Suppose data=(ui,vi) , i= 1,...n . and we want to estimate theta=corr(ui,vi) . My question is , does it matter if the u and v pair are obtained from some population or calculated for example from the following models;
ui=yhat1 ; yhat1=a0+a1X1;
vi=yhat2; yhat2= b0+b1X1+b2X2;
i.e. fitted values from two regression models with the same response but different predictors.In fact I would like to know , do I need to estimate the model and the use the result for bootstrapping, or I can just use output of the models and sample them for bootstrapping.In other words, should the bootstrapping procedure contains the estimation of the models or not?
Bonjour,
I agree with Noel if a0, a1, b0 b1 and b2 are deterministic. However, i guess that they are random
In that case, Yhat1 and Yhat2 are random linear combinations of X1 and X2 and one should read
ui=yhat1 ; yhat1=a0hat+a1hat X1;
vi=yhat2; yhat2= b0hat +b1hat X1+b2hat X2; i=1;...n.
So if you want to use bootstrap to estimate the correlation between ui and vi
You need to repeat K times the following three steps
1) You drawn with replacement a sample of size n
i.e. you draw n values of (X1_i, X2_i)
2) you estimate with this sample a0hat, a1hat, b0hat, b1hat, b2hat
and compute ui, vi
3) you compute theta=corr(ui, vi)
At the end, you will have K estimates of theta whose distribution will be closed to the sample distribution of \hat theta
Cheers
Since Y1 and Y2 are just linear combinations of X1 and X2, the correlation between Y1 and Y2 can be easily computed based on the variances of X1 and X2 and their correlation. No need to perform any bootstrapping.
Bonjour,
I agree with Noel if a0, a1, b0 b1 and b2 are deterministic. However, i guess that they are random
In that case, Yhat1 and Yhat2 are random linear combinations of X1 and X2 and one should read
ui=yhat1 ; yhat1=a0hat+a1hat X1;
vi=yhat2; yhat2= b0hat +b1hat X1+b2hat X2; i=1;...n.
So if you want to use bootstrap to estimate the correlation between ui and vi
You need to repeat K times the following three steps
1) You drawn with replacement a sample of size n
i.e. you draw n values of (X1_i, X2_i)
2) you estimate with this sample a0hat, a1hat, b0hat, b1hat, b2hat
and compute ui, vi
3) you compute theta=corr(ui, vi)
At the end, you will have K estimates of theta whose distribution will be closed to the sample distribution of \hat theta
Cheers
Hi, thank you all for answering.
So I need to estimate the model which generates ui and vi . Cant I estimate them once and draw samples with replacement from (ui,vi) pair?
I guess the underlying question, Amir, is what you want the bootstrapping to do for you. If you want it to tell you how correlated the u and v are when sampling from the X1 and X2 population and using the best model for the data sampled then you would need to follow procedure Didier recommends. If you want to view the a and b coefficients as fixed by a single instance of a regression fit then it seems to me that Noel's suggestion applies. I would bet that repeating the regression (as Didier details) better fits the goals of your work.
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