 0.0105 0.6130Model 7 0.0000 -0.2638 0.0329 8.0356 0.0123 0.0075 0.0068 -1.2984 0.6226Model 8 0.0000 -0.1975 0.0351 8.2313 0.0135 0.0080 0.0050 -1.4089 0.0115 0.631015-INTERNATIONAL BUSINESS MACHINES
IBM O0 O1 O2 E0 E1 E2 E3 E4 E5 R2
Model 1 0.9832 -0.0092 -0.0076 -0.0113 0.4104 0.2812Model 2 1.0009 -0.0092 -0.0075 -0.0103 0.1364 -0.0016 0.2867Model 3 0.0000 0.9832 -0.0092 -0.0076 -0.0113 0.4104 0.2812Model 4 0.0000 1.0011 -0.0092 -0.0075 -0.0103 0.1375 -0.0016 0.2867Model 5 0.0000 -0.6426 1.5795 -0.0110 -0.0030 -0.0079 0.2092 0.3467Model 6 0.0000 -0.6361 1.5908 -0.0111 -0.0030 -0.0070 -0.0300 -0.0015 0.3510Model 7 0.0000 -0.6643 0.0081 3.1913 -0.0113 -0.0028 -0.0074 0.1624 0.3601Model 8 0.0000 -0.6571 0.0084 3.2470 -0.0113 -0.0028 -0.0065 -0.0921 -0.0015 0.3651
35
Table 3: Averages of the Parameter Estimates from the Equity Model Regression.
Table 3 contains the average parameter estimates ),,,,,,,,( 543210210 ββββββλλλ across the months inthe observation period from the equity model regressions. They are presented for each company and foreach model type, separated by industries. Models 1 and 2 have no default. Models 3 and 4 have a constantdefault intensity. Models 5 and 6 have the default intensity dependent on the spot rate of interest. Models7 and 8 have the default intensity dependent on the spot rate of interest and a market index. The number ofobservations per regression is 48. The number of regressions in the average is 23. The average R^2 isgiven.
36
FINANCIALS1-BANKERS TRUST NY
bt O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 5.5380** 1.8227** 2.7373** -0.4994 -2.9552** 0.00000.0000 0.0984 0.0197 0.2014 0.0021
Model 2 5.5066** 1.7420** 2.5609** -0.6517 -2.8412** 0.7640 0.46760.0000 0.1121 0.0317 0.1895 0.0028 0.1877
Model 3 0.0000 5.3627** 1.7718** 2.6048** -0.4847 -2.1031** 0.47240.4038 0.0000 0.1009 0.0237 0.2073 0.0165
Model 4 0.0000 5.2426** 1.6834** 2.4404** -0.6423 -2.5027** 0.4001 0.47390.4038 0.0000 0.1148 0.0359 0.1924 0.0067 0.2791
Model 5 0.0000 0.6128 4.2349** 1.8387** 2.2223** -0.6554 -2.0834*/ 0.90890.4038 0.2245 0.0000 0.0855 0.0395 0.1937 0.0172
Model 6 0.0000 0.5264 4.0700** 1.7479** 2.0596** -0.7770 -2.4501** 0.3534 0.51050.4038 0.2484 0.0000 0.0988 0.0547 0.1709 0.0078 0.2929
Model 7 0.0000 0.5056 1.0319 -0.2236 1.6901** 2.0711** -0.5235 -2.1125** 0.89510.4038 0.2515 0.1376 0.2542 0.1037 0.0469 0.1970 0.0160
Model 8 0.0000 0.4130 1.0412 -0.2304 1.6001* 1.9175** -0.6569 -2.4987** 0.3974 0.47030.4038 0.2760 0.1360 0.2443 0.1138 0.0624 0.1837 0.0067 0.2795
2-MERRILL LYNCH & COmer O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 5.6185** 1.0160 1.2179 1.5668* -11.3822** 0.00000.0000 0.1314 0.1194 0.1021 0.0000
Model 2 5.3521** 0.7323 0.7375 0.9959 -11.2463** 1.7342** 0.10080.0000 0.1932 0.2084 0.1824 0.0000 0.0350
Model 3 0.0000 5.3299** 0.9922* 1.1646 1.4045 -9.8093** 0.10480.4038 0.0000 0.1357 0.1264 0.1151 0.0000
Model 4 0.0000 5.0937** 0.7218 0.7247 0.9091 -10.7425** 0.7307 0.09790.4038 0.0000 0.1957 0.2104 0.1902 0.0000 0.1881
Model 5 0.0000 0.2061 4.4580** 1.0111 1.0560 1.2486 -9.7198** 0.38750.4038 0.2188 0.0002 0.1334 0.1585 0.1143 0.0000
Model 6 -0.0010 0.3276 4.2388** 0.7738 0.5850 0.7543 -10.6137** 0.7083 0.09920.4035 0.2390 0.0003 0.1854 0.2069 0.2001 0.0000 0.1935
Model 7 0.0000 0.0880 0.8360 -0.0226 0.9096 0.9599 1.3513 -9.6544** 0.64720.4038 0.2384 0.1704 0.2177 0.1513 0.1667 0.1122 0.0000
Model 8 -0.0178 0.2193 0.9052 -0.1306 0.6707 0.4947 0.8665 -10.5088** 0.7001 0.11410.3983 0.2580 0.1710 0.2499 0.2081 0.2067 0.1925 0.0000 0.1963
FOOD & BEVERAGES3-PEPSICO INC
pep O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 3.7001** -1.468 -1.2377 1.5369* -6.3723** 0.00000.0002 0.0780 0.1051 0.0752 0.1754
Model 2 3.5745** -1.443 -1.2189 1.3207 -6.2572** 0.2765 0.76470.0002 0.0812 0.1090 0.0943 0.1649 0.3086
Model 3 0.0000 3.5882** -1.448 -1.2118 1.4538 -6.1766** 0.76700.4038 0.0002 0.0798 0.1085 0.0830 0.1928
Model 4 0.0000 3.4873** -1.391 -1.1434 1.2911 -6.1510** 0.0942 0.76820.4038 0.0003 0.0859 0.1195 0.0974 0.1678 0.3653
Model 5 0.0000 -0.0075 3.0278** -1.411 -1.1248 1.3777 -6.1028** 0.90470.4038 0.3316 0.0015 0.0833 0.1215 0.0903 0.1922
Model 6 0.0000 -0.0101 2.9128** -1.363 -1.0406 1.2415 -6.0635** 0.0905 0.77110.4038 0.3350 0.0021 0.0887 0.1353 0.1038 0.1671 0.3663
Model 7 0.0000 -0.0579 0.3636 0.2263 -1.472 -1.1621 1.4475 -6.0638** 0.96080.4038 0.3329 0.2211 0.2548 0.0715 0.1123 0.0746 0.1951
Model 8 0.0000 -0.0605 0.3342 0.2320 -1.422 -1.0670 1.3108 -5.9962** 0.0781 0.76670.4038 0.3365 0.2233 0.2558 0.0767 0.1274 0.0872 0.1712 0.3668
37
4-COCA-COLAcce O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 2.2356** -0.5606 -0.0089 0.1999 0.4983 0.00000.0129 0.2425 0.3221 0.3296 0.2371
Model 2 2.1831** -0.4840 -0.0607 0.1582 0.5010 0.5613 0.57720.0153 0.2671 0.3199 0.3253 0.2350 0.2324
Model 3 -0.1474 2.1907** -0.5266 -0.0132 0.2205 0.3934 0.55830.3608 0.0145 0.2493 0.3218 0.3240 0.2756
Model 4 -0.1384 2.1379** -0.4610 -0.0610 0.1784 0.3870 0.5333 0.58520.3634 0.0170 0.2720 0.3206 0.3200 0.2771 0.2358
Model 5 -0.1146 -1.395 2.6271** -0.7688 0.4691 0.6362 0.3002 0.46590.3694 0.0734 0.0041 0.1951 0.2520 0.2160 0.3008
Model 6 -0.1060 -1.367 2.5650** -0.7012 0.4122 0.5851 0.2945 0.5056 0.61050.3720 0.0768 0.0049 0.2096 0.2634 0.2299 0.3029 0.2461
Model 7 -0.0919 -1.345 -1.0574 1.5537* -0.7067 0.6037 0.5807 0.2476 0.46210.3759 0.0809 0.1419 0.0646 0.2007 0.2228 0.2296 0.3172
Model 8 -0.0844 -1.315 -1.0578 1.5432* -0.6278 0.5372 0.5340 0.2473 0.5666 0.56770.3782 0.0846 0.1441 0.0675 0.2264 0.2359 0.2430 0.3173 0.2287
AIRLINES5-AMR CORPORATION
amr O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 4.3989** 1.3297 0.1303 -0.5812 -0.2589 0.00000.0000 0.0852 0.3000 0.2043 0.2963
