DIFFUSION INDICES:A Potentially Fruitful Application of the

Direct Filter Approach

ISF 2005, June 13, San Antonio, TX

Thomas B. FombyDepartment of Economics

Southern Methodist UniversityDallas, TX 75275tfomby@smu.edu

website: faculty.smu.edu/tfomby and

Federal Reserve Bank of Dallas

OUTLINE

       I. REASONS FOR MY INTEREST IN DFA

 

           II.      THE NEED FOR BUSINESS OUTLOOK SURVEYS

 

         III.     OECD DIFFUSION INDEX METHODOLOGY

 

        IV.     SOME ISSUES CONCERNING DIFFUSION INDICES

 

           V.      DFA RESULTS FOR PHILADELPHIA DIFFUSION INDICES

 

        VI.     CONCLUSIONS

                 

                                 I.      Reasons for My Interest in DFA A. Bernd Schips and Marc Wildi’s ISF 2004 Presentation “Signal Extraction: A Direct Filter Approach and Clustering in the Frequency Domain” B. The Application of DFA to Bounded Time Series C. Boundary Problems and Incorrect Imposition of Unit Roots in the Identification of Such Series D. My Consultancy with the Federal Reserve Bank of Dallas on Building a Business Outlook Survey Index

Reasons for My Interestin DFA

The Need for Business Outlook Surveys

                           II. The Need for Business Outlook Surveys A. The Summarization of Qualitative Responses of Survey Respondents B. Some Producers of Diffusion Indices i. Institute of Supply Managers (ISM) (formerly National Association of Purchasing Managers (NAPM)) ii. Federal Reserve Banks of Philadelphia and New York iii. OECD C. Some Example Questions D. The Data is Immediate and Easily Summarized E. In Contrast: Government Statistics – Issues of Timeliness and Revisions i. Delay in Reporting of Government Statistics (e.g. GDP) ii. Inevitable Revisions That Occur in the Data iii. Conflicting Statistics – e.g. Household Survey versus Business Survey of Employment Statistics  

OECD Diffusion Index Methodology

            III. OECD Diffusion Index Methodology A. Concurrent versus Forward Looking Questions B. Respondents Are Asked to “Seasonally Adjust” Their Answers C. Stratification of the Sample: By Industry and Small versus Large Firms D. Some Mathematical Formulas E. Boundedness of Balance (-100, 100) and Diffusion (0, 100) Indices F. Some Plots of Philadelphia Diffusion Indices

AN EXAMPLE SURVEY

Business Outlook Survey

Federal Reserve Bank of Dallas

Conflict Between Job Surveys

Dallas Morning News

“Job Growth Comes Up Short”

Saturday, June 4, 2005

“Job Growth Comes Up Short” 

Dallas Morning News

Saturday, June 4, 2005  

•“The Labor Department reported disappointing jobs numbers Friday, with U.S. businesses adding 78,000 positions in May – about 100,000 short of economists’ expectations.”•“ ‘The herky-jerky pattern in non-farm payrolls has resulted in some very red-faced economists and sizable moves in financial markets on employment Fridays,’ summed up Joseph Abate, a senior economist with Lehman Bros. in New York.”•“That thinking (the economy is not swinging as wildly as the data) is bolstered by the household survey – used to calculate the unemployment rate.”•Two competing surveys produced by the U.S. Labor Department: The business payroll survey and the household survey.•“In any case, the erratic pace of job growth shouldn’t be given too much attention, Brian S. Wesbury, chief investment strategist at Claymore Advisors, wrote in a recent research note. ‘The payroll data is volatile, and is often revised significantly,’ he wrote. ‘Reading too much into the May report would be a mistake.’”

Some Mathematical Formulas

100)1

(/)1

(1 1

k kn

i

n

iik

ikikik

ikk e

pxe

pB

2/)100( kk BDI

Overall Balance Index Across Industries

K

k

K

kkkkov VBVB

1 1

/

2/)100( ovov BDI

Some Issues Concerning Diffusion Indices

                  IV. Some Issues Concerning Diffusion Indices A. Seasonal Adjustment of Time Series i. Conventional Method of Using X12-ARIMA or TRAMO/SEATS ii. Use of Transformed Series iii. DFA Approach B. Validation of Diffusion Indices i. Predictive Content of Diffusion Indices ii. Turning Point Analysis – The 2x2 Business Cycle Contingency Table

