harmonic regression

A regression model containing Fourier terms is often called a harmonic regression because the successive Fourier terms represent harmonics of the first two Fourier terms.

A test for a group of autocorrelations is called a portmanteau test, from a French word describing a suitcase containing a number of items.

One such test is the Box-Pierce test, based on the following statistic$$Q=T\sum _{k=1}^h{r}_k^2,$$where $h$ is the maximum lag being considered and $T$ is the number of observations. If each $r_k$ is close to zero, then $Q$ will be small. If some $r_k$values are large (positive or negative), then $Q$ will be large. We suggest using $h=10$ for non-seasonal data and $h=2m$ for seasonal data, where $m$ is the period of seasonality. However, the test is not good when $h$ is large, so if these values are larger than $T/5$, then use $h=T/5$

A related (and more accurate) test is the Ljung-Box test, based on$$Q^{\ast }=T(T+2)\sum _{k=1}^h(T-k{)}^{-1}{r}_k^2.$$

Again, large values of $Q^{\ast }$ suggest that the autocorrelations do not come from a white noise series.