Model 2 4.8285** 1.2410 0.2359 -0.5693 -0.8111 -1.979** 0.07240.0000 0.1009 0.3043 0.2109 0.2012 0.0239
Model 3 -1.474 4.6431** 1.7082** 0.1479 -0.5779 1.1405 0.30540.0712 0.0000 0.0426 0.2961 0.2109 0.1299*
Model 4 -1.344 5.0234** 1.5838* 0.2451 -0.5637 0.7152 -1.865** 0.08440.0878 0.0000 0.0584 0.2979 0.2170 0.1702 0.0314
Model 5 -1.362 -0.6351 4.2724** 1.5151* 0.3690 -0.3587 1.0031 0.47060.0856 0.2243 0.0000 0.0628 0.2820 0.2445 0.1514
Model 6 -1.1700 -0.9956 4.7755** 1.3184 0.6116 -0.2328 0.4913 -2.039** 0.05470.1134 0.1500 0.0000 0.0911 0.2398 0.2691 0.1932 0.0195
Model 7 -1.277 -0.7757 1.4087 -0.6053 1.3036 0.1653 -0.2499 0.9448 0.42670.0882 0.1885 0.1040 0.1826 0.0839 0.3070 0.2477 0.1501
Model 8 -1.0880 -1.1922 1.641* -0.7034 1.0774 0.3965 -0.0881 0.4186 -2.200** 0.03860.1168 0.1136 0.0735 0.1875 0.1187 0.2825 0.2666 0.2540 0.0132
6-SOUTWEST AIRLINES COluv O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 3.1244** 0.4052 1.5801* 0.2680 -5.6876** 0.00000.0021 0.2537 0.0680 0.2514 0.0011
Model 2 2.9775** 0.3842 1.5162* 0.0848 -5.5512** 0.7684 0.48570.0024 0.2622 0.0742 0.2386 0.0024 0.1946
Model 3 0.0000 3.0197** 0.3995 1.5274* 0.2474 -5.1632** 0.49010.4038 0.0024 0.2556 0.0709 0.2550 0.0050
Model 4 0.0000 2.9118** 0.3761 1.4348 0.0782 -5.4145** 0.3629 0.49050.4038 0.0028 0.2649 0.0783 0.2409 0.0035 0.2914
Model 5 -0.0013 -1.734** 3.5297** 0.1081 2.0636** 0.6809 -5.0609** 0.41650.4034 0.0533 0.0014 0.3019 0.0213 0.2159 0.0041
Model 6 -0.0504 -1.750** 3.4291** 0.0710 1.9750** 0.5186 -5.3451** 0.4474 0.42610.3878 0.0479 0.0014 0.3061 0.0238 0.2527 0.0032 0.2663
Model 7 0.0000 -1.789** 0.4627 0.1497 0.0162 1.9649** 0.7092 -4.6748** 0.50920.4038 0.0444 0.2477 0.3024 0.2832 0.0292 0.2201 0.0046
Model 8 -0.0498 -1.780** 0.3543 0.2420 -0.0166 1.8808** 0.5452 -4.9557** 0.4275 0.44110.3880 0.0427 0.2650 0.2910 0.2842 0.0321 0.2553 0.0034 0.2716
38
UTILITIES7-CAROLINA POWER LIGHT
cpl O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 3.4567** -1.0833 2.7034** 2.2368** -14.7962** 0.00000.0010 0.1433 0.0032 0.0182 0.0049
Model 2 3.4627** -0.9636 2.7031** 2.2443** -14.4037** -0.4180 0.67320.0011 0.1673 0.0034 0.0168 0.0328 0.2714
Model 3 -0.3943 3.4268** -1.0332 2.7089** 2.2326** -14.4417** 0.70190.2855 0.0010 0.1518 0.0032 0.0183 0.0298
Model 4 -0.2057 3.4098** -0.9711 2.6710** 2.2268** -14.2300** -0.0763 0.73780.3396 0.0011 0.1674 0.0037 0.0175 0.0295 0.3170
Model 5 -0.4191 1.2804 2.2976** -0.7998 2.2033** 1.7485** -14.5647** 0.55030.2775 0.1169 0.0240 0.1897 0.0122 0.0380 0.0309
Model 6 -0.1222 1.2578 2.2831** -0.7148 2.2021** 1.7679** -14.3309** -0.1993 0.77150.3654 0.1192 0.0254 0.2013 0.0125 0.0359 0.0306 0.3330
Model 7 -0.3714 1.1803 0.8663 -0.4635 -0.9440 2.0173** 1.8163** -14.3923** 0.55050.2914 0.1273 0.1851 0.2259 0.1542 0.0186 0.0335 0.0336
Model 8 -0.1235 1.1603 0.8323 -0.4348 -0.8570 2.0149** 1.8252** -14.1572** -0.1627 0.80170.3650 0.1298 0.1892 0.2304 0.1730 0.0184 0.0326 0.0340 0.3424
8-TEXAS UTILITIES ELE COtxu O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 0.9406 -1.326 1.0673 1.2746 -1.7042** 0.00000.1593 0.0917 0.1298 0.0941 0.0388
Model 2 0.8906 -1.327 1.0838 1.2506 -1.5870* -0.2301 0.35410.1748 0.0876 0.1256 0.1004 0.0568 0.1408
Model 3 0.0000 0.8989 -1.270 1.0154 1.1923 -0.8431 0.35960.4038 0.1666 0.0987 0.1394 0.1059 0.1626
Model 4 0.0000 0.8536 -1.272 1.0282 1.1698 -0.8125 -0.2261 0.35970.4038 0.1809 0.0945 0.1355 0.1121 0.1720 0.1438
Model 5 0.0000 2.2216** -0.3280 -0.9310 0.2905 0.5115 -0.8254 0.23550.4038 0.0205 0.2801 0.1539 0.2811 0.2478 0.1671
Model 6 0.0000 2.1259** -0.3247 -0.960 0.3215 0.5112 -0.8106 -0.1092 0.44040.4038 0.0240 0.2745 0.1433 0.2861 0.2481 0.1712 0.1766
Model 7 0.0000 2.2160** -0.3591 0.2556 -0.9792 0.2820 0.5005 -0.8157 0.27610.4038 0.0235 0.1969 0.1970 0.1399 0.2885 0.2514 0.1694
Model 8 0.0000 2.1230** -0.3413 0.2426 -1.0004 0.3094 0.4958 -0.8039 -0.0752 0.49400.4038 0.0267 0.2131 0.2077 0.1330 0.2940 0.2526 0.1723 0.1986
PETROLEUM9-MOBIL CORP
mob O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 3.5411** -1.733** 1.3104** 1.4253 0.7283 0.00000.0003 0.0906 0.0911 0.0739 0.1935
Model 2 3.4993** -1.720** 1.2137* 1.4223 0.7497 -0.3968 0.68080.0003 0.0888 0.1070 0.0744 0.1906 0.2749
Model 3 -0.1674 3.4908** -1.714** 1.2877** 1.3995 0.5202 0.73430.3509 0.0003 0.0918 0.0948 0.0764 0.2432
Model 4 -0.0592 3.4460** -1.684** 1.1964* 1.3114 0.6869 -0.1879 0.71370.3850 0.0004 0.0922 0.1095 0.0850 0.2039 0.3189
Model 5 -0.3451 1.0363 2.3898** -1.5360* 0.8073 1.0425 0.7400 0.64190.2979 0.1278 0.0086 0.1118 0.1754 0.1304 0.1986
Model 6 -0.2975 1.0137 2.2930** -1.4576 0.7597 1.0289 0.8665 0.0287 0.68740.3135 0.1320 0.0115 0.1231 0.1857 0.1295 0.1660 0.2995
Model 7 -0.2366 0.9505 0.9329 -0.4518 -1.647** 0.6545 1.1243 0.6698 0.59790.3304 0.1473 0.1584 0.1762 0.0832 0.2126 0.1142 0.2122
Model 8 -0.2515 0.9367 0.9637 -0.4935 -1.570* 0.5917 1.1143 0.8153 0.0377 0.66640.3265 0.1496 0.1551 0.1894 0.0966 0.2208 0.1129 0.1741 0.2928
39
DEPARTMENT STORES10-SEARS ROEBUCK + CO
s O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 4.9673** 1.7257** 2.2308** 0.0558 -9.8357** 0.00000.0000 0.0444 0.0199 0.3239 0.0026
Model 2 4.9109** 1.7029** 2.1942** 0.0370 -9.7048** 0.0896 0.89930.0000 0.0462 0.0209 0.3214 0.0036 0.3633
Model 3 0.0000 4.7882** 1.7013** 2.1421** 0.0522 -9.4036** 0.90040.4038 0.0000 0.0462 0.0220 0.3275 0.0038
Model 4 0.0000 4.7280** 1.6764** 2.1101** 0.0342 -9.2625** 0.0871 0.90150.4038 0.0000 0.0482 0.0231 0.3245 0.0053 0.3643
Model 5 0.0000 -0.1619 4.1210** 1.6494** 2.1066** 0.0649 -9.1747** 0.75240.4038 0.1876 0.0003 0.0527 0.0201 0.3276 0.0037
