est_theta1_MA1 = function(r1){ 
#
# Estimates \theta_1 in the ARIMA(0,d,1) model:
#
# w_t = a_t - \theta_1 a_{t-1} 
#
# using a very simple version of Newton's method.
#
# Inputs: 
#   r1 = autocorrelation of lag 1
#
# Written by:
# -- 
# John L. Weatherwax                2009-04-21
# 
# email: wax@alum.mit.edu
# 
# Please send comments and especially bug reports to the
# above email address.
# 
#-----

# compute the two values of theta; take the theta such that -1 < theta < +1 
#
theta1 = -1/(2*r1) + sqrt( 1/((2*r1)^2) - 1 ) 
theta2 = -1/(2*r1) - sqrt( 1/((2*r1)^2) - 1 ) 

if( abs(theta1) < 1 ){
  res = theta1 
}else{
  res = theta2 
}
res

}