Basic & Econometrics - Misspecified model - pitfalls

import pandas as pd
import numpy as np
import statsmodels.api as sm

$y_t = \beta y_{t-1}+u_t$

$u_t = \rho u_{t-1}+v_t$

Suppose we want to estimate $\beta$ by running an OLS regression of $y_t$ on $y_{t-1}$

T = 100
y = np.zeros([T,1])
u = np.zeros([T,1])
beta = .7
rho = .5
u[0] = np.random.normal()
y[0] = u[0]
for t in range(1,T):
    u[t] = rho*u[t-1]+np.random.normal()
    y[t] = beta*y[t-1]+u[t]

np.random.normal()

-0.09498498374286703

model = sm.OLS(y[1:],y[:-1])
model = sm.OLS(y[2:],np.concatenate([y[:-2],y[1:-1]],axis = 1))

res = model.fit()
res.summary()

OLS Regression Results

Dep. Variable:	y	R-squared (uncentered):	0.764
Model:	OLS	Adj. R-squared (uncentered):	0.759
Method:	Least Squares	F-statistic:	155.4
Date:	Mon, 25 Jan 2021	Prob (F-statistic):	7.95e-31
Time:	10:13:18	Log-Likelihood:	-140.81
No. Observations:	98	AIC:	285.6
Df Residuals:	96	BIC:	290.8
Df Model:	2
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
x1	-0.2794	0.099	-2.825	0.006	-0.476	-0.083
x2	1.1097	0.099	11.257	0.000	0.914	1.305

Omnibus:	0.819	Durbin-Watson:	2.050
Prob(Omnibus):	0.664	Jarque-Bera (JB):	0.866
Skew:	0.074	Prob(JB):	0.649
Kurtosis:	2.564	Cond. No.	3.67

Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.

rho = beta-0.2794
-rho*beta = -(beta-0.2794)*beta = 1.1097
0.2794-beta^2 = 1.