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�The simple linear regression models�

  • Regression analysis in simple language attempts to establish the nature of relationship between variables – that is, to study the functional relationship between variables and thereby provide a mechanism for prediction or forecasting.
  • Regression analysis is very useful and it is used by firms at the micro level and government at the national level to determine the relationship between various economic variables of importance.
  • Constructing a relationship between X and Y gives us a model of the form: Yi = a + bXi + ei, where ei is the error term.

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  • Note that the mean value of the error or random term is zero. This implies E(Y) = a + bX.
  • This linear part can be used to predict the values of Y from the values of X.
  • Exx
  • (a) Give 4 examples of a simple regression model in the following areas:
  • (i) Microeconomics
  • (ii) Macroeconomics
  • (b) Identify the independent and dependent variables in each model.

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�Assumptions about the population model�

  • Consider the regression model of the form:
  • Y = a + bX + e, where a and b are the population intercept and slope to be estimated and e is the population error term,
  • (i) The relationship between y and x is linear.
  • (ii) The variance of the probability distribution of e is constant for all values of X, i.e σ2
  • (iii) The error term is normally distributed for all values of X. this means that the mean of e is 0 and variance σ2
  • (iv) The error term is independent of X.

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�Derivation of the formula for finding the parameters�

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�Example 1�

  • The table below shows the price and the quantity supply of a commodity.
  • (a) Find the regression of quantity supply (Y) on price (X)
  • (b) Plot the actual points and the regression line on the same graph.
  • (c) What will be the quantity supply at a price of 90 cedis?
  • (d) What will be the price if the quantity supply is 300kg.

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Table

Price (X)

Quantity (Y)

61

105

62

120

63

120

65

160

65

120

68

145

69

175

70

160

72

185

75

210

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Soln

  •  

Price (X)

Quantity (Y)

X2

XY

61

105

3721

6405

62

120

3844

7440

63

120

3969

7560

65

160

4225

10400

65

120

4225

7800

68

145

4624

9860

69

175

4761

12075

70

160

4900

11200

72

185

5184

13320

75

210

5625

15750

∑X=670

∑Y=1500

∑X2 = 45078

∑XY=101810

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�Partitioning the sum of squares�

  • The linear model/regression equation can be divided into two parts: the linear part and the error component.
  • The regression equation can therefore be divided into the sum of squares of the linear component and the sum of squares of the errors.
  • Recall our regression equation Y = -316.86 + 6.97X, we can estimate Y1 for any value of X. E.g if X = 61, the corresponding value of Y1 is 108.19.
  • The difference between the observed value of Y and the estimated Y1, Y-Y1 is called the residual. See table below for the computation of the residuals.

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N

Price (X)

Quantity (Y)

Y1

Residual Y-Y1

1

61

105

108.31

-3.31

2

62

120

115.28

4.72

3

63

120

122.25

-2.25

4

65

160

136.19

23.81

5

65

120

136.19

-16.19

6

68

145

157.1

-12.1

7

69

175

164.07

10.93

8

70

160

171.04

-11.04

9

72

185

184.98

0.02

10

75

210

205.89

4.11

Sum

670

1500

1501.3

-1.3

Mean

67

150

150.13

-0.13

Variance

20.89

1155.56

1014.8

141.31

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  •  

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  • The table below summarizes the relationship between the actual value and the deviations as:

​

​

​

​

​

  • Where RSS is the regression sum of squares and ESS is the error sum of squares

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Observed

Mean

Deviation from mean due to regression, RSS

Error Part, ESS

Y

​

​

(Y - Y1)

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  • Equation 6 says that the sum of squares of Y can be divided into two proportions, that due to regression and that due to error.
  • In other words the sum of squares of Y (total variation) is broken down into the explained variation (regression sum of squares) and the unexplained variation (error or residual sum of squares).
  • The two proportions must add to 1

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Exx

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  • Q1. Consider our example involving quantity supplied and price. Calculate the regression and error sum of squares from the data.
  • Q2. The table below shows the weight (X) and height (Y) of a sample of students.
  • (ii) Find the regression of X on Y.
  • (ii) Find the total sum of squares (TSS), error sum of squares (ESS) and the regression sum of squares (RSS).
  • (iii) Show that the proportion of ESS and the RSS add up to 1.

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