1. Undergraduated Econometrics
Chapter 6
The Simple Linear Regression Model: Reporting the Results and choosing the Functional Form
Undergraduated Econometrics
Chapter 6: The Simple Linear Regression Model: Reporting the Results and choosing the Functional Form
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2. Undergraduated Econometrics
Chapter Contents
6.1 The Coefficient of Determination
6.2 Reporting Regression Results
6.3 Choosing a Functional Form
6.4 Are the Residuals Normally Distributed
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3. The Coefficient of Determination
6.1
The Coefficient of Determination
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4. There are two major reasons for analyzing the model
6.1
The Coefficient of Determination
There are two major reasons for analyzing the model
（6.1.1）
to explain how the dependent variable (yi) changes as the independent variable (xi) changes
to predict y0 given an x0
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5. 6.1
The Coefficient of Determination
Closely allied with the prediction problem is the desire to use xi to explain as much of the variation in the dependent variable yi as possible.
In the regression model Eq we call xi the ‘‘explanatory’’ variable because we hope that its variation will ‘‘explain’’ the variation in yi
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6. E(yi) is the explainable or systematic part
6.1
The Coefficient of Determination
To develop a measure of the variation in yi that is explained by the model, we begin by separating yi into its explainable and unexplainable components.
E(yi) is the explainable or systematic part
ei is the random, unsystematic and unexplainable component
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7. Analogous to Eq. 4.8, we can write: （6.1.3）
The Coefficient of Determination
Analogous to Eq. 4.8, we can write:
（6.1.3）
Subtracting the sample mean from both sides:
（6.1.4）
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8. Figure 6.1 Explained and unexplained components of yi
The Coefficient of Determination
Figure 6.1 Explained and unexplained components of yi
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9. 6.1
The Coefficient of Determination
Squaring and summing both sides of Eq , and using the fact that we get:
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10. These are called ‘‘sums of squares’’
6.1
The Coefficient of Determination
Eq decomposition of the ‘‘total sample variation’’ in y into explained and unexplained components
These are called ‘‘sums of squares’’
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11. Specifically: 6.1 The Coefficient of Determination
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12. Thus Eq. 6.1.5 becomes: 6.1 The Coefficient of Determination
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13. 6.1
The Coefficient of Determination
Let’s define the coefficient of determination, or R2 , as the proportion of variation in y explained by x within the regression model:
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14. 6.1
The Coefficient of Determination
We can see that:
The closer R2 is to 1, the closer the sample values yi are to the fitted regression equation
If R2 = 1, then all the sample data fall exactly on the fitted least squares line, so SSE = 0, and the model fits the data ‘‘perfectly’’
If the sample data for y and x are uncorrelated and show no linear association, then the least squares fitted line is ‘‘horizontal,’’ and identical to y, so that SSR = 0 and R2 = 0
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15. 6.1
The Coefficient of Determination
When 0 < R2 < 1 then R2 is interpreted as ‘‘the proportion of the variation in y about its mean that is explained by the regression model’’
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16. For the food expenditure example, the sums of squares are:
6.1
The Coefficient of Determination
6.1.1
Analysis of Variance Table and R2 for Food Expenditure Example
For the food expenditure example, the sums of squares are:
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17. 6.1
The Coefficient of Determination
6.1.1
Analysis of Variance Table and R2 for Food Expenditure Example
Therefore:
We conclude that 31.7% of the variation in food expenditure (about its sample mean) is explained by our regression model, which uses only income as an explanatory variable
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18. The correlation coefficient ρxy between x and y is defined as:
6.1
The Coefficient of Determination
6.1.2
Correlation Analysis
The correlation coefficient ρxy between x and y is defined as:
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19. r= cov ^ (𝑋,𝑌) var ^ (𝑋) var ^ (𝑌) (6.1.9)
The Coefficient of Determination
6.1.2
Correlation Analysis
Given a sample of data pairs (xt, yt), t=1,…, T, the sample correlation coefficient is obtained by replacing the covariance and variances in (6.1.8) by their sample analogues:
r= cov ^ (𝑋,𝑌) var ^ (𝑋) var ^ (𝑌) (6.1.9)
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20. Where So we can write r as 6.1 The Coefficient of Determination 6.1.2
Correlation Analysis
Where
So we can write r as
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21. Two relationships between R2 and rxy: r2xy = R2
6.1
The Coefficient of Determination
6.1.3
Correlation Analysis and R2
Two relationships between R2 and rxy:
r2xy = R2
