1

The gross value of products that can be manufactured from a tree is the starting 

point for a residual-value appraisal of a forest operation involving the harvest of trees 

suitable for making forest products. The amount of detail in a model of gross product 

value will affect the statistical properties of the estimate and the amount of ancillary 

information that is provided. Seven data sets from forest product recovery studies 

of western conifers were used in the evaluation of three models of gross product 

value. The evaluation of these models was based on the need for information and the 

statistical properties of the estimators. The most detailed method provided additional 

information, but at some loss in the precision and accuracy of the prediction of gross 

value of products from a log.

Keywords: Residual-value appraisal, log value, alternative approaches.

Residual value is a common approach used to estimate the value of forest operations 

involving the removal of trees suitable for making wood products (Duerr 1993). The 

residual-value approach begins with an estimate of the gross value of products that 

can be produced from the logs of a tree. This approach was once used by the USDA 

Forest Service for appraising timber sales but was abandoned in favor of a transactions-evidence appraisal system. Residual-value methods are especially helpful for 

understanding the components of costs and revenues associated with a timber appraisal and are therefore useful in research applications.

The purpose of this study was to evaluate three models for estimating the gross product value that is required for using the residual-value approach. All three models were 

based on data from empirical lumber recovery studies but use different amounts of 

detail in deriving the estimates. The evaluation of the models was based on the need 

for information and the statistical properties of the estimators.

The seven empirical lumber recovery data sets used in this evaluation come from 

lumber recovery studies (Stevens and Barbour 2000). Each data set provides (1) an 

estimate of gross product value per cubic foot of log for a species (or species group) 

for a particular type of lumber product and set of product prices; (2) the proportion of 

the volume of the log ending up in three products: lumber, chips, and residues; (3) the 

proportion of lumber that falls in each lumber grade; and (4) diameters and lengths of 

the logs. These data are available for each of the logs in the data set. These values 

and transformations of them are used in regression equations to estimate gross product value.

1 Rao V. Nagubadi was a forester, Roger D. Fight is a principal 

economist, and R. James Barbour is a research forest products 

technologist, Forestry Sciences Laboratory, P.O. Box 3890, Portland, OR 97208-3890. Rao V. Nagubadi is currently a post-doctoral fellow with the School of Forestry and Wildlife Sciences, Auburn 

University, 108 M. White Smith Hall, Auburn, AL 36849-5418.

Abstract

United States

Department of

Agriculture

Forest Service

Pacifi c Northwest

Research Station

Research Note

PNW-RN-541

February 2003

Valuing a Log:

Alternative Approaches

Rao V. Nagubadi, Roger D. Fight, and R. James Barbour1

Introduction
2

A description of the database used in this study is available in Stevens and Barbour 

(2000). Logs for this study include ponderosa pine (Pinus ponderosa Dougl. ex Laws) 

from the intermountain West and Douglas-fi r (Pseudotsoga menziesii Pseudotsoga menziesii (Mirb.) Franco), (Mirb.) Franco), 

western hemlock (Tsuga heterophylla (Raf.) Sarg.), and true fi r species ( (Raf.) Sarg.), and true fi r species (Abies) from 

western Oregon and Washington. Table 1 provides a description of the data used in 

the study. The logs studied in this analysis are from small-diameter trees with an average small-end diameter ranging from 8.8 to 10.5 inches, average taper ranging from 

0.12 to 0.17 inch per foot of log, and average log length ranging from 14.7 to 23.7 feet. 

We believe that these three variables are the most important in determining the value 

of a log. Although log grade and knots are key variables in determining the value of 

logs, we could not include these variables in our models because relevant data are 

not available in the data sets nor would it be available in potential applications of the 

models.

In pricing lumber to determine the gross product values for logs in the database, volumes in each of the lumber grades were multiplied by the 1999 prices. For the detailed 

method, lumber grades were combined into a workable number of grade groups as 

shown in table 1 and weighted average prices were applied for these grade groups 

(Chmelik et al. 1999). For ponderosa pine, 1999 average prices were for dry surfaced 

lumber from the inland region (WWPA 2000b). For Douglas-fi r, 1999 average prices 

were for green-surfaced lumber from the coast region (WWPA 2000a). For western 

hemlock and true fi rs (hem-fi r), prices were for dry hem-fi r lumber from the coast 

region (WWPA 2000a).

Lumber volumes were measured in board feet, and the corresponding prices in dollars per board foot were applied. The chips and residues (sawdust and planer shavings) portions produced from these logs were measured in cubic feet of solid wood 

equivalent (SWE), and the corresponding prices in dollars per cubic foot were used in 

all three methods. Details of the calculation of the gross product value of logs in the 

database are shown below.

The gross product value of logs in dollars per cubic foot of log volume was calculated 

by summing values of detailed lumber grades, chips, and residues portions as follows:

Gross value =

���

� � �

�

� �

�

��

� �� � � � � � � � � � �

�

�

�

�

�

�

�

�� , (1)

where �

�

� �

��� ���

� is the sum of value of lumber in various grades; i = 1 to 48 for = 1 to 48 for 

ponderosa pine, and 1 to 16 for Douglas-fi r, western hemlock, and true fi rs; C × Pc is 

the value of chip portion; R × Pr is the value of residues portion in a log; is the value of residues portion in a log; r Li is the quan- is the quan- i

tity of lumber in board feet in the i-th grade of lumber; C and R are quantities of chips are quantities of chips 

and residues in cubic feet SWE; Pi is price of lumber in dollars per board foot for the is price of lumber in dollars per board foot for the i

i-th grade of lumber; Pc and Pr are prices of chips and residues in dollars per cubic foot are prices of chips and residues in dollars per cubic foot r

SWE; and CFLOG is the total volume of the log in cubic feet. These values were calculated for each log in the seven data sets and were the basis of comparison between 

the alternative methods of estimation of gross product value of logs.

