The Spearman Rank Correlation Test (Numerical Example)

Consider the following data for the illustration of the detection of heteroscedasticity using the Spearman Rank correlation test. The Data file available to download.

Y	X2	X3
11	20	8.1
16	18	8.4
11	22	8.5
14	21	8.5
13	27	8.8
17	26	9
14	25	8.9
15	27	9.4
12	30	9.5
18	28	9.5

The estimated multiple linear regression model is:

$$Y_i = -34.936 -0.75X_{2i} + 7.611X_{3i}$$

The Residuals with data table are:

Y	X2	X3	Residuals
11	20	8.1	-0.63302
16	18	8.4	0.575564
11	22	8.5	-2.16954
14	21	8.5	0.076455
13	27	8.8	1.317102
17	26	9	3.040825
14	25	8.9	0.047951
15	27	9.4	-1.2497
12	30	9.5	-2.74881
18	28	9.5	1.743171

We need to find the rank of absolute values of $u_i$ and the expected heteroscedastic variable $X_2$.

$Y$	$X_2$	$X_3$	Residuals		Rank of |$u_i$|	Rank of $X_2$	$d$	$d^2$
11	20	8.1	-0.633		4	2	2	4
16	18	8.4	0.576		3	1	2	4
11	22	8.5	-2.170		8	4	4	16
14	21	8.5	0.076		2	3	-1	1
13	27	8.8	1.317		6	7.5	-1.5	2.25
17	26	9	3.041		10	6	4	16
14	25	8.9	0.048		1	5	-4	16
15	27	9.4	-1.250		5	7.5	-2.5	6.25
12	30	9.5	-2.749		9	10	-1	1
18	28	9.5	1.743		7	9	-2	4
							0	70.5

Let us compute the Spearman’s Rank correlation

\begin{align}
r_s&=1-\frac{6\sum d^2}{n(n-1)}\\
&=1-\frac{6\times 70.5)}{100(100-1)}=0.5727
\end{align}

Let perform the statistical significance of $r_s$ by t-test

\begin{align}
t&=\frac{r_s \sqrt{n}}{\sqrt{1-r_s^2}}\\
&=\frac{0.5727\sqrt{8}}{\sqrt{1-(0.573)^2}}=1.977
\end{align}

The value of $t$ from the table at 5% level of significance at 8 degrees of freedom is 2.306.

Since $t_{cal} \ngtr t_{tab}$, there is no evidence of the systematic relationship between the explanatory variables, $X_2$ and the absolute value of the residuals ($|u_i|$) and hence there is no evidence of heteroscedasticity.

Since there is more than one regressor (it example is from the multiple regression model), therefore, Spearman’s Rank Correlation test should be repeated for each of the explanatory variables.

As an assignment perform the Spearman Rank Correlation between |$u_i$| and $X_3$  for the data above. Test the statistical significance of the coefficient in the above manner to explore evidence about heteroscedasticity.

Read about Pearson’s Correlation Coefficient