45-733 PROBABILITY AND STATISTICS I



XIX. Relationships With Prob-Stat II


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Dependent Variable: GNP                                               
Method: Least Squares                                                 
Date: 02/25/99   Time: 15:49                                          
Sample: 1915 1988                                                     
Included observations: 74                                             
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     Variable      CoefficientStd. Errort-Statistic  Prob.            
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         C           3.025348   0.547696   5.523772   0.0000 (1)
      MILMOB         3.697863   0.547363   6.755782   0.0000          
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R-squared (2)        0.387966    Mean dependent var 3.061841 (3)
Adjusted R-squared   0.379466    S.D. dependent var 5.980692 (4)
S.E. of regression   4.711230    Akaike info criter 5.964430          
Sum squared resid    1598.089    Schwarz criterion  6.026702          
Log likelihood (5)  -218.6839    F-statistic        45.64060 (6)
Durbin-Watson stat   1.266900    Prob(F-statistic)  0.000000 (7)
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  1. This is the two-tail P-Value for the null hypothesis in the hypothesis test:
    
         H0: b0 = 0
         H1: b0 ¹ 0
          where
                  Ù        Ù
            (b0 - b0)/(VAR(b0)1/2
    
    has at t-distribution with n-k-1 degrees of freedom (k = number of independent variables excluding the constant). The test statistic here is just the coefficient value divided by the standard error:

    Test Statistic = 3.025348/.547696 = 5.523772

    If you issued the EViews command:

    Scalar PVal=@TDIST(5.523772,72)

    you would get the two-tail P-Value = .000000499.

    Hence, in EViews -- as is the case with almost all stat packages -- the column labeled "Prob" is simply the two-tail P-Values for the null hypothesis that the corresponding coefficient is equal to zero. It is then up to you to interpret this substantively!

  2. R-squared. This is literally the squared correlation coefficient:
                    Ù                 Ù
      r2 = COV(Y,Y)2/[VAR(Y)VAR(Y)],  where
                    Ù
      Y = GNP  and  Y is the estimated GNP based on the above equation:
       Ù
      GNP = 3.025348 + 3.697863*MILMOB
    
  3. Mean dependent var. This is literally the sample mean discussed in class:
          _ 
          Yn = Si=1,nYi/n
    
       Note that the sample size, n, here is equal to 74.
  4. S.D. dependent var: This is the unbiased estimator formula discussed in class:
                          _  
       sy = {[åi=1,n (yi - Yn)2]/(n -1)}1/2
    
  5. Log likelihood. This is the value of the log of the likelihood function:

    L(e1 , e2 , ... , en | b0, b1) = ln{f(e1 , e2 , ... , en | b0, b1)}

    where in this example

    ei = yi - b0 - b1xi and ei ~ N(0, s2)

    Here y = GNP and x = MILMOB. The idea is to find estimates of the coefficients -- the bs -- that maximize the likelihood function.

  6. F-statistic. This is the overall F-Statistic of the regression. It is the ratio:

    [r2/k]/[(1 - r2)/(n - k - 1)]

    Where r2 is the R-squared of the regression explained in point (1) above; k = number of independent variables (excluding the constant or intercept term); and n = sample size.

  7. This is the upper-tail P-Value for the F-Statistic.