The question involves a matrix form of the Ordinary Least Squares (OLS) model, written as:

$y = X\beta + \mu$

where:

• $y$ is a vector of observed dependent variables,
• $X$ is a matrix of independent variables (with each column corresponding to a predictor, and each row to an observation),
• $\beta$ is a vector of coefficients to be estimated, and
• $\mu$ is a vector of errors.

To show why OLS cannot estimate this model, we need to prove it using matrix algebra. Here's the detailed explanation:

Step 1: Normal Equation for OLS

OLS estimates are derived by minimizing the residual sum of squares, which leads to the normal equation: $\hat{\beta} = (X^T X)^{-1} X^T y$ This formula requires that the matrix $X^T X$ (the product of the transpose of $X$ and $X$) must be invertible. If $X^T X$ is not invertible (i.e., singular), the OLS estimates cannot be computed.

Step 2: Checking $X^T X$

From the matrix in the question, we see that the first column of $X$ consists of 1's (this represents the intercept term), and the rest of the columns represent the independent variables $X_{ij}$.

For OLS estimation to work, the matrix $X^T X$ must be full-rank (i.e., the columns of $X$ must be linearly independent). If the columns of $X$ are linearly dependent, then $X^T X$ will be singular, and the inverse $(X^T X)^{-1}$ will not exist, making OLS estimation impossible.

Step 3: Linearly Dependent Columns

In the matrix provided, notice that there is a column of 1's (for the intercept), and there may be other independent variables. However, if any of the independent variables are collinear (i.e., one variable can be expressed as a linear combination of others), this leads to multicollinearity.

In particular, if any of the columns of $X$ are perfectly collinear with others, then $X^T X$ will not be invertible. This situation could arise if, for example:

• Some variables are duplicates of others,
• Some variables are linear transformations of others,
• Or the number of variables exceeds the number of observations (i.e., more columns than rows in $X$).

Conclusion:

In this case, the OLS estimator $\hat{\beta} = (X^T X)^{-1} X^T y$ cannot be computed because $X^T X$ is singular due to linear dependence in the columns of $X$. Hence, OLS cannot estimate the model.

Would you like me to go deeper into any part of this proof or elaborate on any step?

Related Questions:

1. What are some methods to detect multicollinearity in real datasets?
2. How can we remedy the problem of multicollinearity in a regression model?
3. Can you explain why $X^T X$ needs to be invertible in more detail?
4. What are some alternative estimation methods when OLS fails due to singularity of $X^T X$?
5. How does the inclusion of an intercept affect the rank of $X^T X$?

Tip:

When dealing with multicollinearity, techniques like Ridge Regression or Principal Component Analysis (PCA) can help mitigate the issue by reducing the impact of linearly dependent variables.