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Compositional data analysis in the study of carcass composition
of beef cattle
a a ,
*
bD. Muldowney , J. Connolly
, M.G. Keane
aDepartment of Statistics, University College Dublin, Belfield, Dublin 4, Ireland b
Teagasc, Grange Research Centre, Dunsany, Co. Meath, Ireland
Received 10 June 1999; received in revised form 12 October 1999; accepted 7 May 2000
Abstract
Allometric regression (AR) has been widely used to model changes in the body composition of animals. However, predicted body component proportions based on AR equations do not necessarily sum to 1 and this discrepancy is confounded with treatment effects on components of the composition. Predicted component proportions are not bounded to lie between 0 and 1. An alternative method, compositional data analysis (CDA), which avoids these difficulties is proposed for beef carcass dissection data. For a composition consisting of D components (e.g. muscle, fat and bone) a new set of
D21 variables is created based on the logarithm of the ratios of components to one of the components (e.g. log(muscle / bone) and log(fat / bone)). Any statistical analysis can be applied on this scale, subject to the assumptions for that method of analysis being true. Regression models with simple interpretations in terms of animal development can be fitted to these logratio variables. Some inferences and interpretations are best made on the scale of component proportions. Predictions made from the models on the logratio scale may be back-transformed to give compositions on the proportional scale which obey the constraints that the component proportions sum to 1 and individually cannot exceed 1. The method generalises readily to multiple regression models involving factors and variables. CDA provides a fully multivariate framework for dealing with carcass dissection data within which questions on the effects of treatments and covariates on component composition and the differences between components can be addressed. It is a more natural vehicle than AR for analysing part–part relationships as it respects the symmetry between the components being compared. A simple relationship between CDA and AR models is developed. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Beef cattle; Compositional data analysis; Allometric regression; Linear regression; Carcass composition
1. Introduction way, have for long been sought. Huxley (1924,
1932) related the growth of a part of the body (w ,i
Patterns of tissue growth in animals have been where i is the ith body part) to growth of the whole widely studied and methods of describing them (w) by the allometric equation:
mathematically, but also in a biologically relevant
E(log(w ))i 5ci1d log(w)i (1)
*Corresponding author. Tel.:1353-1-706-7103; fax:1
353-1-where E(log(w )) is the expected value of log(w ) for
706-1186. i i
E-mail address: [email protected] (J. Connolly). a given value of log(w), c is the intercept and d isi i
the slope of the relation. It was claimed that this ponent proportions sum to different totals for two equation is frequently suitable for the description of treatments (e.g. 1.02 and 0.99) then, in the com-organ (or part) to whole body relationships based on parison of treatments for a particular component (e.g. the assumption that the changes in the size of the muscle with mean estimated proportions 0.62 and organs or parts during growth is more dependent on 0.65 for the two treatments), how much of the the absolute size of the whole than on the time taken difference (0.65–0.62) between treatment means is to reach that size (Berg and Butterfield, 1976). due to the treatment effect and how much is due to Examination of the effects of rate of growth separ- the bias arising from the non-summation of the ately from that of size has been described by predicted component proportions to unity?
Seebeck (1983). Attempts to provide a rationale for A second difficulty with AR is that individual observed allometric rules of scaling within organisms predicted proportions of composition are not con-continue (Banaver et al., 1999; West et al. 1999). strained to lie between 0 and 1. This can be seen Many studies of the development of beef animals more clearly by rewriting the AR equation in terms have used allometric regression (AR) to examine of proportions, by subtracting log(w) from each side, how changes in body composition through time are as:
affected by levels of various factors, for example
E(log(w /w))i 5log(x )i 5ci1(di21)(log w) (2) breed, nutrition or weight (Berg et al., 1978a,b;
Keane et al., 1990; Keane, 1994). Linear regression In this equation, x is the weight of the ith com-i
has also been used (e.g. Nour et al., 1981). Allomet- ponent as a proportion of the total weight, and so a ric regression, in this context, generally describes the predicted proportion should lie between 0 and 1. relationship between the logarithm of the weight of Predicted proportions are obtained using Eq. (2) by components of a body (log(w )) and the logarithm ofi taking the exponential of values predicted from the the weight of the whole body (log(w)). It is not equation. Eq. (2) is linear in log(w), and unless necessary for w to be the weight of the whole animal di51 predictions from it are unbounded. For di.1, or carcass and AR has been used to describe both log(x ) increases unboundedly as w increases. Thei
part to whole and part to part relationships (Huxley, exponential of these values, the predicted proportions 1932). For example, the growth of individual carcass x , are also unbounded and not constrained to liei
muscles can be described (e.g. psoas, semiten- between 0 and 1 as is required for a proportion. dinosus) can be described relative to the growth of Thus, depending on the estimated parameter values
ˆ
total carcass muscle (w) or to each other. ˆc and d , and the value of w, a predicted componenti i
Two logical difficulties arise with the AR ap- proportion greater than 1 could occur. Note that proach. Firstly, for a carcass of weight w the weights although the first difficulty does not arise when linear of the component parts sum to w exactly. It is rather than allometric regression is used the second desirable that predicted weights of carcass com- one does.
