A general periodic model is of the form, *y*(*j*) = μ + *M* · cos(2π[*j*/*k* + φ]), where *j* is integers indexing time. The mean, *μ,* shifts the function up and down, *M* changes the amplitude, and φ shifts (sometimes called the phase) the location of the maximum and minimum. The function repeats itself every *k* periods. For annual data, *k* would be 12 for monthly data and 52 for weekly data. Figure 10.1 displays the function *y*_{1}(*j*) = 2 + 8 · cos(2π[*j*/12 + .25]) in solid lines and *y*_{2}(*j*) = 4 + 4 · cos(2π[*j*/24 + .70]) in dashed lines.

The most basic periodic models are of the form *y*(*j*) = μ + *M* · cos(2π[*j*/*k* + φ]) + ϵ_{j}, where the errors are white noise. However, the time series may be more complex requiring a linear combination of periodic functions and the error may not be white noise.

For this most basic model, it is assumed that *k* is known, and *μ, M*, and φ must be estimated. The problem min ∑[*y*(*j*) − μ − *M* · cos(2π[*j*/*k* + φ])]^{2}, where *μ*, *M*, and φ are unknown, is not an OLS (ordinary least squares) problem. Why (hint: take the partial derivatives and set them equal to zero)?

However, there is a trick that allows this to be solved as an OLS problem (Bloomfield, 2000). Let *y*(*j*) = μ + *B* · cos(2π*j*/*k*) + *C* · sin(2π*j*/*k*) where *B* = *M* · cos(2πφ) and *C* = −*M* · sin(2πφ). The ...

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