Detrending and the Variate Differencing Method: Student, Pearson and Their Critics

4.1 The differencing approach to detrending time series proposed by Hooker (1905) and Cave-Brown-Cave (1905) (§2.9) was reconsidered some years later by ‘Student’ (1914) in rather more formal fashion.1 Student began by assuming that y _t and x _t were randomly distributed in time and space, by which he meant that, in modern terminology, E(y _t y _t−i ), E(x _t x _t−i ) and E(y _t x _t−i ), i ≠ 0, were all zero if it was assumed that both variables had zero mean. If the correlation between y _t and x _t was denoted r _yx = E(y _t x _t )/σ _y σ _x , where <math display='block'> <mrow> <msubsup> <mi>&#x03C3;</mi> <mi>y</mi> <mn>2</mn> </msubsup> <mo>=</mo><mi>E</mi><mrow><mo>(</mo> <mrow> <msubsup> <mi>y</mi> <mi>t</mi> <mn>2</mn> </msubsup> </mrow> <mo>)</mo></mrow> </mrow> </math>$$\sigma _{y}^{2}=E\left( {y_{t}^{2}} \right)$$ and <math display='block'> <mrow> <msubsup> <mi>&#x03C3;</mi> <mi>x</mi> <mn>2</mn> </msubsup> <mo>=</mo><mi>E</mi><mrow><mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>t</mi> <mn>2</mn> </msubsup> </mrow> <mo>)</mo></mrow> </mrow> </math>$$\sigma _{x}^{2}=E\left( {x_{t}^{2}} \right)$$, Student showed that the correlation between the dth differences of x and y was the same value.