Real Estate Appraisal and Transaction Price: An
Empirical Evaluation of Alternative Theories
Carl R. Gwin
Baylor University
Seow-Eng Ong
National University of
Singapore
Andrew C. Spieler
Hofstra University
November 28, 2005
Abstract
Mortgage appraisals are often required before a loan is approved. When
information on the transaction price is available and when lenders (lenders)
compensate appraisers for mortgage appraisals, a principal-agent problem may
arise. The effect is that appraisers tend to overstate the true value of a
property because they have an incentive to set the appraised value to be equal
to the transaction price. An earlier paper by Gwin and Maxam (2002) examines
this principal-agent problem. This paper offers an alternative theory that
provides a different prediction. The alternative theory is predicated on the
updating appraisal process ala Quan and Quigley (1991), and a signaling
modification to Gwin and Maxam (2002). In addition, an empirical test to the
theoretical moral hazard model postulated by Gwin and Maxam (2002) and the
alternative theory is carried out using appraisal and transaction data from a
lending institution in Singapore.
Acknowledgements:
We wish to thank comments from delegates at the 2005 American Real Estate, 2001
AREUEA, 2001 World Valuation Congress and 2000 ENHR/AREUEA International
conferences, especially David Ling, Joe Albert, Philip White, Soo Chin Lim and Tee
Geok Ng. We also wish to thank an anonymous referee for many excellent comments
and suggestions and Len Zumpano for his comments and guidance. We also thank
Steven Pyhrr and the ARES manuscript awards committee. Last but not least, we are
indebted to Melissa Yam and Alan Teo for providing the data.
Real Estate Appraisal and Bid Price: An Empirical
Evaluation of Alternative Theories
Introduction
Daly (2001) finds that the main priority of valuers in UK, Ireland and
Australia is to confirm the bid price. As one anonymous valuer puts it, “The
mortgagee valuation is a confirmation of bid price.”
Do appraisers acting on behalf of lenders exercise independent judgment in
ascertaining the appraised value, or are they influenced by the lenders? This question
is relevant since lenders require appraisals before mortgages are approved. The
purpose of mortgage appraisal, among others, 1 is to ensure the value of the real estate
meets or exceeds a minimum loan to value ratio. The appraiser knows this and a
moral hazard problem can arise if the lender rewards the appraiser with future
business for successful appraisals, i.e., those that result in a loan being made.
This principal-agent problem has been highlighted in Lentz and Wang (1998).
If a representative of the lender is compensated based on loans generated, then the
lender may put pressure on an appraiser to value the real estate at the price agreed
upon by the seller and buyer. On the other hand, a lender concerned about the number
of defaults may be more likely to pressure an appraiser to undervalue real estate.
Smolen and Hambleton (1997) find in a survey that nearly 80% of appraisers had
been pressured by lenders to alter their appraisals. Smolen and Hambleton argue for
regulations to protect appraisers from zealous lenders. Worzala, Lenk, and Kinnard
(1998) find in a survey of appraisers that most of the respondents had experienced
lender pressure to overestimate real estate value. However, they find no evidence that
1
In addition to collateral valuation, mortgage appraisals also deter fraud prevention
(e.g. to determine that the property actually exists).
2
lender size, the value of the change to the appraisal, or both have any influence on
appraisers.
Bid or transaction prices are the product of negotiation between sellers, real
estate brokers, and buyers who may have interests inconsistent with those of a lender.
For example, home buyers/borrowers have a strong incentive to see appraisals at their
maximum value in order to qualify for as large a loan as possible and as independent
verification of a fair price. Sellers and brokers would happily accept high appraisals to
close the sale and avoid the costs of further marketing the property. These incentives
contribute to the possibility that bid price can be significantly greater than true value.
Unfortunately, a lender may face losses if the borrower defaults on the mortgage and
there is insufficient collateral to recover the face value of the loan. Consequently,
lenders rely on independent appraisers to verify the true value of the real estate, but
can they?
Gwin and Maxam (2002) examine an appraiser’s incentives in conducting an
appraisal. They find that a moral hazard problem can arise if the lender rewards the
appraiser with future business for successful appraisals, i.e., those that result in a loan
being made. An appraiser may be willing to overstate the value of a property if the
lender wants him to do so. This moral hazard problem can lead to real estate
appraisals being equal to bid price.