Model 6 0.0000 -0.1539 4.0702** 1.6157* 2.0685** 0.0499 -9.0161** 0.0543 0.92390.4038 0.1898 0.0003 0.0558 0.0214 0.3271 0.0051 0.3734
Model 7 0.0000 -0.2593 0.7696 -0.0110 1.5306* 2.0042** 0.1646 -9.1732** 0.85600.4038 0.2019 0.1777 0.2868 0.0590 0.0235 0.3336 0.0041
Model 8 0.0000 -0.2459 0.7624 -0.0142 1.5069* 1.9642** 0.1412 -9.0127** 0.0891 0.90630.4038 0.2045 0.1789 0.2869 0.0615 0.0252 0.3344 0.0057 0.3663
11-WAL-MART STORES, INCwmt O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 3.1272** 1.2819 -0.6456 -2.572** -9.2198** 0.0000
0.0014 0.1077 0.2311 0.0551 0.0211Model 2 2.7181** 0.9742 -1.1116 -3.062** -9.6119** 1.8077** 0.1709
0.0033 0.0897 0.1967 0.0136 0.0298 0.0649Model 3 0.0000 3.0208** 1.2611 -0.6182 -2.439** -8.9624** 0.1738
0.4038 0.0017 0.1096 0.2335 0.0579 0.0282Model 4 -0.6139 2.6613** 0.8482 -1.2069 -3.000** -9.0132** 1.2357 0.1610
0.2442 0.0038 0.0827 0.1873 0.0128 0.0305 0.1350Model 5 0.0000 0.5900 2.2622** 1.3330 -0.7718 -2.517** -8.8660** 0.3364
0.4038 0.2387 0.0135 0.1168 0.2125 0.0340 0.0286Model 6 -0.5753 0.4987 2.0223** 0.9222 -1.2935 -3.006** -8.9137** 1.1797 0.1628
0.2517 0.2567 0.0200 0.0945 0.1762 0.0085 0.0311 0.1394Model 7 0.0000 0.6416 -0.2896 0.6808 1.3632 -0.7242 -2.588** -8.7461** 0.6029
0.4038 0.2313 0.2388 0.2216 0.1230 0.2074 0.0230 0.0292Model 8 -0.5160 0.5585 -0.2656 0.6192 0.9610 -1.2475 -3.068** -8.7769** 1.1252 0.1647
0.2611 0.2449 0.2418 0.2300 0.1110 0.1743 0.0046 0.0311 0.1433
TECHNOLOGY12-EASTMAN KODAK COMPANY
ek O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 1.9389** 0.2073 1.0854 0.8540 -2.7011** 0.00000.0339 0.2574 0.1179 0.1670 0.0035
Model 2 2.0149** 0.4374 1.0978 1.0228 -3.3936** 1.9362** 0.08390.0299 0.1974 0.1178 0.1306 0.0006 0.0287
Model 3 0.0000 1.9053** 0.2044 1.0683 0.8174 -2.0608** 0.08730.4038 0.0354 0.2590 0.1204 0.1746 0.0173
Model 4 0.0000 1.9793** 0.4266 1.0820 0.9571 -2.9651** 1.6122* 0.08060.4038 0.0314 0.2004 0.1199 0.1428* 0.0022 0.0477
Model 5 0.0000 0.6431 1.3105 0.3193 0.8355 0.5699 -2.0841** 0.33210.4038 0.2236 0.1016 0.2356 0.1519 0.2329 0.0159
Model 6 0.0000 0.8733 1.2703 0.5862 0.7713 0.6561 -3.0307** 1.6784** 0.06900.4038 0.1669 0.1076 0.1761 0.1556 0.2099 0.0015 0.0416
Model 7 0.0000 0.6581 -0.6918 0.8899 0.3416 0.9062 0.5329 -2.0016** 0.72800.4038 0.2171 0.1706 0.1578 0.2568 0.1584 0.2439 0.0194
Model 8 0.0000 0.8607 -0.5499 0.7469 0.5628 0.8132 0.6548 -2.9376** 1.6747** 0.06850.4038 0.1675 0.1778 0.1689 0.1926 0.1604 0.2114 0.0018 0.0422
40
13-XEROX CORPxrx O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 3.9747** 2.6915** 0.5044 -1.345 -7.4071** 0.00000.0001 0.0062 0.2487 0.0828 0.0924
Model 2 3.9863** 2.7132** 0.6278 -1.2157 -7.3710** -0.6178 0.55690.0001 0.0057 0.2189 0.1009 0.0930 0.2243
Model 3 0.0000 3.8503** 2.6399** 0.4938 -1.253 -7.4183* 0.56120.4038 0.0002 0.0070 0.2515 0.0934 0.1370
Model 4 0.0000 3.8603** 2.6622** 0.6164 -1.1457 -7.3582* -0.5981 0.56310.4038 0.0002 0.0065 0.2217 0.1106 0.1445 0.2291
Model 5 0.0000 -0.8179 3.6854** 2.4331** 0.7337 -0.9470 -7.4297* 0.72000.4038 0.1766 0.0004 0.0097 0.1926 0.1433 0.1380
Model 6 0.0000 -0.9099 3.7158** 2.4669** 0.9008 -0.7936 -7.3848* -0.7223 0.48170.4038 0.1556 0.0003 0.0089 0.1559 0.1769 0.1459 0.1968
Model 7 0.0000 -0.7837 0.0539 0.6998 2.3763** 0.6995 -0.9338 -7.3462* 0.94140.4038 0.1851 0.2520 0.1730 0.0115 0.2005 0.1482 0.1378
Model 8 0.0000 -0.8680 0.0232 0.7494 2.4067** 0.8562 -0.7916 -7.2940* -0.6746 0.51020.4038 0.1652 0.2629 0.1769 0.0105 0.1645 0.1794 0.1457 0.2082
14-TEXAS INSTRUMENTStxn O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 3.2322** 1.6837** 0.7490 0.8613 -6.5269** 0.00000.0016 0.0542 0.1798 0.1744 0.0282
Model 2 3.0068** 1.7743** 0.8591 0.6572 -6.4342** 1.0504 0.34540.0035 0.0479 0.1540 0.2095 0.0346 0.1367
Model 3 0.0000 3.1307** 1.6620** 0.7249 0.8029 -6.0184** 0.35020.4038 0.0020 0.0558 0.1824 0.1846 0.0390
Model 4 0.0000 2.9500** 1.7392** 0.8207 0.6207 -6.2837** 0.7552 0.34880.4038 0.0040 0.0499 0.1586 0.2170 0.0425 0.1890
Model 5 0.0000 -0.7220 3.0184** 1.4743 0.9169 0.9733 -5.8678** 0.68170.4038 0.1923 0.0022 0.0725 0.1615 0.1506 0.0405
Model 6 0.0000 -0.5880 2.7928** 1.5605* 0.9558 0.7813 -6.0817** 0.7052 0.38030.4038 0.2276 0.0052 0.0633 0.1479 0.1921 0.0442 0.2011
Model 7 0.0000 -0.6335 -1.0707 1.6681** 1.6152** 1.0611 0.9064 -5.9923** 0.74470.4038 0.2164 0.1667 0.0701 0.0532 0.1488 0.1675 0.0406
Model 8 0.0000 -0.4735 -1.1588 1.7173** 1.7248** 1.1163 0.6716 -6.2476** 0.7683 0.33260.4038 0.2589 0.1580 0.0618 0.0437 0.1426 0.2024 0.0445 0.1838
15-INTERNATIONAL BUSINESS MACHINESibm O0 O1 O2 E0 E1 E2 E3 E4 E5 F-Test
Model 1 2.0200** -1.465 -1.2198 -2.072** 0.5476 0.00000.0220 0.0876 0.1405 0.0340 0.2403
Model 2 2.0352** -1.459 -1.1911 -1.764** 0.2812 -0.5997 0.55340.0208 0.0894 0.1488 0.0505 0.2215 0.2233
Model 3 0.0000 1.9613** -1.444 -1.2010 -2.012** 0.3351 0.55800.4038 0.0253 0.0893 0.1425 0.0383 0.2764
Model 4 -0.0011 1.9720** -1.438 -1.1723 -1.705** 0.1600 -0.5885 0.56080.4035 0.0243 0.0911 0.1507 0.0565 0.2597 0.2262
Model 5 0.0000 -1.921** 2.7590** -1.793** -0.4595 -1.3827 0.1929 0.27020.4038 0.0378 0.0033 0.0569 0.2748 0.1054 0.2897
Model 6 0.0000 -1.884** 2.7464** -1.780** -0.4477 -1.1595 0.0377 -0.5491 0.58840.4038 0.0414 0.0034 0.0592 0.2703 0.1349 0.2732 0.2375
Model 7 0.0000 -1.970** -0.0235 0.5644 -1.793** -0.4291 -1.2641 0.1591 0.30460.4038 0.0285 0.2053 0.2128 0.0459 0.2898 0.1055 0.2859
Model 8 -0.0041 -1.931** -0.0288 0.5652 -1.777** -0.4140 -1.0401 -0.0028 -0.5818 0.56640.4025 0.0314 0.2014 0.2081 0.0478 0.2865 0.1393 0.2682 0.2285
41
Table 4: T-Scores and Average P-values for the Estimated Parameters from theEquity Model Regression
