Estimation: The Ordinary Least Squares (OLS) Method.

The scattergram examined earlier contains a discussion of both the Problem Description and the Data used in deriving
the results presented here. The estimation method is the classical Ordinary Least Squares (OLS) which is programmed into the
SPSS/win statistical package. The Linear Regression Model (LRM) has the form
where Y
is the DV (in this case, **annual Family Food Expenditure**), X is the IV (in this case, **annual
Family Income**), and E
is the **random error** term; it is a proxy for all the uncertain factors
that may also affect family food expenditure. In regression analysis, all of the Classical Assumptions of the LRM basically
apply to the error term. A and
B are the regression
parameters whose numerical values we seek to estimate; and in so doing, we will have
succeeded to estimate the underlying **Population
Regression Line (PRL)** using the OLS method. By using the
command sequence presented earlier, SPSS/win automatically implements this method.

**Discussion of the Outputs/Results and Related Tests
The results will be discussed in the order in which
SPSS/win generates the outputs. These outputs are presented in the tables below. For
instance, the discussion in part I pertains to the DESCRIPTIVE
STATISTICS table, followed by part II which pertains to
the CORRELATIONS table, and so on. This approach permits a critical analysis of the results and
their implications. **

Mean | Std. Deviation | N | |
---|---|---|---|

Annual Food Expenditure ($000) | 7.965 | 4.664 | 20 |

Annual Income ($000) | 45.50 | 23.96 | 20 |

Annual Food Expenditure ($000) | Annual Income ($000) | ||
---|---|---|---|

Pearson Correlation | Annual Food Expenditure ($000) | 1.000 | .946 |

Annual Income ($000) | .946 | 1.000 | |

Sig. (1-tailed) | Annual Food Expenditure ($000) | . | .000 |

Annual Income ($000) | .000 | . | |

N | Annual Food Expenditure ($000) | 20 | 20 |

Annual Income ($000) | 20 | 20 |

Variables | R | R Square | Adjusted R Square | Std. Error of the Estimate | Durbin-Watson | ||
---|---|---|---|---|---|---|---|

Model | Entered | Removed |
|||||

1 | Annual Income ($000)(c,d) | . | .946 | .894 | .888 | 1.559 | 2.834 |

a Dependent Variable: Annual Food Expenditure ($000) | |||||||

b Method: Enter | |||||||

c Independent Variables: (Constant), Annual Income ($000) | |||||||

d All requested variables entered. |

Model | Sum of Squares | df | Mean Square | F | Sig. | |
---|---|---|---|---|---|---|

1 | Regression | 369.573 | 1 | 369.573 | 151.975 | .000(b) |

Residual | 43.773 | 18 | 2.432 | |||

Total | 413.346 | 19 | ||||

a Dependent Variable: Annual Food Expenditure ($000) | ||||||

b Independent Variables: (Constant), Annual Income ($000) |

Unstandardized Coefficients | Standardized Coefficients | t | Sig. | |||
---|---|---|---|---|---|---|

Model | B | Std. Error | Beta |
|||

1 | (Constant) | -.412 | .764 | -.539 | .596 | |

Annual Income ($000) | .184 | .015 | .946 |
12.328 | .000 | |

a Dependent Variable: Annual Food Expenditure ($000) |

**I. Descriptive Statistics Table
1.**

a)

**II. Correlations Table
**This table contains the

**III. Model Summary Table
**This table contains the necessary summary statistics for assessing the
accuracy of the estimated sample regression line (SRL) , where '

Before examining the meaning of the summary statistics, some remarks about the appended footnotes are in order. Footnotes

'Enter' simply means that annual Family Food Expenditure (DV) is regressed on both the constant term ‘a’ and the income (IV) using the OLS method. For those students who might take

**IV. ANOVA Table
**The summary measures reported here are used in the partitioning of the the
total variation in the DV according to the identity relation

: Some authors use RSS (regression sum of squares) instead of ESS (explained sum of squares), and ESS (error sum of squares) instead of RSS (residual sum of squares) so that the identity is stated as TSS = RSS + ESS. So pay attention to how these acronyms are defined.

From the table, under the df column,

**V. Coefficients Table**

This table contains the estimated regression coefficients (a = -.412, b = .184), and hence
the estimated SRL/equation written as **ý _{i} = -.412 + .184X_{i}**.
These interpretations follow:

1.

2.

3.

4. S

As part of investigating the accuracy of the fitted SRL, it is often useful to verify both the statistical significance and the economic significance (i.e., the sign) of the regression parameter/coefficient B. For statistical significance, the null hypothesis is stated as

An interesting variation of the t-test is to verify the economic significance of the parameter with respect to the direction of causality of the associated IV. In this case, the null is phrased as

Suppose a typical or

**Copyright© 1996, Ebenge Usip, all rights reserved.
Last revised: Wednesday, July 10, 2013.**