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### 45-733 PROBABILITY AND STATISTICS I

#### XIX. Relationships With Prob-Stat II

```
============================================================
Dependent Variable: GNP
Method: Least Squares
Date: 02/25/99   Time: 15:49
Sample: 1915 1988
Included observations: 74
============================================================
Variable      CoefficientStd. Errort-Statistic  Prob.
============================================================
C           3.025348   0.547696   5.523772   0.0000 (1)
MILMOB         3.697863   0.547363   6.755782   0.0000
============================================================
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)
============================================================
```
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.