What is the full form of GLS?

Introduction

Generalized Least Squares (GLS) is an extended method used to determine the parameters in a regression model. The ordinary least squares (OLS) approach, which assumes that the errors in the model are uncorrelated and have a constant variance.

The model's faults, however, may not be independent and may have significant variances in numerous real-world settings. In order to correct for the correlation and heteroscedasticity in the errors, GLS incorporates a weighting matrix into the regression model. This makes it possible to estimate the parameters more precisely and improve the fit of the data. In econometrics, finance, and other domains where regression analysis is crucial for data analysis, GLS is commonly utilised.

Assumptions of Generalized Least Squares

To guarantee that the approach generates trustworthy and accurate parameter estimates, Generalized Least Squares (GLS) is predicated on a number of premises. Here are some of the key assumptions of GLS −

• Linearity − The dependent variable and the independent variables have a linear relationship.

• Normality − The model's mistakes have a normal distribution

• Homoscedasticity − The variance of the model's mistakes is constant.

• Independence − The model's flaws are unrelated to one another.

• Stationarity − The mistakes' mean and variance remain constant throughout time.

• No Multicollinearity − The independent variables have a low level of correlation

• No Endogeneity − There is no correlation between the independent variables and the model's faults.

Derivation of Generalized Least Squares Eliminator

A technique used to estimate the parameters of a linear regression model with associated errors is known as the Generalized Least Squares Eliminator (GLSE). Its foundation is the Generalized Least Squares (GLS) approach, which makes use of a weighting matrix to account for heteroscedasticity and correlation in the errors.

We begin with a linear regression model to obtain the GLSE −

$\mathrm{Y\:=\:Xβ\:+\:ε}$

Where X is a matrix of independent variables, Y is a vector of dependent variables, ε is a vector of errors, and β is a vector of unknown parameters.

We assume that the errors ε have a known covariance matrix and are correlated Ω. The

covariance matrix is expressed as follows −

$\mathrm{Ω \:=\: E\:[εε'\:]}$

Where E[] denotes the expected value. The variance of the mistakes may alternatively be expressed as −

$\mathrm{Var(ε) \:=\: σ²Ω}$

Where, σ² is a scalar representing the variance of the errors.

To estimate the parameters β, we must identify the values that reduce the total squared errors −

$\mathrm{SSE \:=\:ε'ε}$

Assuming the mistakes are correlated and have a given covariance matrix, the following sentence may be written −

$\mathrm{SSE\:=\:(\:Y\:-\:Xβ\:)\:'Ω^\:(-1)\:(\:Y - Xβ\:)}$

The values that minimize SSE may then be determined by differentiating this equation with regard to β −

$\mathrm{dSSE\:/\:dβ\:=\:-2X'Ω^\:(\:-1\:)\:(\:Y - Xβ\:)}$

We get the GLS estimator by solving for β and setting this equation equal to zero −

$\mathrm{β_hat\:=\:(\:X'Ω^\:(\:-1\:)\:X\:)\:^\:(-1)\:X’Ω^\:(\:-1\:)\:Y}$

This estimation, nevertheless, relies on our knowledge of the covariance matrix Ω. In reality, we often don't know Ω and must infer it from the evidence. Utilizing the residuals from an initial OLS estimate to estimate Ω is one method of doing this.

The residuals may be expressed as −

$\mathrm{\:e\:=\:Y\:-\:Xβ_O\:lS}$

Where, β_OLS is the OLS estimator of β. We can then estimate Ω as −

$\mathrm{Ω_\:hat\:=\:e'e\:/\:(\:n - k\:)}$

Where, n is the sample size and k is the number of independent variables.

Substituting Ω hat into the GLS estimator, we get the GLSE −

$\mathrm{β\:_\:hat\:=\:(\:X'e'eX\:)\:^\:(\:-1\:)\:X'e'eY}$

The computed residual covariance matrix Ω hat is used by the GLSE to correct for error correlation. Despite the correlated and heteroscedastic nature of the errors, it is a reliable estimator of the real values β.

Applications of Generalized Least Squares

A popular technique in statistics and econometrics with several applications is generalized least squares (GLS). Here are some examples of applications of GLS −

Time series analysis: Time series data with associated errors, such as financial or economic indicators, may be modeled using GLS. GLS may provide more precise and effective parameter estimates than other techniques by taking into account the autocorrelation and heteroscedasticity in the errors.

• Panel Data Analysis − Panel data, which consists of observations on the same variables across a range of time periods and/or cross-sectional units, may be modeled using GLS. GLS can estimate the parameters more precisely and effectively by taking into consideration the correlation between the errors for each unit and time period.

• Spatial Analysis − GLS may be used to model geographical data, including geospatial and climatic data. GLS can estimate the parameters more precisely and effectively than other techniques since it can take into consideration the spatial correlation in the errors.

• Survey Data Analysis − Survey data with intricate sampling schemes and linked errors may be analyzed using GLS. GLS may give more precise and effective estimates of the population parameters by taking into consideration the correlation between the sample units.

• Financial Modeling − In financial modeling, GLS may be used to calculate asset prices, volatility, and risk. GLS may provide more precise and effective parameter estimates than other techniques by taking into account the autocorrelation and heteroscedasticity in the errors.

Conclusion

In conclusion, the strong statistical and econometric technique known as generalized least squares (GLS) can manage heteroscedastic and/or correlated errors in data. GLS is able to provide more precise and effective parameter estimates than other techniques by calculating the variance-covariance matrix of the errors and utilizing it to weight the data. Time series analysis, panel data analysis, geographical analysis, survey data analysis, and financial modeling are just a few of the areas in which GLS is used. Researchers and analysts that deal with complicated and connected data might benefit from its flexibility and adaptability.

FAQs

Q 1: How is the variance-covariance matrix of the errors estimated in GLS?

Various techniques, like the maximum likelihood, the generalized method of moments, and practicable generalized least squares, may be used to estimate the variance- covariance matrix of the errors.

Q 2: What are the advantages of using GLS over Ordinary Least Squares (OLS)?

When the errors in the data are correlated and/or heteroscedastic, GLS may provide parameter estimates that are more accurate and efficient than OLS.

Q 3: In what types of data analysis is GLS commonly used?

Time series analysis, panel data analysis, geographical analysis, survey data analysis, and financial modeling are all typical applications of GLS.