Seasonal Adjustmentof Diffusion Indices

Various Approaches:

X12-TRAMO/SEATS (MBA)

Log Transformation

Direct Filter Approach (DFA)

New York FedSeasonal Adjustment Process

Consider the log-ratio of the unadjusted diffusion index 

We work with the log-ratio of the Diffusion index because the Diffusion index has a natural range of 0 to 100 and the log-ratio is an ideal transformation to take the Diffusion into the real line, a natural metric for seasonal analysis. (No similar transformation exists for Balance indices that range in value from –100 to +100). After transformation, one can use X12-ARIMA or some other seasonal adjustment program to produce seasonally adjusted log-ratios 

un

unun DI

DIz

1ln

adj

adjadj DI

DIz

1ln

It follows that  

 

 

Therefore,

 

is the formula that allows a translation of the seasonally adjusted log-ratio to the seasonally adjusted Diffusion index.

adj

adjadj DI

DIz

1)exp(

)exp()1( adjadjadj zDIDI

)exp())exp(1( adjadjadj zzDI

)exp(1

)exp(

adj

adjadj z

zDI

Ideally, if one wants to produce a seasonally adjusted overall Diffusion index

one would first seasonally adjust each industry Diffusion index producing

and then use the following formula to produce a seasonally adjusted overall Diffusion index 

adjovD ,

adjkD ,

K

k

K

kkadjkkadjov VDVD

1 1,, /

Graphs of Some PhiladelphiaDiffusion Indices

0

10

20

30

40

50

60

70

80

90

100

Mar-68 Aug-73 Feb-79 Aug-84 Jan-90 Jul-95 Jan-01

gacdfna

Figure 1A: Current General Activity Non-Seasonally Adjusted

0

10

20

30

40

50

60

70

80

90

100

Mar-68 Aug-73 Feb-79 Aug-84 Jan-90 Jul-95 Jan-01

gafdfna

Figure 1B: Future General Activity Non-Seasonally Adjusted

0

10

20

30

40

50

60

70

80

90

100

Mar-68 Aug-73 Feb-79 Aug-84 Jan-90 Jul-95 Jan-01

nocdfna

Figure 2A: Current New Orders Non-Seasonally Adjusted

0

10

20

30

40

50

60

70

80

90

100

Mar-68 Aug-73 Feb-79 Aug-84 Jan-90 Jul-95 Jan-01

nofdfna

Figure 2B: Future New Orders Non-Seasonally Adjusted

Validation of Diffusion Indices

1) Predictive Content vis-à-vis

Transfer Function Modeling

2) Turning Point Analysis

2x2 Business CycleContingency Table

IFO Germany

DFA Results for Philadelphia Diffusion Indices

        V. DFA Results for Philadelphia Diffusion Indices (Marc Wildi and Associates) A. Marc Wildi’s Experiment Involving 85 series from the Philadelphia FED Diffusion Index Database B. DFA versus MBA: Difference in Mean-square Filter Errors

Philadelphia Diffusion Index Database

168 Diffusion and Balance Indices

Over Multiple Questions

Both NSA and SA

Comparison of DFA with MBA

85 Philadelphia Series

Criterion: One-Step Ahead

Mean Square Revision (Filter) Error

DFA Asymmetric FilterTransfer Function

(Implemented by Symmetric MA(120) Filter)

1 if 0 | | / 9

/ 7 | |( ) for /9<| |< /7

/ 7 / 90 for | |> /7

DFA Versus MBA Results• The Mean Square Revision Error increased by 33%

in the mean (over all 85 series) when using the MBA instead of the efficient DFA

• Equally Weighted Combination of MBA and DFA produced 8% higher MSE (over all 85 series) than DFA.

• DFA outperformed the MBA for 79 out of 85 series (93% of series)

• DFA outperformed the Equally Weighted Combination for 67 out of 85 series (79% of series)

• The above results suggest that DFA encompasses MBA

Conclusions

             VI. Conclusions A. Trend Extraction Results Look Promising B. Prediction Comparisons should consider Different Forecast Horizons, Different Loss Functions, and Concentrate on the Non- seasonally adjusted series C. Consider More Sophisticated Combination Methods D. What about Logarithmic Transformation (NY Fed) technique? E. Compare Turning Point Prediction Errors of Competing Methods (e.g. IFO Cycle Diagram)

Thank You

CongratulationsISF

on your 25th Anniversary!