Data and Methods

Calculated Gross 

Product Value of Logs
3

The three alternative methods are (1) direct method: value was estimated directly 

by regressing the actual value of log per cubic foot on small-end diameter in inches, 

taper in inches per foot of log length, and log length; (2) intermediate method: value 

per cubic foot was estimated by using regressions for lumber recovery factor (board 

feet per cubic foot of log) for lumber (LRF), average price per board foot, and recovery 

proportions for chips and residues (in cubic feet of SWE per cubic foot of log); and (3) 

detailed method: value per cubic foot was estimated by using regressions for detailed 

grade recovery equations, LRF, and recovery proportions of chips and residues. In all 

three methods, a value of the log in dollars per cubic foot was estimated for comparison. These three methods are described below. 

The simplest model of gross product value for logs was based on a regression of the 

value per cubic foot of log as a function of log small-end diameter, taper (computed 

from log diameters and length), and length of log. This provided a single regression 

equation for a log manufactured into the specifi ed lumber product; for example, visually graded Douglas-fi r Dimension lumber. This will be referred to as the direct method. 

In this method, the value of a log was estimated by using an equation of the following 

form:

Value (direct) = f(D, T, H) , (2)

where D is small-end diameter in inches, T is taper in inches per foot of log, and is taper in inches per foot of log, and H is 

length of log in feet.

The dependent variable used in the equation came from the gross product value of 

logs calculated in equation (1). Various forms of small-end diameter (D, D2, 1/D, and 

1/D2), taper (T and T2), and length of log (H, H2, 1/H, and 1/H2) were tried in a stepwise regression equation, and the equation with the highest R2 was selected, (the 

p-values selected for entry and retention of independent variables are 0.15 and 0.10, 

respectively). If more than one form of a variable were retained in the fi nal selection 

of the step-wise regression, two forms for the small-end diameter, and one form each 

Direct Method

a There was a small amount of volume in some of these studies that was originally graded into grades that are no longer 

commonly used, and that volume was reassigned to the current most nearly equivalent grade. 

b SED = small-end diameter.

Table 1—Details species, products, logs, lumber grades, and average values of variables used 

in the study

No. of Average value of variables

lumber

Tree species and lumber type No. of logs grades a SED b (D) Taper (T) Length (H)

- - - - Inches - - - - Feet

Ponderosa pine:

Old growth, Dimension 66 48 10.44 0.14 15.56

Old growth, Appearance 266 48 10.48 .15 14.49

Young growth, Dimension 178 48 8.79 .16 15.11

Young growth, Appearance 418 48 8.93 .17 14.68

Douglas-fi r: Dimension 1,517 16 9.65 .13 18.89

Western hemlock: Dimension 321 16 8.96 .12 23.64

True fi rs: Dimension 1,663 16 10.00 .17 15.49
4

for taper and length of log were selected based on highest partial R2. In this method, 

there is only one prediction equation. By defi nition, the mean value of log estimated 

from this direct method is always equal to the mean of the gross product value of log 

calculated in the previous section. In this method, the only information predicted is the 

value of log in dollars per cubic foot based on the predictive regression equation. No 

other information is derived from this method.

An approach intermediate in complexity was based on regressions of the average 

value per board foot of lumber from the log and the proportion of log volume going into 

lumber, chips, and residues (as in the previous method). These regressions also were 

based on log small-end diameter, taper, and length. These estimates were combined 

with prices for lumber, chips, and residues to get gross product value per cubic foot of 

log. This method does not involve equations for the proportion of lumber in each grade 

or grade group and is therefore unable to provide any information about the quality 

of lumber other than its average value per board foot. This will be referred to as the 

intermediate method.

This method uses predictive equations for LRF, recovery proportion for residues (Rrp ), 

price in dollars per board foot of lumber (Pbf ), and recovery proportion of lumber (Lrp ), 

which were estimated as follows:

LRF = f(D, T, H), (3a)

Rrp = f(D, T, H), (3b)

Pbf = f(D, T, H), and (3c)

Lrp = f(D, T, H), (3d)

where LRF is the lumber recovery factor (board feet of lumber per cubic foot of log is the lumber recovery factor (board feet of lumber per cubic foot of log 

volume), D is small-end diameter in inches, T is taper in inches per foot of log, is taper in inches per foot of log, H is the is the 

length of log in feet, Rrp is proportion of the log that ends up as residues, Pbf is price of bf is price of bf

lumber in dollars per board foot, and rplumber in dollars per board foot, and rpLrp is the proportion of log that ends up as lumber 

(CF/CF of log volume). As in the direct method, various forms of (CF/CF of log volume). As in the direct method, various forms of rpD, , and H were tried were tried 

in a step-wise regression, and the equation with the highest R2 value was selected. 