ponents from models of carcass composition should The objective of this study is to explore the also share this property to avoid confounding effects importance of difficulties in the AR approach and to of treatments with artifacts due to model specifica- evaluate an alternative method of analysis for carcass tion. From the AR equations the ith component can dissection data, based on compositional data analysis
ˆ
be predicted for a given weight w as wi5 (CDA) (Aitchison, 1986). This serves the same
ˆ
ˆci1d log(w)i ˆ
ˆ
e , where c and d are the estimates of thei i purposes as AR when used on carcass dissection parameters of the AR equation. However, there is data, but constrains predicted component proportions nothing in the method that will constrain the pre- to sum to 1, and to lie between 0 and 1. Although dicted components to sum to the total w. This means CDA does not appear to have been applied to carcass
ˆ ˆ
1995), the composition of sand (Johnsson, 1990), of weight and, for bi,0, it decreases relative to the Dth fossils (Reyment and Kennedy, 1991) as well as proportion for increased carcass weight.
several examples cited in Aitchison (1986). The relationship between two components, neither In subsequent sections of this paper the CDA of which is the reference component, can also be theory is developed for application to the study of evaluated. How the ith component proportion carcass composition, it is then applied to a set of data changes by comparison with the jth (rather than the and the results compared with those from AR. Dth) component proportion as total carcass weight
Various aspects of the use of CDA are discussed. changes is assessed by examining the sign of the estimate of (bi2b ); positive, negative and zeroj
values being interpreted as above. This follows readily from the CDA equations for the two com-ponent proportions:
2. Material and methods
E( y )i 5ai1b log(w)i
CDA is first described in general, then for a specific case where interest is focused on a single
E( y )j 5aj1b log(w)j
components) represent the component proportions of 5a 2a 1(b 2b ) log(w) (4)
i j i j
D
some whole, where oi51xi51. The component
proportions can be transformed to produce D21 new The intercept for the ratio of the ith to the jth variables y , . . . , y1 D21 using the logratio transforma- proportion is estimated by aˆi2a and the coefficientˆj
ˆ ˆ
tion: yi5log x /xs i Dd for i51, . . . ,D21. This trans- of log(w) is estimated by bi2b .j
formation uses the Dth or final component propor- The CDA equations can be used to predict com-tion as the denominator for the transformacom-tion, but position at a given carcass weight. For a given total the method will yield the same results irrespective of weight, predictions (y ) are made from the CDAˆi
which proportion is taken as the denominator (Ait- models for i51, . . . ,D21 and these predicted chison, 1986). Once the data have been transformed values are back-transformed to the proportional (x) to this y scale, provided that the relevant assumptions scale using the back-transformation:
hold (usually that the y values follow a multivariate
ˆyi e
normal distribution), any statistical modelling can be
ˆ ]]]]]]]
xi5 yˆ yˆ
1 D21 done on the y scale. The simplest CDA equation that e 1 ? ? ? 1e 11 may be fitted relating the ith logratio ( y ) to the logi
ˆ ˆ ˆ
proportion to lie between 0 and 1. The predicted model, a is the intercept and b measures how thei i
component proportions also sum to 1 as is shown
ith proportion changes relative to the Dth proportion
below. The sum of the predicted proportions other as log(w) changes. If b equals 0 the ratio of the ithi
than the Dth is: to the Dth component proportion remains constant as
ˆ ˆ interpretation as the coefficient of log(w) being equal
to 1 in AR). For bi.0 the ith proportion increases which is less than 1. The final predicted proportion,
ˆ
ˆ ˆ ˆ dissection of 162 beef carcasses described by Keane 1
]]]]]]]
5 ˆy1 yˆD21 (6) et al. (1990), but excluding the data from the 27
e 1 ? ? ? 1e 11
animals in the pre-finishing slaughter group. The experiment was replicated with animals born in three and so summing all D predicted proportions (Eqs. 5
consecutive years using 18 Hereford3Friesian (HE), and 6) gives 1.
