## MODEL FOR ESTIMATING GENERAL AVIATION OPERATIONS AT NON - TOWERED AIRPORTS

USING TOWERED AND NON - TO WERED AIRPORT DATA

July 2001

Prepared for: Statistics and Forecast Branch Office of Aviation Policy and Plans Federal Aviation Administration

Prepared by: GRA, Inc. 115 West Avenue, Suite 201 Jenkintown, PA 19046

### i 

## TABLE OF CONTENTS Page List of Tables ................................ ................................ ................................ ............ ii List of Figures ................................ ................................ ................................ .......... iii Executive Summary ................................ ................................ ................................ iv Section 1. Introduction ................................ ................................ ................................ . 1 2. Methodology and Results from Prior Study ................................ ........... 2 3. New Variables That May Affect Aviation Activity at Small Towered and Non - Towered Airports ................................ ...................... 3 4. New Approaches to Estimating Aviation Activity Levels at Small Towered and Non - Towered Airports ................................ ................................ ....... 6 4.1 Models of GA

Operations ................................ ................................ ..... 6 

### 4.2 Model of GA Operations per Based Aircraft ................... 17 5. Applying the Estimated Model to Small Non - Towered GA Airports Nationwide ................................ ................................ ................................ .. 18 6. Conclusion ................................ ................................ ................................ ... 22 Appendix A: An Explanation of the Chow Test ................................ ................ 24 Appendix B: Complete Data Set Used for Regression Analysis ..................... 26

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## LIST OF TABLES

Number Page Table 1: Selected Equations from Regression Analysis in Hoekstra (2000) ... 

### 2 Table 2: Categories of New Independent Variables Used in

Regression Analysis ................................ ................................ ................ 6 Table 3: GA Operations Regression Equations — Data for Small Towered GA Airports Only ................................ ................................ ................... 8 Table 4: GA Operations Regression Equations — Data for Small Towered GA Airports and Non - Towered GA Airports — No Dummy Variable ................................ ................................ ................................ ..... 12 Table 5: GA Operations Regression Equations — Data for Small Towered GA Airports and Non - Towered GA Airports — With Dummy Variable ................................ ................................ ................................ ..... 13 Table 6: GA Operations Regression Equations Small Towered GA Airports and Truncated Non - Towered GA Airports Data Set using Dummy Variable to Examine Fit to Non - Towered Airports ................................ ................................ .......... 14 Table 7: Best Equation from Minitab Stepwise Regression for Small Towered Airport Operations per Based Aircraft ............................... 17 Table 8: Summary State Data for 2,780 Non - Towered GA Airports .............. 19 Table 9: M odel Goodness of Fit Measures for Towered GA Airports, State - Estimated Non - Towered GA Airports, and Form 5010 Non - Towered GA Airports ................................ ................................ ... 22

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## LIST OF FIGURES

Number Page Figure 1: State Estimated and Equation Estimated Annual Operations at

Non - Towered Airports ................................ ................................ ........ 9 Figure 2: Fit of Out - of - Sample Estimates for 13 Non - Towered Airports

Excluded from Estimating Data for Equation 16 ............................. 14 Figure 3: Comparison of Tower Counts and State Estimates to Model

Estimates (Equation 9) for Small Towered GA Airports and Non - Towered Airports — Model Without Tower Dummy Variable ................................ ................................ ................... 15 Figure 4: Com parison of Tower Counts and State Estimates to Model

Estimates (Equation 13) for Small Towered GA Airports and Non - Towered Airports — Model with Tower Dummy Variable ................................ ................................ .................. 16 Figure 5: Form 5010 Estimates and Model Estimates for Annual

GA Operations ................................ ................................ ....................... 20 Figure 6: Variability of Differences between Form 5010 Annual

Operations Estimates and Model Estimates of Annual Operations for U.S. Non - Towered GA Airports .............................. 21

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## EXECUTIVE SUMMARY This report develops a model for estimating general aviation (GA) operations at non - towered airports. It builds on previous research done for the FAA Office of Policy and Plans, Statistics and Forecast Branch. In the previous research, statistical relationships were developed between GA operations at small towered airports and the characteristics of these airports. Models based on these relationships were used to est imate GA operations for a set of non - towered airports for which state estimates (derived primarily from counter and survey estimation procedures) were available. The model estimates of operations for the non - towered airports tended to exceed the state est imates. In this report, a new model was estimated that augments the previous research by using additional variables for population, airport regional prominence, and certificated flight schools. In addition, the model uses a combined data set for small to wered and non - towered GA airports and incorporates a dummy variable to distinguish the two airport types. The new model produced operation estimates for non - towered airports that were unbiased relative to the state estimates based primarily on sampling pr ocedures. The new model was also applied to a large set of 2,789 non - towered GA airports. This data set includes all of the non - towered GA airports in the Terminal Area Forecast (TAF) not included in the development of the model estimation. For the majority of these airports, operation estimates reported on DOT Form 5010 are roughly developed and are not based on sampling procedures. For these airports, model estimates of annual GA activity were compared to the Form 5010 estimates. On average, the model estimates tended to be unbiased relative to the reported Form 5010 estimates. In the future, research efforts can focus on testing and refining the model for specific subsets of the large data set of non - towered GA airports.

### 1 1. INTRODUCTION This report presents results from a GRA study aimed at refining and improving recent efforts by the FAA Office of Policy and Plans (APO) to model aviation activity at non - towered airports. This study was undertaken as part of Task Order 16, Task 1 under FAA contract DTFA01 - 98 - C - 00096. In a previous report 1 

### , FAA identified several characteristics of small towered airports that have a statistically significant relationship to operations at these airports. The results quant ify observations made in prior APO - funded research 2 

### which noted that aviation activity at individual airports is highly dependent on “local factors.” It remained clear, however, that additional research could improve the set of possible “local factors” co nsidered and could refine the equations used to model the relationships between these factors and airport activity. A principal purpose of the research that is contained in this report has been to develop models relating “local factors” to airport activi ty that could be used to estimate operations at non - towered airports for which the only counts of activity available are at best cursory. Since the FAA uses data on current and projected airport activity to guide its airport capital investment decisions, there is always a pressing need to improve FAA’s ability to estimate activity at these airports and to assess the forecasts made by airport sponsors to support investment or facility upgrade requests. The remainder of this report is organized as follows. Section 2 summarizes the methodology and results from Hoekstra (2000) and discusses the avenues for improving and refining the results applied in this report. Section 3 identifies and discusses the new data and variables that are used for the current an alysis and suggests some refinements to the models used in Hoekstra (2000). Section 4 reports on the effectiveness of these model refinements, including the closeness of fit between the model estimates of airport activity — expressed either as annual operat ions or as average annual operations per based aircraft — and the observed values (which may be tower counts from small towered airports in the sample or estimates of airport activity based on an explicit sampling procedure). Section 5 applies the equations developed in Section 4 to a large set of small non towered GA airports throughout the United States. These results are examined at a very broad level to assess the equation’s value for estimating airport activity. Section 6 summarizes the report finding s and identifies possible next steps for

1 Mark Hoekstra, “Model for Estimating General Aviation Operations at Non - Towered Airports” prepared for FAA Office of Aviation Policy and Plans, April 2000. 2 GRA, Inc. “Review of TAF Methods,” Final Report, prepared for FAA Office of Aviation Policy and P lans under Work Order 45, Contract No. DTFA01 - 93 - C - 00066, February 25, 1998.