The recovery proportion of chips (Crp ) in the log was estimated by subtracting recovery proportions of lumber (Lrp ) and residues ( rp 

) and residues ( rp Rrp ) from 1 (Crp = 1 - Lrp- Rrp ). Here, the 

distinction between LRF and rp LRF and L

rp 

rp should be noted. The unit of LRF is the number of rp 

 should be noted. The unit of LRF is the number of rp rp should be noted. The unit of LRF is the number of rp rp should be noted. The unit of LRF is the number of rp rp  should be noted. The unit of LRF is the number of rp 

board feet per cubic foot of the part of log volume that is made into lumber, whereas rpboard feet per cubic foot of the part of log volume that is made into lumber, whereas 

rp

Lrp is the proportion of log volume in cubic feet that is converted into lumber from the 

total volume of log in cubic feet. The total volume of log is composed of total volume of log in cubic feet. The total volume of log is composed of rpLrp, Crp, and 

Rrp.

In this method, the value of a log in dollars per cubic foot was estimated as:

Value (intermediate) = LRF × Pbf (Lumber portion)

 + Rrp × Pr (Residues portion)

 + Crp× Pc (Chips portion) , (4)

where LRF is lumber recovery factor in board feet per cubic foot of log volume is lumber recovery factor in board feet per cubic foot of log volume, Pbf

is price of lumber in dollars per board foot, Rrp and Crp are recovery proportions of 

residues and chips in the total volume of log, and rpresidues and chips in the total volume of log, and rpPr and 

rp

 and r Pc are prices for residues are prices for residues c

and chips in dollars per cubic foot SWE.

Intermediate Method
5

Detailed Method

In this method, there are four predictive equations; i.e., for LRF, recovery proportion , recovery proportion 

of residues (Rrp), price of lumber per board foot (Pbf), and recovery proportion of lum- bf), and recovery proportion of lum- bf

ber (Lrp). Thus, in addition to the estimated value of a log, information can be generated on LRF; recovery proportions of lumber, residues, and chips; and price of lumber 

in dollars per board foot based on the predictive equations.

The most complex model of gross product value was based on regressions of the 

proportion of lumber in each of several grade groups and the proportion of log volume 

going into lumber, chips, and residues. These regressions were based on log smallend diameter, taper, and length. These estimates were combined with prices for chips, 

residues, and each grade group of lumber to get gross product value per cubic foot 

of log. This method provided information not available from the other methods but 

required many more equations. This will be referred to as the detailed method.

In this method, the volume of lumber in board feet for each of the grade groups for 

each log is estimated as:

Gi = f(D, T, H) for all i (i = 1, 2, ..., L) = 1, 2, ..., L) (5a)

LRF = f(D, T, H) (5b)

Li = Gi × LRF × CF × LRF × CFLOG × LRF × CFLOG for all i (i = 1, 2, ..., L) , = 1, 2, ..., L) , (5c)

where Gi are the proportions of lumber in lumber grade groups, LRF is the lumber re- is the lumber recovery factor (board feet per cubic foot of log) for each log, Li

 is the volume of lumber is the volume of lumber i

in board feet in the log in the i-th grade, CFLOG is the total volume of a log in cubic feet, 

and L is the total number of lumber grades produced from the logs.

The value of the log in the detailed method in dollars per cubic foot was estimated as:

Value (detailed) = 

���

� � �

�

� �

�

��

� ��� � � � � � � � � �

�

�

�

�

�

�

�

�� , (6)

where �

�

� �

��� ���

� is the lumber portion, C × Pc is the chips portion, R × Pr

 is the resi- is the resi- r

dues portion, and CFLOG is the log volume, Pi are the weighted average prices of are the weighted average prices of i i-th 

lumber grade group in the detailed method, and Pc and Pr are prices for chips and are prices for chips and r

residues in dollars per cubic foot SWE.

The detailed method provides extensive information on the volumes, lumber grades, 

and values of all the products derived from the logs manufactured from each tree 

based on the predictive equations and the prices provided. This detailed information 

can often be useful for various purposes. This method was adopted in the fi nancial 

evaluation of ecosystem management activities (FEEMA) and FEEMA WS (westside) software developed by the USDA Forest Service, Pacifi c Northwest Research 

Station (Fight and Chmelik 1998). This software and documentation is available at 

http://www.fs.fed.us/pnw/data/soft.htm.
6

The value of a log in dollars per cubic foot was estimated for each log in all of these 

methods for ponderosa pine, Douglas-fi r, western hemlock, and true fi rs. The estimated values obtained in the three models were compared with the actual gross value 

of the log with a specifi ed set of market prices. For this purpose, statistical measures 

such as average deviation and average absolute deviation of the predicted values in 

the direct, intermediate, and detailed methods from the actual gross value of logs were 

derived. Average deviation indicated the estimated bias, and average absolute deviation measured the estimated accuracy (Max et al. 1985) and were derived as:

Average deviation = , (7a)

Average absolute deviation = , and (7b)

Average deviation squares = ,

 

(7c)

where is the actual value of a log; is the estimated value of a log in the direct, 

intermediate, or detailed methods; and n is the number of observations.

These three models also were compared by model selection criteria, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC). These criteria are similar to R2 in that these reward good fi t but also penalize the loss of degrees of freedom 

(Greene 1993). The difference between these two criteria is that BIC imposes a larger 

penalty for the extra coeffi cients. The information criteria are estimated as:

AIC = (8a)

BIC = , (8b)

where is RSS/n, RSS is the estimated residual sum of squares obtained by summing the squared residuals between actual value that was estimated under calculated 

gross product value and the predicted values obtained in the different methods (i.e., 

direct, intermediate, and detailed methods), n is the number of observations, and k is 

the number of estimated parameters in the entire model. Because there were differing numbers of equations in each method, the number of parameters (k) including the ) including the 

intercept was counted for each method. From a statistical standpoint, the model with 

the lowest value of AIC or BIC is preferred as these criteria select a model with lowest 

RSS consistent with keeping the risk of including spurious correlations in the model at 

an acceptable level. 

