Sociology Mind
2012. Vol.2, No.3, 282-288
Published Online July 2012 in SciRes (http://www.SciRP.org/journal/sm) http://dx.doi.org/10.4236/sm.2012.23037
282
The Body Mass Index (BMI) and TV Viewing in a
Co-Integration Framework
Anastasia Victoria Lazaridi
Department of Nutrition and Dietetics, Technological Education Institute (T.E.I.) of Larissa, Larissa, Greece
Received February 2nd, 2012; revised March 3rd, 2012; accepted April 2nd, 2012
Many techniques are met in the literature, trying to investigate the effect of TV watching hours on BMI.
However, we haven’t traced any empirical study with co-integration analysis, as it is applied here. With
this in mind, we present in this paper the proper methodology, based on the co-integration analysis for a
detailed justification of the effect of TV viewing hours together with some minor changes in life style of
participants on BMI. Apart from finding and testing an acceptable co-integration relation, we further for-
mulated an error correction model to determine the coefficient of adjustment. All findings, which are fully
justified, are presented in details in the relevant sections. It should be pointed out, that we haven’t met this
type of analysis in the relevant literature.
Keywords: BMI; Integration; Co-Integration; Error Correction Model; Coefficient of Adjustment;
Spurious Regression
Introduction
It is worthy to mention that BMI has received particular at-
tention in many research works (see for instance Ahn et al.,
2011; Bridger et al., 2011; Browning et al., 2011; Eek et al.,
2009; Farkas et al., 2005; Harbin et al., 2006; Komlos et al.,
2006; Lee et al., 2008; Manios et al., 2004; Stea et al., 2008;
Stenhammmar et al., 2010, among others). In most cases the
effect of various factors on BMI was the main object. Besides,
in many papers (see for instance Danner et al., 2008; Davison et
al., 2006; Henderson, 2007; Lazarou et al., 2009, among others)
an effort is made to investigate the effect of TV viewing hours
on the BMI. Almost in all cases, cross-sectional data are used
and the standard statistical methods are applied. Apart from
simply presenting the results obtained from commercial com-
puter packages (usually SPSS), no attempt is made to analyze
the mathematics involved even for a plain justification of the
results obtained. It seems that the computer output plays the
role of the “follow me” car in the airports, driving the authors
through a specific direction, without any explanation that this
direction is the right one and why. Instead, a lot of redundant
descriptive details are sited; no any specific and well-formu-
lated model is presented, although the term “model” is exten-
sively used. Regardless of the robustness of these methods, we
haven’t seen in the relevant literature an empirical study with
co-integration analysis, as it is applied here. With this in mind,
we made an attempt to jump to the other side of the fence, con-
sidering time series data and applying co-integration analysis in
order to analytically investigate the effect of TV watching
hours on BMI. This analysis requires that the series used must
be integrated, of the same order, so that we can finally obtain a
stationary linear combination (that can be a static regression)
which constitutes the necessary and sufficient condition to con-
clude that the series used are co-integrated and we don’t face
the case of spurious regression. Further, we formulated an error
correction model to determine the coefficient of adjustment. It
is worthy to mention, that we haven’t met this type of analysis
in the relevant literature.
Data Used
Initially, 50 volunteers from the greater region of Thessalo-
niki, Greece, agreed to report regularly (twice per month) via an
e-mail, their weight and height, together with some major re-
marks, if necessary. Finally we obtained a continuous stream of
such data from only 32 individuals (40% males and 60% fe-
males), for almost 3 years, which means 72 reports (2 per
month), by each participant1. At the very beginning, the age of
participants vary between 17 and 21 years. As mentioned above,
we received individual reports every 15th day (i.e. two reports
per month). Each time we calculated the BMI of each partici-
pant and finally2 we computed the mean BMI, regarding the 32
cases. The same calculations hold for the mean hours of TV
viewing during the period of 15 days. These 72 mean values
refer to series {BMIi}, {TVHi} (i = 1, 2, 3, ···, 72), which are
presented in appendix. It should be noted however, that the
majority of the participants stated that from the 62nd observa-
tion (i.e. from middle of the 31st month) and onwards, they
increase there seating hours, for various reasons (decreasing
physical exercise, working from home, etc.). To capture this
structural change in participants lifestyle, we introduced a
dummy (binary) variable (i), which takes the value 1 for the
last 11 observations (from 62 up to 72), and the value of zero
elsewhere.
d
It is noted that we denote by Y the natural logs of BMI [i.e.
Yi = ln(BMIi)] and by X the natural logs of TV hours [i.e. Xi =
ln(TVHi), i = 1, 2, 3, ···,72]. Since the latter variables (i.e. Y
1Last report received beginning of 2012.
2It is recalled that the formula used to compute BMI is:



2
Weightin kg
BMI Height inm 

A. V. LAZARIDI
and X) are used in this study, it should be recalled that the
regression parameters represent constant elasticities. The
time-paths of variables Yi and Xi are presented in Figure 1.
It should be mention at this point, that we found an analo-
gous formulation with group means in Johnston (1984: p. 406).
Further, according to Green (2008: p. 189), the loss of informa-
tion that may occur through using group means, as we have
done here, might be relatively small. Also the estimators to be
obtained in such cases would be consistent if the number of
periods (T) is big enough, which complies with our data.
The Long-Run Relationship. The Concept of
Co-Integration
Considering the series {Yi}, {Xi}, and applying the Dic-
key-Fuller (DF/ADF) tests (Dickey & Fuller, 1979, 1981; Har-
ris, 1995: pp. 28-47), we found that both series are not station-
ary, but they are integrated of order 1, that is I(1). This implies
that differencing the initial series we get stationary ones. In
other words the series
ΔYi = Yi – Yi−1 and ΔΧi = Xi – Xi−1
are stationary, that is I(0). Additionally, a fairly new and much
easier test proposed by Lazaridis (2008) has been also applied,
in order to detect stationarity which can be easily verified from
Figure 1.
It should be recalled that if two series are not stationary but
they have common trends so that there exist a linear combina-
tion of these series that is stationary, which means that they are
integrated in a similar way and for this reason they are called
co-integrated, in the sense that one or the other of these vari-
ables will tend to adjust so as to restore a long run equilibrium.
And this is the case of the two series {Yi}, {Xi}, considered here,
as we’ll see in what follows.
The next step of this procedure is to specify and estimate a
long-run linear relationship for the I(1) series, which in this
particular case has the form
12 3iii
YbbXbd u
i

 (1)
which is a static linear model.
The OLS (Ordinary Least Squares) estimates from fitting (1)
to the data presented in appendix are as follows (standard errors
in brackets).
Yi Xi
ΔYi ΔXi
Figure 1.
he I(1) series {Yi}, {Χi} and the stationary ones {ΔΥi}, {ΔΧi}. T

A. V. LAZARIDI
 
0.013644 0.0044550.00533
ˆ2.3751910.2226990.04106
174.08 49.993 7.708
val. 0.0 0.0 0.0
Hansen 0.21
iii
YXd
t
p
 


