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Give data in $t_n$ denoted by $[x_1^n, x_2^n, ... x_d^n]$ and label $y_n$ to be predicted. We can just train a $d$-dimensional linear regression $y_n=\sum b_ix_i^n$ to make a prediction. However, I think the data in $t_{n-1}$ can also be helpful to predict $y_n$. So my way is to concatenate data in $t_{n-1}$ and $t_n$ by $[x_1^n, x_2^n, ... x_d^n, x_1^{n-1}, x_2^{n-1}, ... x_d^{n-1}]$ (i.e., multi-periods data) and use a $2d$-dimensional linear regression to predict $y_n$.

The problems of my method are: 1st $[x_1^n, x_2^n, ... x_d^n]$ and $[x_1^{n-1}, x_2^{n-1}, ... x_d^{n-1}]$ seems to be much correlated, so feature can be very redundant. 2nd, some normalization may be conducted on $[x_1^n, x_2^n, ... x_d^n]$ and $[x_1^{n-1}, x_2^{n-1}, ... x_d^{n-1}]$ to form a good concatenation, but how to do? 3rd, if we think that data in $t_{n-100}$ is also help to predict $y_n$, then $100d$ dimension is very high.

So could anyone solve my above problems? or have other ways to use multi-periods data.

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To solve your multicollinearity problem I would first perform a regularization technique such as Ridge or Elastic Net. If you choose Ridge for example, once you have tuned your hyperparameter through cross validation (for time series a forward walk approach is preferable) you can fit after your simple OLS by choosing the predictors with the biggest coefficients from your Ridge regression.

However, I am guessing that what you are trying to do is to build some kind of a hidden state model, wanting your model to know "where you are right now". For that purpose, if you wanted to use 100 previous steps as features, I would rather use a recurrent neural network such as LSTM.

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