UNIVERSITY OF CALIFORNIA, LOS ANGELES
Department of Economics

Economics 143 (Cameron) - Applied Regression Analysis

Classroom Handout #8: Guided Tour of Regression Output


If you are looking at this handout using a Web browser, I have set it up so that as you move the mouse pointer over a link, a box should pop up in the upper left corner of your screen which contains a brief explanation of each underlined element of the SHAZAM output. This is accomplished via JavaScript. WARNING: We have noticed that after a while, an error message may start to appear ("hintwin has no properties"). We haven't managed to trace the source of this error, but exiting the browser and starting over again seems to clear it. We'll keep working on this. SUGGESTION: If the pop-up window is too small to show the entire text of the description, you can right click on the main window to keep the pop-up from disappearing, then move to the pop-up box and resize or move it as desired.

THE SCENARIO: We have 14 observations on INCOME (the income of a college student) and EXPEND (the consumption expenditures of each student). Suppose we are interested in knowing, on average, the approximate relationship beween EXPEND and the INCOME that makes it possible. Thus EXPEND is the dependent (Y) variable and INCOME is the explanatory (X) variable. First, we will want to be able to express E[Y] as a linear function of X. This requires using OLS to find the parameters b1 and b2 such that E[Y] = b1 + b2(X). Or, we could refer to E[EXPEND] = b1 + b2 (INCOME).

Once the model is estimated, we might be interested in some hypotheses.

1. If the true but unknown slope B2 (for which b2 is an estimate) is actually zero, then changes in income X have no effect on the expected value of consumption expenditure Y. If B2 is zero, then, there is no linear relationship between X and Y and they are linearly independent.

2. If the true but unknown value of the intercept B1 is zero, then the expected value of Y is zero when X is zero.

3. We might want to know if, on average, when a student has one more dollar, they spend ALL of it on consumption. This would be a test of the hypothesis that the true slope B2 is exactly one. 

 |_sample 1 14
 |_read(mpc.dat) income expend
 UNIT 88 IS NOW ASSIGNED TO: mpc.dat
    2 VARIABLES AND       14 OBSERVATIONS STARTING AT OBS       1
 
 |_stat / pcor
 NAME        N   MEAN        ST. DEV      VARIANCE     MINIMUM      MAXIMUM
 INCOME      14   155.00       41.278       1703.8       100.00       220.00
 EXPEND      14   125.57       35.224       1240.7       80.000       180.00
 
  CORRELATION MATRIX OF VARIABLES -       14 OBSERVATIONS
 
 INCOME     1.0000
 EXPEND    0.84385       1.0000
              INCOME       EXPEND
 
 |_ols expend income
 
 REQUIRED MEMORY IS PAR=     1 CURRENT PAR=   500
  OLS ESTIMATION
       14 OBSERVATIONS     DEPENDENT VARIABLE = EXPEND
 ...NOTE..SAMPLE RANGE SET TO:      1,     14
 
  R-SQUARE =   0.7121     R-SQUARE ADJUSTED =   0.6881
 VARIANCE OF THE ESTIMATE-SIGMA**2 =   387.00
 STANDARD ERROR OF THE ESTIMATE-SIGMA =   19.672
 SUM OF SQUARED ERRORS-SSE=   4644.0
 MEAN OF DEPENDENT VARIABLE =   125.57
 LOG OF THE LIKELIHOOD FUNCTION = -60.4950
 
                      ANALYSIS OF VARIANCE - FROM MEAN
                       SS         DF             MS                 F
 REGRESSION        11485.          1.        11485.                29.678
 ERROR             4644.0         12.        387.00               P-VALUE
 TOTAL             16129.         13.        1240.7                 0.000
 
                      ANALYSIS OF VARIANCE - FROM ZERO
                       SS         DF             MS                 F
 REGRESSION       0.23224E+06      2.       0.11612E+06           300.052
 ERROR             4644.0         12.        387.00               P-VALUE
 TOTAL            0.23688E+06     14.        16920.                 0.000
 
