Scale of OBSERVED=AVERAGE Values

The average values are assumed to be expressed in the time units defined by the FROM= or TO= option. That is, the product of the average value for an interval and the width of the interval is assumed to equal the total value for the interval. For purposes of interpolation, OBSERVED=AVERAGE values are first converted to OBSERVED=TOTAL values using this assumption, and then the interpolated totals are converted back to averages by dividing by the widths of the output intervals.

For example, suppose the options FROM=MONTH, TO=HOUR, and OBSERVED=AVERAGE are specified. Since FROM=MONTH is specified, each input value is assumed to represent an average rate per day such that the product of the value and the number of days in the month is equal to the total for the month. The input values are assumed to represent a per-day rate because FROM=MONTH implies SAS date ID values that measure time in days, and therefore the widths of MONTH intervals are measured in days. If FROM=DTMONTH is used instead, the values are assumed to represent a per-second rate, because the widths of DTMONTH intervals are measured in seconds.

Since TO=HOUR is specified, the output values are scaled as an average rate per second such that the product of each output value and the number of seconds in an hour (3600) is equal to the interpolated hourly total. A per-second rate is used because TO=HOUR implies SAS datetime ID values that measure time in seconds, and therefore the widths of HOUR intervals are measured in seconds.

Note that the scale assumed for OBSERVED=AVERAGE data is important only when converting between AVERAGE and another OBSERVED= option, or when converting between SAS date and SAS datetime ID values. When both the input and the output series are AVERAGE values, and the units for the ID values are not changed, the scale assumed does not matter.

For example, suppose you are converting gross domestic product (GDP) from quarterly to monthly. The GDP values are quarterly averages measured at annual rates. If you want the interpolated monthly values to also be measured at annual rates, then the option OBSERVED=AVERAGE works fine. Since there is no change of scale involved in this problem, it makes no difference that PROC EXPAND assumes daily rates instead of annual rates.

However, suppose you want to convert GDP from quarterly to monthly and also convert from annual rates to monthly rates, so that the result is total gross domestic product for the month. Using the option OBSERVED=(AVERAGE,TOTAL) would fail, because PROC EXPAND assumes the average is scaled to daily, not annual, rates.

One solution is to rescale to quarterly totals and treat the data as totals. You could use the options TRANSFORMIN=( / 4 ) OBSERVED=TOTAL. Alternatively, you could treat the data as averages but first convert to daily rates. In this case you would use the options TRANSFORMIN=( / 365.25 ) OBSERVED=AVERAGE.

Results of the OBSERVED=DERIVATIVE Option

If the first value of the OBSERVED= option is BEGINNING, TOTAL, or AVERAGE, the result is the derivative of the spline curve evaluated at first-of-period ID values for the output observation. For OBSERVED=(MIDDLE,DERIVATIVE), the derivative of the function is evaluated at output interval midpoints. For OBSERVED=(END,DERIVATIVE), the derivative is evaluated at end-of-period ID values.