18 Friesian (FR) and 18 Charolais3Friesian (CH) The full range of multiple regression models,
steers per year. The animals were all reared together including factors and variates and their interactions
from shortly after birth to about 18 months of age can be used in CDA models on the y scale. In
and 400 kg live weight. They were then allocated to particular, interactions of treatment factors with
either a high (High) or a medium (Medium) energy log(w) implies different rates of development
(differ-level diet, offered ad libitum, until slaughter at one ent b values) of a component for different treatments.
of three target carcass weights: light (260 kg for the Treatments may also affect the size of intercepts.
three breeds), normal (300 kg for HE and FR, and 320 kg for CH) and heavy (340 kg for HE and FR, 2.2. Development of a single component
and 380 kg for CH). This gave a 3 (breed types)32 (dietary energy concentrations)33 (slaughter So far the exposition has focused on part–part
weights) factorial arrangement of treatments with relationships among the components of the
com-nine animals per treatment group. position. If a particular component (the ith) is of
Carcasses were dressed, and the left side of each primary importance and the composition of the
was jointed and dissected into the muscle, bone, fat remainder is of secondary or no importance then it is
and ‘other’ tissue (connective tissue and ligamentum simpler to consider just the relationships for that
nuchae) components, as described by Williams and component relative to the other components
com-¨
Bergstrom (1977). The ‘other’ tissue component was bined. The logratio transformation can be taken with
added to the bone component in the analysis. respect to the sum of the other components to give a
single response variable on the logratio scale as:
2.3.2. Analysis of experimental data
In the experimental data, there were D53
pro-xi ]]
yi5log
S
12xD
(7) portions, the muscle, bone and fat components asi
fractions of the side of the carcass. For easier reference these will be represented by x , x and x , and the coefficient b of the CDA regression:i m b f
respectively. Two new variables were formed: y15
log(x /x ) and y 5log(x /x ) and these were taken
E( y )i 5ai1b log(w)i m b 2 f b
as the two response variables in the analysis of the can be interpreted as measuring the relative change data. A model consisting of breed type, levels of in the component within the whole. Thus, if bi50, dietary energy concentration and their interaction, the component maintains the same relative size in the and the logarithm of side weight (logside) as a body of which it is a component as log(w) changes, covariate was fitted to y and y . The model may be1 2
while bi.0 (bi,0) means that it is becoming an written as:
increasing (decreasing) fraction of the body as log(w) y 5m 1a 1d 1ad 1b log(w)1e (8)
ijkl i ij ik ijk i ijkl
increases. If it is also of interest to study
concentration (diet), adijk is the interaction term for joint distribution of y1 and y2 and using the delta the jth breed and the kth diet, b is the regressioni method based on Taylor series expansion of the coefficient for the covariate log(w), the natural back-transform (Kendall and Stuart, 1977)
logarithm of side weight andeijkl is the residual error term. This was taken as the most appropriate model
for the data because there was no significant inter- 3. Results
action between the covariate and either of the
factors, breed and dietary energy concentration, for 3.1. Compositional data analysis
y1 or y . The analysis is somewhat simplified for2
expository purposes by the omission of year effects. The assumption of multivariate normality of y1
This model was estimated using analysis of co- and y2 was not rejected by the Anderson–Darling variance or multiple regression in the statistical test (Payne et al., 1993).
package Genstat (Payne et al., 1993). Predictions of Analysis of covariance (Eq. (8)) with the factors, treatment means for factors and interactions between breed, diet, their interaction and the covariate, factors can be made from the estimated model on the logarithm of side weight (logside) was performed for
y scale and back-transformed to the x or proportional response variables y and y . Breed and diet affected1 2
scale using the following back-transformations: log(muscle / bone) and log(fat / bone), with P values of,0.001,,0.001,,0.001 and 0.002, respectively.
