
<PAGE>   1
 
                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549
                             ---------------------
                                 SCHEDULE 13E-3
                        RULE 13E-3 TRANSACTION STATEMENT
(PURSUANT TO SECTION 13(E) OF THE SECURITIES EXCHANGE ACT OF 1934 AND RULE 13E-3
                          (SEC. 240.13E-3) THEREUNDER)
 
   
                                AMENDMENT NO. 3
    
                             ---------------------
                                BRAD RAGAN, INC.
                              (Name of the Issuer)
 
                                BRAD RAGAN, INC.
                       THE GOODYEAR TIRE & RUBBER COMPANY
                      (Name of Person(s) Filing Statement)
 
                    COMMON STOCK, PAR VALUE $1.00 PER SHARE
                         (Title of Class of Securities)
 
                                   750626103
                     (CUSIP Number of Class of Securities)
                             ---------------------
 
<TABLE>
<S>                                                 <C>
                WILLIAM P. BROPHEY                                 GEORGE E. STRICKLER
      PRESIDENT AND CHIEF EXECUTIVE OFFICER                           VICE PRESIDENT
                 BRAD RAGAN, INC.                           THE GOODYEAR TIRE & RUBBER COMPANY
          4404-G STUART ANDREW BOULEVARD                         1144 EAST MARKET STREET
         CHARLOTTE, NORTH CAROLINA 28217                            AKRON, OHIO 44316
                  (704) 521-2100                                      (330) 796-2121
</TABLE>
 
(Name, Address and Telephone Number of Person Authorized to Receive Notices and
            Communications on Behalf of Person(s) Filing Statement)
                             ---------------------
      THE COMMISSION IS REQUESTED TO SEND COPIES OF ALL COMMUNICATIONS TO:
 
<TABLE>
<S>                                                 <C>
                GARZA BALDWIN, III                                   ROBIN L. HINSON
      WOMBLE CARLYLE SANDRIDGE & RICE, PLLC                 ROBINSON, BRADSHAW & HINSON, P.A.
           3300 ONE FIRST UNION CENTER                      101 NORTH TRYON STREET, SUITE 1900
             301 SOUTH COLLEGE STREET                              CHARLOTTE, NC 28246
       CHARLOTTE, NORTH CAROLINA 28202-6025
</TABLE>
 
                               CLARENCE W. WALKER
                  KENNEDY COVINGTON LOBDELL & HICKMAN, L.L.P.
                       100 NORTH TRYON STREET, SUITE 4200
                      CHARLOTTE, NORTH CAROLINA 28202-400
 
This statement is filed in connection with (check the appropriate box):
 
a.     [X]     The filing of solicitation materials or an information statement
               subject to Regulation 14A, Regulation 14C, or Rule 13e-3(c) under
               the Securities Exchange Act of 1934.
 
b.     [ ]     The filing of a registration statement under the Securities Act
               of 1933.
 
c.     [ ]     A tender offer.
 
d.     [ ]     None of the above.
 
Check the following box if the soliciting materials or information statement
referred to in checking box (a) are preliminary copies:  [X]
                             ---------------------
 
                           CALCULATION OF FILING FEE
 
<TABLE>
<CAPTION>
------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------
              TRANSACTION VALUATION*                               AMOUNT OF FILING FEE
------------------------------------------------------------------------------------------------------
<S>                                                 <C>
                   $20,745,419                                          $4,149.08
------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------
</TABLE>
 
* Pursuant to Rule 0-11(c)(1), the transaction valuation is based on the amount
  of cash to be received by the issuer's public shareholders in connection with
  the proposed share exchange.
 
[ ]  Check box if any part of the fee is offset as provided by Rule 0-11(a)(2)
     and identify the filing with which the offsetting fee was previously paid.
     Identify the previous filing by registration statement number or the Form
     or Schedule and the date of its filing.
 
                                   $4,149.08
                             Amount Previously Paid
 
                                  LISTED ABOVE
                                  Filing Party
 
                                 SCHEDULE 13E-3
                            Form or Registration No.
 
                                  MAY 12, 1998
                                   Date Filed
<PAGE>   2
 
                                BRAD RAGAN, INC.
 
                             CROSS REFERENCE SHEET
 
     The information contained in the Preliminary Proxy Statement of Brad Ragan,
Inc. (the "Preliminary Proxy Statement") is incorporated by reference in answer
to the items of this Amended Rule 13e-3 Transaction Statement on Schedule 13E-3
(the "Transaction Statement"). This cross reference sheet shows the location of
such information required to be included in the response to the items of the
Transaction Statement. The Preliminary Proxy Statement is being filed
concurrently herewith as Exhibit 17(d) to this Transaction Statement.
 