R2 can also be computed as the square of the sample correlation coefficient between yi and
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22. Reporting Regression Results
6.2
Reporting Regression Results
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23. The key ingredients in a report are: the coefficient estimates
6.2
Reporting Regression Results
The key ingredients in a report are:
the coefficient estimates
the standard errors (or t-values)
an indication of statistical significance R2
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24. 6.2
Reporting Regression Results
One way to summarize the regression results is in the form of a “fitted” regression equation:
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25. Possible effects of scaling the data:
6.2
Reporting Regression Results
6.2.1
The Effects of Scaling the Data
Possible effects of scaling the data:
Changing the scale of x: the coefficient of x must be multiplied by c, the scaling factor
When the scale of x is altered, the only other change occurs in the standard error of the regression coefficient, but it changes by the same multiplicative factor as the coefficient, so that their ratio, the t-statistic, is unaffected
All other regression statistics are unchanged
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26. Possible effects of scaling the data (Continued):
6.2
Reporting Regression Results
6.2.1
The Effects of Scaling the Data
Possible effects of scaling the data (Continued):
Changing the scale of y: If we change the units of measurement of y, but not x, then all the coefficients must change in order for the equation to remain valid
Because the error term is scaled in this process the least squares residuals will also be scaled
This will affect the standard errors of the regression coefficients, but it will not affect t-statistics or R2
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27. Possible effects of scaling the data (Continued):
6.2
Reporting Regression Results
6.2.1
The Effects of Scaling the Data
Possible effects of scaling the data (Continued):
Changing the scale of y and x by the same factor: there will be no change in the reported regression results for b2 , but the estimated intercept and residuals will change
t-statistics and R2 are unaffected.
The interpretation of the parameters is made relative to the new units of measurement.
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28. Choosing a Function Form
6.3
Choosing a Function Form
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29. 6.3
Choosing a Function Form
By “linear” in the parameters” we mean that the parameters are not multiplied together, divided, squared, cubed, and so on. The variables, however, can be transformed in any convenient way, as long as the resulting model satisfies assumptions SR1-SR5 of the simple linear regression model.
Economic theory does not always imply that there is a linear relationship.
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30. 6.3
Choosing a Function Form
Figure 6.2 A nonlinear relationship between food expenditure and income
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31. Reciprocal: if x is a variable, then its reciprocal is 1/x
6.3
Choosing a Function Form
6.3.1
Some commonly used Functional Forms
Choosing an algebraic form for the relationship means choosing transformations of the original variables. The variable transformations that we begin with are:
Natural logarithm: If x is a variable, then its natural logarithm is ln(x)
Reciprocal: if x is a variable, then its reciprocal is 1/x
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32. Figure 6.3 Alternative functional forms
Choosing a Function Form
Figure 6.3 Alternative functional forms
6.3.1
Some commonly used Functional Forms
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33. Summary of three configurations:
6.3
Choosing a Function Form
6.3.1
Some commonly used Functional Forms
Summary of three configurations:
The model that is linear in the variables describes fitting a straight line to the original data, with slope β2 and point elasticity β2xt/yt The slope of the relationship is constant, but the elasticity changes at each point.
The reciprocal model takes shapes shown in Figure 6.3a. As x increase, y approaches the intercept, its asymptote, from above or below depending on the sign of β2.
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34. Summary of three configurations (continue):
6.3
Choosing a Function Form
6.3.1
Some commonly used Functional Forms
Summary of three configurations (continue):
3. In the log-log model all values of y and x must be positive. The slopes of these curves change at every point, but the elasticity is constant and equal to β2.
4. In the log-linear model only the dependent variable is transformed by the logarithm.
5. In the linear-log model the variable x is transformed by the natural logarithm.
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35. The reciprocal model is
6.3
Choosing a Function Form
6.3.2
Examples Using Alternative Functional Forms
Suppose that in the food expenditure model we wish to choose a functional form that is consistent with figure 6.2 .
The reciprocal model is
The linear-log model is
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36. The reciprocal model is
6.3
Choosing a Function Form
6.3.2
Examples Using Alternative Functional Forms
Suppose that in the food expenditure model we wish to choose a functional form that is consistent with figure 6.2 .