The model in Gwin and Maxam (2002) lends itself to empirical testing in that
the prediction of the incentive to set an appraised value equal to the bid/transaction
price is demarcated under different market conditions. Specifically, the prediction is
for a lower incentive to set an appraised value equal to the bid/transaction price under
a bear market. Under a bear market, the appraiser is more likely to set an appraised
value lower than the bid/transaction price. This paper, in contrast, draws on the Quan
3
and Quigley (1991) framework and reworks the moral hazard model in Gwin and
Maxam (2002) to formulate an alternative theory that predicts an increased incentive
to set an appraised value equal to the bid/transaction price under a bear market. In
addition, this paper provides an empirical test of the competing theories by way of a
logit model that examines the effect of a bear market on the likelihood to set an
appraised value equal to the bid/transaction price.
The next section outlines the theoretical models and provides a signaling
modification to the moral hazard model in Gwin and Maxam (2002). In the following
section, a testable hypothesis is formulated to evaluate the competing models. The
data from a loan cooperative in Singapore is then used to empirically test the theories
and the results are reported. The empirical findings provide support for an alternative
theory predicated on the Quan and Quigley (1991) appraisal updating framework and
a signaling modification to the Gwin and Maxam (2002) moral hazard model.
Theoretical Models
A Moral Hazard Model
The theoretical model postulated by Gwin and Maxam (2002) is a moral
hazard model where the appraiser considers the effects of his appraisal on his
opportunity to continue to do business with the lender. This in turn depends on (1) the
likelihood of securing a mortgage for the lender and (2) the probability of future
default. 2 The lender will penalize the appraiser for losses of business and/or if the
borrower defaults within a reasonably short time period.
2
A recent paper by Ong, Neo and Spieler (Real Estate Economics¸ forthcoming)
shows that the premium paid at purchase has a positive and significant effect on
foreclosures. An appraiser is supposed to provide an independent assessment of the
value of the property to guard against overpayment, and hence the likelihood of
default.
4
The appraiser evaluates the probability of default conditioned on the appraised
value and the state of the real estate market. The appraiser compares the probability of
default if the appraised value (A) is equal to the transaction price (Po) and that if A <
Po. Interestingly, the state of the real estate market, which is indicative of the future
price trend, has a significant impact on the appraiser's evaluation. If the trend of
property prices is rising, then the probability of a default by the borrower is low. So
the appraiser finds it optimal to choose A = Po. If the market is on a downtrend, then
the appraiser is likely to choose A < Po in order to protect against future loss in
business with the lender due to borrower defaults.
The above predictions provide a testable hypothesis - in a bear market, the
probability of appraisal being less than the transaction price is likely to increase and
that the probability of appraisal being equal to the transaction price is likely to
decrease.
An Alternative Model
Gwin and Maxam (2002) build on a theoretical model in Quan and Quigley
(1991) that offers an explanation of appraisal smoothing. Gwin and Maxam allow for
an appraiser's subjective valuation of the property in his formulation of the appraisal.
An alternative hypothesis on the relationship between appraisals and market
conditions can be drawn from the original modeling framework of Quan and Quigley
(1991). Quan and Quigley (1991) show that the appraised value in period t (At) is a
weighted average of the transaction price in period t (Pt) and the appraised value in
period t-1 (At-1) of comparable properties. In other words,
At = KPt + (1 − K ) At −1 ,
(1)
5
where K is a function of the variances of the market ( σ η2 ) and transaction prices ( σ ν2 )
σ η2
. Quan and Quigley (1991) show that appraisal smoothing
given by K = 2
σ η + σ ν2
(i.e., At = At −1 ) can result if the variance or noise of transaction prices is high which
means than K → 0 .
Although Quan and Quigley (1991) is silent on the formation of the appraised
value under different market conditions, we contend that in a bearish market, the prior
appraised value is not a good indicator of true value. The motivation is as follows:
When the market is bearish, the appraiser knows with a high degree of certainty that
the prior appraised value (in period t-1) is too high. In addition, if the proposed
transaction value Pt is lower than the prior appraised value, At-1, it would be difficult
for the appraiser to attach a positive weight to At-1. In contrast, given that that buyer
has agreed to purchase at Pt, the appraiser would attach a higher weight to the agreed
price as being the true market value in a bearish market.