In each cell under the columns ),,,,,,,,( 543210210 ββββββλλλ the first number is the t-score for thecorresponding average parameter estimate in Table 3. This t-score is adjusted for the fact that theregressions contain overlapping time intervals. The second entry is the average P-score obtained from thet-tests of the individual regression coefficients. The P-score from an individual t-test corresponds to theprobability of rejecting the null hypothesis that the coefficient is zero when it is true.
The number of observations per regression is 48. The number of regressions in the average is 23.
The F-test column contains the average P-score where the P-scores are obtained from the F-tests of theindividual regressions. The P-score from an individual F-test corresponds to the probability of rejecting thenull hypothesis when it is true. The first row corresponds to the null hypothesis( 043210 ===== βββββ ). The F-tests for models 3, 5 and 7 test for the joint hypothesis that all
default parameters are zero, i.e. 00 =λ , 010 == λλ , and 0210 === λλλ respectively. The F-tests
from models 2,4,6 and 8 test the hypothesis that 05 =β .
**Significant at 10% level*Significant at 15% level
42
O0 O1 E0 E1 E2 E3
O0 1.0000 0.4685 0.4580 -0.4072 0.2607 0.2896O1 0.4685 1.0000 0.9934 -0.1277 -0.3539 0.1141E0 0.4580 0.9934 1.0000 -0.1449 -0.3765 0.1060E1 -0.4072 -0.1277 -0.1449 1.0000 -0.5153 0.0850E2 0.2607 -0.3539 -0.3765 -0.5153 1.0000 -0.0267E3 0.2896 0.1141 0.1060 0.0850 -0.0267 1.0000
Table 5: Correlation Matrix for Selected Independent Variables in the EquityModel Regression
43
FINANCIALS1-BANKERS TRUST NY
bt O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -0.9764 -0.1956 0.1940 -2.3448 -2.87471.0000 1.0000 1.0000 1.0000 0.1000
Model 2 -0.9197 -0.1575 0.2039 -2.5004 -1.8053 -0.71301.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 3 0.0000 -0.9764 -0.1956 0.1940 -2.3448 -2.87470.0000 1.0000 1.0000 1.0000 1.0000 0.1000
Model 4 0.0000 -0.9197 -0.1575 0.2039 -2.5004 -1.8053 -0.71300.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 5 0.0000 -1.4962 -0.8350 -0.2339 0.1465 -2.7305 -3.02080.0000 1.0000 1.0000 1.0000 1.0000 0.1000 0.0500
Model 6 0.0000 -1.4379 -0.8066 -0.1934 0.1321 -2.9241 -1.7725 -0.62720.0000 1.0000 1.0000 1.0000 1.0000 0.1000 1.0000 1.0000
Model 7 0.0000 -1.8304 -1.6891 -1.7076 -0.4335 0.0025 -2.4839 -3.40570.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0500
Model 8 0.0000 -1.7123 -1.6322 -1.6584 -0.4097 -0.0266 -2.6264 -2.3034 -0.93600.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2-MERRILL LYNCH & COmer O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -1.8044 -2.5992 0.6786 -0.6976 -1.46641.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -1.8015 -2.8419 0.5674 -0.8651 -1.4884 -2.24371.0000 0.1000 1.0000 1.0000 1.0000 1.0000
Model 3 0.0000 -1.8044 -2.5992 0.6786 -0.6976 -1.46640.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 0.0000 -1.8015 -2.8419 0.5674 -0.8651 -1.4884 -2.24370.0000 1.0000 0.1000 1.0000 1.0000 1.0000 1.0000
Model 5 0.0000 -1.6218 -1.7119 -2.8309 0.1653 -0.5027 -1.36340.0000 1.0000 1.0000 0.1000 1.0000 1.0000 1.0000
Model 6 0.0000 -1.6699 -1.7360 -2.6198 -0.0671 -0.6240 -1.5128 -2.47270.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 7 0.0000 -1.3984 -1.3719 -1.3164 -2.5243 0.1545 -0.7797 -1.37990.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 8 0.0000 -1.4781 -1.7128 -1.6553 -2.9046 -0.2053 -0.8250 -1.4810 -3.47880.0000 1.0000 1.0000 1.0000 0.1000 1.0000 1.0000 1.0000 0.0500
FOOD & BEVERAGES3-PEPSICO INC
pep O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -1.6564 -1.9001 -1.5938 -1.1898 -1.06791.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -1.9527 -1.8873 -1.5587 -1.2552 -1.6008 -2.26661.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 3 0.0000 -1.6564 -1.9001 -1.5938 -1.1898 -1.06790.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 0.0000 -1.9527 -1.8873 -1.5587 -1.2552 -1.6008 -2.26660.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 5 0.0000 -2.0347 -2.3032 -1.7796 -1.6912 -0.8229 -1.10090.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 6 0.0000 -2.1952 -2.6696 -1.7677 -1.6824 -0.8476 -1.6320 -2.24620.0000 1.0000 0.1000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 7 0.0000 -2.0021 -2.7571 -2.7523 -1.6610 -1.5638 -0.9005 -1.13530.0000 1.0000 0.1000 0.1000 1.0000 1.0000 1.0000 1.0000
Model 8 0.0000 -2.1351 -2.6417 -2.6566 -1.6316 -1.5402 -0.9475 -1.5579 -1.99250.0000 1.0000 0.1000 0.1000 1.0000 1.0000 1.0000 1.0000 1.0000
44
4-COCA-COLAcce O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -2.1647 -0.6833 -1.3978 -1.5127 -2.67371.0000 1.0000 1.0000 1.0000 0.1000
Model 2 -1.6736 -1.1557 -1.1092 -0.7673 -2.9315 -0.60021.0000 1.0000 1.0000 1.0000 0.1000 1.0000
Model 3 0.0000 -2.2525 -0.6907 -1.4518 -1.5257 -3.25660.0000 1.0000 1.0000 1.0000 1.0000 0.0500
Model 4 0.0000 -1.7552 -1.2372 -1.1623 -0.8589 -3.3752 -0.55060.0000 1.0000 1.0000 1.0000 1.0000 0.0500 1.0000
Model 5 0.0000 -1.3895 -3.8563 -1.8759 -1.3204 -2.6105 -3.23110.0000 1.0000 0.0100 1.0000 1.0000 1.0000 0.0500
Model 6 0.0000 -1.3997 -2.8237 -1.4950 -1.2313 -1.6649 -3.3572 -0.34650.0000 1.0000 0.1000 1.0000 1.0000 1.0000 0.0500 1.0000
Model 7 0.0000 -1.5156 -2.5315 -2.5069 -0.7772 -1.4147 -2.7426 -3.31480.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.1000 0.0500
Model 8 0.0000 -1.5541 -2.5734 -2.5540 -1.3759 -1.3767 -1.9648 -3.4587 -0.78030.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0500 1.0000
AIRLINES5-AMR CORPORATION
