Consider the $ARMA(1,1)$ model $$ X_t - 0.5X_{t-1} = W_t + 0.5W_{t-1}. $$

In this question, we will investigate recursive forecasting.

The following code snippet generates a sequence of length $n=50$ drawn from the above model.

```{r}

set.seed(5209)

n <- 50

wn <- rnorm(n)

xt <- arima.sim(model = list(ar = 0.5, ma = 0.5), innov = wn, n = n)

```

a. Fill in the following code snippet using equation for recursively compute forecasts using the ARMA(p,q) formula to generate a sequence $wn_hat$.

```{r} wn_hat <- rep(0, n)

wn_hat[[1]] <- xt[[1]]

for (i in 2:n)

{ # FILL IN }

```

b. What consequence does this have for truncated forecasts?

c.Compute the truncated forecast for $X_{53}$.