1. EC220 - Introduction to econometrics (chapter 1)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 1)
Slideshow: deriving linear regression coefficients
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 1). [Teaching Resource]
© 2012 The Author
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Available in LSE Learning Resources Online: May 2012
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2. DERIVING LINEAR REGRESSION COEFFICIENTS
True model: Y X
This sequence shows how the regression coefficients for a simple regression model are derived, using the least squares criterion (OLS, for ordinary least squares) 1

3. DERIVING LINEAR REGRESSION COEFFICIENTS
True model: Y X
We will start with a numerical example with just three observations: (1,3), (2,5), and (3,6). 2

4. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line: Y b2 b1 X
Writing the fitted regression as Y = b1 + b2X, we will determine the values of b1 and b2 that minimize RSS, the sum of the squares of the residuals. ^ 3

5. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line: Y b2 b1 X
Given our choice of b1 and b2, the residuals are as shown. 4

6. SIMPLE REGRESSION ANALYSIS
The sum of the squares of the residuals is thus as shown above. 5

7. SIMPLE REGRESSION ANALYSIS
The quadratics have been expanded. 6

8. SIMPLE REGRESSION ANALYSIS
Like terms have been added together. 7

9. SIMPLE REGRESSION ANALYSIS
For a minimum, the partial derivatives of RSS with respect to b1 and b2 should be zero. (We should also check a second-order condition.) 8

10. SIMPLE REGRESSION ANALYSIS
The first-order conditions give us two equations in two unknowns. 9

11. SIMPLE REGRESSION ANALYSIS
Solving them, we find that RSS is minimized when b1 and b2 are equal to 1.67 and 1.50, respectively. 10

12. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line: Y b2 b1 X
Here is the scatter diagram again. 11

13. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line: Y
1.50
1.67 X
The fitted line and the fitted values of Y are as shown. 12

14. DERIVING LINEAR REGRESSION COEFFICIENTSY
True model: X1 Xn X
Now we will do the same thing for the general case with n observations. 13

15. DERIVING LINEAR REGRESSION COEFFICIENTSY
True model:
Fitted line: b2 b1 X1 Xn X
Given our choice of b1 and b2, we will obtain a fitted line as shown. 14

16. DERIVING LINEAR REGRESSION COEFFICIENTSY
True model:
Fitted line: b2 b1 X1 Xn X
The residual for the first observation is defined. 15

17. DERIVING LINEAR REGRESSION COEFFICIENTSY
True model:
Fitted line: b2 b1 X1 Xn X
Similarly we define the residuals for the remaining observations. That for the last one is marked. 16

18. DERIVING LINEAR REGRESSION COEFFICIENTS
RSS, the sum of the squares of the residuals, is defined for the general case. The data for the numerical example are shown for comparison.. 17

19. DERIVING LINEAR REGRESSION COEFFICIENTS
The quadratics are expanded. 18

20. DERIVING LINEAR REGRESSION COEFFICIENTS
Like terms are added together. 19

21. DERIVING LINEAR REGRESSION COEFFICIENTS
Note that in this equation the observations on X and Y are just data that determine the coefficients in the expression for RSS. 20

22. DERIVING LINEAR REGRESSION COEFFICIENTS
The choice variables in the expression are b1 and b2. This may seem a bit strange because in elementary calculus courses b1 and b2 are usually constants and X and Y are variables. 21

23. DERIVING LINEAR REGRESSION COEFFICIENTS
However, if you have any doubts, compare what we are doing in the general case with what we did in the numerical example. 22

24. DERIVING LINEAR REGRESSION COEFFICIENTS
The first derivative with respect to b1. 23

25. DERIVING LINEAR REGRESSION COEFFICIENTS
With some simple manipulation we obtain a tidy expression for b1 . 24

26. DERIVING LINEAR REGRESSION COEFFICIENTS
The first derivative with respect to b2. 25

27. SIMPLE REGRESSION ANALYSIS
Divide through by 2. 26

28. SIMPLE REGRESSION ANALYSIS
We now substitute for b1 using the expression obtained for it and we thus obtain an equation that contains b2 only. 27

29. SIMPLE REGRESSION ANALYSIS
The definition of the sample mean has been used. 28

30. SIMPLE REGRESSION ANALYSIS
The last two terms have been disentangled. 29

31. SIMPLE REGRESSION ANALYSIS
Terms not involving b2 have been transferred to the right side. 30

32. SIMPLE REGRESSION ANALYSIS
To create space, the equation is shifted to the top of the slide. 31

33. SIMPLE REGRESSION ANALYSIS
Hence we obtain an expression for b2. 32

34. SIMPLE REGRESSION ANALYSIS
In practice, we shall use an alternative expression. We will demonstrate that it is equivalent. 33

35. SIMPLE REGRESSION ANALYSIS
Expanding the numerator, we obtain the terms shown. 34

36. SIMPLE REGRESSION ANALYSIS
In the second term the mean value of Y is a common factor. In the third, the mean value of X is a common factor. The last term is the same for all i. 35

37. SIMPLE REGRESSION ANALYSIS
We use the definitions of the sample means to simplify the expression. 36

38. SIMPLE REGRESSION ANALYSIS
Hence we have shown that the numerators of the two expressions are the same. 37

39. SIMPLE REGRESSION ANALYSIS
The denominator is mathematically a special case of the numerator, replacing Y by X. Hence the expressions are quivalent. 38

40. DERIVING LINEAR REGRESSION COEFFICIENTSY
True model:
Fitted line: b2 b1 X1 Xn X
The scatter diagram is shown again. We will summarize what we have done. We hypothesized that the true model is as shown, we obtained some data, and we fitted a line. 39

41. DERIVING LINEAR REGRESSION COEFFICIENTSY
True model:
Fitted line: b2 b1 X1 Xn X
We chose the parameters of the fitted line so as to minimize the sum of the squares of the residuals. As a result, we derived the expressions for b1 and b2. 40

42. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line:
Typically, an intercept should be included in the regression specification. Occasionally, however, one may have reason to fit the regression without an intercept. In the case of a simple regression model, the true and fitted models become as shown. 41

43. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line:
We will derive the expression for b2 from first principles using the least squares criterion. The residual in observation i is ei = Yi – b2Xi. 42

44. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line:
With this, we obtain the expression for the sum of the squares of the residuals. 43

45. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line:
Differentiating with respect to b2, we obtain the first-order condition for a minimum. 44

46. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line:
Hence, we obtain the OLS estimator of b2 for this model. 45

47. DERIVING LINEAR REGRESSION COEFFICIENTS
True model:
Fitted line:
The second derivative is positive, confirming that we have found a minimum. 46

48. Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author.
The content of this slideshow comes from Section 1.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre
Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course
EC212 Introduction to Econometrics
or the University of London International Programmes distance learning course
20 Elements of Econometrics