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future research. Appendix A describes a statistical procedure used in the report, and Appendix B contains the data for small towered and non - towered GA airports used in the regression analysis.

2. METHODOLOGY A ND RESULTS FROM PRIOR STUDY Hoekstra (2000) developed a sizeable set of airport characteristics from varied sources and used these data to estimate regression models explaining annual general aviation (GA) operations and annual GA operations per based air craft, using a data set of 127 small towered U.S. airports. These airports had fewer than 100,000 GA operations in FY 1999. 3 

### Extensive documentation on these airports and the variables collected for them is contained in the Hoekstra report. After analys is of these data using the Minitab software and its Stepwise Regression procedure, the “best” equations — in terms of proportion of variance explained (R 2 

### ), — were reported. These equations are shown in Table 1 below (which is identical to Hoekstra’s Table 11). Table 1 Selected Equations from Regression A nalysis in Hoekstra (2000) Hoekstra Equation Number Equation 4 R 2

H - 7 H - 11 OPS = 813.5 + 417 BA + 0.80 PCI – 0.63 BA2 – 11,683 W ST – 21,752 AAL – 7,072 FAR139 + 4.0 EMP

(0.12) (7.46) (3.74) ( - 4.40) ( - 3.75) ( - 2.86) ( - 2.11) (1.68) OPSBA = 581.3 – 138.5 BAE100 – 125.9 W ST – 326.1 AAL + 113.1 R12

(18.37) ( - 3.31) ( - 2.50) ( - 2.51) (2.10) .7296 .2556 

### The variables that appear in Equations H - 7 and H - 11 in Table 1 include: OPS — Annual GA Operations at an airport (FY1999) OPSBA — Annual GA Operations per Based Aircraft (BA) at an airport BA — Total Based Aircraft at an airport (FY1999)

BA2 — Based Aircraft squared, which is included since airport operations tend to increase as the number of based aircraft increases, but at a slower and slower rate

3 Hoekstra (2000) began with 138 such airports, but data considerations reduced this figure to 127 airports susceptible to analysis. The current report uses the same set of 127 s mall towered GA airports. 4 T - statistics are given in parentheses under each coefficient.

### 3 PCI — Per Capita Income in the county in which the airport is located ($1999)

EMP — Non - agricultural Employment in the airport’s county (1999) FAR139 — Categorical variable, 1 if airport is certificated for commercial

air carrier service, 0 otherwise WST — Categorical variable, 1 if airport is located in FAA Western Region (excluding Alaska), 0 otherwise

AAL — Categorical variable, 1 if airport is located in Alaska, 0 ot herwise R12 – Categorical variable, 1 if airport is located in FAA New England Region or FAA Eastern Region, 0 otherwise

BAE100 — Categorical variable, 1 if airport based aircraft is 100 or greater, 0 otherwise Hoekstra us ed the equations in Table 1 to assess the results from sampling - based counting and extrapolation procedures 5 

### used by nine states at 129 small non towered airports. It was found that the equations tended to produce higher annual operations estimates than t he state estimates of annual operations at these airports. In addition, the model of Equation H - 11 produces modestly better estimates of activity at the non - towered airports than that of Equation H - 7, even though H - 7 has a “tighter” fit to the small tower ed airport data (in terms of R 2 

### ) than does H - 11. In addition, Equation H - 11 is based entirely on categorical data about the FAA region the airport is located in and on the airport having greater or fewer than 100 based aircraft. 3. NEW VARIABLES THAT M AY AFFECT AVIATION ACTIVITY AT SMALL TOWERED AND NON - TOWERED AIRPORTS For this analysis several additional types of data were developed for consideration as “local factors” affecting activity at small towered and non - towered airports. The set of airports used for the analysis is identical to that used in Hoekstra (2000), with the exception of a reduction in the set of non - towered airports considered, for reasons given below. In some cases these new variables are similar to demographic variables developed in Hoekstra (2000), and in others the variables represent new types of “local factors.” Population: County population (POP) and county employment (EMP) are significant demographic variables in Hoekstra (2000), but counties vary in size and airports vary in position within a county. A n airport in a small county may serve those in nearby
5 A variety of counting procedures were cited, including acoustic counters, machine counts, surveys, and operations logs, at these airports.

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counties as well, and an airport located near a county boundary may provide some or most of its services to the residents of other counties. To capture this type of demo graphic effect, GRA

calculated the 1998 population within 100 miles, 50 miles, and 25 miles from each airport in the Hoekstra (2000) data set that also appears in the FA A Terminal A rea Forecast (TA F). 6 It is expected that these variables will be positivel y correlated with airport annual operations. GRA also calculated the ratio of population within 25 miles of the airport to population within 100 miles of the airport (which will be a value between 0 and 1) as a proxy for relative population density around the airport. While this seems to be a reasonable candidate for being a “local factor” influencing airport activity, GRA had no prior belief on the direction of its influence. Ai rport Prominence -- The Proportion of Based A ircraft in Region: GRA calculated, for each towered and non - towered TA F airport in Hoekstra (2000), its proportion of based aircraft among all TA F GA airports within 50 miles and within 100 miles. This variable will be a number between 0 and 1. If the airport has all the based aircraft within the radius in question, the variable will take the value 1. GRA ’s initial beliefs about this “airport prominence” proxy were that it would have a positive influence on airport operations, since if an airport were more prominent (had a greater percentage of based aircraft) in its region, it would be a more attractive or popular site for aviation, and therefore have more operations. A

