Deriving the long-horizon predictive regression and hypothesis testing

I am working on the long-horizon regression,

$$y_{t,K}=\mu+\beta_1x_{1,t-1}+...+\beta_nx_{n,t-1}+e_{t}$$, where $$y_{t,K}=y_{t}+y_{t+1}+...+y_{t+K-1}$$ and there can be multiple x's.

So I am trying to joint hypothesis test, say that that a $$\beta_1 = \beta_2 = 0$$,

I need to show this in matrix form so that I can code it up.

I know it has something to do with the Kronecker product, and the vectorisation of these equations.

However, I don't know how to define the 'hypothesis matrix', so that I am only testing that $$\beta_1 = \beta_2 = 0$$

To be specific if this was a standard regression equation, to test the hypothesis that $$\beta_1 = \beta_2 ... = \beta_n = 0$$, I would just write an identity matrix.

Thanks in advance, I've been struggling with this.

• wouldn't it be the first two rows of that identity matrix? Jul 18 '18 at 19:43

Generally an $F$-test hypothesis in a linear regression is expressed as $$A \beta = c$$ for some matrix $A$ and vector $c$ (typically with $c$ the zero vector). In your case you want $A$ to be the first two rows of the identity matrix, or $e_1^{\top}$ stacked on top of $e_2^{\top}$, and $c$ to be the zero vector.