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 , ... , e_n | b₀, b₁) = ln{f(e₁ , e₂ , ... , e_n | b₀, b₁)}
   
   where in this example
   
   e_i = y_i - b₀ - b₁x_i and e_i ~ 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.