ˆ ˆ
y1 y2
e e There was a significant (P,0.05) breed by diet
ˆ ]]]] ˆ ]]]] ˆ
xm5 ˆy1 yˆ2 , xf5 yˆ1 yˆ2 and xb
interaction for log(muscle / bone), but not for log(fat /
e 1e 11 e 1e 11
bone). The covariate, logside had a significant effect
ˆ ˆ
512(xm1x )f (9)
on both response variables, with P values of 0.001
ˆ ˆ
where y and y are predicted means for a treatment1 2 and ,0.001, respectively.
from the models for the first and second logratio, Predicted intercepts for breed and diet and the
ˆ
respectively. The bone proportion is taken as the estimated regression coefficients (bi values) are common divisor in the creation of y and y but any1 2 given in Table 1 (Although there is interaction in the of the proportions can be taken as the common model we present the main effects here for simplicity divisor depending on what particular ratios are of of exposition). The regression coefficients for the interest (Aitchison, 1986). muscle / bone and fat / bone ratios are 0.124 and Computation of standard errors of difference for 1.475, respectively. These coefficients are both posi-these back-transformed proportions is approximate tive indicating that the muscle / bone ratio and the and their computation is complex, relying on the fat / bone ratio increased as the side weight increased.
Table 1
The intercepts for breed and diet and the regression coefficients for the covariate, logside (Eq. (7)) from the compositional analysis of y and1
y2
ˆ
Variable Breed Intercept Diet Intercept b (S.E.)i
y15log(x /x )m b
HE 26.939 Medium 27.216 1.475 (0.108)
b b
Table 3
A coefficient for log(muscle / fat) is estimated by
Predicted composition (g / kg) of muscle, bone and fat for 111 and
ˆ ˆ
b12b250.124–1.4755 21.351 (see Eq. (4)). This 202 kg side weights for each of the three breeds using com-negative coefficient indicates that the muscle / fat positional data analysis (CDA)
ratio decreased as the side weight increased. The
Side weight (kg) Component FR HE CH S.E.D.
intercepts for breed or diet for log(muscle / fat) can
111 Muscle 654 640 689 41
be estimated similarly to the estimation of the
Bone 192 179 192 43
coefficient for log(muscle / fat). For example, the
Fat 154 180 119 18
intercept for log(muscle / fat) for HE is estimated as
202 Muscle 555 528 607 33
0.6902(26.939)57.629.
Bone 151 137 157 39
The intercepts and regression coefficients in Table
Fat 294 334 236 10
1 can be used to predict the muscle, bone and fat proportions for a breed, averaged over diet or for
diet, averaged over breed, for various side weights. (Table 1) for 111 and 202 kg carcass side weights For example, to predict component proportions for a for each of the three breeds averaged over dietary 150-kg side weight (logarithm55.011) Hereford3 energy level (Table 3). These were the minimum and Friesian animal, averaged over diet, first predict maximum side weights in the data set. The heavier
ˆ ˆ ˆ
y and1 y2 as y (log(muscle / bone))1 50.6910.124 animals had lower predicted proportions of muscle
ˆ
(5.011)51.3114 and y (log(fat / bone))2 5 26.9391 and bone and higher proportion of fat for all breeds. 1.475(5.011)50.4522. These predicted values can be Muscle proportion was higher in CH than the other back-transformed (Eq. (9)) to give predicted muscle, two breeds and the advantage over HE was accen-bone and fat proportions (g / kg) of 591, 159 and 250, tuated for the heavier weight. Bone fraction was respectively, which sum to 1000. similar for all breeds at a given weight and fat was Comparable parameter estimates for the data lower for CH, the differential over HE being greater analysed using AR are given in Table 2. The muscle, at the high side weight.