amr O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -2.1111 -1.2734 -0.2029 -1.1066 -1.69841.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -1.5740 -0.7720 0.0508 -1.1016 -1.3715 -1.17901.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 3 -1.3683 -1.9135 -1.3683 0.0229 -1.0519 -1.35551.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 -1.2518 -2.0367 -0.9144 0.2460 -1.0351 -1.3042 -1.17571.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 5 -1.4385 -1.5381 -1.4438 -1.5302 0.0553 -1.2708 -1.43581.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 6 -1.1049 -1.4427 -1.6237 -1.0025 0.2433 -1.1367 -1.1633 -1.00071.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 7 -2.0052 -1.2685 -1.5561 -1.6531 -2.5145 -1.4114 -1.4388 -1.82041.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 8 -1.8654 -1.0650 -1.1887 -1.2565 -2.6418 -1.2913 -1.3543 -1.5888 -1.54011.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
6-SOUTWEST AIRLINES COluv O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -2.5515 -2.4563 -1.3384 -0.5948 -0.86491.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -2.4818 -2.3921 -1.6884 -1.7186 0.6084 -0.44781.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 3 0.0000 -2.5515 -2.4563 -1.3384 -0.5948 -0.86490.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 0.0000 -2.4818 -2.3921 -1.6884 -1.7186 0.6084 -0.44780.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 5 0.0000 -2.2728 -2.7330 -2.0402 -1.6303 -0.4166 -1.18950.0000 1.0000 0.1000 1.0000 1.0000 1.0000 1.0000
Model 6 -0.3034 -2.2441 -2.5860 -1.9505 -1.8682 -1.5994 0.4581 -1.61601.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 7 0.0000 -2.0347 -1.4362 -1.4271 -1.5945 -0.8459 -0.5960 -1.17860.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 8 0.3699 -2.0782 -1.0755 -1.1434 -1.5043 -1.0738 -1.7079 0.1297 -1.84461.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
45
UTILITIEScpl O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -1.9304 -0.7972 -1.3362 -1.7369 1.37251.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -2.0161 -0.6128 -0.8560 -1.3217 -0.2680 0.54491.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 3 0.2274 -2.0285 -0.9322 -1.0208 -1.4247 -0.02921.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 -2.0463 -2.1034 -0.6339 -0.7756 -1.2563 -0.0636 0.01201.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 5 -0.0877 -1.5565 -1.9946 -0.9748 -1.0222 -1.2944 -0.12481.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 6 -2.0708 -1.5423 -1.9760 -0.5728 -0.8452 -1.0033 -0.1694 0.18611.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 7 -1.4368 -1.3969 -0.9864 -1.5207 -2.0064 -1.5470 -1.2689 0.57541.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 8 -2.2387 -1.3525 -1.5290 -1.5753 -1.7238 -1.3482 -1.0674 0.4431 -0.22121.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
8-TEXAS UTILITIES ELE COtxu O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -1.4443 -1.1776 -1.7224 -1.8343 -1.17171.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -1.3025 -1.7012 -1.3403 -1.6853 0.2702 -0.06031.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 3 0.0000 -1.4443 -1.1776 -1.7224 -1.8343 -1.17170.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 0.0000 -1.3025 -1.7012 -1.3403 -1.6853 0.2702 -0.06030.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 5 0.0000 -1.0179 -1.2998 -1.4862 -1.5740 -3.2214 -1.48430.0000 1.0000 1.0000 1.0000 1.0000 0.0500 1.0000
Model 6 0.0000 -1.0993 -1.3021 -2.3873 -1.2184 -3.0371 -0.2424 -0.09740.0000 1.0000 1.0000 1.0000 1.0000 0.0500 1.0000 1.0000
Model 7 0.0000 -1.1936 -1.2772 -1.2804 -2.8968 -1.6201 -4.1395 -1.64060.0000 1.0000 1.0000 1.0000 0.1000 1.0000 0.0100 1.0000
Model 8 0.0000 -1.2810 -1.4044 -1.3937 -3.7238 -1.1971 -4.5241 -0.4112 0.01230.0000 1.0000 1.0000 1.0000 0.0500 1.0000 0.0100 1.0000 1.0000
PETROLEUM9-MOBIL CORP
mob O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -3.7745 0.3537 -2.5355 -2.9919 -1.21550.0100 1.0000 1.0000 0.1000 1.0000
Model 2 -3.8691 0.4386 -2.2897 -2.9402 -1.6751 -1.84040.0100 1.0000 1.0000 0.1000 1.0000 1.0000
Model 3 -2.4550 -4.0767 0.3685 -2.4724 -2.9841 -1.42811.0000 0.0100 1.0000 1.0000 0.1000 1.0000
Model 4 -3.2957 -3.8322 0.4396 -2.3019 -3.1874 -1.5903 -2.03240.0500 0.0100 1.0000 1.0000 0.0500 1.0000 1.0000
Model 5 -2.0824 -2.4252 -4.6528 0.2912 -2.7907 -2.8600 -1.37841.0000 1.0000 0.0100 1.0000 0.1000 0.1000 1.0000
Model 6 -1.5063 -2.2661 -4.2032 0.3549 -2.7386 -3.0979 -1.1897 -1.58311.0000 1.0000 0.0100 1.0000 0.1000 0.0500 1.0000 1.0000
Model 7 -1.9623 -1.9135 -1.3217 -1.3460 -0.7097 -2.9329 -2.0424 -1.60801.0000 1.0000 1.0000 1.0000 1.0000 0.1000 1.0000 1.0000
Model 8 -1.3236 -2.1904 -1.2938 -1.3074 -0.5497 -2.6168 -2.0970 -1.7960 -1.07361.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
46
DEPARTMENT STORES10-SEARS ROEBUCK + CO
s O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -1.0297 -1.4707 -2.1606 -3.0088 -4.35691.0000 1.0000 1.0000 0.0500 0.0100
Model 2 -1.0036 -1.4768 -2.2597 -3.1240 -4.3468 -3.37411.0000 1.0000 1.0000 0.0500 0.0100 0.0500
Model 3 0.0000 -1.0297 -1.4707 -2.1606 -3.0088 -4.35690.0000 1.0000 1.0000 1.0000 0.0500 0.0100
Model 4 0.0000 -1.0036 -1.4768 -2.2597 -3.1240 -4.3468 -3.37410.0000 1.0000 1.0000 1.0000 0.0500 0.0100 0.0500
Model 5 0.0000 -0.5587 -0.7063 -1.9479 -1.2209 -3.6236 -4.28920.0000 1.0000 1.0000 1.0000 1.0000 0.0500 0.0100
Model 6 0.0000 -0.5490 -0.6905 -2.0209 -1.2771 -3.7958 -4.2805 -3.44060.0000 1.0000 1.0000 1.0000 1.0000 0.0100 0.0100 0.0500
Model 7 0.0000 -0.5480 -2.9682 -2.8470 -2.6887 -1.4592 -4.5630 -4.29860.0000 1.0000 0.1000 0.1000 0.1000 1.0000 0.0100 0.0100
Model 8 0.0000 -0.5374 -2.9776 -2.8584 -2.8270 -1.5286 -4.6116 -4.2873 -3.37100.0000 1.0000 0.1000 0.1000 0.1000 1.0000 0.0100 0.0100 0.0500
11-WAL-MART STORES, INCwmt O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -2.5906 -1.6957 -1.4014 -1.6853 0.9565