s the regression results below indicate, however, this belief was incorrect — it turns out that the grea ter the airport’s “regional prominence” the smaller the number of annual operations. This result can be rationalized in the following way: If a single airport is “prominent” in its region — has a relatively high percentage of based aircraft — that airport li kely has the lion’s share of the region’s operations as well, but it also indicates that the region itself does not support many operations overall. It seems that if the airport is “the only game in town” or “ the major game in town” for a region then the region itself is not very active. The “regionally prominent” airport seems to be the big frog in a small pond, but not a big frog overall. Complexity of the A irport’s Based A ircraft: This variable is the ratio of the airport’s single engine piston base d aircraft to all the based aircraft at the airport. The higher this ratio, the less complex (and less costly) are the airport’s based aircraft. It is not clear how this variable might be expected to influence airport operations, although the variable ma y be related to the flight school variable, since lower cost single engine piston aircraft might be the preferred aircraft for flight school use. Presence of Certificated Flight School: In Hoekstra (2000) it was found that nearly every airport in the dat a set had flight instruction available, although it was unclear what level of complexity or comprehensiveness characterized these training facilities from airport to airport. Because flight instruction was virtually ubiquitous across the airports examined , the variable provided no meaningful information about airport activity levels. 6 There were 24 non - tower ed airports that did not appear in the TAF, and for which the “population within X miles” could not be calculated. These non - towered airports are not included in the analyses and equations that follow, since a complete set of new variables could not be de veloped for them. These airports are: (1H3) Linn State, MO; (1N0) Ridgley Airpark, MD; (2W2) Clearview Airpark, MD; (6MO) Washington County, MO; (75K) Bethany Memorial, MO; (7S3) Stark’s Twin Oaks, OR; (8W2) New Market, VA; (AOV) Ava Bill Martin, MO; ( C16) Frasca Field, IL; (CJR) Culpepper Regional, VA; (DMW) Carroll County, MD; (H17) Buffalo Muni, MO; (HRI) Hermiston Muni, OR; (IGQ) Lansing Muni, IL; (K52) Monroe City, MO; (M17) Bolivar Muni, MO; (MO3) Stockton Muni, MO; (OXB) Ocean City Muni, MD; (RPD ) Rice Lake Regional, WI; (S49) Miller Memorial, OR; (SPB) Scappose Industrial, OR; (UAO) Aurora State, OR; and (VCB) Nut Tree, CA. In addition, Frederick Municipal in Maryland (FDK) has an unusually large number of based aircraft, and is omitted from est imations as an outlier.

### 5

For this update, GRA

examined the FA

A VITA LS database, maintained by the FA A Flight Standards Service. This database contains information on individuals and entities that ar e certificated under various FA R Parts. FA R Part 141 covers requirements for flight school certification, and the VITA LS data for Part 141 certification identified those certificate holders, including the number of employees at each certificated flight sc hool. In addition, some airports have more than one certificated flight school on site. Therefore, GRA developed data on the presence of Part 141 certificated flight schools at each of the Hoekstra (2000) data set airports, constructing three specific va riables: the presence or absence of a certificated flight school at an airport, the number of these flight schools at each of these airports, and the number of flight school employees at each of these airport certificated flight schools. 7 Pacific Coast St ates: The regional variable WSTA K used in Hoekstra (2000) combined states with a Pacific coast boundary (California, Oregon, Washington, and Al aska) with more inland western states such as Idaho, Montana, Utah, and others. These inland states are more sp arsely populated than the coastal states (with the exception of Al aska). Because models estimated using the WSTA K variable often resulted in estimates of negative annual operations for airports in these inland states, a new categorical regional variable w as created for those four Pacific coast states for this analysis. This new variable — called WA CA ORA K — turns out to have greater explanatory power than the WSTA K variable in the Minitab stepwise regression procedure. Table 2 shows the new independent vari ables used to complement and refine the analysis in Hoekstra (2000). Data used in the regression analysis in this report are contained in A ppendix B.

7 The number of flight school employees is of interest as a proxy for the relative size of each school, or for the relative size of certificated flight instruction at each airport for those airports with more than one certificated flight school.

### 6 Table 2 Categories of New Independent Variables Used In Regression An alysis Variable Name and Definition Me asurement/Units Source Pop25 1998 Population w/i 25 m iles Pop50 Wi thin 50 m iles Pop100 Wi thin 100 m iles Number of people By census tract, U.S. Census Pop25/100 Ratio of Pop25 to Pop100 Proportion, between 0 and 1 By census tract, U.S. Census Se BA /BA Single engine based Aircraft/All based aircraft Proportion, between 0 and 1 Terminal Area Forecast (TAF) TOWDUM %in50mi Percentage of based aircraft am ong based aircraft at GA airports within 50 m iles %in100mi Percentage of based aircraft among based aircraft at GA airports within 100 miles VITFS Presence or absence of FAR141 certificated pilot school VITFSnum Number of FAR141 certificated pilot schools on airport VITFSemp Em ploy ees of FAR141 certificated pilot schools at airport WA CA ORA K 1 if towered airport, 0 otherwise Proportion, between 0 and 1 Proportion, between 0 and 1 1 if FAR141 certificated pilot school present, 0 otherwise Number of FAR141 certificated pilot schools on airport Number of employ ees 1 if state is CA, OR, W A, or AK, 0 otherwise TAF TAF and Mapinfo software TAF and Mapinfo software FAA Flight Standards VITALS database FAA Flight Standards VITALS database FAA Flight Standards VITALS database Categorical/geographical



### 4. NEW APPROACHES TO ESTIMATING AVIATION ACTIVITY LEVELS

AT SMALL TOWERED AND NON - TOWERED AIRPORTS

4.1 Models of GA Operations The approach taken by GRA for enhancing the Hoekstra (2000) report was to use the final models specified in that report as the starting point for our investigations.

### 7

Therefore, the new variables developed as potential inputs for modeling small er airport activity were added to the list of variables included in the final Hoekstra (2000) equations, and stepwise regression was used to identify those variables that most strongly contributed to explaining airport activity, expressed as either annual operations or annual operations per based aircraft. These new variables were identified in Section 3 above. Table 3 contains the results of this stepwise regression. Note that many of the variables developed by Hoekstra remain in the final regression eq uation, and that some of the newly defined variables do not contribute enough to explanatory power to enter the equations in Table 3 (or into subsequent sets of equations). Since the number of based aircraft remains a highly significant and positive reg ressor, it is important to note that the inverse relationship between annual operations and the square of based aircraft also continues from the earlier results in Hoekstra (2000). Other things equal, it seems that the number of operations grows as the nu mber of based aircraft increases, but at an increasingly slower rate. This “slowdown” is governed by the negative coefficient on the square of based aircraft. 8 As Equation 9 of Table 3 indicates, towered airport operations are positively related to the number of based aircraft, to the population within 100 miles of the airport, and to the ratio of the population within 25 miles of the airport to the population within 100 miles of the airport. This last result seems to imply that the denser the populatio n is nearby the airport (relative to the total population within 100 miles), the greater number of operations that occur. Equation 9 also indicates that other things equal, airport activity is negatively related to an airport’s presence in the Pacific Coa st states of Washington, California, Oregon or A laska, to the square of the number of based aircraft, to the airport’s proportion of based aircraft among all based aircraft for GA airports within 100 miles 9 and to FA R 139 carrier certification at the airpo rt.

