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I have defined the model as follows. Let

$$y_1 = 0$$

and

$$ y_i = \alpha + \beta y_{i-1} + \epsilon_i $$

for $i_2\ldots i_T$, where $\alpha$ and $\beta$ are the estimated coefficients and $\epsilon$ is the bootstrap residual series.

My question is: should we take $y_1 = 0 $ or should we take $y_1$ from original $y$ series?

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It can be checked that the mean and variance for an AR(1) process are respectively $\alpha/(1-\beta)$ and $\sigma^2_\epsilon/(1-\beta^2)$ model. Assuming that $\epsilon_t \sim NID(0, \sigma^2_\epsilon)$ you could set $y_1$ as a random draw from the Normal distribution with the mean and variances of the AR(1) process.

Nevertheless, bootstrapping does not generally involve the restriction that a value that has been chosen cannot be chosen again. In other words, resampling of residuals is done with replacement. Having $T-1$ residuals due to the lagged variable is therefore not troublesome. You can generate a series of bootstrapped disturbances $\epsilon_t^*$ of length $T$ by resampling with replacement.

Notice that resampling with replacement does not necessarily lead to a series of $T-1$ resampled residuals plus one additional value that will be a duplicate from this set of $T-1$ residuals. Replacement is done for every draw and, hence, there may be more than one duplicates or even a given residual $\hat{\epsilon}_t$ may show up more than two times in the resampled series.

Edit

Looking again at your question I'm not sure you are concerned with the fact that one residual observation is missed due to the lagged variable. Regardless of the size of the bootstrapped residuals, you need an initial $y_1$ to initialize the AR recursions that will generate the resampled series.

My answer can therefore stick to the first paragraph above. For a general AR(p) model or for ARMA models, where the mean and variance may not be so simple expressions, you could to initialize the first "p" values as random values taken from the standard Gaussian distribution, N(0, 1).

Alternatively, for small samples it may be safer to generate a resampled series $y_t^*$ of length, for example, $T+p\times 4$ and then remove the first $p\times 4$ values. In this way, the first values will be a kind of warming observations; after them, the effect of the initial observations will be very small and hence it won't be so critical how they were initialised.

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