Essentials of Time Series for Financial Applications

(4.32)

where ${\mathit{x}}_t$ may contain the intercept and a polynomial time trend. KPSS test is then an Lagrange multiplier (LM) test:

${\mathrm{KPSS}}_T\equiv \frac{{{\sum }_{t=1}^T\left({\sum }_{i=1}^t{\hat{u}}_i\right)}^2}{T^2{\sum }_{i=[-T^{2/9}]}^{\left[T^{2/9}\right]}\frac{1}{T-i}\sum _{t=i+1}^T{\hat{\epsilon }}_t{\hat{\epsilon }}_{t-i}},$ (4.33)

(4.33)

where the denominator is an estimator of the residual spectrum at frequency 0, in this case written already to reflect a Newey–West automatic, data-driven bandwidth choice as in Example 4.3.¹⁰ Example 4.4 shows the PP and KPSS tests at work.

Example ...