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Regression coefficients represent the mean change in the response variable for one unit of change in the predictor variable while holding other predictors in the model constant. This statistical control that regression provides is important because it isolates the role of one variable from all of the others in the model.

Moreover, the key to understanding the coefficients is to think about them as slopes and invariably they are often called slopes.

Let’s use an example below to explain.

Table 2

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Coefficients

Term Coefficient S-E

Coefficient

T

Statistics P

Constant −3.4601 18.4421 −1.6472 0.000 Height M 102.6422 16.0041 1.6632 0.000

The fitted line plot shows the same regression results graphically.

Figure 1

𝑊𝑊𝑒𝑒𝑖𝑖𝑒𝑒ℎ𝑑𝑑 𝑘𝑘𝑒𝑒 = 102.6422 ℎ𝑒𝑒𝑖𝑖𝑒𝑒ℎ𝑑𝑑 𝑀𝑀.

The equation shows that the coefficient for height is 102.6422 kilograms. The coefficient indicates that for every additional meter in height you can expect weight to increase by an average of 102.6422 kilograms.

From the graph above, a fitted line graphically shows the same information. For example if you move left right along the x axis by an amount that denotes a one meter change in height, the fitted line rises or fall by 102.6422 kilograms.

However, these heights are from middle –school aged boys and ranges from 1.3𝑑𝑑 𝑑𝑑𝑑𝑑 1.7𝑑𝑑. The relationship is only valid within this data range, so we could not actually shift up or down the line by a full meter in this case.

In the fitted lines were flat (a slope coefficient of zero), the expected value for weight would not change no matter how far up and down the line you may go.

Therefore, when we have low p-value, it shows that theslope is not zero, but

1.7 𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥 𝑥𝑥

Height M Weight

Fittted Line Plot

70 60 50 40 30

1.3 1.4 1.5 1.6

𝑥𝑥 𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥 𝑥𝑥 𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑆𝑆= 7.8432 𝑅𝑅 − 𝑆𝑆𝑒𝑒 50.5%

𝑅𝑅 − 𝑆𝑆𝑒𝑒 (𝑎𝑎𝑑𝑑𝑖𝑖) = 55.5%

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indicates that the changes in the predictor variable are associated with changes in the response value.

Let us then take a full Simple Linear Regression:

A model is specified as: 𝐺𝐺𝑅𝑅𝑇𝑇 =𝑓𝑓(𝐺𝐺𝐻𝐻𝐸𝐸) ⇒ 𝐺𝐺𝑅𝑅𝑇𝑇= 𝛽𝛽₀ + 𝛽𝛽_₁𝐺𝐺𝐻𝐻𝐸𝐸 Simple regression result

Dependent variable: GDP Method:- Least Square Year of analysis: 1980 −2013 Included observations – 33

Variable Coefficient Std Error T - Statistics P C 0.53101 0.39431 0.64311 0.000 GHE 1.23421 0.66401 1.36101 0.000

R Square 0.66400 F test 23.43202

Where the model specification is given as;

GDP = f(GHE) ... (1) GDP = ao + a1GHE + e ...(2)

Where GDP is the Gross domestic product (proxy for economy development) GHE is the Government health expenditure in Nigeria and ‘e’ is the error term

Hypothesis

Null Hypothesis (Ho): Government health expenditure has no significant effect on economy development in Nigeria.

Alternative Hypothesis (H1): Government health expenditure has a significant effect on economy development in Nigeria.

From the table above, we can say that we have a simple linear regression result.