Evaluation Criteria
7

Results and 

Discussion

The purpose of this research was to evaluate alternative models of valuing a log so 

that its value would be as close as possible to the calculated gross value in dollars 

per cubic foot considering the amount of information provided by each model. As 

mentioned before, the gross value was estimated by using the quantities of lumber 

manufactured from a log and prices of all available lumber grades. The gross value 

estimated in equation (1) was the base with which the estimated values obtained in 

different methods were compared.

Tables 2 through 4 show the predictive equations for estimating the value of a log in 

the direct, intermediate, and detailed methods. The signs of the coeffi cients are along 

expected lines in most cases. The number of equations was reduced from a total of 61 

in the detailed method to 28 in the intermediate method, and to 7 in the direct method. 

Note, however, that these equations were used to predict different things in each of 

these three models as explained in the “Data and Methods” section. 

Table 5 provides a comparative summary of R2 values from the three methods. Note 

that the R2 values in the direct method are higher than the lowest R2 values in the 

other two methods. The R2 values were higher in the direct and intermediate methods 

than the highest R2 values in the detailed method for all ponderosa pine data sets. The 

highest R2 values in the detailed method were higher than the highest R2 values in the 

direct and intermediate methods for the equations for other species.

The coeffi cient of variation (standard deviation as a percentage of the mean) provides a measure of variation in the estimated values in each particular method. The 

coeffi cient of variation in the estimated values of logs was highest in the intermediate method and lowest in the detailed method for all data sets except the Douglas-fi r 

and western hemlock data sets (table 6). The coeffi cient of variation for actual gross 

values of log, however, was higher than those in all three methods.

The values of logs in direct, intermediate, and detailed methods were estimated by 

using equations (2), (4), and (6), respectively. Mean values and other statistics for the 

three different models for ponderosa pine, Douglas-fi r, western hemlock, and true fi rs, 

for comparison, are shown in table 7. The estimated mean values of logs in the direct 

and intermediate methods were close, and the difference in value of logs between 

these two methods did not exceed $0.02 per cubic foot. The difference between the 

direct and detailed methods ranged from $0.12 to $0.63 indicating a deviation of 4.56 

to 21.95 percent from the actual value of logs. The detailed method underestimated 

the value of logs for three data sets (old- and young-growth Dimension ponderosa 

pine and Douglas-fi r) and overestimated for the remaining four data sets. 

As already mentioned, the gross calculated value and the value estimated by the 

direct method are equal by defi nition, and consequently the average deviation, which 

measures the average bias, from the calculated (observed) gross value was zero for 

the direct method. Between the two remaining methods, average deviation from the 

gross calculated value was less for the intermediate than for the detailed method for 

all the data sets.

Average absolute deviations, which measure the estimated accuracy, were about the 

same in the direct and intermediate methods for all data sets. Average absolute deviation was higher for the detailed method for all cases. Mean deviation squares were 

also higher for the detailed method than for the direct and intermediate methods for all 

data sets without exception.
8

Table 2—Estimated equations for the direct method

Small-end diameter (D)

Tree species and Number of

lumber type Intercept (1) (2) Taper (T) Length (H) R2 observations

Ponderosa pine:

Old growth, Dimension 2.19 a 0.01(D2)

a — — -88.20*(1/H2)c 0.57 66

Old growth, Appearance -3.15 a .51(D)a 13.93(1/D)a -7.56(T2)

a — .63 266

Young growth, Dimension 1.97 a .008(D2)

a — 1.06(T)c -57.96(1/H2)a .46 278

Young growth, Appearance -.45. 32(D)a 17.57(1/D2)

c -2.76(T) a .0017(H2)a .52 418

Douglas-fi r, Dimension 2.24 a .11(D)a -.70(1/D) -2.68(T)a — .23 1,517

Western hemlock, Dimension 2.60 a .04(D)a — -2.14(T)a -33.65(1/H2)

b .12 321

True fi rs, Dimension 4.76 a -21.11(1/D)a 50.95(1/D2)

a -2.46(T)a -59.48(1/H2)a .29 1,663

— = variable not entered or retained in the step-wise regression; variables were selected by step-wise regression with signifi cance level to 

enter = 0.15 and signifi cance level to retain = 0.10; when more than one form of the same variable was selected in the step-wise regression, 

two forms of D, and one form for each of T and H with the highest partial R2 were retained in the fi nal estimated equations. a Signifi cance at 1-percent level.

b Signifi cance at 5-percent level.

c Signifi cance at 10-percent level. Significance at 10-percent level. c

The results indicate that the mean values of logs in the direct and intermediate methods are closer to the actual value. In the detailed method, the number of equations 

involved in the prediction was higher than in the intermediate method. The reason for 

the higher deviations in the detailed method is that it involved more equations with low 

R2 values. Although the detailed method used fewer lumber grade groups, the intermediate method used more lumber grades with the corresponding prices.

The AIC and BIC values were highest for the detailed method in all cases and lowest 

for the direct method. Because of the higher number of equations and more estimated 

parameters involved in the estimation, the detailed method resulted in higher AIC and 

BIC values. This also resulted in higher average deviation and average absolute deviation from the actual gross values in the detailed method. Hence the detailed method 

does not give any better estimates than those given by the direct and intermediate 

methods. However, the value of the detailed method lies in the information about the 

quantity and quality of lumber grade groups it generated.

