Download Lurking Variables in Econometric Models: Understanding Synergistic Effects and more Study notes Statistics in PDF only on Docsity!
Lurking Variable in
Econometric Models
LURKING VARIABLES IN ECONOMETRIC MODELS
Frank G. Landram, West Texas A & M University
Amjad Abdullat,West Texas A & M University
ABSTRACT
Lurking variables are omitted variables that should be included in the
regression model. If the lurking variable is part of a synergistic combination, the
effects it has on a regression model are magnified. This paper illustrates the
seriousness of omitting a variable that is part of a synergistic combination. When this
happens, synergistic variables in the regression model act as if they are unrelated with
the dependent variable. This drastically reduces the model's effectiveness and can
lead to misleading results. An awareness of synergistic variables and their possible
effect on underspecified regression models enable analysts to become more proficient
in econometric modeling.
INTRODUCTION
Little research is available concerning the possible effects of excluding
relevant variables which are part of a synergistic combination [4]. Although these
effects are known in many circles by oral tradition and intuition, they have not been
documented. Moreover, the results of excluding these types of variables are usually
misleading.
Using obvious notations, this paper illustrates the seriousness of lurking
variables in regression [10]. If the lurking variable is synergistic, the seriousness is
magnified. Synergism in regression enables multiple R
2
to become greater than the
sum of the simple r
2
coefficients; R
2
> Γr .j. These regressors which are seemingly
unrelated with Y become significant when combined with other variable(s). Using
empirical data, an example is given which illustrates the misleading results when
synergistic variables are omitted from the regression model. Concluding remarks are
given which will enhance ones ability in model building.
Benefits
This paper benefits readers by showing (a) the possible misleading results of
lurking variables and underspecified models. This encourages analysts to become
more diligent in their preliminary research procedures. (b) Readers are given a
strong awareness that they should not always accept statistical tests as being
definitive. If intuition and a knowledge of the subject matter indicate that an
insignificant regressor is in fact related with Y, search for additional influential
variables. The regressor in question may be a synergistic variable needing another
regressor to complete the synergistic combination. (c) Variable selection algorithms
can not replace a knowledge of the subject matter and must be used with caution.
When the regression model is underspecified, variable selection algorithms usually
magnify the problem. (d) A knowledge of synergism also enhances ones skills in
Southwestern Economic Review
regression analysis. Multicollinearity is desirable for synergistic variables but
becomes a problem for other variables [3]. This concept is explained below.
SYNERGISTIC VARIABLES
A synergistic variable is defined as a variable whose partial r
2 value is
greater than its simple r 2 value [4]. A variable is also considered synergistic if it
possesses a significant partial F value and an insignificant simple r
2 value.
Synergistic variables must be used in a combination with other variables. Several
articles have illustrated synergism in regression [4][9]. Kendall and Stuart [5], show
that given r (^) y.jr (^) y.k > 0, if variables X (^) j and X (^) k are inversely related (r (^) j.k < 0) or if
r (^) j.k > 2r (^) y.jr (^) y.k /(r (^) .j + r (^) .k), (1)
then the variables are synergistic; R^2 > r (^) .j + r (^) .k. They also give the conditions for
identifying synergism when r (^) y.jr (^) y.k < 0.
Daniel and Wood [2] and Freund [3] show that synergism in regression is a
function of multicollinearity. As multicollinearity increases among the regressors, the
importance of the regressors may also increase; this increases the partial F values and
decreases the residual mean square. This concept is illustrated in Figure 1 for the
regression model
Y
^ = b 0 + b 1 X 1 + b (^) 2X (^) 2. (2)
Multiple R^2 is measured on the vertical axis and the correlation or multicollinearity
between X 1 and X 2 (r (^) 1.2) is measured on the horizontal axis. Given specific values for
r (^) y.1 and r (^) y.2 and starting at the extreme negative point for r (^) 1.2 , multiple R^2 decreases as
r (^) 1.2 increases between X 1 and X (^) 2. This continues throughout the permissible range for
r (^) 1.2 until R^2 reaches its minimum after which it increases. Hence, multicollinearity is
desirable if X 1 and X 2 are inversely related (r (^) 1.2 < 0) or if r (^) 1.2 is greater than (1)
given r (^) y.1r (^) y.1 > 0. These values are where X 1 and X 2 become synergistic.
FINANCIAL ANALYSIS EXAMPLE
Stock prices (Y) are a function of annual return on investment and
anticipated growth. The data in Table 1 was obtained from a sample of 35 companies
in Dun's Review. The types of variables employed in describing stock prices are
listed below.
X (^) 1: yield = (Dividend + Price Change)/Current Price X2: Dividends
X (^) 3: Earnings per share X (^) 4: Sales X5: Income
X (^) 6: Return on sales X 7 : Return on equity (ROE) X (^) 8: Exchange
traded
Southwestern Economic Review
ANALYSIS
Table 2 gives the computer output for two scenarios. Assume the analyst
excludes variable X and computes the initial least squares regression model with the
following variables:
Y = f(X X 3 X 4 X 5 X 6 X 7 X 8 ) R
2 = 0.53 (3)
Using stepwise regression with significant level at 0.10, the above model reduces to
Y = f(X 3 X (^) 4) R
2 = 0.46 (4)
By excluding X, both of the above models are underspecified.
Table 2 Partial F Measures In Regression Models
(a) Y = f(X 1 X 2 X 3 X 4 X 5 X 6 X 7 X) R = 0.