By this reasoning, the appraiser tends to attach a low weight on the appraised
value from the previous period and a higher weight to the agreed transaction price; so
(1 − K ) → 0 when the market is bearish. As such, At = Pt (to maintain consistency
with the previous notation, A = Po ). This tendency does not hold when the market is
stable or bullish. In a bullish market, K → 0 and thus (1 − K ) → 1 . Thus, appraisal
smoothing, or At = At −1 , occurs most often in a bullish market. Consequently, the
alternative hypothesis is in contrast with that in Gwin and Maxam (2002).
A Signaling Modification
6
Although at first sight, the alternative model ala Quan and Quigley (1991)
may produce conflicting results from Gwin and Maxam (2002), it can be shown that
the model in Gwin and Maxam (2002) can be modified to produce a consistent result.
The insight is that the appraised value can be regarded as a signal of value to the
buyer. In particular, we need to consider how a buyer would act when he receives a
report (signal) that the appraised value is less than the intended transaction price when
the market is bearish.
Following the same notation as in Gwin and Maxam (2002), suppose A < P0
when the market is bearish. Po is the price that the seller and buyer have negotiated.
When the buyer observes A < P0 , his reaction would be that he has overpaid for the
property. In most countries, the appraisal (A) is revealed to the buyer before the final
completion of the purchase. At this point in time, all the buyer has paid is an option to
purchase and he has the right not to proceed with the purchase (Ong, 1999). Given the
signal that the true value is lower than the contracted price, the buyer has every
incentive to break off the purchase, or negotiate for a lower price.
In either case, the mortgage application is unlikely to proceed, which results in
a loss of business for the lender. As noted in Gwin and Maxam (2002), if the appraisal
is less than the price, then the lender rejects the loan application. The appraiser knows
this and consequently, the loss of business for the lender may be severe enough to
offset against a future but uncertain loss should the borrower default.
This reasoning would change the original prediction in Gwin and Maxam
(2002) in that it would be optimal for the appraiser to choose A = P0 even in a bearish
market environment.
7
Methodology and Hypotheses Testing
To test the two theoretical models, this paper posits a logit model that defines
the probability of appraisal being equal to transaction price. The dependent binary
variable yi, which can be either 0 (failure) or 1 (success), depends on a vector of
independent variables, denoted as xi. The dependent variable is defined as taking a
value of 1 if the appraisal value equals transaction price. So y1i = 1 if A=Po.
Among the explanatory variables is a dummy variable to indicate a bear
market. Under the first hypothesis ala Gwin and Maxam (2002), we expect the
coefficient on the bear market dummy variable to be negative, but under the
alternative hypothesis, we expect the bear market dummy variable coefficient to be
positive.
In addition, we define a second dependent variable that takes the value of 1 if
the appraised value is less than the transaction price. So y2i = 1 if A<Po. We expect the
coefficient on the bear market dummy variable in the regression on y2i to be positive
under the hypothesis postulated by Gwin and Maxam (2002).
A general specification is that the probability of observing 1 for yi is
Pr( y i = 1) = F ( β ' x i ),
(2)
for i=1, 2,..., N and F is an appropriate distribution function.
We shall define a logit specifications for F, by specifying F=Λ where
Λ ( x) =
exp( β ' x)
.
1 + exp( β ' x)
(3)
It is well accepted that the logit model can be estimated by maximizing the
likelihood function:
8
N
L = ∏ [F ( β ' x i )] i [1 − F ( β ' x i )]
y
1− yi
.
(4)
i =1
Data
The data for this study comes from a major a cooperative association in
Singapore that has been issuing mortgages since 1983. Although mortgage financing
comprises a relatively small part of the association’s loan portfolio, it exercises very
prudent lending requirements. For example, the cooperative association will issue
loans only if the loan is no more than 5 times the borrowers’ total annual income. In
comparison, some local banks issue loans much higher than 5 times annual income. In
addition, the cooperative association tends to exercise more restraint in the setting of
its mortgage rates while other banks tend to be extremely aggressive in lowering rates
to gain a competitive edge. Consequently, their loan portfolio consists mainly of
“genuine” homebuyers instead of speculators.
