Does seasonality change the relationship?

Try the residual-spectrum comparison

Fit monthly housing starts on building permits from January 2000 through December 2024. Run these two lines to display the comparison directly.

Housing=SERIES_BETWEEN(LIST(HOUSTNSA,PERMITNSA),20000101,20241231)
Fit=SEASONAL_REGRESSION(Housing,"spectrum",4)

In “Compare residual cycles”, read across each row for Durbin–Watson, lag-one residual autocorrelation and holdout RMSE. Row 0 is the baseline. Rows 1 through 4 add sine/cosine pairs in order of residual spectral power. Each of the four expanding training windows finds its own frequencies. Holdout predictions use observed permits.

To inspect two pairs, append Model=EX(Fit,"model:2") for the regression or ACF=EX(Fit,"acf:2") for the residual autocorrelation chart. Other spectrum extracts include summary, coefficients, periods and validation.

Try a fixed period on the same sample

Fixed=SEASONAL_REGRESSION(Housing,8)
Results=EX(Fixed,"summary")

Here 8 specifies an eight-month period. With the default of one harmonic, the model adds one sine/cosine pair and compares the fit with the baseline on the same dates.

Why both cosine and sine?

A pair allows the seasonal peak to occur anywhere in the cycle. One harmonic gives one smooth annual wave; two add a half-year wave. Individual coefficient significance depends on the time origin; consider the pair jointly.

Period

Optional period defaults to an annual cycle: 12 for monthly, 4 for quarterly, 365.2425/7 for weekly, 365.2425 days for daily or unrecognized frequency. Annual data require an explicit multiyear period. Optional harmonics default to 1, must be integers from 1 to 20, and require period greater than twice the harmonic count.

Time uses UTC dates relative to 1970, not compressed row positions. Monthly and quarterly coordinates use calendar bins. Daily/weekly coordinates use elapsed time. Existing alignment may convert frequency; inspect the aligned frequency before choosing the period.

Read the comparison

Shared coefficients appear side by side with seasonal-minus-baseline differences. Added terms are shown separately. These differences are descriptive, not significance tests. R-squared normally rises when predictors are added; adjusted R-squared and AIC/BIC account for model size. Neither is proof of better forecasts or causality.

DW near 2 is consistent with little first-order residual autocorrelation. A move toward 2 can accompany a better seasonal specification, but DW alone cannot determine which model is correct. Inspect residual plots and autocorrelation; HAC inference uses retained-row lag counts.

Fixed-period extracts

baseline=EXTRACT(Fixed,"regression")
seasonal=EXTRACT(Fixed,"seasonalregression")
EXTRACT(seasonal,"hac")

Keep predictors few relative to observations; use several cycles when possible. Constant or duplicate predictors and aliased seasonal terms can make a fit singular. This comparison does not select a model automatically or provide a test of coefficient differences.