In Figure I.4 of the Introduction we sketched the anatomy of econometric modeling. Now that we have presented the results of regression analysis of our consumption-income example in (5.11.1), we would like to question the adequacy of the fitted model. How "good" is the fitted model? We need some criteria with which to answer this question.

First, are the signs of the estimated coefficients in accordance with theoretical or prior expectations? A priori, ft, the marginal propensity to consume (MPC) in the consumption function, should be positive. In the present example it is. Second, if theory says that the relationship should be not only positive but also statistically significant, is this the case in the present application? As we discussed in Section 5.11, the MPC is not only positive but also statistically significantly different from zero; the p value of the estimated t value is extremely small. The same comments apply about the intercept coefficient. Third, how well does the regression model explain variation in the consumption expenditure? One can use r2 to answer this question. In the present example r2 is about 0.96, which is a very high value considering that r2 can be at most 1.

Thus, the model we have chosen for explaining consumption expenditure behavior seems quite good. But before we sign off, we would like to find out

5. Two-Variable Regression: Interval Estimation and Hypothesis Testing

CHAPTER FIVE: TWO VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING 147

whether our model satisfies the assumptions of CNLRM. We will not look at the various assumptions now because the model is patently so simple. But there is one assumption that we would like to check, namely, the normality of the disturbance term, ui . Recall that the t and F tests used before require that the error term follow the normal distribution. Otherwise, the testing procedure will not be valid in small, or finite, samples.

Although several tests of normality are discussed in the literature, we will consider just three: (1) histogram of residuals; (2) normal probability plot (NPP), a graphical device; and (3) the Jarque-Bera test.

Histogram of Residuals. A histogram of residuals is a simple graphic device that is used to learn something about the shape of the PDF of a random variable. On the horizontal axis, we divide the values of the variable of interest (e.g., OLS residuals) into suitable intervals, and in each class interval we erect rectangles equal in height to the number of observations (i.e., frequency) in that class interval. If you mentally superimpose the bell-shaped normal distribution curve on the histogram, you will get some idea as to whether normal (PDF) approximation may be appropriate. A concrete example is given in Section 5.13 (see Figure 5.8). It is always a good practice to plot the histogram of the residuals as a rough and ready method of testing for the normality assumption.

Normal Probability Plot. A comparatively simple graphical device to study the shape of the probability density function (PDF) of a random variable is the normal probability plot (NPP) which makes use of normal probability paper, a specially designed graph paper. On the horizontal, or x, axis, we plot values of the variable of interest (say, OLS residuals, ui ), and on the vertical, or y, axis, we show the expected value of this variable if it were normally distributed. Therefore, if the variable is in fact from the normal population, the NPP will be approximately a straight line. The NPP of the residuals from our consumption-income regression is shown in Figure 5.7, which is obtained from the MINITAB software package, version 13. As noted earlier, if the fitted line in the NPP is approximately a straight line, one can conclude that the variable of interest is normally distributed. In Figure 5.7, we see that residuals from our illustrative example are approximately normally distributed, because a straight line seems to fit the data reasonably well.

MINITAB also produces the Anderson-Darling normality test, known as the A2 statistic. The underlying null hypothesis is that the variable under consideration is normally distributed. As Figure 5.7 shows, for our example, the computed A2 statistic is 0.394. The p value of obtaining such a value of A2 is 0.305, which is reasonably high. Therefore, we do not reject the

Normality Tests

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Testing

148 PART ONE: SINGLE-EQUATION REGRESSION MODELS

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