2
2, 69
0.24 0.13
0.987, 0.012, 1.4, 2853.8,
Condition number21.17
RsDWdF
CN
 

(1a)
It is recalled that Hansen (1992) statistics, reveal that the
model coefficients, individually considered, are stable for all
conventional levels of significance (α = 0.01, 0.05, 0.10). The
low value of the condition number (CN) indicates that no mul-
ticollinearity problems exist (Lazaridis, 2007). Finally from the
p-values we conclude that all coefficients are significant. This
implies that hours of TV viewing are (positively) associated
with BMI. This is in line with the findings of Danner (2008: p.
1101). Obviously, the same applies for the change of lifestyle
of participants from 62nd observation and onwards. Also, the
joined effect of both variables (X and d) is significant, accord-
ing to the large value (2853.8) of F statistic. It may be worthy
to mention at this point, that Danner (2008) using pooled data
estimates a model where the only explanatory variable is a
categorical one [time and (time)2, p. 1103], taking arbitrary
values, so that one may argue that the correct specification of
the model is questionable. Besides no model testing results are
presented and no explanation is provided about the subscript i
(i.e. where this subscript refers to). Next (same page), the au-
thor presents the estimated results of two models (fixed effects
and random effects) without even a vague explanation of what a
random effects model is, what estimation process is applied in
this case and how the random effects are computed, if they are
needed. Mainly, there is a lack of any proper testing (see
Hausman, 1978), in order to justify as to whether the fixed
effects or the random effects model is more suitable for the data
used. In other cases (see for instant Lazarou et al., 2009), one
reads the term “regression models” (page 71), but no such a
model is explicitly presented in the typical form, as the one
seeing in (1a). Henderson (2007), speaks about “modeling la-
tent growth curves” (p. 547), without any sufficient and ana-
lytical explanation of what it is and whether any conventional
model tests have been applied. In other cases, misleading statis-
tical measures are presented as in Browning et al. (2011: p.
1383), where the plain arithmetic mean (30.6 Kg), of a sample
where weight varies from 2.5 to 117.8 Kg is reported. In similar
cases with such a large range, a sort of weighted mean, where
the weights will be related somehow to the age, or at least the
median (even if classification of the data is necessary), may be
more representative measures of central tendency.
Finally, from the small value (0.012) of the standard error of
estimate (s) seeing in (1a), we may conclude that uncertainty is
rather limited when forecasting the values of the dependent
variable. Note that in such a case, a measure of uncertainty is
the width of the relevant confidence interval.
According to Granger and Newbold (1974), high t-values,
high 2
R
(i.e. the adjusted coefficient of determination 2
R
),
together with low values of the DW d statistic, when a static
regression of two independent random walks is estimated [as in
(1a)], then we are facing the case of spurious regression, with
undesired properties. However, a static regression between
non-stationary variables will not be a spurious regression, if the
variables have common trends, for instance if they are inte-
grated of first order and the OLS residuals are stationary. Then
the series [here {Yi} and {Xi}] are integrated in a similar way
and for this reason they are called co-integrated. This implies
that although the series under consideration are non-stationary,
there exists a stationary linear combination of these variables.
To show that {Yi} and {Xi} are co-integrated, we compute
the OLS residuals {} from:
ˆi u
ˆ
ˆii uYY
i

 (2)
ˆ0.2226990.04106 2.375191
iii i
uYX d (2a)
Then we run the following regression without constant
term, since for the OLS residuals
0
ˆ.
i
u

11 1
1
ˆˆ ˆ
q
ii jij
j
ububui




 
 (3)
The value of q is set such that, after excluding the terms with
insignificant coefficients, the noises εi to be white, as it will be
explained in details later on. With q = 0, the estimation results
are as follows:

1
0.004455
ˆˆ 0.727716
6.35
val. 0.0
ii
uu
t
p




(3a)
DW d = 2, AIC (Akaike’s Information Criterion)3 = −6.105.
With q = 1 and 2, we get the following estimation results.
 
11
0.14473 0.11991
ˆˆ ˆ 5.9505 0.173559
4.111 1.447
val. 0.0 0.152
DW d = 2, AIC =6.0955
ii
uu
t
p
i
u






(3b)
 
11
0.146112 0.1549430.12268
ˆˆ ˆ 0.582877 0.1935220.008113
3.552 1.249 0.0661
val. 0.0 0.216 0.947
DW d = 1.9
ii i
uu u
t
p
2
ˆ
i
u



 

, AIC = 6.065
(3c)
(It is recalled that when the value of DW d statistic is close to
2, then we may conclude that no problem of first order auto-
correlation exists).
Although in all cases there is no any problem of autocorrela-
tion and heteroscedasticity, we considered model (3a), since in
the other two models- and for higher values of q the additional
explanatory variables are insignificant, as it is verified from the
corresponding p-values. Hence the value of t-statistic (−6.35),
which refers to the estimated coefficient of , will be taken
into consideration next. 1
ˆi u
Since the series


ˆi
u is the result of specific calculations
and in the simplest case are the OLS residuals, it is not advis-
able (Harris, 1995: pp. 54-55) to apply the Dickey-Fuller (DF/
ADF) test, as we do with any variable in the initial data set. We
have to compute the tu statistic from McKinnon (1991: pp.
267-276) critical values (see also Harris, 1995: Table A6, p.
158). This statistic (tu = Φ∞ + Φ1/Τ + Φ2/Τ2, where T denotes
the sample size) for the static relationship seen in (1a) is:
3It is noted that in usual applications, when competing models are consid-
ered, the one with the smallest value of AIC is preferred.
284

