Question

Are these equations written in the general linear regression model?

Y_i = B₀ + B₁X_1i + B₂ log(X_2i) + B₃X_1i² + e_i

Y_i = e_i exp(B₀ + B₁X_1i + B₂ log(X_2i) + B₃X_3i)

Y_i = B₀ exp(B₁X_1i) + e_i

Answer 1

Y_i = B₀ + B₁X_1i + B₂ log(X_2i) + B₃X3_i + e_i

_{This can be re written as}

Y_i = B₀ + B₁X_1i + B₂ X'_2i + B₃X_1i² + e_i

_where X'_{2i =} log(X_2i)

and X3_{i =} X_1i²

Hence it is a general linear model ( although it will have some collinearity )

Y_i = e_i exp(B₀ + B₁X_1i + B₂ log(X_2i) + B₃X_3i)

Taking log

log(Y_i ) = log( e_i exp(B₀ + B₁X_1i + B₂ log(X_2i) + B₃X_3i) )

log(Y_i ) = log( e_i ) + B₀ + B₁X_1i + B₂ log(X_2i) + B₃X_3i)

Y'_i = e'_i + B₀ + B₁X_1i + B₂ X'_2i + B₃X_3i

_where, Y'_i  = log(Y_i )

e'_i = log( e_i )

X'_{2i =} log(X_2i)

Hence it is a general linear model  

Y_i = B₀ exp(B₁X_1i) + e_i

_{Taking log}

_log(Y_i ) = log ( B₀ exp(B₁X_1i) + e_i)

_{This can not be further become a general linear model}