When computing the least-squares estimator, we are computing some \hat{{\boldsymbol \beta }} which minimizes the error
\min _{{\boldsymbol \beta }\in \mathbb {R}^ p} \| \mathbf Y- \mathbb {X}{\boldsymbol \beta }\| _2^2
where \| v\| _2 is the Euclidean norm.
Let n be the number of samples and let each \mathbf X_ i be p-dimensional. (For example, n might be the number of patients, and p-1 is the number of covariates that we are trying to study - e.g. height, weight, age and blood pressure as in the previous problem.)
Recall that by employing the same technique of computing the gradient (with respect to the components of {\boldsymbol \beta }) and setting it equal to zero, we can show that \hat{{\boldsymbol \beta }} must satisfy the score equation
\mathbb {X}^ T \mathbb {X} \hat{{\boldsymbol \beta }} = \mathbb {X}^ T \mathbf Y.
We would like to isolate \hat{{\boldsymbol \beta }} by multiplying by (\mathbb {X}^ T \mathbb {X})^{-1} from the left. Which of the following conditions, each on its own, guarantees that \mathbb {X}^ T \mathbb {X} is invertible? Choose all that apply.
Hint: Use the fact from linear algebra that \mathrm{rank}(A^ T A) = \mathrm{rank}(A). What are the dimensions of \mathbb {X}^ T \mathbb {X}?
There are at least as many observations as covariates (i.e. n \geq p).
There are at least as many covariates as observations (i.e. n \leq p).
\mathbb {X} has rank n, where n is the number of samples.
\mathbb {X} has rank p, where p is the number of covariates.
There are at least p distinct samples amongst the X's, so that \mathbb {X} has at least p distinct rows.
There are at least p distinct values amongst the Y.
The conditions that guarantee that \mathbb {X}^ T \mathbb {X} is invertible are:
1) There are at least as many observations as covariates (i.e. n \geq p).
2) \mathbb {X} has rank p, where p is the number of covariates.
These conditions ensure that the matrix \mathbb {X}^ T \mathbb {X} has full rank and is therefore invertible.
To isolate \(\hat{{\boldsymbol \beta}}\) by multiplying by \((\mathbb{X}^T\mathbb{X})^{-1}\) from the left, we need to guarantee that \(\mathbb{X}^T\mathbb{X}\) is invertible.
To determine the conditions that guarantee this, we can consider the dimensions and rank of \(\mathbb{X}^T\mathbb{X}\).
The dimensions of \(\mathbb{X}^T\mathbb{X}\) are \(p \times p\), where \(p\) is the number of covariates.
The conditions that guarantee \(\mathbb{X}^T\mathbb{X}\) to be invertible are:
1. There are at least as many observations as covariates (i.e. \(n \geq p\)). This condition ensures that there are enough data points to estimate the parameters accurately.
2. \(\mathbb{X}\) has rank \(p\), where \(p\) is the number of covariates. This condition ensures that the covariates are linearly independent and not redundant.
Therefore, the correct answers are:
1. There are at least as many observations as covariates (i.e. \(n \geq p\)).
4. \(\mathbb{X}\) has rank \(p\), where \(p\) is the number of covariates.