Given the above tests and comparisons, valuing a log can be simplifi ed by the direct 

and intermediate methods by using fewer equations and incorporating more detailed 

lumber grade prices into the process of prediction than in the detailed method. The 

value of logs predicted by the direct and intermediate methods was closer to the calculated gross value than that predicted by the detailed method. 

The direct method is the simplest method and is therefore desirable when estimates 

of the value of the log are all that are desired. When some information on the relative 

proportions of log value that are derived from lumber, chips, and residues is of interest, 

then the intermediate method offers a good alternative that is still computationally easy 

to derive. When specifi cs about the change in the proportions of grade groups with 

increasing log diameter are of interest, the detailed method is needed. The detailed 

method allows users to infer information about which grade groups increase and which 

decrease when the average lumber value changes. This is not always obvious from 
9

Table 3—Estimated equations for the intermediate method

Tree species Small-end diameter (D)

and lumber type Number of

recovery variablesd Intercept (1) (2) Taper (T) Length (H) R2 observations

Ponderosa pine:

 Old growth, Dimension:

 LRF

 Rrp

 Pbf

 Lrp

 Old growth, Appearance:

 LRF

 Rrp

 Pbf

 Lrp

 Young growth, Dimension:

 LRF

 Rrp

 Pbf

 Lrp

 Young growth, Appearance:

 LRF

 Rrp

 Pbf

 Lrp

Douglas-fi r, Dimension:

 LRF

 Rrp

 Pbf

 Lrp

Western hemlock, Dimension:

 LRF

 Rrp

 Pbf

 Lrp

True fi rs, Dimension:

 LRF

 Rrp

 Pbf

 Lrp

10.85a

.10

.48a

.83a

12.56a

.09a

.32a

.24a

7.48a

.05a

.39a

.47a

3.34a

.09a

.32a

 -.45

5.10a

.06a

.40a

.36a

9.57a

.08a

.38a

.69a

12.19a

.07a

.43a

.61a

-28.31(1/D2)

c

-.006(D)c

-.05(D)a

-.88(1/D)a

-74.79(1/D)a

.0006(D)

.002(D2)

a

.04(D)a

.12(D)a

.00015(D2)

a

-.03(D)a

.01(D)a

.29(D)a

.0004(D)

.001(D2)

a

.32(D)a

.29(D)a

 .002(D)a

.00001(D2)

.02(D)a

-10.80(1/D)a

-.05(1/D)c

-.23(T)a

-.82(1/D)a

-43.28(1/D)a

.001(D)a

-1.06(1/D)a

.007(1/D)b

 —

0.0004(D2)

a

.004(D2)

a

 —

177.00(1/D2)

a

- .18(1/D)c

2.46(1/D2)

b

-.74(1/D2)

 —

.15(1/D2)

c

.002(D2)

a

 —

-3.46(1/D2)

-.15(1/D)

-.16(T)a

17.57(1/D2)

c

 —

 —

-.13(1/D)b

 —

 —

 —

 —

 —

100.41(1/D2)

a

 —

2.70(1/D2)

a

-.69(1/D2)

a

-5.77(T)a

-.17(T2)

 —

-.34(T)b

-2.34(T)b

-.03(T)b

-.23(T)a

-.22(T)b

 —

 —

 —

-4.09(T)a

-.07(T2)

a

 —

-2.76(T)a

-4.29(T)a

-.04(T)a

-.45(T2)

a

-.32(T)a

 —

-.07(T)a

-.23(T)a

 —

-6.36(T)a

-.06(T)a

-.07(T)a

-.42(T)a

-261.66(1/H2)

b

-1.80(1/H2)

c

-6.78(1/H2)

-18.98(1/H2)

b

 —

.00007(H2)

a

 —

 —

-151.59(1/H2)

a

 —

-3.49(1/H2)

b

-9.42(1/H2)

a

.003(H2)

a

.00005(H2)

a

.0001(H2)

b

.002(H2)

a

 —

 —

 —

 —

-16.72(1/H)a

.16(1/H2)

-.00001(H2)

a

-1.14(1/H)a

-167.27(1/H2)

a

-.19(1/H)a

-2.06(1/H2)

a

-13.23(1/H2)

a

0.28

.41

.65

.33

.42

.24

.48

.51

.13

.21

.52

.19

.33

.15

.44

.52

.23

.13

.09

.24

.10

.08

.24

.10

.26

.18

.11

.22

66

66

66

66

266

266

266

266

178

178

178

178

418

418

418

418

1,517

1,517

1,517

1,517

321

321

321

321

1,663

1,663

1,663

1,663

— = variable not entered or retained in the step-wise regression; variables were selected by step-wise regression with signifi cance level to 

enter = 0.15 and signifi cance level to retain = 0.10; when more than one form of the same variable was selected in the step-wise regression, 

two forms of D, and one form for each of T and H with the highest partial R2 were retained in the fi nal estimated equations. a Signifi cance at 1-percent level.

b Signifi cance at 5-percent level.

c Signifi cance at 10-percent level Significance at 10-percent level c

d LRF = lumber recovery factor, R LRF = lumber recovery factor, R d rp = recovery portion of residues, Pbf = price of lumber in dollars per board foot, L = price of lumber in dollars per board foot, L bf rp = recovery portion of 

lumber.
10

Table 4—Estimated equations for the detailed method

Small-end diameter (D)

Tree species Number of

and lumber grade Intercept (1) (2) Taper (T) Length (H) R2 observations

Ponderosa pine:

