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.