1.0000 1.0000 1.0000 1.0000 1.0000Model 2 -3.4204 -1.6772 -1.1618 -2.0535 -0.3851 -0.0348
0.0500 1.0000 1.0000 1.0000 1.0000 1.0000Model 3 0.0000 -2.5906 -1.6957 -1.4014 -1.6853 0.9565
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000Model 4 0.0000 -3.2170 -1.6569 -1.0688 -1.7034 -0.4774 0.0399
0.0000 0.0500 1.0000 1.0000 1.0000 1.0000 1.0000Model 5 0.0000 -0.7145 -2.3224 -1.7492 -1.2352 -1.7707 0.6077
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Model 6 0.0000 -0.8002 -2.3219 -1.7276 -0.9053 -1.8412 -0.9136 -0.1099
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Model 7 0.0000 -0.6980 -2.6088 -2.8188 -1.7389 -1.4355 -1.7835 0.9402
0.0000 1.0000 1.0000 0.1000 1.0000 1.0000 1.0000 1.0000Model 8 0.0000 -0.6311 -2.7416 -3.1955 -1.7860 -1.2001 -1.8131 -0.9533 -0.2517
0.0000 1.0000 0.1000 0.0500 1.0000 1.0000 1.0000 1.0000 1.0000TECHNOLOGY
12-EASTMAN KODAK COMPANYek O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -1.4011 -1.4881 -0.7744 -1.8138 -2.14681.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -1.2754 -0.7538 -0.6254 -1.7628 -1.9005 -1.67911.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 3 0.0000 -1.4011 -1.4881 -0.7744 -1.8138 -2.14680.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 0.0000 -1.2754 -0.7538 -0.6254 -1.7628 -1.9005 -1.67910.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 5 0.0000 -0.8765 -1.1756 -1.4546 -0.7668 -2.4451 -1.91100.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 6 0.0000 -1.0561 -1.1040 -0.8045 -0.5021 -2.1474 -2.5583 -1.44980.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 7 0.0000 -0.9972 -1.5574 -1.5271 -2.5255 -1.1717 -2.3949 -1.48180.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 8 0.0000 -1.1817 -1.7634 -1.7094 -1.4249 -0.7765 -2.1491 -3.1830 -2.54230.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0500 1.0000
47
13-XEROX CORPxrx O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -3.0813 -1.5729 -3.3620 -1.8060 -1.09470.0500 1.0000 0.0500 1.0000 1.0000
Model 2 -3.3063 -1.5752 -2.9492 -1.8295 -1.1821 -1.55380.0500 1.0000 0.1000 1.0000 1.0000 1.0000
Model 3 0.0000 -3.0813 -1.5729 -3.3620 -1.8060 -1.09470.0000 0.0500 1.0000 0.0500 1.0000 1.0000
Model 4 0.0000 -3.3063 -1.5752 -2.9492 -1.8295 -1.1821 -1.55380.0000 0.0500 1.0000 0.1000 1.0000 1.0000 1.0000
Model 5 0.0000 -1.3030 -2.6851 -1.7088 -2.6812 -2.5357 -1.14980.0000 1.0000 0.1000 1.0000 0.1000 1.0000 1.0000
Model 6 0.0000 -1.2457 -2.7942 -1.6345 -2.3857 -2.8406 -1.2639 -2.02910.0000 1.0000 0.1000 1.0000 1.0000 0.1000 1.0000 1.0000
Model 7 0.0000 -1.3376 -1.6667 -1.7139 -1.8625 -3.2238 -2.2269 -1.16070.0000 1.0000 1.0000 1.0000 1.0000 0.0500 1.0000 1.0000
Model 8 0.0000 -1.2860 -1.8549 -1.9222 -1.6742 -2.6293 -2.4606 -1.2751 -2.03940.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
14-TEXAS INSTRUMENTStxn O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -1.7698 -2.1816 -0.6294 -1.8927 -2.69541.0000 1.0000 1.0000 1.0000 0.1000
Model 2 -1.8448 -1.9029 -0.3307 -1.2398 -2.7338 -1.77051.0000 1.0000 1.0000 1.0000 0.1000 1.0000
Model 3 0.0000 -1.7698 -2.1816 -0.6294 -1.8927 -2.69540.0000 1.0000 1.0000 1.0000 1.0000 0.1000
Model 4 0.0000 -1.8448 -1.9029 -0.3307 -1.2398 -2.7338 -1.77050.0000 1.0000 1.0000 1.0000 1.0000 0.1000 1.0000
Model 5 0.0000 -3.3683 -1.8460 -2.0587 -0.8084 -2.0900 -2.68350.0000 0.0500 1.0000 1.0000 1.0000 1.0000 0.1000
Model 6 0.0000 -3.3449 -1.9997 -1.9387 -0.4283 -1.3448 -2.7251 -1.83320.0000 0.0500 1.0000 1.0000 1.0000 1.0000 0.1000 1.0000
Model 7 0.0000 -2.9372 -1.5236 -1.5675 -2.3295 -1.8402 -1.9619 -2.68280.0000 0.1000 1.0000 1.0000 1.0000 1.0000 1.0000 0.1000
Model 8 0.0000 -2.3104 -1.5293 -1.6025 -2.2092 -1.3018 -1.2370 -2.7074 -1.85750.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.1000 1.0000
15-INTERNATIONAL BUSINESS MACHINESibm O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 -0.4822 -1.1758 0.0008 -0.6878 -0.36191.0000 1.0000 1.0000 1.0000 1.0000
Model 2 -0.6719 -1.2155 -0.1199 -0.2635 -0.6033 -4.63821.0000 1.0000 1.0000 1.0000 1.0000 0.0100
Model 3 0.0000 -0.4822 -1.1758 0.0008 -0.6878 -0.36190.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 4 0.0000 -0.6804 -1.2161 -0.1201 -0.2666 -0.6204 -4.63110.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0100
Model 5 0.0000 -1.7948 -1.1128 -1.1613 0.3665 -1.1714 -0.33990.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 6 0.0000 -1.8823 -1.2496 -1.1790 0.2745 -1.1204 -0.6372 -3.78500.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0100
Model 7 0.0000 -1.6365 -1.2616 -1.2433 -1.3287 0.1635 -1.2780 -0.31070.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Model 8 0.0000 -1.7297 -1.2780 -1.2680 -1.3544 0.0675 -1.2810 -0.6505 -3.68360.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0500
48
Performance Across all Companies
O0 O1 O2 E0 E1 E2 E3 E4 E5
Model 1 2/15 0/15 0/15 1/15 2/15 0/15Model 2 2/15 1/15 0/15 1/15 3/15 2/15Model 3 2/15 0/15 0/15 1/15 3/15 0/15Model 4 2/15 1/15 0/15 1/15 3/15 2/15Model 5 2/15 4/15 1/15 1/15 3/15 4/15 0/15Model 6 1/15 4/15 0/15 1/15 3/15 3/15 2/15Model 7 2/15 3/15 3/15 2/15 1/15 3/15 4/15 0/15Model 8 0/15 4/15 3/15 3/15 0/15 3/15 3/15 3/15
Table 6: Unit Root Tests
There are two entries under the ),,,,,,,,( 543210210 ββββββλλλ columns. The first entry is themodified Dickey-Fuller (DF) test statistic for a unit root test. The modified DF test statistic is given as thet-statistic of the 1a coefficient in the linear regression: t1t10t yaay ε∆ ++= − where yt represents the time
t value of each parameter, 1ttt yyy −−=∆ , and tε is an error term. The null hypothesis for a unit root is
0a1 = . The second entry is the P-score for the DF test statistic, i.e. it is the probability of rejecting the unitroot when it is true.