The c is the constant and it is also called the intercept and in regression result we do not normally interpret this, but the model shows the impact of government health expenditure on economy development in Nigeria. So let us start by interpreting the other parameters. We can say that there is a direct/positive relationship between Government health expenditure and economy development in Nigeria that is 1unit increase in economy

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development will lead 1.23421 unit increases in government health expenditure.However, the t calculated from the result is given to be 1.3101 while the t calculated using 5% (0.05) level of significance which is called the t tabulated is given to be 1.96 from the t statistical table. The decision rule for t test is that since the t tabulated (1.96) is greater than the t calculated, we accept the null hypothesis and reject the alternative hypothesis, then conclude that Government health expenditure has a significant effect on economy development in Nigeria and also conclude that the parameter government health expenditure is a good explanatory variable. More so the f test for the overall significance of the parameters, and the f calculated from the result is 23.4320 while the f tabulated using 5% (0.05) level of significance is 12.70 from the f statistical table. The decision rule is that since the f calculated is greater than the f tabulated, we accept the alternative hypothesis and reject the null hypothesis, and then we conclude that the overall parameter of the model is statistically significant.Finally, the coefficient of multiple determinations (R square) is 0.66400 from the result above which is 66.4%. This means that about 66.4% of the dependent variable (GDP) has been explained by the explanatory variable (GHE) while the remaining 33.6% are not present in the model or are outside the model.

SELF ASSESSMENT EXERCISE

Explain what steps you will take in interpreting the predictor for value in a regression analysis result.

4.0 CONCLUSION

How to interpret the 𝑝𝑝 𝑑𝑑𝑎𝑎𝑑𝑑𝑢𝑢𝑒𝑒𝑑𝑑 and the coefficient of a regression analysis has shown an important tool in predicting to the future of analysis that may prove difficult when looking through its theoretical analysis. However, regression model take care of the situation of research problem and analyse it completely by running the analysis of dependent and independent variables on one another to get a suggested value for policy maker and planning analysis for the present and thefuture.

5.0 SUMMARY

This unit has briefly touch the application of simple linear regression analysis to statistical analysis of computer software such as stata, e-views and SPSS (Special Package for Social Science) to run an analysis that will leads to getting close to solution of a certain policy problems in the present and future justification.

6.0 TUTOR MARKED ASSIGNMENT

1. Define the following terms with detailed examples:

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(a) Predictor Value

(b) Coefficient of a regression analysis (c) Model specification

2. Given the p values below, interpret the implications.

P values 0.000 0.042 0.001 0.032

3. Given the regression result below with the model is given as:

𝐺𝐺𝐷𝐷𝑃𝑃 = 𝑓𝑓(𝐼𝐼𝑁𝑁𝐹𝐹) 𝐺𝐺𝐷𝐷𝑃𝑃 = 𝛽𝛽0 + 𝛽𝛽1𝐼𝐼𝑁𝑁𝐹𝐹

The parameter is looking at the impact of inflationary rate on economic growth.

Hypothesis

Null hypothesis (Ho): Inflationary rate has no significant impact on economic growth in Nigeria

Alternative hypothesis (H1): Inflationary rate has a significant impact on economic growth in Nigeria

Dependent variable: - GDP Year of analysis: - 1980-2013 Included variable: - 33 Method: Least Square

Variable Coefficient T stat. Prob.

C 0.43260 0.6431 0.000

INF −1.36471 1.6612 0.000

R-Square 46.4201 F test 21.64320

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You are required to interpret the regression result above and test for statistical significance of t and f test.

4. Given the regression result below with the model is given as:

𝐺𝐺𝐷𝐷𝑃𝑃 = 𝑓𝑓(𝑀𝑀𝑆𝑆) 𝐺𝐺𝐷𝐷𝑃𝑃 = 𝛽𝛽₀ + 𝛽𝛽₁𝑀𝑀𝑆𝑆

The parameter is looking at the impact of money supply on Economic growth.

Hypothesis

Null hypothesis (Ho): Money Supply has no significant impact on economic growth in Nigeria

Alternative hypothesis (H1): Money Supply has a significant impact on economic growth in Nigeria

Dependent variable: - GDP Year of analysis: - 1990-2013 Included variable: - 23 Method: Least Square

Variable Coefficient T stat. Prob.

C 0.42311 0.78843 0.000

INF 𝟐𝟐.𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟖𝟐𝟐 1.94222 0.000

R-Square 0.88211 F test 18.88972

You are required to interpret the regression result above and test for statistical significance of t and f test.

7.0 REFERENCES/FURTHER READING

Olomeko, O. A. (2012).Regression Analysis, 1^st edition, pg 88 Millworld Publication Limited, Lagos.