 Old growth, Dimension—

 No. 2 and better

 Utility

 Economy

 2 Commons and better

 3 Commons

 4 Commons

 Chips

 Residues

 Lumber recovery factor

 Old growth, Appearance—

 Moulding and better

 No. 1 Shop

 No. 2 Shop

 No. 3 Shop

 2 Commons and better

 3 Commons

 4 Commons

 5 Commons

 Chips

 Residues

 Lumber recovery factor

 Young growth, Dimension—

 No. 2 and better

 Utility

 Economy

 2 Commons and better

 3 Commons

 4 Commons

 Chips

 Residues

 Lumber recovery factor

 Young growth, Appearance:

 Moulding and better

 No. 1 Shop

 No. 2 Shop

 No. 3 Shop

 2 Commons and better

 3 Commons

 4 Commons

 5 Commons

 Chips

 Residues

 Lumber recovery factor

-36.52

 23.17a

-14.85a

27.27a

25.04b

10.77c

10.65 b

9.99a

10.85a

-6.54 b

-.02

11.68 a

-27.04a

59.33a

65.07a

7.29a

.49 

68.69a

9.34a

12.56a

16.76

18.54a

-15.69a

15.88a

12.53b

-7.18a

7.46

4.95a

7.48a

.59

-1.20c

4.61

-29.74a

30.66a

20.53

-2.31

.26 

66.14a

8.99a

3.34a

29.30(D)c

-1.55(D)a

 —

-7.58(D)a

-6.39(D)b

81.01(1/D)a

 —

-.65(D)c

-28.31(1/D2)

c

.06(D2)

a

.02(D2)

a

-3.12(D)a

3.02(D)a

-.13(D2)

a

-.14(D2)

a

 —

.01(D)

-3.81(D) a

.06(D)

-74.79(1/D)a

16.46(D)a

-.05(D2)

c

54.99(1/D)a

-4.85(D)a

-3.83(D)a

31.71(1/D)a

377.89(1/D)a

.02(D2)

a

.12(D)a

.06(D)

 —

-1.53(D)b

.25(D2)

a

-.099(D2)

a

9.64(D)a

344.84(1/D2)

a

-.02(D)

-3.81(D)a

.04(D)

.29(D)a

-1.58(D2)

a

 —

 —

.49(D2)

a

.43(D2)

a

 —

 —

.04(D2)

a

 —

293.24(1/D2)

a

 —

.21(D2)

a

278.25(1/D)c

 —

-142.32(1/D)c

-.02(D2)

b

 —

 —

-17.98(1/D)c

177.00(1/D2)

a

-.99(D2)

a

 —

 —

.34(D2)

a

.30(D2)

a

 —

-1066.2(1/D2)

b

14.55(1/D2)

c

 —

 —

 —

0.12(D2)

a

131.17(1/D)a

 —

-0.56(D2)

a

 —

 —

 —

-15.1(1/D)

-3.46(1/D2)

 —

 —

 —

 —

 —

 —

97.84(T)a

-16.56(T2)

a

-5.77(T)a

 —

 —

 —

 —

-85.00(T)a

62.57(T)a

72.37(T2)

a

-

46.17(T)a

-3.33(T)b

-2.34(T)b

 —

 —

 —

 —

55.64(T2)

b

14.73(T)c

218.71(T2)

a

 —

 —

 —

3.20(T)c

14.18(T2)

b

-9.99(T)b

-51.53(T)a

36.76(T)a

17.38(T)b

 —

63.47(T)a

-7.01(T2)

a

-4.09(T)a

-3387.0(1/H2)

b

—

4108.1(1/H2)

a

—

—

-.95(H)a

2286.3(1/H2)

a

-180.41(1/H2)

c

-261.66(1/H2)

b

—

—

—

—

—

—

—

—

—

.007(H2)

a

—

-2714.9(1/H2)

a

—

214.67(1/H)a

—

—

1294.6(1/H2)

a

—

—

-151.59(1/H2)

a

—

.007(H2)

b

—

—

125.81(1/H)

-1658.1(1/H2)

a

1053.9(1/H2)

a

—

—

.005(H2)

a

.003(H2)

a

0.44

.12

.27

.53

.36

.37

.41

.41

.28

.08

.08

.43

.30

.19

.11

.05

.00

.49

.24

.42

.27

.02

.11

.43

.42

.22

.22

.21

.13

.00

.02

.20

.49

.13

.08

.10

.00

.41

.15

.33

66

66

66

66

66

66

66

66

66

266

266

266

266

266

266

266

266

266

266

266

178

178

178

178

178

178

178

178

178

418

418

418

418

418

418

418

418

418

418

418
11

— = variable not entered/retained in the step-wise regression; variables were selected by step-wise regression with signifi cance level to enter 

= 0.15 and signifi cance level to retain = 0.10; when more than one form of the same variable were selected in the step-wise regression, two 

forms of D, and one form for each of T and H with the highest partial R2 were retained in the fi nal estimated equations. a Signifi cance at 1-percent level.

b Signifi cance at 5-percent level.

