GLS (General Least Square Model) Assumptions
Tanaka and Huba (1989) discover that if the third or
fourth criteria of the normal linear regression model are violated, i.e., if
the random components do not have constant variance or correlate, the
generalized least squares method (GLS) is utilized. This is indicative of a
heterogeneous population made up of incredibly dissimilar units.
The
fundamental distinction between the normal generalized regression model and the
U is a random component covariance matrix. The identity matrix C 0 is
considered to be equivalent to the identity matrix in the typical model. The residuals'
covariance (thus, variance and correlation) is presumed to be arbitrary in the
generalized model; hence, the matrix C 0 might have arbitrary values. This is
the essence of the normal model's generalization.
The
results of a generalized regression model using classic (conventional) OLS are
consistent and unbiased. However, these computations become ineffective. As a
result, the parameters of the generalized model are estimated using generalized
least square models.
A
normal linear regression model's initial premise is that the explanatory
variables x j (j = 1; m) are deterministic (non-stochastic). Cook and Weisberg (1994) assert that this implies that
explanatory variables would remain static if the regression analysis were
repeated. The dependent variable y value will change when the random component
values in the new sample vary.
Other assumptions are:
•
If all of a model's equations are correctly
identified, it is said to be accurately identified.
•
If there is at least one unidentified model
among the model's equations, the model is considered unidentified.
•
If there is at least one overidentified model
among the model's equations, the model is termed overidentified.
•
An equation is said to be precisely recognized
if the coefficients of the simplified model can find the structural parameter
estimates uniquely (uniquely).
•
An equation is overidentified if more than one
numerical value can be derived for some structural parameters.
•
If estimations of an equation's structural parameters
can't be found, it's considered unidentified.
Transforming
Variable to Linear
The
structural form of the model describes a real phenomenon or process. Most
often, natural phenomena or processes are so complex that systems of
independent or recursive equations are not suitable for their description.
Therefore, they resort to systems of simultaneous equations. The parameters of
the structural form are called structural parameters or coefficients. MacKinnon and Magee (1990) find that some of the
structural form equations can be represented in the form of identities, that
is, equations of a given form with known parameters.
It is easy to move from the structural form to the so-called
reduced form of the model. The reduced form of the model is a system of
independent equations in which all the current endogenous variables of the
model are predefined.
R procedures for
Linear Regression
Linear regression using R comes in two flavors multiple and
simple. Below is an example of the simple linear regression using R and R
studio:
Linear regression calculation required only an atomic
variable (independent).
Step A: R should be used to load the data.
For each dataset, follow these four steps: Go to File >
Import dataset > From Text in RStudio (base).
summary (imported data)
When we call this function, we get a table in our console
with a numeric summary of the data because both of our variables are
quantitative. This gives us the independent variable's (var1) and dependent
variable's (var2) lowest, median, mean, and maximum values.
Step B: Ensure the data assumptions are valid
We may use R to see if our data meets the four fundamental
linear regression assumptions.
Observational independence (aka no autocorrelation)
Jajo (2005)
finds that there is no need to evaluate for hidden relationships among
variables because there is only one independent variable and one dependent
variable. Do not use a simple linear regression if autocorrelation is required
within variables, for instance, numerous observations of the same study
participant. Instead, use a structured model, such as a linear mixed-effects
model.
Use the hist () function to see if the dependent variable
has a normal distribution.
hist ("your data")
Histogram of simple regression
It is safe to proceed with the linear regression if the
results show a bell curve with more observations in the distribution center and
fewer on the tails.
Step C: Construct a linear regression model.
Perform a linear regression analysis to evaluate specific
associations between variables (independent - dependent) if the data match the
assumptions.
To see if the observed data matches our model assumptions,
run plot(var):
Step D is the final step. To see how the results of
the simple linear regression can be visualized, simply use the ggplot package
and plot the data points on a graph.
References
Cook, R. D., & Weisberg, S. (1994).
Transforming a response variable for linearity. Biometrika, 81(4), 731-737.
Jajo,
N. K. (2005). A review of robust regression and diagnostic procedures in linear
regression. Acta Mathematicae Applicatae
Sinica, 21(2), 209-224.
MacKinnon,
J. G., & Magee, L. (1990). Transforming the dependent variable in
regression models. International Economic
Review, 315-339.
Tanaka, J., & Huba, G. (1989). A general
coefficient of determination for covariance structure models under arbitrary
GLS estimation. British Journal of
Mathematical and Statistical Psychology, 42(2), 233-239.
