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    Bootstrap




    Hey,

    I am new to this forum and I have a question about Bootstrapping.
    I want to calculate some critical values. The data generating process is assumed to be

    Y(t+1)=alpha + u1(t+1)
    X(t+1)=mu + roh*X(t) + u2(t+1)

    where roh is estimated by OLS using sample observations. I then have to generate 10,000 bootstrapped time series by drawing with replacement from the residuals.

    I want to implement this in MATLAB, however I first would like to understrand what I need to do. There is nothing mentioned about u1 and u2 but I assume those are the residuals? I think I understand the data generation of X(t+1), this seems to be a simple linear regression, but how can I generate Y(t+1) with just a constant ?

    I really hope somebody can help me. Thank you all!

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    Re: Bootstrap

    To me, the answer is obvious. That is, generate u1 and add (the constant) alpha to each value of u1 to create Y.....Of course, I could be missing something here, but I don't know what it would be.

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    Re: Bootstrap


    I thought of this too, but I need to resample from the residuals and this would make Y(t+1) kind of needless because I could have just used the generated random numbers right? Maybe I am missing something aswell. This is the original text from the paper I am using (Goyal and Welche, 2008):

    Bootstrap : Our bootstrap follows Mark (1995) and Kilian (1999) and
    imposes the NULL of no predictability for calculating the critical values.
    In other words, the data generating process is assumed to be
    yt+1 = α + u1t+1
    xt+1 = μ + ρ xt + u2t+1.
    The bootstrap for calculating power assumes the data generating process
    is
    yt+1 = α + β xt + u1t+1
    xt+1 = μ + ρ xt + u2t+1,
    where both β and ρ are estimated by OLS using the full sample of
    observations, with the residuals stored for sampling. We then generate
    10,000 bootstrapped time series by drawing with replacement from
    the residuals. The initial observation—preceding the sample of data
    used to estimate the models—is selected by picking one date from the
    actual data at random. This bootstrap procedure not only preserves
    the autocorrelation structure of the predictor variable, thereby being
    valid under the Stambaugh (1999) specification, but also preserves the
    cross-correlation structure of the two residuals.

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