bone and fat components were analysed using a
model with the same explanatory factors and vari- 3.2. Comparison of compositional data analysis ables as above. with allometric regression
In further predictions of composition, the
propor-tional contributions of muscle, bone and fat com- To compare the two approaches (AR and CDA), ponents were predicted using the CDA equations the muscle, bone and fat component weights were
Table 2
The intercepts for breed and diet and the regression coefficients for the covariate, logside when muscle, bone and fat components were analysed using allometric regression
ˆ
Variable Breed Intercept Diet Intercept b (S.E.)i
a
Muscle HE 0.757 Medium 0.803 0.743 (0.0267)
b
Bone HE 0.067 Medium 0.147 0.619 (0.034)
b
Fat HE 26.872 Medium 27.069 2.094 (0.0877)
b
Table 4
Predicted components weights (kg) of muscle, bone and fat for 111 and 202 kg side weights for each of the three breeds using allometric regression (AR) and compositional data analysis (CDA)
Side weight Component Friesian Hereford3Friesian Charolais3Friesian S.E.D. (CDA) (kg)
AR CDA AR CDA AR CDA
111 Muscle 73.03 72.62 70.45 71.04 77.92 76.48 4.56
Bone 21.41 21.29 19.75 19.92 21.66 21.26 4.79
Fat 17.19 17.09 19.87 20.03 13.50 13.25 2.02
Total 111.63 111 110.07 110.99 113.08 110.99
202 Muscle 113.97 112.18 109.94 106.68 121.6 122.64 6.73
Bone 31.00 30.51 28.61 27.76 31.36 31.63 7.87
Fat 60.25 59.31 69.63 67.56 47.33 47.73 2.01
Total 205.22 202 208.18 202 200.29 202
predicted for each of the three breeds using the two accounted discrepancies when estimating differences methods for a 111- and 202-kg carcass side weight between treatments.
(Table 4). For CDA, the predicted weights of each It may be objected that these comparisons were component summed to the correct total (apart from a carried out at the extremes of weight but these were minimal effect of rounding) for all breeds and both chosen to examine the limits of the consequences of side weights, as expected from the theory. For AR, bias which would be expected to occur at one or the total of predicted weights for the lighter carcass other of the extremes. However, were the predictions differed from 111 by 0.63, 20.93 and 2.08 kg for made out at the mean log(side weight) of 5.004, FR, HE and CH, respectively, and for the heavier corresponding to a side weight of 149 kg, the carcass the differences were 3.22, 6.18, and 21.71 discrepancies in the AR prediction of total side kg for the three breeds, respectively. These dis- weight would be 20.57, 20.80, and 21.01 kg for crepancies are not the same for each breed and so Friesian, Hereford and Charolais, respectively, not they will be confounded with breed differences in inconsiderable relative to the size of breed differ-components. They are not negligible in size when ences in composition. Predictions at the three compared with estimated treatment differences or the weights shows that there is no consistent pattern in S.E.D. between treatments. For example, the differ- the direction or magnitude of the error in the sum of ence in muscle weight between FR and HE predicted the predicted components from AR, but these dis-by AR for the heavy carcass is 4.03 kg (113.97– crepancies can be sizeable when compared with 109.94) whereas the difference between predicted treatment effects and with the S.E.D. between treat-total side weights is 22.96 kg (205.22–208.18). ments.
Since muscle contributes about 60% of the weight of
these animals most of the discrepancy is likely to 3.3. Development of a single component lodge in the estimates of muscle and so the
dis-crepancy is likely to be about half of the size of the The data were re-analysed using CDA but with estimated treatment difference. Furthermore, the only two components, muscle and non-muscle. The discrepancy is appreciable compared with the size of bone and fat components were combined to form the S.E.D. between treatments (6.73 for CDA but non-muscle, which as a proportion of side weight is similar measures of variability would apply for AR.). 12x . One y variable was created:m
For fat weight, the allocation of the discrepancy x
m ]]
would tend to reduce the size of the fat difference y5log
S
D
12xmestimated by AR by about 0.9 kg, not negligible
(logarithm of carcass side weight) as a covariate was transformed proportions or components of total side fitted. The resulting regression coefficient 20.676 weights as in Table 4 poses a problem when using (60.069) shows that the muscle / non-muscle ratio either AR or CDA. In both cases, standard errors and declines as side weight changes. The estimated standard errors of difference are available for the model allows prediction of the muscle proportion for predicted values on the transformed scale, but it is different side weights for each breed3diet combina- debatable how best to estimate those associated with tion as above. the back-transformed values as in Table 4. The method proposed here for computing approximate S.E.D. values for the back-transformed predicted
4. Discussion component proportion can also be used for AR
back-transformed estimates.
