Gauss-Markov Models

Two famous scientists discovered Gauss-Markov models, named Andrey Markov and Carl Friedrich Gauss. This model demonstrates the relationship of ordinary least squares. This probabilistic model is designed to make complicated dynamical probabilistic models easier. Under this model, Carl Friedrich Gauss discovered many assumptions based on normality and independence. But, lately, Andrey Markov has changed his assumptions by reducing unreliable facts. And the statement passed by Andrey Markov is the final statement that we are currently studying in our curriculum. Alexander Aitken also customized the model for dealing with non-spherical errors. 

When analysing or computing regression-based models, it is indispensable to understand the difference between biased and unbiased estimators for acquiring precise results. So, the Gauss Markov model provides a good insight for understanding this difference correctly. 

Gauss Markov Theorem

The Gauss Markov theorem can be separated as the Gauss theorem or the Markov theorem. The Gauss Markov theorem demonstrates that OLS (ordinary least squares) based estimators consist of minor sampling variance between the classes based on linear unbiased estimators.

The errors are not required to be settled down or not needed to become independent and identically distributed. It applies when they are not related to the linear regression model and consist of equal variances. 

Gauss Markov model assumptions

Some common assumptions based on the Gauss Markov models are cataloged here:

• In Gauss Markov models, the random error variable consists of a mean zero. E(E’) = 0.

• In Gauss Markov models, the random error variables are also homoscedastic. So, all the variables contain finite equal variance.

• In gauss Markov models, all the unique and different error terms are not related. 

Proof of Gauss Markov Theorem

To prove the Gauss Markov theorem, we will first consider B’ = Cy as one more linear estimator of B. This extra linear estimator consists of a C = (X’X)¯¹X’ + D). In this equation, D equals a K × n, which is a non-zero matrix.

In this proof, we consider less variance of minimum mean squared error because we ignore unbiased estimators. So, the main aim of this proof is to depict that the estimator’s variance is not too much less than the value of the OLS estimator. 

On computing, 

E[B’]  = E[Cy]

          = E[((X’X)¯¹X’ + D) (XB + ε)]

          = ((X’X)¯¹X’ + D) XB + ((X’X)−¹X’ + D) E[ɛ]

          = ((X’X)¯¹X’ + D) XB       ( E[ε] = 0) 

          = (X’X)-¹X’ XB + DXB

          = (IK + DX)B

In the above computation, B is not observable. Conversely, if DX = 0, if B’ will be unbiased. 

Let’s calculate further:

Var (B’) = Var(Cy)

             = C Var(y) C’

             = o² CC’

             = o² ((X’X)-¹ X’ + D) (X(X’X)¯¹ + D’)

             = o² ((X’X)-¹ X’X (X’X)−¹ + (X’X)-¹ X’D’ + DX (X’X)-¹ + DD’) 

            = o²(X’X)-¹ + o² (X’X)-¹ (DX)’ + o²DX (X’X)-¹ + o²DD’

            = o² (X’X)-¹ + o² DD’             DX = 0

            = Var (B’) + o² DD’             o² (X’X)-¹ = Var(B’) 

So, DD’ can be represented as a positive semidefinite matrix. 

Gauss Markov Process 

The Gauss Markov process is also described as a  stochastic process. This process is the integratory process that defines both Gaussian and Markov processes. The process is called an Ornstein–Uhlenbeck process. The most significant property of the Gauss-Markov process is that along with Gaussian and Markov, it also satisfies the equations of Langevin. 

Some typical properties of the Gauss Markov process

The Gauss Markov process is represented as an X(t), which holds the three significant properties. These properties are:

• In this case, when h(t) becomes a non-zero scalar function of t. Then, the Gauss-Markov process becomes,

Z(t) = h(t)X(t) 

• In this case, when f(t) becomes the non-decreasing scalar function of t. Then, the Gauss-Markov process becomes,

Z(t) = X(f(t)) 

• If the Gauss-Markov process becomes mean-square continuous and non-degenerated, the condition changes. Two unique functions, strictly increasing scalar function f(t) and non-zero scalar function h(t), will be considered in this condition. It will be written as a, 

 X(t) = h(t)W(f(t)

W(t) can be defined as a standard Wiener process in the above equation.

Gauss Meter 

One Gauss meter is equated to the 10-4 Tesla. Along with this, the Gauss meter unit was also demonstrated as one Maxwell per square centimeter. It can be represented as a 10-4 weber per square meter. The Gauss meter unit is used to measure and calculate the direction and intensity of the magnetic fields. A scientific device named Gauss meter is used for measuring the direction or intensity of magnetic field in terms of Gauss meter. 

Conclusion 

The Gauss Markov model is an introductory study of mathematical science. All its assumptions and equations provide the guarantees to validate OLS (ordinary least squares). Hence, it helps estimate or compute the regression coefficients. Apart from this, the Gauss Markov model is also helpful in rectifying the problem related to inaccurate regression coefficients. Hence, from this article, we learn much more about the Gauss Markov model, its theorem, its process, and its unit.