8 It may be worth noting that the coefficients on BA and BA2 in Equation 9 imply that other things equal, the number of operations at an airport begins to decrease once the number of based aircraft exceeds 653 (which is what 421 divided by 0 .64 approximately equals). 9 The negative parameter estimate for the airport’s percentage of based aircraft among all based aircraft at airports within 100 miles is somewhat counterintuitive. It might be thought that an airport with more of an area’s base d aircraft would have more operations, other things equal, but this seems to be true only relatively. What seems to be more important is that if a single airport is dominant (in terms of based aircraft) within the surrounding 100 miles, then the area itse lf has relatively low levels of GA activity, even if that dominant airport has the bulk of it.

### 8 Table 3 GA Operations Regression Equations — Data for Small Towered GA A irports Only Equation Number 

### Equation 10 

### R 2 1 2 3 4 5 6

7

8

9 OPS = 21,555 + 242 BA

(8.11) (12.52) OPS = 18,606 + 211 BA + 0.002 Pop100 (7.55) (11.46) (5.37) OPS = 18,718 + 229 BA + 0.001 Pop100 – 10,059 W ACAORAK (7.85) (12.2) (5.46) ( - 3.11) OPS = 7,495 + 425 BA + 0.001 Pop100 – 12,501 W ACAORAK – 0.56 BA2 (2.04) ( 7.95) (5.26) ( - 4.00) ( - 3.88) OPS = 12,598 + 453 BA + 0.001 Pop100 – 12,956 W ACAORAK - 0.62 BA2 – 19,958 %in50mi (3.19) (8.60) (2.80) ( - 4.39) ( - 4.39) ( - 2.96) OPS = 10,422 + 462 BA + 0.001 Pop100 – 13,754 W ACAORAK - 0.68 BA2 – 23,481 %in50mi (2.61) (8.60) (3.07) ( - 4.58) ( - 4.84) ( - 3.46) + 18, 587 Pop25/100 (2.33) OPS = 9,142 + 449 BA + 0.001 Pop100 – 13,292 W ACAORAK - 0.67 BA2 – 8,448 %in50mi (2.29) (8.69) (3.23) ( - 4.47) ( - 4.81) ( - 0.83) + 32,823 Pop25/100 - 44,094 %in100mi

(3.07) ( - 1.97) OPS =7,954 + 440 BA + 0.001 Pop100 – 13,024 W ACAORAK - 0.65 BA2 (2.13) (8.73) (3.87) ( - 4.41) ( - 4.74) + 36,362 Pop25/100 - 58,055 %in100mi (3.71) ( - 3.93) OPS =14,449 + 421 BA + 0.001 Pop100 – 12,452 W ACAORAK - 0.64 BA2 (2.73) (8.25) (3.69) ( - 4.22) ( - 4.66) + 31,361 Pop25/100 – 52,130 %in100mi - 5,528 FAR13 9 (3.09) ( - 3.47) ( - 1.72) .5564 .6402 .6664 .7031 .7231 .7351

.7435

.7420

.7482

One of the principal objectives of this research study is to develop equations by which airport “local conditions” can be used to estimate annual operations at other airports, in particular non - towered airports for which activity data may be of low or uncertain quality. In Hoekstra (2000) a set of non - towered airports from selected states for which “reasonable” state estimates of GA operations activity were available was used to assess the accuracy of models developed from activity and “local conditions” data at small towered airports for estimating non - towered airport acti vity. A similar assessment was used for this report, for a smaller set of non - towered airports as explained above. For this purpose, the model represented by Equation 9 in Table 3 was applied to data for 105 non - towered airports. Figure 1 shows the comp arison between the annual operations estimates at the non - towered airports, as extrapolated from the state counter and survey programs used at the airports, and the annual operations estimated at each non - towered airport, based on 10 T - statistics are given in parentheses under each coefficient.

### 9

Equation 9 in Table 3. F or 94 airports, or 90 percent of the 105 non - towered airports, the operations estimates produced by the equations exceed the state estimates, and for 57 airports, or 54 percent, the number of annual operations estimated using Equation 9 is more than double the state estimates. 

### Figure 1

State Estimated and Equation Estimated A nnual Operations

at Non - Towered A irports



### Thus, as in Hoekstra (2000), a model of GA annual activity developed from local conditions at small towered airports explains GA activ ity at the towered airports quite well (in terms of R 2 

### ), but tends to produce higher estimates than the state estimation procedures for a set of non - towered airports. This suggests that fundamental distinctions may exist between the determinants of activi ty at these two types of airports (or that there are systematic underestimations of activity occurring in the count programs at these non towered airports). It is possible to “bridge” this apparent divergence between the characteristics of data from sma ll towered GA airports and data from non towered GA airports, and to confirm statistically that it exists. The tendency of the Equation 9 model to produce higher estimated operations than the state

0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000

105 Non-Towered GA AirportsAnnual Estimated GA Operations

State Estimates Model Estimates

### 10

estimates at non - towered GA airports is certainly appare nt in Figure 1. This tendency may be an expression of the value that a tower adds to an airport’s effectiveness and attractiveness to users. Alternately, the tendency may simply reflect the effectiveness of FAA establishment criteria for towers, since th e tendency seems to imply that towers have been placed at airports that, other things equal, have more activity than comparable non - towered airports. Finally, the tendency of the model may in fact imply that the state estimate programs tend to underestima te annual activity levels. Regardless of the reason for their differences, bridging the divergence between the two data sets can be done by combining them into a single data set outright or by combining them in a way that continues to treat the two gro ups of airports somewhat differently. Grouping them together outright overlooks the fact that significant differences may exist between the two groups of airports, and linking them together in a more discriminating way provides a compromise that recognize s that the two groups are in some ways dissimilar even though both groups provide information that may be useful for modeling and estimating activity at less well documented non - towered airports. To test the hypothesis that there is a statistically signi ficant difference between the estimating equations for small towered and non - towered GA airports, a common test procedure, known as the Chow test, was used. The Chow test “is the most popular way of testing whether the parameter values associated with on e data set (say, Period 1 [or with the set of small towered GA airports]) are the same as those associated with another data set (say, Period 2 [or with the set of GA non - towered airports]).” 11 

