Question

An investigator estimates the following equation using Ordinary Least Squares from time series data: Y1 = a0 + a1X1t + a2X2t + Ut The following scenarios represent violations of the CLRM assumptions. For each scenario, state the underlying assumption being referred to and state the null hypothesis of that assumption. i- There is another important variable, which has been omitted. ii- X1 and X2 are correlated iii- Ui is correlated with its own past values. iv- The mean value of the residual from the regression is equal to 1 Please help

Answer 1

i) there is another important variable which has been omitted. say that is X3

i.e. the true model will be y_t=a₀+a₁x_1t+a₂x_2t​​​​+a₃x_3t​​​​+u_t

But if we will use the previous model, then E(u_t)is not equal to zero, so that assumption will be violated.

here E(u_t)=E( y_t-a₀+a₁x_1t+a₂x_2t​​​​)=E( y_t)-a₀+a₁x_1t+a₂x_2t​​​​= a₀+a₁x_1t+a₂x_2t​​​​+a₃x_3t -(a₀+a₁x_1t+a₂x_2t​​​​)=a₃x_3t​​​​

which is not equal to zero as x₃ is a important variable.

null hypothesis will be H₀:a3=0

ii) x1 and x2 are correlated. i.e. these two regressors can be expressed by each other.

i.e. there is a problem of multicollinearity involved in the data,so that assumption is violated in this case.

null hypothesis will be H₀:cov(x1,x2)=0

iii)Ui is correlated with its own past values. that means Ui's are not independent .

i.e. there is a problem of auto correlation in the data. that assumption is violated.

null hypothesis will be H₀:cov(ui,uj)=0,for all i ≠ k

iv) E(u_t)=1

but we know mean value of residual should be 0; that assumption is violated here.

null hypothesis will be H₀:E(ut)=0,