The reciprocal model is
The linear-log model is
These results are consistent with the idea that a high incomes, and large food expenditures, the effect of an increase in income on food expenditure is small.
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37. Some other economic models and functional forms
6.3
Choosing a Function Form
6.3.2
Examples Using Alternative Functional Forms
Some other economic models and functional forms
1. Demand models: statistical models of the relationship between quantity demanded (yd) and price (x) very frequently taken to be linear in the variables.
2. Supply models: if ys is the quantity supplies, then its relationship to price is often assumed to be linear, creating a linear supply curve.
3. Production functions: another basic economic relationship that can be investigated using the simple linear statistical model is a production function that relates the output of a good produced to the amount of a variable input.
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38. Some other economic models and functional forms (continued)
6.3
Choosing a Function Form
6.3.2
Examples Using Alternative Functional Forms
Some other economic models and functional forms (continued)
4. Cost functions: a family of cost curves, which can be estimated using the simple linear regression model, is based on a “quadratic” total cost curve.
5. The Phillips curve: this important relationship conjectures a systematic relationship between changes in the wage rate and changes in the level of unemployment.
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39. Figure 6.4 Scatter plot of wheat yield over time
6.3
Choosing a Function Form
Figure 6.4 Scatter plot of wheat yield over time
6.3.3
Empirical Issues
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40. One problem with the linear equation
6.3
Choosing a Function Form
6.3.3
Empirical Issues
One problem with the linear equation
is that it implies that yield increases at the same constant rate β2, when, from Figure 6.4, we expect this rate to be increasing
The least squares estimated equation is:
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41. Figure 6.5 Predicted, actual and residual values from straight line
6.3
Choosing a Function Form
Figure 6.5 Predicted, actual and residual values from straight line
6.3.3
Empirical Issues
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42. Perhaps a better model would be:
6.3
Choosing a Function Form
6.3.3
Empirical Issues
Perhaps a better model would be:
But note that the values of x t3 can get very large
This variable is a good candidate for scaling. Define zt = x t3/
The least squares fitted line is:
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43. Figure 6.6 Bar chart of residuals from straight line
6.3
Choosing a Function Form
Figure 6.6 Bar chart of residuals from straight line
6.3.3
Empirical Issues
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44. Are the Residuals Normally Distributed?
6.4
Are the Residuals Normally Distributed?
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45. Are they normally distributed?
6.4
Are the Residuals Normally Distributed?
Hypothesis tests and interval estimates for the coefficients rely on the assumption that the errors, and hence the dependent variable y, are normally distributed
Are they normally distributed?
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46. We can check the distribution of the residuals using: A histogram
6.4
Are the Residuals Normally Distributed?
We can check the distribution of the residuals using:
A histogram
Formal statistical test
Merely checking a histogram is not a formal test
Many formal tests are available
A good one is the Jarque–Bera test for normality
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47. 6.4
Are the Residuals Normally Distributed?
Figure 6.8 EViews output: residuals histogram and summary statistics for food expenditure
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48. Skewness refers to how symmetric the residuals are around zero
6.4
Are the Residuals Normally Distributed?
The Jarque–Bera test for normality is based on two measures, skewness and kurtosis
Skewness refers to how symmetric the residuals are around zero
Perfectly symmetric residuals will have a skewness of zero
The skewness value for the food expenditure residuals is 0.397
Kurtosis refers to the ‘‘peakedness’’ of the distribution.
For a normal distribution the kurtosis value is 3
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49. The Jarque–Bera statistic is given by:
6.4
Are the Residuals Normally Distributed?
The Jarque–Bera statistic is given by:
where
N = sample size
S = skewness
K = kurtosis
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50. 6.4
Are the Residuals Normally Distributed?
When the residuals are normally distributed, the Jarque–Bera statistic has a chi-squared distribution with 2 degrees of freedom
We reject the hypothesis of normally distributed errors if a calculated value of the statistic exceeds a critical value selected from the chi-squared distribution with two degrees of freedom
The 5% critical value from a χ2-distribution with 2 degrees of freedom is 5.99.
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51. For the food expenditure example, the Jarque–Bera statistic is:
6.4
Are the Residuals Normally Distributed?
For the food expenditure example, the Jarque–Bera statistic is:
Because < 5.99 there is insufficient evidence from the residuals to conclude that the normal distribution assumption is unreasonable at the 5% level of significance
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