(Solved): Consider the multivariate linear regression model assuming full ideal conditions \[ Y=X \beta+\vare ...

Consider the multivariate linear regression model assuming full ideal conditions \[ Y=X \beta+\varepsilon, \] where \( Y \) is a \( n \times 1 \) vector, \( X \) a \( n \times k \) matrix, \( \beta \) a \( k \times 1 \) parameter vector and \( \varepsilon \) a \( n \times 1 \) error term vector. We denote the OLS estimator of \( \beta \) by \( \hat{\beta}_{U} \). Let \( \tilde{\beta} \) be an arbitrary estimator of \( \beta \) satisfying the following condition: \[ R \tilde{\beta}=r, \] where \( R \) is a \( m \times k \) known matrix, \( m \leq k \), and \( r \) a known \( m \times 1 \) vector of coefficients defining the linear restrictions. a) The Restricted Least Squares estimator \( \left(\hat{\beta}_{R}\right) \) of \( \beta \), fulfilling (1) is given by \[ \hat{\beta}_{R}=\hat{\beta}_{U}+\left(X^{\prime} X\right)^{-1} R^{\prime}\left[R\left(X^{\prime} X\right)^{-1} R^{\prime}\right]^{-1}\left(r-R \hat{\beta}_{U}\right) \] i) Write down the formula for \( \hat{\beta}_{U} \). ii) Interpret the above relationship briefly. 4 marks b) You want to test a linear restriction like \( R \beta=r \). What test do you use? What is the null hypothesis? 2 marks c) Explain which impact imposing restrictions has on the properties of \( \hat{\beta}_{R} \) ? 4 marks