Why is the intercept different in an AR(1) model compared to a lagged endogenous variable model?

Question

In other words, why is that when estimating in EViews

y = c ar(1)

yields a different coefficient for c when compared to

y = c y(-1)

although the coefficient for ar(1) and y(-1) are the same?

Answer 1

Consider the model \begin{align*}y_t&=\psi+\tilde{y}_t\\ \tilde{y}_t&=\rho \tilde{y}_{t-1}+u_t \end{align*} Rearrange this to $y_t-\psi=\tilde{y}_t$, insert and solve for $y_t$ to get $$ y_t=\psi(1-\rho)+\rho y_{t-1}+u_t $$ Compare this with the model $$ y_t=c+\rho y_{t-1}+u_t. $$ I do not have access to eviews, but I believe the first model corresponds to approach y c ar(1) (estimating $\psi$) and the second to y c y(-1).

I haven't checked thoroughly, but a similar discrepancy also seems to be inherent in the following approaches in R:

library(nlme)
library(dynlm)
y <- 2 + arima.sim(list(ar=0.8), n=10000)
gls(y~1, corr=corAR1(0.5,form=~1))
coef(dynlm(y~L(y)))

Output:

> gls(y~1, corr=corAR1(0.5,form=~1))
Generalized least squares fit by REML
  Model: y ~ 1 
  Data: NULL 
  Log-restricted-likelihood: -14269.74

Coefficients:
(Intercept) 
   2.044563 

Correlation Structure: AR(1)
 Formula: ~1 
 Parameter estimate(s):
      Phi 
0.7951246 
Degrees of freedom: 10000 total; 9999 residual
Residual standard error: 1.661907 

> coef(dynlm(y~L(y)))
(Intercept)        L(y) 
  0.4189489   0.7949498 

In particular, $2.044563\cdot(1-0.7951246)\approx0.4189489$ (discrepancies arise from gls using an iterative ML approach).