Variable X 1 X 2 X 3 X 4 X 5 X 6 X 7 X
partial F 6.15 30.86 25.50 2.45 1.67 4.67 0.24 1. p-value 0.020 0.001 0.001 0.130 0.208 0.040 0.626 0.
(b) Y = f(X 1 X 2 X 3 X 4 X) R = 0.
Variable X 1 X 2 X 3 X 4 X 6
partial F 7.05 34.86 24.01 2.47 3. p-value 0.013 0.001 0.001 0.127 0.
(c) Y = f(X 1 X 2 X3) R = 0.
Variable X 1 X 2 X 3
partial F 12.39 36.5 23. p-value 0.0014 0.0001 0.
(d) Y = f(X 2 X 3 X 4 X 5 X 6 X 7 X) R = 0.
Variable X 2 X 3 X 4 X 5 X 6 X 7 X 8
partial F 0.26 17.49 2.88 1.81 1.20 1.33 1. p-value 0.617 0.001 0.101 0.190 0.283 0.259 0.
Lurking Variable in
Econometric Models
When X is included, the stepwise regression algorithm reduces the initial model
Y = f(X X X X X X X X) R = 0.79 (5) to Y = f(X 1 X 2 X (^) 3) R^2 = 0.73 (6)
The adjusted R^2 value for (5) and (6) are 0.722 and 0.70. Table 3 reveals that
variables (X X X X) possess significant simple r-value. However, a variable is
deleted from a regression model because it is either unrelated with Y or it is
multicollinear with other X-variables (partial duplication of data) and not needed.
The latter reason is why variables X and X are deleted from (5). Although variables
X and X possess significant simple r-values, their multicollinearity causes them to
become insignificant and deleted from the model.
Table 3 Correlation Coefficient Matrix (Bottom Diagonal)
Variable Y X1 X2 X3 X4 X5 X6 X
Y 1.000 xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx
X1 - 0.442 1.000 xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx
X2 0.191 0.597 1.000 xxxxx xxxxx xxxxx xxxxx xxxxx
X3 0.633 0.033 0.348 1.000 xxxxx xxxxx xxxxx xxxxx
X4 0.425 0.033 0.443 0.320 1.000 xxxxx xxxxx xxxxx
X5 0.383 0.132 0.512 0.309 0.967 1.000 xxxxx xxxxx
X6 -0.096 0.256 0.048 -0.107 -0.270 -0.089 1.000 xxxxx
X7 0.073 -0.260 -0.289 -0.105 -0.118 -0.082 0.289 1.
X8 0.034 0.184 0.311 -0.083 0.204 0.221 -0.192 -0.
The Least Significant Absolute Value for simple r equals 0. (LSV-r = 0.254). All r-values below 0.254 are insignificant.
By employing the stepwise regression algorithm, (6) is obtained from (5).
By employing the all possible regressions algorithm on (5), the following model is
obtained;
Y = f(X X X X X) R = 0.76 (7)
The adjusted R
2 value for (7) is 0.721. The partial F-value for X is marginal at 2.47;
p-value = 0.13. However, X is a synergistic variable and only becomes significant
when X 2 and X 4 are included in the model. Thus, the identification of synergistic
combinations and a knowledge of the subject matter is used in selecting regression
model (7) over model (6).
Lurking Variable in
Econometric Models
times autocorrelation and violations such as normality and homoskedasticity may be
correctly viewed as underspecification error. Kraner [7], also Maddala [8] have
excellent expositions on misspecification errors. Belsley [1] argues for the use of
prior information in specification analysis. Certainly, econometric models must be
developed by people well grounded in economic theory and a firm knowledge of the
subject matter.
Finally, in the preliminary research phase, always obtain a further knowledge
of the subject matter. Take time to identify influential variables. In the analysis
phase of the research, if intuition strongly indicates that a seemingly unrelated
variable should be related to Y, return to the preliminary research phase. Search for
additional variables so that the proper synergistic combination is included in the
regression model.
REFERENCES
Besley, D.A. (1986), Model Selection in Regression Analysis, Regression Diagnostics
and Prior Knowledge," International Journal of Forecasting 2, 41-6, and commentary 46-52.
Daniel, C., & Wood F.S. (2000). Fitting Equations to Data , 3nd.ed., New York: John
Wiley.
Freund, R.J. (1988), "When is R
2
r (^) y
2 .1 + r^ y
2 .2 (Revisited),"^ The American Statistician , 42, 89-90.
Hamilton, David (1987), "Sometimes R
2
r (^) y
2 .1 + r^
2 y.2: Correlated Variables Are Not Always Redundant," The American Statistician , 41, 129-132. Kendall, M.G. and Stuart, A. (1973), The Advanced Theory of Statistics , (Vol. 2, 3rd. ed.). New York: Hafner Publishing.
Kennedy, Peter (1998), A Guide To Econometrics, 4th ed., Cambridge, Mass., The
MIT Press
Kramer, w. (1985), "Diagnostic Checking in Practice", Review of Economics and
Statistics 67, 118-23.
Maddala, G.S. (1995), "Specification Tests in Limited Dependent Variable Models",
Advances in Econometrics and Quantitative Economics , Oxford, Blackwell, 1-49.
Mitra, S. (1988), "The relationship between the multiple and the zero- order
correlation coefficients," The American Statistician , 42, 89. Ryan, T.P. (1997), Modern Regression Methods , New York, Wiley.
Southwestern Economic Review