<TABLE>
<CAPTION>
ITEM           CAPTION IN SCHEDULE 13E-3             LOCATION IN PRELIMINARY PROXY STATEMENT
----           -------------------------             ---------------------------------------
<S>   <C>                                          <C>
 1.  ISSUER AND CLASS OF SECURITY SUBJECT TO THE
     TRANSACTION
      1(a).......................................  Cover Page; "Summary -- Parties to the
                                                   Exchange Agreement"
      1(b).......................................  Cover Page; "Summary -- Votes Required";
                                                     "General Information -- Voting
                                                     Procedures"
      1(c) and (d)...............................  "Summary -- Market Prices; Dividends"; "The
                                                     Exchange -- Terms of the
                                                     Exchange -- Conduct of the Business of
                                                     the Company"
      1(e) and (f)...............................  *
 2.  IDENTITY AND BACKGROUND
           This Transaction Statement is being filed by Brad Ragan, Inc., the issuer of the
      class of equity securities which is the subject of the Rule 13e-3 transaction (the
      "Company"), and The Goodyear Tire & Rubber Company, an Ohio corporation ("Goodyear").
      The address of Goodyear's principal executive offices is 1144 East Market Street, Akron,
      Ohio 44316. Goodyear currently owns approximately 74.5% of the outstanding shares of
      common stock, par value $1.00 per share, of the Company (the "Common Stock"). The
      transaction (the "Exchange") will be a share exchange between the Company and Goodyear
      pursuant to which all outstanding shares of Common Stock, other than shares already held
      by Goodyear, will be acquired by Goodyear, and each such share (other than shares held
      by dissenting shareholders) will be converted into the right to receive $37.25 in cash
      from Goodyear.
      2(a)-(d)...................................  "Corporate Governance-Nominees for
                                                   Directors"; "Corporate
                                                     Governance -- Executive Officers";
                                                     "Information Concerning Directors and
                                                     Executive Officers of Goodyear"
      2(e).......................................  Neither the Company nor Goodyear nor any of
                                                     their respective executive officers,
                                                     directors or controlling persons have
                                                     been convicted during the past five years
                                                     in a criminal proceeding (excluding
                                                     traffic violations or similar
                                                     misdemeanors)
      2(f).......................................  Neither the Company nor Goodyear nor any of
                                                     their executive officers, directors or
                                                     controlling persons have been a party
                                                     during the last five years to a civil
                                                     proceeding of a judicial or
                                                     administrative body of competent
                                                     jurisdiction and as a result of such
                                                     proceeding were or are subject to a
                                                     judgment, decree or final order enjoining
                                                     further violations of, or prohibiting
                                                     activities subject to, federal or state
                                                     securities laws or finding any violation
                                                     of such laws.
</TABLE>
<PAGE>   3
 
<TABLE>
<CAPTION>
ITEM           CAPTION IN SCHEDULE 13E-3             LOCATION IN PRELIMINARY PROXY STATEMENT
----           -------------------------             ---------------------------------------
<S>   <C>                                          <C>
      2(g).......................................  "Corporate Governance -- Nominees for
                                                     Directors"; "Corporate
                                                     Governance -- Executive Officers";
                                                     "Information Concerning Directors and
                                                     Executive Officers of Goodyear"
 3. PAST CONTACTS, TRANSACTIONS OR NEGOTIATIONS
      3(a).......................................  "Compensation Committee Interlocks and
                                                   Insider Participation"; "Special
                                                     Factors -- Background of, and Reasons
                                                     for, the Exchange"
      3(b).......................................  *
 4. TERMS OF THE TRANSACTION
      4(a).......................................  Cover Page; "Summary -- The Exchange"; "The
                                                     Exchange -- Terms of the Exchange"
      4(b).......................................  Goodyear will not exchange its shares of
                                                   Common Stock pursuant to the Exchange
                                                     Agreement
 5.  PLANS OR PROPOSALS OF THE ISSUER OR
     AFFILIATE
      5(a).......................................  *
      5(b) and 5(c)..............................  "Special Factors -- Purpose of Exchange;
                                                   Plans for the Company"
      5(d).......................................  *
      5(e)-(g)...................................  "Special Factors -- Purpose of Exchange;
                                                   Plans for the Company"
 6. SOURCE AND AMOUNTS OF FUNDS OR OTHER
    CONSIDERATION
      6(a) and (b)...............................  "Summary -- Termination; Fees and
                                                   Expenses"; "Special Factors -- Estimated
                                                     Fees and Expenses"; "The
                                                     Exchange -- Terms of the
                                                     Exchange -- Termination; Fees and
                                                     Expenses"; "The Exchange -- Source and
                                                     Amount of Funds"
      6(c) and (d)...............................  *
 7. PURPOSE(S), ALTERNATIVES, REASONS AND EFFECTS
      7(a)-(c)...................................  "Special Factors -- Background of, and
                                                   Reasons for, the Exchange"; "Special
                                                     Factors -- Purpose of Exchange; Plans for
                                                     the Company"
      7(d).......................................  "Summary -- Parties to the Exchange; The
                                                     Exchange"; "Special Factors -- Purpose of
                                                     the Exchange; Plans for the Company";
                                                     "Special Factors -- Certain Tax
                                                     Consequences of the Exchange"
 8. FAIRNESS OF THE TRANSACTION
      8(a).......................................  "Summary -- Recommendation of the Board;
                                                     Fairness of the Exchange"; "Special
                                                     Factors -- Fairness of the Exchange"
</TABLE>
<PAGE>   4
 