c Signifi cance at 10-percent level Significance at 10-percent level c

Douglas-fi r: Dimension—

 Select structural

 No. 2 and better

 No. 3 and Utility

 Economy

 Chips

 Residues

 Lumber recovery factor

Western hemlock: Dimension—

 Select structural

 No. 2 and better

 No. 3 and Utility

 Economy

 Chips

 Residues

 Lumber recovery factor

True fi rs: Dimension—

 Select structural

 No. 2 and better

 No. 3 and Utility

 Economy

 Chips

 Residues

 Lumber recovery factor

31.04a

-44.20a

-14.03a

6.12 c

58.55a

5.78a

5.10a

-3.76

-43.94a

-55.10a

-19.97a

26.30a

9.17a

9.57a

6.44a

-66.37a

-17.13a

-2.34b

40.50a

6.61a

12.19a

-6.24(D)a

.62(D2)

a

.12(D2)

a

-3.14(D)a

-2.31(D)a

.19(D)a

.29(D)a

0.18(D2)

a

.77(D2)

a

.33(D2)

a

.12(D2)

a

90.42(1/D)a

-5.68(1/D)b

-10.80(1/D)a

0.10(D2)

a

.56(D2)

a

.12(D2)

a

.04(D2)

a

-1.04(D)a

.11(D)a

-43.28(1/D)a

.50(D2)

a

-69.69(1/D)

 —

.22(D2)

a

 —

 —

 —

 —

 —

 —

 —

 —

 —

 —

 —

216.34(1/D2)

b

 —

 —

185.69(1/D2)

a

 —

100.41(1/D2)

a

-69.44 (T)a

274.15(T2)

a

127.94(T2)

a

35.70(T)a

-3.81(T)a

-4.29(T)a

-50.74(T)a

93.28(T)b

200.1(T)a

72.27(T)a

 —

-6.08(T)a

 —

-76.20(T2)

a

8.69(T)b

6.71(T)b

48.05(T)a

-5.59(T)a

-6.36(T)a

 —

3.33(H)a

.02(H2)

a

.01(H2)

a

 —

 —

 —

.01(H2)

a

.07(H2)

a

.05(H2)

a

.014 (H2)

a

901.41(1/H2)

a

-.03(H)a

-16.72(1/H)a

-130.16(1/H)a

3.82(H)a

1.18(H)a

.02(H2)

a

1448.71(1/H2)

a

-18.74(1/H)a

-167.27(1/H2)

a

.25

.63

.28

.20

.23

.13

.23

.31

.68

.66

.43

.09

.10

.10

.18

.67

.24

.08

.22

.18

.26

1,517

1,517

1,517

1,517

1,517

1,517

1,517

321

321

321

321

321

321

321

1,663

1,663

1,663

1,663

1,663

1,663

1,663

Table 4—Estimated equations for the detailed method (continued)

 Small-end diameter (D)

Tree species Number of

and lumber grade Intercept (1) (2) Taper (T) Length (H) R2 observations

Table 5—Summary of R2 values for estimated equations in the different methods

Intermediate method R2 Detailed method R2

Tree species and Direct method

lumber type R2 Lowest Highest Lowest Highest

Ponderosa pine:

Old growth, Dimension 0.57 0.28 0.65 0.12 0.53

Old growth, Appearance .63 .24 .51 .00 .49

Young growth, Dimension .46 .13 .52 .02 .43

Young growth, Appearance .52 .15 .52 .00 .49

Douglas-fi r: Dimension .23 .09 .24 .13 .63

Western hemlock: Dimension .12 .08 .24 .09 .68

True fi rs: Dimension .29 .11 .26 .08 .67
12

Table 6—Coeffi cient of variation of values of logs in the different methods a

Gross Direct Intermediate Detailed

Tree species and lumber type value method method method

Ponderosa pine:

Old growth, Dimension 33.97 25.63 29.34 19.13

Old growth, Appearance 47.63 37.75 38.67 29.57

Young growth, Dimension 33.49 22.67 23.90 20.14

Young growth, Appearance 41.80 30.19 30.44 21.87

Douglas-fi r: Dimension 28.43 14.48 14.23 17.70

Western hemlock: Dimension 25.04 8.56 8.11 8.57

True fi rs: Dimension 33.50 17.99 18.39 13.06

a Coeffi cient of variation = (standard deviation/mean) × 100.

Table 7—Comparison of mean values of log under different methods

 Direct Intermediate Detailed

Item method method method

Ponderosa pine:

Old growth, Dimension—

 Mean 3.13 3.12 2.90

 Average deviation .00 .01 .23

 Average absolute deviation .54 .53 .56

 Average deviation squares .48 .46 .61

 Number of lumber grades 48 48 6

 Number of regression equations 1 4 9

 Number of parameters 3 17 29

 AICa -.23 .18 .66

 BICb -.24 .13 .58

Ponderosa pine: 

Old growth, Appearance—

 Mean 3.44 3.44 3.63

 Average deviation .00 -.01 -.19

 Average absolute deviation .72 .72 .80

 Average deviation squares .99 1.00 1.16

 Number of lumber grades 48 48 8

 Number of regression equations 1 4 11

 Number of parameters 4 17 35

 AICa .03 .13 .33

 BICb .03 .15 .38

Ponderosa pine: 

Young growth, Dimension—

 Mean 2.56 2.55 2.29

 Average deviation .00 .00 .27

 Average absolute deviation .50 .50 .55

 Average deviation squares .40 .40 .48

 Number of lumber grades 48 48 6

 Number of regression equations 1 4 9

 Number of parameters 4 13 30

 AICa -.36 -.26 .02

 BICb -.35 -.24 .06
13

Table 7—Comparison of mean values of log under different methods 

(continued)

 Direct Intermediate Detailed

Item method method method

Ponderosa pine: 