In the last table, the entries under the columns correspond to the number of companies for the relevantcoefficient where the null hypothesis of a unit root is rejected at the 90 percent level. There are 15 totalcompanies – tests for a unit root.
49
FINANCIALS1-BANKERS TRUST NYbt Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0027 0.0488 0.0048 0.4873Model 2 0.0027 0.0490 0.0048 0.4901Model 3 0.0028 0.0494 0.0000 0.1367 0.0000 0.0048 0.4873Model 4 0.0029 0.0496 0.0000 0.2771 0.0000 0.0048 0.4901Model 5 0.0029 0.0497 0.0066 0.1386 0.0074 0.0048 0.4933Model 6 0.0030 0.0499 0.0059 0.2846 0.0065 0.0048 0.4954Model 7 0.0030 0.0495 -0.0483 0.1513 -0.0721 0.0048 0.5101Model 8 0.0031 0.0498 -0.0518 0.2914 -0.0781 0.0048 0.51192-MERRILL LYNCH & COmer Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0038 0.0585 0.0027 0.8292Model 2 0.0037 0.0572 0.0027 0.8379Model 3 0.0040 0.0592 0.0000 0.1353 0.0000 0.0027 0.8292Model 4 0.0039 0.0579 0.0000 0.3100 0.0000 0.0027 0.8379Model 5 0.0041 0.0594 0.0027 0.1367 0.0026 0.0027 0.8319Model 6 0.0041 0.0581 0.0044 0.3208 0.0050 0.0027 0.8406Model 7 0.0042 0.0591 -0.0277 0.1539 -0.0480 0.0027 0.8377Model 8 0.0041 0.0579 -0.0355 0.3295 -0.0483 0.0027 0.8459
FOOD & BEVERAGES3-PEPSICO INCpep Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0026 0.0483 0.0018 0.5877Model 2 0.0027 0.0487 0.0018 0.5889Model 3 0.0027 0.0489 0.0000 0.1556 0.0000 0.0018 0.5877Model 4 0.0029 0.0493 0.0000 0.3531 0.0000 0.0018 0.5889Model 5 0.0029 0.0494 -0.0001 0.1591 -0.0002 0.0018 0.5884Model 6 0.0030 0.0499 -0.0001 0.3659 -0.0002 0.0018 0.5895Model 7 0.0030 0.0497 -0.0121 0.1706 -0.0198 0.0018 0.5944Model 8 0.0031 0.0502 -0.0113 0.3801 -0.0193 0.0018 0.59594-COCA-COLAcce Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0054 0.0693 0.0156 0.1675Model 2 0.0056 0.0697 0.0156 0.1764Model 3 0.0056 0.0701 0.0000 0.2571 0.0343 0.0156 0.1714Model 4 0.0058 0.0705 0.0000 0.2608 0.0329 0.0156 0.1800Model 5 0.0056 0.0692 -0.0212 0.2557 0.0031 0.0156 0.2154Model 6 0.0058 0.0697 -0.0209 0.2596 0.0018 0.0156 0.2227Model 7 0.0057 0.0687 0.0737 0.2696 0.0210 0.0156 0.2470Model 8 0.0059 0.0692 0.0746 0.2735 0.0170 0.0156 0.2556
AIRLINES5-AMR CORPORATIONamr Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0044 0.0627 0.0015 0.3543Model 2 0.0042 0.0607 0.0015 0.4100Model 3 0.0044 0.0618 0.3620 0.2443 0.2994 0.0015 0.3920Model 4 0.0042 0.0600 0.3237 0.2385 0.2721 0.0015 0.4407Model 5 0.0045 0.0621 0.3314 0.2496 0.2770 0.0015 0.3999Model 6 0.0043 0.0598 0.2713 0.2423 0.2322 0.0015 0.4569Model 7 0.0045 0.0611 0.2129 0.2621 0.1266 0.0015 0.4350Model 8 0.0042 0.0583 0.1402 0.2522 0.0525 0.0015 0.4979
50
6-SOUTWEST AIRLINES COluv Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0113 0.1005 -0.0069 0.4663Model 2 0.0116 0.1007 -0.0069 0.4796Model 3 0.0118 0.1017 0.0000 0.2302 0.0000 -0.0069 0.4663Model 4 0.0122 0.1020 0.0000 0.5092 0.0000 -0.0069 0.4796Model 5 0.0115 0.0989 -0.0374 0.2256 -0.0442 -0.0069 0.5093Model 6 0.0118 0.0990 -0.0092 0.5111 -0.0172 -0.0069 0.5244Model 7 0.0119 0.0997 -0.0790 0.2623 -0.1080 -0.0069 0.5143Model 8 0.0123 0.0998 -0.0419 0.5345 -0.0632 -0.0069 0.5286
UTILITIES7-CAROLINA POWER + LIGHTcpl Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0017 0.0386 -0.0055 0.8275Model 2 0.0017 0.0389 -0.0055 0.8294Model 3 0.0017 0.0389 0.0381 0.0877 0.0366 -0.0055 0.8281Model 4 0.0018 0.0393 0.0311 0.1526 0.0301 -0.0055 0.8292Model 5 0.0017 0.0384 0.0495 0.0877 0.0489 -0.0055 0.8348Model 6 0.0018 0.0388 0.0288 0.1520 0.0293 -0.0055 0.8357Model 7 0.0018 0.0384 0.0035 0.1000 -0.0117 -0.0055 0.8401Model 8 0.0019 0.0388 -0.0105 0.1594 -0.0251 -0.0055 0.84058-TEXAS UTILITIES ELE COtxu Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0017 0.0393 0.0024 0.2101Model 2 0.0018 0.0393 0.0024 0.2278Model 3 0.0018 0.0397 0.0000 0.1584 0.0000 0.0024 0.2101Model 4 0.0019 0.0398 0.0000 0.1591 0.0000 0.0024 0.2278Model 5 0.0017 0.0379 0.0182 0.1523 0.0204 0.0024 0.2988Model 6 0.0017 0.0380 0.0176 0.1533 0.0197 0.0024 0.3100Model 7 0.0017 0.0379 0.0267 0.1612 0.0194 0.0024 0.3169Model 8 0.0018 0.0381 0.0267 0.1631 0.0207 0.0024 0.3257
PETROLEUM9-MOBIL CORPmob Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0010 0.0302 0.0115 0.3943Model 2 0.0011 0.0305 0.0115 0.4077Model 3 0.0011 0.0305 0.0192 0.1162 0.0188 0.0115 0.4018Model 4 0.0011 0.0308 0.0109 0.1947 0.0106 0.0115 0.4094Model 5 0.0011 0.0305 0.0472 0.1192 0.0466 0.0115 0.4170Model 6 0.0011 0.0308 0.0688 0.2117 0.0657 0.0115 0.4234Model 7 0.0011 0.0302 -0.0028 0.1257 -0.0199 0.0115 0.4408Model 8 0.0012 0.0305 0.0214 0.2161 0.0048 0.0115 0.4478
DEPARTMENT STORES10-SEARS ROEBUCK + COs Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0036 0.0571 0.0007 0.7250Model 2 0.0038 0.0577 0.0007 0.7251Model 3 0.0038 0.0578 0.0000 0.1192 0.0000 0.0007 0.7250Model 4 0.0040 0.0584 0.0000 0.1217 0.0000 0.0007 0.7251Model 5 0.0040 0.0580 -0.0024 0.1208 -0.0033 0.0007 0.7289Model 6 0.0041 0.0587 -0.0024 0.1237 -0.0032 0.0007 0.7289Model 7 0.0041 0.0581 -0.0422 0.1407 -0.0586 0.0007 0.7341Model 8 0.0043 0.0589 -0.0423 0.1435 -0.0590 0.0007 0.7342
51
11-WAL-MART STORES, INCwmt Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0036 0.0564 -0.0180 0.7309Model 2 0.0034 0.0546 -0.0180 0.7770Model 3 0.0037 0.0570 0.0000 0.1319 0.0000 -0.0180 0.7309Model 4 0.0035 0.0548 0.0000 0.3243 0.1501 -0.0180 0.7761Model 5 0.0039 0.0573 0.0072 0.1345 0.0079 -0.0180 0.7342Model 6 0.0037 0.0552 0.0058 0.3286 0.1474 -0.0180 0.7779Model 7 0.0040 0.0576 0.0140 0.1526 0.0083 -0.0180 0.7368Model 8 0.0038 0.0556 0.0078 0.3404 0.1330 -0.0180 0.7805