<TABLE>
<CAPTION>
ITEM           CAPTION IN SCHEDULE 13E-3             LOCATION IN PRELIMINARY PROXY STATEMENT
----           -------------------------             ---------------------------------------
<S>   <C>                                          <C>
      8(b).......................................  "Summary -- Recommendation of the Board;
                                                     Fairness of the Exchange"; "Summary --
                                                     Opinion of the Financial Advisor";
                                                     "Special Factors -- Background of, and
                                                     Reasons for, the Exchange"; "Special
                                                     Factors -- Opinion of Financial Advisor"
      8(c).......................................  "Summary -- Votes Required"; General
                                                     Information -- Voting Procedures";
                                                     "Special Factors -- Background of, and
                                                     Reasons for, the Exchange"
      8(d).......................................  "Special Factors -- Fairness of the
                                                   Exchange"
      8(e).......................................  "Summary -- Recommendation of the Board;
                                                     Fairness of the Exchange"; "Special
                                                     Factors -- Background of, and Reasons
                                                     for, the Exchange"; "Special
                                                     Factors -- Fairness of the Exchange"
      8(f).......................................  *
 9. REPORTS, OPINIONS, APPRAISALS AND CERTAIN
    NEGOTIATIONS
      9(a) and (b)...............................  "Summary -- Opinion of Financial Advisor";
                                                     "Special Factors -- Background of, and
                                                     Reasons for, the Exchange"; "Special
                                                     Factors -- Opinion of Financial Advisor"
      9(c).......................................  "Special Factors -- Opinion of Financial
                                                   Advisor"
10.  INTEREST IN SECURITIES OF THE ISSUER
      10(a)......................................  "Summary -- Votes Required"; "Summary --
                                                     Interests of Certain Persons in the
                                                     Exchange"; "Special Factors -- Interests
                                                     of Certain Persons in the Exchange";
                                                     "Security Ownership of Management and
                                                     Certain Beneficial Owners"
      10(b)......................................  "Corporate Governance -- Executive Officers
                                                   of the Company"; "Information Concerning
                                                     Directors and Executive Officers of
                                                     Goodyear"
11. CONTRACTS, ARRANGEMENTS OR UNDERSTANDINGS
    WITH RESPECT TO THE ISSUER'S SECURITIES......  "Summary -- Votes Required"; "Summary --
                                                     Interests of Certain Persons in the
                                                     Exchange"; "General Information -- Voting
                                                     Procedures"; "Special
                                                     Factors -- Interests of Certain Persons
                                                     in the Exchange"; "Special Factors --
                                                     Background of, and Reasons for, the
                                                     Exchange"; "The Exchange -- Proposal to
                                                     Amend the Articles"; "Security Ownership
                                                     of Management and Certain Beneficial
                                                     Owners"
12. PRESENT INTENTION AND RECOMMENDATION OF
    CERTAIN PERSONS WITH REGARD TO THE
    TRANSACTION
      12(a)......................................  "Summary -- Votes Required"; "General
                                                     Information -- Voting Procedures"
      12(b)......................................  Goodyear has not made any recommendation in
                                                     support of or opposed to the Exchange
</TABLE>
<PAGE>   5
 