A. V. LAZARIDI
α0.01 α0.05 α0.10
3.52 2.9 2.59
uuu
ttt


Since t = −6.35 < tu, the series


ˆi
u
ˆ u
is stationary for α =
(0.01, 0.05, 0.10). Recall that the null [i ~ I(1)] is rejected in
favor of H1 [~ I(0)], if t < tu. Hence (1) concerns a co-inte-
grating relationship and can be considered as a longrun equilib-
rium relationship. The estimation results seeing in (1a) are quite
satisfactory.
ˆi u
A comparatively simple way (Lazaridis, 2008) to test that for
the noises ei in (3), with q = 0, we don’t have any problem of
autocorrelation of higher order, is to compute the residuals
from (3a) and to consider the corresponding Ljung-Box Q sta-
tistics and particularly their p-values, which should be greater
than 0.1 to say that no autocorrelation is present. For this par-
ticular case, the corresponding Q statistics (column 4) together
with p-values are presented in Table 1. From this table, we see
that for all k (column 1) the corresponding p-values (column 5)
are greater than 0.1, so we can conclude that there is no need to
increase the value of q, to get Equations (3b) and (3c) in order
to face autocorrelation problems. As far as heteroscedasticity is
concerned, a practical way to trace it, is to find the explanatory
variable which yields the smallest p-value for the corresponding
Spearman’s correlation coefficient (
ˆi e
s
r), or the larger Z* statistic
(absolute value). Since in (3a) there is only one explanatory
variable, we found: rs = 0.075, p = 0.48 and Z*= 0.63. From the
p-value we conclude that we have to accept the null of homo-
scedastic disturbances. When larger samples are considered,
then Z* statistic is used, which is computed from:


*1
1
1
s
s
r
Zr
T


T
We accept the null, if |Z*| < Zα/2.
From the table of standard normal distribution we find:
α2α2α2
α0.01 α0.05 α0.10
2.5758 1.96 1.6449ZZZ


Hence we accept the null for all conventional levels of sig-
nificance.
It may be useful to mention that we reach to the same results
when model (3b) and (3c) are considered. We pointed out
however, why these models have been dropped.
Having in mind some confusing remarks on this point, we
underline once more that (1), which is a static regression, is a
co-integrating relation, iff4 the residuals , computed from
(1a), (2a) are stationary. Apart from the proper test shown
above, the stationarity of
ˆi u


ˆi
u may be detected from Figure 2
presented above.
From the estimated co-integration regression (1a), we can tell
that the dummy di, used as a proxy to capture the change in
lifestyle of the participants, had a significant effect on BMI.
We see also that the elasticity of the TV viewing hours is
0.222699. It is worthy to mention here, that taking into account
that the quotient of sample standard deviations of Yi and Xi is
0.107777/0.433222 = 0.248, the estimated long-run response of
0.222699 in (1a) is reasonable close, which is another verifica-
tion that this co-integration analysis has resulted to an accept-
able model for investigating the relations between BMI and
Table 1.
Residuals : Autocorrelations, PAC, Q statistics and p-values.
ˆi
e
Values
of k AC PAC
Q_Stat
(L-B) p value Q_Stat (B-P) p value
1 −0.041 −0.0410.127 0.720 0.1220.726
2 0.1700.169 2.320 0.313 2.1940.333
3 −0.0070.005 2.324 0.507 2.1980.532
4 0.1620.138 4.378 0.357 4.0830.394
5 −0.090 −0.0835.014 0.414 4.6590.458
6 0.0570.004 5.275 0.508 4.8910.557
7 −0.133 −0.1146.729 0.457 6.1660.520
8 0.0430.008 6.884 0.549 6.3000.613
9 −0.132 −0.0778.353 0.498 7.5470.580
10 −0.100 −0.1319.209 0.512 8.2630.603
11 −0.103 −0.04710.12 0.518 9.0180.620
12 −0.055 −0.05510.40 0.580 9.2390.682
13 −0.0270.036 10.46 0.655 9.2920.750
14 −0.060 −0.04410.80 0.701 9.5510.794
15 −0.033 −0.02110.90 0.759 9.6310.842
16 0.0830.092 11.56 0.773 10.120.859
17 0.0790.078 12.17 0.789 10.570.877
18 0.0390.031 12.32 0.829 10.690.907
19 0.0670.017 12.78 0.849 11.010.923
20 0.0590.009 13.13 0.871 11.260.939
21 0.2300.197 18.63 0.608 15.020.821
22 −0.132 −0.17120.49 0.552 16.270.801
23 −0.079 −0.18121.17 0.570 16.720.822
24 −0.094 −0.11422.15 0.570 17.350.833
25 −0.061 −0.10522.58 0.602 17.620.857
26 0.0020.143 22.58 0.656 17.620.889
27 −0.115 −0.08024.14 0.621 18.560.885
Figure 2.
The stationary series