TECHNOLOGY12-EASTMAN KODAK COMPANYek Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0052 0.0683 0.0099 0.2610Model 2 0.0050 0.0662 0.0099 0.3036Model 3 0.0055 0.0691 0.0000 0.1783 0.0000 0.0099 0.2610Model 4 0.0053 0.0670 0.0000 0.2040 0.0000 0.0099 0.3036Model 5 0.0057 0.0695 0.0098 0.1803 0.0108 0.0099 0.2732Model 6 0.0054 0.0671 0.0129 0.2052 0.0144 0.0099 0.3204Model 7 0.0058 0.0695 0.0850 0.2000 0.0376 0.0099 0.2892Model 8 0.0056 0.0671 0.0743 0.2214 0.0339 0.0099 0.335513-XEROX CORPxrx Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0027 0.0486 0.0042 0.6048Model 2 0.0028 0.0489 0.0042 0.6125Model 3 0.0028 0.0491 0.0000 0.1459 0.0000 0.0042 0.6048Model 4 0.0029 0.0495 0.0000 0.1517 0.0000 0.0042 0.6125Model 5 0.0029 0.0493 -0.0088 0.1483 -0.0101 0.0042 0.6135Model 6 0.0030 0.0495 -0.0101 0.1546 -0.0116 0.0042 0.6230Model 7 0.0030 0.0496 -0.0269 0.1628 -0.0380 0.0042 0.6167Model 8 0.0031 0.0499 -0.0249 0.1694 -0.0344 0.0042 0.625914-TEXAS INSTRUMENTStxn Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0096 0.0926 0.0089 0.5997Model 2 0.0098 0.0923 0.0089 0.6083Model 3 0.0101 0.0937 0.0000 0.2214 0.0000 0.0089 0.5997Model 4 0.0102 0.0934 0.0000 0.3076 0.0000 0.0089 0.6083Model 5 0.0104 0.0942 -0.0149 0.2272 -0.0173 0.0089 0.6056Model 6 0.0107 0.0941 -0.0121 0.3132 -0.0139 0.0089 0.6130Model 7 0.0104 0.0930 0.1230 0.2528 -0.0053 0.0089 0.6226Model 8 0.0106 0.0928 0.1337 0.3318 -0.0009 0.0089 0.631015-INTERNATIONAL BUSINESS MACHINESIBM Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0066 0.0769 0.0059 0.2812Model 2 0.0069 0.0775 0.0059 0.2867Model 3 0.0070 0.0778 0.0000 0.2304 0.0000 0.0059 0.2812Model 4 0.0072 0.0784 0.0000 0.2341 0.0003 0.0059 0.2867Model 5 0.0067 0.0752 -0.0318 0.2238 -0.0372 0.0059 0.3467Model 6 0.0069 0.0758 -0.0315 0.2276 -0.0368 0.0059 0.3510Model 7 0.0068 0.0753 0.0045 0.2423 -0.0340 0.0059 0.3601Model 8 0.0071 0.0759 0.0061 0.2460 -0.0326 0.0059 0.3651
52
Average Avg GCV Avg RMSE O se(O) 1ydf Avg Y values R2
Model 1 0.0044 0.0597 0.0026 0.5018Model 2 0.0045 0.0595 0.0026 0.5174Model 3 0.0046 0.0603 0.0280 0.1776 0.0259 0.0026 0.5051Model 4 0.0047 0.0600 0.0244 0.2711 0.0331 0.0026 0.5197Model 5 0.0046 0.0599 0.0237 0.1781 0.0208 0.0026 0.5261Model 6 0.0047 0.0596 0.0219 0.2752 0.0293 0.0026 0.5409Model 7 0.0047 0.0598 0.0203 0.1949 -0.0135 0.0026 0.5397Model 8 0.0048 0.0595 0.0178 0.2882 -0.0066 0.0026 0.5548
Table 7: Summary Statistics for Model Performance
Given are the average Generalized Cross Validation statistics (GCV), the average Root Mean SquaredError (RMSE), the average default intensity (λ), the average standard error of the default intensity (se(λ)),the average 1-year default probability (1ydf), the average value for the dependent variable in the equitymodel regression, and the average R^2.
The number of observations per regression is 48. The averages are taken across all months from the equitymodel regressions. The number of regressions in the average is 23.
53
Avg(Od(t)) Avg(Oe(t)) Avg(t-score)
amr N/A N/A N/A N/A N/A
bt N/A N/A N/A N/A N/A
cce N/A N/A N/A N/A N/A
cpl N/A N/A N/A N/A N/A
ek N/A N/A N/A N/A N/A
ibm 0.0103 -0.0315 0.18260.0069 0.2271
luv 0.0104 -0.0092 0.04990.0064 0.4995
mer 0.0100 0.0048 0.01640.0029 0.3201
mob 0.0036 0.0069 -0.01480.0009 0.2192
pep 0.0082 -0.0001 0.02290.0075 0.3649
s 0.0086 0.0023 0.04820.0074 0.1233
txn 0.0083 -0.0121 0.0760.0089 0.3003
txu 0.0061 0.0176 -0.07730.0067 0.1531
wmt 0.0036 0.171 -0.5830.0045 0.3221
xrx 0.0056 -0.0101 0.11760.0057 0.1507
Table 8: Test for the Equivalence Between the Default Intensities based on DebtPrices versus Equity Prices
In the second and third columns, he first element in each cell is the average default intensity while thesecond element in each cell is the average standard error. The fourth column contains the average t-scorefor the difference in the two estimates. The debt estimates for only a zero recovery rate are reported.The equity estimation is from 31-May-95 to 31-March-97 for all the companies. The debt estimation timeperiod is contained in Table 1. Table 8 represents the intersecting dates from both experiments.
54
K0 K1 R2 Namr NaN NaN NaN NaN
NaN NaN
bt NaN NaN NaN NaN
NaN NaN
cce NaN NaN NaN NaN
NaN NaN
cpl NaN NaN NaN NaN
NaN NaN
ek NaN NaN NaN NaN
NaN NaN
ibm -0.0028 -0.5263 0.2459 21-0.1383 -2.4888*
luv 0.0557 -0.5834 0.3402 210.6878 -3.1299*
mer -0.3765 -0.2308 0.0573 21-0.3917 -1.0745
mob -0.0587 0.2514 0.0905 8-1.6168 0.7726
pep 0.0035 -0.2031 0.0413 210.0022 -0.9046
s -0.2742 -0.3202 0.1025 14-0.3799 -1.1706
txn 0.0259 -0.6219 0.3893 210.1985 -3.4804*
txu 0.0416 -0.6067 0.3740 210.4789 -3.3691*
wmt 0.0261 -0.4829 0.2418 210.0240 -2.4616*
xrx 0.0308 -0.1826 0.0333 210.2359 -0.8095
Table 9: A Test for a Constant Recovery Rate
This table estimates the regression:
( ) ( )( )( )
( ) ( )( )( )( ) ( )( )
( )( ) ( )( )
( ) tie
iid
ie
iid
10
ie
id
ie
iid
2t
2t12t
1t
1t11t
1t
1t11t
t
t1t
ελ
δλλ
δληη
λδλ
λδλ
+
−−−−
−−
−−−+
=−
−−−−
−
In the second and third columns, the first number in each cell is the parameter estimate and the secondnumber is the t-score. The R2 and the number of observations in each regression (N) is also provided.
*Significant at 10 percent level