maximum likelihood estimator of regression coefficient

Question

Consider the following simple linear regression model:

$$ y_i = (-1)^i \cdot \beta_1 + \epsilon_i \quad \text{where} \quad i = 1, \ldots, n $$

here $\epsilon_i$'s are i.i.d. $N(\beta_0, 1) \text{ random variables. If } \beta_0 = 2\beta_1, = \beta, \text{ we need to find } \hat{\beta}, \text{ the maximum likelihood estimator of } \beta, \text{ E}(\hat{\beta}) \text{ and } \text{V}(\hat{\beta}) $

My approach:

For finding maximum likelihood estimator of $\beta$ we need to minimize the $ \sum_{i=1}^{n} \left( y_i - (-1)^i \frac{\beta}{2} \right)^2 $

Derivating the above expression with respect to $\beta$ and equating it to zero, i got $\sum_{i=1}^{n} \left( (-1)^i y_i - \beta (-1)^i \right) = 0 $ which then comes to two results:

1. when n = odd:

$\sum_{i=1}^{n} (-1)^i y_i - 1\cdot \beta = 0$ $\implies \beta = \sum_{i=1}^{n} (-1)^i y_i$

2. when n = even:

$\sum_{i=1}^{n} (-1)^i y_i - 0\cdot\beta = 0$ $\implies \beta$ can be anything

I am new to regression so I am not sure is this approach correct? If not what is the correct approach to solve such problems?

Answer 1

you can view this as repeated measurements at $x_\text{odd}=-1$ and $x_\text{even}=1$. I would rewrite your expression splitting sums over even and odd and using $n_\text{even}$, $n_\text{odd}$ and $\bar{y_\text{even}}=\frac{1}{n_\text{even}}\sum_{i=1}^{n_\text{even}}y_{2i}$, and $\bar{y_\text{odd}}$

so you want to minimise $$\frac{1}{2}\sum_{i=1}^{n_o}(y_{2i-1}-\frac{1}{2}\beta)^2 + \frac{1}{2}\sum_{i=1}^{n_e}(y_{2i}-\frac{3}{2}\beta)^2$$

and then use some term for the means over the odd/even $y_i$s