# true parameters
gamma =  3
beta  =  1
theta = -1
delta =  2

# explanatories
x     = rnorm(250)
z     = x * gamma + rnorm(250)

# errors
err   = rnorm(250)

# dgp1
y     = z * theta + x * beta + err

lm(y ~ z + x - 1) # good estimate of theta

# dgp2
y     = (z - x * gamma) * theta + x * beta + err

lm(y ~ z + x - 1) # equivalent estimate of theta


# dgp3

y     = z * theta + x * z * delta + x * beta + err

lm(y ~ z + x + z * x - 1) # good estimates of theta and delta



# dgp4

y     = (z - x * gamma) * theta + x * (z - x * gamma) * delta + x * beta + err

lm(y ~ z + x + z * x - 1) # good estimate of theta, bad estimate of delta