<TABLE>
<CAPTION>
ITEM           CAPTION IN SCHEDULE 13E-3             LOCATION IN PRELIMINARY PROXY STATEMENT
----           -------------------------             ---------------------------------------
<S>   <C>                                          <C>
13. OTHER PROVISIONS OF THE TRANSACTION
      13(a)......................................  "Summary -- Dissenters' Rights"; "The
                                                     Exchange -- Dissenters' Rights"
      13(b)......................................  *
      13(c)......................................  *
14. FINANCIAL INFORMATION
      14(a)......................................  The information set forth in the Company's
                                                     Annual Report on Form 10-K for the year
                                                     ended December 31, 1997 is incorporated
                                                     herein by reference
      14(b)......................................  *
15. PERSONS AND ASSETS EMPLOYED, RETAINED OR
    UTILIZED
      15(a)......................................  "General Information -- Proxy Solicitation"
      15(b)......................................  *
16.  ADDITIONAL INFORMATION......................  Included throughout the Preliminary Proxy
                                                     Statement
17. MATERIAL TO BE FILED AS EXHIBITS
</TABLE>
 
   
<TABLE>
<CAPTION>
      EXHIBIT
        NO.     DESCRIPTION                              METHOD OF FILING
      -------   -----------                              ----------------
<S>   <C>       <C>                                      <C>
      17(b)(1)  Opinion of Interstate/Johnson Lane
                  Corporation dated February 13,
                  1998.................................  Previously filed
      17(b)(2)  Opinion of Interstate/Johnson Lane
                  Corporation dated May 4, 1998........  Previously filed
      17(b)(3)  Opinion of Interstate/Johnson Lane
                  Corporation dated                ,
                  1998.................................  To be included as Annex II to the
                                                           Preliminary Proxy Statement, filed as
                                                           Exhibit 17(d) hereto
      17(b)(4)  Financial analysis presentation
                  materials prepared by
                  Interstate/Johnson Lane Corporation
                  in connection with providing its
                  opinion to the Special Committee on
                  November 14, 1997....................  To be filed by amendment
      17(b)(5)  Letter, dated December 4, 1997, from
                  Mario J. Gabelli to Robin L.
                  Hinson...............................  Previously filed
      17(b)(6)  Letter, dated December 5, 1997, from
                  Jane E. Mautner to Robin L. Hinson...  Previously filed
      17(b)(7)  Materials prepared by Credit Suisse
                  First Boston ("CFSB") and provided to
                  Goodyear concerning valuation
                  methodology and prior transactions...  Filed herewith
      17(b)(8)  Materials prepared by Securities Data
                  Company and provided to Goodyear by
                  CFSB concerning prior transactions...  To be filed by amendment
</TABLE>
    
<PAGE>   6
 
   
<TABLE>
<CAPTION>
      EXHIBIT
        NO.     DESCRIPTION                              METHOD OF FILING
      -------   -----------                              ----------------
<S>   <C>       <C>                                      <C>
      17(b)(9)  Summary of conversations between
                  representatives of Goodyear and
                  representatives of CFSB during late
                  September and early October 1997.....  Filed herewith
      17(c)     Agreement and Plan of Exchange.........  See Annex I to the Preliminary Proxy
                                                           Agreement, filed as Exhibit 17(d)
                                                           hereto
      17(d)     Preliminary Proxy Statement............  Filed herewith
      17(e)     Article 13 of the North Carolina
                  Business Corporation Act.............  See Annex IV to the Preliminary Proxy
                                                           Statement, filed as Exhibit 17(d)
                                                           hereto
</TABLE>
    
 
---------------
 
* Item not applicable or answer is negative.
<PAGE>   7
 
                                   SIGNATURES
 
     After due inquiry and to the best of our knowledge and belief, the
undersigned certify that the information set forth in this Transaction Statement
is true, complete and correct.
 
                                          BRAD RAGAN, INC.
 
                                          By:        /s/ RONALD J. CARR
 
                                            ------------------------------------
                                          Name: Ronald J. Carr
 
                                              ----------------------------------
                                          Title: Vice President
 
                                             -----------------------------------
   
Date: October 15, 1998
    
 
                                          THE GOODYEAR TIRE & RUBBER COMPANY
 
                                          By:     /s/ GEORGE E. STRICKLER
 
                                            ------------------------------------
                                          Name: George E. Strickler
 
                                              ----------------------------------
                                          Title: Vice President
 
                                             -----------------------------------
   
Date: October 15, 1998
    