The mortgage data from the cooperative association comprises the transaction
price (Po), appraised value (A), loan amount (LOAN), term of mortgage (TERM),
household income (HHINC), age of oldest borrower (AGE), number of borrowers
(NB) and the transaction date (PDATE). The data set spans a period from 1983
through 1999. Most of the loans are issued in the last 5 years through 1999. All the
loans were originated for purchase rather than refinancing since almost all mortgages
in Singapore are adjustable rate mortgages. 3
The real estate market in Singapore, after growing an average of 5% per
annum in the mid-1990s, underwent a severe bear spell from 1996 through late 1998
3
Since adjustable rate mortgages dominate, there is little refinancing due to interest
rate movements over the sample period (there is more refinancing activity from 2002
9
as a consequence of anti-speculation measures and the Asian economic crisis. Over
this period, the official property price index published by the Urban Redevelopment
Authority fell more than 40% across all sectors (multi-family and single-family
dwellings). In 1999, real estate prices rebounded between 15 to 20%.
The Asian economic crisis period created a natural experiment to test our
hypothesis. A bear market dummy variable (BEAR) is created that takes the value of 1
if the transaction is originated over this period, 0 otherwise. 1997 Another definition
of a bear market is two quarters of consecutive negative price changes, consistent with
the definition of economic recession (Ong, et al, 2005). The 1996/1998 period satisfy
this alternative definition as well.
Exhibit 1 summarizes the data. A total of 766 residential mortgages were
sampled. This represents almost half of the cooperative association’s mortgage loan
portfolio. 4 The average age of the oldest owner (AGE) is 38. The average annual
household income (HHINC) is S$103,000, but the range is rather wide. In terms of
loan-specific information, we see that the average loan amount (LOAN) is S$360,000.
The loan-to-value ratio (LV) averages 0.57, with a maximum of 0.99. The mean loan
term (TERM) is 23 years; some terms can be as short as 5 years and the maximum is
30 years. 5 The annual household income-to-property price ratio (HHINCOV) averages
17.3%.
Of the 766 observations, 477 loans (62%) were made when the property
market is bearish. In the full sample, the appraisal is equal to the transaction price in
through 2005 when teaser rates were aggressively offered, but our sample period ends
in 1999).
4
Mortgages comprising the other half of the loan portfolio (that is omitted for this
analysis) were issued within less than 6 months of the date of data extraction.
5
A 30-year maximum loan term is the norm in Singapore.
10
more than 73.4% of the loans. The appraisal is higher than the transaction price in 61
cases, and lower than the transaction price in the remaining 142 cases.
Exhibit 2 shows the margin (computed as the difference between the
transaction price and appraised value the normalized by the price) plotted against the
loan-value ratio (LV). The distribution shows that there is a lot of clustering around a
margin of zero. 6 In other words, in a majority of cases, the appraised value is equal or
nearly equal to the transaction price. This result can be viewed as another validation
of the observation that appraisers often set the appraised value close or equal to the
transaction price.
Empirical Results
The distribution of appraised values relative to transaction prices is
summarized in Exhibit 3. The distribution is also computed under bear market
conditions, and non-bear market conditions. When the market is bearish, the
proportion of cases in which the appraised value is equal to the transaction price is
0.815, compared to 0.734 in the full sample. By contrast, this proportion falls to 0.599
in non-bear markets. It is clear from the results in Exhibit 3 that prima facie evidence
exists to support the hypothesis that appraisers are more likely to appraise at the
transaction price in a bear market.
Exhibit 4 shows the results of the logit model where the dependent variable is
y1i where it takes the value of 1 if A=Po, 0 otherwise. A logit model is preferred when
using indicator variables (Maddala, 1983). 7 The independent variables are BEAR, NB,
LV, HHINCOV and AGE. The main interest is the effect of a bear market (BEAR), and
6
A very small number of outliers (3 observations) can be seen from Exhibit 2, but the
results of the logit model are unchanged even when the outliers were excluded.
11
to a lesser extent, loan-to-value (LV). To guard against omitted variables, we also
included borrower characteristics (NB, HHINCOV and AGE) as instrumental variables
(model 1). We also estimated the logit models without borrower characteristics
(model 2).
To differentiate between the different theoretical models, the variable of
interest is BEAR for reasons highlighted in the previous section. The coefficient on
BEAR is positive and significant in both models 1 and 2. This means that the
probability of observing an appraisal that is equal to the transaction price increases
when the market is bearish. This result is supportive of the alternative theory that
appraised value tends to be equal to the transaction price in a bear market.
Exhibit 5 shows the results of the logit model where the dependent variable is
y2i where it takes the value of 1 if A<Po, 0 otherwise. The coefficient on BEAR is
consistently negative and significant in both models. Again, the result is contrary to
the prediction in Gwin and Maxam (2002), but is consistent with the alternative
theory and the signaling modification to the moral hazard model.
