Question

2. In a model to predict weekly sales of soft drinks, your hypothesis is that the average temperature in the same week helps to predict sales. Your model is SALES, = α + 3TEM Pit ei, and you assume that each ei is an independent N(0, σ*) randon variable. You also assume that the average temperature in week i, TEMP, is non-random like to estimate. n, and that a and B are unknown parameters that you would (a) Find a and b to satisfy the following statement: SALESN(0,1) b) Suppose you believe that 3 is zero, but a is not. Derive the least-squares estimator for a

Answer 1

$a.\\ Since~TEMP~is~non-random~so~and~SALES~is~a~linear~combination~of~e\\ so,~SALES_i\sim N(\alpha+\beta TEMP_i,~\sigma^2).\\ Hence~a=\alpha+\beta TEMP_i,~b=\sigma\\ b.\\ If~\beta=0~then~the~model~is\\ y_i=\alpha+e_i\\ where,~y_i=SALES_i$

$where~\alpha~is~obtained~by~minimizing~\sum_{i=1}^n(y_i-\alpha)^2~w.r.t.~\alpha.\\ And~we~know~\sum_{i=1}^n(y_i-\alpha)^2~is~minimum~when~\alpha=\bar{y}=\frac{1}{n}\sum_{i=1}^ny_i.\\ Hence~least~square~estimator~of~\alpha~when~\beta=0~:\hat{\alpha}=\bar{y}.\\ (c)~Suppose~\alpha=0~then~the~model~is\\ y_i=\beta x_i+e_i\\ where,~x_i=TEMP_i$

$For~estimating~\beta~minimize~\Delta =\sum_{i=1}^n(y_i-\beta x_i)^2~w.r.t.~\beta.\\ Now,~\frac{\mathrm{d} \Delta }{\mathrm{d} \beta}=-2\sum_{i=1}^n(y_i-\beta x_i)x_i=0\Rightarrow \hat{\beta}=\frac{\sum_{i=1}^nx_iy_i}{\sum_{i=1}^nx_i^2}\\ Hence~least~square~estimator~of~\beta~when~\alpha=0=\hat{\beta}=\frac{\sum_{i=1}^nx_iy_i}{\sum_{i=1}^nx_i^2}\\ (d)~E(\hat{\beta})=\frac{\sum_{i=1}^nx_iE(y_i)}{\sum_{i=1}^nx_i^2}=\beta\frac{\sum_{i=1}^nx_i^2}{\sum_{i=1}^nx_i^2}=\beta.$