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Consider a following model of lenders and borrowers:$$\triangle Loan_{ij}=\alpha+L_{i}+\beta_{j}+\epsilon_{ij}$$ where $L_{i}$ is a dummy variable for lender i , $\beta$ j is borrower j and the LHS is the change in loan between lenders and borrowers from one time period to the next. Now, let us consider the following argument. The RHS consists of constant term, a full set of dummy variables for each lender and a full set of dummy variables for each borrower. Of course, the $X'X$ matrix is rank deficient (by 2). Let us drop the constant. We still have linear dependence, as the sum of all lender dummy variables = sum of all borrower dummy variables=1. So, we drop another arbitrary dummy variable. (Say Lender 1). Now, we have estimated the full set of dummy variables (-1) subject to two constraints:

  1. Constant term is 0

  2. One of the coefficients on the dummy variables is set to 0.

What implication does this have for the rest of the estimates? How can I interpret them?

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  • $\begingroup$ This is a standard issue in ANOVA. The $\alpha$ term is the grand mean, which isn't what is typically meant by "constant". This situation is typically dealt with via "reference level coding" (here is a search of the site that should help you). $\endgroup$ – gung - Reinstate Monica Mar 10 '16 at 20:30
  • $\begingroup$ FWIW, I don't see anything too unclear about this question. $\endgroup$ – gung - Reinstate Monica Mar 10 '16 at 20:31
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Probably better to set up your equation as $Y=X\boldsymbol{\beta}$, and then use something like $y_i = \beta_0 + L_i \beta_{i1} + bor_i \beta_{i2} + \epsilon_i$ and drop your $j$ parameter. Set your data up with the dependent and all independents in each row of the dataset, using only the $i$ subscript for each term per row. Use of $j$ is more of an ANOVA setup (which is regression), so make a choice of an ANOVA question or multiple linear regression question. Also, make sure you have replicate (numerous) observations per table cell so that the mean $\mu_{ij}$ is reliable. If you have only one value for each cell, then your model will be overparametrized, and $(\mathbf{X}^T\mathbf{X})^{-1}$ will not be positive-definite, but rather positive-semidefinite with some eigenvalues, $\lambda_j=0$, and hence, an ill-conditioned problem.

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