Constructing least-squares polynomial smoothers

A linear-regression construction

A least-squares smoother is built as a linear-regression fit. The fit chooses coefficients so the squared vertical gaps between observed prices and the fitted values are minimized. All error is assigned to the price axis and none to the time axis.

Polynomial order is the degree of that local fitted curve. The construction requires evenly spaced sampling and a series that is continuous enough to treat as a smooth curve.

Moving-average and exponential-smoothing baselines

Place this construction beside two baselines that use the same ordered observations. A moving average averages a block of those observations. When that block is long, it tends to flatten sharp peaks and troughs and, relative to the least-squares construction, a long moving average reduces extreme peaks and troughs.

Exponential smoothing applies decaying, asymmetric weights. Relative to a symmetric least-squares fit, exponential smoothing and other asymmetric weighting schemes introduce some shape distortion and shift the shape of turning points.

An integer-weight window

The fitted midpoint can be obtained by multiplying a window of observations by a fixed integer weight set and dividing by a published normalizing factor. That integer-weight window is a compact implementation of the polynomial fit.

Tabulated sets cover second- through fifth-order polynomials and windows from 5 to 25 observations. Second- and third-order sets form one tabulated family, and fourth- and fifth-order sets form another.

The five-point second- and third-order weight set is symmetric about its midpoint and expands from the folded table as -3, 12, 17, 12, and -3.

Five-point and 25-point windows on weekly yields

On weekly intermediate-term bond yields from 6 March 1987 through 12 January 1990, a five-point second- and third-order smoother kept the main highs and lows nearly in phase with the raw series, including the week-34 peak tied to the 19 October 1987 market reaction.

The matching 25-point second- and third-order smoother delayed tall, sharp peaks and troughs by about two to three weeks. That later appearance of the same turns is phase delay. The longer window also spread the week-34 peak toward later observations and left shallow turns near weeks 66 and 110 without that delay.

What the residual series removed

The residual series is the pointwise difference between the raw observations and the smoothed series. It shows how much variation was removed and whether leftover structure remains.

The five-point residual removed about plus or minus 0.1 percent yield at the week-34 spike, confined that burst to about four weeks, and otherwise left background fluctuations near plus or minus 0.03 percent yield plus a roughly regular plus or minus 0.05 percent jump every 17 to 25 weeks.

The 25-point residual removed about plus or minus 0.3 percent yield at the same spike, raised background residual amplitude to about plus or minus 0.1 percent yield, stretched that peak residual toward week 50, and showed a larger leftover oscillation near 30-week spacing.

Sizing the window from leftover structure

Window length is a construction choice that should follow inspection of leftover structure in the residual. After that inspection, the smoother can sit in front of later analysis as a way to decide how much variation, and of what kind, to remove.