### Additional information on the Chow test is contained in Append ix A. In both approaches to estimating joint equations that include data from both small towered airports and non - towered airports (or equations with no dummy variable for “towered” airports and one with that dummy variable), the stepwise regression proce dure was used to choose the best fitting set of variables. To perform this test, the stepwise procedure was used to find the best estimating equation for the entire data set of towered airports and non - towered airports. Because this estimate “forces” t he model coefficients to be the same for small towered GA airports and non - towered airports, it represents a “restricted” regression. The unrestricted regressions occur when the same variables are used to estimate separate equations for the small towered GA airport set and the non towered airport set. The difference in the regression sums of squares for these two scenarios (“restricted” and “unrestricted”) can be examined using an F distribution, and a significant test result indicates that the two data s ets are

11 Peter Kennedy, A Guide to Econometrics , Cambridge, MA: The MIT Press, 1979, p. 87.

### 11

sufficiently different as to give rise to different model parameters — they cannot be combined into a statistically homogeneous single data set. A second functional form was also estimated in an attempt to relate operations at small towered airport s and those at non - towered airports. An equation that included a dummy variable for “towered airports” was used — a variable — denoted TOWDUM — that took the value 1 if the airport was one of the 127 small towered airports and took the value 0 otherwise. Usin g an equation of this form to estimate a joint model that included both small towered and non towered airports has the effect of letting each group have its own intercept term, while restricting all other coefficients to be equal on other parameters. The validity of this method can also be assessed using the Chow test mentioned above. Tables 4 and 5 show the equations that resulted from these stepwise regressions, for the “no dummy variable” and the “with dummy variable” cases respectively. (In Table 4, POP is the estimated 1999 population of the county in which the airport is located.) In each table, the calculated F - statistic is shown, along with the critical F - statistic at the 0.01 level of significance. In both cases, we must reject the hypothesi s that the small towered airport data set and the non - towered airport data set are from similar distributions. (This is so even though a comparison of the two tables shows that the strategy of using a dummy variable for distinguishing between towered and non - towered airports in the joint data set provides a modestly improved R 2 

### .)

### 12 Table 4 GA Operations Regression Equations — Data for Small Towered GA A irports and Non - Towered GA

A irports — No Dummy Variable Equation Number Equation 12 (No Dummy Variable Used f or “Towered A irports”) R 2

10 All 232 Airports

11 127 Towered Airports

12 105 Non - towered Airports OPS = 9,309 + 391 BA – 0.47 BA2 + 3.1 POP + 7,095 FAR139 + 0.001 Pop100 + 5,190 VITFSnum

(1.49) (9.19) ( - 3.69) (2.62) (3.22) (3.32) (2.45)

- 11,091 W ACAORAK – 11,220 seBA/BA Regression SS: 1.15679E+11 ( - 4.39) ( - 1.57) OPS = 13,256 + 386 BA – 0.55 BA2 + 0.33 POP – 7,711 FAR139 + 0.001 Pop100 + 3,764 VITFSnum (1.92) (7.20) ( - 3.96) (0.28) ( - 2.28) (4.37) (1.67)

- 12,272 W ACAORAK + 4,997 seBA/BA Regression SS: 5.23207E+10 ( - 3.81) (0.64) OPS = 21,905 + 216 BA – 0.07 BA2 + 27.3 POP + 20,641 FAR139 + 0.001 Pop100 + 3,701 VITFSnum (1.66) (2.62) ( - 0.27) (2.95) (3.40) (1.08) (0.91)

- 5,697 W ACAORAK – 21,853 seBA/BA Regression SS: 3.11064E+10 ( - 1.50) ( - 1.44) Chow Test result : F - statistic calculated from data: 9.19, distributed as F(232,9), with critical value 4.33 at 1% significance level: Reject h ypothesis that the Towered and Non - Towered data come from similar distributions as represented by Equation 10. .7170

.7270

.6480



### While the Chow test results indicate that there are basic differences between the towered and non - towered airp ort data sets, it may remain practical to use the joint models, with the dummy variable TOWDUM to distinguish between towered and non - towered characteristics. This is especially true because activity levels for the non - towered airports in the data set are themselves based on state estimates, rather than actual counts as with the small towered airports. To see how effectively the dummy variable approach might be used for estimating activity at non - towered airports outside the sample set, GRA made two compa risons.

12 T - statist ics are given in parentheses under each coefficient.

### 13 Table 5 GA Operations Regression Equations Data for Small Towered GA A irports and Non - Towered GA A irports — With Dummy Variable Equation Number Equation 13 ( Dummy Variable for “Towered” Used) R 2

13 All 232 Airports, Dummy Variable for Towered 14 127 Towered Airports 15 105 Non Towered Airports OPS = - 571 + 355 BA – 0.46 BA2 – 40,510 %in100mi + 3,795 VITFSnum + 0.001 Pop100

( - 0.25) (8.41) ( - 3.83) ( - 2.79) (1.87) (3.48)

- 8,587 W ACAORAK + 24,102 Pop25/100 + 13,674 TOW DUM Regression SS: ( - 3.61) (2.67) (6.44) 1.19818E+11 OPS = 7,731 + 428 BA – 0.65 BA2 – 54,681 %in100mi + 3,485 VITFSnum + 0.001 Pop100 (2.08) (8.47) ( - 4.77) ( - 3.69) (1.63) (4.07)

- 12,856 W ACAORAK + 34,958 Pop25/100 Regression SS: 5.377 2E+10 ( - 4.38) (3.58) OPS = 775 + 241 BA – 0.14 BA2 + 31,478 %in100mi + 5,577 VITFSnum + 0.001 Pop100 (0.22) (2.52) ( - 0.46) (0.85) (1.23) (1.80)

- 3,736 W ACAORAK + 12,121 Pop25/100 Regression SS: 2.7327E+10 ( - 0.89) (0.65) Chow Test result : F - statistic calculated from data: 11.5, distributed as F(232,9), with critical value 4.33 at 1% significance level: Reject hypothesis that the Towered and Non - Towered data come from similar distributions as represent ed by Equation 13. .7430

