The Development of Leading and C

The Development of Leading and Coincident Indices

The Leading and Coincident Indices are calculated on a monthly basis for the "Washington-Arlington-Alexandria, DC-VA-MD-WV Metropolitan Statistical Area" as defined by the U.S. Office of Management and Budget. The area consists of the following jurisdictions: District of Columbia, DC, Calvert County, MD, Charles County, MD, Frederick County, MD, Montgomery County, MD, Prince George's County, MD, Arlington County, VA, Clarke County, VA, Fairfax County, VA, Fauquier County, VA, Loudoun County, VA, Prince William County, VA, Spotsylvania County, VA, Stafford County, VA, Warren County, VA, Alexandria city, VA, Fairfax city, VA, Falls Church city, VA, Fredericksburg city, VA, Manassas city, VA, Manassas Park city, VA, Jefferson County, WV.

Preliminary Steps

Variable Selection—It is tempting to use as many variables as possible in the development of an index in order to achieve "completeness." However, index development generally benefits from incorporating no more variables than are absolutely necessary. Since different variables respond differently to economic stimuli, having too many of them could result in the effects of one being offset by the effects of another.

The Washington Area Leading Index is constructed from 5 variables: residential building permits, consumer expectations, initial claims for unemployment insurance, help wanted advertising index, and durable goods retail sales. The Washington Area Coincident Index is based on 4 variables: domestic air passenger boardings, wage and salary employment, consumer confidence in the present, and nondurable goods retail sales.

Seasonal Adjustment—Several adjustments to the data may be necessary prior to their inclusion in an index. Some series may contain regular fluctuations that are not the result of changes in the economy, but reflect normal seasonal changes. In order to uncover any real trend in a variable, the original data should be deseasonalized, that is, the seasonal component should be removed from the series. The following table illustrates the deseasonalization of total monthly Washington Area Domestic Airline Boardings. The seasonal adjustment factors represent the seasonal variation in the original series. A value of 1 represents the absence of seasonal variation. Values less than 1 indicate a seasonal "dip" in sales while values greater than 1 indicate a seasonal "spike."

Date	Original Series ('000)	Seasonal Adjustment Factor	Seasonally Adjusted Series ('000)
Jan-06	1875.9	0.81142	2311.9
Feb-06	2157.5	.082733	2607.8
Mar-06	2274.0	1.03403	2199.2
Apr-06	2256.5	1.06863	2111.6
May-06	2276.0	1.07364	2119.9
Jun-06	2313.1	1.10082	2101.3
Jul-06	2274.3	1.09371	2079.4
Aug-06	2187.6	1.06462	2054.8
Sep-06	1906.4	0.93806	2032.3
Oct-06	2230.0	1.05273	2118.3
Nov-06	2205.0	1.00190	2200.8
Dec-06	2144.1	0.93310	2297.8

Data Estimation—Every attempt is made to use original up-to-date series where possible. There are, however, occasions when this is not possible and it is necessary to estimate missing values.

The series may, for one reason or another, be late. While this is a rare occurrence, it is necessary to estimate the missing value for inclusion in an index. The usual procedure for estimating a missing value is the ARIMA (Autoregressive Integrated Moving Average) approach. This estimated value is replaced when the missing value becomes available.
Some series are available only during certain months. For example, the Consumer Price Index for the Washington-Baltimore Region is available only during odd months (January, March, etc.). Interim months are interpolated based on surrounding values.
Census Bureau estimates of retail sales for the Washington Metropolitan Statistical Area were discontinued beginning in January 1997. Durable and nondurable goods sales are central to both indices and must be estimated for inclusion. Estimates are prepared using ARIMA.

Index Construction

Month-To-Month Changes—Indices are designed to reflect month-to-month changes in the values of the components. It is tempting to consider percent changes as the preferred method. However, conventional percent changes are not symmetric with respect to increases and decreases. For example, the conventional percent change for the values 200 to 220 would be:

100[(220 – 200)/200] = +10.0%.

Conversely, the conventional percent change from 220 to 200 would be:

100[(200 – 220)/220] = -9.09%.

Applying a symmetric percent change to these same values would yield:

200[(220 – 200)/(220 + 200)] = 9.52% and,

200[(200 – 220)/(200 + 220)] = -9.52%.

Accordingly, the symmetric percent change approach is used to account for month-to-month changes in the components of the indices.

Weighting The Variables—There is likely to be substantial differences in the volatilities of the month-to-month symmetric percent changes of the components of an index. The standard deviation (s ) of the symmetric percent changes for the Help Wanted Advertising Index is 3.6078 and the standard deviation of the symmetric changes for building permits is 29.4480. Thus, building permits are about 8 times as variable the help wanted index. Such wide discrepancies would result in the more variable components dominating the movements of the index. The solution is to equalize the volatility of the components. The following table summarizes the steps to calculate the weights for the 5 components of the Washington Area Leading Index.

	Operation	Building Permits	Expectations	Help Wanted Index	Initial Claims	Durable Goods Sales	Totals
1	Std. Dev. (s )	29.4480	7.8576	3.6078	24.5815	9.5107
2	Reciprocal (1/s )	0.0340	0.1273	0.2772	0.0407	0.1051	0.5888
3	Normalized Weight	0.0581	0.2178	0.4744	0.0696	0.1800	1.0000

Calculate the standard deviation for the symmetric changes of each index component.
Take the reciprocal of these standard deviations and sum these values.
Normalize the reciprocals by dividing each by the total. The resulting values are the weights.
Multiply each symmetric percent change value in a series by its weight.

Calculate The Indices—The calculation of the leading and coincident indices is divided into 5 distinct steps.

Sum the weighted symmetric percent changes separately for the components of the coincident and leading indices.
Equalize the volatility of the two indices. Even though the volatility of the separate components has been equalized, the volatility of the two indices may be substantially different. The standard deviations of the leading and coincident indices are, respectively, 4.2699 and 1.2963, indicating that the coincident index is about 30% (1.2963 / 4.2699 = 0.3036) as variable as the leading index. Multiplying each symmetric percent change value in the leading index by 0.3036 equalizes the standard deviation of the two indices. Note: This procedure does not change the interpretation of the indices because the symmetric percent change values of the leading index are multiplied by a constant.

The next step is to compute preliminary indices. This is accomplished by setting the initial value (I1) of each index equal to 100.0. The second month’s value is calculated using a variation of the symmetric percent change formula. These calculations for the first 6 months of the leading index are shown in the following table.

Date	Adjusted Symmetric Percent Changes	Equalized Symmetric Percent Changes	Calculation Of The Preliminary Index
Month 1	N.A.	N.A.	100.0
Month 2	8.66	2.63	100.0(200 + 2.63)/(200 - 2.63) = 102..7
Month 3	5.41	1.64	102.1(200 + 1.64)/(200 - 1.64) = 104.4
Month 4	1.54	0.47	103.8(200 + 0.47)/(200 - 0.47) = 104.9
Month 5	-2.07	-0.63	104.1(200 + 0.63)/(200 - 0.63) = 104.2
Month 6	2.32	0.71	103.3(200 + 0.71)/(200 - 0.71) = 104.9

The final step consists of rebasing the indices to some convenient reference year. For the Washington indices, the reference year is 1987, i.e., the average for 1987 = 100. Multiplying each value in the indices by 100 and dividing by the average index value for 1987 accomplishes this. This step is analogous to converting current dollars to constant dollars.