content_image/312931.jpg content_image/312932.jpg content_image/312933.jpg content_image/312934.jpg content_image/312935.jpg content_image/312936.jpg content_image/312937.jpg content_image/312938.jpg
individual when there are r treatments remaining. Then 

using ∑ s δ$_{sq}$ = ∑ s δ$_{ps}$ = 1. 

Lemma 4.2. Take the vector of fixed effects β to be 

Then we have the following result 

<img src='content_image/312931.jpg'>

<img src='content_image/312932.jpg'>

<img src='content_image/312933.jpg'>

where 0$_{m,n}$ is a matrix of zeroes of dimension m × n , and 

<img src='content_image/312934.jpg'>

<img src='content_image/312935.jpg'>

<img src='content_image/312936.jpg'>

<img src='content_image/312937.jpg'>

Proof. Denote the columns of X$_{s}$$_{ri}$ by 

<img src='content_image/312938.jpg'>

Thus Π$_{jr}$ is the column corresponding to the period effect π$_{j}$ , T$_{ds}$$_{ri}$ to the treatment effect τ$_{d}$ , and 1$_{r}$ to the intercept µ$_{0}$ . Using this representation, and Lemma 4.1, we have 