.7480

.5690



### In the first, one - seventh of the non - towered airports were randomly excluded from the joint data set. 14 

### Using this combined but “truncated” data set of 127 towered airports and 90 non - towered airports, G RA re - estimated the model (using the dummy variable approach). Then, using this re - estimated equation, the expected operations at the omitted non - towered airports were calculated. Table 6 shows the equation that resulted from this estimation for the trun cated data set (note that the equation has the same functional form as Equation 13 in Table 5 above). Figure 2 shows the state estimates and equation estimates for this “out of sample” estimation of annual GA operations for the 15 non - towered airports ran domly excluded from the data set. In Figure 2, the airports are listed, left to right, in ascending order of state estimated annual GA operations, for ease of viewing. The data shown in Figure 2 depict a closer fit of model estimates to state estimates t han does Figure 1 above. For Figure 2, only six, or 40 percent of the model estimates exceed the state estimates (compared to 90 percent in Figure 1), and only one, or seven percent of the model estimates are more than double the state estimates (compared to 54 percent in Figure 1).

13 T - statistics are given in parentheses under each coefficient. 14 Airports w ere randomly excluded from the list of 105 non - tow ered GA airports by ordering the 105 airports alphabetically by LOCID and the n excluding every seventh airport from this list. The LOCIDS of the 15 excluded airports are (in order of state estimated annual GA operations, as along the horizontal axis of Figure 2) RIF, BUM, AJG, GLY, M58, FWC, OKV, O61, RBG, GGW, 7S5, CBE, 0W3, JYO, and ESN.

### 14

### 15 Table 6 GA Operations Regression Equations Small Towered GA

A irports and Truncated Non - Towered GA A irports Data Set using Dummy Variable to Examine Fit to Non - Towered A irports Equation Number Equation 15 ( Dummy Variable for “Towered” Used) R 2

16 All Airports Dummy Variable for Towered, 15 Non - Towered Airports Excluded OPS = - 1,215 + 370 BA – 0.49 BA2 – 48,631 %in100mi + 3,663 VITFSnum + 0.001 Pop100

( - 0.50) (8.45) ( - 4.01) ( - 3.21) (1.75) (3.04) 7,793 W ACAORAK + 26,801 Pop25/100 + 13,791 TOW DUM ( - 3.21) (2.92) (6.17) .7450



### Figure 2 Fit of Out - of - Sample Estimates for 15 Non - Towered A irports Excluded from Estimating Data for Equation 16

-10,000 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000

Non-Towered GA Airports, Ranked by State Estimated Annual GA OperationsState and Model Estimated Annual GA Operations

State Ops Estimate Model Ops Estimate 

### Visual examples can be seen in Figures 3 and 4. Both figures show the annual operations tower counts (on the l eft) and annual operations state

15 T - statistics are given in parentheses under each coefficient.

### 16

estimates (on the right), each ranked from least to greatest. Thus, on the horizontal axis of both Figure 3 and Figure 4 is a sequence of the 232 airport data set, first the 127 small towered GA airports, ranked from fewes t annual operations to greatest, and then the 105 non - towered GA airports, ranked from fewest state estimated operations to greatest. In Figure 3, model estimates for both the towered airports and the non towered ones are shown using Equation 9 from Tabl e 3, the best fitting equation based on the towered airport data alone. On the left side of the Figure, the equation estimates are evenly distributed around the towered airport data set, but on the right side one can clearly see the tendency of this equat ion to produce higher estimates relative to the state estimates of activity at the non - towered airports. (Note: The right side of Figure 3 shows the same comparison of model estimates and state operations estimates as Figure 1.) Figure 3 Comparison of Tower Counts and State Estimates to Model Estimates (Equation 9) for Small Towered GA A irports and Non - Towered A irports — Model Without Tower Dummy Variable



### In Figure 4, ho wever, the model estimates are based on Equation 13, the best - fitting equation for the joint set of 232 towered and non - towered GA airports (using the TOWDUM dummy variable to separate some of the towered and non - towered effects). In this figure, the mode l estimates are evenly distributed around both the towered airport tower count data and the non 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000

Towered GA Airports, Ranked by --- Non-Towered GA Airports, Ranked by

Annual GA Ops (Left) State Estimated Annual GA Ops (Right)Annual GA Operations

Tower Counts/State Estimates Model Estimates

### 17

towered airport state estimates. This equation can be regarded as utilizing information about small towered GA airports to model small non - towered GA airports, while maintaining some unique factors of these non - towered facilities (in a way that Equation 9, based solely on towered airport data does not). Figure 4 Comparison of Tower Counts and State Estimates to Model Estimates (Equation 13) f or Small Towered GA A irports and Non - Towered Ai rports — Model with Tower Dummy Variable



### The out of sample procedure and the value of information from the towered airport data for modeling estimates of annual operations in the non towered airport se t suggest a productive next step for this modeling and research analysis. One use of the model equations shown above, in particular, Equation 13, is to estimate operations for a large data set of U.S. non - towered airports contained in the Terminal Area Fo recast (TAF). For the majority of these airports, operation estimates reported on DOT Form 5010 are roughly developed and are not based on sampling procedures. This step would allow a rough assessment to be made of the plausibility of these 5010 data, an d would potentially provide APO staff with a means of assessing claims about airport operations at these types of poorly documented non - towered airports.

0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 160,000

Towered GA Airports, Ranked by --- Non-Towered GA Airports, Ranked by

Annual GA Ops (Left) State Estimated Annual GA Ops (Right)Annual GA Operations

Tower Counts/State Estimates Model Estimates

### 18 4 .2 Model of GA Operations per Based Aircraft Hoekstra (2000) also estimated models of operations per based aircraft at the small towered GA airports in the data set; these models generally had poorer explanatory power than the models of GA operations per se. Also worth noting is the fact that the model of GA operations per based aircraft that was mos t effective — shown as Equation H - 11 in Table 1 above — actually included no airport “local factors” other than the range of based aircraft and some dummy variables for the airport’s FAA region. For this study, GRA used the Minitab Stepwise Regression procedu re to modestly improve the model of Equation H - 11 by including the four categorical variables included in H - 11 and the new variables developed for this report. The results of this procedure are reported in Table 7. In Equation 17, operations per based ai rcraft are reduced (i.e., have negative coefficients) for airports with greater than 100 based aircraft, for those in the Western or Alaska regions, for those that are “regionally prominent” in the sense that they have a large proportion of all based aircr aft at nearby airports, and for those with relatively higher populations within 50 miles. Operations per based aircraft are modestly increased (i.e., have positive coefficients) for those airports with relatively higher populations within 100 miles. Tab le 7 Best Equation from Minitab Stepwise Regression for Small Towered A irport Operations per Based A ircraft Equation Number Equation 16 R 2 17 OPSBA = 595.2 – 164 BAE100 – 325 %BA100 - 107 W ST – 244 AAL

