Reading 11, 3.5 Five Steps to test Regression Coefficient (Part A)

Level 2, Volume 1, Quantitative methods, Reading 11, Correlation & Regression

In a previous section (2.6) hypothesis testing was used to test the significance of the correlation coefficient.

Hypothesis testing will again be used to test the hypotheses concerning the population values of the intercept (b0) or slope coefficient (b1) of a regression model. (remember intercept & slope coefficient is called the regression coefficients).

Step 1: Obtain the relevant information

1)   the estimated parameter value, bˆ0 or bˆ1;

2)   a confidence interval around the estimated parameter. (A confidence interval is an interval of values that we believe includes the true parameter value, b1, with a given degree of confidence. To compute a confidence interval, we must select the significance level for the test and know the standard error of the estimated coefficient); and

3)   the hypothesized value of the parameter, b0 or b1

Step 2:  State the hypothesis

Ho: b1 (or b0) = A certain value AND bˆ1   = estimate for b1

Ha: b1 (or b0) < > A certain value AND bˆ1   < > estimate for b1

(Because b1 (or b0) = A certain value, we are testing for a two tailed test!)

Step 3: Obtain the critical t value using Student’s t-distribution

Calculate degrees of freedom = n(sample size) – 2

Determine the level of significance. This is typically set at the 5% and/or 1% level.

Find the corresponding answer on the t table.

Step 4: Calculate test

Where

bˆ1  = estimated parameter value

tc = critical t value

sbˆ1  = standard error of the estimated parameter value

Step 5: Conclude

If answer of Step 4 < > than hypothesized value, Reject Ho AND accept Ha.

Example – Testing the hypotheses concerning the population values of the intercept (b0) or slope coefficient (b1) of a regression model (BHP Revenue & Earnings per ordinary share)

Relevant information:

1. Data
Year	2010	2009	2008	2007	2006
X: Revenue US $m	52798	50211	59473	47473	39099
Y: Earnings per ordinary share (diluted) (US sent)	227.8	105.4	274.8	228.9	172.4

Step 3: Calculate Yi based on the regression formula
	The regression formula is: Yi = b0 + b1Xi + ei
	Using this formula we can calculate the dependent variable, being Earnings per share

	Year		Actual Earnings (For Info)	Calculated Earnings per regression model (Yi)	bo	Plus	b1	Multiply Xi (Revenue, independent variable))
	2010		227.8	215.22	-20.9637		0.004473	52798
	2009		105.4	203.65	-20.9637		0.004473	50211
	2008		274.8	245.08	-20.9637		0.004473	59473
	2007		228.9	191.40	-20.9637		0.004473	47473
	2006		172.4	153.94	-20.9637		0.004473	39099

Step 1: Obtain the relevant information

	Intercept	Slope
The estimated parameter value (from above)	-20.9637	0.004473
A confidence interval (Selected)	5%	1%
Standard error (randomly selected)	1	0.00002
Hypothesized value of the parameter	-19	0.005

Step 2:  State the hypothesis

Ho: b0 = -19 AND bˆ0   = estimate for b0

Ha: b0 < > -19 AND bˆ0  < > estimate for b0

(Because b0 = -19, we are testing for a two tailed test!)

Ho: b1 = 0.005 AND bˆ1   = estimate for b1

Ha: b1 < > -0.005 AND bˆ1   < > estimate for b1

(Because b1 = 0.005, we are testing for a two tailed test!)

Step 3: Obtain the critical t value using Student’s t-distribution

The sample size for both b0 & b1 = 5. Therefore n-2 = 3

	Intercept	Slope
N	5	5
Therefore n-2	3	3
Confidence interval	5%	1%
Therefore critical t	3.182	5.841

Step 4: Perform calculation

For bˆ0

= -20.9637  +/- 3.182(1)

= -17.7817 to -24.1457

For bˆ1

= 0.004473 +/- 5.841(0.00002)

= 0.00459 to 0.004356

Step 5: Conclude

For bˆ0:

Ho: b0 = -19

Because -19 falls within the calculated range (-17.7817 to -24.1457) we accept the H0 hypothesis.

For bˆ1:

Ho: b1 = 0.005

Because 0.005 falls outside the calculated range (0.00459 to 0.004356) we reject the H0 hypothesis. (and accept the Ha hypothesis)