 
 VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
   NAME    COEFFICIENT   ERROR      12 DF   P-VALUE CORR. COEFFICIENT  AT MEANS
 INCOME    0.72009     0.1322       5.448     0.000 0.844     0.8438     0.8888
 CONSTANT   13.957      21.15      0.6599     0.522 0.187     0.0000     0.1112

 |_test income=0
 TEST VALUE =  0.72009     STD. ERROR OF TEST VALUE  0.13218
 T STATISTIC =   5.4477715     WITH   12 D.F.    P-VALUE= 0.00015
 F STATISTIC =   29.678214     WITH    1 AND   12 D.F.  P-VALUE= 0.00015
 WALD CHI-SQUARE STATISTIC =   29.678214     WITH    1 D.F.  P-VALUE= 0.00000
 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.03369

 |_test income=1
 TEST VALUE = -0.27991     STD. ERROR OF TEST VALUE  0.13218
 T STATISTIC =  -2.1176290     WITH   12 D.F.    P-VALUE= 0.05577
 F STATISTIC =   4.4843528     WITH    1 AND   12 D.F.  P-VALUE= 0.05577
 WALD CHI-SQUARE STATISTIC =   4.4843528     WITH    1 D.F.  P-VALUE= 0.03421
 UPPER BOUND ON P-VALUE BY CHEBYCHEV INEQUALITY = 0.22300

 |_stop

THE FINDINGS: Above, we identified three interesting hypotheses that might be tested using these data. What have we determined by our regression analysis?

1. The sample evidence strongly suggests that there IS a relationship between income and consumption expenditures, and it is a positive relationship. We can easily reject the hypothesis of a zero slope, since the t-test statistic value is far greater than the critical value of the t-distribution with 12 degrees of freedom. We can look at the tables in the back of the text to determine that this critical value is 2.179, and our test value is 5.448. However, you can see that the "P-VALUE" for this test statistic gives you exactly the information you need without having to go to the t-tables in the textbook. RECIPE: if the P-VALUE in the OLS output is LESS THAN 0.05, then your test statistic is further out in the tails than the symmetric 5% cutoff values of the appropriate t-distribution, and you can REJECT the null hypothesis of a zero coefficient at the 5% level of significance. (If the P-VALUE was, say, 0.0821, then you could reject at the 10 percent level, but not at the 5% level. In fact, you would "just reject" at the 8.21% level of significance.) Notice that we get tests of the hypotheses "coefficient is zero" automatically with every regression. We can also ask for them explicitly with the TEST command.

2. The sample evidence suggests that if income is zero, consumption expenditures are also zero. We cannot reject the hypothesis of a zero intercept for this regression model. The t-test statistic is less than one in absolute value, which makes it a perfectly plausible value of a t distribution with 12 degrees of freedom if the true intercept is zero. The sample evidence about B1 is consistent with the hypothesis. We cannot reject the hypothesis that the true B1 is zero at any normal level of significance. The P-value tells us that in order to reject, we would have to go to the 52.2% level of significance...that means we would have to be willing to accept a 52.2% chance of making a mistake by electing to reject the hypothesis that B1=0. However, all this is probably actually moot, because there are no observations anywhere near X=0 (INCOME=0). Wherever the regression line goes as it crosses the vertical axis is just an accident of using a straight-line to fit the data. We have no evidence in the sample at all about actual behavior when income gets very small. Intuitively, there must be some subsistence consumption necessary even if no income is present.

3. The sample evidence suggests that giving a student and extra dollar will result in ALL of that dollar being spent on consumption (implicitly, no savings). Algebraically, this translates to the slope of the regression line being one. The point estimate for the slope is less than one (at 0.72). However, this estimate is a random variable with a distribution. Hypotheses about the slope and intercept coefficients OTHER THAN the basic zero hypotheses are NOT produced automatically by every OLS regression. However, you can certainly use the TEST command to perform tailor-made hypothesis tests about the coefficients in a regression model. Each coefficient is referred to in the TEST command by using the variable it modifies (including the CONSTANT which is a variable that always takes on a value of 1). Testing whether true slope B2=1 is accomplished by using TEST INCOME=1. The t-test statistic ends up being pretty far from zero--its expected value if the hypothesis is true--but not far enough to reject the null hypothesis at the 5% level. The P-VALUE for the test is .05577, which means that we can only reject at the 5.577% level, but not at the usual 5% level. In words, then, it is possible that for every extra dollar you give a college student, they will spend all of it on consumption.
Updated: 8:23 AM 10/20/98; Prepared by: Trudy Ann Cameron; Site Index