(15.2) ( - 3.84) ( - 1. 56) ( - 2.17) ( - 1.85)

+ 0.00002 Pop100 – 0.00002 Pop50 (2.51) ( - 1.79) .3072 

### The modeling approach represented by Equation 17, which estimates annual GA operations per based aircraft, has not been pursued further in this report for three reasons. First, the goodness of fit to the data, based on the model R 2 

### , is significantly lower than that in the models that estimate annual GA operations alone. Second, the explanatory variables that are significant for the regression equation are mainly categorical or regional variables, and thus say little about sources of variability from airport to airport (which may include the number of based aircraft at an airport, a factor that is only included in the categorical variable BAE100 in equation 17). Finally, for many of the non towered airports that are the focus of this report, based aircraft may be a more
16 T - statistics are given in parentheses under each coefficient.

### 19

reliable estimate than annual operations. This circumstance makes a model that treats an airport's number of ba sed aircraft as an important factor in explaining annual GA operations somewhat more compelling. 5. APPLYING THE ESTIMATED MODEL TO SMALL NON - TOWERED GA AIRPORTS NATIONWIDE The estimating equation shown as Equation 13 of Table 5 was used to develop mod el estimates for 2,789 non - towered GA airports across the country and its territories. This data set includes all of the non - towered GA airports in the TAF not included in the development of the model estimation. For the majority of these airports the op eration estimates reported on DOT Form 5010 are not based on sampling procedures. The demographic and other “local factors” data were gathered, and the model equation applied. This section discusses some broad initial results from this effort. Table 8 s ummarizes data for 2,780 of the non - towered GA airports that are located in U.S. states (the remainder are located in U.S. territories such as the Virgin Islands). For each state, the data include the number of non - towered TAF airports in the state (exclu ding those non - towered airports that were part of the modeling process in Hoekstra (2000) and the prior sections of this report), the maximum number of annual GA operations per airport reported in the state and the minimum number reported. Some airports ( 13, or 0.5%) report 0 annual GA operations on their 5010 report, and 225, or eight percent, report fewer than 1,000 annual GA operations. Also shown in the table are average absolute percentage deviations between model estimates (based on Equation 13) and the airport 5010 estimate. 17 

### The state - by - state data shown in Table 8 indicates a wide range of reported annual operations by non - towered GA airports in most states. Since these data are based on airport - specific Form 5010 estimates, it is not clear how much of this range should be attributed to variability in airport usage and how much to variability in Form 5010 reporting and estimation procedures. The table also suggests that a closer look, on a state - by - state basis, at relationships between airport o perations and the other demographic or “local factor” data could add to APO’s understanding of the determinants of airport activity by GA users of the National Airspace System.

17 No percentage absolute deviation is reported for Alaska because several Alaska airports report 0 annual operations, w

hich (even if changed to 1 operation) make the absolute deviation from the model estimate quite large in percentage terms. For other stat es, those airports reporting 0 or very small numbers of annual operations are omitted from the average absolute percentage deviation.

### 20 Table 8 Summary State Data for 2,780 Non - Towered GA A irports Maximum Minimum Average NonTowered Annual GA Annual GA Absolute State GA Airports Ops (5010) Ops (5010) Deviation (%) AK 226 61,000 0 ---- AL 62 90,000 672 49% AR 74 90,000 200 67% AZ 33 64,000 700 68% CA 145 100,000 100 88% CO 47 95,990 259 59% CT 10 58,656 1,345 128% DE 3 42,000 35,000 45% FL 54 97,085 0 72% GA 85 73,000 1,000 55% HI 9 73,950 100 580% IA 73 47,408 1,000 38% ID 32 85,710 5,787 46% IL 75 90,980 950 62% IN 62 63,622 2,483 40% KS 80 87,473 1,300 39% KY 44 49,909 800 31% LA 41 63,772 500 49% MA 22 97,105 6,922 45% MD 17 91,100 1,295 64% ME 22 60,500 1,400 62% MI 92 86,534 1 59% MN 75 68,400 2,420 44% MO 57 33,128 1,091 41% MS 66 60,000 400 59% MT 68 62,300 1 54% NC 61 65,624 2,300 45% ND 40 63,681 100 56% NE 63 65,000 225 35% NH 11 93,381 5,730 34% NJ 29 90,000 2,635 46% NM 41 67,483 10 164% NV 25 77,000 50 115% NY 73 90,000 0 73% OH 83 81,915 2,100 56% OK 69 93,504 50 56% OR 48 74,267 1,572 52% PA 62 80,310 3,000 43% RI 4 34,390 9,008 80% SC 52 92,266 1,000 56% SD 44 52,806 300 54% TN 61 67,704 576 37% TX 175 97,000 600 38% UT 34 94,583 607 68% VA 42 93,882 1,000 52% VT 10 33,800 400 66% WA 44 96,215 2,700 51% WI 82 84,391 2,000 45% WV 21 48,356 110 124% WY 32 29,927 650 43%

### 21 Figu re 5 depicts the Form 5010 report of annual GA operations estimates for the entire set of 2,789 non - towered GA airports. For ease of viewing, the airports are ranked, from left to right on the horizontal axis of the figure, from fewest 5010 operations to greatest. Thus, the horizontal axis of the figure is made up of 2,789 individual airport data filings, which are shown increasing in a somewhat exponential curve from least to greatest. For each of these airports, the model estimate of annual GA operatio ns is also shown. These estimates vary in their accuracy, and appear as dots around the increasing line depicting the ranked 5010 operations reports. The model produces higher estimates of annual GA operations, relative to the Form 5010 estimate, for 973 airports (or about 35 percent), and produces lower estimates for 1,816 airports (about 65 percent). For 84 airports (or about three percent) the model produces estimates of negative annual operations. This occurs predominantly in Alaska and other sparse ly populated northwestern states, in which an airport may be quite prominent within the surrounding 100 miles, leading its model estimate to be strongly affected by the large negative coefficient on local prominence. Figure 5 Form 5010 Estimates and Mode l Estimates for A nnual GA Operations at 2,789 Non - Towered GA A irports

