Can negative autocorrelation at lags 1 and 2 happen? Just doing a couple of mind games going through my stats notes...
I've seen ACFs around with negative values at lags 1 and 2 - I might be having a mind blank here, but wouldn't a high negative AC at lag 1 imply a series like (-1,1,-1,1,...), and as such we'd be expecting the AC to alternate between positive and negative as well? 
If I'm completely wrong here - is there an easy made up example where we have strong negative AC for both lags 1 and 2?
Thank you!
 A: The following DGP, an MA($2$) process, has negative autocorrelation at lags 1 and 2:
$$
Y_t=10-.5\cdot u_{t-1}-.25\cdot u_{t-2}+u_t
$$
Here's some R code to simulate the DGP and see the ACF for yourself:
library(stats)
library(forecast) # for the Acf() function

# number of "observations"
n<-500 
# initialization periods
j<-1000

# choose parameters
alpha<-10
theta<-c(-.5,-.25)
Q<-length(theta)

# generate iid disturbances
u<-rnorm(n+j,0,2)

# define the DGP and generate data series iteratively
y<-rep(alpha,n+j)
for(k in (Q+1):(n+j)){
  y[k]<-alpha + sum(theta*u[k-c(1:Q)]) + u[k] 
}

# get rid of the initialization periods
Y<-y[-c(1:j)]

# confirm the parameters
arima(Y,c(0,0,Q))

#   Call:
#   arima(x = Y, order = c(0, 0, Q))
#   
#   Coefficients:
#             ma1      ma2  intercept
#         -0.4763  -0.2546     9.9979
#   s.e.   0.0448   0.0485     0.0246
#   
#   sigma^2 estimated as 4.124:  log likelihood = -1064.03,  aic = 2134.05

# look at the ACF/PACF
par(mfrow=c(2,1))
Acf(Y)
pacf(Y)


