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.
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!
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.
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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