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Your problem stems from the simple fact that $\log X_i^2 = 2\log X_i$. Thus, your model reduces to: $\log Y_i = \alpha + \beta_1 \log X_i + 2 \beta_2 \log X_i + u_i = \alpha + (\beta_1 + 2 \beta_2) \log X_i + u_i$. Due to the perfect collinearity, $\beta_1$ and $\beta_2$ cannot be distinguished. As @whuber mentioned in a comment: Maybe the second ...

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In almost every textbook or publication that I have read it is said that MLR is not applicable if we have more variables than samples, because the inverse of the $\mathbf X′\mathbf X$ matrix, where $\mathbf X$ is the predictor block, doesn't exist. You'll want to look up classical (aka ordinary) vs. inverse models here. In chemometric (or chemistry), a ...

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A stepwise method should not be better. To a first approximation, stepwise regression should never be used (cf., Algorithms for automatic model selection). You have multicollinearity. The rule of thumb that you don't have problematic multicollinearity until the VIFs are >10 is just a rule of thumb. Your variables are still strongly correlated with each ...

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PLSR or partial least squares regression is a dimension reduction technique that shares similarities with principal component analysis. In principal component regression you seek to obtain a set of new variables (the principal components) that maximize the variance of $X$ and that are uncorrelated to each other. In PLSR you seek to obtain a set of new ...

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You are correct in relation to your understanding, but I will detail a little more what would be multicollinearity and interaction. Multicollinearity: within the context of regression, multicollinearity is when there is a correlation between two or more continuous independent variables. When the objective of your analysis consists only of prediction, there ...

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In regression there is a danger of omitted-variable bias. In linear regression, if you omit a predictor that is associated both with outcome and with a predictor that is in the model, your results will be biased. With A and B highly correlated, your results are what you thus might expect. Model 1 attributes to A both its own contribution to outcome and a ...

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