ˆi u, computed from (2a). 4If and only if.

A. V. LAZARIDI
TV viewing hours. Thus we may conclude that an increase by
10% of TV viewing hours, may result to an increase of about
2.2% of BMI.
The Short Run Relationship
The lagged values of the residuals computed from the static
long-run equation (i.e. 1), serve as an error correction me-
chanism in a short-run dynamic relationship, where the addi-
tional explanatory variables may appear in first differences and
lagged first differences. All variables in this equation, known
also as error correction model (ECM), are stationary so that,
from the statistical point of view it is a standard single equation
model, where all the classical tests are applicable. It should be
noted, that the lag structure should be so selected in order to
eliminate autocorrelation and to obtain at the same time a sig-
nificant adjustment coefficient, which refers to 1. With this
in mind the short-run ECM corresponding to (1a) may have the
form:
ˆi
u
ˆi u
 
1 1
0.00131 0.02160.020.116
ˆˆ
0.001986 0.12430.0430.2858
val. 0.13 0.0 0.025 0.016
Hansen 0.23 0.14 0.398 0.23
RESET F0
i iii
YXX
p
u


 


2
3, 66
.879 (val.0.46)
0.33,0.009, DW d2.25,12.5,
Condition number2.03
p
Rs F

 

(4)
According to Hansen statistics, all coefficients-individually
considered seem to be stable for α = (0.01, 0.05). The condition
number reveals that there is no any multicollinearity problem.
That no autocorrelation of higher order exists can be detected
from the table which is analogous to Table 1 and refers to the
residuals of Equation (4), where all the p-values corresponding
to the Ljung-Box Q statistics are greater than 0.1. In models
like the one seeing in (4), the term 1 usually produces the
smallest p-value, regarding the t-statistic of the corresponding
Spearman’s correlation coefficient
ˆi
u

s
r, or the larger Z* statis-
tic (absolute value). In this particular case we have: rs = 0.16, t
= 1.336, p = 0.156 and Z* = 1.329, which means that no hetero-
scedasticity problems exist. Finally the RESET test (Ramsey,
1969), shows that there is no any specification error.
Before interpreting the estimation results, a further comment
should be made about the sign of the coefficient of adjustment
(−.2858). It is recalled that the residuals have been com-
puted from (2), that is . However, if we compute
these residuals from
ˆi u
ˆ
ˆii uYY
i
i
1i
u

ˆ
ˆii uYY (5)
Then Equation (4) will take the form:
 