Interestingly, the results in Exhibit 4 show that the likelihood of appraised
value being equal to the transaction price is increasing in the loan-value ratio (LV).
Put differently, the higher the loan amount, the higher the tendency for the appraiser
to match the transaction price. Age and household income are not significant
variables. A consistent effect is observed in Exhibit 5, in that the likelihood of
observing lower appraisals is lower when loan-value ratios are higher. This additional
insight holds even when instrumental variables are used to control for omitted
variables.
7
Results for the probit model are qualitatively the same as that for the logit model.
12
Finally, we wish to evaluate if there is an increased tendency to set appraised
value less than transaction price for higher LV loans in a bear market when the stakes
for the lender is higher. We created an interactive variable to capture mortgages with
LV of greater than 0.75 in a bear market. The results (not reported) show that loans
with higher LVs lead to a higher probability that the appraised value is less than
transaction price, but the coefficient is insignificant. This evidence provides some
support that appraisers tend to be more conservative when the LV is higher, which is
somewhat consistent with the moral hazard model in Gwin and Maxam (2002).
Conclusion
In conclusion, this paper is an attempt to empirically test the likelihood of an
appraiser setting an appraised value that equals the bid or transaction price. The extant
literature has highlighted this issue (Smolen and Hambleton, 1997; Rudolph, 1998;
Lentz and Wang, 1998; Worzala, et al. 1998; Finch, et al., 1999). Gwin and Maxam
(2002) provide an interesting moral hazard model that examines the appraiser's
incentives in conducting an appraisal. In particular, the focus is on an appraiser that
acts on behalf of the lender and is thus remunerated by the lender for current and
future businesses (see Lentz and Wang, 1998).
The model in Gwin and Maxam (2002) lends itself to empirical testing in that
predictions of the incentive to set an appraised value equal to the bid/transaction price
is demarcated under different market conditions. Specifically, the prediction is for a
lower incentive to set an appraised value equal to the bid/transaction price under a
bear market. This paper, in contrast, draws on the Quan and Quigley (1991)
framework and reworks the moral hazard model in Gwin and Maxam (2002) to
formulate an alternative theory that predicts an increased incentive to set an appraised
value equal to the bid/transaction price under a bear market.
13
Utilizing mortgage data from a loan cooperative in Singapore, we empirically
test the likelihood of appraised value being equal to the transaction price in a logit
model. The results find support for the alternative theory. The implications are that
while there is evidence that appraisers are influenced by the agreed transaction price,
they do so not necessarily to protect future business. Rather, the empirical results
support the price formation process of Quan and Quigley (1991). In addition, the
results are also consistent with the idea that the appraised value can be used as a
signal of value for buyers who may renege or renegotiate should the mortgage
appraisal provides a value lower than the agreed price.
References
Daly, J. (2001) “Economic Sustainability in Real Estate Markets: Implications no a
Federal State” Fourth Sharjah Urban Planning Sympsium.
Finch, J. Howard; Fogelberg, Larry; and Weeks, Shelton. (April 1999). “The Role of
Professional Designations as Quality Signals.” The Appraisal Journal, 67(2), pp. 14346.
Gwin, Carl R. and Maxam, Clark L. (2002) “Why do Real Estate Appraisals Nearly
Always Equal Offer Price? A Theoretical Justification.” Journal of Property
Investment & Finance, Vol. 20, No. 3, pp. 242-253.
Lentz, George H. and Wang, Ko. (1998). “Residential Appraisal and the Lending
Process: A Survey of Issues.” Journal of Real Estate Research, 15(1/2), pp. 11-39.
Maddala, G. S. (1983) Limited Dependent and Qualitative Variables in Econometrics,
Cambridge University Press.
Ong, S. E. (1999). “Aborted Property Transactions: Seller Under–compensation in the
Absence of Legal Recourse,” Journal of Property Investment and Finance, 17(2), 126
– 144.
Ong, S. E., Lusht, K., and Mak, C. Y., (2005) “Factors Influencing Auction
Outcomes: Bidder Turnout, Auction Houses and Market Conditions,” Journal of Real
Estate Research, 27 (2), 177 - 191.
Ong, S. E., Neo, P. H. and Spieler, A., (2006) “Price Premium and Foreclosure Risk,”
Real Estate Economics, forthcoming.
