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14 following nonfinancial markets have close relationships that can be found using regression analjsis. MULTIVARIATE APPROXIIVLTIONS Regression analjsis is most often used in complex economic models to find the combination of two or more independent variables that best explain or forecast prices. A simple application of annual production and distribution of sojlseans will determine whether these factors are significant in determination of sojlsean prices. Because the demand for sojlseans and its products is complex, a high correlation should not be expected between the data. However, there is no way of knowing how much impact other factors have on the prices over a long term. Using the method of least squares, which was enployed for a simple linear regression, the new equation is As in the linear approximation, the solution to this problem will be found by minimize ing the sum of the squares of the errors at each point The solution to the multivariate problem ot two independem variables x, and requires the fbllobint; three leastsquares equations: The procedure for solving the three simultaneous equations is the same as the curvilinear method of coefficient elimination. The sums are calculated in Table 36, then subatituted into the last three equations: The coefficient matrix solution is4 to tiie multivariate rTtleni Appenli:: 3 contains tiie c. luputer progi TABLE 36 Totals for Multivariate Solution Soybeans Supply Demand
n 239 1V22/ The results show = I 641. t> of Ihe price 1 I 641 + .970 , + .8183.11 of tl tl tl t iry wl n a ll. answ<?r In h examfil , ..upp yantltl and fisu pi hue I  h . upply and inflation 41 " To nn<J out wh ch ts of  ul COEiapurec] lUmhe ( 111 Ik iilpul as pulenlially rcrliv: I of supply Is alwa>s four cimes greattr daia rclailonililps r* in 3mp" problem of supply and n that fixes tb relatlonslilp 1 prnhl 1 IS tbe primary fectur Neural Selecting Data for an S&P Model The stock maitet is driven by a healthy economy and by lower interest rates. A good economy means thai there IS high employment and consumers are actively bujing homes, durable goods, and frivolous items. Low interest rates result in more co orate profitability, which is also influenced by confrolled growth and low inflation, both delicate issues orchestrated by the Cenfral Bant. To create a robust S&P model, whether using multiple regression or, as we will discuss later, neural networks, it is necessary to select the most meaningful data. The following was suggested by Lincoln to be used for a 6month S&P forecast:5 1. S&Ppricesthe closing prices of the Standard & Poors 500 cash index 2. Co orate bondsTreasurj bondsthe BAA co orate bond yield divided by the 30year Treasurj bond yield normalized by subtracting the historical mean 3. Annual change in the dollarthe 12month change in the dollar, minus 1, which might be based on the Dollar Index traded on the New York Futures Exchange, or a comparable weighting of major currencies 4. Annual change in Federal funds ratethe 12month change in the Fed funds rate, minus 1 5. Federal funds ratcdiscount ratethe Fed funds rate divided by the discount rate, normalized by subtraaing the historical mean
6. Money 51 1  1 money simply, not seasonally adjusted 7. Annual Consumer Price Indexthe 12month change in the CPl, minus 1 8. Inflation/disinflation indexthe annualized 1month change in the CPl divided by the 12month change in the CPL minus 1 9. Leading economic indicatorsthe 12month change in the leading economic indicators, minus 1 10. Onemonth versus 10month oscillator for the S&P cadi indexthe difference between the monthly average and the past 10 months, approximately 200 dajs 11. Inflationadjusted commercial loansthe inflationadjusted 12month growth in commercial loans Lincoln used 20 years of monthly values to forecast the S&P price 6 months ahead. Some of this data is available on a weekly basis and might be adapted to a shorter time frame; however, it is unreasonable to think that an accurate daily forecast is possible using weekly or monthly data. It is also likely that the use of more frequent S&P price data is inconsistent with the monthly statistics and will infroduce more noise and make the results less reliable. One element that is missing from the 11 items listed is a volatility adjustment. There has been a 600*> increase in the S&P price over 20 years, and volatility is clearly much higher. A simple percentage relationship may not describe this change adequately, but may be used until a better one is subatituted. Therefore, those items that become more or less volatile as prices move higher and lower must be corrected using a volatilitynormalizing factor. For 1 , if we consider the initial volatility when the S&P was 100 at the beginning of the data as normal, then when the price is 800 we will divide the current volatility by 8 to normalize, or even take the square root if the increase in volatility is nonlinear. This tjpe of adjustment would apply to nearly all the items except money supply and the inflation/disinflation index. Dont forget that, where applicable, yields must always be used, not prices. Th.JuasH Lincoln, "Time SenesF.TeciHug AlJJjH:," TecLuical Aualysis ..f c>..ck.andCumodities .tember 1991) Generalized Multivariate in general, the relationship between n independent variables is expressed as
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