REGRESI SEDERHANA BAGIAN 2 Ekonometrika 1 Al Muizzuddin F

REVIEW BAGIAN 1 Menentukan nilai koefisien bo dan b1 Asumsi dalam regresi OLS

A Measure of “Goodness of Fit” THE COEFFICIENT OF DETERMINATION (R2) we shall find out how “well” the sample regression line fits the data Figure A : Venn diagram

In this figure the circle Y represents variation in the dependent variable Y and the circle X represents variation in the explanatory variable X. When there is no overlap, R2 is obviously zero, but when the overlap is complete, R2 is 1, since 100 percent of the variation in Y is explained by X. As we shall show shortly, R2 lies between 0 and 1.

Analysis of Variance The total variability in a regression analysis, SST, can be partitioned into a component explained by the regression, SSR, and a component due to unexplained error, SSE

With the components defined as, Total sum of squares : Error sum of squares : Regression sum of squares :

Measure of Coefficient of Determination, R2 The Coefficient of Determination for a regression equation is defined as

Some Examples EXAMPLE 1 : HYPOTHETICAL DATA ON WEEKLY FAMILY CONSUMPTION EXPENDITURE (Y) AND WEEKLY FAMILY INCOME (X)

The estimated regression line is The value of β2 = 0.5091, which measures the slope of the line, shows that, within the sample range of X between $80 and $260 per week, as X increases, say, by $1, the estimated increase in the mean or average weekly consumption expenditure amounts to about 51 cents. Y = 24,4545 + 0,5091 Xi R2 = 0,9621

The value of β1 = 24.4545, which is the intercept of the line, indicates the average level of weekly consumption expenditure when weekly income is zero. The value of R2 of 0.9621 means that about 96 percent of the variation in the weekly consumption expenditure is explained by income. Since R2 can at most be 1, the observed R2 suggests that the sample regression line fits the data very well.

EXAMPLE 2 : FOOD EXPENDITURE IN INDIA The data relate to a sample of 55 rural households in India. The regressand in this example is expenditure on food and the regressor is total expenditure, a proxy for income, both figures in rupees. The data in this example are thus cross-sectional data.

FoodExpi = 94,20 + 0,43 TotalExpi If total expenditure increases by 1 rupee, on average, expenditure on food goes up by about 44 paise (1 rupee = 100 paise). If total expenditure were zero, the average expenditure on food would be about 94 rupees. The R2 value of about 0.37 means that only 37 percent of the variation in food expenditure is explained by the total expenditure.

EXAMPLE 3 : THE RELATIONSHIP BETWEEN EARNINGS AND EDUCATION The data relating average hourly earnings and education, as measured by years of schooling. Using that data, if we regress average hourly earnings (Y) on education (X), we obtain the following results. Yi = -0,0144 + 0,7241 Xi

As the regression results show, there is a positive association between education and earnings, an unsurprising finding. For every additional year of schooling, the average hourly earnings go up by about 72 cents an hour. The intercept term is positive but it may have no economic meaning. The R2 value suggests that about 89 percent of the variation in average hourly earnings is explained by education. For cross-sectional data, such a high R2 is rather unusual.

Basis for Inference About the Population Regression Slope Let 1 be a population regression slope and b1 its least squares estimate based on n pairs of sample observations. Then, if the standard regression assumptions hold and it can also be assumed that the errors i are normally distributed.

Excel Output for Retail Sales Model The regression equation is Y Retail Sales = 1922 + 0.382 X Income

Tests of the Population Regression Slope If the regression errors i are normally distributed and the standard least squares assumptions hold (or if the distribution of b1 is approximately normal), the following tests have significance value : To test either null hypothesis against the alternative the decision rule is

2. To test either null hypothesis against the alternative the decision rule is

3. To test the null hypothesis Against the two-sided alternative the decision rule is

F test for Simple Regression Coefficient We can test the hypothesis against the alternative By using the F statistic The decision rule is We can also show that the F statistic is For any simple regression analysis.

Conclusion Coefficient intercept and slope R-squared F-test t-tes

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