14
Quan, Daniel C. and Quigley, John M. (June 1991). “Price Formation and the
Appraisal Function in Real Estate Markets.” Journal of Real Estate Finance and
Economics, 4(2), pp. 127-46.
Rudolph, Patricia M. (1998). “Will Mandatory Licensing and Standards Raise the
Quality of Real Estate Appraisals? Some Insights from Agency Theory.” Journal of
Housing Economics, 7, pp. 165-79.
Smolen, Gerald E. and Hambleton, Donald Casey. (January 1997). “Is the Real Estate
Appraiser’s Role Too Much to Expect?” The Appraisal Journal, 65(1), pp. 9-17.
Worzala, Elaine M.; Lenk, Margarita M.; and Kinnard, William N. (October 1998).
“How Client Pressure Affects the Appraisal of Residential Property.” The Appraisal
Journal, 66(4), pp. 416-27.
15
Exhibit 1: Descriptive Statistics
The variables are: transaction price (Po), appraised value (A), bear market dummy variable
(BEAR), number of borrowers (NB), loan amount (LOAN), loan to value ratio (LV), annual
household income (HHINC), annual household income-to-property price ratio (HHINCOV) and
age of borrower (AGE). Number of observations: 766
Variable
Po ($)
A ($)
BEAR
NB
LOAN ($)
LV
HHINC ($)
HHINCOV
AGE (years)
Mean
Std Deviation
Minimum
Maximum
643,347
352,495
136,000
6,060,000
639,527
0.6227
2.0118
360,212
0.5657
102,885
0.1725
38
345,678
0.4850
0.5279
221,876
0.1581
81,823
0.1949
7
120,000
0.0000
1.0000
47,000
0.0562
9,800
0.0176
22
6,000,000
1.0000
6.0000
3,680,000
0.9993
1,235,461
4.6250
64
Exhibit 2: Distribution of Margin by Loan-Value Ratio
MARGPCT = (Po-A)/Po
LV = LOAN/Po
.3
MARGPCT
.2
.1
.0
-.1
-.2
-.3
-.4
-.5
-.6
.0
.2
.4
.6
LV
.8
1.0
1.2
16
Exhibit 3: Distribution of Appraised Values relative to Transaction Prices (proportions)
Variables are: Transaction price (Po) and Appraised value (A),
A=Po
A<Po
A>Po
Sample size
Full Sample
Bear Market
Non-bear market
0.7339
0.1852
0.0809
766
0.8155
0.1300
0.0545
477
0.5986
0.2768
0.1245
289
Exhibit 4: Likelihood of Appraised Value equal to Transaction Price
Dependent variable y1i =1 if A=Po; 0 otherwise. Independent variables are bear market
dummy variable (BEAR), number of borrowers (NB), loan to value ratio (LV), annual
household income-to-property price ratio (HHINCOV) and age of borrower (AGE).
Variable
Constant
BEAR
LV
HHINCOV
AGE
NB
Coefficient
-0.3398
1.1074**
1.3137*
0.0008
0.0007
-0.0036
Log likelihood function
Number
of
observations
Model 1
Std Error
0.3292
0.1714
0.5276
0.0005
0.0012
0.0067
p-value
0.3020
0.0000
0.0128
0.1163
0.5458
0.5915
-417.55
766
Coefficient
-0.3243
1.1234
1.2546
Model 2
Std Error
0.3246
0.1701**
0.5247*
p-value
0.3178
0.0000
0.0168
-419.87
766
Exhibit 5: Likelihood of Appraised Value less than Transaction Price
Dependent variable y2i =1 if A<Po; 0 otherwise. Independent variables are bear market
dummy variable (BEAR), number of borrowers (NB), loan to value ratio (LV), annual
household income-to-property price ratio (HHINCOV) and age of borrower (AGE).
Variable
Constant
BEAR
LV
HHINCOV
AGE
NB
Log likelihood function
Number
of
observations
Coefficient
0.0034
-0.9626**
-1.7431**
-0.0011*
-0.0010
0.0038
-346.977
766
Model 1
Std Error
0.3574
0.1936
0.5865
0.0005
0.0012
0.0071
p-value
0.9924
0.0000
0.0030
0.0426
0.4098
0.5967
Coefficient
-0.0219
-0.9867**
-1.6451**
Model 2
Std Error
0.3522
0.1913
0.5815
p-value
0.9505
0.0000
0.0047
-350.798
766
17