The discrepancies in total weights predicted by The CDA method is fully multivariate and is AR are the most serious issue in its use. The appropriate for the multivariate nature of composi-alternative provided by CDA is a more natural tional data; hypotheses can be tested not only framework for analysing compositional data, but between treatments for a particular response variable there are some difficulties in calculating the S.E.D. y but also across response variables and, at least
between predictions on the proportional or original approximately, for comparisons of treatments for weight scale when implementing this approach. linear combinations of the back-transformed pro-There are similar difficulties with the AR method. portions. For example, one could check whether the The CDA is a more complete framework for infer- difference between muscle and fat proportions was ence as it is based on a multivariate model rather constant across treatments. This is not possible for than a series of univariate analyses. It is a more the usual implementation of AR which is a series of natural framework than AR for modelling part–part univariate analyses, although it is possible to extend relationships as it preserves the symmetry among this to a multivariate framework.
components. Although links exist between CDA and Several multivariate generalisations of AR have AR the CDA equations are not uniquely determined been proposed. Joliceur (1963) proposed using the by them. first principal component of the covariance matrix of CDA overcomes difficulties associated with AR as the logarithm of the component variables to define a a tool in the analysis of carcass dissection data. It series of pairwise relationships among the D com-does not suffer from the discrepancies between ponents of the composition. The characterisation predicted totals and the total of the body being depended on D parametersui, i51, . . . ,D on which predicted that arise in AR (i.e. predicted component was based a D3D array with typical membera 5ij
phase. It is not obvious from the mathematical models and CDA as detailed in Appendix A. How-development that, were the separate phases to be ever, this link does not simplify inferences when estimated for a particular body measure, the com- analysis is performed solely on the AR scale. ponents would sum to the body measure, as would The AR coefficients derived in the analysis for the be desirable for reasons advanced above. current paper (0.743, 0.619, and 2.094 for muscle, CDA has the added advantage that it is par- bone and fat, respectively) differ slightly from those ticularly suitable for the study of part–part relation- of Keane et al. (1990) for the same data set (0.726, ships between carcass components. Traditional AR 0.625, and 2.057 for muscle, bone and fat, respec-deals with this symmetric idea asymmetrically, with tively). In the earlier analysis fat included cod fat one component being nominated the ‘x’ (indepen- (scrotal fat) and the components were regressed on dent) variable and the other the ‘y’ (dependent) actual side weight. In contrast, in the present study variable, without any compelling rationale for which carcass side weight was taken as the sum of the component is so denominated. Were the roles re- dissected muscle, bone (including other tissue) and versed a different regression equation would emerge, fat (subcutaneous and intermuscular) components which is unsatisfactory. CDA, through the logratio from the joints, which excludes cod fat. These small transformation, models the difference between the differences between the earlier and present analysis logarithms of the components as a function of other do not affect the interpretation. In both analyses the variables which may include total body size, time estimated regression slope coefficients for the muscle and / or other variables and deals with the relationship and bone components are both less than 1 indicating in a single equation. that these components decreased as proportions of An original argument by Haldane against the side as the carcass side weight increased. The allometric relationship (cited in Huxley, 1932) was coefficient for the fat component (2.094) is greater that if each of the parts of the organism (e.g. each than 1 indicating that fat increased as a proportion of muscle) bears an allometric relation to the whole side, as side weight increased.
organism, then the aggregate of a number of parts (say all the component muscles aggregated to give
total muscle) cannot have an allometric relation with 5. Conclusion
the whole. Despite this, AR appears to work very
approximation based on Taylor series expansions giving a variance–covariance matrix of estimated being used here. parameters.
The models are not essentially identical, however, there are an infinity of AR models equivalent to a given CDA model since addition of a constant to all
Acknowledgements
c and a different constant to all d will lead to thei i
same CDA model. The CDA model could apply Thanks are due to the perceptive and helpful
without the AR being appropriate. Any model of the comments of an anonymous referee. This work was
form: carried out by DM as part of investigations leading
to an MA degree. The support of Teagasc through a
E(log(w ))5c 1d log(w)1g
two-year Walsh Fellowship is gratefully acknowl- i i i
edged.
where g is some constant function, will lead to the same CDA model but the model just described may not be an AR model.
Appendix A. Relationship between AR and CDA
There is a simple relationship between the AR and References CDA models. Using the notation developed in the
paper suppose that the AR equations are, for i5
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ˆ ˆ
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