Young growth, Appearance—

 Mean 2.64 2.62 2.85

 Average deviation .00 .02 -.21

 Average absolute deviation .57 .58 .64

 Average deviation squares .58 .593 .66

 Number of lumber grades 48 48 8

 Number of regression equations 1 4 10

 Number of parameters 5 19 39

 AICa -.21 -.14 .01

 BICb -.20 -.11 .07

Douglas-fi r, Dimension—

Mean 2.87 2.85 2.23

Average deviation .00 .02 .63

Average absolute deviation .57 .57 .80

Average deviation squares .57 .57 .98

Number of lumber grades 16 16 4

Number of regression equations 1 4 7

Number of parameters 4 13 27

AICa -.24 -.23 .03

BICb -.24 -.22 .05

Western hemlock, Dimension—

Mean 2.63 2.62 2.75

Average deviation .00 .01 -.12

Average absolute deviation .46 .46 .49

Number of lumber grades 16 16 4

Average deviation squares .39 .39 .44

Number of regression equations 1 4 7

Number of parameters 4 14 26

AICa -.38 -.32 -.19

BICb -.37 -.30 -.15

True fi rs, Dimension—

Mean 2.42 2.40 2.57

Average deviation .00 .02 -.16

Average absolute deviation .54 .54 .56

Average deviation squares .48 .48 .52

Number of lumber grades 16 16 4

Number of regression equations 1 4 6

Number of parameters 5 19 27

AICa -.31 -.30 -.25

BICb -.31 -.28 -.23

aAIC = Akaike Information Criterion.

bBIC = Bayesian Information Criterion.
14

Acknowledgments

Metric Equivalents

the average, and understanding how the proportion of lumber grade groups change 

in relation to each other is sometimes important in understanding the effects of forest 

management on resource value. 

Our results show that the mean values per cubic foot of log were close in the direct 

and intermediate methods but diverged in the detailed method from the actual value 

by either overestimating or underestimating the value of logs. This suggests that users 

can select the least complex computational method that gives them the desired information and feel confi dent that the value of log estimated by using either of the other 

methods would be relatively close.

These methods refl ect the tradeoff between simplicity and the amount of information developed. The detailed method provides extensive information on the volumes 

of different lumber grades as well as a means to do a complete analysis of forest 

management activities; the other two methods do not. If users wish to make a simple 

calculation of value of a forest stand based on the small-end diameter and taper of 

logs, the direct method would suffi ce. If more detailed information is needed, such as 

an estimate of the value of a forest stand based on the LRF, recovery proportions of 

chips and residues, and price in dollars per board foot, the intermediate method would 

give the most accurate estimate. Although the detailed method enables us to get more 

information pertaining to recovery of lumber in different grade groups, it does not enable us to get a better estimate of log value than the direct and intermediate methods 

as it tends to overestimate or underestimate the values.

In this study, three alternative approaches for predicting the value of a log were 

examined. Taking the calculated gross value of logs obtained by applying all detailed 

grades and available prices as the base method, the estimated value of logs in three 

models—direct, intermediate, and detailed—were compared with the calculated gross 

value of logs. The average deviations, average absolute deviations, average deviation squares, and model selection criteria showed that the direct and intermediate 

models produced predictive values of logs closer to the calculated gross value for 

most samples. Direct and intermediate models also have the advantage of relying on 

fewer predictive equations and more detailed grades and prices than does the detailed 

method. Although the detailed model uses fewer lumber grade groups and respective 

prices, it enables us to derive extensive information on the values and volumes of lumber products from a forest. The detailed model, however, produced values that deviated more from the calculated gross value than did the other two models. Depending 

on the need for detailed information, any of these models can be used in valuing logs 

in a forest stand. The simplest one is the direct model in that it gives only the value of 

the forest stand quickly and the detailed model is more sophisticated because it gives 

an array of detailed information. Although the estimated value in the intermediate 

model is closer to the calculated gross value, it enables us to get limited information 

on proportions of lumber, chips, and residues.

The authors gratefully acknowledge the critical comments received from Timothy Max, 

Xiaoping Zhou, and David Nicholls on an earlier draft. 

When you know Multiply by To get:

Inches 2.540 Centimeters

Feet 0.305 Meters

Conclusions
15

Chmelik, J.T.; Fight, R.D.; Barbour, R.J. 1999. Softwood lumber prices for 

evaluation of small-diameter timber stands in the intermountain West. Res. 

Note. FPL-RN-0270. Madison, WI: U.S. Department of Agriculture, Forest 

Service, Forest Products Laboratory. 4 p.

Duerr, W.A. 1993. Introduction to forest resource economics. New York: McGrawHill. 485 p.

Fight, R.D.; Chmelik, J.T. 1998. Analysts guide to FEEMA for fi nancial analysis of 

ecosystem management activities. Gen. Tech. Rep. FPL-GTR-111. Madison, WI: 

U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 5 p.

Greene, W.H. 1993. Econometric analysis. Englewood Cliffs, NJ: Prentice Hall. 791 p.

Max, T.A.; Cahill, J.M.; Snellgrove, T.A. 1985. Validation of a butt log volume 

estimator for Douglas-fi r. Forest Science. 31(3): 643–646.

Stevens, J.A.; Barbour, R.J. 2000. Managing the stands of the future based on the 

lessons of the past: estimating western timber species product recovery by using 

historical data. Res. Note. PNW-RN-528. Portland, OR: U.S. Department 

of Agriculture, Forest Service, Pacifi c Northwest Research Station. 8 p.

Western Wood Products Association. 2000a. Coast F.O.B. price summary. 

December, 1999. Portland, OR. 8 p.

Western Wood Products Association. 2000b. Inland F.O.B. price summary. 

December, 1999. Portland, OR. 13 p.

Literature Cited


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