1
0.00131 0.02160.020.116
ˆˆ
0.001986 0.12430.0430.2858
iii
YXX

  (6)
In other words, the sign of the coefficient of adjustment de-
pends upon the relation (2 or 5) used to compute the residuals
entered in the ECM with one period lag. In any case, this coef-
ficient is significant at α = 0.05. From Equations (4) and (6) we
see that the adjustment coefficient (0.2858) gives a satisfactory
percentage, regarding the BMI convergence towards a long run
equilibrium. Besides, the significance of the coefficient of dis-
equilibrium error (i.e. the coefficient of adjustment), in-
dicates that in the long run there is a causality effect from TV
viewing hours to BMI. Also, the coefficient of ΔXi is significant,
so that we may conclude that changes in TV viewing hours
influence BMI. In the short-run, a change of the hours viewing
TV by 1%, results to an increase of BMI by about 124% from
one period to the next.
1
ˆi u
Another point which deserves further attention refers to the
condition number reported here, although we don’t see it in
other studies (see for instance Nguyen, 1987; Ouyang et al.,
2000, among others). In many applications, particularly when
the variables are in logs, then in the corresponding statistical
models the value of CN is extremely high revealing in most
cases a severe multicollinearity problem. And it seems that this
is the main reason, for not reporting this statistic in relevant
applications. However, Lazaridis (2008) has shown that in such
cases, usually we have spurious multicollinearity.
Conclusion
We applied detailed co-integration analysis to investigate the
effect of TV viewing hours on BMI. For this particular case, we
found that the elasticity of TV viewing hours will not exceed
0.223, which implies that if these hours increase by 10%, than
the expected increase of BMI will be about 2.2%. It has been
verified that TV viewing hours influence BMI together with the
lifestyle change of the participants, captured by the dummy
variable di. It is also verified that changes of the TV viewing
hours influence BMI and a 1% change of the hours watching
TV, may results to a change of BMI by about 0.124% from one
period to the next. These findings do not mean that TV viewing
hours might be inefficient in the long run, since from
co-integration analysis we found that the adjustment coefficient
is significant and it will be expected to fluctuate around 0.28,
giving thus a satisfactory percentage, regarding the BMI con-
vergence towards a long run equilibrium. Ceteris paribus, BMI
is expected to undergo significant effects in the long run, from
TV viewing hours.
It should be emphasized however, that this analysis is based
upon self-reported data. Also the sample size (32 cases), to
compute relevant means may be considered insufficient. This
gives rise to further research by increasing the number of par-
ticipants and mainly to have available authorized data. Addi-
tionally, if the number of participants may be considerably
increased, separately estimation results could be obtained for
mails and females, to better trace possible similarities and dif-
ferences. Also, an effort should be made to capture the change
in life-style of participants, by more official indices instead of a
plain binary variable used here. Finally, if the number of peri-
ods (T) can be further increased, then we may reach to more
detailed and robust results, regarding the effect of TV viewing
hours and changes of life-style on BMI, separately for males
and females. In fact, in this research work we didn’t pay par-
ticular attention regarding data collection, since the main aim of
the study was to analytically present a new approach to face
similar tasks, which has not met in the relevant literature.
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A. V. LAZARIDI
Appendix
Data Used
Observ BMI TVH Observ BMI TVH
1 18.57775 12.79032 37 21.87022 24.85286
2 18.68586 11.82927 38 21.82463 25.14042
3 19.00319 12.33256 39 21.98319 25.82406
4 18.51739 12.36122 40 22.84051 26.51400
5 18.74946 11.81716 41 22.19137 26.96486
6 18.79981 11.81716 42 22.41486 31.30866
7 18.72497 12.19349 43 22.61118 28.23884
8 18.84913 12.52188 44 22.24890 28.40504
9 18.95579 12.50572 45 22.43692 29.79380
10 19.15903 14.00346 46 22.18516 29.43506
11 19.35127 14.06992 47 22.40386 27.61750
12 19.35512 14.02559 48 22.37713 28.25546
13 19.27584 13.07618 49 22.32228 27.24924
14 19.53847 14.16653 50 22.39915 28.64808
15 19.75065 14.35446 51 22.64464 28.78764
16 20.37244 18.42744 52 22.80503 28.16436
17 20.55135 17.55791 53 23.04032 28.86788
18 20.72399 18.54403 54 23.16660 30.62664
19 21.23958 19.61646 55 23.38699 31.59010
20 21.09851 20.65146 56 23.58326 32.30628
21 21.21332 21.17182 57 23.44378 32.53032
22 21.11734 21.53726 58 23.68647 33.94054
23 20.84162 19.90122 59 23.88441 32.91348
24 20.87866 18.99290 60 24.39455 37.12680
25 21.00906 21.11854 61 24.7638 37.85142
26 20.99755 19.85708 62 25.22743 41.84732
27 20.82740 20.49084 63 25.29593 10.96966
28 20.76497 19.74730 64 25.49729 42.10952
29 20.84590 20.45662 65 25.62897 41.76908
30 20.91438 19.62188 66 25.95255 44.76924
31 21.01190 21.48892 67 26.38734 46.28160
32 21.07538 20.04534 68 26.67996 48.56144
33 20.78198 18.54911 69 24.24756 48.88870
34 20.71410 17.90580 70 27.35022 53.80172
35 20.87725 18.86783 71 27.79341 55.47396
36 21.56734 23.14602 72 27.91538 58.71166
288