I would like to know whether analytical or closed-form expressions could be obtained for the orthogonalization of a set of vectors in the following setting.
Let $x_t$ be a vector indexed as a time series observed at times $t=1,2,\dots,n$. Upon this vector, the following vectors are defined:
$$ \begin{eqnarray} \begin{array}{l} y_{1,t} = (1 + L + L^2 + L^3) x_t \,, \\ y_{2,t} = -(1 - L + L^2 - L^3) x_t \,, \\ y_{3,t} = -(1 - L^2) x_t \,, \\ y_{4,t} = y_{3,t-1} \,, \end{array} \end{eqnarray} $$
where $L$ is the lag operator such that $L^i x_t = x_{t-i}$. These are the so-called HEGY regressors [1]. An alternative representation of these vectors is given in [2]:
$$ \begin{eqnarray} \begin{array}{l} y_{1,t} = \sum_{i=0}^3 x_{t-i} \,, \\ y_{2,t} = \sum_{i=0}^3 \cos \left[ (i+1)\pi \right] x_{t-i} \,, \\ y_{3,t} = \sum_{i=0}^3 \cos \left[ (i+1) \pi/2 \right] x_{t-i} \,, \\ y_{4,t} = -\sum_{i=0}^3 \sin \left[ (i+1) \pi/2 \right] x_{t-i} \,. \end{array} \end{eqnarray} $$
The vectors $y_{1,t}$, $y_{2,t}$, $y_{3,t}$, $y_{4,t}$ can be computed independently of each other using the data in the vector $x_t$. That is, in order to obtain for example $y_{2,t}$, the remaining vectors need not be computed.
I want to find a set of orthogonal vectors that generate the same subspace as the vectors $y_{1,t}$, $y_{2,t}$, $y_{3,t}$, $y_{4,t}$. I can compute the four vectors $y_{1,t}$, $y_{2,t}$, $y_{3,t}$ and $y_{4,t}$ and apply a numerical algorithm, for example the Gram-Schmidt procedure or Householder transformations. However, I'm interested in analytical expressions that could allow me to compute one of the orthogonal vectors without allocating memory space for the remaining vectors.
In a given application, I want to limit the amount of required memory space. I don't need to compute the whole set of orthogonal vectors at the same time because they are used at different steps. That's why I'm interested in analytical expressions that could allow me to save memory space.
Is it possible to obtain analytical expressions for a set of orthogonal vectors that span the same subspace as that spanned by $y_{1,t}$, $y_{2,t}$, $y_{3,t}$, $y_{4,t}$? If so, could those expressions be written as a function of only the vector $x_t$?
References
[1] Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., Shin, Y. (1992). Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root. Journal of Econometrics 54(1-3), pp. 159-178, DOI: 10.1016/0304-4076(92)90104-Y.
[2] Smith, R.J., Taylor, A.M.R., del Barrio Castro, T. (2009). Regression-based seasonal unit roots. Econometric Theory 25, pp. 527-560. DOI: 10.1017/S0266466608090166. Granger Centre Discussion Paper No. 07/05 http://core.ac.uk/download/pdf/6245791.pdf.