-60000.0 -40000.0 -20000.0 0.0 20000.0 40000.0 60000.0 80000.0 100000.0 120000.0

2,789 Nontowered GA Airports Ranked by 5010 Ops EstimatesEstimated Annual Ops

Model Estimate Airport 5010 Estimate

### 22 Figure 6 illustrates the pattern of variability that exists between Form 5010 and model - based operations estimates. In Figure 6, the horizontal axis again contains the 2,789 non - towered GA airports ranked, from left to right, by Form 5010 GA operations estimates. For each of these airports, the chart displays the absolute percentage deviation of the model estimate from the airport’s 5010 estimate of annual operations. Also shown in the chart is a trend line, which is constructed by averaging the 25 average percentage deviation calculations to the right and left of each airport. Such a trending procedure, called a “moving average,” smoothes the airport - to - airport variability out of the absolute percentage deviation data to make general patterns more visible. This figure shows that for most of the 2,789 airports (and in particular after Form 5010 estimated GA annual operations exceed 3,000) absolute percentage deviation remains fa irly stable at between 40 and 55 percent. Note, however, that Figure 5 indicates in addition that with growing estimates of annual operations (that is, moving from left to right in the figure), model estimates tend to be less than those reported in Form 5 010 estimates. Figure 6 Variability of Differences between Form 5010 A nnual Operations Estimates and Model

Estimates of A nnual Operations for U.S. Non - Towered GA A irports

0% 50% 100% 150% 200% 250% 300%

Nontowered GA Airports Ranked by 5010 Ops EstimatesAbsolute Percentage Deviation, Form 5010 Operations Estimates andModel Estimates

% Absolute Deviation Moving Average Trend

### 23 The "flattening out" of the moving average trend in Figure 6 provides evi dence that the model developed in this paper provides a consistent approach to estimating non - towered GA airport operations using demographic and "local factor" data. Table 9 reports the average absolute deviations and the average absolute percentage devi ations for the small towered GA airports (Equations 9 and 13), for the non - towered GA airports with state operation estimates based primarily on sampling procedures (Equations 9 and 13), and for the large set of non - towered GA airports (Equation 13 only, f or all 2, 789 airports, for those 2,628 airports reporting more than 500 annual GA operations, and for those 2,469 airports reporting more than 2,000 annual GA operations). These values indicate that the joint Equation 13, which links the towered and non - towered GA airport data, provides a model with improved estimating consistency for the 105 non towered airports with state estimates based primarily on sampling procedures. Table 9 Model Goodness of Fit Measures for Towered GA A irports, State - Estimated Non - Towered GA A irports, and Form 5010 Non - Towered GA A irports Set of Airports Equation Average Absolute Deviation Standard

Deviation of Absolute Deviation Average

Absolute Percentage Deviation Standard

Deviation of Absolute Percentage Deviation 127 Sma ll Towered GA Airports Equation 9 (no dummy) 9,537.0 7,213.8 25.2 % 30.82 %

Equation 13 (w/ dummy) 10,121.2 6987.8 28.4 % 39.7 % 105 Non Towered GA Airports Equation 9 (no dummy) 20,029.8 11,648.3 236.5 % 427.8 %

Equation 13 (w/ dummy) 10,011.7 10,64 0.4 80.0 % 190.1 % 2,789 Non Towered GA Airports Eq. 13, all 2,789 airports 8,634 11,063 2,114 % 38,180 %

Eq. 13, Est. Ops > 500 8,949 11,235 60 % 103 % Eq. 13, Est. Ops > 2,000 9,540 11,482 50 % 62 %



### 6. CONCLUSION The work described in this r eport has refined the modeling technique developed in Hoekstra (2000) by adding new explanatory variables that are related to the “local factors” that influence aviation activity at small airports and by developing new functional approaches that allow join t equations to link the small towered airport data set and the non - towered airport data set. This linkage addresses and remedies to some extent the tendency of models based solely on towered airport data to exceed state - estimated activity (with estimates based on

### 24

other non - econometric methods) at non - towered airports. The modeling has also quantified important differences between small towered GA airports and non towered ones, giving an improved view of the impact a tower might have on aviation activity. Statistical testing indicates that there are fundamental differences between the characteristics of activity at small towered airports and activity at their non - towered counterparts. In spite of these differences, the report has also shown that a joint m odel of towered GA airports and non towered ones can provide more unbiased estimates for activity at these non towered airports, compared to a model based on towered airport activity alone. These improvements suggest that the joint models may be used, in combination with “local factors” and demographic data for a broader set of non towered airports, to begin a rough assessment of activity at these non - towered airports. This report has provided a preliminary estimate of this data for 2,789 small non - towere d GA airports. A more detailed examination of these airports and of the relationship between the activity levels estimated for them in 5010 data and their demographic and other characteristics may provide new tools for characterizing and categorizing thes e airports. Such an assessment tool for small GA airports could provide APO analysts with an improved ability to assess the reliability of master plans and activity forecasts at these smaller airports.

### 25 

## APPENDIX A THE CHOW TEST FOR CHANGING PARAMETER VALUES The Chow test (named after the noted econometrician Gregory Chow) is a commonly used test for determining whether a single regression equation can be applied to a data set. The test compares results from a regression run over the entire data set of interest — the “restricted” or “constrained” regression, since the parameters are “restricted” or “constrained” to be equal for the entire data set — with results from an “unrestricted” or “unconstrained” regression which allows the data set to be broken in to two pieces for which there is some plausible rationale. Examples of such rationales might be breaking a time series of economic data into a pre - World War II set and a post - World War II set, or breaking a set of small airport data into two subsets, one for small towered airports and one for small non - towered airports. For the defined subsets, the regression equation identical to that used for the “restricted” regression (in some or all parameters) is used for each of the subgroups of data. The test s tatistic, which is distributed as an F - distribution, is based on the regression sums of squares from the “restricted” and “unrestricted” estimations. Test statistic parameters are the number of data points in the total data set — N — (that used for the “rest ricted” estimation), the number of parameters being “restricted” to being held constant in the restricted estimation — R, and the number of regressor variables in the “unrestricted” estimation — K. The test statistic is calculated as



#     )

/( ) ( / ) ( ) ( ( K N ed unrestrict SSE R ed unrestrict SSE restricted SSE



 and is distributed as an F - statistic with R and (N - K) degrees of freedom. If the test statistic exceeds the critical F(R,N - K) value of choice, we conclude that the two sub - data sets should not be regarded as coming from the same population.

## APPENDIX B COMPLETE DATA SET USED FOR REGRESSION ANALYSIS